We start with a brief introduction to artificial gauge fields in ultracold atomic gases, motivating our theoretical study. Then, we provide a description of the classical field Monte Carlo methods, used to obtain the main, new results of the thesis, presented in chapter three and four. Our conclusions together with perspectives for future work are summarized in chapter five. CHAPTER 0. OVERVIEW Nous commençons ce manuscrit avec une brève introduction aux champs de jauge artificiels dans les atomes froids qui appuient notre étude théorique. Nous fournissons ensuite une description des méthodes Monte Carlo basées sur champs classiques utilisées pour obtenir les résultats nouveaux et principaux de cette thèse qui seront présentés durant les Chapitre III et IV. Nos conclusions ainsi que des perspectives pour de prochaines études sont résumées Chapitre V. CHAPTER 0. OVERVIEW

Les oiseaux de passage La chanson des gueux (1876) de Jean Richepin Poème interprété par Georges Brassens

The very first person I wish to thank is the director of this thesis, Markus Holzmann. I feel very lucky as I had the privilege of working alongside him, grasping every answer he gave to my too often naive questions. Markus truly is a remarkable supervisor, inside and outside the office, and I hope that he will continue to share his knowledge in his pleasant, passionate and easy way.

I would also like to thank all the members of the jury, for their expertise, time and kindness. My very last PhD physics discussion, during the defense, will be a lasting memory that I will enjoy remembering thanks to each member of the jury.

I thank the president of the jury Simone Fratini for his perfect french accent among many others things, as a representative of my university the Communauté Université Grenoble Alpes. I would like also to deeply thank the two rapporteurs Jordi Boronat and Patrizia Vignolo for their time and work. Their comments were especially important to me. Indeed, while writing the manuscript a PhD student needs to focus on all its defects aiming on how to improve the thesis. Their constructive and positive feedbacks were then very precious to me. I would like to thank the two examiners Stefano Giorgini and Tommaso Roscilde with whom I already had very interesting discussions on the topic of this thesis.

My gratitude goes also to my former internship supervisor Anna Minguzzi who is the current director of the laboratory LPMMC as she introduced me both to the laboratory and to Markus. The very first director of the laboratory that I met was Frank Hekking who was also my teacher, I would particularly like to thank him.

Lastly I wish to thank the University UGA for giving me the opportunity of doing my PhD at the LPMMC. I wish here to thank all the members of this University that were very helpful from my very first year in Grenoble, in particular Arnaud Ralko and Thierry Klein.

Je voudrais également remercier toute l'équipe pédagogique d'informatique de l'UPMF (puis UGA) qui m'a permis d'enseigner durant ces trois années de thèse. Ce fût une expérience très enrichissante, remplie de belles rencontres comme Nathalie Denos et Dominique Vaufreydaz.

Je voudrais ensuite chaleureusement remercier l'ensemble de mes collègues de laboratoire enseignants, gestionnaires, chercheurs, ingénieurs de recherche, informaticiens, stagiaires, post-docs ou doctorants, en particulier Malo, Aleksandr, Guillaume, Van-Duy, Jose Maria, Natalia, Ivan, Katarina, Antton, Steven, Li-Jing, Florian et Davide. Merci pour nos belles discussions et pour m'avoir accompagné tout au long de ces années. Questa tesi deve anche molto ai "vecchi" membri del laboratorio che sono oggi grandi amici. Auguro un bel in bocca al lupo a Daniel e Maria che ringrazio per la loro amicizia. Grazie anche al mitico trio degli italiani Angelo, Marco e Francesco. Il filosofo Marco e lo spericolato Francesco mi hanno trasmesso innumerevoli consigli già utili per il seguito della tesi. Merci aussi à la nouvelle génération de doctorants Evrard-Ouicem, Etienne et Nicolas avec qui j'ai partagé la dernière partie de ma thèse. Je vous souhaite le meilleur pour la suite et que tous vos projets se réalisent. Enfin j'ai été accompagné durant ces années par les gestionnaires du laboratoire Habiba et Claudine que je tiens à remercier. Merci Habiba pour ces longues discussions et rires, durant lesquelles ta gentillesse et ta spontanéité m'ont permis de rester moi-même.

Je remercie mes parents pour leurs encouragements et leur foi en moi, j'ai été très heureux et ému de partager ce moment symbolique avec vous, au nez et à la barbe des tumultes de la vie. En parlant de famille, un grand merci à mon frère de pensée et ami philosophe Benjamin Le Roux à qui je souhaite tout ce qu'il y a de meilleur pour la suite. Océane, tu es présente et m'accompagnes à chaque page de cette thèse et il m'est impossible de résumer ma gratitude envers toi. Je me contenterai de te remercier pour le sourire que tu me donnes chaque jour.

En conclusion, je remercie mon pays et ses habitants pour l'université qui m'a été offerte, le CROUS qui m'a accompagné de mille façons et la CAF qui m'a permis d'avoir un logement durant mes études pour ne citer que quelques exemples. Enfin, je voudrais terminer ces remerciements en exprimant ma reconnaissance envers toutes celles et tous ceux sans qui je n'aurais jamais pu vivre cette aventure. Merci pour tout et à très bientôt.

Chapter 0

Overview S PIN-ORBIT coupling links a particle's velocity to its quantum-mechanical spin, and it is essential in numerous condensed matter phenomena. Recently, in ultracold atomic systems [START_REF] Dalibard | Colloquium[END_REF], highly tunable synthetic spin-orbit couplings have been engineered enabling unique features and new physical phenomena. Spin-orbit coupled Bose gases present a notable example raising fundamentally new questions [START_REF] Lin | Spin-orbit-coupled boseeinstein condensates[END_REF][START_REF] Jiménez-García | Tunable spin-orbit coupling via strong driving in ultracold-atom systems[END_REF][START_REF] Galitski | Spin-orbit coupling in quantum gases[END_REF][START_REF] Wu | Realization of two-dimensional spin-orbit coupling for bose-einstein condensates[END_REF]. For instance, a pioneering experiment at NIST achieved a spin-orbit coupled Bose gas and Bose-Einstein condensation [START_REF] Lin | Spin-orbit-coupled boseeinstein condensates[END_REF]. Indeed realization of a pseudo-spin one-half Bose gas was achieved by selecting two internal states of the atoms and by coupling them through Raman processes.

At the mean-field level, spin-orbit coupling (SOC) introduces degenerate ground states expected to enhance fluctuation effects and giving rise to new, exotic quantum phases. However, the occurrence and nature of finite temperature transitions in bosonic systems have not yet been fully established [START_REF] Ozawa | Stability of ultracold atomic bose condensates with rashba spin-orbit coupling against quantum and thermal fluctuations[END_REF][START_REF] Ozawa | Condensation transition of ultracold bose gases with rashba spin-orbit coupling[END_REF][START_REF] Zhai | Degenerate quantum gases with spin-orbit coupling: a review[END_REF][START_REF] Chao | Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons[END_REF][START_REF] Liao | Spin-orbitcoupled bose gases at finite temperatures[END_REF].

In this thesis, we determine the finite-temperature phase diagram of a twodimensional interacting Bose gas with two hyperfine (pseudospin) states coupled via a Rashba-Dresselhaus spin-orbit interaction using classical field Monte Carlo calculations.

First, we review the results of mean-field calculations [START_REF] Zhai | Degenerate quantum gases with spin-orbit coupling: a review[END_REF][START_REF] Chao | Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons[END_REF][START_REF] Liao | Spin-orbitcoupled bose gases at finite temperatures[END_REF][START_REF] Ho | Bose-einstein condensates with spin-orbit interaction[END_REF][START_REF] Wang | Spin-orbit coupled spinor bose-einstein condensates[END_REF] that indicate a Bose condensed ground state strongly dependent on the anisotropy of the interparticle interactions. At zero temperature, we expect exotic ground states formed by either a single plane wave with non-vanishing momentum or a linear superposition of two plane waves with opposite momenta, called plane wave state (PW) and stripe phase (SP), respectively. For spin-independent interaction between atoms, PW and SP remain degenerate at the mean-field level.

We then explore the phase diagram using classical field Monte Carlo calculations, and present the main results of the thesis. Classical field Monte Carlo simulations provide a numerical method to accurately describe continuous phase transitions of Bose gases at finite temperatures. We have adapted this method to perform simulations of interacting Bose gases with SOC. In two spatial dimensions, we show that for anisotropic SOC, the systems undergoes a Kosterlitz-Thouless phase transition from a normal to superfluid state. In the superfluid state, the single particle density matrix decays algebraically and directly reflects the PW/SP character of the mean-field ground state. In the limit of isotropic interparticle interaction, the PW/SP degeneracy is unaffected by the transition and fragmentation of the condensate occurs [START_REF] Kawasaki | Finite-temperature phases of twodimensional spin-orbit-coupled bosons[END_REF].

In the case of isotropic SOC, we show that the transition temperature decreases with increasing system size due to the increasing number of degenerate mean-field ground states and eventually vanishes in the thermodynamic limit. Our simulations show that the circular degeneracy of the single-particle ground state destroys the algebraic ordered phase. No superfluid transition is then expected in the thermodynamic limit.

Introduction et Résumé

L E COUPLAGE spin-orbite, essentiel dans de nombreux phénomènes en matière condensée, fait intéragir la vitesse d'une particule avec son propre spin. Récemment dans le domaine des atomes froids [START_REF] Dalibard | Colloquium[END_REF], un couplage spin-orbite finement réglable a été conçu. Il a depuis ouvert la voie à des dispositifs uniques et à la découverte de nouveau phénomènes physiques. Les gaz de Bose avec couplage spin-orbite sont un exemple majeur de ces nouvelles pistes qui soulèvent des questions inédites [START_REF] Lin | Spin-orbit-coupled boseeinstein condensates[END_REF][START_REF] Jiménez-García | Tunable spin-orbit coupling via strong driving in ultracold-atom systems[END_REF][START_REF] Galitski | Spin-orbit coupling in quantum gases[END_REF][START_REF] Wu | Realization of two-dimensional spin-orbit coupling for bose-einstein condensates[END_REF]. Par exemple, une expérience pionnière au laboratoire NIST a permis d'obtentir un gaz de Bose avec couplage spin-orbit ainsi qu'une condensation de Bose-Einstein dans ce système [START_REF] Lin | Spin-orbit-coupled boseeinstein condensates[END_REF]. Le gaz de bosons avec des pseudo spins 1/2 a été réalisé en sélectionnant deux états internes propre à chaque atome et en les couplant à travers un processus Raman.

Selon l'approximation champ moyen, le couplage spin-orbite introduirait une forte dégénérescence de l'état fondamental qui donnerait lieu à des phases quantiques exotiques et qui permettrait également d'envisager un rôle prépondérant des effets des fluctuations. Pourtant, l'apparition, l'existence et la nature de transitions à température finie dans un système de bosons avec couplage spin-orbite ne sont pas encore réellement établies [START_REF] Ozawa | Stability of ultracold atomic bose condensates with rashba spin-orbit coupling against quantum and thermal fluctuations[END_REF][START_REF] Ozawa | Condensation transition of ultracold bose gases with rashba spin-orbit coupling[END_REF][START_REF] Zhai | Degenerate quantum gases with spin-orbit coupling: a review[END_REF][START_REF] Chao | Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons[END_REF][START_REF] Liao | Spin-orbitcoupled bose gases at finite temperatures[END_REF].

L'objectif de cette thèse est de déterminer le diagramme de phase à température finie d'un gaz de Bose bidimensionnels interagissant avec deux états hyperfins (pseudospins) couplés à travers une interaction spin-orbite Rashba-Dresselhaus en utilisant des calculs Monte Carlo type champs classiques.

Tout d'abord, nous examinons les résultats des calculs type champ moyen [START_REF] Zhai | Degenerate quantum gases with spin-orbit coupling: a review[END_REF][START_REF] Chao | Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons[END_REF][START_REF] Liao | Spin-orbitcoupled bose gases at finite temperatures[END_REF][START_REF] Ho | Bose-einstein condensates with spin-orbit interaction[END_REF][START_REF] Wang | Spin-orbit coupled spinor bose-einstein condensates[END_REF] qui indiquent un état fondamental, du gaz de Bose condensé, fortement déterminé par l'anisotropie des interactions interparticules. A température nulle, des phases exotiques sont attendues formant soit un état type onde plane avec une impulsion non-nulle soit une superposition linéaire de deux ondes planes avec deux impulsions opposées, chacune appelée respectivement onde plane (PW) et état de bande (SP). Pour des interactions indépendantes du spin de chaque atome, les états PW et SP restent dégénérés dans le cadre de la théorie champ moyen.

Nous explorons alors le diagramme de phase en utilisant les calculs Monte Carlo basés sur champs classiques pour ensuite présenter les résultats principaux de cette thèse. Les simulations Monte Carlo avec champs classiques fournissent une méthode décrivant précisemment les transitions de phases continues dans les gaz CHAPTER 0. OVERVIEW de Bose à températures finies. Nous avons adapté cette méthode afin d'effectuer des simulations d'un gaz de Bose interagissant avec couplage spin-orbite. En deux dimensions spatiales, nous montrons que, pour un coupalge spin-orbite anisotrope, le système subit une transition de phase type Kosterlitz-Thouless qui sépare une phase dite normale d'une phase superlfuide. Dans la phase superlfuide, la matrice densité à un corps décroît algébriquement et reflète directement le caractère type PW/SP en écho avec les prédictions d'état fondamental provenant de la théorie champ moyen. Dans la limite d'interactions interparticules totalement isotropes, la dégénérescence entre les états PW et SP n'est pas affectée par la transition. Une fragmentation du quasi-condensat s'opère dans ce cas [START_REF] Kawasaki | Finite-temperature phases of twodimensional spin-orbit-coupled bosons[END_REF].

Dans le cas d'un couplage spin-orbite isotrope, nous montrons que la température de transition diminue avec la taille du système à cause du nombre croissant d'états fondamentaux décrivant le minimum dégénéré d'énergie champ moyen. Cette température tend alors vers zéro à la limite thermodynamique. Nos simulations montrent que la dégénérescence circulaire de la courbe de dispersion de l'énergie de chaque particule détruit l'ordre algébrique et donc la phase ordonnée. Aucune transition vers une phase superfluide n'est attendue dans ce cas à température finie à la limite thermodynamique. new chapter in atomic and molecular physics, in which particle statistics and their interactions, rather than the study of single atoms or photons, are at center stage."

Chapter 1

Extracted from [START_REF] Bloch | Many-body physics with ultracold gases[END_REF].

Why simulate gauge fields in ultra-cold atoms ?

Trapped cold atoms usually have a neutral charge. However, magnetic phenomena are of a particular interest in quantum mechanics. They appear for example in spin Hall effects [START_REF] Kato | Vortex pairs in a spin-orbitcoupled bose-einstein condensate[END_REF][START_REF] König | Quantum spin hall insulator state in hgte quantum wells[END_REF], spin-orbit coupling [START_REF] Yu | Oscillatory effects and the magnetic susceptibility of carriers in inversion layers[END_REF][START_REF] Dresselhaus | Spin-orbit coupling effects in zinc blende structures[END_REF], Aharonov-Bohm effect, Hofstadter butterfly physics and topological insulators [19,[START_REF] Bernevig | Quantum spin hall effect and topological phase transition in hgte quantum wells[END_REF][START_REF] Hsieh | A topological dirac insulator in a quantum spin hall phase[END_REF] when a charged particle interacts with a magnetic field. In order to better understand these effects in situations where particle statistics and interactions are important, various propositions to enable such physics in ultra-cold gases were explored. Very recently new ways of creating artificially synthetic magnetic couplings have been found [START_REF] Lin | Synthetic magnetic fields for ultracold neutral atoms[END_REF]. This thesis focuses on the interplay between collective behavior and single particle magnetic phenomena in ultra-cold gases. In particular, we study the effect of a spin-orbit coupling in a homogeneous Bose gas with repulsive hardcore two-body interaction.

In the following we will first sketch how magnetic phenomena and artificial gauge fields can be created in single atoms, then we will describe the collective behavior of homogeneous bosonic gases in the presence of this new coupling. In the next chapters we will then focus on the interplay between the two-body interaction and the single particle spin-orbit coupling (SOC) spectrum.

CHAPTER 1. INTRODUCTION

Artificial gauge fields

A free non-relativistic particle of mass m with a charge q coupled to a magnetic field B = ∇ × A with A being the vector potential, is described by the Hamiltonian,

H = 1 2m p - q c A 2 (1.1)
where c is the speed of light and p is the canonical momentum operator. In most experimental systems, trapped ultra-cold atoms are neutral (q = 0) and do not naturally couple to electromagnetic fields. In the next section we will very briefly introduce how to create artificial gauge fields that mimics the effect of electromagnetic fields on neutral atoms.

Rotating gas

Artificial gauge fields were studied very early in trapped gases [START_REF] Ho | Local spin-gauge symmetry of the bose-einstein condensates in atomic gases[END_REF]. A standard way of creating strong artificial magnetic field is by rotating a neutral particle system which is equivalent to placing them in a magnetic field proportional to the rotation vector Ω with the appearance of additional terms.

p 2 2m -Ω(r × p) = p -mΩ × r 2 2m - 1 2 m(Ω × r) 2 (1.2)
The artificial magnetic field produced through rotation is necessarily uniform. This idea has shown to be very effective to study the creation of vortex lattices in BECs [START_REF] Bloch | Many-body physics with ultracold gases[END_REF].

Raman induced gauge field

Here we consider a toy model of a three-level atom coupled to two lasers in order to give a little insight on how induced transitions can simulate artificial gauge fields. This toy model is presented from the reference [START_REF] Dalibard | Colloquium[END_REF], which contains a more accurate and detailed description of artificial gauge field in ultracold atoms. As in the rotational gas case we will aim to cast the Hamiltonian in a form like Eq. (1.1), we will however not give an accurate explanation of the experimental realization itself. section. The transfer of momentum can be realized with no change in energy for ω a = ω b , such that the two ground states of energies E 1 and E 2 can be equally populated and considered degenerate, E 1 = E 2 , as it will be the case in the rest of this thesis.

In the rotating frame we can extract an effective Hamiltonian that does not depend on the excited state but only on |g 1 〉 and |g 2 〉. Raman transitions do not necessary populate the excited state (for large detuning ∆ e ) which is very convenient in cold atoms experiments. The reduced Hamiltonian for this effective two-level system is then a 2 × 2 matrix and writes in this basis,

H = ħ 2 ∆ κ * κ -∆ (1.3)
The effective Rabi frequency and the Raman mismatch write

κ = κ a κ * b 2∆ e ∆ = ħ(ω a -ω b ) -(E 2 -E 1 ) (1.4)
with κ a and κ b the Rabi frequencies corresponding to the two lasers of frequencies ω a and ω b . The light intensity is proportional to |κ| 2 . In order to extract the gauge field term appearing in the Hamiltonian, we write Eq. (1.3) in the a general form

H = ħ 2 cos(θ) e -i φ sin(θ) e i φ sin(θ) -cos(θ) (1.5)
with the generalized Rabi frequency Ω = ∆ 2 + |κ| 2 1/2 , the mixing angle tan(θ) = |κ|/∆ and the phase φ from κ = |κ|e i φ . In the basis defined by the eigenvalues |ψ -〉 CHAPTER 1. INTRODUCTION and |ψ + 〉, the two energies write E ± = ± ħΩ 2 .

|ψ -〉 = sin(θ/2) -e -i φ sin(θ/2)

|ψ + 〉 = cos(θ/2) e i φ sin(θ/2) (1.6)
Up to now Eq. (1.3) only described the internal degree of freedom of an atom. In order to describe the spatial degree of freedom of the whole atom we treat the internal (electronic) state in the adiabatic approximation.

In the adiabatic approximation Ψ(r, t ) = φ -(r, t ) |ψ -(r)〉 the equation of evolution for an atom in its internal ground state E -can be written in the form like Eq. (1.1)

i ħ ∂φ - ∂t = (p -A -(r)) 2 2M + E -(r) + ν -(r) φ -(r, t ) (1.7)
Where ν -(r) = ħ 2 8M (∆θ) 2 + sin 2 θ(∆φ) 2 and the vector potential is defined by

A -(r) = i ħ 〈ψ -|∇ψ -〉 (1.8)
Creating an artificial magnetic field, this causes a shearing of the atomic cloud and allows the entry of vortices into the BEC as presented in figure 1.2. In contrast to the case of a rotational gas which was equivalent to a uniform magnetic field, the coupling κ in Eq. (1.3) may depend on the position of the atom and on the shape of the laser beam. The two plane waves created by the two lasers can be each tuned independently described as κ a/b (r) = κ 0 e i 2k a/b •r where 2k a/b is the momentum transfered to the atom. During a Raman process, the single atom then acquires in total the difference of the two momenta carried from the two photons.

CHAPTER 1. INTRODUCTION

Raman induced SOC

"Thus, as first put forward by Higbie and Stamper-Kurn [START_REF] Higbie | Periodically dressed bose-einstein condensate: A superfluid with an anisotropic and variable critical velocity[END_REF], Raman transitions can provide the required velocity-dependent link between the spin [the internal state of the atom] and momentum: because the Raman lasers resonantly couple the spin states together when an atom is moving, its Doppler shift effectively tunes the lasers away from resonance, altering the coupling in a velocity-dependent way."

Extracted from [START_REF] Galitski | Spin-orbit coupling in quantum gases[END_REF] It is then possible to simulate different shapes of the vector potential A and tune each component differently. When the vector potential components A = A x , A y , A z do not commute, the gauge potential is called non-abelian. One particular example of non commuting components are Pauli matrices which give rise to SOC terms.

Generating non-abelian gauge potential is of a higher level of difficulty and it relies on multipod configuration (multiple low energy states).

In summary, it is possible to generate a vector potential A such that,

H = p -A 2 2M (1.9)
where the components of A do not necessary commute. In particular, in this thesis, we will study the case A ∝ σ x , σ y , 0 T where σ x and σ y are the x and y Pauli spin matrices acting on a two level atom (figure 1.1) [START_REF] Jiménez-García | Tunable spin-orbit coupling via strong driving in ultracold-atom systems[END_REF].

Ultra-cold SOCed systems have been of a great interest recently from both experimental and theoretical point of view [START_REF] Jiménez-García | Tunable spin-orbit coupling via strong driving in ultracold-atom systems[END_REF][START_REF] Lin | Synthetic magnetic fields for ultracold neutral atoms[END_REF][START_REF] Lin | Bose-einstein condensate in a uniform light-induced vector potential[END_REF]. Indeed extremely tunable SOC are a good playground for new accessible questions to arise. Some examples of phenomena that do not exist in standard condensed matter systems are listed here below.

• What is the collective behavior of SOCed bosons?

• How does SOC change when the spin is S > 1/2?

• What is the interplay between interactions and SOC for bosons?

After a more precise definition of our SOC Hamiltonian, we will study the single particle spectrum of SOCed atoms and then ask again the questions arising from these new features.

CHAPTER 1. INTRODUCTION

Rashba-Dresselhaus spin-orbit coupling

Spin-orbit coupling in condensed matter

For electrons in atoms or solid, the spin-orbit coupling is naturally present due to relativistic corrections. When a charged particle is moving in an electric field, the particle in its reference frame sees a magnetic field which couples to the internal magnetic moment (spin).

Different physical realizations produce different types of SOC with different symmetries. In two dimensions they are generally regrouped in two classes called Rashba [START_REF] Yu | Oscillatory effects and the magnetic susceptibility of carriers in inversion layers[END_REF] and Dresselhaus [START_REF] Dresselhaus | Spin-orbit coupling effects in zinc blende structures[END_REF] SOC. These couplings were originally studied in the context of two dimensional semiconductors.

In the context of ultra-cold gases, the SOC is not a relativistic correction to the electronic energy levels but arises as an effective description for the hyperfine states of the atom and its coupling to the laser fields via the atomic momentum. The form and the strength of the SOC are experimentally tunable and therefore their study is distinct and often far from the SOC studies based on electrons in standard condensed matter. We quickly define below the terminology used in the context of ultra-cold atoms where different types of SOC are experimentally achievable [START_REF] Jiménez-García | Tunable spin-orbit coupling via strong driving in ultracold-atom systems[END_REF][START_REF] Campbell | Realistic rashba and dresselhaus spin-orbit coupling for neutral atoms[END_REF].

Rashba and Dresselhaus couplings

The general Rashba-Dresselhaus coupling is defined as

H RD = ħ 2 κ m p x σ x + η soc ħ 2 κ m p y σ y
where κ corresponds to the strength of the SOC. The real scalar η soc characterizes the anisotropy/isotropy of the SOC in the x-y plane which can be experimentally controlled by coupling lasers with different amplitudes in the x-y plane. As a definition, the pure Rashba case corresponds to an isotropic SOC with η soc = 1. Otherwise for 0 ≤ η soc < 1 the anisotropic coupling is called Rashba-Dresselhaus. In the next section we will draw the energy spectrum of the free particle Hamiltonian including the general SOC and discuss its dependency on the parameter η soc .

CHAPTER 1. INTRODUCTION

Non interacting SOCed Bose gases

Before discussing the implications of including SOC in the Hamiltonian for interacting many body systems, let us first study the single particle spectrum of the ideal (non interacting) SOCed atoms. We introduce in this section the eigenbasis of the non interacting Hamiltonian given by a linear superposition of spin up and down states.

Diagonal form

In this thesis we will consider bosons with two internal degrees of freedom coupled by a Rashba-Dresselhaus SOC in a homogeneous system. In the second quantization basis the field creator operator writes

Ψ † (r) = 1 V k e i k•r Ψ † k (1.10)
We define û † k and d † k as creation operator of a spin up and down particle of momentum k.

Ψ † k = û † k , d † k (1.11)
In a homogeneous system and in presence of a SOC, the Hamiltonian takes the form

Ĥ0 = k Ψ † k ħ 2 k 2 + ħ 2 κ 2 2m I 2 + ħ 2 κ m k x σ x + η soc ħ 2 κ m k y σ y Ψk (1.12)
with I 2 the 2 × 2 identity matrix. The constant term ħ 2 κ 2 2m is added for convenience in order to set the absolute minimum of the energy to zero. The following results are of course completely independent of it. In matrix form, the Hamiltonian Ĥ0 writes Ĥ0 = 1

V k d r Ψ † k M (k) Ψk (1.13) CHAPTER 1. INTRODUCTION
For simplicity we set ħ = m = 1 in the following. The matrix M (k) writes

M (k) = k 2 /2 + κ 2 /2 κ(k x -i η soc k y ) κ(k x + i η soc k y ) k 2 /2 + κ 2 /2 = k 2 /2 + κ 2 /2 κk ⊥ e -i θ k κk ⊥ e +i θ k k 2 /2 + κ 2 /2 (1.14)
where we have written the off-diagonal terms as

κ(k x ± i η soc k y ) = κk ⊥ e ±i θ k (1.15)
or

e ±i θ k = k x ± i η soc k y k ⊥ k ⊥ = k 2 x + η 2 soc k 2 y (1.16)
Diagonalizing M (k), we obtain

Ĥ0 = k,σ=+,- σ (k) Φσ † k Φσ k (1.17)
where the energies of the two branches write

± (k) = k 2 /2 + κ 2 /2 ± |κk ⊥ | = (k ⊥ ± κ) 2 + 1 -η 2 soc k 2 y + k 2 z 2 (1.18)
The new field operators write

Φ± k = ûk ± e i θ k dk 2 (1.19)
The energy eigenfunctions is composed out of the two spin states of the corresponding momentum with equal amplitude but momentum dependent phase. Figure 1.3 shows the single particle energy spectrum of the ideal gas for two different values of η soc .

CHAPTER 1. INTRODUCTION

Energy spectrum

These last years in the context of ultra-cold atoms [START_REF] Galitski | Spin-orbit coupling in quantum gases[END_REF][START_REF] Zhai | Degenerate quantum gases with spin-orbit coupling: a review[END_REF], both fermionic and bosonic systems were studied in presence of these energy spectra.

In the case of bosons, at low temperature a large fraction of particles occupy the lowest energy state and can give rise to the phase transition known as Bose Einstein condensation (BEC). The absolute minimum of the energy is the key feature for the physics of bosons at low temperature which is strongly affected by the presence of a SOC term. For the pure Rashba gauge potential η soc = 1 the dispersion curve minimum is a constant nonzero radius in the momentum space as shown in the left figure 1.3. This corresponds to a massive degeneracy of the single-particle ground level. In this particular case, as we will see, Bose-Einstein condensation does not necessary occur and a highly correlated ground state may be expected in the presence of a contact interaction.

The existence of this degeneracy determines strongly the behavior of such bosonic systems. The simplest way to break explicitly this symmetry is to change the intensity of the two Raman lasers e.g. changing the amplitude of their plane-waves in the x-y plane leading to η soc = 1, as shown in the right figure 1.3.

What happens to the BEC scenario in presence of a SOC?

Ideal Bose gas in 3D

In three dimensions the degeneracy of the single particle ground state critically changes the BEC scenario. We study, in this section, the effect of SOC term in a homogeneous three dimensional Bose gas. The impact of a SOC on a BEC was already pointed out in recent theoretical studies [START_REF] Ozawa | Condensation transition of ultracold bose gases with rashba spin-orbit coupling[END_REF][START_REF] Cui | Enhancement of condensate depletion due to spinorbit coupling[END_REF].

Ideal Bosons in absence of SOC : κ = 0 We briefly recall the standard BEC phase transition in a three dimensional homogeneous system without any coupling or interaction. The number of particles at temperature T = (k B β) -1 and chemical potential µ ≤ 0 in the energy eigenstate (k) is simply given by the Bose distribution,

N k = (exp(β[ k -µ)] -1) -1
, and the density of the excited states writes [START_REF] Baym | Condensation of bosons with rashbadresselhaus spin-orbit coupling[END_REF]. Two dimensional dispersion of a homogeneous SOCed system. The two branches ± touch at the origin. The left graph corresponds to the pure Rashba term whereas the right graph is plotted for η soc = 0.7. We see in this last case that only two minima appear in p x = ±κ. Measured location of energy minimum or minima, where as a function of laser intensity the characteristic double minima of SOC dispersion move together and finally merge. c) Dispersion measured in 6 Li.

n ex ≡ 1 V k = 0 N k =

CHAPTER 1. INTRODUCTION

where

g α (z) = ∞ l =1 z l
l α is the Riemann Zeta function and

λ T = 2πħ 2 mk B T (1.21)
is the thermal de Broglie wavelength. The BEC phase transition occurs at µ = 0 corresponding to a critical temperature T C where the density of particles in the excited state, Eq.(1.20) saturates and the number of particles in the ground state, N 0 , becomes extensive. The condensed fraction N 0 /N remains non-zero even in the thermodynamic limit. In this case the system has formed a Bose-Einstein condensate in the zero momentum mode k = 0.

Impact of SOC : κ = 0 We now consider a bosonic gas with two internal degrees of freedom. As we have seen in the previous section, in presence of the SOC the dispersion relation ± (k) is not minimal at k = 0 any more. In the basis of energy eigenstates, the total number of particles is given by the sum over both energy branches. The excited state density now writes

n ex = d 3 k (2π) 3 1 e β( + (k)-µ) -1 + 1 e β( -(k)-µ) -1 (1.22)
Depending on the value of η soc , two very different effects occur.

Rashba-Dresselhaus term η soc = 0 In this case, the two energy branches simply correspond to a shift of the energy dispersion in the k x direction

(2π) 3 n ex = ∞ -∞ d k z ∞ -∞ d k y ∞ -∞ d k x   e β (k x +κ) 2 +k 2 y +k 2 z 2m -µ -1   -1 + ∞ -∞ d k z ∞ -∞ d k y ∞ -∞ d k x   e β (k x -κ) 2 +k 2 y +k 2 z 2m -µ -1   -1 (1.23)
Shifting the integration variable, we recover the ideal gas expression for the density of excited particles in each energy branch. Below the transition at µ = 0, both eigenstates with k x = ±κ become occupied. The condensate is therefore built out of two exactly degenerate modes.

CHAPTER 1. INTRODUCTION

Pure Rashba η soc = 1 Whereas η soc < 1 is qualitatively similar to the scenario with η soc = 0 leading to a two-fold degeneracy of the condensate, fully isotropic SOC, η soc = 1, is essentially different. It was experimentally achieved and explored [START_REF] Lin | Spin-orbit-coupled boseeinstein condensates[END_REF][START_REF] Jiménez-García | Tunable spin-orbit coupling via strong driving in ultracold-atom systems[END_REF]. For η soc = 1 the minimum of the single particle energies is infinitely degenerate.

(2π) 2 n ex = ∞ -∞ d k z ∞ 0 k ⊥ d k ⊥ e β (k ⊥ +κ) 2 +k 2 z 2m -µ -1 -1 + ∞ -∞ d k z ∞ 0 k ⊥ d k ⊥ e β (k ⊥ -κ) 2 +k 2 z 2m -µ -1 -1 (1.24)
This expression does not simplify as in the pure Rashba case, but can be written as

n ex = n + + n -= 2 g 3 2 (e βµ-βκ 2 2m
)

λ 3 T + κ λ 2 T ∞ n=1 e nβµ n erf   βn 2m κ   (1.25)
where erf(x) = 1 π x -x e -t 2 d t is the error function. The last term in this expression prevents the occurrence of BEC since it diverges in the limit of vanishing chemical potential. Any arbitrary high density is therefore accessible without the need of macroscopically occupying any single particle state. In contrast to η soc < 1, the ideal Bose gas with isotropic SOC does not have a BEC phase transition at finite temperature.

As pointed out by reference [START_REF] Baym | Condensation of bosons with rashbadresselhaus spin-orbit coupling[END_REF], the absence of a BEC at finite temperature can be explained by the infinite degeneracy of the single particle ground state in the pure Rashba case (η soc = 1). At very low energies the density of states is constant like for non-interacting particles in two dimensions

n(E ) = d 3 k (2π) 3 δ E --(p) ∼ κ 2π (1.26)
Therefore, at low temperatures, our system behaves similar to a two dimensional Bose gas without SOC where Bose condensation is suppressed by the higher density of states.

Ideal Bose gas in 2D

In two spatial dimensions, a Kosterlitz-Thouless phase transition occurs for an interacting homogeneous Bose gas without SOC, so that we expect also the SOCed Bose gas to be particularly affected by presence of interactions. These systems are also of great interest for experimental groups [START_REF] Wu | Realization of two-dimensional spin-orbit coupling for bose-einstein condensates[END_REF]. Before focusing on the interacting system in the next chapters, we study quantitatively in this section the impact of the SOC on a two dimensional non interacting gas.

Ideal Bose gas without SOC: κ = 0 Similar to Eq.(1.22), we calculate the density of non-condensed particles

n = π 4π 2 2mT ħ 2 ∞ 0 d x e -x+βµ 1 -e -x+βµ (1.27) = - mT 2πħ 2 log[1 -e βµ ] (1.28)
Using the thermal wave length defined in Eq.(1.21 the density writes

nλ 2 T = -log[1 -e βµ ] (1.29)
As anticipated, Bose-Einstein condensation is absent, since the density diverges for vanishing chemical potential. Nevertheless, the number of bosons in the ground state, N 0 , becomes large at low temperatures,

N 0 T |µ| e nλ 2 (1.30)
though never macroscopic at any finite temperature, N 0 /V = 0 in the thermodynamic limit.

CHAPTER 1. INTRODUCTION

Isotropic (pure Rashba) SOC Since we have seen that no BEC occurs in two dimensions, we limit ourself to study the impact of a pure Rashba SOC, η soc = 1, where the single particle ground state is infinitely degenerate. In the two dimensional system, we get for the density

2πn = ∞ 0 k ⊥ d k ⊥ e β (k ⊥ +κ) 2 2m -µ -1 -1
(1.31)

+ ∞ 0 k ⊥ d k ⊥ e β (k ⊥ -κ) 2 2m -µ -1 -1 (1.32)
which can be simplified by changing the integration variables

2πn = 2 ∞ 0 r d r e β r 2 2m -µ -1 -1 + 2κ κ 0 d r e β r 2 2m -µ -1 -1 (1.33) (1.34)
or

nλ 2 = -log[1 -e βµ ] + 2κ
mT κ 0 d r 1 e β r 2 2m -µ -1 (1.35)
The additional term on the density due to SOC is always positive, so that the density of particles per spin at constant chemical potential is higher than without SOC.

Although BEC does not occur for η soc = 1 in two and three dimensions at finite temperatures for ideal Bosons, this does not exclude the occurrence of a phase transition in the interacting case. Therefore we investigate the effect of interparticle interactions within the mean field approximation in the next section.

What is the effect of the interactions ?

CHAPTER 1. INTRODUCTION

Interacting Bose gas: Mean Field Approximation

In a dilute system, interactions between the particles are dominated by two-body collisions. At low energies, the two-body interaction can be effectively described by a single parameter, the s-wave scattering length a, independently of the details of the two body potential. In the following, we will use a contact pseudo-potential g δ(r) for the interaction where the interaction strength g is related to the scattering length a through g = 4πħ 2 a/m [START_REF] Pitaevskii | International Series of Monographs on Physics[END_REF]. In the case of a Bose gas with two internal (spin) states, three coupling parameter are in general needed, g σ σ proportional to the scattering amplitudes between different the various hyperfine states as σ and σ . In second quantization, the interaction part of the Hamiltonian is then given by

Ĥint = d x σ,σ =↑,↓ g σ σ Ψ † σ (x) Ψ † σ (x) Ψσ (x) Ψσ (x) (1.36)
with g ↑↓ = g ↓↑ . Together with the single particle Hamiltonian, Eq.(1.17), the total Hamiltonian then writes

Ĥ = Ĥ0 (µ, κ, η soc ) + Ĥint (g ↑↑ , g ↓↓ , g ↑↓ ) (1.37)
At high density and low temperature the energy is dominated by interactions proportional to the density squared, Eq. (1.36). We may expect fluctuations of the density to be strongly suppressed. Replacing the density operator by its expectation value, we obtain a mean-field description neglecting flucuation effects.

As we will show, within mean-field, the low temperature phases are selected by the single combination

g = 2g ↑↓ -g ↑↑ -g ↓↓ (1.38)
In the other limit of high temperatures and low densities, the interparticle interaction can be considered as a perturbation to the non-interacting kinetic energy. This regime is again correctly described in leading order by a mean field theory. In the next section we will study the two mean field prediction at high and low temperature.

CHAPTER 1. INTRODUCTION

High temperature Mean Field Hartree approximation

In this section to review the derivation of the high temperature mean field approximation following closely the reference of J-P Blaizot and G.Ripka, Quantum theory of finite systems [START_REF] Blaizot | Quantum Theory of Finite Systems[END_REF] where the interaction leads to a simple shift of the effective chemical potential. For simplicity, we only consider an averaged interaction strength with g ↑↑ = g ↓↓ = g ↑↓ , or g = 0. Corrections due to an anisotropy g = 0 of the couplings g ↑↑ , g ↓↓ , g ↑↓ can be included without difficulty, but are negligible for all coupling parameters considered in this thesis.

The free energy of the system is defined as,

F = E -T S -µN = -k B T log(Z ) (1.39)
where Z = Tr[exp(-β(H -µN ))] is the grand canonical partition function.

The mean field approximation is based on the following inequality

F ≤ 1 β 〈log ρ 0 〉 0 + 〈 Ĥ 〉 0 (1.40)
where ρ 0 = exp[-H 0 ]/Z 0 is any trial density matrix corresponding to a Hamiltonian H 0 and Z 0 is the corresponding partition function,

〈 • 〉 0 ≡ Tr { • ρ 0 }
Our ansatz for the trial density matrix is based on the non-interacting part of the Hamiltonian with an additional mean-field shift of all energies, such that the Hamiltonian of our system can be separated as

Ĥ = Ĥ0 + Ĥ1 (1.41) Ĥ0 = k,σ=± (k ⊥ + κ) 2 + k 2 z 2m -µ + ξ Φ † k,σ Φ k,σ (1.42) Ĥ1 = g ↑↑ 2V p, k, q Φ † k+ q Φ † p-q Φ k Φp -ξ k,σ=± Φ † k,σ Φ k,σ (1.43)
We obtain

F ≤ 1 β 〈log ρ 0 〉 0 + 〈 Ĥ0 〉 0 + 〈 Ĥ1 〉 0 = F 0 + 〈 Ĥ1 〉 0 (1.44)
where F 0 (T, µ, ξ) = -T log Z 0 is the free energy of our reference system. Using Wick's theorem we get

〈 Ĥ1 〉 0 = g ↑↑ V k 〈 Φ † k Φ k 〉 0 p 〈 Φ † p Φp 〉 0 -ξ k,σ=± 〈 Φ † k,σ Φ k,σ 〉 0 (1.45) CHAPTER 1. INTRODUCTION
Introducing the density of our mean-field approximation

n(ξ) = 1 V k 〈 Φ † k Φ k 〉 0 = - T V d d ξ F 0 (1.46)
the variational expression for the free energy writes

F ≤ F 0 (ξ) + V g ↑↑ [n(ξ)] 2 -ξV n(ξ) (1.47)
Since we have

- ∂F 0 (ξ) ∂ξ = ∂F 0 (ξ) ∂µ = βV n(ξ) (1.48)
the derivative of the variational free energy with respect to ξ leads to the following condition for its minimum

-V n(ξ) + 2V g ↑↑ n(ξ)n (ξ) -V n(ξ) -ξV n (ξ) = 0 (1.49)
which determines the mean-field energy shift self-consistently

ξ = 2g ↑↑ n(ξ) (1.50)
Explicitly, we then obtain a set of equations for the quasi-particle energies and density

M F ± (k) = (k ⊥ ± κ) 2 + k 2 z 2m + 2g ↑↑ n M F (1.51) n M F (µ) = σ=± d 3 k (2π) 3 1 e β( M F σ (k)-µ) -1 (1.52)
which has to be solved self-consistently. For fixed chemical potential, the single particle energies are shifted by a constant 2g ↑↑ n M F . We see that the mean field approach at high temperature leads to qualitatively similar conclusions as for the ideal gas.

We will need this leading order approximation at high temperature to match our classical field calculations in Chapter II.

CHAPTER 1. INTRODUCTION

Mean Field ground states: Plane-Wave & Stripe phase

Our ideal gas calculations have shown that isotropic SOC may suppress Bose condensation at finite temperatures, and our previous mean field calculation only introduces a rigid shift of all energy levels.

Still, approaching zero temperature, the occupation of the lowest energy modes dominate and the formation of a condensate at zero temperature is expected. Here, we study possible phases of the ground state within the mean field approximation.

Let us rewrite the interaction energy, Eq. (1.36),

E int = 1 2 d r σ,σ =↑,↓ g σ σ 〈n σ (r)n σ (r)〉 (1.53)
using the density operator n σ (r). In a homogeneous (translationally invariant) system the tendency of the interaction is to flattened the coupled densities in real space.

Neglecting density fluctuations, we expect two different situations to minimize the interaction energy:

Case g ↑↑ , g ↓↓ > g ↑↓ Each densities n ↑ and n ↓ should be constant in space.

Case g ↑↓ > g ↑↑ , g ↓↓ The sum of the two densities is constant but densities of opposite spin, n ↑ and n ↓ , avoid each other spatially.

Based on this heuristic considerations, we will now write down a mean field ansatz for the different ground states of isotropic and anisotropic SOCed Bosons.

Variational calculation

In Fourier space, using Eq. (1.11), the interaction Hamiltonian, Eq. (1.36), writes

Ĥint = k 1 +k 2 =k 3 +k 4 g ↑↑ 2V û † k 1 û † k 2 ûk 3 ûk 4 + g ↓↓ 2V d † k 1 d † k 2 dk 3 dk 4 + g ↑↓ V û † k 1 d † k 2 ûk 3 dk 4 (1.54)
where we have explicitly written out all three couplings. For our variational calculations of the ground state energy, we start with the simplest, less symmetric case of bosons with anisotropic SOC.

CHAPTER 1. INTRODUCTION

Anisotropic SOC, η soc < 1 In this case the single particle spectrum has two degenerate minima in k = κ = (±κ, 0, 0) corresponding to the single particle states

Φ- † ±κ |0〉 = 1 2 û † ±κ ∓ d † ±κ |0〉 (1.55)
Introducing the angle φ that described the linear superposition of the two minimal states (in a single particle), our mean field ansatz for the ground state then writes

|Φ -(φ)〉 = cos(φ) Φ- † κ + sin(φ) Φ- † -κ N N ! |0〉 (1.56)
with N the number of particles in the system. We then evaluate the interacting energy, Eq.(1.54),

〈Φ -(φ)| Ĥint |Φ -(φ)〉 (1.57)
The non zero components of this expression can be decomposed into three different situations :

(1) all the particles carry a κ momentum, (2) all the particle carry a -κ momentum, (3) the two particles carry different momenta ±κ and exchange a momentum 2κ. In this last case, four different arrangement of momenta ±κ are possible. When the two interacting particle carry a different spin, the negative sign in Eq.

(1.55) has to be taken and it introduces a term proportional to

-g ↑↓ | cos(φ)| 2 | sin(φ)| 2 .
By explicitly calculating each non zero configuration and using the normalization condition we obtain,

〈Φ -| Ĥint |Φ -〉 = N (N -1) 8V g ↑↑ + g ↓↓ + 2g ↑↓ + | sin(2φ)| 2 g ↑↑ + g ↓↓ -2g ↑↓ (1.58)
Minimizing the interaction energy, Eq. (1.58), we distinguish three cases depending on the sign of the anisotropy of the interaction, g .

Case g ↑↑ + g ↓↓ > 2g ↑↓ i.e. g < 0 : The mimimum of the energy corresponds to φ = {0, π/2}. In this case the particle populate a single momentum ±κ and the state is called Plane Wave state.

Case 2g ↑↓ > g ↑↑ + g ↓↓ i.e. g > 0 : The mimimum of the energy corresponds to φ = π/4. The particles are in a superposition of two opposite momenta ±κ e.g. in a superposition of two Plane Wave states. This standing wave forms stripes in the real space and is called Stripe Phase.

CHAPTER 1. INTRODUCTION

Case g ↑↑ + g ↓↓ = 2g ↑↓ i.e. g = 0 : The Plane Wave state and Stripe Phase are degenerate. This degeneracy is present within the mean field approximation and it is not expected to be robust beyond mean field.

Isotropic SOC, η soc = 1 As we have seen in the non-interacting section, when the spin-orbit coupling is isotropic in the xy plane the single-particle ground state is infinitely degenerate along a momentum ring of radius κ. In order to minimize the energy, we can choose a mean-field ansatz where we choose to populate only one direction, so that we essentially recover the scenario above. Allowing for a combination of more momenta in the ansatz, the calculation also involves the angles k 1 , k 2 and k 3 , k 4 in Eq. (1.54). This approach was studied in reference [START_REF] Ozawa | Condensation transition of ultracold bose gases with rashba spin-orbit coupling[END_REF] and numerical calculation done by reference [START_REF] Ho | Bose-einstein condensates with spin-orbit interaction[END_REF] produced consistent with the results based on a single direction shown in figure 1.5.

Mean field phase diagram and ground state wave function

We have seen that in the presence of interactions the mean field ground state breaks the symmetry of the degenerate single particle energy levels selecting one direction |κ| = κ. The wave function of the mean field ground state, Eq. (1.56), writes

ψ M F κ (r) = ψ M F ↑,κ (r) ψ M F ↓,κ (r) = 1 2 cos(φ)e i κ•r 1 -e i θ κ + sin(φ)e -i κ•r 1 e i θ κ
(1.59) up to total phase e i θ . Without loss of generality, we can choose κ = κ(1, 0, 0). For φ = 0 we have a pure plane-wave state :

ψ M F κ (r) = 1 2 e i κx , -e i κx .
The wave-function is described by a single plane-wave and the density of each spin component is therefore flat in space. For φ = π/4 we describe the stripe phase:

ψ M F κ (r) = 1 2
(cos(κx), i sin(κx)). The total density is also flat but the density of each component fluctuates spatially with a defined wavelength κ e.g. appearance of stripes.

However, since mean field usually overestimates ordering, it is questionable if these exotic mean field ground states are really stable at zero temperature, since the large degeneracy of the single particle spectrum may significantly enhance fluctuation effects.

What are the main theories/results beyond the mean approach? CHAPTER 1. INTRODUCTION

Fluctuations & Open questions

Fluctuations beyond mean-field We propose in this section to review different approaches that have been applied to address questions beyond mean field.

• Variational wave function Fragmentation is a main example of a ground state not captured by the mean field approach. Despite being particularly fragile, references [START_REF] Stanescu | Spin-orbit coupled bose-einstein condensates[END_REF][START_REF] Zhou | Fate of a bose-einstein condensate in the presence of spin-orbit coupling[END_REF] propose fragmented states as possible ground states. Based on very dilute limit arguments, reference [START_REF] Tigran | Composite fermion state of spin-orbit-coupled bosons[END_REF] propose a fermonized manybody state by composite fermion construction as a ground state of interacting bosons.

• Effective theory At extremely low temperature, only low energy excitations are populated and the degrees of freedom of the system are therefore expected to be reduced. A more simple, effective theory based on minimal fluctuations around the mean field solution has been used to describe the system. Following this approach we can, for instance, consider only the lower branch of the energy spectrum Φ -(k) in Eq. (1.19). One standard approximation is for instance, by defining the wave-function as a phase and a density component, to integrate out density fluctuations and to consider only phase fluctuations. The Plane Wave state energy is then described by only one phase. On the other hand the Stripe Phase breaks the translational symmetry and is therefore described by two phases [START_REF] Chao | Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons[END_REF]. Within these approaches, references [START_REF] Chao | Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons[END_REF] and [START_REF] Liao | Spin-orbitcoupled bose gases at finite temperatures[END_REF] propose a qualitative phase diagram at finite temperature of a SOCed Bose gas in two dimension. In the third chapter of this thesis, we will draw a significantly different phase diagram based on a classical field theory which is expected to be valid at finite temperature around the phase transition.

• Renormalized T-matrix approach In the pure Rashba case η soc = 1 the interacting term Eq. (1.54) couples any momentum relying on the ring minimum k = κ. As we showed in the previous section, in the case of an fully isotropic interaction g = 0 the ground state is degenerate. In order to lift this degeneracy, reference [START_REF] Ozawa | Ground-state phases of ultracold bosons with rashba-dresselhaus spin-orbit coupling[END_REF] consider renormalized interactions to determine the absolute ground state. Using the T-matrix formalism they consider a renormalized contact interaction that depends on the angle between the two momenta g k,k ,σ,σ . The effective interaction is stronger when k and k are in the same direction, therefore indicating a stripe phase as an absolute ground state.

CHAPTER 1. INTRODUCTION

• Bogoliubov approach The Bogoliubov approach is suited for studying fluctuations on top of the mean field prediction. In our case, this study is particularly interesting in the case of isotropic contact interaction g = 0 when no single ground state is selected within the mean field approximation. By an order by disorder mechanism, reference [START_REF] Barnett | Order by disorder in spin-orbit-coupled bose-einstein condensates[END_REF] finds that the absolute ground state corresponds to a Plane Wave state and results from a competition of the thermal and quantum fluctuations. By studying the depletion of the condensate and the impact of the finite temperature excitations, both references [START_REF] Barnett | Order by disorder in spin-orbit-coupled bose-einstein condensates[END_REF] and [START_REF] Ozawa | Condensation transition of ultracold bose gases with rashba spin-orbit coupling[END_REF] draw a finite temperature phase diagram based on the Bogoliubov approximation, in particular in three dimensions (since the thermal fluctuations diverges in two dimensions).

• Experimental challenge The Stripe Phase is predicted to have supersolid properties by breaking both the gauge and the translational symmetries. Very fragile in experiments, many propositions were made to increase the stability of the stripe phase [START_REF] Martone | Visibility and stability of superstripes in a spin-orbit-coupled bose-einstein condensate[END_REF][START_REF] Martone | Approach for making visible and stable stripes in a spin-orbit-coupled bose-einstein superfluid[END_REF][START_REF] Ozawa | Striped states in weakly trapped ultracold bose gases with rashba spin-orbit coupling[END_REF]. The predicted density modulation of the stripes were observed in 2017 by reference [START_REF] Li | A stripe phase with supersolid properties in spin-orbit-coupled bose-einstein condensates[END_REF]. The role of an harmonic trap was also considered [START_REF] Sinha | Trapped two-dimensional condensates with synthetic spin-orbit coupling[END_REF][START_REF] Ramachandhran | Half-quantum vortex state in a spin-orbit-coupled bose-einstein condensate[END_REF]and the breaking of the stripe phase because of vortices was also carefully studied [START_REF] Kato | Vortex pairs in a spin-orbitcoupled bose-einstein condensate[END_REF]. Because of the SOC term, the wave-function can change sign by either rotating the relative phase or by flipping spin. Halfquatum vortices are therefore naturally present in the system as non standard topological defects [START_REF] Nikolić | Vortices and vortex states in rashba spin-orbit-coupled condensates[END_REF][START_REF] Chao | Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons[END_REF][START_REF] Fetter | Vortex dynamics in a spin-orbit-coupled bose-einstein condensate[END_REF].

• Superfluidity Because of the breaking of Galilean invariance in SOCed system, the superfluidity is expected to be strongly affected by the presence of the SOC. In particular Landau's criterion for the critical velocity cannot be defined independently of the reference frame [START_REF] Zhu | Exotic superfluidity in spin-orbit coupled bose-einstein condensates[END_REF]. Reference [START_REF] Stringari | Diffused vorticity and moment of inertia of a spin-orbit coupled bose-einstein condensate[END_REF] suggests that the normal density does not vanish at zero temperature, a strong reminder of the distinction between superfluidity and BEC.

Despite these efforts, the nature of the quantum ground state of interacting bosons with Rashba SO coupling remains an open issue. It is also a strong motivation for the experimental realization of such a SO coupling in cold atom systems, where a strongly correlated quantum state can be expected.

CHAPTER 1. INTRODUCTION

Outline of the thesis

In the following, we wish to establish the phase diagram of a two-dimensional homogeneous gas of Rashba-Dresselhaus spin-orbit-coupled bosons. This work has being inspired by both fundamental investigation and experimental progress, we will especially insist on a quantitative prediction of the phase diagram and on experimental implications of our results.

Chapter II : The method In order to focus on the finite temperature description of the bosonic gas, we selected a completely different approach from the methods presented in the previous section : the classical field approximation. This approximation is based on the field character description of the bosons that essentially occupy only the very low energy modes which become very highly populated. We then perform classical field Monte Carlo calculations which are expected to correctly describe the finite-temperature behavior close to a possible phase transition.

Chapter III : BKT phase transition After discussing the best numerical tools to correctly evaluate observables of interest like the condensed fraction and the density, we first show that the system undergoes a Kosterlitz-Thouless phase transition from a normal to superfluid state in presence of the SOC term. The thermodynamic limit behavior strongly depends on the anisotropy η soc and in particular, we show that for η soc = 1 a crossover occurs for finite systems at similar phase-space densities, but no superfluid transition is expected for infinite sizes.

Chapter IV : SP/PW orders We then characterize the low temperature phases and we show that the spin order of the quasicondensate is driven by the anisotropy of the interparticle interaction. In particular in the superfluid state, we study the singleparticle density matrix that decays algebraically and directly reflects the PW or SP character of the mean-field ground state. We show that in the case of an anisotropy g = 0 spins exhibit a quasi-long-range order corresponding to the KT transition.

In the case of a fully isotropic interparticle interaction, we show that the PW or SP degeneracy is unaffected by the transition.

Introduction

In this chapter we present the main methods we used to establish the finitetemperature phase diagram of two dimensional Bose gases in presence of SOC and inter particle interactions. We will concentrate on weakly interacting systems at low temperature region. From the absence of BEC in an ideal (or mean-field) gas at any finite temperature, invoking continuity, we can expect the critical temperature of any possible phase transition to approach zero temperature in the limit of vanishing interaction. Approaching zero temperature, bosons essentially occupy only the very low energy modes which become very highly populated. In this regime the field character of the quantum particles dominates and the description in terms of a classical field theory becomes quantitatively accurate [START_REF] Baym | The transition temperature of the dilute interacting bose gas[END_REF][START_REF] Holzmann | Condensate density and superfluid mass density of a dilute bose-einstein condensate near the condensation transition[END_REF][START_REF] Giorgetti | Semiclassical field method for the equilibrium bose gas and application to thermal vortices in two dimensions[END_REF][START_REF] Prokof | Two-dimensional weakly interacting bose gas in the fluctuation region[END_REF][START_REF] Holzmann | Superfluid transition of homogeneous and trapped two-dimensional bose gases[END_REF]. Beyond the weakly interacting region, classical field theory is still capable to describe the universal behavior around a continuous phase transition, a well known result from the theory of critical phenomena [START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF]. In this thesis, we have established for the first time the phase diagram of interacting SOCed bosons, based on classical field Monte Carlo calculations.

Roughly speaking, the classical field description emerges by replacing the occupation of a "quantum mode" of energy ε, given by the Bose distribution (exp[ε/k B T ] -1), with that of the classical field k B T / , and, further, neglecting the non-vanishing commutators of the quantum fields. Both approximations becomes exact for the low energy modes in the limit of T → 0 and provide the starting point to a quantitative description of weakly interacting Bose gases.

Still, due to interactions, we cannot explicitly diagonalize the Hamiltonian even within the classical field approximation. However, the calculation of static observables directly maps to the calculation of classical probability distributions, well known in classical statistical mechanics. In our work, we have used Monte Carlo methods to numerically sample the classical field distribution.

The weight of each classical field configuration is given by a Boltzmann distribution according to its energy. However, two technical issues arise. First, in order to well define the energy of a classical field theory, we have to regularize its behavior at high energies ("ultraviolett divergencies"). Within numerical Monte Carlo calculations, this is naturally taken into account by discretizing the fields on a lattice. Second, SOC formally introduces imaginary terms in the action of the two complex fields representing the two spin states. So we first have to show that we really obtain a probability distribution, i.e the discretized action stays real for any configuration of the fields. Then, we can correctly sample classical field configurations by proposing efficient Monte Carlo algorithms ensuring the ergodicity of the system.

CHAPTER 2. METHODS

Eventually, we have to correct the raw densities of our classical field calculations which depends on the lattice discretization and to take into account the correct behavior of high energy modes which are Bose distributed. However, since these low occupied modes are only weakly affected by the interaction, the mean field approximation provides an accurate description for them. Using this correction, we can in principle match any observable from classical field calculations to provide quantitative predictions for our systems which can be directly compared to experiment [START_REF] Baym | The transition temperature of the dilute interacting bose gas[END_REF]. CHAPTER 2. METHODS

Classical field approximation

As discussed in Chapter I, in this thesis we are interested into establishing the phase diagram in the limit of small interaction strength, mg ↑↑ 1 and small spin-orbit coupling, κλ T 1 where λ T = 2πħ 2 /mk B T is the thermal wave length at temperature T .

In this limit, the leading order corrections to mean-field are captured within classical field theory where the occupation of low energy modes is high such that commutators like [ Ψ † (r), Ψ(r )] can be neglected. In this approximation, the field operator, Ψ(r), can be replaced by two complex fields, Ψ(r) ≡ (Ψ ↑ (r), Ψ ↓ (r)), one for each spin.

Starting from the ideal gas results of two dimensional systems presented in chapter I, we first introduce the classical field approximation in the absence of a SOC. Calculating the phase-space density for particles in the low-energy modes with k < k B T ,

n < λ 2 ≡ λ 2 β k <1 d 2 k (2π) 2 N k = log 1 -e βµ-1 1 -e βµ = nλ 2 + log[1 -e βµ-1 ] ≈ nλ 2 -log e e -1 (2.1)
we notice that at high degeneracy, nλ 2 , the occupation of higher energy modes becomes negligible. At the same level of accuracy, we may also replace the Bose occupation N k by the occupation of classical field,

N k ≈ N c f k ≡ 1 β( k -µ) (2.2)
which gives,

λ 2 β k <1 d 2 k (2π) 2 N c f k = log 1 + |βµ| |βµ| ≈ nλ 2 + |βµ| (2.
3)

The classical field distribution with a simple cut-off Λ = 2mk B T /ħ 2 therefore quantitatively describes the leading order behavior of the density up to corrections of order (nλ 2 ) -1 . The corresponding energy distribution of the classical fields writes

H 0 c f -µN = k<Λ ( k -µ)α * k α k ≈ r - ħ 2 2m |∇ψ(r)| 2 -µ|ψ(r)| 2 (2.4)
where α k are complex numbers which describe the classical field. Their Fourier transform, ψ(r ), defines their real space distribution on a lattice with minimum distance CHAPTER 2. METHODS ∼ Λ -1 . The theory is therefore naturally regularized by discretizing space on a lattice of linear extension L. The probability distribution for a given field configuration, ψ(r ), is then

p[ψ(r )] = Z -1 c f e -β(H c f -µN ) (2.5)
where Z c f = Dψ(r )e -βH c f is the partition function, where D indicates the summation over all discrete field configurations. For our non-interacting system, H c f = H 0 c f and we can explicitly perform the Gaussian integral to obtain Z 0 c f . The classical field description of the ideal Bose gas with SOC does not pose additional difficulties using the eigenmode basis.

For interacting fields, we simply add their interaction energy to the non-interacting system

H c f = H 0 c f -µN + r σσ g σσ 2 |ψ σ (r)| 2 |ψ σ (r)| 2 (2.6)
where we have explicitly written out possible spin dependence of the interaction.

Validity range

In order to study the validity range of the classical field distribution we Taylor expand the nominator of the Bose distribution for small energies,

N k = 1 exp(β( k -µ)) -1 = ∞ n=1 β n ( k -µ) n n! -1
(2.7)

The leading order term is precisely the classical field distribution,

N k → k B T E k -µ under the condition k -µ ≤ k B T (2.8)
Outside this energy range, classical field theory cannot be taken literally. However, interaction corrections to high energy modes k -µ ≥ k B T can usually be treated perturbatively in the limit of weak interactions. We can then match the classical field results with accurate high energy behavior obtained perturbatively.

At very low temperatures, the classical field distribution approaches the configuration which minimizes the energy. From the variation of the fields, we obtain the time independent Gross-Pitaevskii equation in the limit of vanishing temperature. Therefore, the classical field approximation merges continuously the Gross-Pitaevskii CHAPTER 2. METHODS theory. However, quantum corrections, as contained for example in the Bogoliubov approximation, are not included. At high temperatures and weak interactions, the classical field as well as the full quantum field theory are both accurately described by their corresponding mean-field approximation, so that the difference between quantum and classical theory can be worked out analytically.

Between these two regimes, the classical field approximation captures the leading order thermal corrections to mean-field for the strongly degenerate, low energy states. In particular, it is capable to detect and correctly describe any possible finite temperature continuous phase transitions. For systems without SOC classical field theory correctly describes the BEC transition in three dimensions as well as the Kosterlitz-Thouless phase transition in two dimensions (e.g. the XY-model).

CHAPTER 2. METHODS

Markov Chain Monte Carlo

Even after applying the classical field approximation, we cannot explicitly integrate over the distribution p[Ψ(r)] Eq. (2.5) because of the interaction term from Eq. (1.36) that needs to be included. Indeed, the interaction term and the single-particle terms are both diagonal in different basis : real space and Fourier space, respectively.

We will therefore calculate the observables as numerical integrals. Multi-dimensional integrals like Eq. (2.5) can be evaluated using Monte Carlo methods. For numerical efficiency of the algorithm we aim to use a local form of the Hamiltonian. Since the Laplacian and derivative operators remain local around a point r in the real space, we write down the effective total action for a given field configuration in real space.

Effective action S

We can write the probability p[Ψ(r)] of a given classical field configuration as proportional to exp(-S[Ψ(r)]). Summing the kinetic energy, the SOC term and the contact interaction, the total local action S writes

S[Ψ(r)] = a 2 k B T r σ=↑,↓ -Ψ * σ (r) ħ 2 ∇ 2 D 2m Ψ σ (r) -µ|Ψ σ (r)| 2 + ħ 2 κ m Ψ * ↑ (r) -i ∂ D x -η soc ∂ D y Ψ ↓ (r) + ħ 2 κ m Ψ * ↓ (r) -i ∂ D x + η soc ∂ D y Ψ ↑ (r) + 1 2 σ,σ =↑,↓ g σσ |Ψ σ (r)| 2 |Ψ σ (r)| 2 (2.9)
where a is the lattice spacing, µ is the chemical potential, and ∇ D and ∂ D α are finite difference expressions approximating the derivatives and we sum over all positions r of the lattice. Provided that the expression of S[Ψ(r)] is real for any configuration of the field we can sample the distribution by Monte Carlo methods. Below we show explicitly that this discrete action is real.

CHAPTER 2. METHODS

Kinetic term

r Ψ * σ (r)∇ 2 D Ψ σ (r) = r Ψ R σ (r) -i Ψ I σ (r) ∇ 2 D Ψ R σ (r) + i Ψ I σ (r) (2.10) = r Ψ R σ (r)∇ 2 D Ψ R σ (r) + r Ψ I σ (r)∇ 2 D Ψ I σ (r) (2.11) + i r Ψ R σ (r)∇ 2 D Ψ I σ (r) - r Ψ I σ (r)∇ 2 D Ψ R σ (r) (2.12)
with Ψ R σ and Ψ I σ the real and imaginary part of the two component complex field Ψ having spin σ. Using the following finite difference expression of the Laplacian,

∇ 2 D Ψ ↑ (r) = a -2 d i Ψ ↑ (r + ai) + Ψ ↑ (r -ai) -2Ψ ↑ (r) (2.13)
where i denotes the unit vector pointing to the nearest neighbors on the lattice and a the lattice spacing. For simplicity we consider in this section the case a = 1. Together with periodic boundary conditions, we write

r Ψ R σ (r)∇ 2 D Ψ I σ (r) ≡ r Ψ R σ (r)Ψ I σ (r -1) + Ψ R σ (r)Ψ I σ (r + 1) -2Ψ R σ (r)Ψ I σ (r) (2.14) ≡ r Ψ R σ (r + 1)Ψ I σ (r) + Ψ R σ (r -1)Ψ I σ (r) -2Ψ R σ (r)Ψ I σ (r) (2.15) ≡ r Ψ I σ (r)∇ 2 D Ψ R σ (r) (2.16)
Therefore the imaginary part of the kinetic energy identically vanishes and the corresponding action is real for any configuration of the fields.

SOC term

Writing out the Pauli matrices in the SOC part of the action, we have

S SOC = x Ψ * ↑ (r) i ∂ D x + η∂ D y Ψ ↓ (r) + Ψ * ↓ (r) i ∂ D x -η∂ D y Ψ ↑ (r) (2.17)
Let us concentrate on the terms involving ∂ D x first

Ψ * ↑ (r)∂ D x Ψ ↓ (r) + Ψ * ↓ (r)∂ D x Ψ ↑ (r) = Ψ R ↑ (r)∂ D x Ψ R ↓ (r) + i Ψ R ↑ (r)∂ D x Ψ I ↓ (r) -i Ψ I ↑ (r)∂ D x Ψ R ↓ (r) + Ψ I ↑ (r)∂ D x Ψ I ↓ (r) + Ψ R ↓ (r)∂ D x Ψ R ↑ (r) + i Ψ R ↓ (r)∂ D x Ψ I ↑ (r) -i Ψ I ↓ (r)∂ D x Ψ R ↑ (r) + Ψ I ↓ (r)∂ D x Ψ I ↑ (r) (2.18)
Again, using the finite difference expression together with periodic boundary condi-

CHAPTER 2. METHODS tions r Ψ R ↑ (r)∂ D x Ψ R ↓ (r) + Ψ R ↓ (r)∂ D x Ψ R ↑ (r) = r ∂ D x Ψ R ↑ (r)Ψ R ↓ (r) = 0 (2.19) r Ψ I ↓ (r)∂ D x Ψ I ↑ (r) + Ψ I ↑ (r)∂ D x Ψ I ↓ (r) = r ∂ D x Ψ I ↑ (r)Ψ I ↓ (r) = 0 (2.20)
and we obtain r

Ψ * ↑ (r)∂ D x Ψ ↓ (r) + r Ψ * ↓ (r)∂ D x Ψ ↑ (r) ≡ 2i r Ψ I ↓ ∂ D x Ψ R ↑ + Ψ I ↑ ∂ D x Ψ R ↓ (2.21)
After similar manipulations of the terms involving ∂ D y , we obtain

S SOC = 2 r Ψ I ↓ (r)∂ D x Ψ R ↑ (r) + Ψ I ↑ (r)∂ D x Ψ R ↓ (r) + 2η soc r Ψ R ↓ (r)∂ D y Ψ R ↑ (r) + Ψ I ↓ (r)∂ D y Ψ I ↑ (r) (2.22)
Again, the SOC action is real for any field configurations, as well as the interaction energy.

S(Ψ(r)) ∈ R ∀Ψ(r) in a periodic system (2.23)
Therefore, exp(-S[Ψ(r)]) is non negative and we can interpretate weight as a probability of a given field configuration suitable for Monte Carlo sampling.

Monte Carlo algorithms: Metropolis, Heat bath and Fourier moves

In this section, we briefly present few numerical algorithms that we have found essential to correctly sample the distribution of the fields using Monte Carlo methods based on a Markov process. Standard algorithms are typically based on Metropolis' rule for acceptation or rejection of changes in the field configuration.

The Markov chain is constructed by a random walk in configuration space, where the transition probability from one configuration R (note that R in our context labels the values of all fields at each lattice site) to another one R' satisfies

R' T (R → R') = 1 (2.24) R T (R → R') = 1 (2.25) CHAPTER 2. METHODS
together with so called detailed balance condition

π(R)T (R → R') = π(R')T (R' → R) (2.26)
where π(R) ∝ exp[-S(R)] is the probability distribution which we aim to sample.

Metropolis algorithm is a particularly simple solution for the transition where one proposes an arbitrary change of the configuration R which is accepted with probability

T (R → R') = min 1, e -S(R') e -S(R) (2.27)
Whenever rejected, we remain at the same configuration. One can explicitly verify that Metropolis rule satisfies the detailed balance condition.

Since our action, Eq. (2.9) is local, we can efficiently compute S(R') -S(R) for local changes avoiding the calculating of total action at each step.

However, standard Metropolis algorithm leads to a slow convergence for our purposes. Two main problems appear when decreasing the temperature. First, the acceptance of the moves decreases since Metropolis steps are completely random and most of the moves at low temperature lead to high energy changes which are highly unlikely. Secondly, different degenerate ground states of the ideal (or mean-field) system are separated from each other in the sense that they are not connected by local moves. In practice, local moves in real space do not ensure the ergodicity of the sampling for large systems where the probability to tunnel from one ground state to another gets exponentially small.

During the next sections, we propose few ways to tackle these problems. First, we will study the action itself and propose efficient changes instead of total random ones leading eventually to the Heat Bath, Gaussian, and Fourier space algorithms.

A priori probabilities To increase the acceptance probability of the simple Metropolis algorithm, we decompose the transition probability

T (R → R') = A (R → R') p (R → R') (2.28)
into the a-priori probability A (R → R') and the final acceptance rate p (R → R'). Our strategy will be to choose an a-priori probability which is easy to compute, typically CHAPTER 2. METHODS a Gaussian, and which increases the final acceptance rate given by

p (R → R') = min 1, e -S(R') A (R' → R) e -S(R) A (R → R') (2.29)
Instead of proposing completely random moves, we therefore try to orientate changes.

In order to propose the most efficient local move, let us consider the action keeping all the fields fixed except one component Ψ α (r). We will then propose an optimized change of Ψ α (r), the value of the α field at r. The part of the action without spin-orbit coupling involving Ψ α (r) writes

- 1 2 d i [Ψ α (r + ai) + Ψ α (r -ai))] Ψ α (r) + -µ + κ 2 2 + d Ψ α (r) 2 + g ↑↑ 2 N β=1 Ψ β (r) 2 2 (2.30)
where we have assumed symmetric interactions for simplicity. Note that the change of the action is diagonal in the different field components

Ψ α (r) = {Ψ R ↑ (r), Ψ I ↑ (r), Ψ R ↓ (r), Ψ I ↓ (r)
}, whereas the spin-orbit interaction couples different component of the fields.

∆S SOC Ψ R ↑ (r) ∝ const (2.31) ∆S SOC Ψ I ↑ (r) ∝ κ ∂ D x Ψ R ↓ (r) (2.32) ∆S SOC Ψ R ↓ (r) ∝ κ ∂ D x Ψ R ↑ (r) + ∂ D y Ψ I ↑ (r) (2.33) ∆S SOC Ψ I ↓ (r) ∝ κ ∂ D y Ψ R ↑ (r) (2.34)
Thus, changes in the total action involving Ψ α (r) can be written in the general form

∆S(Ψ α (r)) ∝ bΨ α (r) -aΨ α (r) 2 - g ↑↑ 2 Ψ α (r) 4 (2.35) with b = 1 2 d i [Ψ α (r + i) + Ψ α (r -i))] + ∆S SOC (Ψ α (r)) (2.36) a = -µ + κ 2 2 + d + g ↑↑ β =α Ψ β (r) 2 (2.37)
where ∆S SOC (r)) is linear in the field components Ψ β (r) with β = α containing the contributions from the SOC.

CHAPTER 2. METHODS

Heat bath algorithm

The heat bath algorithm provides an exact sampling at high temperatures where interaction effects are small. Neglecting the anharmonic term, ∝ g 0 Ψ α (r) 4 , the change of the action, Eq. (2.35), becomes a quadratic form. The distribution exp(-∆S[Ψ(r)]) is therefore Gaussian centered around the minimum given by

2aΨ α (r) = b (2.38)
or

Ψ α (r) = b 2a = d i [Ψ α (r)(r + i) + Ψ α (r -i))] + 2∆S SOC (r) 2(-2µ + κ 2 + 2d ) (2.39)
for g ↑↑ = 0. Explicitly, we have

∆S ∝ --µ + κ 2 /2 + d Ψ α (r) -Ψ α (r) 2 (2.40)
We can sample exactly the distribution using

Ψ α (r) → Ψ α (r) = Ψ α (r) + δΨ (2.41)
where δΨ is sampled from a normal distribution of variance

σ 2 = 1 -2µ + κ 2 + 2d (2.42)
The corrections needed for the acceptance rate then writes log

A (R' → R) A (R → R') = -Ψ α (r) -Ψ α (r) 2 -δΨ 2 2σ 2 (2.43)
We can further improve the acceptance using Eq. (2.37) for g 0 = 0 which takes into account the local interactions with the other field components.

Gaussian algorithm

The Heat Bath algorithm is based on an essentially exact sampling of the noninteracting system. However, for interacting systems with g ↑↑ > 0, we have shown in Chapter I section 1.3.1 that the effective chemical potential is shifted by the mean field interactions. The chemical potential µ can become positive, and the variance of the Gaussian sampling is not any more guaranteed to be positive.

CHAPTER 2. METHODS

In order to adapt to this situation, we make a general Gaussian ansatz for the a-priori probability

A (R' → R) = exp - Ψ(R) -Ψ(R') + f (Ψ(R')) 2 2σ 2 (2.44)
We determine the mean f and variance σ 2 of this Gaussians such that the acceptance rate gets close to one. We have

log e -S(R') A (R' → R) e -S(R) A (R → R') = -S Ψ + S (Ψ) - Ψ -Ψ + f 2 2σ 2 + Ψ -Ψ + f 2 2σ 2 = -S Ψ + S (Ψ) - f Ψ -Ψ σ 2 + f Ψ -Ψ σ 2 + f 2 2σ 2 - f 2 2σ 2 (2.45)
Assuming small changes in the field, we can expand the action

S (Ψ) = S Ψ + ∂S ∂Ψ Ψ -Ψ + Θ Ψ -Ψ 2 and approximate the acceptance rate log e -S(R') A (R' → R) e -S(R) A (R → R') = (∂ Ψ S + ∂ Ψ S) Ψ -Ψ 2 - ( f + f ) Ψ -Ψ σ 2 + f 2 -f 2 2σ 2 + Θ Ψ -Ψ 2 (2.46)
Up to first order, the acceptance of the moves p (R → R') → 1 is maximized by

f (Ψ) = σ 2
2 ∂ Ψ S, and we have log e -S(R') A (R' → R)

e -S(R) A (R → R') = σ 2 8 (∂ Ψ S) 2 -(∂ Ψ S) 2 + Θ Ψ -Ψ 2 (2.47)
From the second order terms, we can determine the variance σ 2 . However, a simpler solution is to consider σ 2 as an external parameter of our Monte Carlo algorithm, which we adapt for different temperatures to maximize the efficiency of the moves.

Fourier algorithm

Still, at low energy, ergodic sampling of the configurations is challenging. Local moves in the real space can easily change the high momenta k of the energy spectrum. However, it is very difficult at low temperature to equally sample the degenerate energy minima of SOCed bosons.

In order to sample efficiently these minima, we have implemented Metropolis moves in the Fourier space around the minima of the non-interacting energy spectrum. The calculation of the action for this moves scales worse in the system size, N log(N ) where N the number of discretized points, using fast Fourier transform. Below we present the main steps of our Fourier space Metropolis algorithm.

• Calculate the Fourier transforms Φ + k and Φ - k as defined in Chapter I or later in Eq. 2.62.

• Propose a change of mainly Φ - k around the mimina of the energy i.e the most populated momenta

Φ - k → Φ - k = Φ - k + z z ∈ C (2.48)
where z is a random, Gaussian distributed, complex variable. To gain efficiency, we simultaneously compute

u k = u k + e -i θ k ⊥ z d k = d k -z (2.49)
We then choose a momemtum k around the minimum of the energy selecting it from the distribution,

f (k) ∝ e -k 2 2 +κ k 2 x +η 2 soc k 2 y α α ∈ R (2.50)
where α is arbitrarily chosen to fit the density of state in function of the temperature.

• If the change is accepted, we calculate the inverse Fourier transform and update the fields.

Algorithms interplay

In order to optimize the efficiency and reduce the time consumption of the computation, we switch between different algorithms during a single simulation run. At each Monte Carlo step, we randomly select, according to a externally selected probability, one of the different algorithms. Using Metropolis algorithm we obtain a much larger variance and incertitude on the observable than using the Heat Bath and Gaussian algorithm.

CHAPTER 2. METHODS

Partition function

We have presented different algorithms to correctly sample the distribution p[ψ(r )],

p[ψ(r )] = Z -1 c f e -β(H c f -µN ) (2.51)
where Z c f = Dψ(r )e -βH c f is the partition function. One of our central observables in the following is total density as a function of the chemical potential. Discretizing the system size L on N sites, it is given by

n c f = α N i =1 Dψ α (i ) |ψ α (i )| 2 e -βH c f (ψ α (1),ψ α (2),...,ψ α (N )) Z c f (2.52)
Together with the density of particles in the energy minimum state, we will be able to to draw the phase diagram of the interacting SOCed bosons in Chapter III. However, as we have seen above, predictions of the classical field theory may systematically differ from those of interacting bosons, in particular in the high temperature limit.

In the next section, we will show how to reduce this differences in order to make quantitative predictions for dilute Bose gases.

Density matching

Procedure

Within classical field theory, the occupation of eigenmodes of energy is given by the equipartition theorem instead of the full Bose distribution. For weakly interacting systems at high energy, mean-field theory correctly describes the leading order interaction corrections [START_REF] Pitaevskii | International Series of Monographs on Physics[END_REF] and the occupation of energy eigenstates asymptotically approaches their mean-field values at high energy. Therefore, we can correct the densities of our classical field calculations to account for the correct ultraviolet behavior adding the difference

∆n = 1 L 2 k,α=± n B (ε m f ,B kα -µ) -n c f (ε m f ,c f kα -µ) (2.53)
where the single particle mean-field energies are given by ε

m f ,B /c f kα = ε B,c f kα +2 α g αα n m f ,B /c f α
, where ε B,c f kα are the eigen energies of the ideal SOC gas (see below). The corresponding mean-field densities,

n m f ,B /c f α = L -2 k n c f /B (ε m f kα -µ), have to be determined self-consistently as presented in figure 2.3.
Note that the Bose distribution of the occupation numbers merges the classical field occupation for low energies, as we have shown before. Therefore, the low energy modes do not contribute to the density difference, Eq. (2.53), the difference only arises from the different ultraviolet behavior.

Lattice expressions for ideal and mean-field classical fields

In the notation above we have indicated one further subtlety arising from the regularization of our classical field theory. The eigenmodes of our classical field theory on the lattice in general differ from those of the Bose gas for high momenta already for an ideal gas.

Let us therefore calculate explicitly the eigenmodes, ε c f kα and the occupation numbers, n c f (ε c f kα -µ), of the non interacting classical field system as presented in figure 2.4. The action of the ideal system is diagonal in the Fourier space with k i , j = 2π L i , j . The action can then be written

Ψ ↑ (r) = 1 L 2 L i , j =1 u k i , j e i k i , j •r Ψ ↓ (r) = 1 L 2 L i , j =1 d k i , j e i k i , j •r (2.
S U = 1 L 2 i , j k 2 i j ,U 2 -µ + κ 2 2 u -k i , j u k i , j + d -k i , j d k i , j + κ L 2 i , j k x,U -i η soc k y,U u -k i , j d k i , j + κ L 2 i , j k x,U + i η soc k y,U d -k i , j u k i , j (2.55)
where k x,U and k 2 U are the Fourier transform of the finite difference expressions approximating the derivatives. Using the lowest order finite difference expressions of the Laplacian, we have

∇ 2 U Ψ ↑ (r) = a -2 d i Ψ ↑ (r + ai) + Ψ ↑ (r -ai) -2Ψ ↑ (r) (2.56) k 2 U ≡ a -2 d i [2 -2 cos (k(i )a)] = k 2 + O (a 2 k 4 ) (2.57)
The derivatives and the corresponding wave vectors in Fourier space write

∂ x,U Ψ ↑ (r) = 1 2a Ψ ↑ (r + ai x ) -Ψ ↑ (r -ar x ) (2.58) k x,U ≡ 1 a sin (k x a) = k x + O (a 6 k 3 x ) (2.59)
Diagonalizing the action similar to the continuous system studied in section I.1.2, we obtain the corresponding lattice expressions

S U = 1 L 2 i , j k 2 i j ,U + κ 2 2 -µ + κ k 2 x i ,U + η 2 soc k 2 y j ,U Φ + -k Φ + k + 1 L 2 i , j k 2 i j ,U + κ 2 2 -µ -κ k 2 x i ,U + η 2 soc k 2 y j ,U Φ - -k Φ - k (2.60)
With the new basis vectors

Φ + (r) = e i θ k Ψ ↑ (r) + Ψ ↓ (r) 2 Φ -(r) = e i θ k Ψ ↑ (r) -Ψ ↓ (r) 2
(2.61)

Φ + k = 1 L 2 r Φ + (r)e -i k•r Φ - k = 1 L 2 r Φ -(r)e -i k•r (2.62)
where CHAPTER 2. METHODS Since these fields are non interacting, we can then explicitly calculate the density Eq.

e i θ k = (k x,U + i η soc k y,U )/ k 2 x,U + η 2 soc k 2 y,U CHAPTER 2. METHODS
(2.52) as a Gaussian integral

n c f (ε c f kα -µ) = 1 2L 2 i , j     1 k 2 i j ,U 2 -µ + κ 2 2 + κ k 2 x i ,U + η 2 soc k 2 y j ,U     + 1 2L 2 i , j     1 k 2 i j ,U 2 -µ + κ 2 2 -κ k 2 x i ,U + η 2 soc k 2 y j ,U     (2.63) with k x = 2πi N and k y = 2π j N .
The analytical results of the non-interacting classical field on the lattice already presents an important benchmark of our numerical Monte Carlo calculation in the non-interacting limit. In order to include the mean field corrections, it is rather straightforward to use these results and solve the self-consistent mean-field equation for the density n c f (ε

m f ,c f kα -µ).
CHAPTER 2. METHODS

Convergence and scale of energy

.5 presents, at each step of an algorithm, the density as a function of the corresponding instantaneous value of the action. We see that at high temperature i.e low density, the action is only determined by the chemical potential and fluctuations of the density are strong. At low temperature, fluctuations around the mean density are highly suppressed. We comment in more detail of this feature in the Appendix. In this strongly degenerate regime the convergence of the algorithm towards the correct distribution can be rather slow and must be checked for each observable separately.

Studying the correlation of the observable with the value of the action can give important insight into its convergence properties.

• I. Density n In all regimes, the density usually converges fast since density changes are strongly correlated with the action. As shown in figure 2.2, the density converges rapidly towards its mean value determined by the chemical potential µ.

• II. Condensed fraction and momentum distribution n(k) As shown in figure 2.2, the condensed fraction converges significantly slower than the density.

• III. SP and PW states n κ 0 The correct balance between the population of degenerate momenta is typically converging slower than the condensate fraction. Since in our later study we focus on the regime of very small anisotropy g , PW and SP states are always very close in energy. At low temperature, the local minima are ubiquitous and the distribution is converging too slowly for purely local moves. Global changes like the Fourier algorithm described in the previous section are needed to reach convergence. CHAPTER 2. METHODS

Résumé

Au cours de ce chapitre nous avons présenté les principales méthodes utilisées au cours de cette thèse pour établir le diagramme de phase à température non nulle d'un gaz de bosons bidimensionnels en présence d'un couplage spin-orbite et d'interactions interparticules. Nous nous sommes intéressés à des systèmes avec de faibles interactions et à des températures basses. En se basant sur l'absence d'un condensat de Bose-Einstein dans un gaz idéal (ou dans le cas des théories champ moyen) des arguments de continuité tenderaient vers une température critique d'une possible transition repoussée à zéro dans la limite d'une interaction également infiniment proche de zéro.

En approchant le zéro absolu, la vaste majorité des bosons occupent les états de très basse énergie qui sont donc fortement peuplés. Dans ce régime, le caractère ondulatoire des particules quantiques domine et la description en tant que champs classiques devient quantitativement correcte [START_REF] Baym | The transition temperature of the dilute interacting bose gas[END_REF][START_REF] Holzmann | Condensate density and superfluid mass density of a dilute bose-einstein condensate near the condensation transition[END_REF][START_REF] Giorgetti | Semiclassical field method for the equilibrium bose gas and application to thermal vortices in two dimensions[END_REF][START_REF] Prokof | Two-dimensional weakly interacting bose gas in the fluctuation region[END_REF][START_REF] Holzmann | Superfluid transition of homogeneous and trapped two-dimensional bose gases[END_REF]. Au-delà du régime de faibles interactions, la théorie de champs classiques reste capable de décrire des comportements universels autour d'une transition continue, un phénomène très connu des théories critiques [START_REF] Zinn-Justin | Quantum Field Theory and Critical Phenomena[END_REF]. Au cours de cette thèse et en se basant sur des calculs de champs classiques, nous avons établi pour la première fois le diagramme de phase de bosons interagissant avec couplage spin-orbite.

De manière schématique, la description type champs classiques apparaît en remplaçant l'occupation d'un état quantique donnée par la distribution de Bose (exp[ε/k B T ] -1) avec une énergie ε, par l'occupation d'un champ classique k B T / tout en négligeant les commutateurs des champs quantiques. Ces deux approximations deviennent exactes pour les états de basse énergie dans la limite T → 0 et elles fournissent le point de départ d'une description quantitativement correcte d'un gaz de Bose faiblement interagissant.

Pourtant, à cause des interactions, il n'est pas possible de diagonaliser explicitement l'Hamiltonien même après avoir appliqué l'approximation type champs classiques. Toutefois, le calcul des observables statiques correspond directement à un calcul de distribution de probablilités, bien connu en physique statistique classique. Au cours de cette étude et en se basant sur ce constat, nous avons utilisé les méthodes Monte Carlo pour échantillonner numériquement la distribution des champs classiques.

Le poids de chaque configuration du champ classique est donné par la distribution de Boltzmann en fonction de son énergie. Cependant, deux problèmes techniques CHAPTER 2. METHODS apparaissent en préambule de notre calcul. Premièrement, afin de correctement définir l'énergie d'une théorie de champs classiques, il faut régulariser son comportement à hautes énergies ("divergences ultraviolettes"). Au travers du calcul numérique Monte Carlo, cet obstacle est naturellement résolu par la discrétisation des champs sur réseau. Deuxièmement, le couplage spin-orbite introduit des termes imaginaires dans l'action des deux champs complexes représentant les deux états de spin. Nous devons donc montrer que l'on obtient réellement une distribution de probabilité, c'est à dire que l'action discrétisée est maintenue réelle quelque soit la configuration des champs. Ensuite seulement, il est possible de correctement échantillonner les configurations des champs classiques en proposant des algorithmes Monte Carlo efficaces qui assurent l'ergodicité du système.

Enfin, nous avons corrigé les densités brutes provenant des calculs de champs classiques qui dépendent de la discrétisation sur réseau en tenant compte des comportements à hautes énergie des modes distribués selon la statistique de Bose. Cependant, ces états étant peu peuplés, ils sont peu affectés par les interactions et l'approximation type champ moyen en fournit donc une description très pertinente. En utilisant cette correction, il est en principe possible de corriger chaque observable à partir des calculs de champs classiques et de fournir des prédictions quantitatives pour nos systèmes alors directement comparables aux expériences [START_REF] Baym | The transition temperature of the dilute interacting bose gas[END_REF].

Introduction

In this chapter, we explore the phase diagram of a two dimensional SOCed Bose gas based on the methods presented in the last chapter. Here, we will first establish the presence or absence of a finite temperature phase transition in the interacting system and provide quantitative predictions for the phase diagram. In the next chapter, we will study and characterize the (quasi-) ordering of the different phases at low temperature.

According the Mermin-Wagner theorem [START_REF] Hohenberg | Existence of long-range order in one and two dimensions[END_REF][START_REF] Mermin | Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic heisenberg models[END_REF][START_REF] Coleman | There are no goldstone bosons in two dimensions[END_REF], no long range order can occur at finite temperatures. Still, in the absence of SOC, a Berenzinskii-Kosterlitz-Thouless phase (BKT) transition from the normal to a superfluid phase occurs for interacting Bose gases [START_REF] Holzmann | Superfluid transition of homogeneous and trapped two-dimensional bose gases[END_REF] where the low temperature superfluid phase is characterized by algebraic (quasi-long range) order.

Our numerical studies clearly establish that the weakly interacting Bose gas still undergoes a BKT phase transition for anisotropic SOC, η soc < 1. In the low temperature phase, the condensate fraction decays algebraically with system size and the gas becomes superfluid. However, for isotropic SOC, η soc = 1, our calculations show a cross-over behavior at finite systems, with strong evidence for the absence of a finite temperature phase transition in the thermodynamic limit.

Bose gas without SOC: Berezinskii, Kosterlitz and

Thouless phase transition

Let us first briefly review the Berezinskii, Kostelitz and Thouless transition of the interacting two-dimensional Bose gas without SOC described by the Hamiltonian

H = d 2 r Ψ † (r ) - ħ 2 2m ∆ + g 0 2 Ψ † (r )Ψ(r ) Ψ(r ) (3.1)
where g 0 is the interaction strength. At low temperatures, density fluctuations are strongly suppressed. Keeping only phase fluctuations, Ψ(r ) = ρe i θ(r ) , the Hamiltonian can be reduced to the so-called XY model [START_REF] Fröhlich | The kosterlitz-thouless transition in twodimensional abelian spin systems and the coulomb gas[END_REF].

In the XY-model, a Berenzinskii-Kosterlitz-Thouless transition takes place [START_REF] Berezinski | Destruction of Long-range Order in One-dimensional and Twodimensional Systems Possessing a Continuous Symmetry Group. II. Quantum Systems[END_REF][START_REF] Kosterlitz | Ordering, metastability and phase transitions in two-dimensional systems[END_REF][START_REF] Kosterlitz | The critical properties of the two-dimensional xy model[END_REF] where the superfluid density, n S , jumps from n s = 0 at high temperatures to n S = 2mk B T /πħ 2 (or n S λ 2 = 4) at the transition temperature T C . Below T C , the firstorder correlation function, g 1 (r ) = 〈Ψ † (r )Ψ(0)〉, algebraically decays, g 1 (r ) ∼ r -η(T ) characterized by a temperature dependent exponent η(T

) = 1/n S λ 2 . At T C , η(T C ) = 1/4
, and the exponent decreases with decreasing temperature. Thus, the algebraic decay is quite slow, so that for any finite size system we can expect a significant condensate fraction

n 0 = N 0 N = 1 N d 2 r g 1 (r ) ∼ L 2-η L 2 ∼ N -η/2 (3.2)
Although the condensate fraction vanishes in the thermodynamic limit, numerical simulations as well as many experimental systems will be affected by strong finite size effects. Experiments on ultra cold atomic gases typically involve mesoscopic system sizes, e.g. N ∼ 10 4-8 , where the condensate fraction at T C still plays a dominating role, n 0 ∼ 10 -1 .

As presented in Chapter I, in a SOCed two dimensional system the dispersion relation of the single particle is very different from Eq. (3.1), and quasi-long range order may be destroyed in certain parameter regimes [START_REF] He | Instability of a two-dimensional bose-einstein condensate with rashba spin-orbit coupling at finite temperature[END_REF].

What is the effect of the SOC on the BKT scenario ?

In the following, we study the condensate and superfluid fraction for different SOC anisotropy, η soc = 0, 0.5, 0.9, and 1, for signatures of a possible finite temperature phase transition.

Bose gas with SOC: condensate fraction

For a single component Bose gas without SOC, the single particle density matrix depends only on distance and decays algebraically in the low temperature superfluid phase [START_REF] Berezinski | Destruction of Long-range Order in One-dimensional and Twodimensional Systems Possessing a Continuous Symmetry Group. II. Quantum Systems[END_REF][START_REF] Kosterlitz | Ordering, metastability and phase transitions in two-dimensional systems[END_REF][START_REF] Kosterlitz | The critical properties of the two-dimensional xy model[END_REF]. The same algrebraic decay propagates to the condensate fraction.

In the case of a two-component Bose gas with SOC, the single particle density matrix further depends on the spin-projection, G σ,σ (r, r ). Quasi-long range order occurs in the distribution of the modes and the single particle density matrix gets dominated by one or few highly occupied modes. We therefore project G σ,σ (r, r ) over all degenerate PW mean-field ground states, e.g. we sum over all the minima of the single particle spectrum

n κ 0 = k=(±κ,0) σσ d rd r L 2 ψ M F kσ (r)G σ,σ (r, r )ψ M F * kσ (r ) (3.3)
to estimate the condensate fraction n κ 0 /n where n = σ G σσ (r, r) is the total particle density. It is important to notice that n κ 0 is a direct indicator for a phase transition. However, it does not fully describe the character of the low temperature phase, in particular it does not distinguish between PW or SP order.

In figure 3.1 we show n κ 0 as a function of density for a finite system of extension L/a = 80 where a = ħ/ mk B T is the minimal distance on our lattice. The condensate fraction grows rapidly around a cross-over density which decreases from η soc = 0 to η soc = 1. However, no differences are visible changing the sign of our small anisotropic interaction from negative to positive g . We therefore expect that the cross-over/transition temperature is a smooth, continuous function of g around g = 0.

In two dimensional systems of infinite size, the condensate density is expected to vanish at any finite temperature. However, as explained in the previous section, huge finite size effects are expected. In order to determine a possible sharp phase transition in the thermodynamic limit, we have to determine the behavior of the condensate fraction increasing the system size. The occurrence of a BKT phase then shows up in the algebraic scaling of the condensate fraction with system size, n κ 0 /n ∼ L -η(T ) . ] -1 for a finite system of length L/a = 80. The cross-over from normal to condensed phase slightly lowers with increasing SOC anisotropy, η soc . Although the PW/SOC character of the condensate depends essentially on the sign of the anisotropic interaction mg = ±π/100, differences in n κ 0 between g ≥ 0 and g < 0 for equal SOC are beyond our resolution. The colored zones indicate our estimates for the Kosterlitz-Thouless transition in the thermodynamic limit from finite-size-scaling of the condensate fraction described in the text. Condensate fraction, n κ 0 /n, as a function of inverse volume in presence of the SOC but in absence of interactions. As we will precisely detail in the paragraph dedicated to the isotropic SOC, we notice that the absolute value of the condensed density depends on the sum introduced in Eq. 3.4, describing the number of degenerate points corresponding to the fundamental state |k| = κ. The condensate density decreases with the volume for any density, no transition occurs.

CHAPTER 3. INTERACTING BOSONS WITH SOC IN 2D: PHASE DIAGRAM

Non-interacting system

Let us start discussing finite size effects for non-interacting bosons. In figure 3.2 we show the behavior of the condensate fraction in the non-interacting case. The exact expressions of this limiting case not only provide a benchmark for our simulation, but also serve to illustrate the finite size scaling of the condensate in the normal phase. We then obtain

n = k B T L 2 i , j     1 k 2 i j ,U 2 -µ + κ 2 2 + κ k 2 x i ,U + η 2 soc k 2 y j ,U     + k B T L 2 i , j     1 k 2 i j ,U 2 -µ + κ 2 2 -κ k 2 x i ,U + η 2 soc k 2 y j ,U     n κ 0 = |k|=κ k B T L 2 µ (3.4)
In the expression of the condensate density, we explicitly sum over all degenerate ground states with |k| = κ. It is important to keep in mind that the number of degenerate minima depends on the isotropicity of the SOC. For η soc < 1, we have two degenerate minima, whereas for isotropic SOC, η soc = 1, we have an infinite degeneracy, a circle in momentum space in the thermodynamic limit. Finite size effects introduce qualitative changes for η soc = 1, where the additional symmetry of the underlying lattice strongly reduces the degeneracy to a finite number. These effects are present in both figures 3.4 and 3.5.

Nevertheless, in both cases, we see that the condensate density decreases with the system size as

n κ 0 n ∝ 1 L 2 .
This exponent of this algebraic decay simply reflects that the condensate fraction decays as 1/L 2 for all densities and no phase transition occurs.

Interacting system with anisotropic SOC

We now turn to the interacting systems, summarizing our results of the classical field Monte Carlo calculations. Whereas in the high temperature, normal phase the condensate density decreases with the volume, η = 2, the exponent changes rapidly around the temperature where condensation occurs in the finite system. At lower temperatures, the exponent almost vanishes. This behavior is consistent with a Berenzinskii-Kosterlitz-Thouless transition [START_REF] Berezinski | Destruction of Long-range Order in One-dimensional and Twodimensional Systems Possessing a Continuous Symmetry Group. II. Quantum Systems[END_REF][START_REF] Kosterlitz | Ordering, metastability and phase transitions in two-dimensional systems[END_REF][START_REF] Kosterlitz | The critical properties of the two-dimensional xy model[END_REF]. Assuming the transition to be within the Kosterlitz-Thouless class, the critical temperature can be estimated to , for anisotropic SOC bosons with η soc = 0 at different phase space densities and anisotropic interaction, g > 0. We observe a transition between the high temperature regime, where the condensate density decreases with the volume, and the low temperature regime where the exponent depends on the density and eventually vanishes. The solid black line indicates the critical exponent η = 1/4. Our simulations indicate that for anisotropic SOC, η soc < 1, the Berenzinskii-Kosterlitz-Thouless phase occurs at finite temperature, independent of the sign of the anisotropy g of the interaction (see figure 4.1 for η soc = 0). Further, the limit of isotropic interaction, g = 0, is approached smoothly from both sides, g > 0 and g < 0, so that the critical temperature is continuous around g = 0. This latter behavior is in contradiction with the discontinuity predicted in reference [START_REF] Chao | Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons[END_REF].

Interacting system with isotropic SOC

For isotropic SOC, η soc = 1, the system behaves qualitatively different. As shown in figure 3.3 for g < 0, we do not observe the onset of quasi-long range order over the whole density regime and system sizes we considered.

For g > 0, we observe a cross-over similar to η soc < 1, but this time the onset of algebraic order strongly depends on the number of degenerate mean-field ground states. For our finite simulation box, only 4 or 8 minima are strictly degenerate for the system sizes we considered. The circular degeneracy only occurs after performing the thermodynamic limit. As shown in figure 3.5, the behavior of the condensate fraction is qualitatively and quantitatively affected by the number of degenerate states. In particular, the onset of algebraic order is shifted towards considerable higher densities, i.e lower temperatures, increasing the degeneracy from 4 to 8 degenerate modes. For η soc = 1 and infinity system sizes, the transition will therefore be shifted to zero temperature and no finite temperature transition with algebraic order in the single particle channel should occur.

It is important to point out that the BKT transition is absent within the classical field calculation. Therefore, the transition is suppressed by purely classical fluctuations in strong contrast to prediction of reference [START_REF] Liao | Spin-orbitcoupled bose gases at finite temperatures[END_REF] i.e. quantum fluctuations do not play an essential role. , for isotropic SOC bosons with η soc = 1 at different phase space densities and anisotropic interaction, g < 0. We do not observe any quasi-long range order. Dashed lines corresponds to finite systems with 8 degenerate minima where the algebraic behavior, n κ 0 ∼ L -η with η > 1/4, at high phase space density is suppressed.

CHAPTER 3. INTERACTING BOSONS WITH SOC IN 2D: PHASE DIAGRAM

Let us illustrate in more detail the degeneracy of the finite size simulation for isotropic SOC. In our simulations, we have chosen the value of the SOC strength κ = 2π 40 , commensurable with κ ≡ k = 2πi

L for system sizes L = 40, 80, 120 where we have four minima at k = (±κ, 0) and k = (0, ±κ). Instead, for L = 81, 144, we have eight minima as in the scheme shown below. Notice, that for g < 0, where mean field predicts PW, we have four or eight possibilities for the direction. Instead, for g > 0, the mean field ground state is a superposition of two opposite PW phases, so that we only have half of the possibilities for the unsigned momentum direction, two or four possibilities in our case. This simple picture may already explain the main qualitative different behavior of the condensate fraction depending on the sign g of the interparticle interaction we have observed in our simulations.

Intermezzo: XY vs Heisenberg model

In the thermodynamic limit, for η soc = 0 and isotropic interaction, we can eliminate the SOC via a gauge transformation, and we obtain a Bose gas with two internal spin component and isotropic interaction. This model is equivalent to a field theory with N = 4 internal components. Only the model with N = 2 maps to the XY model giving rise to a BKT transition. For N > 2, a Kosterlitz-Thouless phase transition is absent [START_REF] Polyakov | Interaction of goldstone particles in two dimensions. applications to ferromagnets and massive yang-mills fields[END_REF][START_REF] Brézin | Spontaneous breakdown of continuous symmetries near two dimensions[END_REF], in accordance with the absence of a phase transition in the 2D Heisenberg model.

However, in the case of η soc = 0, the thermodynamic limit is singular. For any finite system, the above argument only applies for situations where κ is commensurate with the boundary conditions. Here, we address the limit η soc → 0 continuously connected to non vanishing η soc > 0, which corresponds to situation with noncommensurate values of κ of any finite system, so that the BKT transition survives even in the limit of infinite system sizes. The situation η soc = 0 is therefore singular.

CHAPTER 3. INTERACTING BOSONS WITH SOC IN 2D: PHASE DIAGRAM

Superfluidity

In systems without SOC, the quasi-long range order in the low temperature phase also implies superfluidity, and one of the most striking prediction of the Kosterlitz-Thouless transition is the occurrence of a universal jump of the superfluid density at the critical temperature [START_REF] Nelson | Universal jump in the superfluid density of two-dimensional superfluids[END_REF][START_REF] Kosterlitz | The critical properties of the two-dimensional xy model[END_REF][START_REF] Agnolet | Kosterlitz-thouless transition in helium films[END_REF]. In SOC systems, Galilean invariance is broken with important consequences for the superfluid phase. In such systems peculiar features have been predicted like the appearance of spatial anisotropic superlfuidty, BEC with zero superfluid fraction at zero temperature [START_REF] Stringari | Diffused vorticity and moment of inertia of a spin-orbit coupled bose-einstein condensate[END_REF], and a critical velocity which is not uniquely defined [START_REF] Zhu | Exotic superfluidity in spin-orbit coupled bose-einstein condensates[END_REF].

In thermal equilibrium, the superfluid mass density ρ S can be directly related to the phase stiffness ρ S = ∂ 2 F (θ) ∂θ 2 where F (θ) is the free-energy density where the momentum operator p is replaced by pθ in the Hamiltonian [START_REF] Baym | The microscopic description of superfluidity[END_REF][START_REF] Holzmann | Condensate superfluidity and infrared structure of the single-particle green's function: The josephson relation[END_REF][START_REF] Josephson | Relation between the superfluid density and order parameter for superfluid he near tc[END_REF][START_REF] Pollock | Path-integral computation of superfluid densities[END_REF].

We have therefore further calculated the superfluid and normal mass density, ρ n = mn -ρ s from the phase stiffness. For η soc = 0 and isotropic interactions, we have

ρ n = 1 k B T L 2 〈 P tot x + ħκS tot x 2 〉 (3.5)
where P tot is the total momentum and S tot = d rS(r) the total magnetization of the system. Deviations from a Boltzmann distribution of [P tot x + ħκS tot x ] 2 /(2mnL 2 ) are directly connected to the quantization of the center of mass motion in the x direction. For general η soc > 0 or anisotropic interactions g = 0, quantum effects may modify superfluid properties [START_REF] Zhang | Superfluid density of a spin-orbit-coupled bose gas[END_REF], but we can still use Eq. (3.5) to study the universal behavior of the normal density around a superfluid phase transition.

Our results for the normal/superfluid density (see inset of figure 3.1 for η soc = 0.5, g < 0) confirm the conclusions drawn above from the finite-size analysis of the condensate fraction. Consistent with the prediction of Berenzinskii, Kosterlitz and Thouless, the low temperature, algebraically ordered phase is superfluid for η soc < 1. The transition temperature is roughly independent of the sign of g and decreases with increasing η soc . It vanishes for isotropic SOC with increasing degeneracy. This absence of a transition for isotropic SOC is consistent with recent hydrodynamical results predicting the appearance of rigid flow at zero temperature in three spatial dimensions [START_REF] Stringari | Diffused vorticity and moment of inertia of a spin-orbit coupled bose-einstein condensate[END_REF].

CHAPTER 3. INTERACTING BOSONS WITH SOC IN 2D: PHASE DIAGRAM

Conclusion and phase diagram

In conclusion we have drawn the finite temperature phase diagram of a two dimensional SOCed Bose gas. We have shown the signature of a KT transition in the case of η soc < 1 with the presence of superfluidity in the low temperature phase. By scaling the condensed fraction with the system size we have predicted the critical densities corrected by the matching presented in Chapter II. In the particular case of a pure Rashba SOC η soc = 1, we have shown that a crossover occurs for finite systems at similar phase-space densities, but no superfluid transition is expected in the thermodynamic limit. Beyond mean field, the existence and appearance of such exotic phases are still unclear and open questions remain. As an example, reference [START_REF] Gopalakrishnan | Universal phase structure of dilute bose gases with rashba spin-orbit coupling[END_REF] proposes a phase diagram based on qualitative argument in the very dilute limit, which does not depend on the sign of anisotropy g , in contrast to the mean field predictions. In the following, we will investigate in detail, the character of the low temperature phases observed in our classical field simulation.

In particular, we will study the magnetic ordering of the atoms/spins at low temperature and link it to the BKT phase transition studied in Chapter III. We show that in the case of an anisotropy g = 0 the spin correlations exhibit quasilong-range order induced by the KT transition in contrast to prediction of long range order from reference [START_REF] Su | Hidden long-range order in a spin-orbit-coupled twodimensional bose gas[END_REF]. We therefore find a phase diagram in strong connexion to the Kosterlitz-Thouless transition. We also investigated predictions of zero momentum transition by [START_REF] Chen | Quantum and thermal fluctuations in a raman spin-orbit-coupled bose gas[END_REF][START_REF] Yu | Ground-state phase diagram and critical temperature of twocomponent bose gases with rashba spin-orbit coupling[END_REF], or the appearance of bosons pairs by references [START_REF] Gopalakrishnan | Universal phase structure of dilute bose gases with rashba spin-orbit coupling[END_REF][START_REF] Chao | Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons[END_REF] but we did not find any indications of such exotic phases and behavior.

In the case of isotropic interactions, g = 0, mean field calculations do not select an unique ground state, SP and PW states remain degenerate. In this case, one may expect that thermal and quantum fluctuations break this degeneracy and select an unique ground state. The classical field approximation allows us to address the question of thermal fluctuations in a direct and explicit way. Here, we show that the system undergoes a KT transition without selecting a unique ground state. Instead, our calculations predict a fractionalization of the condensate where SP and PW remain degenerate.

Parameter values in the simulations

In order to study the competition between SOC and interparticle interaction, we have fixed σσ =↑↓ mg σσ /4 = κ/ mk B T = π/20 with mg = 0 to address isotropic interaction and mg = ±π/100 to slightly break the spin isotropy of scattering particles.

CHAPTER 4. LOW TEMPERATURE STATES: PLANE WAVE AND STRIPE PHASE

Reduced single body density matrix

In the previous chapter, we have observed a Kosterlitz-Thouless transition for η soc < 1. The low temperature phase was characterized by the occurrence of a quasi-condensate, a large, though non extensive, occupation of the two degenerate single particle ground states. More precise information on the character of the quasicondensate can be expected from the full calculation of the reduced single-particle density matrix given by

G σ,σ (r, r ) = 〈 Ψ † σ (r ) Ψσ (r)〉 (4.1)
In the high temperature regime, the one-body density matrix decays to zero over a distance given by the thermal de Broglie length λ T . At the transition, quasi-long range order appears, characterized by an algebraic decay. In momentum space, the reduced one-body density matrix writes

G σ,σ (k, k ) = 〈 Φσ † k Φσ k 〉 (4.2)
and the distinction between quasi and true long range order is directly connect to the occurrence of a quasi-condensate or full BEC in momentum space.

BEC

The momentum distribution for a system revealing BEC exhibits a singular behavior at the minimum of the energy spectrum. In the case of a three dimensional ideal Bose gas without SOC, only the zero momentum state is macroscopically occupied. In real space, the one-body density matrix at large distances saturates to a finite value set by the condensate fraction of the gas. The system is then said to show off diagonal long-range order, i.e. finite values in G(r, r ) for r = r . The criterion for BEC of a macroscopic occupation given by Penrose and Onsager [START_REF] Penrose | Bose-einstein condensation and liquid helium[END_REF] is thus equivalent to the existence of a long range order.

In the case of ideal SOCed Bosons in three dimensions, the ground state is degenerate leading to a macroscopic occupation of all modes with |k| = κ. The Fourier transform of these modes will lead to oscillations of the one-body density matrix in real space.

Although possible, the observation of true-long range order gets more involved in the real space density matrix than in Fourier space. In general, BEC corresponds to a macroscopic occupation of one (or more) eigenmodes of the single-particle density matrix. In the case of SOC, these eigenmodes in general couple spin and momentum degrees of freedom.

BKT phase

In the case of a quasi-condensate below the BKT transition in two dimensions, the single particle density matrix in real space is algebraically decaying, with additional oscillations for systems with SOC corresponding to peaks of the momentum distribution at non-vanishing momenta. The spin-structure of the quasi-condensate can be obtained from the dominating modes after diagonalization of the single particle density matrix.

Since the occupation number is strongly peaked for k = κ with |κ| = κ we have calculated the reduced single particle density matrix only for momenta at the minimum of the single particle energy spectrum, G σ,σ (κ, κ ). For η soc < 1, we only need to consider (±κ, 0). The resulting 4 × 4 matrix can be calculated using classical field Monte Carlo.

Explicitly, using the definition 1.11, this reduced density matrix writes 〈M (κ)〉 with

M (κ) = u * κ d * κ u * -κ d * -κ ⊗      u κ d κ u -κ d -κ      (4.3)
Population of momenta ±κ In the last Chapter we have shown that for η soc < 1 the system undergoes a BKT transition. There are only two minima in the single particle energy spectrum ±κ = (±κ, 0) and their population n(±κ) depends on the sign of the interaction g . In order to study the matrix M (k) the two directions ±κ are of course equivalent.

For isotropic SOC, η soc = 1, the minimum of the energy spectrum is a full ring of radius |k| = κ in the thermodynamic limit. However, for our numerical calculation on a finite system, we have only a small number of degenerate single particle ground states, typically four or eight. However, we numerically observe that for finite simulation time, only one direction κ is selected. The two corresponding momenta ±κ are almost macroscopically populated, whereas the other momenta on the ring have a negligeable occupation at low temperature when the density is large nλ 2 >∼ 1.

Averaging over different initial conditions reestablishes the symmetry between all directions κ. , for anisotropic SOC bosons with η soc = 0 at different phase space densities and anisotropic interaction, g > 0. Dashed lines show the corresponding maximal occupation number after diagonalizing the single body density matrix (not ensemble averaged). In the normal phase at low phase space density, we have n κ 0 ∼ L -2 and two degenerate modes, whereas in the superfluid phase at high phase space density we have n κ 0 ∼ L -η with η < 1/4, the degeneracy is broken, and only one mode contributes to the quasi-condensate.

Link between low temperature order and the BKT transition for η soc < 1 and g = 0 In the last chapter, we have determined the BKT from the appearance of a quasi-condensate density, averaged over the two momentum states which minimize the single particle energy. From the diagonalization of the single particle density matrix, we obtain additional information.

Above the critical temperature, we obtain two degenerate modes within our numerical accuracy -both minima in momentum space are equally populated within a single Monte Carlo run. Below the critical temperature, the system spontaneously chooses one mode which dominates. The two minima are only equivalent after ensemble averaging different Monte Carlo calculations. As shown in figure 4.1, for g = 0, in the BKT phase, the single particle density matrix is dominated by a single, highly occupied mode.

PW and SP phase

Since the low temperature phase of the system is dominated by a single mode of the reduced density matrix, the spin-structure of the corresponding eigenstate characterizes the spin-structure of the BKT phase, e.g. an algebraically decaying PW or SP quasi-condensate.

Diagonalizing the matrix M (k), we numerically obtain the eigenvectors that describe the appearing order. However, in order to interpret better these results, let us first analyze the structure of the matrix assuming PW or SP order.

Our mean-field ground state introduced in Chapter I Eq. (1.55), we based on the following

|Φ -(φ)〉 = cos(φ) Φ- † κ + sin(φ) Φ- † -κ N N ! |0〉 (4.4)

CHAPTER 4. LOW TEMPERATURE STATES: PLANE WAVE AND STRIPE PHASE

The resulting structure of the matrix M for pure PW (φ = 0) or SP (φ = π/4) states is then

Plane Wave : φ = 0 M (κ) ∝ (1 -1 0 0) ⊗      1 -1 0 0      (4.5) Stripe phase : φ = π 4 M (κ) ∝ (1 -1 1 1) ⊗      1 -1 1 1      (4.6)
We recover these features in our numerical calculation and we extracted numerically the angle φ. At high temperature, where we have two degenerate modes, the angle φ is not correctly defined. Approaching the transition point, when one mode starts to dominate, the value of angle becomes well defined. Its mean value depends on the anisotropy g > 0 or g < 0 (SP and PW respectively), fluctuations around it are strongly suppressed.

Our analysis of the eigenmodes and their occupations clearly connects the quasicondensate structure with the spin-ordering in the BKT phase showing PW or SP depending on the sign of g . Although there is clearly one mode dominating, the occupation is slowly decaying with system size. Therefore, also in spin space, there is no true, but only quasi-long range order. In the next section, we will study the spin order in a more intuitive and experimentally better accessible way, calculating correlation functions of the local spin density.

CHAPTER 4. LOW TEMPERATURE STATES: PLANE WAVE AND STRIPE PHASE

Local Spin Density

The PW and SP character of the reduced one-body density matrix computation naturally propagates to various observables and higher order correlation function.

In the following we will analyze the spin-order in terms of the local spin density.

Spin Density Since we labeled the two hyperfine states of the atoms as spin up and spin down we can use the formalism of magnetism to study the ordered phases. For instance, counting the difference of the number of atoms in the two hyperfine states is equivalent to compute the magnetic moment in the direction z

σ z (r) ≡ Ψ † (r)σ z |Ψ(r) Ψ † (r)Ψ(r) = |ψ ↑ (r)| 2 -|ψ ↓ (r)| 2 |ψ ↑ (r)| 2 + |ψ ↓ (r)| 2 (4.7)
The colors of figures 4.2, 4.3 and 4.4 represent the value of the local spin density σ z (r) of the field at each point of space r. These quantities are not averages but they are rather obtained at single step of the algorithm. For g > 0 we can clearly identify stripes at low temperature whereas for g < 0 the density is constant in space.

Spin projections

We can then generalize the spin formalism to the other directions of the spin. Note that the gas is confined in two dimensions but the spin degree of freedom is three-dimensional. Then using the Pauli matrices,

σ x = 0 1 1 0 σ y = 0 -i i 0 σ z = 1 0 0 -1 (4.8)
we can then compute the local spin density of the field in the three directions

σ x (r) = Ψ † (r)σ x Ψ(r) Ψ † (r)Ψ(r) = ℜ[ψ † ↑ (r)ψ ↓ (r)] |ψ ↑ (r)| 2 + |ψ ↓ (r)| 2 (4.9) σ y (r) = Ψ † (r)σ y Ψ(r) Ψ † (r)Ψ(r) = ℑ[ψ † ↑ (r)ψ ↓ (r)] |ψ ↑ (r)| 2 + |ψ ↓ (r)| 2 (4.10) σ z (r) = Ψ † (r)σ z Ψ(r) Ψ † (r)Ψ(r) = |ψ ↑ (r)| 2 -|ψ ↓ (r)| 2 |ψ ↑ (r)| 2 + |ψ ↓ (r)| 2 (4.11)
and by this way obtain all the information about the spin structure of the field.

CHAPTER 4. LOW TEMPERATURE STATES: PLANE WAVE AND STRIPE PHASE

CHAPTER 4. LOW TEMPERATURE STATES: PLANE WAVE AND STRIPE PHASE

Mean-field predictions It is instructive to explicitly write down the mean-field spin density for PW and SP using the mean-field wave function

ψ M F κ (r) = ψ M F ↑,κ (x, y) ψ M F ↓,κ (x, y) = 1 2 cos(φ)e i κx 1 -1 + sin(φ)e -i κx 1 1
For g < 0 the mean-field ground state corresponds to φ = 0, a Plane Wave state, with

σ x (r) = 1 σ y (r) = 0 σ z (r) = 0 (4.12)
All the spins are aligned and point in the direction κ = (κ, 0) of the minimum.

For g < 0, the minimum of the energy corresponds to φ = π 4 , the Stripe Phase state, where we have

σ x (r) = 0 σ y (r) = sin(2κx) σ z (r) = cos(2κx) (4.13)
Now the spins direction rotates around the x-axis with periodicity 2κ.

Numerical simulation From figures 4.2 and 4.3 we observe that the spin density of our classical field simulation reflects the mean-field ground state at very low temperature. However, at slightly higher temperature, thermal excitations mask the state. Further, we also observe vortices due to thermal excitations, typical for BKT physics. ), at phase space density 1/nλ 2 0.026 for η soc = 0.9 where n κ 0 /n ∼ 40%. For g < 0, M x (x, 0) shows quasilong range order indicating PW, whereas M y (x, 0) is short ranged. For g > 0 we obtain SP where the amplitude of the oscillations of M y (x, 0) decays algebraically and no order is present in M x (x, 0).

CHAPTER 4. LOW TEMPERATURE STATES: PLANE WAVE AND STRIPE PHASE

Spin density correlation functions

In the last section, we have shown the local spin density of an instantaneous field configuration. For a more quantitative study, we calculated the spin-density correlation function averaged over many field configurations

M α (r) = 〈 Ŝα (r) Ŝα (0)〉 with Ŝα (r) = Ψ † (r)σ α Ψ(r) (4.14)
As shown in figure 4.5, the spin structure of this condensate mode is directly reflected in the spin correlation function M α (r).

Quasi-long range order Quasi-long range stripe order is reflected in slowly decaying oscillations of period 2κ in M y (x, 0). For g < 0, M x (x, 0) develops quasi-long range order. In both cases, the exponent of the algebraic decay is given by the scaling exponent of n κ 0 and compatible with η(T ) obtained from the superfluid density. Therefore, the quasi-long range spin order results from the spin structure of the underlying quasi-condensate.

Anisotropy

We also notice that the correlation functions, M x (x, 0) and M y (x, 0), remain short ranged in the SP and PW state, respectively. The system therefore exhibits a strong anisotropy between the direction κ and the one orthogonal to it. Reference [START_REF] Stringari | Diffused vorticity and moment of inertia of a spin-orbit coupled bose-einstein condensate[END_REF] addresses this feature, focusing in particular on the possibility of anisotropic superfluidity.

Isotropic interaction: Fragmented condensate

Let us now study the case of isotropic interparticle interaction g = 0. As shown in Chapter I, mean field calculations do not select an unique ground state between SP and PW states.

Since SP and PW degeneracy may only reflect the insensitivity of the mean field ansatz for the ground state, many studies focused on looking for the true absolute ground state [START_REF] Yu | Ground-state phase diagram and critical temperature of twocomponent bose gases with rashba spin-orbit coupling[END_REF][START_REF] Barnett | Order by disorder in spin-orbit-coupled bose-einstein condensates[END_REF][START_REF] Ozawa | Ground-state phases of ultracold bosons with rashba-dresselhaus spin-orbit coupling[END_REF][START_REF] Ozawa | Stability of ultracold atomic bose condensates with rashba spin-orbit coupling against quantum and thermal fluctuations[END_REF]. In these approaches the symmetry between the SP and PW phases is broken by different physical mechanisms, for example by introducing quantum fluctuations [START_REF] Barnett | Order by disorder in spin-orbit-coupled bose-einstein condensates[END_REF] or by renormalization procedures [START_REF] Ozawa | Ground-state phases of ultracold bosons with rashba-dresselhaus spin-orbit coupling[END_REF].

However, as shown in figure 4.6 for two fundamentally distinct cases η soc = 0 and η soc = 1, in the limit of isotropic interaction, g = 0, we always obtain two highly occupied modes of the single particle density matrix, degenerate within our The top and bottom plots correspond to two opposite SOC anisotropies η soc , respectively η soc = 0 and η soc = 1. Each point represents a single long run of a computation at extremely low temperature in a regime numerically challenging within our approximation. In the superfluid regime and for any density, we observe a strong signature of fractionalization. Black points show the corresponding eigenvalues in the case of an anisotropic interaction g = 0 when only one eigenvalue λ 1 is non zero (λ 2 ∼ 10 -2 λ 1 ).

λ 2 λ 2 /2π λ 1 λ 2 /2π λ 1 =λ 2 η SOC =1, 1/nλ 2 =0.006 η SOC =1, 1/nλ 2 =0.013
CHAPTER 4. LOW TEMPERATURE STATES: PLANE WAVE AND STRIPE PHASE numerical precision. By analyzing the spin correlation functions as in the previous sections, we find that both M x (x, 0) and M y (x, 0) become quasi-long ranged and indicate simultaneous PW and SP characters.

Therefore, for our great surprise, PW and SP remain degenerate and robust against thermal, critical fluctuations. In case of macroscopically occupied modes, this phenomena corresponds to a fractionalized condensate [START_REF] Mueller | Fragmentation of bose-einstein condensates[END_REF]. Since we do not have true long range order, we observe for the first time a fractionalized quasi-condensate.

This unusual behavior indicates that although two modes are extremely populated, the phase φ between them is not locked and the spin does not prefer any particular direction. However, classical field description takes only into account thermal fluctuations. From the Bogoliubov approximation around the T = 0 mean-field ground states, we expect that quantum fluctuations lift the degeneracy and favor the PW character decreasing the temperature without further phase transition [START_REF] Barnett | Order by disorder in spin-orbit-coupled bose-einstein condensates[END_REF].

Low temperature wave-function

References [START_REF] Gopalakrishnan | Universal phase structure of dilute bose gases with rashba spin-orbit coupling[END_REF][START_REF] Chao | Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons[END_REF] predicted the occurrence of a paired condensate. Our classical field Monte Carlo have not shown any evidence for the occurrence of this phase. However, it is not clear which observable would best show up such a phase. Here, we present some of our analysis done on the whole field distribution.

Figure 4.8 shows the distribution of the field along the real (x-axis) and imaginary axis (y-axis) at high and low temperature. At a single single Monte Carlo step, we plot the field Ψ σ (r) at every point in space r for both σ =↑ and σ =↓.

At high temperature we recognize the Gaussian regime centered on 〈Ψ σ (r)〉 = 0. At low temperature, we recover the mean-field predictions in addition to a broadening due to thermal fluctuations .

CHAPTER 4. LOW TEMPERATURE STATES: PLANE WAVE AND STRIPE PHASE

Bosons pairs

In this distribution, we can notice a great difference between the SP and the other phases. Indeed the average 〈Ψ 2 〉 is non zero in the case of a SP wave-function and zero otherwise.

High temperature Gaussian regime

The average of a squared Gaussian distributed function is zero.

Plane Wave

The average of a squared Plane Wave is also zero.

〈 e i kx 2 〉 = 〈e i 2kx 〉 = 0 (4.15)
Stripe Phase The square of a Stripe Phase wave-function is not zero 〈si n(2κx) 2 〉 = 0.5 (4.16)

The results discussed in last chapter, indicated that for isotropic SOC, η soc = 1, no standard BKT transition occurs. In this particular region of the phase diagram and for g > 0, in the Stripe Phase, condensation of pairs of bosons were predicted by references [START_REF] Gopalakrishnan | Universal phase structure of dilute bose gases with rashba spin-orbit coupling[END_REF][START_REF] Chao | Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons[END_REF].

We analyzed how this observable for pairing scales with the system size. Figure 4.10 shows that it decreases with the system size L 2 depending on the discrete number of degenerate minima as the condensate fraction observable studied before. Therefore we do not find any indications of a phase transition to a pairing phase as proposed in reference [START_REF] Chao | Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons[END_REF]. However, we cannot exclude the possibility of pair superfluidity at considerably lower densities or much larger system sizes. .9: Imaginary part of the wave-function as a function of its real part at every point in space r. The system size is L/a = 80, the SOC anisotropy η soc = 0, the space density 1/nλ 2 = 0.0126 and the contact interaction anisotropy is set to g < 0 i.e Plane Wave state. At low temperature, in addition to a broadening due to thermal fluctuations, we recover the mean-field prediction of ψ M F κ (r) ≡ e i κx 1 -1 . g = 2g ↑↓g ↑↑g ↓↓ l'état fondamental par champ moyen est alors décrit soit comme une onde plane (PW) soit comme une superposition de deux ondes planes avec impulsions opposées appelée état de bande (SP) [START_REF] Gopalakrishnan | Universal phase structure of dilute bose gases with rashba spin-orbit coupling[END_REF].

Au-delà de l'approximation type champ moyen, l'existence et l'apparition de ces phases exotiques n'est pas établi et beaucoup de questions restent ouvertes. Par exemple, la référence [START_REF] Gopalakrishnan | Universal phase structure of dilute bose gases with rashba spin-orbit coupling[END_REF] propose un diagramme de phase basé sur des arguments qualitatifs dans la limite du régime fortement dilué. Les différentes phases ne dépenderaient plus alors du signe de l'anisortopie g , en fort contraste avec les prédictions en champ moyen. Nous avons, à notre tour, caractérisé les différentes phases à basse température observées au sein de nos simulations de champs classiques.

En particulier, nous avons étudié l'ordre magnétique des atomes/spins à basse température que nous avons relié à la transition BKT étudiée au cours du chapitre précédent. Nous avons également montré que dans le cas d'une anisotropie g = 0 les corrélations de spin présentent un ordre de quasi-longue portée induit par la transition KT en opposition avec les prédictions de ordre longue portée de la part de la référence [START_REF] Su | Hidden long-range order in a spin-orbit-coupled twodimensional bose gas[END_REF]. Nous observons donc un diagramme de phase fortement déterminé par la transition Kosterlitz-Thouless. Nous avons également sondé différentes prédictions, en particulier celle d'une transition à impulsion nulle [START_REF] Chen | Quantum and thermal fluctuations in a raman spin-orbit-coupled bose gas[END_REF][START_REF] Yu | Ground-state phase diagram and critical temperature of twocomponent bose gases with rashba spin-orbit coupling[END_REF] ainsi que l'apparition de paires de bosons [START_REF] Gopalakrishnan | Universal phase structure of dilute bose gases with rashba spin-orbit coupling[END_REF][START_REF] Chao | Paired superfluidity and fractionalized vortices in systems of spin-orbit coupled bosons[END_REF]. Toutefois, nous n'avons pas trouvé de trace de ces comportements et phases très exotiques.

Dans le cas d'interactions isotropes, g = 0, l'approximation type champ moyen ne sélectionne pas d'état fondamental unique, les états SP et PW restent alors Our numerical studies clearly establish that the weakly interacting Bose gas undergoes a BKT phase transition for anisotropic SOC, η soc < 1. In the low temperature phase, the condensate fraction decays algebraically with system size and the gas becomes superfluid. However, for isotropic SOC, η soc = 1, our calculations shows a cross-over behavior at finite systems, with strong evidence for the absence of a finite temperature phase transition in the thermodynamic limit.

We have further characterized superfluid many body states for η soc < 1 as a function of a vanishing or small spin-anisotropy of the interparticle interaction, g = 2g ↑↓g ↑↑g ↓↓ , of positive or negative sign. In particular, we have shown that in the case of an anisotropy g = 0 the spin correlations exhibit quasi-long-range order and that the magnetic ordering of the atoms/spins at low temperature is linked to the BKT phase transition. Our calculations confirm mean field predictions for the character of the quasi-condensate in the superfluid state, i.e. PW or SP order depending on the sign of g . For isotropic interactions, g = 0, we obtained a fractionalized quasi-condensate with two degenerate modes at the transition showing both, PW and SP character.

Originally motivated by the mean field prediction of the degeneracy between the SP/PW states, the system of isotropically interacting bosons, g = 0, with Rashba spin-orbit coupling, η soc = 1, has attracted considerable attention [START_REF] Ozawa | Condensation transition of ultracold bose gases with rashba spin-orbit coupling[END_REF][START_REF] Barnett | Order by disorder in spin-orbit-coupled bose-einstein condensates[END_REF]. Fluctuations and correlations beyond mean field were expected to break this degeneracy. Using classical field Monte Carlo calculations, we directly addressed the role of thermal fluctuations. The stability of the SP/PW degeneracy leading to a fractionalized quasi-condensate in our calculations came out unexpectedly. However, within classical field theory, quantum effects due to non-vanishing commutators of the quantum fields are neglected.

Close to zero temperature, the Bogoliubov approach is suited for studying quantum fluctuations around the mean field state. Reference [START_REF] Barnett | Order by disorder in spin-orbit-coupled bose-einstein condensates[END_REF] shows that, within the Bogoliubov approximation, the three dimensional isotropic SOCed Bose gas condenses into a single-momentum state of the Rashba spectrum, thus resulting in order by disorder.

In two dimensions, thermal fluctuations destabilize the system at any finite temperature. Nevertheless, decreasing temperature, the analogous calculation of reference [START_REF] Barnett | Order by disorder in spin-orbit-coupled bose-einstein condensates[END_REF] predicts quantum fluctuations to lift the degeneracy and favor the PW character for η soc < 1 and g = 0. However, the exact transition from a fractionalized quasi-condensate at the critical temperature to the broken degeneracy at zero temperature is unclear and still an open question.

Addressing numerically the full quantum system is extremely challenging due to the presence of the SOC which introduces a sign problem into all known quantum Monte Carlo algorithms. Similar to the fermionic sign problem, the error of such a calculation increases exponentially with system size, inverse proportional to temperature. Similar, already for classical field calculations, SOC prevents the use of the Worm algorithm [START_REF] Prokof | Worm algorithms for classical statistical models[END_REF] to speed up our computation.

As an outlook, we want to include quantum fluctuations, as described in the Bogoliubov theory, within our classical field approach, with the hope to quantitatively descrive both thermal and quantum fluctuations for a weakly interacting SOCed Bose gas.

CHAPTER 5. CONCLUSION AND PERSPECTIVES

Conclusion et Perspectives

A U COURS de cette thèse nous avons déterminé le diagramme de phase à température finie d'un gaz de Bose bidimensionnel avec deux états hyperfins (pseudospin) couplés au travers d'une interaction spin-orbite Rashba-Dresselhaus en utilisant des calculs Monte-Carlo basés sur champs classiques.

Nos études numériques établissent clairement qu'un gaz de Bose intéragissant subit une transition de phase de type BKT en présence d'un couplage spin-orbite anisotrope η soc < 1. La phase à basse température présente une fraction condensée qui décroit algébriquement avec la taille du système et le gaz devient alors superfluide. Au contraire, dans le cas d'un couplage spin-orbite isotrope, η soc = 1, nos calculs pointent l'absence d'une transition de phase à température finie et à la limite thermodynamique. Une transition lisse (cross-over) subsiste pour des systèmes à taille finie.

Nous avons ensuite étudié plus en détail les différents états fondamentaux à plusieurs corps pour η soc < 1 en fonction d'une anisotropie des interactions interparticules g = 2g ↑↓g ↑↑g ↓↓ nulle, positive ou négative. En particulier, nous avons montré que dans le cas d'une fine anisotropie g = 0 les correlations de spins présentent un ordre quasi-longue portée et que l'ordre magnétique des atomes/spins à basse température est lié à la transition de phase BKT. Nos calculs confirment les prédictions de type champ moyen à propos de la nature du quasi-condensat dans la phase superluide c'est à dire la sélction des ordres SP et PW en fonction du signe de g . Dans le cas d'interactions isotropes, g = 0, nous obtenons un quasi-condensat fractionnalisé avec deux modes dégénérés à la transition et qui présente deux ordres simultanément PW et SP.

Une attention toute particulière de la part de la communauté [START_REF] Ozawa | Condensation transition of ultracold bose gases with rashba spin-orbit coupling[END_REF][START_REF] Barnett | Order by disorder in spin-orbit-coupled bose-einstein condensates[END_REF] est portée sur le système composé de bosons interagissant de manière isotrope, g = 0, avec un couplage spin-orbite également isotrope, η soc = 1 provoquée à l'origine par la prédiction d'une dégénérescence entre les états SP et PW dans le cadre des théories de type champ moyen. Il serait attendu que cette dégénérescence soit levée par les fluctuations au-delà du champ moyen. En utilisant les calculs de champs classiques par Monte Carlo, nous avons pour notre part abordé directement le rôle des fluctuations thermiques. C'est alors que la stabilité de la dégénérescence entre les ordres SP et PW entraînant une fractionalisation du quasi-condensat est apprarue dans nos calculs de façon inattendue. Toutefois, au sein d'une théorie de champs classiques, les effects quantiques dûs aux commutateurs des champs quantiques CHAPTER 5. CONCLUSION AND PERSPECTIVES sont négligés.

Au contraire, en se rapprochant de la limite à température nulle, l'approche Bogoliubov est particulièrement adaptée pour l'étude des fluctuations quantiques autour de l'état fondamental provenant du champ moyen. En particulier la référence [START_REF] Barnett | Order by disorder in spin-orbit-coupled bose-einstein condensates[END_REF] montre que, au sein de l'approximation Bogoliubov, le gaz de Bose tridimensionnel avec couplage spin-orbite isotrope condense dans un état du spectre Rashba avec une unique impulsion, ceci se traduisant alors dans un processus de ordre par le désordre.

En deux dimensions, les fluctuations thermiques déstabilisent le système pour n'importe quelle température non nulle. Néanmoins, en diminuant la tempéraure, des calculs analogues à ceux de la référence [START_REF] Barnett | Order by disorder in spin-orbit-coupled bose-einstein condensates[END_REF] prédisent que les fluctuations quantiques lèvent la dégénéréscence et favorise la nature PW du système pour η soc < 1 et g = 0. Pourtant, la transition exacte entre un quasi-condensat fractionnalisé autour de la température critique vers une dégénéréscence brisée à tempérautre nulle, est incertaine et plusieurs questions restent ouvertes.

Chapter 6

Contents 1 . 1 7 1. 1 . 1 9 1. 1 . 2 1 . 2 1 . 3 1 . 5 1 . 6 1 . 1

 117119121213151611 Spin-orbit coupling (SOC) in ultra-cold atoms . . . . . . . . . . . . Artificial gauge fields . . . . . . . . . . . . . . . . . . . . . . . . Rashba-Dresselhaus spin-orbit coupling . . . . . . . . . . . . Non interacting SOCed Bose gases . . . . . . . . . . . . . . . . . . . 1.2.1 Diagonal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Ideal Bose gas in 3D . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Ideal Bose gas in 2D . . . . . . . . . . . . . . . . . . . . . . . . Interacting Bose gas: Mean Field Approximation . . . . . . . . . . 1.3.1 High temperature Mean Field Hartree approximation . . . . 1.3.2 Mean Field ground states: Plane-Wave & Stripe phase . . . . 1.4 Fluctuations & Open questions . . . . . . . . . . . . . . . . . . . . . Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin-orbit coupling (SOC) in ultra-cold atoms "The achievement of Bose Einstein condensation (BEC) (Anderson et al. 1995; Bradley et al. 1995; Davis et al. 1995) and of Fermi degeneracy (DeMarco and Jin, 1999; Schreck at al., 2001; Truscott et al. 2001) in ultra-cold dilute gases has opened a

Figure 1 .Figure 1 . 1 :

 111 Figure1.1 shows the induced transitions of a schematic three-level atom placed in two counter propagating laser beams. A Raman transition corresponds to the absorption of a single photon from one laser beam and its stimulated re-emission into the second. The momentum carried by each of these photons can be quite large compared to the typical ultra-cold atom setup. It is possible to coherently couple the state and the momentum of the cold atoms as we will see in the next

Figure 1 . 2 :

 12 Figure 1.2: Figure extracted from [22]. Vortices created in a BEC of 87 Rb coupled to an artificial magnetic field.

g 3 2 (e βµ ) λ 3 TFigure 1 . 3 :

 2313 Figure 1.3: Figures extracted from[START_REF] Baym | Condensation of bosons with rashbadresselhaus spin-orbit coupling[END_REF]. Two dimensional dispersion of a homogeneous SOCed system. The two branches ± touch at the origin. The left graph corresponds to the pure Rashba term whereas the right graph is plotted for η soc = 0.7. We see in this last case that only two minima appear in p x = ±κ.

Figure 1 . 4 :

 14 Figure 1.4: Figure extracted from [4] a) Typical level diagram. b) Minima location.Measured location of energy minimum or minima, where as a function of laser intensity the characteristic double minima of SOC dispersion move together and finally merge. c) Dispersion measured in6 Li.

Figure 1 . 5 :

 15 Figure 1.5: Mean field phase diagram from ref[START_REF] Ho | Bose-einstein condensates with spin-orbit interaction[END_REF] α = g ↑↓ /g , β = (g ↑↑g ↓↓ )/g and g = (g ↑↑ + g ↓↓ )/2. Using the formalism of this thesis, the critical value of α is predicted for α c = 1 and the indices p + and p -correspond to the directions k x = ±κ. Region I is a superposition of p + and p -e.g. the stripe phase. Region II and III are states in which only p + and p -are populated respectively.

Figure 2 . 1 :

 21 Figure 2.1: Qualitative scheme of the validity range of the theories in function of the temperature.

  algorithm Condensed density, Gaussian algorithm Condensed density, Metropolis algorithm

Figure 2 . 2 :

 22 Figure 2.2: Example of convergence of different algorithms. The plot represents the condensed fraction starting from a random set in function of the steps of the different algorithms. Using Metropolis algorithm we obtain a much larger variance and incertitude on the observable than using the Heat Bath and Gaussian algorithm.

Figure 2 . 3 :

 23 Figure 2.3: Density as a function of the chemical potential.We have corrected the densities of our classical field calculations to account for the correct ultraviolet behavior. We see especially at low density that the Hartree mean field approach developed in Chapter I recovers the numerical integration. At low temperature the density is proportional to the chemical potential as expected c.f Chapter I section 1.3.

Figure 2 . 4 :

 24 Figure 2.4: Density as a function of the chemical potential for non interacting SOCed bosons. It is interesting to notice that n Id,MC as a constant deviation from the exact solution n Id,SOC . As expected from the non interacting results in Chapter I the SOC term increases the density at a fixed chemical potential.

2 Figure 2 . 5 :

 225 Figure 2.5: Density n as a function of the action S for a finite system length L/a = 40 with small anisotropy g = π/100, spin-orbit coupling κ = π/20 and anisotropy η soc = 0. Each point corresponds to one Monte Carlo iteration step.We observe that at high temperature µ = -2 the density is small, the fluctuations are strong and its value is strongly correlated to the value of the action S. Indeed, in this regime every point of Ψ(r) is almost independent and the action is determined by the chemical potential µ. For increasing chemical potential, i.e decreasing temperature, we observe a decorrelation of the density and the action. As presented in the Appendix, at low temperature, fluctuations of the density are strongly suppressed and the value of the action is determined by the many body state i.e interactions and density of state.

CHAPTER 2. METHODS Chapter 3 InteractingContents 3 . 1 59 3. 2 61 3. 3 62 3. 4 69 3. 5 70 3. 6

 331592613624695706 Bosons with SOC in 2D: Phase Diagram Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bose gas without SOC: Berezinskii, Kosterlitz and Thouless phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bose gas with SOC: condensate fraction . . . . . . . . . . . . . . . . Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion and phase diagram . . . . . . . . . . . . . . . . . . . . . Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

CHAPTER 3 .

 3 INTERACTING BOSONS WITH SOC IN 2D: PHASE DIAGRAM 3.2. Bose gas without SOC: Berezinskii, Kosterlitz and Thouless phase transition Page 61

CHAPTER 3 . 2 η 40 Figure 3 . 1 :

 324031 Figure 3.1: Condensate fraction n κ0 /n as a function of inverse phase space density [nλ 2 ] -1 for a finite system of length L/a = 80. The cross-over from normal to condensed phase slightly lowers with increasing SOC anisotropy, η soc . Although the PW/SOC character of the condensate depends essentially on the sign of the anisotropic interaction mg = ±π/100, differences in n κ 0 between g ≥ 0 and g < 0 for equal SOC are beyond our resolution. The colored zones indicate our estimates for the Kosterlitz-Thouless transition in the thermodynamic limit from finite-size-scaling of the condensate fraction described in the text.

Figure 3 . 2 :

 32 Figure 3.2: Condensate fraction, n κ0 /n, as a function of inverse volume in presence of the SOC but in absence of interactions. As we will precisely detail in the paragraph dedicated to the isotropic SOC, we notice that the absolute value of the condensed density depends on the sum introduced in Eq. 3.4, describing the number of degenerate points corresponding to the fundamental state |k| = κ. The condensate density decreases with the volume for any density, no transition occurs.

CHAPTER 3 .

 3 INTERACTING BOSONS WITH SOC IN 2D: PHASE DIAGRAM

Figure 3 . 3 :

 33 Figure 3.3: Condensate fraction, n κ 0 /n, as a function of the inverse volume, L -2, for anisotropic SOC bosons with η soc = 0 at different phase space densities and anisotropic interaction, g > 0. We observe a transition between the high temperature regime, where the condensate density decreases with the volume, and the low temperature regime where the exponent depends on the density and eventually vanishes. The solid black line indicates the critical exponent η = 1/4.

2 Critical exponent η=1/ 4 Figure 3 . 4 :

 2434 Figure 3.4: Exponent η(T ) as a function of the density. The exponent is obtained by scaling the condensate fraction from figure 3.3, with the system size n κ 0 /n ∼ L -η(T ) . By scaling the condensed fraction for different SOC anisotropies η soc we obtain the phase diagram showed in figure 3.7.

Figure 3 . 5 :

 35 Figure 3.5: Condensate fraction, n κ 0 /n, as a function of the inverse volume, L -2, for isotropic SOC bosons with η soc = 1 at different phase space densities and anisotropic interaction, g < 0. We do not observe any quasi-long range order.

Figure 3 . 6 :

 36 Figure 3.6: Solid lines: Condensate fraction, n κ 0 /n, as a function of inverse volume for isotropic SOC, η soc = 1, of finite systems with 4 degenerate minima and g > 0.Dashed lines corresponds to finite systems with 8 degenerate minima where the algebraic behavior, n κ 0 ∼ L -η with η > 1/4, at high phase space density is suppressed.

Figure 3 . 7 :

 37 Figure 3.7: Critical densities as a function of the SOC anisotropy η soc . The KT phase transition from normal to superfluid phase takes place at slightly higher densities with increasing SOC anisotropy, η soc . No finite temperature phase transition occurs for isotropic SOC η soc = 1.

CHAPTER 4 .

 4 LOW TEMPERATURE STATES: PLANE WAVE AND STRIPE PHASE

Figure 4 . 1 :

 41 Figure 4.1: Solid lines: Condensate fraction, n κ 0 /n, as a function of the inverse volume, L -2, for anisotropic SOC bosons with η soc = 0 at different phase space densities and anisotropic interaction, g > 0. Dashed lines show the corresponding maximal occupation number after diagonalizing the single body density matrix (not ensemble averaged). In the normal phase at low phase space density, we have n κ 0 ∼ L -2 and two degenerate modes, whereas in the superfluid phase at high phase space density we have n κ 0 ∼ L -η with η < 1/4, the degeneracy is broken, and only one mode contributes to the quasi-condensate.

Figure 4 . 2 :

 42 Figure 4.2: Spin local projection at extremely low temperature for anisotropic interaction g < 0 i.e Plane Wave state. The arrows' directions represent the local spin σ x and σ y projection on the x-y plane. Colors represent the value of the local spin density σ z . As expected for the Plane Wave state, spins' projections point in average in the same direction. We observe vortices typical of the KT physics.

Figure 4 . 3 :

 43 Figure 4.3: Spin local projection at extremely low temperature for anisotropic interaction g > 0 i.e Stripe Phase state. The arrows' directions represent the local spin σ x and σ y projection on the x-y plane and colors represent the value of the local spin density σ z . We observe the signature of a SP order : spins rotate with periodicity 2κ. The direction of the rotation is determined by the direction of the two populated momenta ±κ.

Figure 4 .Figure 4 . 4 :Figure 4 . 5 :

 44445 Figure 4.4 also shows that few thermal excitations are enough to significantly modify the stripe order of the local spin density. In the following, we will use spin density correlations for a more quantitative study to characterize spin ordering.

CHAPTER 4 .

 4 LOW TEMPERATURE STATES: PLANE WAVE AND STRIPE PHASE

Figure 4 . 6 :

 46 Figure 4.6: Second largest eigenvalue λ 2 of the reduced density matrix M (κ) as a function of the largest eigenvalue λ 1 for isotropic interactions g = 0. The top and bottom plots correspond to two opposite SOC anisotropies η soc , respectively η soc = 0 and η soc = 1. Each point represents a single long run of a computation at extremely low temperature in a regime numerically challenging within our approximation. In the superfluid regime and for any density, we observe a strong signature of fractionalization. Black points show the corresponding eigenvalues in the case of an anisotropic interaction g = 0 when only one eigenvalue λ 1 is non zero (λ 2 ∼ 10 -2 λ 1 ).

Figure 4 . 7 :

 47 Figure 4.7: Imaginary part of the wave-function as a function of its real part at every point in space r. The system size is L/a = 80, the SOC anisotropy η soc = 0 and the space density 1/nλ 2 = 0.240. As expected, at high temperature the density is Gaussian distributed.

Figure 4 . 8 :

 48 Figure 4.8: Imaginary part of the wave-function as a function of its real part at every point in space r. The system size is L/a = 80, the SOC anisotropy η soc = 0, the space density 1/nλ 2 = 0.0126 and the contact interaction anisotropy is set to g > 0 i.e Stripe Phase state. At low temperature, in addition to a broadening due to thermal fluctuations, we recover the mean-field prediction of ψ M F κ (r) ≡ cos(κx) -i sin(κx) .

CHAPTER 4 .

 4 LOW TEMPERATURE STATES: PLANE WAVE AND STRIPE PHASE

Figure 4

 4 Figure 4.9: Imaginary part of the wave-function as a function of its real part at every point in space r. The system size is L/a = 80, the SOC anisotropy η soc = 0, the space density 1/nλ 2 = 0.0126 and the contact interaction anisotropy is set to g < 0 i.e Plane Wave state. At low temperature, in addition to a broadening due to thermal

Figure 4 . 10 : 7

 4107 Figure 4.10: Pairs of bosons < Ψ 2 > as a function of the inverse volume, L -2, for isotropic SOC with η soc = 1 at different phase space densities and anisotropic interaction, g > 0 i.e Stripe Phase. We recognize the dependency of this observable on the discrete number of degenerate minima as the condensate fraction studied before. We do not find any indication of a finite temperature phase transition.

CHAPTER 4 . 5

 45 LOW TEMPERATURE STATES: PLANE WAVE AND STRIPE PHASE dégénérés. Dans ce cas spécifique, il serait attendu que les fluctuations thermiques et quantiques lèvent cette dégénérescence et qu'un unique état fondamental soit sélectionné. L'approximation type champs classiques nous permet d'aborder la question des fluctuations thermiques de façon directe et explicite. Nous avons montré que le système subit une transition de phase sans sélectionner d'état fondamental unique. Nos calculs prédisent dans ce cas une fractionalisation du condensat, les états PW et SP se maintenant dégénérés. Dans le cadre de notre étude et pour étudier la compétition entre couplage spin-orbite et interactions interparticules, nous avons fixé σσ =↑↓ mg σσ /4 = κ/ mk B T = π/20 avec mg = 0 pour étudier le cas d'une interaction isotrope et mg = ±π/100 pour étudier le cas d'une fine brisure de symmétrie entre les diffusions des différents spins. CHAPTER 4. LOW TEMPERATURE STATES: PLANE WAVE AND STRIPE PHASE Chapter Conclusion and Perspectives I N THIS THESIS we have determined the finite-temperature phase diagram of a two-dimensional interacting Bose gase with two hyperfine (pseudospin) states coupled via Rashba-Dresselhaus spin-orbit interaction using classical field Monte Carlo calculations.

  très haute température l'effet des interactions correpond à un décalage effectif du potentiel chimique[START_REF] Blaizot | Quantum Theory of Finite Systems[END_REF] égal à 2g ↑↑ n M F avec g ↑↑ l'interaction moyenne entre deux particules et n M F la densité champ moyen déterminée de façon auto-cohérente. Celui-ci correspond au terme connu de Hartree. Deux phases distinctes[START_REF] Ho | Bose-einstein condensates with spin-orbit interaction[END_REF] sont alors apparues en fonction de l'intensité des interactions entre particules de même (g ↑↑ et g ↓↓ ) et différent spin (g ↑↓ ). Pour g ↑↑ + g ↓↓ > 2g ↑↓ , l'état fondamental correspond à une onde plane avec impulsion non nulle (PW). Dans le cas opposé, 2g ↑↓ > g ↑↑ + g ↓↓ , chaque particule est décrite comme une superposition de deux ondes planes avec impulsions opposées. Ce dernier état est appelé état de bande (SP) car des bandes apparaissent dans la densité en espace réel.Enfin nous avons étudié les enjeux et avancées actuelles autour de ce type de système en insistant sur le rôle prépondérant des fluctuations[START_REF] Ozawa | Condensation transition of ultracold bose gases with rashba spin-orbit coupling[END_REF][START_REF] Barnett | Order by disorder in spin-orbit-coupled bose-einstein condensates[END_REF] qui dépassent les prédictions souvent trop schématiques des théories type champ moyen.

	1.6. Résumé
	1.6 Résumé
	Au cours du premier chapitre de cette thèse nous avons introduit le concept de
	champ de jauge artificiel. Afin de simuler des effets magnétiques dans les systèmes
	d'atomes neutres et ultra-froids, des techniques basées sur les transitions Raman entre deux états hyperfins des atomes ont été développées ces dernières années [1, 2, 3]. A l'aide de faisceaux lasers, ces différents champs peuvent être manipulés et contrôlés pour donner naissance en particulier à des vecteurs potentiels effectifs Chapter 2
	dont les components ne commutent pas.
	Our calculations then indicate a
	fractionalized quasicondensate where the mean-field degeneracy of the two states
	remains robust against critical fluctuations. We conclude on new prospects and
	motivations beyond the classical field approximation.
	CHAPTER 1. INTRODUCTION CHAPTER 1. INTRODUCTION

Un célèbre exemple, inclus dans ce dernier cas, est le couplage spin-orbite (SOC) dont nous avons sélectionné l'écriture en tant que interaction Rashba-Dresselhaus

[START_REF] Yu | Oscillatory effects and the magnetic susceptibility of carriers in inversion layers[END_REF][START_REF] Dresselhaus | Spin-orbit coupling effects in zinc blende structures[END_REF]

: les amplitudes du couplage selon les deux directions spatiales x et y peuvent être différentes et nous avons donc défini le nombre 0 ≤ η soc ≤ 1 décrivant cette anisotropie.

Nous considérons ensuite le cas d'une particule isolée pour obtenir son spectre d'énergie en présence du terme SOC. Après avoir correctement transformé la base décrivant les deux états couplés de l'atome, nous avons mis en évidence la nouvelle dégénérescence de l'état fondamental induite par le couplage SOC. En effet dans le cas d'une anisotropie η soc < 1 le spectre d'énergie présente deux minima. Dans le cas d'un couplage SOC isotrope, le minimum de l'énergie correspond à un anneau dans l'espace des impulsions. Dans ce dernier cas et pour un système de bosons non interagissants, nous avons également montré l'absence de transition de phase telle que la condensation de Bose-Einstein (BEC).

Nous avons alors introduit une première approche pour décrire les effets des interactions interparticules dans les gaz de bosons avec couplage SOC : l'approximation champ moyen. Deux régimes peuvent en principe être correctement décrits par cette méthode, celui d'une température infinie et celui d'une température nulle. A A très basse température nous avons procédé par méthode variationnelle en écrivant un Ansatz de la fonction d'onde minimisant l'énergie d'une particule simple. Nous CHAPTER 1. INTRODUCTION avons alors déterminé les paramètres libres en minimisant l'énergie d'interaction. Methods Contents 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2 Classical field approximation . . . . . . . . . . . . . . . . . . . . . . 38 2.3 Markov Chain Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.1 Effective action S . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3.2 Monte Carlo algorithms: Metropolis, Heat bath and Fourier moves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.3.3 Partition function . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.4 Density matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.4.1 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.4.2 Lattice expressions for ideal and mean-field classical fields . 50 2.4.3 Convergence and scale of energy . . . . . . . . . . . . . . . . . 55 2.5 Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.6 Résumé Au

  cours de ce chapitre, nous explorons le diagramme de phase d'un gaz de bosons bidimensionnel avec couplage spin-orbite en utilisant les différentes méthodes définies dans le chapitre précédent. Nous avons établi spécifiquement la présence ou l'absence d'une transition de phase à température finie dans le système avec interactions interparticules. Nous avons également proposé des prédictions quantitatives pour le diagramme de phase. Au cours du prochain chapitre nous proposons d'étudier et correctement caractériser le (quasi-) ordre des différentes phases à basse température.Selon le théorème de Mermin-Wagner[START_REF] Hohenberg | Existence of long-range order in one and two dimensions[END_REF][START_REF] Mermin | Absence of ferromagnetism or antiferromagnetism in one-or two-dimensional isotropic heisenberg models[END_REF][START_REF] Coleman | There are no goldstone bosons in two dimensions[END_REF], aucun ordre à longue portée ne peut s'établir à température finie. Cependant, en absence de couplage spin-orbite, une transition de phase Berenzinskii-Kosterlitz-Thouless (BKT) est possible dans un gaz de bosons interagissant entre une phase normale et une phase superfluide. Cette dernière est caractérisée par un ordre algébrique de quasi-longue portée.Nos études numériques établissent que une transition BKT a toujours lieu dans le gaz de bosons faiblement interagissant en présence d'un couplage spin-orbite anisotrope η soc < 1. Dans la phase basse température, la fraction condensée décroit de manière algébrique en fonction de la dimension du système et le gaz devient alors superfluide. Au contraire, pour un couplage spin-orbite isotrope, η soc = 1, nos calculs indiquent une absence de transition de phase à température finie et à la limite thermodynamique. Une transition lisse (cross-over) subsiste pour des systèmes à taille finie. La figure 3.7 présente le diagramme de phase de notre système, détaillant les densités critiques en fonction de l'anisotropie du couplage spin-orbite η soc . CHAPTER 3. INTERACTING BOSONS WITH SOC IN 2D: PHASE DIAGRAM 4.1. Introduction and motivations the mean-field ground state is either given by a single Plane Wave State (PW), or by a linear superposition of two Plane Waves with opposite momenta, the Stripe Phase (SP).

1.2. Non interacting SOCed Bose gases

1.3. Interacting Bose gas: Mean Field Approximation

Résoudre numériquement le système quantique complet est extrêmement difficile à cause de la présence du couplage spin-orbite qui introduit un problème de signe abondamment connu dans tous les algorithmes de Monte Carlo quantique. De façon semblable au problème du signe fermionique, l'erreur provenant de ce type de calcul croît de façon exponentielle avec la taille du système et de manière inversement proportionnelle à la température. De manière analogue dans le cadre des calculs de champs classiques que nous avons développés, le terme traduisant le couplage spin-orbite empêche l'utilisation des algorithmes type Worm[START_REF] Prokof | Worm algorithms for classical statistical models[END_REF] pour accélérer nos calculs.Comme projet futur et comme proposition d'ouverture, nous souhaitons inclure les fluctuations quantiques, telles celles décritent dans la théorie de Bogoliubov, au sein de notre approche se basant sur des champs classiques avec l'espoir de décrire de manière quantitativement correcte simultanément les fluctuations quantiques et les fluctuations thermiques dans un gaz de Bose faiblement interagissant.CHAPTER 5. CONCLUSION AND PERSPECTIVES

Acknowledgments

Chapter 4

Low temperature states: Plane Wave and Stripe Phase

Introduction and motivations

In Chapter III, we have studied the phase diagram of a two-dimensional SOCed Bose gas. For anisotropic SOC, we have identified a low and high temperature phases separated by a transition that we have shown to be within the Kosterlitz-Thouless class. As we have seen in Chapter I, mean field theory predicts exotic many body ground states, in particular, depending on the strength of interactions between same and different spins (defined in Eq. (1.38))

Fluctuations of the density at low temperature

As shown in figure 2.5, at low temperature, the interaction energy, g n 2 /2, dominates over the kinetic energy, ∝ nλ 2 . For simplicity, let us consider the system in absence of SOC, κ = 0, where the density distribution at low temperature is approximately given by

Therefore, fluctuations of the density around its mean value, n = 〈n〉 ≡ µ/g , are Gaussian distributed, with mean-square fluctuations

and highly suppressed for large phase space density, ∆n 2 /n 2 ∼ [g nλ 2 ] -1 0, in contrast to the non interacting case. Approaching zero temperature, density fluctuations smoothly vanish. At high temperature, fluctuations around the density are much larger, since the kinetic energy can never be neglected. In this regime, the Fourier modes of the fields are Gaussian distributed leading to qualitatively different density fluctuations, 〈n 2 〉 = 2n 2 .

Abstract

In this thesis, we theoretically study the occurence of exotic phases in a dilute two component (spin) Bose gas with artificial spin-orbit coupling (SOC) between the two internal states.

Including spin-orbit coupling in classical field Monte Carlo calculations, we show that this method can be used for reliable, quantitative predictions of the finite temperature phase diagram. In particular, we have focused on SOCed bosons in two spatial dimensions and established the phase diagram for isotropic and anisotropic SOC and interparticle interactions.

In the case of anisotropic SOC, the system undergoes a Berenzinskii-Kosterlitz-Thouless transition from a normal to a superfluid state at low temperature. The spin order of the quasicondensate in the low temperature superfluid phase is driven by the spin dependence of the interparticle interaction, favoring either the occurence of a single plane wave state at non-vanishing momentum (PW) or a linear sperposition of two plane waves with opposite momenta, called stripe phase (SP). For spin-independent interparticle interaction, our simulations indicate a fractionalized quasicondensate where PW and SP remain degenerate. For isotropic SOC, our calculations indicate that no true phase transition at finite temperature occurs in the thermodynamic limit, but a cross-over behavior remains visible for large, but finite number of atoms.

Résumé

Cette thèse est dédiée à l'étude théorique de phases exotiques dans un gaz dilué de bosons avec deux composantes (spins) en présence d'un couplage spin-orbite (SOC) entre ces deux états internes. En ajoutant ce dernier couplage à une description de type champs classiques de notre système, nous montrons que cette méthode permet de prédire le diagramme de phase à température finie de manière quantitative, efficace et fiable. Notre étude porte en particulier sur un système de bosons bidimensionnels avec SOC dont nous dessinons le diagramme de phase en fonction de l'anisotropie du SOC ainsi que des interactions.