Twofold correlation spreading in a strongly correlated lattice Bose gas Julien Despres, Louis Villa, and Laurent Sanchez-Palencia CPHT, Ecole Polytechnique, CNRS, Université Paris-Saclay, F-91128 Palaiseau, France (Dated: December 12, 2018) We study the spreading of correlations in the Bose-Hubbard chain, using the time-dependent matrix-product state approach. In both the superfluid and the Mott-insulator phases, we find that the time-dependent correlation functions generally display a universal twofold cone structure characterized by two distinct velocities. The latter are related to different microscopic properties of the system and provide useful information on the excitation spectrum. The twofold spreading of correlations has profound implications on experimental observations that are discussed.

In the last decades, simultaneous progress of the manybody quantum theory and the experimental control of quantum matter in condensed matter and atomic, molecular, and optical physics has given dramatic momentum to the understanding of the out-of-equilibrium dynamics of correlated quantum systems [START_REF] Polkovnikov | Colloquium: Nonequilibrium dynamics of closed interacting quantum systems[END_REF][START_REF] Eisert | Quantum many-body systems out of equilibrium[END_REF][START_REF] Langen | Ultracold atoms out of equilibrium[END_REF][START_REF] Lewenstein | Ultracold atomic gases in optical lattices: Mimicking condensed matter physics and beyond[END_REF][START_REF] Bloch | Quantum simulations with ultracold quantum gases[END_REF]. The spreading of quantum correlations governs many fundamental phenomena, including the propagation of information and entanglement, thermalization, and the area laws for entanglement entropy. For lattice systems with local interactions, the existence of Lieb-Robinson (LR) bounds implies the emergence of a causal light cone beyond which the correlations are exponentially suppressed [START_REF] Lieb | The finite group velocity of quantum spin systems[END_REF][START_REF] Bravyi | Lieb-Robinson bounds and the generation of correlations and topological quantum order[END_REF][START_REF] Hastings | Spectral gap and exponential decay of correlations[END_REF]. So far, light-cone-like spreading of correlations has been reported in short-range interacting models [START_REF] Barmettler | Propagation front of correlations in an interacting Bose gas[END_REF][START_REF] Cheneau | Lightcone-like spreading of correlations in a quantum many-body system[END_REF][START_REF] Carleo | Light-cone effect and supersonic correlations in one-and two-dimensional bosonic superfluids[END_REF][START_REF] Manmana | Time evolution of correlations in strongly interacting fermions after a quantum quench[END_REF] as well as long-range models [START_REF] Jurcevic | Quasiparticle engineering and entanglement propagation in a quantum many-body system[END_REF][START_REF] Richerme | Nonlocal propagation of correlations in quantum systems with longrange interactions[END_REF][START_REF] Hauke | Spread of correlations in longrange interacting quantum systems[END_REF][START_REF] Eisert | Breakdown of quasilocality in long-range quantum lattice mod-els[END_REF][START_REF] Cevolani | Protected quasi-locality in quantum systems with long-range interactions[END_REF][START_REF] Schachenmayer | Dynamics of correlations in two-dimensional quantum spin models with long-range interactions: A phase-space Monte-Carlo study[END_REF][START_REF] Buyskikh | Entanglement growth and correlation spreading with variable-range interactions in spin and fermionic tunneling models[END_REF][START_REF] Cevolani | Spreading of correlations in exactly solvable quantum models with longrange interactions in arbitrary dimensions[END_REF][START_REF] Frérot | Multispeed prethermalization in quantum spin models with power-law decaying interactions[END_REF][START_REF] Cevolani | Universal scaling laws for correlation spreading in quantum systems with short-and long-range interactions[END_REF] where weaker LR bounds exist [START_REF] Hastings | Spectral gap and exponential decay of correlations[END_REF][START_REF] Foss-Feig | Nearly linear light cones in long-range interacting quantum systems[END_REF]. However, many questions remain open. For instance, it is still debated whether a non-linear cone emerges in generic long-range systems, for which different results point towards either super-ballistic, ballistic or sub-ballistic spreading. It was recently proposed that these apparently conflicting results can be reconciled by the coexistence of several signals governed by different scaling laws [START_REF] Cevolani | Universal scaling laws for correlation spreading in quantum systems with short-and long-range interactions[END_REF]. This behavior may be related to the non-linearity of the quasiparticle excitation spectrum, and may also appear in systems with short-range interactions. In the later case, it is expected that both the signals spread ballistically but with different velocities. However, this picture relies on meanfield theory, which ignores potentially important dynamical effects, such as quasiparticle collisions and finite lifetime.

In this work, using an exact many-body approach beyond meanfield theory, we demonstrate the emergence of a universal twofold dynamics in the spreading of correlations for a generic short-range, strongly correlated quantum model. Specifically, we consider the one-dimensional Bose-Hubbard model and use time-dependent tensor network techniques based on matrix product states. Spanning the phase diagram, we almost always find a twofold structure of the space-time correlation pattern, characterized by two distinct velocities, essentially irrespective of the correlation function. Exceptions, discussed below, only appear for particular cases. In the superfluid meanfield regime and in the Mott insulator phase, the two spreading velocities are readily interpreted from the properties of the corresponding excitation spectra, which are known. In the strongly correlated superfluid regime, only the sound velocity is known. There, our results show beyond Luttinger liquid behavior and provide useful information about the excitation spectrum beyond the phonon branch. The emergence of a universal twofold spreading of correlations has profound implications on experimental observations, which we discuss, including with a view towards extensions to longrange systems.

Model and approach.-The Hamiltonian of the onedimensional (1D) Bose-Hubbard (BH) model, considered throughout this work, reads as

Ĥ = -J R â † R âR+1 + h.c. + U 2 R nR (n R -1), (1)
where âR and â † R are the bosonic annihilation and creation operators on site R, nR = â † R âR is the occupation number (filling), J is the hopping amplitude, U > 0 is the repulsive on-site interaction energy, and the lattice spacing is fixed to unity (R ∈ Z). At equilibrium and zero-temperature, the phase diagram of the 1D BH model is well known [START_REF] Sachdev | Quantum Phase Transitions[END_REF][START_REF] Cazalilla | One dimensional bosons: From condensed matter systems to ultracold gases[END_REF], and sketched on Fig. 1(a). It comprises a superfluid (SF) and a Mott insulator (MI) phase, determined by the competition of the hopping, the interactions, and the average filling n (or, equivalently, the chemical potential µ). For commensurate filling, n ∈ N * the SF-MI (Mott-U ) transition is of the Berezinskii-Kosterlitz-Thouless type, at the critical value u c 3.3 for unit filling (n = 1) in 1D [START_REF] Kühner | One-dimensional Bose-Hubbard model with nearest-neighbor interaction[END_REF][START_REF] Kashurnikov | Exact diagonalization plus renormalization-group theory: Accurate method for a onedimensional superfluid-insulator-transition study[END_REF][START_REF] Ejima | Dynamic properties of the one-dimensional bose-hubbard model[END_REF][START_REF] Rombouts | Loop updates for quantum Monte Carlo simulations in the canonical ensemble[END_REF]. For incommensurate filling, the Bose gas is a SF for any value of U/J. The commensurate-incommensurate (Mott-δ) transition, of the meanfield type, is then driven by doping when n approaches a positive integer value for sufficiently strong interactions.

We study the out-of-equilibrium dynamics of the BH model by applying a sudden global quench [START_REF] Barmettler | Propagation front of correlations in an interacting Bose gas[END_REF][START_REF] Carleo | Light-cone effect and supersonic correlations in one-and two-dimensional bosonic superfluids[END_REF][START_REF] Manmana | Time evolution of correlations in strongly interacting fermions after a quantum quench[END_REF][START_REF] Calabrese | Time dependence of correlation functions following a quantum quench[END_REF][START_REF] Kollath | Quench dynamics and nonequilibrium phase diagram of the Bose-Hubbard model[END_REF][START_REF] Moeckel | Interaction quench in the Hubbard model[END_REF][START_REF] Roux | Finite-size effects in global quantum quenches: Examples from free bosons in an harmonic trap and the onedimensional bose-hubbard model[END_REF][START_REF] Navez | Emergence of coherence in the Mott insulator-superfluid quench of the Bose-Hubbard model[END_REF][START_REF] Krutitsky | Propagation of quantum correlations after a quench in the Mott insulator regime of the Bose-Hubbard model[END_REF], as can be realized in ultracold-atom experiments [START_REF] Cheneau | Lightcone-like spreading of correlations in a quantum many-body system[END_REF][START_REF] Greiner | Collapse and revival of the matter wave field of a Bose-Einstein condensate[END_REF][START_REF] Langen | Local emergence of thermal correlations in an isolated quantum many-body system[END_REF][START_REF] Geiger | Local relaxation and light-cone-like propagation of correlations in a trapped one-dimensional Bose gas[END_REF]. We start from the ground state for some initial value of the interaction parameter (U/J) 0 and let the system evolve with a different value of U/J. In the following, we consider a variety of quenches, spanning the phase diagram, see arrows on Fig. 1(a). We study the spreading of both phase and density fluctuations, via the connected correlation functions G 1 (R, t) = â † R (t)â 0 (t) -â † R (0)â 0 (0) and G 2 (R, t) = g(R, t) -g(R, 0) with g(R, t) = nR (t)n 0 (t) -nR (t) n0 (t) . Both can be measured in experiments using time-of-flight and fluorescence microscopy imaging, respectively [START_REF] Cheneau | Lightcone-like spreading of correlations in a quantum many-body system[END_REF][START_REF] Langen | Local emergence of thermal correlations in an isolated quantum many-body system[END_REF][START_REF] Geiger | Local relaxation and light-cone-like propagation of correlations in a trapped one-dimensional Bose gas[END_REF][START_REF] Trotzky | Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas[END_REF].

All the results presented below are obtained using densitymatrix renormalization group simulations within the timedependent matrix-product state (t-MPS) representation [START_REF] Schollwöck | The density-matrix renormalization group[END_REF][START_REF] Schollwöck | The density-matrix renormalization group in the age of matrix product states[END_REF][START_REF] Dolfi | Matrix product state applications for the ALPS project[END_REF]. A careful analysis of the numerical cut-offs (high-filling cut-off and bond dimension) has been systematically performed to certify the convergence of the results in all the considered cases. This is particularly critical for quenches in the SF phase where the numerical requirements are most binding [43].

Meanfield regime.-We first consider the meanfield regime in the SF phase, where the numerical results can be compared to analytic predictions. This regime is characterized by a small Lieb-Liniger parameter, γ ≡ U/2J n 1. Figure 2(a) displays the t-MPS result for the G 2 correlation function versus distance (R) and time (t) for a quench from (U/J) 0 = 0.2 to U/J = 0.1 and n = 5, i.e. from γ 0 = 0.02 to γ = 0.01 [see red arrow on Fig. 1(a)]. It clearly shows a spike-like structure, characterized by two different velocities. On the one hand, a series of parallel maxima and minima move along straight lines corresponding to a constant propagation velocity V m (the dashed blue lines show fits to two of these minima). On the other hand, the various local extrema start at different activation times t (R). The latter are aligned along a straight line with a different slope (solid green line), corresponding a constant velocity V CE . The latter defines the correlation edge (CE) beyond which the correlations are suppressed. Similar results are obtained for all the other quenches in the meanfield regime, as well as for the

G 1 function [44].
This twofold structure near the CE is readily interpreted using the quasiparticle picture, which we briefly outline here (for details, see Ref. [START_REF] Cevolani | Universal scaling laws for correlation spreading in quantum systems with short-and long-range interactions[END_REF]): the G 1 and G 2 correlation functions are expanded onto the elementary excitations of the system. In the meanfield regime of the BH model, the latter are Bogoliubov quasiparticles with the quasimomentum k ∈ [-π, +π] and the dispersion relation

E k ε k (ε k + 2nU ), (2)
where

ε k = 4J sin 2 (k/2
) is that of the free-particle tightbinding model. A correlation between two points at a distance R is seeded when two correlated, counter-propagating quasiparticles emanating from the center reach the two points, see Fig. 1(b). The fastest ones are those with the maximum group velocity, V g = max k -1 ∂E k /∂k . It yields the activation time t (R) = R/2V g and the CE velocity V CE = 2V g , consistently with the expected Lieb-Robinson bound [START_REF] Lieb | The finite group velocity of quantum spin systems[END_REF][START_REF] Calabrese | Time dependence of correlation functions following a quantum quench[END_REF]. More precisely, the correlation at a distance R and a time t is built as a coherent superposition of the contributions of the various quasiparticles. In the vicinity of the CE, only the fastest quasiparticles, i.e. those with a quasimomentum k close to k , contribute. It creates a sine-like signal at the driving spatial frequency k , whose extrema move at twice the phase velocity [START_REF] Cevolani | Universal scaling laws for correlation spreading in quantum systems with short-and long-range interactions[END_REF]. The dispersion around k then modulates the sine-like signal by an envelope moving at the CE velocity V CE , see Fig. 1(c). This behavior is reminiscent of the propagation of a coherent wave packet in a dispersive medium [START_REF] Brillouin | Wave propagation and group velocity[END_REF][START_REF] Lighthill | Group velocity[END_REF][START_REF] Born | Principles of Optics[END_REF].

V ϕ (k) = -1 E k /k with k = k , i.e. V m = 2V ϕ
To test this picture quantitatively, we have extracted the velocities V m and V CE from the t-MPS results for G 2 (R, t) by tracking, respectively, the local extrema and the activation time. The results, displayed on Fig. 2(b), show excellent agreement with the theory, i.e. V CE 2V g and V m 2V ϕ within the fitting errorbars. This cross-validates the t-MPS results in the most-demanding SF, meanfield regime on the one hand and the quasiparticle picture above on the other hand. Note that the t-MPS results are numerically exact and include effects beyond the Bogoliubov approximation, such as quasiparticle collisions. The solid green and dashed blue lines correspond to fits to the CE and extrema, respectively. Note that on panel (b2), the fits to the maxima are shown as dashed white lines for clarity. (d) Spreading velocities VCE (green diamonds) and Vm (blue disks), as extracted from fits to the t-MPS data, and comparison to the characteristic velocities 2V g (solid green lines) and 2V ϕ (dashed blue lines), as found from the dispersion relations in the SF [Eq. ( 2)] and MI [Eq. ( 3)] regimes. All the quenches are performed from the initial values (U/J)0 = 1 for the SF regime and (U/J)0 = ∞ for the MI regime.

Strongly correlated regime at unit filling.-We now turn to the strongly correlated regime γ ∼ 1, where the correlation functions cannot be systematically computed. We first scan the after-quench interaction parameter U/J from the SF to the MI, along the Mott-U transition at unit filling [n = 1, see magenta arrows on Fig. 1(a)]. Note that each quench is performed in a unique phase: for U/J < u c 3.3 (SF regime), we use the initial interaction strength (U/J) 0 = 1 while for U/J > u c (MI regime), we start from (U/J) 0 = ∞. Figure 3 shows typical results for the spreading of the G 1 (upper row) and G 2 (lower row) correlations for quenches to the SF regime [U/J = 0.5, Fig. 3(a)], and to the MI regime, both slightly beyond the transition [U/J = 8, Fig. 3(b)], and deep in the MI regime [U/J = 24, Fig. 3(c)]. In all cases, at the notable exception of G 2 deep in the MI phase [Fig. 3(c2), see discussion below], we find a twofold spike-like structure. The velocities V m and V CE , extracted as before, are plotted on Fig. 3(d), showing similar results for G 1 and G 2 . This is consistent with the prediction that these velocities are characterized by the spectrum, irrespective of the observable [START_REF] Cevolani | Universal scaling laws for correlation spreading in quantum systems with short-and long-range interactions[END_REF].

In the SF regime, U/J < u c , the results compare very well with the predictions 2V ϕ and 2V g as found from the Bogoliubov dispersion relation [START_REF] Eisert | Quantum many-body systems out of equilibrium[END_REF] [see, respectively, the dashed blue and solid green lines on Figs. 3(d1) and (d2)]. Quite surprizing, the agreement is fair up to the critical point where γ c 1.6, far beyond the validity condition of the Bogoliubov theory (γ 1). In fact, when U/J increases from the meanfield regime, the momentum k decreases down to the phonon regime, k π, and the precise k-dependence of the dispersion relation beyond this regime becomes irrelevant. Moreover, the physics being dominated by long wavelength excitations, the lattice discretization in Eq. ( 1) may be disregarded and the BH model maps onto the con-tinuous Lieb-Liniger model [43]. The latter is integrable by Bethe ansatz (BA) [START_REF] Lieb | Exact analysis of an interacting Bose gas. I. The general solution and the ground state[END_REF][START_REF] Lieb | Exact analysis of an interacting Bose gas. II. The excitation spectrum[END_REF]. It yields the sound velocity V s 2n

√ γ 1 -√ γ/4π , to lowest order in the weak-γ expansion. Up to the critical point, the beyond-meanfield correction, √ γ/4π, is less than 10%, which explains the good agreement between the numerics and the analytic formula. At the critical point, the numerical results for V m and V CE are consistent with the exact BA value 2V s 4.6 [START_REF]Close to the Mott-U critical point at U/J = 3.5, we find Vm VCE 4.7 (5.3)[END_REF].

The spreading velocities V m and V CE are continuous at the Mott-U transition, and do not show any critical behavior. Right beyond the critical point, they are still nearly equal and we can hardly distinguish two features from the numerics up to U/J 6. Deeper in the MI phase, however, we recover two distinct features and two different velocities. Contrary to the SF regime, here we find V m > V CE . These results are readily interpreted from the quasiparticle picture. Deep enough in the MI phase, U/J 6, the low-energy excitations are doublon-holon pairs, characterized by the dispersion relation [START_REF] Barmettler | Propagation front of correlations in an interacting Bose gas[END_REF][START_REF] Ejima | Characterization of Mottinsulating and superfluid phases in the one-dimensional Bose-Hubbard model[END_REF]

2E k [U -2J(2n+1) cos(k)] 2 + 16J 2 n(n+1) sin 2 (k).
(3) The comparison between the spreading velocities V m and V CE fitted from the t-MPS results and the characteristic values 2V ϕ and 2V g , found from Eq. (3), yields a very good agreement, within less than 5% for G 1 and 9% for G 2 [see Figs. 3(d1) and (d2) respectively]. The quantitative agreement between the t-MPS results and the theoretical predictions for the G 1 correlations persists up to arbitrary values of U/J. This validates the quasiparticle analysis also in the strong-coupling regime.

Yet, the G 2 correlations behave differently. For intermediate interactions, 6 U/J 9, we find a twofold structure consistent with that found for G 1 . The signal for G 2 blurs when entering deeper in the MI regime, and we are not able to identify two distinct features for U/J 9. To understand this behavior, one may resort on a strong-coupling (U J) expansion of the correlation functions. In contrast to G 1 , the G 2 function cannot be cast into the generic form analyzed in Ref. [START_REF] Cevolani | Universal scaling laws for correlation spreading in quantum systems with short-and long-range interactions[END_REF]. Instead, combining Jordan-Wigner fermionization and Fermi-Bogoliubov theory [START_REF] Barmettler | Propagation front of correlations in an interacting Bose gas[END_REF]43], one finds

G 2 (R, t) -2|g 2 (R, t)| 2 with g 2 (R, t) ∝ J U R t ˆ+π -π
dk 2π e i(2E k t+kR) + e i(2E k t-kR) .

(4) For U 2(2n+1)J, the doublon-holon pair dispersion relation (3) reduces to 2E k U -2(2n + 1)J cos(k). Owing to the square modulus in the formula G 2 (R, t) -|g 2 (R, t)| 2 , we immediately find that the Mott gap U becomes irrelevant and we are left with the effective dispersion relation 2 Ẽk -2(2n + 1)J cos(k). On the one hand, the group velocity is not affected and we find the maximum value 2V g 2(2n + 1)J/ at k π/2. The value 2V g = 6J/ found for n = 1 is in excellent agreement with the value of V CE fitted from the G 2 function deep in the MI phase, see Fig. 3(d2). On the other hand, the corresponding effective phase velocity vanishes, 2 Ṽ ϕ 0. This is consistent with the disappearance of the spike-like structure observed in the t-MPS calculations for G 2 deep in the MI phase [START_REF]we find that in the vicinity of the CE both the real and imaginary parts of g2 display a series of static local maxima[END_REF]. In addition, the first-order correction to the leading strong-coupling term, relevant for moderate values of U/J, sustains a double structure with V g = V ϕ . The latter is consistent with the observation of two distinct spreading velocities, V CE = V m , closer to the Mott-U transition [43].

Strongly interacting superfluid regime.-We finally consider the strongly interacting regime of the SF phase, corresponding to γ 1 and n / ∈ N. In this regime, the Tomonaga-Luttinger liquid (TLL) theory accurately describes the lowenergy physics of the BH model at equilibrium, including the Mott-δ transition, see for instance Refs. [START_REF] Cazalilla | One dimensional bosons: From condensed matter systems to ultracold gases[END_REF][START_REF] Haller | Pinning quantum phase transition for a Luttinger liquid of strongly interacting bosons[END_REF][START_REF] Boéris | Mott transition for strongly interacting one-dimensional bosons in a shallow periodic potential[END_REF]. The TLL theory considers an effective harmonic fluid, characterized by a single characteristic velocity, namely the sound velocity V s .

In contrast, our t-MPS simulations in the strongly interacting SF regime clearly show beyond TLL physics. We have computed the spreading of correlations for a large value of the after-quench interaction parameter, U/J = 50, and varying the filling n up to the Mott-δ transition at n = 1 [see pink arrow on Fig. 1(a)]. The spreading velocities V CE (green diamonds) and V m (blue disks), found from fits to the two-body correlation function G 2 (R, t), are shown on Fig. 4. They show clear deviations from twice the sound velocity of the BH model in the strongly interacting limit, 2V s (4J/ ) sin(πn) 1 -(8J/U ) cos(πn) (orange dotted line and squares) [START_REF]The sound velocity Vs has been computed by mapping the BH model to an equivalent spinless Fermi model[END_REF]. Moreover, the emergence of two different characteristic velocities, V CE = V m , indicates that the TLL approach is insufficient to describe the spreading of correlations, even upon renormalization of the effective TLL parameters. Note that the two velocities become nearly equal in the vicinity of the Mott-δ transition and reach the value V CE V m 6J/ . This is consistent with the disappearance of the twofold structure and the value found for V CE deep in the MI phase at n = 1, see Fig. 3(d).

Conclusions.-In summary, working within the case study of the Bose-Hubbard chain and using a numericallyexact many-body approach, we have presented evidence of a universal twofold dynamics in the spreading of correlations. The latter is characterized by two distinct velocities, corresponding to the spreading of local maxima on the one hand and to the CE on the other hand. This has been found in all the phases of the model. Exceptions appear only in a few cases, for instance (i) for specific observables in specific regimes, or (ii) when the two velocities happen to be equal, as found at the Mott critical points for instance.

Our predictions are directly relevant to quench experiments on ultracold Bose gases in optical lattices, where the dynamics of one-body and two-body correlation functions can be observed on space and time scales comparable to our simulations [START_REF] Lewenstein | Ultracold atomic gases in optical lattices: Mimicking condensed matter physics and beyond[END_REF][START_REF] Bloch | Quantum simulations with ultracold quantum gases[END_REF][START_REF] Cheneau | Lightcone-like spreading of correlations in a quantum many-body system[END_REF][START_REF] Trotzky | Probing the relaxation towards equilibrium in an isolated strongly correlated one-dimensional Bose gas[END_REF]. Importantly, while in most experiments and numerics the CE is infered from the behavior of the correlation maxima, our results show that the two must be distinguished. This is expected to be a general feature of shortrange systems and should be relevant to models other than the sole BH model. Moreover, our study may be extended to long-range systems, such as spin models as realized in trapped-ion experiments [START_REF] Jurcevic | Quasiparticle engineering and entanglement propagation in a quantum many-body system[END_REF][START_REF] Richerme | Nonlocal propagation of correlations in quantum systems with longrange interactions[END_REF]. While the notions of a maximum group velocity and phase velocity may break down in such systems, the meanfield theory also predicts a twofold dynamics [START_REF] Cevolani | Universal scaling laws for correlation spreading in quantum systems with short-and long-range interactions[END_REF]. In this case, it is characterized by the coexistence of superballistic and sub-ballistic signals. The results of the present paper suggest that the twofold structure of the correlation function may survive in strongly correlated regimes also for long-range systems. The demonstration of this effect would shed light on the still debated scaling of the light cone in longrange systems.

This research was supported by the European Commission FET-Proactive QUIC (H2020 grant No. 641122). The numerical calculations were performed using HPC resources from CPHT and GENCI-CCRT/CINES (Grant No. c2017056853), and make use of the ALPS library [START_REF] Dolfi | Matrix product state applications for the ALPS project[END_REF]. We are grateful to the CPHT computer team for valuable support. present at equilibrium. For the SF strongly correlated regime at n = 1 [Fig. 3(a)], we used χ = 300 for both correlation functions. A similar value of χ was considered for moderate values of U/J in the MI phase at n = 1 [Fig. 3(b)]. Deep in the MI phase [Fig. 3(b)], the bond dimension can be significantly decreased and we consider χ = 100. Finally, in the SF strongly interacting regime at U/J = 50, we found that the value χ = 100 is enough.

S2. ONE-BODY CORRELATION FUNCTION G1(R, t) IN THE MEANFIELD REGIME

In the analysis of the SF meanfield regime reported in the main paper, we focused on the two-body correlation function G 2 (R, t). We have also studied the one-body correlation G 1 (R, t) using the same t-MPS simulations. We found that the dynamics of the G 1 function shows a spike-like structure, similar to that found for the G 2 function. The values of the correlation edge (V CE ) and maxima (V m ) velocities agree with those found for the G 2 function within less than 10%. Figure S1 shows an example, for the quench from (U n/J) 0 = 1 to U n/J = 0.5, and n = 5. The fits to the correlation edge and to the maxima yield the velocities V CE = (4.4 ± 0.3) J/ and V m = (3.3 ± 0.2) J/ , in excellent agreement with the corresponding values found from the dynamics of the G 2 function, see Fig. 2(b).

The agreement between the spreading velocities for different correlation functions was found in all regimes, see for instance Figs. 3(d1) and (d2). It is consistent with the prediction that these velocities are characteristic of the excitation spectrum and not on the details of the correlation function [START_REF] Cevolani | Universal scaling laws for correlation spreading in quantum systems with short-and long-range interactions[END_REF]. Note, however, that the full space-time dependence of the signal depends on the correlation function. In general, we found that the signal for G 1 is less sharp than for G 2 . This may be attributed to the long-range phase correlations present in the initial state, which blur the correlation function [START_REF] Bravyi | Lieb-Robinson bounds and the generation of correlations and topological quantum order[END_REF].

S3. MAPPING ON THE 1D LIEB-LINIGER MODEL

In the long-wave length regime, the lattice discretization of the Bose-Hubbard (BH) may be disregarded. The BH model then maps onto the continuous-space Lieb-Liniger (LL) model,

Ĥ = 2 2m   - N i=1 ∂ 2 ∂x 2 i + c i =j δ(x i -x j )   . (S2)
It describes a one-dimensional gas of N bosons of mass m with contact interactions, characterized by the interaction strength c > 0. The correspondance between the parameters of the BH and LL models is found by discretizing the LL model, Eq. (S2), on the length scale defined by the lattice spacing a. It yields J = 2 /2ma 2 and U = 2 c/ma. The density of the LL model is ρ ≡ N/L = n/a, where n is the number of bosons per lattice site (filling) and L is the system size. The LL Hamiltonian is exactly solvable by Bethe ansatz [START_REF] Lieb | Exact analysis of an interacting Bose gas. I. The general solution and the ground state[END_REF][START_REF] Lieb | Exact analysis of an interacting Bose gas. II. The excitation spectrum[END_REF]. All the thermodynamic quantities at zero temperature can be written as universal functions of the Lieb-Liniger parameter γ = c/ρ and the dimensionless quantity e(γ) = E 0 /N n 2 , where E 0 is the ground state energy. For instance, the macroscopic sound velocity [START_REF] Lieb | Exact analysis of an interacting Bose gas. II. The excitation spectrum[END_REF] reads as

v s ≡ L mρ ∂ 2 E 0 ∂L 2 N,S = ρ m 3e(γ) -2γe (γ) + 1 2 γ 2 e (γ). (S3)
Using the small γ expansion, e(γ) = γ 1 -(4/3π) √ γ , one then finds

v s = ρ m √ γ 1 - √ γ/4π , (S4)
valid in the weakly-interacting regime, γ 1. Finally, using the correspondance between the parameters of the BH and LL models, one finds

V s ≡ v s /a = 2Jn √ γ 1 - √ γ/4π (S5)
and γ = U/2Jn.

S4. TWO-BODY CORRELATION FUNCTION G2(R, t) IN THE MOTT-INSULATING PHASE

In order to explain the suppression of the twofold structure for the two-body correlations deep in the Mott insulator phase (MI; U J and n = 1), we compute the function G 2 (R, t), working along the lines of Ref. [START_REF] Barmettler | Propagation front of correlations in an interacting Bose gas[END_REF]. Considering the manifold of doublon-holon pairs and mapping the resulting Hamiltonian into a fermionic one, the two-body correlation function may be written as

G 2 (R, t) -2 |g 2 (R, t)| 2 + |ḡ 2 (R, t)| 2 , (S6) with g 2 (R, t) ∼ J U R t ˆ+π -π dk 2π e i(2E k t+kR) + e i(2E k t-kR) , (S7) ḡ2 (R, t) ∼ J U 2 ˆ+π -π dk 2π sin 2 (k) e i(2E k t-kR) + e -i(2E k t+kR) (S8)
and the excitation spectrum is 2E k [U -2J(2n+1) cos(k)] 2 + 16J 2 n(n+1) sin 2 (k), see Eq. (3).

Quench deep into the Mott insulator phase.-For a quench, very deep in the MI phase, U J, the second right-hand-side term in Eq. (S6) is much smaller than the first one and the former can be neglected. Using Eq. (S7), it yields explicitly for

G 2 (R, t) -2|g 2 (R, t)| 2 , G 2 (R, t) ∼ -2 J U 2 R t 2 ˆπ -π dk 2π e i(2E k t+kR) + e i(2E k t-kR) 2 (S9)
Moreover, the excitation spectrum may be expanded in powers of J/U . Up to first-order, it yields 2E k U -2J(2n + 1) cos(k). The gap term e iU t can then be factorized in the two terms under the integral in Eq. (S9) and disappears due to the square modulus. Introducing the effective excitation spectrum 2 Ẽk = -2J(2n + 1) cos(k), we then find

G 2 -2|g 2 (R, t)| 2 with g 2 (R, t) ∼ J U R t ˆπ -π dk 2π e i(2 Ẽk t+kR) + e i(2 Ẽk t-kR) . (S10)
The integral may be evaluated using the stationary phase approximation. In the infinite time and distance limit along the line R/t = cst, the integral in Eq. (S10) is dominated by the momentum contributions with a stationary phase (sp), i.e. ∂ k (2 Ẽk t ± kR) = 0 or, equivalently, 2 Ṽg (k sp ) = ±R/t where Ṽg = ∂ k Ẽk is the group velocity of the effective excitation spectrum. Since the latter is upper bounded by the value Ṽ * g = max( Ṽg ) = J(2n + 1), it has a solution only for R/t < 2 Ṽ * g . We then find with σ = sgn ∂ 2 k E ksp and E k the excitation spectrum given at Eq. (3). Using the same argument as above, we find that ḡ2 (R, t) shows a twofold structure characterized by, now, the CE velocity 2V * g = 2max (∂ k E k ) but the velocity of the maxima 2V * ϕ = 2E k * /k * = 0. Since there is a single contribution here, the quantity |ḡ 2 (R, t)| 2 displays a twofold structure with the same characteristic velocities. More precisely, both the length and time scales of the oscillations are divided by two but the velocities are not affected.

g 2 (R, t) ∼ J U Ṽg (k sp ) |∂ 2 k Ẽksp |t 1/2 cos 2 Ẽksp t -k sp R + σ π 4 + i sin 2 Ẽksp t -k sp R + σ π 4 . (
For a quench into the MI phase at a moderate value of U/J, both |g 2 (R, t)| 2 and |ḡ 2 (R, t)| 2 contribute to the two-body correlation function G 2 (R, t). While the |g 2 (R, t)| 2 contribution is characterized by the sole CE velocity 2V * g , the |ḡ 2 (R, t)| 2 contribution provides the double structure observed on G 2 for 6 < U/J < 10 in the t-MPS calculations.

Figure 1 .

 1 Figure 1. Quantum quench in the Bose-Hubbard model. (a) Schematic phase diagram as a function of the inverse interaction strength and chemical potential, comprising a MI phase (pink lobes at integer fillings n) and a SF phase. The Mott-U transition at unit filling is indicated by the dashed pink line and the Mott-δ transition by the vertical line. The arrows indicate the various quenches considered in this work. (b) Generation of correlations between two points at a distance R by pairs of counter-propagating quasiparticles emitted at the mid-point R/2. The first correlation is generated by the fastest quasiparticles at the activation time t = R/2V g . (c) Correlation spreading in the vicinity of the correlation edge (CE). The correlation function forms a periodic series of maxima moving at the velocity Vm = 2V ϕ , with an envelope moving at the velocity VCE = 2V * g .

Figure 2 .

 2 Figure 2. Spreading of correlations in the meanfield regime, see red arrow on Fig. 1(a). (a) t-MPS result of G2(R, t) for a quench to U/J = 0.1, together with ballistic fits to the CE (solid, green line) and minima (dashed, blue lines). (b) Velocities of the CE (VCE, green diamonds) and minima (Vm, blue disks), found from the fits, versus the interaction strength, and comparison to twice the group velocity 2V * g (solid green line) and twice the phase velocity 2V * ϕ (dashed blue line). All the quenches are performed with n = 5 from (U/J)0 = 0.2, except for the points at U n/J = 1 where U/J = 0.2 and we use a different initial value, (U/J)0 = 0.4 (open points).

Figure 3 .

 3 Figure 3. Spreading of the G1 (upper row) and G2 (lower row) correlations in both the SF and MI phases for n = 1, scanning the after-quench interaction U/J along the Mott-U transition, see pink dashed line and magenta arrows on Fig. 1(a): (a) SF regime with U/J = 0.5; (b) MI regime near the critical point with U/J = 8; (c) deep MI regime with U/J = 24.The solid green and dashed blue lines correspond to fits to the CE and extrema, respectively. Note that on panel (b2), the fits to the maxima are shown as dashed white lines for clarity. (d) Spreading velocities VCE (green diamonds) and Vm (blue disks), as extracted from fits to the t-MPS data, and comparison to the characteristic velocities 2V g (solid green lines) and 2V ϕ (dashed blue lines), as found from the dispersion relations in the SF [Eq. (2)] and MI [Eq. (3)] regimes. All the quenches are performed from the initial values (U/J)0 = 1 for the SF regime and (U/J)0 = ∞ for the MI regime.

Figure 4 .

 4 Figure 4. Twofold spreading of the G2 correlations in the strongly interacting SF regime for U/J = 50 and 0 < n < 1. Shown are the spreading velocities VCE (green diamonds) and Vm (blue disks) fitted from the t-MPS simulations, together with twice the sound velocity 2Vs of the BH model as found from Bose-Fermi mapping (dashed orange line) and from MPS calculations (orange squares)[START_REF]The sound velocity Vs has been computed by mapping the BH model to an equivalent spinless Fermi model[END_REF]. Filled symbols correspond to the initial interaction parameter (U/J)0 = 1 and open symbols to (U/J)0 = 40. The crosses are linear extrapolations of VCE and Vm to the Mott-δ transition at n = 1.

Figure S1 .

 S1 Figure S1. Spreading of the one-body correlation function G1(R, t) for a global quench in the SF meanfield regime from (U/J)0 = 0.2 to U/J = 0.1 and n = 5. The solid-green and dashed-blue lines are fits to the CE and maxima, respectively.

  S11) with σ = sgn ∂ 2 k Ẽksp . For both the real and imaginary parts of g 2 (R, t), the correlations are activated ballistically at the time t = R/2 Ṽ * g . It defines a linear correlation edge (CE) with velocity V CE = 2 Ṽ * g . In addition, Eq. (S11) also yields a series of local maxima, defined by the equation 2 Ẽksp t -k sp R = cst. In the vicinity of the CE cone, these maxima (m) propagate at the velocity V m = 2 Ṽ * ϕ = 2 Ẽk * /k * , i.e. twice the phase velocity at the maximum of the group velocity, k * . Hence, the real and imaginary parts of g 2 (R, t) both display a twofold structure with a CE velocity 2 Ṽ * g = 2J(2n + 1) and a velocity of the maxima 2 Ṽ * ϕ = 0, as shown on Figs. S2(a) and (b). In contrast, G 2 (R, t), does not display the twofold structure. This is because it is the sum of the squares of the two latter contributions [see Eq. (S11)], which are shifted by half a period and cancel each other. It thus gives a single cone structure, characterized by the sole CE velocity 2 Ṽ * g , as shown on Fig. S2(c).

Figure S2 . 2 sin 2 (k sp ) |∂ 2 k E ksp |t 1 / 2

 S222212 Figure S2. Analysis of the space-time correlation pattern of G2(R, t) via g2(R, t) [see Eq. (S10)] at n = 1 for a global quench confined deep into the Mott-insulating phase starting from a pure Mott state (U/J)0 → ∞. Analytical expression, owing to prefactors, of (a) -2 [g2(R, t)] (b) -2 [g2(R, t)] (c) sum of the two contributions shown at Fig. (a) and (b). The solid green line corresponds to the theoretical CE velocity characterized by 2 Ṽ * g = 2J(2n + 1). On Fig. (c), the first extremum propagates with the same velocity as the one associated to the CE. Quench into the Mott insulator phase for moderate U/J.-For moderate values of U/J, still in the MI phase, the second term in the right-hand-side of Eq. (S6), |ḡ 2 (R, t)| 2 , becomes relevant. Using again the stationary-phase approximation for ḡ2 (R, t), we find ḡ2 (R, t) ∼ J U 2 sin 2 (k sp ) |∂ 2 k E ksp |t 1/2 cos 2E ksp t -k sp R + σ π 4 (S12)

Supplemental Material for

Twofold correlation spreading in a strongly correlated lattice Bose gas

In this supplemental material, we give more details about several points discussed in the main paper. In Sec. S1, we discuss the time-dependent matrix-product state (t-MPS) simulations and the choice of the numerical parameters to ensure the convergence of the numerical results in all the regimes of the Bose-Hubbard model considered in the main paper. In Sec. S2, we present t-MPS results for the spreading of the one-body correlation function G 1 in the meanfield superfluid (SF) regime. Section S3 briefly outlines the mapping from the 1D Bose-Hubbard model to the Lieb-Liniger model and gives the correspondance of the parameters. Finally, in Sec. S4 we discuss the strong-coupling expansion of the correlation function G 2 for unit filling, n = 1, and discuss the suppression of its twofold structure deep in the Mott insulator (MI) phase.

S1. TIME-DEPENDENT MATRIX-PRODUCT STATE SIMULATIONS

The numerical results reported in the main paper are all obtained using the time-dependent density-matrix renormalization group approach (DMRG) with the matrix-product state representation (t-MPS approach) [START_REF] Schollwöck | The density-matrix renormalization group[END_REF][START_REF] Schollwöck | The density-matrix renormalization group in the age of matrix product states[END_REF][START_REF] Dolfi | Matrix product state applications for the ALPS project[END_REF]. It yields numerically-exact results on both equilibrium and out-of-equilibrium properties of low dimensional lattice models. The approach resorts on the Schmidt expansion of the many-body wave function and permits to reduce the Hilbert space to a finite, relevant subset, provided the entanglement entropy remains sufficiently small. Owing to the area law [START_REF] Wolf | Area laws in quantum systems: Mutual information and correlations[END_REF][START_REF] Eisert | Colloquium: Area laws for the entanglement entropy[END_REF], it is optimal for 1D lattice models with a finite local Hilbert space in gapped phases, the entanglement of which remains finite in the thermodynamic limit. It also applies to gapless phases, although with more stringent numerical parameters (high-filling cut-off and the bond dimension). To validate the accuracy of out results in all phases of the BH model, a systematic study of the effect of these parameters has been performed.

Truncation of the local Hilbert space.-For the BH model considered in this work, the local Hilbert space is spanned by the Fock basis of number states, |n R , where n R ∈ N, which is infinite. However, the probability distribution of the latticesite occupation n R decays faster than exponentially in both the SF and MI phases. Accurate results can thus be obtained by cutting off the local Hilbert space to some value n max . It is important to note that, in some cases, the value of n max needs to be significantly much larger than the average filling n and its fluctuations. This observation is consistent with analyses of truncated Bose-Hubbard models in quantum Monte Carlo simulations [START_REF] Kashurnikov | Zero-point phase transitions in the one-dimensional truncated bosonic Hubbard model and its spin-1 analog[END_REF].

The SF meanfield regime, which corresponds to a high filling factor n and the gapless dispersion relation, has the most binding criteria. We found that a good estimator for n max is given by the condition 1 -nmax n=0 P (n) 10 -2 , where P (n) is the probability that n bosons occupy a given lattice site. In the SF meanfield regime, the probability distribution is nearly Poissonian, P (n) nn e -n /n!. For instance, for the filling factor n = 5 used for the data of Fig. 2, it yields n max 12. For the strongly correlated SF regime at n = 1 considered for Fig. 3(a), the density fluctuations are significantly suppressed and using the same condition as previously leads to n max = 5. For the MI phase at n = 1 and moderate values of U/J (15 ≥ U/J ≥ u c ) considered for Fig. 3(b), we kept n max = 5. Deep in the MI phase (U/J ≥ 15), truncating the local Hilbert space to n max = 2, as used for Fig. 3(c) turns out to be sufficient. Finally, the strongly interacting SF regime is the easiest case from a numerical point of view. Owing to the low filling factor n < 1 and the large value of the interaction parameter U/J, the above condition also yields n max = 2, as used for Fig. 4. In all cases, we have checked that the numerics are converged for these values of n max .

Bond dimension.-Within the MPS approach, the many-body state for a M -site lattice is represented in the tensor network form

where n j spans a local Hilbert space basis. For the BH model, it corresponds to a Fock basis truncated at n max . For each value of n j , the quantity A nj [j] is a χ j-1 × χ j matrix, where χ j is the rank associated to the Schmidt matrix when applying the j-th singular value decomposition [START_REF] Schollwöck | The density-matrix renormalization group in the age of matrix product states[END_REF]. The bond dimension χ is defined as the maximum rank, χ = max j (χ j ) , j ∈ [0 . . . M ].

Note that for open-boundary conditions, the quantities A n1 [START_REF] Polkovnikov | Colloquium: Nonequilibrium dynamics of closed interacting quantum systems[END_REF] and A n M [M ] are actually a row vector and a column vector, respectively, i.e. χ 0 = χ M = 1.

In the numerics, the maximum value of χ is chosen sufficiently large so that the truncation does not affect the results. In practice, the calculations are run for several values of χ up to convergence of the correlation function G 1 (R, t) or G 2 (R, t). The required value of χ significantly depends on the regime and on the observable. In the following, we give the values used for the final results presented in the paper.

For the SF meanfield regime [Figs. 2(a) and S1], we used the values χ = 300 and χ = 450 for the G 2 and G 1 functions, respectively. The bond dimension used for G 1 is higher than the one for G 2 due to the long-range phase correlations already


Joachim Asch

Alain Joye

Lower Bounds on the Localisation Length of Balanced Random Quantum Walks

We consider the dynamical properties of Quantum Walks defined on the d-dimensional cubic lattice, or the homogeneous tree of coordination number 2d, with site dependent random phases, further characterised by transition probabilities between neighbouring sites equal to 1/(2d). We show that the localisation length for these Balanced Random Quantum Walks can be expressed as a combinatorial expression involving sums over weighted paths on the considered graph. This expression provides lower bounds on the localisation length by restriction to paths with weight 1, which allows us to prove the localisation length diverges on the tree as d 2 . On the cubic lattice, the method yields the lower bound 1/ ln(2) for all d, and allows us to bound the localisation length from below by the correlation length of self-avoiding walks computed at 1/(2d).

Introduction

The last decade has seen Quantum Walks, (QWs for short), deterministic or random, play a growing role in several scientific fields, starting with the modeling of quantum dynamics in various physical situations. Let us simply mention that QWs provide a simple description of the electronic motion in the Quantum Hall geometry [START_REF] Chalker | quantum tunnelling and the integer Hall effect[END_REF][START_REF] Kramer | Random network models and quantum phase transitions in two dimensions[END_REF][START_REF] Asch | Dynamical Localization of the Chalker-Coddington Model far from Transition[END_REF], while their effectiveness in quantum optics as discrete models to study atoms trapped in time periodic optical lattices or polarized photons propagating in networks of waveguides has found experimental confirmation [ZKGS+, SVAM+]. Theoretical quantum computing is another field in which QWs prove to be useful as tools in the elaboration or assessment of quantum algorithms, see [Sa, MNRS, P]. From a mathematical perspective, QWs have been considered as non commutative analogs of classical random walks, see e.g. [Ko, SKJ, GNVW], or as subject of study in the spectral analysis of (random) unitary operators, in the line of [START_REF] Bourget | Spectral Analysis of Unitary Band Matrices[END_REF][START_REF] Cantero | Five-Diagonal Matrices and Zeros of Orthogonal Polynomials on the Unit Circle[END_REF][START_REF] Joye | Density of States and Thouless Formula for Random Unitary Band Matrices[END_REF][START_REF] Hamza | Dynamical Localization for Unitary Anderson Models[END_REF][START_REF] Damanik | Dynamics of unitary operators[END_REF]. See the reviews and book [START_REF] Joye | Dynamical Localization for d-Dimensional Random Quantum Walks[END_REF][START_REF] Simon | Orthogonal polynomials on the unit circle[END_REF][START_REF] Asch | Spectral Stability of Unitary Network Models[END_REF] for more applications, mathematical results, and references.

The present paper addresses the mathematical (de)localisation properties of certain Random Quantum Walks (RQWs for short) defined on infinite graphs, for the discrete time random quantum dynamical system defined by the iterates of the RQW unitary operator. Anderson localisation is ubiquitous in the Physics of disordered systems and has been the object of intense research since its discovery in the fifties [A-CAT], see e.g. [CL, Ki, AW]. Delocalisation of Quantum particles in presence of disorder, although expected to be present in high dimensional lattices, is however difficult to exhibit; it has been proven to hold for the Anderson model on the tree, or mild variants thereof [Kl, W]. In this context, certain RQWs viewed as substitutes of the random evolution operators generated by the discrete Anderson Hamiltonians, have proven to be mathematically more tractable and to display analogous behaviours regarding the transport properties: these RQWs describe the motion of a particle with internal degree of freedom, or quantum walker, hopping on the sites an infinite underlying graph G d , standing for the cubic lattice Z d or T 2d , the homogeneous tree of coordination number 2d, in a static random environment. The internal degree of freedom, or coin state, lives in C 2d and the deterministic part of the walk is defined as follows. Given a natural basis in C 2d , the one time step unitary evolution U (C) is obtained by the action of a unitary matrix C ∈ U (2d) on the coin state of the particle, followed by the action of a coin state conditioned shift S which moves the particle to its nearest neighbours on G d . Static disorder is introduced in the model via i.i.d. random phases used to decorate the coin matrix C in such a way that the unitary coin state update becomes site-dependent and random. The coin matrix C of the resulting random unitary operator U ω (C), called the skeleton, is a parameter which, in a sense, monitors the strength of the disorder; see the next section for precise definition.

Restricting attention to RQWs constructed this way, we recall some of their properties. RQWs U ω (C) defined on the one dimensional lattice exhibit dynamical localisation for all non-trivial choice of skeleton C, [JM, ASW]. Moreover, RQWs that have skeleton close to certain permutation matrices are known to exhibit dynamical localisation on the cubic lattice Z d , d > 1 [J2, J3]; a result analogous to the strong disorder regime in the Anderson model framework. To complete the analogy, RQW's U ω (C) defined on homogeneous trees are proven to undergo a localisation-delocalisation spectral transition in [START_REF] Hamza | Spectral Transition for Random Quantum Walks on Trees[END_REF], when the skeleton C moves from neighbourhoods of certain permutation matrices to neighbourhoods of other permutation matrices, akin to the spectral transition occurring for the Anderson model on the tree as a fonction of the strength of the disorder [W]. See also [START_REF] Asch | Chirality induced Interface Currents in the Chalker Coddington Model[END_REF] for delocalisation results of a topological nature for the Chalker Coddington model.

The goal of the present paper is to analyse the unitary operator U ω (C) in case the skeleton C lies as far as possible from the permutation matrices, for which localisation results hold, and to investigate the possible delocalisation properties of those RQWs. Therefore we consider RQWs associated to balanced skeletons C ∈ U (2d) characterised by the fact that in the natural basis the moduli of its entries are all equal to 1/ √ 2d. This amounts to saying that the quantum mechanical transition probabilities induced by U ω (C) between neighbouring sites are all equal to 1/(2d), irrespective of the realisation of the random phases decorating the balanced skeleton. Such RQWs are called Balanced Random Quantum Walks, BRQWs for short. That BRQWs are likely to exhibit delocalisation properties on the cubic lattice is not granted. However, numerics obtained for the Chalker-Coddington model suggest delocalisation for values of the parameters of the model that correspond, essentially, to the balanced behaviour introduced above, see [KOK]; hence, the analogy is tempting. When considered on the tree T 2d , BRQWs are thus neither related to localisation, nor to delocalisation, a priori, and correspond to values of the parameter C that are not covered by the analysis of [START_REF] Hamza | Spectral Transition for Random Quantum Walks on Trees[END_REF].

Our results concern lower bounds on the localisation length of BRQWs U ω (C) defined on the cubic lattice Z d or the tree T 2d , making use of two characteristic specificities of BRQWs operators: uniform transition probabilities between neighbouring sites and uniform distribution of the iid random phases. The localisation length L we consider is defined, informally, as the inverse of the largest α > 0 such that lim sup n→∞ E( e α|X|/2 U n ω (C) ψ 0 2 ) < ∞, for all initial state ψ 0 with finite support on G d , and |X| is the multiplication operator by the norm of the position on G d . The fact that RQWs couple nearest neighbours on G d only allows for a head on approach of the localisation length.

Since BRQWs are essentially parameter free models, we investigate the large d limit of L, that we will dub (improperly for the tree case) the large dimension limit, a regime in which delocalisation is intuitively more likely to occur, so that L is more likely to be large. Our first contribution to this question is a reformulation of the problem in terms of a combinatorics problem on the underlying graph G d , presented in Section 3 . Within this framework, lower bounds on the localisation length are obtained via the analysis of partition functions defined on certain subsets of paths on G d , with weights related to the phases of the entries of the balanced skeleton in U ω (C). Using this framework for the homogeneous tree T 2d in Section 3.2, we get the large d behaviour L ≥ 2d 2 (1 + O(1/d)) in Theorem 3.11. We discuss this approach in the case of the cubic lattice Z d in Section 3.3. We cannot show that the localisation length diverges with the dimension d, but get the lower bound L ≥ 1/ ln(2), for all d ≥ 1, see Theorem 3.14. From a more general perspective, we relate the partition functions used to bound the localisation length to partition functions or susceptibilities considered in the study of self-avoiding walks in Z d in the closing Section 4. In particular, we show in Proposition 4.3 that that the localisation length satisfies L ≥ ξ d (1/(2d)), where ξ d (1/(2d)) is the correlation length of self-avoiding walks in Z d , at the critical value of simple random walks. The large dimension analysis of this expression remains to be performed which, according to experts in the field, represents a nontrivial task. see [START_REF] Hamza | Spectral Transition for Random Quantum Walks on Trees[END_REF][START_REF] Hamza | Spectral Properties of Non-Unitary Band Matrices[END_REF] 2.1 Random quantum walks on T 2d Let T 2d denote the homogeneous tree of degree 2d of the free group F {a 1 ,...,a d } generated by the alphabet

A 2d = {a 1 , . . . , a d , a -1 1 , . . . , a -1 d }, (1)
with a j a -1 j = a -1 j a j = e, j = 1, . . . , d, e being the neutral element of the group. A vertex of T 2d denoted by e is chosen to be the root of the tree. Each vertex x = x 1 x 2 . . . x n , n ∈ N of T 2d is a reduced word of finitely many letters from the alphabet A 2d and an edge of T 2d is a pair of vertices (x, y) such that xy -1 ∈ A 2d . Any pair of vertices x and y can be joined by a unique set of edges, or path in T 2d and the number of nearest neighbours of any vertex in T 2d is thus 2d. We identifying T 2d with its set of vertices, and define the configuration Hilbert space of the walker by

l 2 (T 2d ) = ψ = x∈T 2d ψ x |x s.t. ψ x ∈ C, x∈T 2d |ψ x | 2 < ∞ , (2)
where |x denotes the element of the canonical basis of l 2 (T 2d ) which sits at vertex x.

The coin Hilbert space (or spin Hilbert space) of the quantum walker on T 2d is C 2d . The elements of the ordered canonical basis of C 2d are labelled by letters in the alphabet A 2d = {a 1 , . . . , a d , a -1 1 , . . . , a -1 d }, and the total Hilbert space is

K d = l 2 (T 2d ) ⊗ C 2d with canonical basis x ⊗ τ ≡ |x ⊗ |τ , x ∈ T 2d , τ ∈ A 2d . (3)
The quantum walk on the tree is defined as the composition of a unitary update of the coin (or spin) variables in C 2d followed by a coin state dependent shift on the tree. Let C be a unitary matrix on C 2d , i.e. C ∈ U (2d). The unitary update operator given by I ⊗ C acts on the canonical basis of K d as

(I ⊗ C)x ⊗ τ = |x ⊗ |Cτ = σ∈A 2d C στ x ⊗ σ, (4)
where

{C στ } (σ,τ )∈A 2 2d
denote the matrix elements of C. The coin state dependent shift S on K d is defined by

S = τ ∈A 2d S τ ⊗ |τ τ | (5)
where for all τ ∈ A 2d the unitary operator S τ is a shift that acts on l 2 (T 2d ) as

S τ |x = |xτ , ∀τ ∈ A 2d , ∀x ∈ T 2d . (6)
Note that S -1 τ = S * τ = S τ -1 . A homogeneous quantum walk on T 2d is then defined as the one step unitary evolution operator on

K d = l 2 (T 2d ) ⊗ C 2d given by U (C) = S(I ⊗ C) = τ ∈A 2d ,x∈T 2d |xτ x| ⊗ |τ τ |C, (7)
where C ∈ U (2d) is a parameter. Consider now a family of coin matrices C = {C(x) ∈ U (2d)} x∈T 2d , indexed by the vertices x ∈ T 2d . We generalise the construction to quantum walks with site dependent coin matrices by means of the definition

U (C) = τ ∈A 2d ,x∈T 2d |xτ x| ⊗ |τ τ |C(x). (8)
In order to deal with RQWs, we introduce a probability space Ω = T T 2d ×A 2d , T being the torus, with σ algebra generated by the cylinder sets and measure

P = ⊗ x∈T 2d τ ∈A 2d
dν where dν is a probability measure on T. Let {ω x,τ } x∈T 2d ,τ ∈A 2d be a set of i.i.d. random variables on the torus T with common distribution dν. We will note Ω ω = {ω x,τ } x∈T 2d ,τ ∈A 2d .

The RQWs we consider are constructed by means of families of site dependent random coin matrices. Let C ω = {C ω (x) ∈ U (2d)} x∈T 2d be the family of random coin matrices depending on a fixed matrix C ∈ U (2d), the skeleton, where, for each x ∈ T 2d , C ω (x) is defined by its matrix elements

C ω (x) τ σ = e iωxτ,τ C τ σ , (τ, σ) ∈ A 2d × A 2d . (9)
The site dependence appears only in the random phases of the matrices C ω (x), which have a fixed skeleton C ∈ U (2d). The RQWs considered depend parametrically on C ∈ U (2d) and are defined by the random unitary operator

U ω (C) := U (C ω ) on K = l 2 (T 2d ) ⊗ C 2d . ( 10
)
Observe that defining a random diagonal unitary operator on K d by

D ω x ⊗ τ = e iωx,τ x ⊗ τ, ∀(x, τ ) ∈ T 2d × A 2d , (11)
we have the identity

U ω (C) = D ω U (C) on K d , (12)
which confirms that U ω (C) is manifestly unitary. Note also that the transition amplitudes induced by U ω (C) are nonzero for nearest neighbours on the tree only and

x ⊗ τ |U ω (C)y ⊗ σ = e iωx,τ x ⊗ τ |U (C)y ⊗ σ . ( 13
)
Remark 2.1 RQWs of a similar flavour defined on trees with odd coordination number are studied in [START_REF] Hamza | Spectral Transition for Random Quantum Walks on Trees[END_REF].

Random quantum walks on Z d

The definition of a RQW U ω (C) on Z d instead of T 2d is quite similar: the sites x ∈ T 2d are replaced by x ∈ Z d so that the configuration space l 2 (T 2d ) is replaced by l 2 (Z d ) but the coin space remains the same, i.e. C 2d . Hence the complete Hilbert space is

K d = l 2 (Z d ) ⊗ C 2d
and the update operator I ⊗ C is the same as on the tree. The definition of the shifts S τ in 5), needs to be slightly adapted. We associate the letters τ of the alphabet A 2d with multiples of the canonical basis vectors {e 1 , . . . , e d } of R d as follows

S = τ ∈A 2d S τ ⊗ |τ τ |, see (
a τ ↔ e τ , a -1 τ ↔ -e τ , τ ∈ {1, . . . , d}, (14)
and define the action of S τ on l 2 (Z 2 ) accordingly: for any

x = (x 1 , . . . , x d ) ∈ Z d S aτ |x = |x + e τ , S a -1 τ |x = |x -e τ , τ ∈ {1, . . . , d}. (15)
We shall sometimes abuse notations and write for short

S τ |x = |x + τ , τ ∈ A 2d .
The random quantum walk is then defined by U ω (C), as in ( 12).

The use of the same symbol K d for the total Hilbert space is no coincidence: we deal with the cases of the tree and cubic lattice in parallel in what follows, denoting by G d the graph corresponding either to Z d or T 2d . We shall use the notations introduced for the case of the tree only, with the understanding that the replacements just described yield the corresponding statements for the lattice case.

Localisation Length

We will say that averaged dynamical localisation occurs when there exists an α > 0 such that for all normalised

ψ 0 ∈ K d with compact support on G d , sup n∈Z E( e α|X|/2 U n ω (C)ψ 0 2 ) < ∞, (16)
where X denotes the position operator on G d , and |X| is the multiplication operator given by either the distance to the root of the point of T 2d considered, or the l ∞ or l 1 norm of the point considered in Z d . Averaged localisation may happen on certain spectral subspaces of U ω (C) only, in which case ψ 0 is restricted to these subspaces in ( 16). This notion, weaker than exponential dynamical localisation, see Remark 2.3 below, turns out to be useful in defining the quantitative criterion we need in our investigation of the delocalisation properties of RQWs. We define the (dynamical) localisation length of a RQW on G d as follows.

Definition 2.2 The localisation length L of the model is given by L = 1/α s , where

α s = sup{α ≥ 0 | s.t. (16) holds} (17)
and with the convention ∞ = 1/0.

Considering ψ 0 = e ⊗ τ 0 , we will thus analyse the large n behaviour of the expectation of

e α|X|/2 U n ω (C) e ⊗ τ 0 2 = x,τ e |x|α | x ⊗ τ |U n ω (C) e ⊗ τ 0 | 2 , ( 18
)
where τ ∈A 2d | x ⊗ τ |U n ω e ⊗ τ 0 | 2 denotes the quantum mechanical probability to find the quantum walker on site x ∈ G d , knowing it started at time zero on the root e, with coin state τ 0 . Hence, if the expectation of the right hand side of equation ( 18) is finite uniformly in n ∈ N, for some α > 0, it means that the limiting averaged distribution of the position of the quantum walker has moments of exponential order and that the localisation length L satisfies L < 1/α.

Our aim is to exhibit an α c ≥ 0, as small as possible, such that α ≥ α c implies

sup n∈Z E( e α|X|/2 U n ω (C) e ⊗ τ 0 2 ) = ∞, (19)
for certain choices of coin matrix C and phase distributions, that we will call balanced. Hence, if averaged dynamical localisation takes place for some α, then α < α c , so that the localisation length of the model is bounded below:

L ≥ 1/α c . Moreover, (19) implies that for all α ≥ α c , E(sup n∈Z e α|X|/2 U n ω (C) e ⊗ τ 0 2 ) = ∞, (20)
which we interpret as a step towards delocalisation, all the more pertinent that α c is small, i.e. L is large.

Remark 2.3 If exponential dynamical localisation holds, characterised by the existence of a dynamical localisation length 0 < 1/µ < ∞ and a constant c < ∞ such that

E(sup n∈Z | x ⊗ τ |U n ω (C) e ⊗ τ 0 | 2 ) ≤ ce -µ|x| (21)
see [AW, HJS], then averaged dynamical localisation (16) holds, for all α < µ. The existence of α c such that (19) holds thus also implies the lower bound 1/µ > 1/α c .

From BRQW to Combinatorics

We focus on balanced random quantum walks on Z d and T 2d that we now define.

Definition 3.1 A balanced random quantum walk (BRQW) on G d is characterised by a uniform distribution of random phases

dν(θ) = dθ 2π , (22)
and by a balanced skeleton matrix C b ∈ U (2d) whose elements satisfy

C b τ,τ = e iα τ,τ √ 2d , α τ,τ ∈ R, ∀(τ, τ ) ∈ A 2 2d . ( 23
)
Remarks 3.2 i) There exist balanced skeleton matrices in all dimensions. An example is provided by the unitary matrix related to the discrete Fourier transform such that C b j,k = e -iπjk/d / √ 2d, where j, k ∈ {0, . . . , 2d -1}. Another example is provided by Hadamard matrices whose coefficients are equal to ±1/ √ 2d, which exist for d = 2 k , k ∈ N, at least. ii) BRQW are essentially parameter free models, besides the dimension d.

For a BRQW U ω (C b ), we have thanks to (13)

x ⊗ τ |U ω (C b ) y ⊗ σ = e i(ωx,τ +ατ,σ) √ 2d δ x,yτ . (24)
Thus, the quantum mechanical transition probabilities between neighbouring sites are all equal to 1/(2d), and zero otherwise, be it on the tree or the cubic lattice. Thus, for BRQWs (noted U below), we have the finite sums, for any n > 1,

x n ⊗ τ n |U n e ⊗ τ 0 = x 1 ,...,x n-1 τ 1 ,...,τ n-1 x n ⊗ τ n |U x n-1 ⊗ τ n-1 • • • x 2 ⊗ τ 2 |U x 1 ⊗ τ 1 x 1 ⊗ τ 1 |U e ⊗ τ 0 = 1 √ 2d n τ 1 ,...,τ n-1 s.t. τ 1 •••τ n-1 τn=xn e i n j=1 (ατ j ,τ j-1 +ωx j ,τ j ) with x j = τ 1 • • • τ j-1 τ j . ( 25
)
The constraint on the τ j s can be expressed by saying that the summation takes place on all paths on the graph G d from the root e to the end point x n , with steps of length one.

Remarks 3.3 i) Given the initial point x 0 and initial coin state τ 0 , it is equivalent to have the sequence of points visited {x 1 , . . . , x n-1 , x n } or to have the sequence of coin states {τ 1 , . . . , τ n-1 , τ n }.

ii) The random phases ω x j ,τ j are random variables attached to the oriented edge (x j , x j-1 ), since x j-1 = x j τ -1 j is defined uniquely by x j and τ j . We adopt this point of view from now on, using the notation ω x j ,τ j = ω(x j , x j-1 ), ∀ j = 1, . . . , n.

By contrast, the deterministic phases α τ j ,τ j-1 depend on the orientation of the pair of edges (x j-1 , x j-2 ) and (x j , x j-1 ), (with a fictitious point x -1 = x 0 τ -1 0 , in case j = 1).

Consequently, the right hand side of (18) reads

1 (2d) n xn,yn δ xn,yn e α|xn|/2 e α|yn|/2 τ 1 ,...,τ n-1 s.t. τ 1 •••τ n-1 τn=xn σ 1 ,...,σ n-1 s.t. σ 1 •••σ n-1 σn=yn e i n j=1
(ατ j ,τ j-1 +ω(x j ,x j-1 )-ασ j ,σ j-1 -ω(y j ,y j-1 )) ,

(27) with x j = τ 1 • • • τ j-1 τ j , y j = σ 1 • • • σ j-1 σ j , ∀j ∈ {1, . . . , n}. (28)
Introducing the notation -→

x n = {e, x 1 , . . . , x n-1 , x n } for paths of length n, with steps of length one, and keeping τ j = x -1 j-1 x j wherever more convenient, we can write the right hand side of (18) using δ xn,yn = x δ x,xn δ x,yn , as

1 (2d) n x e α|x| -→ xn, -→ yn δ x,xn δ x,yn e i n j=1
(ατ j ,τ j-1 +ω(x j ,x j-1 )-ασ j ,σ j-1 -ω(y j ,y j-1 )) ,

with ( 28) again. At this point, we want take the expectation over the i.i.d. uniformly distributed random phases. We need the following Definition 3.4 Given a path in G d of length n starting at e, -→

x n = {e, x 1 , . . . , x n-1 , x n }, we define its phase content , P C( -→

x n ), by

P C( -→ x n ) = {m-→ xn (y, z), (y, z) ∈ G d × G d } where m-→ xn (y, z) = (x k ,x k-1 )⊂ -→ xn δ (x k ,x k-1 ),(y,z) (30)
and we say that two such paths are equivalent if and only if

-→ x n ∼ -→ y n ⇐⇒ P C( -→ x n ) = P C( -→ y n ). ( 31
)
Denoting by [ -→

x n ] the equivalence class of -→

x n and by P e n the set of all paths of length n starting at e we have

P e n = [ -→ xn]∈P e n /∼ [ -→ x n ]. ( 32
)
The integer m-→ xn (y, z) gives the number of times the oriented edge (y, z) is visited by the path -→

x n . Note that m(y, z) = 0 for all but a finite number of edges (y, z), and that P e n is finite.

Lemma 3.5 With the notations above,

E( e α|X|/2 U n ω e ⊗ τ 0 2 ) = 1 (2d) n x∈G d e α|x| [ -→ xn]∈P e n /∼ -→ x ∈[ -→ xn] δ x,xn e i n j=1 ατ j ,τ j-1 2 . ( 33
)
Proof: By independence and the property E(e inω(x j ,x j-1 ) ) = δ n,0 for any j, the expectation of ( 29) is nonzero only if the paths -→

x n and -→ y n over which the summation is performed have the same phase contents. Hence

E( e α|X|/2 U n ω e ⊗ τ 0 2 ) = 1 (2d) n x∈G d e α|x| -→ xn -→ yn s.t. P C( -→ yn)=P C( -→ xn)
δ x,xn δ x,yn e i n j=1 (ατ j ,τ j-1 -ασ j ,σ j-1 ) .

(34) Splitting the sum over -→

x n ∈ P e n into a sum over all distinct equivalence classes [ -→

x n ], and using (31) in the sum over -→ y n , we get (33).

Remarks 3.6 0) The computation of the exponential moments of the position operator averaged over the disorder is thus equivalent to a combinatorial problem.

i) The function e α|x| can be replaced by any other function of the position. The average of the quantum mechanical position moments at time n are obtained with |x| p , p > 0.

ii) For a given equivalence class [ -→

x n ], the sum

-→ x ∈[ -→ xn]
δ x,xn e i n j=1 ατ j ,τ j-1 may vanish, as illustrated by the examples below.

Let us review some general properties of the combinatorial expression

S τ 0 n (α) = 1 (2d) n x∈G d e α|x| [ -→ xn]∈P e n /∼ -→ x ∈[ -→ xn] δ x,xn e i n j=1 ατ j ,τ j-1 2 (35) e x e x
Figure 1: An equivalence class of paths on Z 2 that does not contribute to (35).

For fixed n, S τ 0 n (α) is finite for any α ∈ C, and is equal to one if α = 0. As a function of α ∈ R, S τ 0 n (•) is strictly increasing. Moreover, we have the trivial bounds for α ≥ 0, 1 ≤ S τ 0 n (α) ≤ e nα . More generally, for a non negative functions g on N the contribution in (35) stemming for the summation over

x ∈ G d with |x| ≤ g(n) ≤ n is bounded by 1 (2d) n x∈G d |x|≤g(n) e α|x| [ -→ xn]∈P e n /∼ -→ x ∈[ -→ xn] δ x,xn e i n j=1 ατ j ,τ j-1 2 ≤ e αg(n) . ( 36
)
Note also that we can consider 1 2d τ 0 ∈A 2d S τ 0 n (α), since the existence of a α c satisfying (19) for this sum implies it satisfies (19) for some τ 0 as well. Therefore we drop the superscript τ 0 from the notation.

Two-Dimensional Example:

Consider a BRQW on G 2 driven by a 4 × 4 Hadamard matrix in the basis {a 1 , a 2 , a -1

1 , a -1 2 } 1 2     -1 1 1 1 1 -1 1 1 1 1 -1 1 1 1 1 -1     . (37)
The corresponding quantity -→ x ∈[ -→ xn] δ x,xn e i n j=1 ατ j ,τ j-1 thus belongs to Z, and a minus sign is picked up at each step of the path which doesn't change direction. One readily checks that distinct paths can have the same phase content and are thus equivalent. Moreover, the contribution from a given equivalence class may be zero, see Figures 1 and2.

Lower Bounds from Restrictions

In order to avoid dealing with the combinatorial factor

-→ x ∈[ -→
xn] δ x,xn e i n j=1 ατ j ,τ j-1 , we consider the following subsets of paths in P e n :

Definition 3.7 Let SP n ⊂ P e n be the set of paths the equivalence class of which consists of a single path, i.e.

-

→ x ∈ SP n ⇔ #[ - → x ] = 1. ( 38
)
Let SAW n ⊂ SP n be the set of self avoiding walks on G d , i.e. of paths -→ x = {e, x 1 , . . . , x n } s.t. x j = x k if j = k , and x j = e.

In particular, for n fixed, paths that are sufficiently stretched belong to SP n , whereas self-avoiding walks belong to SP n by construction. Consequently, if

x n ∈ SP n , -→ x ∈[ -→ xn] δ x,xn e i n j=1 ατ j ,τ j-1 = δ x,xn . (39)
Hence, restricting the sum (35) to equivalence classes of such paths, S n (α) is bounded from below by quantities that do not depend on the phases of the balanced coin matrix C b :

S n (α) ≥ 1 (2d) n -→ x ∈SPn e α|xn| ≥ 1 (2d) n -→ x ∈SAWn e α|xn| ≥ 1 2 n . ( 40
)
The last inequality corresponds to the very crude lower bound on the number of self-avoiding walks of length n in G d given by d n . We use the notations

Z SPn (α) = -→ x ∈SPn
e α|xn| , and

Z SAWn (α) = -→ x ∈SAWn e α|xn| , (41)
for those partition functions. The following properties of these functions are derived in a standard fashion, making use of the fact that they are quite similar to partition functions of polymer models, see e.g [IV]. We provide a proof in Appendix for the convenience of the reader.

Lemma 3.8 For all α ∈ R, both functions Z Xn (α), X ∈ {SP, SAW } are strictly logconvex. For α ≥ 0, one has

Λ X (α) = lim n→∞ ln Z Xn (α) n = inf n≥1 ln Z Xn (α) n , ( 42
)
with ln d ≤ Λ X (α) ≤ α + ln(2d). Moreover Λ X is convex, increasing and continuous on [0, ∞), and the convergence is uniform on compact subsets of [0, ∞).

Remarks 3.9 i) One gets therefore Z Xn (α) ≥ e nΛ X (α) , and

lim n→∞ S n (α) = ∞ ∀α > α c ≥ 0, where Λ X (α c ) = ln(2d). ( 43
)
ii) The connective constant of the paths in the set X is given by

lim n→∞ -→ x ∈Xn 1 1/n = e Λ X (0) ∈ [d, 2d]. ( 44
)
The connective constant on Z d for the set SP , respectively SAW , will be denoted by s(d), respectively µ(d).

Let us introduce two auxiliary quantities, the 2-point function and the susceptibility, that appear in the analysis of partition functions in the statistical mechanics of polymers.

Considering paths in X n , with X = SP or X = SAW again, the 2-point function G α (z, x) is defined in our setup by

G α (z, x) = n≥0 z n - → x ∈Xn xn=x e α|x| (45)
and the the susceptibility χ α (z) is given by

χ α (z) = n≥0 z n -→ x ∈Xn e α|xn| = x∈G d G α (z, x). ( 46
)
Note that |x n | ≤ n implies that the sum over x ∈ G d is finite. The foregoing and the root test show that the radius of convergence of

χ α (z) = n≥0 z n Z Xn (α) (47)
is given by the continuous, decreasing log-concave function of α

z c (α) = 1 e Λ X (α) > 0. ( 48
)
Given the interpretation of χ α (z), this characterises z c (α) as a critical point. Moreover, lim

z→z - c (α) χ α (z) = ∞, (49)
Indeed, we get from Remark 3.9 that for z < z c (α)

χ α (z) ≥ n≥0 z n e nΛ X (α) ≥ 1 1 -ze Λ X (α) , (50)
where the lower bound tends to infinity as z → z - c (α). Hence, we can relate the behaviour of Z Xn (α)/(2d) n for n large to that of χ α (z/(2d)) for z close to z c (α):

Lemma 3.10 Let α c be defined by Λ X (α c ) = ln(2d), see (43). We have the characterisation

α < α c ⇔ z c (α) > 1/(2d) ⇔ χ α (1/(2d)) < ∞ ⇔ lim n→∞ Z 1/n Xn (α)/2d = 0.
(51)

SAW on T 2d

The self avoiding walks on the tree are not numerous so that we can compute explicitly the partition function Z SAWn (α) and the quantities that derive from it. Any self avoiding path of length n in T 2d , -→ x ∈ SAW n , is such that |x n | = n, and there are 2d(2d -1) n-1 such paths. Hence Z SAWn (α) = 2d 2d -1 ((2d -1)e α ) n , and Λ SAW (α) = ln(2d -1) + α, on T 2d .

The function Λ SAW is linear, hence convex, and we get a value of α c according to (43), and a lower bound on the localisation length which read

α c = ln 2d 2d -1 = 1 2d + O(d -2 ) ⇒ L ≥ 2d + O(1), when d → ∞, on T 2d . (53)
Since for any given x ∈ T 2d , there exists a unique SAW of length n = |x| from e to x, we have the following explicit expressions for G α (z, x) and χ α (z)

G α (z, x) = (ze α ) |x| , ∀ z ∈ R (54) χ α (z) = 2d (2d -1)(1 -z(2d -1)e α ) , ∀ 0 < z < e -α /(2d -1), (55)
which lead to the same conclusion about α c . We improve the bound (53) in the next Section, discussing the susceptibility χ α (z) instead of the partition function.

3.2 Lower Bound on S n (α) on T 2d via Z SP n (α)

We estimate S n (α) by the sum Z SPn (α) = -→ x ∈SPn e α|xn| on T 2d , taking into account a larger subset of paths in SP n than SAW n , which allows to improve the lower bound on L.

Theorem 3.11 Let U ω be a balanced quantum walk on T 2d , with d ≥ 2. Then

lim sup n→∞ E( e α|X|/2 U n ω e ⊗ τ 0 2 ) = ∞ if α ≥ ln 1 -1/(2d) + 1/(2d) 2 (1 -1/(2d))(1 -1/(2d) 2 ) = 1 2d 2 + O 1 d 3 as d → ∞,
so that the localisation length L satisfies

L > 2d 2 + O (d) , as d → ∞. (56)
Remark 3.12 This result shows that dynamical localisation of a BRQW on a symmetric tree can only hold for larger and larger localisation length for higher and higher coordination number. Again, BRQWs are far from the RQWs known to displaying delocalisation.

Proof: Let -→ x k be one of the 2d(2d -1) k-1 SAW of length k ≥ 1. We shall attach at 1 ≤ j ≤ k + 1 distinct sites of -→
x k , decorations of variable lengths 1 ≤ r i , i ∈ {1, 2, . . . , j} consisting of SAW of length r i that do not intersect -→

x k . These decorations will be run back and forth in sequence, so that such paths belong to SP by construction. Given r i ≥ 1, there are 2(d -1)(2d -1) r i -1 ways to construct these decorations, the first factor taking into account the fact they need to avoid the ingoing and outgoing directions of -→

x k at the site where they are attached. We neglect the extra possibilities at both ends of -→

x k . There are k + 1 j ways to choose the j sites of -→

x k and, given ρ ≥ 1, there are ρ -1 j -1 compositions of ρ with j parts, i.e. distinct ordered sets {r 1 , r 2 , . . . , r j } such that ρ = r 1 + r 2 + • • • + r j , with 1 ≤ r i ≤ ρ. The total length of such a decorated path is thus n = k + 2ρ, so that their contribution to S n (α) reads

S n (α) ≥ 2d 2d -1 ρ≥1,k≥1 1≤j≤k+1 s.t. n=k+2ρ (e α (2d -1)) k 2(d -1) (2d -1) j (2d -1) ρ k + 1 j ρ -1 j -1 . (57)
Note that the summand is non negative, and that the constraints on j and ρ can be taken care of by the binomials. At this point it is convenient to consider the susceptibility χ α (z), see ( 46), which thus admits the lower bound

χ α (z) ≥ c(d) n≥0 δ n,k+2ρ z n ρ≥1,k≥1,j≥1 (e α (2d-1)) k 2(d -1) (2d -1) j (2d-1) ρ k + 1 j ρ -1 j -1 , (58)
where c(d) is an inessential constant that can change from line to line. Performing the sums of k and ρ first, making use twice of the identity

k≥j k j x j = x j (1 -x) j+1 , if |x| < 1, (59)
we get

χ α (z) ≥ c(d) j≥1 ze α (2d -1) 1 -ze α (2d -1) j 2(d -1) (2d -1) j z 2 (2d -1) 1 -z 2 (2d -1) j , (60)
provided z 2 (2d -1) < 1, and ze α (2d -1) < 1.

(61)

Divergence of χ α (z) thus holds if

z 3 e α (2d -1)2(d -1) (1 -ze α (2d -1))(1 -z 2 (2d -1)) ≥ 1 ⇔ e α ≥ 1 -z 2 (2d -1) (1 -z 2 )z(2d -1) . ( 62
)
One checks that the second estimate above is compatible with the second condition ( 61) for all values of z > 0 and d > 1, whereas the first condition holds in particular for z = 1/(2d) for any d ≥ 1. Thus, in the limit of large d, (62) implies divergence of χ α (1/(2d)) for

e α ≥ 1 + 1/(2d 2 ) + O(1/d 3 ), ( 63
)
hence the announced upper bound on α c .

Remark 3.13 Actually, considering paths with decorations of length 1 only is enough to improve the lower bound on L (53) to a constant times d 2 , whereas paths with decorations of fixed length strictly larger than one do not provide such an improvement.

Lower Bound on S n (α) on Z d

We now turn to the cubic lattice case in dimension larger than or equal to two. We have the choice in the norm we use in ( 18), and the critical value α c beyond which lim sup n→∞ E( e α X /2 U n ω e ⊗ τ 0 2 ) = ∞ depends on this choice. If (64)

This is a consequence of the fact that X → E( e α X /2 U n ω e ⊗ τ 0 2 ) is increasing. Thanks to the relations between l p norms on Z d , x p ≤ x q ≤ d 1/q-1/p x p , for any 1 ≤ q < p ≤ ∞, and any

x ∈ Z d , (65)
a critical value α c (∞) obtained for p = ∞ will do for all p ∈ [1, ∞[, while an estimate for α c (1) is easier to get.

We investigate here the dependence on dimension of the quantities involved so far, in order to get a bound on α c (1). In the sequel, it will sometimes be useful to write x = (L, y) ∈ Z d , with L ∈ Z and y ∈ Z d-1 , so that (L, y) 1 = |L| + y 1 . Dropping the subscript SAW and emphasising the dependence on the dimension in the notation, the partition function (41) writes

Z (d) n (α) = L∈Z y∈Z d-1 - → x ∈SAWn xn=(L,y)
n -k = p + q n , (67)
shows that, given a SAW -→ y in Z d-1 , there are n |L| SAWs in Z d from the origin to (L, y) constructed this way. If L = 0, we just consider -→ y in Z d-1 as a SAW in Z d . Hence, restricting attention to this subset of SAWs, we deduce

- → x ∈SAWn xn=(L,y) 1 ≥ n |L| - → y ∈SAW n-|L| yn=y 1 (68) so that Z (d) n (α) ≥ L∈Z y∈Z d-1 e α|L| e α y 1 n |L| - → y ∈SAW n-|L| yn=y 1 ≡ L∈Z e α|L| n |L| Z (d-1) n-|L| (α). (69)
This last estimate yields Theorem 3.14 For the l 1 norm on Z d , we have the estimate for any d ∈ N, α) and the starting point Λ (1) (α) = α, see (52), we get from (69) Λ (d) (α) ≥ ln(e α + e Λ (d-1) (α) ) ≥ ln(de α ).

α c (1) ≤ ln(2). ( 70
) Proof: From Z (d) n (α) ≥ e nΛ (d) (
With the criterion (43), we get α c (1) ≤ ln(2).

Remark 3.15 For the l p norm with ∞ > p ≥ 1, using (L, y) p ≥ |L|+ y p 2 1-1/p , the same argument yields an upper bound on α c (p) which, however, is not as good as ln(2).

For the sake of comparison, it is worth mentioning that for the l ∞ norm, the same strategy also provides an explicit bound. Indeed, using (L, y) ∞ = max(|L|, y ∞ ), we get

Z (d) n (α) ≥ y∈Z d-1 L∈Z |L|≤ y ∞ e α y ∞ n |L| - → y ∈SAW n-|L| yn=y 1 + y∈Z d-1 L∈Z |L|> y ∞ e |L|α n |L| - → y ∈SAW n-|L| yn=y 1. (72)
Now, using e α y ∞ ≥ e α|L| in the first sum yields

Z (d) n (α) ≥ y∈Z d-1 L∈Z e α|L| n |L| - → y ∈SAW n-|L| yn=y 1 ≡ y∈Z d-1 L∈Z e α|L| n |L| Z (d-1) n-|L| (0). (73)
Then, with Z

(d) n-|L| (0) ≥ µ(d -1) n-|L| , see (44), we obtain Z (d) n (α) ≥ (e α + µ(d -1)) n ⇒ Λ (d) (α) ≥ ln(e α + µ(d -1)). (74)
Hence, together with the known asymptotics µ(d) = 2d -1 + O(1/d) in high dimension, see e.g. [MS, BDGS], we obtain

α c (∞) ≤ ln(2d -µ(d -1)) = ln(3) + O(1/d). ( 75
)
Not only is this bound not as good as the one on α c (1) for d large, but, since d ≤ µ(d) ≤ 2d, the method used here to estimate α c (∞) cannot yield a better estimate than ln(2), the bound for α c (1).

Localisation length and correlation length of SAW

In this closing section, we make the link between the quantities involved in the analysis of the localisation length of BRQWs and the correlation length appearing in the analysis of SAWs in Z d . Our concern being in the large d limit, we assume d ≥ 5 in this section. We recall the relevant notions and results to be found in [MS, BDGS], for example. The two point function for SAWs, for paths starting at the origin, is given by ( 45) at α = 0, and similarly for the corresponding susceptibility

G 0 (z, x) = n≥0 z n - → x ∈SAWn
where we stress the dependence on the dimension d in the notation. We provide below alternative expressions of the inverse correlation length m d (z) = 1/ξ d (z). The mass m d enjoys several properties proven in [MS], that we summarise in the next proposition: In order to make the link with the localisation length L, we introduce for all z > 0, G L (z) = Remarks 4.4 i) If one can show that lim d→∞ m d (1/(2d)) = 0, it would prove the divergence with the dimension of the localisation length L. Despite the fact that 1/(2d) is the critical point for simple random walks, to which 1/µ(d) converges, the control of the limit seems to be non trivial according to experts in SAWs, and it is not clear that the limit vanishes.

ii) An uncontrolled prediction on lim d→∞ ξ d (1/(2d)) is however not encouraging: for d ≥ 5 fixed, it is known that ξ d (z) { D 2d ( zc zc-z )} 1/2 , as z → z c = 1/µ(d), where D is the diffusion constant of SAWs, such that 1 + c 1 /d ≤ D ≤ 1 + c 2 /d, for large d, with c 1 , c 2 > 0, see Thm 6.1.5 and Proposition 6.2.11 in [MS]. Ignoring the fact that error terms are a priori not uniform in d, and plugging in the asymptotics µ(d) = 2d -1 + O(1/d), we get ξ d (1/(2d)) 1. However, Theorem 3.14 yields the better bound L ≥ 1/ ln(2).

Figure 2 :

 2 Figure 2: An equivalence class of paths on T 4 that does not contribute to (35).

  • a ≤ ν • b , and α c (a), α c (b) denote the corresponding critical values, we have α c (b) ≤ να c (a).

  L = 0 fixed, we can further restrict summation to SAWs -→ x in Z d of length n from the origin to (L, y) that stem form a SAW -→ y in Z d-1 of length n -|L| from the origin to y in the following way: For r = 1, . . . , n -|L|, one attaches at r of the n -|L| + 1 sites of -→ y a straight segment along the first coordinate axis, of length n i ≥ 1, i = 1, . . . , r, such that r i=1 n i = |L|. We observe that given |L| ≥ 1, and 1 ≤ r ≤ |L| there are |L| -1 r -1 compositions of |L| with r parts, i.e. ways to write |L| = n 1 + n 2 + • • • + n r , where (n 1 , • • • , n r ) is an ordered set, with n j ≥ 1. Since the number of ways to choose the r locations on -→ y is n

  x = 0, both functions have radius of convergence z c (0) = 1/µ(d), for SAWs, see (48). The inverse correlation length or mass for SAWs in Z d is defined for all z > 0 by m d (z) = lim inf L→∞ -ln G 0 (z, (L, 0)) L ,

Proposition 4. 1

 1 For any 0 < z < 1/µ(d), z → m d (z) is real analytic, strictly decreasing, concave in ln z, For z > 1/µ(d), m d (z) = -∞ and, if d ≥ 5, m d (1/µ(d)) = 0.

  generating functions of SAWs from the origin to the plane {x 1 = L}. This generating function is known to satisfy, among other things,Lemma 4.2 For all 0 < z < 1/µ(d), d (z)L ≤ G L (z) ≤ χ 0 (z) 2 e -m d (z)L , if L ≥ 1. (81)This is enough to prove the sought for relationship between localisation length and correlation length:Proposition 4.3 On the cubic lattice Z d , the localisation length of BRQWs is bounded below by the correlation length of SAWs at the critical value of simple random walks:L ≥ ξ d (1/(2d)) ⇔ α c ≤ m d (1/(2d)).(82)

Proof:

  Considering χ α (z) = n≥0 z n -→ x ∈SAWn e α xn ∞ and paths with end points of the form x n = (L, y), we get with (L, y) ∞ ≥ |L|, y) . (83)The last sum above coincides with G L z 2d , so that restricting to L ≥ 1, Lemma 4For the log-convexity property we compute the second derivative of Λ Xn (α) = 1 n ln Z Xn (α):Λ Xn (α) = Z Xn (α) nZ Xn (α) , Λ Xn (α) = Z Xn (α)Z Xn (α) -(Z Xn (α)) 2 nZ 2 Xn (α),

Acknowledgments

This work is partially supported by the ANR grant NONSTOPS (ANR-17-CE40-0006-01), and by the program ECOS-Conicyt C15E10. We acknowledge the warm hospitality and support of the Centre de Recherches Mathématiques in Montréal during the last stage of this work: A.J. was a CRM-Simons Scholar-in-Residence Program, and J.A. was a guest of the Thematic Semester Mathematical Challenges in Many Body Physics and Quantum Information. We wish to thank V. Beffara, L. Coquille, and S. Warzel for several enlightening discussions and G. Slade for correspondance, at various stages of this work.

where Z (p) Xn (α) = -→ x ∈SPn |x n | p e α|xn| , for p ∈ N. Hence Λ Xn (α) > 0 and

To prove the existence of the limit as n → ∞ of Λ Xn (α), we use a subadditivity argument.

For -→ x ∈ X n and -→ y ∈ X m , the concatenated path -→ xy obtained by following -→ x from e to

x n and then -→ y from x n to x n y m does not necessarily satisfy the requirements to belong to

Consequently, {Λ Xn (α) = 1 n ln(Z Xn (α))} n∈N * is a subadditive sequence, which implies (42) for each α ≥ 0. The upper and lower bounds on Λ X (α) are consequences of S n (α) ≤ e nα and (40). Since Λ Xn is convex and increasing for all n ∈ N * , we immediately get that Λ X is convex and non decreasing on [0, ∞). Moreover, Λ Xn converges uniformly to Λ X on any compact set of ]0, ∞[, and Λ X is continuous on ]0, ∞[, see [START_REF] Simon | Convexity: An Analytic Viewpoint[END_REF]. It remains to extend the result to compact sets of the form [0, b], b > 0. Let 0 < α < 1 and write α = 0(1 -α) + 1α. By convexity and monotony of Λ X , we have

so that Λ X is continuous at 0. Since a sequence of monotonous functions that converges to a continuous function on a compact set converges uniformly, this finishes the proof.