Nicolas Rougerie

Qiyun Yang

DIMENSIONAL REDUCTION FOR A SYSTEM OF 2D ANYONS

Anyons with a statistical phase parameter ∈ (0, 2) are a kind of quasiparticles that, for topological reasons, only exist in a 1D or 2D world. We consider the dimensional reduction for a 2D system of anyons in a tight wave-guide. More specifically, we study the 2D magnetic-gauge picture model with an imposed anisotropic harmonic potential that traps particles much stronger in the -direction than in the -direction. We prove that both the eigenenergies and the eigenfunctions are asymptotically decoupled into the loose confining direction and the tight confining direction during this reduction. The limit 1D system for the -direction is given by the impenetrable Tonks-Girardeau Bose gas, which has no dependency on , and no trace left of the long-range interactions of the 2D model.

1. INTRODUCTION 1.1. Motivation. Fundamental particles are sorted into two symmetry classes: bosons (e.g. photons and other force carriers) and fermions (e.g. electrons, quarks and other matter particles). Bosons follow Bose-Einstein statistics, while fermions obey Fermi-Dirac statistics and hence comply with the Pauli exclusion principle. For topological reasons, particles with statistics in between those of bosons and fermions cannot exist except in a one or twodimensional system. Such so-called anyons can thus exist only as emergent quasi-particles.

Recent experiments [START_REF] Bartolomei | Fractional statistics in anyon collisions[END_REF][START_REF] Nakamura | Direct observation of anyonic braiding statistics[END_REF] provided firm evidence that the charge carriers in fractional quantum Hall systems [START_REF] Goerbig | Quantum Hall effects[END_REF][START_REF] Laughlin | Nobel lecture: Fractional quantization[END_REF][START_REF] Jain | Composite fermions[END_REF][START_REF] Rougerie | Some contributions to many-body quantum mathematics[END_REF] behave like anyons. This is due to the strong correlation between electrons under a strong magnetic field, leading to the formation of incompressible quantum liquids such as the Laughlin state. Charge carriers couple to excitations thereof (Laughlin quasi-particles or quasi-holes), and thereby can acquire exotic quantum statistics. In these experiments, anyons are observed via transport along one-dimensional edges of the two-dimensional samples, which provides part of our motivation to study the dimensional reduction for 2D anyons trapped in 1D.

There is essentially a single agreed-upon model for 2D anyons, which was derived formally as an emergent description in fractional quantum Hall physics [START_REF] Arovas | Fractional statistics and the quantum Hall effect[END_REF][START_REF] Rougerie | Emergence of fractional statistics for tracer particles in a Laughlin liquid[END_REF][START_REF] Lambert | On quantum statistics transmutation via magnetic flux attachment[END_REF]. On the other hand, there are several different inequivalent models for 1D anyonic behavior [START_REF] Myrheim | Topological aspects of low dimensional systems[END_REF][START_REF] Ouvry | Anyons and lowest Landau level anyons[END_REF][START_REF] Girardeau | Anyon-fermion mapping and applications to ultracold gases in tight waveguides[END_REF][START_REF] Andrić | Quantum fluctuations of the Chern-Simons theory and dynamical dimensional reduction[END_REF][START_REF] Fresta | Approaching off-diagonal long-range order for 1 + 1-dimensional relativistic anyons[END_REF][START_REF] Moosavi | Exact Dirac-Bogoliubov-de Gennes dynamics for inhomogeneous quantum liquids[END_REF][START_REF] Rabello | 1d generalized statistics gas: A gauge theory approach[END_REF][START_REF] Aglietti | Anyons and chiral solitons on a line[END_REF][START_REF] Kundu | Exact solution of double function Bose gas through an interacting anyon gas[END_REF][START_REF] Batchelor | One-dimensional interacting anyon gas: Lowenergy properties and haldane exclusion statistics[END_REF][START_REF] Santos | Anyon Hubbard model in one-dimensional optical lattices[END_REF][START_REF] Tang | Ground-state properties of anyons in a one-dimensional lattice[END_REF][START_REF] Bonkhoff | Bosonic continuum theory of one-dimensional lattice anyons[END_REF]. Here we prove that, to leading order, 2D anyons confined to a 1D wave-guide always behave as fermions, independently of the original statistical parameter ∈ (0, 2). The different proposed 1D anyons models hence seem to describe each a physics of their own, a priori unrelated to 2D anyons, but accessible to different emergence phenomena [START_REF] Edmonds | Simulating an interacting gauge theory with ultracold bose gases[END_REF][START_REF] Valentí-Rojas | Synthetic flux attachment[END_REF][START_REF] Cardarelli | Engineering interactions and anyon statistics by multicolor lattice-depth modulations[END_REF][START_REF] Chisholm | Encoding a onedimensional topological gauge theory in a Raman-coupled Bose-Einstein condensate[END_REF][START_REF] Frölian | Realizing a 1d topological gauge theory in an optically dressed BEC[END_REF]. Our results were announced in [START_REF] Yang | Anyons in a tight wave-guide and the Tonks-Girardeau gas[END_REF], where the reader can find more physics background and discussion.

1.2. The model. Formally, one might want to describe ≥ 2 anyons by a (multi-valued) complex many-body wave-function Ψ ∈ 2 (ℝ ) in a -dimensional space ℝ satisfying

Ψ(… , , … , , … ) = i Ψ(… , , … , , … ) (1.1)
for some constant . The cases = 0, 1 correspond to usual bosons and fermions respectively, and one can restrict to ∈ [0, 1) by periodicity and complex conjugation. Free, non-relativistic, 2D anyons of statistics parameters ≠ 0, 1 would be described by acting on such wave-functions with a usual Schrödinger Hamiltonian

∑ =1 -Δ + ( ) .
Studying the spectral properties of this operator on an appropriate domain incorporating a symmetry such as (1.1) would require modelling wave-functions as sections of line bundles, a formalism that one often prefers to bypass for analysis purposes [START_REF] Myrheim | Topological aspects of low dimensional systems[END_REF][START_REF] Solovej | Hardy and Lieb-Thirring inequalities for anyons[END_REF]. This means using the so-called "magnetic gauge picture" as we explain next.

Let us consider the 2D system of anyons represented by the Hamiltonian

 2D ∶= ∑ =1 -Δ + ( )
defined on the domain of anyonic wave "functions" [START_REF] Myrheim | Topological aspects of low dimensional systems[END_REF], where, for some > 0

( ) = ( , ) = 2 + 1 2 2 .
We write an anyonic wave "function" Φ as

Φ( 1 , … , ) = ∏ 1≤ < ≤ i arg( -) Ψ( 1 , … , ),
where Ψ is some symmetric (i.e. bosonic) function in 2 sym (ℝ 2 ) and arg is the angle of as a two-dimensional vector in the plane. The multi-valued phase related to this angle can be written as

i arg = | | for = ( , ) and = + i ,
and its derivative with respect to on a branch of is

i arg = i ⟂ | | 2 i arg with ⟂ = (-, ).
With this observation, one can check that acting on Φ with  2D (anyonic gauge-picture) is formally equivalent to acting on the corresponding Ψ with the modified Hamiltonian (magnetic gauge-picture)

2D ∶= ∑ =1 2 + ( ) = ∑ =1 -i + ( ) 2 + ( ) , (1.2)
where

( ) ∶= ∑ ≠ - ⟂ | | -| | 2 ,
i.e. it is equivalent to study the operator 2D defined on 2 sym (ℝ 2 ). Note that is a function on ℝ 2 instead of on ℝ 2 , we write it in this form for the sake of simplicity. The effective gauge vector potential is associated with a magnetic field perpendicular to the plane, of magnitude

curl ∶= ⟂ ⋅ = ∑ ≠ Δ ln | | -| | = 2 ∑ ≠ ( -),
where we denoted ⟂ = (-, ).

Hence, in this model, particles feel one another as carrying Aharonov-Bohm magnetic fluxes of intensity 2 . This generates the long-range magnetic interactions in (1.2). Throughout the paper, we use the above magnetic gauge picture for our system of 2D anyons, as in e.g. [START_REF] Solovej | Hardy and Lieb-Thirring inequalities for anyons[END_REF][START_REF]Local exclusion principle for identical particles obeying intermediate and fractional statistics[END_REF][START_REF]Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics[END_REF]. We investigate the most stable states of the above Hamiltonian, i.e. the low-lying energy spectrum. Letting the parameter in 2D go to 0, intuitively, the trapping potential will force particles together around the one-dimensional line = 0, and the vector potential will keep particles apart. We find that the Tonks-Girardeau gas is the correct limit 1D system on = 0, i.e. we find a model of impenetrable bosons in one dimension, described by the free Hamiltonian

1D ∶= ∑ =1 -2 + 2
but restricted to the domain of symmetric functions vanishing on diagonals,

 1D ∶= ∈ 2 (ℝ ), ≡ 0 on 1D , where 1D ∶= ( 1 , … , ) ∈ ℝ ∶ = for some ≠ .
One can find more details in Section 2 below.

We note that the repulsive effect of the vector potential is well-documented in the literature (see e.g. [START_REF] Solovej | Hardy and Lieb-Thirring inequalities for anyons[END_REF][START_REF]Local exclusion principle for identical particles obeying intermediate and fractional statistics[END_REF][START_REF]Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics[END_REF][START_REF] Larson | Exclusion bounds for extended anyons[END_REF][START_REF] Girardot | A Lieb-Thirring inequality for extended anyons[END_REF] for rigorous results). In our situation of strong confinement along one spatial direction, this repulsion is so strong that it forces a kind of hard-core exclusion upon the particles. This is in some sense our main finding here.

The movement in the direction will be, in the → 0 limit, frozen in the ground state of the harmonic oscillator

HO ∶= -2 + 2 2 , namely ( ) ∶= √ -1 2 - 2 2 .
(1.

3)

The corresponding eigenvalue (ground state energy) is ∶= 1 .

1.3. Main results and discussion. We define our Hamiltonians as Friedrichs extensions of non-negative quadratic forms. Regarding the 2D Hamiltonian, we close the quadratic form starting from the core ∞ ℝ 2 ⧵ 2D , where 2D ∶= ( 1 , … , ) ∈ ℝ 2 ∶ = for some ≠ , and then consider the unique self-adjoint extension 2D whose domain is included in the form closure of the above. Other self-adjoint extensions exist [START_REF] Oddis | Hamiltonians for two-anyon systems[END_REF][START_REF] Correggi | Magnetic perturbations of anyonic and Aharonov-Bohm Schrödinger operators[END_REF] but shall not be considered hereafter. More details on these definitions will be provided in Section 2 below.

Our main results are stated as follows: Theorem 1.1 (Relation between energies). Let 2D and 1D be the -th eigenvalues of 2D and 1D respectively (counting the multiplicity), and let be the ground energy of HO . For any fixed , we have, in the → 0 limit, 2D = + 1D + (1).

Theorem 1.2 (Relation between eigenfunctions).

With the same notation as in Theorem 1.1, let {Ψ 2D } be an orthonormal basis of eigenfunctions corresponding to the eigenvalues { 2D } . Then, after extracting a subsequence, there exists an orthonormal basis of eigenfunctions { 1D } corresponding to eigenvalues { 1D } such that, in the → 0 limit, for any fixed ,

Ψ 2D -1D → 0 strongly in 2 (ℝ 2 )
and → 1D strongly in 1 (ℝ ∖ 1D ), where

( 1 , … , ) = ∏ =1 ( ) (1.4)
with as in (1.3) and

( 1 , … , ) ∶= ∫ ℝ Ψ 2D ( 1 , … , ) i ( 1 ,…, ) ( 1 , … , )d 1 ⋯ d , (1.5) ( 1 , … , ) = ∑ 1≤ < ≤ arctan - - . (1.6)
We emphasize the role of the phase factor

i ( 1 ,…, ) = ∏ < i arctan - - apparent in (1.5).
Extracting it from the many-body wave-functions has the effect of gauging the long-range magnetic interactions away, explaining that our limit model does not depend on . The only remnant of the 2D interactions is the hard-core constraint upon particle encounters imposed in the domain of the 1D Hamiltonian. As we explain in more detail in [START_REF] Yang | Anyons in a tight wave-guide and the Tonks-Girardeau gas[END_REF], failing to perform this singular gauge transformation would lead to a limit 1D model of Calogero-type

Cal ∶= ∑ =1 -2 + | | 2 + 2 2 ∑ 1≤ < ≤ 1 | -| 2 ,
which is also a candidate for the description of 1D anyons. Indeed, the energy of an ansatz

( 1 , … , ) ( 1 , … , )
under 2D is almost the sum of and the energy of under Cal instead of 1D . The correct ansatz actually is of the form

( 1 , … , ) ( 1 , … , ) -i ( 1 ,…, ) (1.7)
with as in (1.6), whose gradient is close to and thus compensates its action on in the above ansatz. One may check that = (1.8)

when ≠ for all ≠ , with a distributional component on the 1D diagonals {∃ ≠ , = } that gives no contribution when combined with the vanishing of trial states on this set. Note that is invariant under the exchange of particle labels, so that (1.7) does have the necessary bosonic symmetry, unlike the function obtained by changing (1.6) into

∑ 1≤ < ≤ arg - .
The proof for Theorem 1.1 will be separated into proving two opposite inequalities in two separate sections, upper and lower bounds for 2D . Theorem 1.2 follows from these sharp energy bounds, as we prove in the last section. In essence we follow a Γ-convergence approach, which is an equivalent way of checking convergence of operators in resolvent sense [START_REF] Maso | An introduction to Γ-convergence[END_REF].

Acknowledgments. Funding from the European Research Council (ERC) under the European Union's Horizon 2020 Research and Innovation Programme (Grant agreement COR-FRONMAT No 758620) is gratefully acknowledged. We had interesting discussions with Douglas Lundholm, Michele Correggi, Per Moosavi and Nikolaj Thomas Zinner.

PRELIMINARIES

We first collect a few general facts. The more technically crucial in the sequel is the Hardy inequalities for 2D anyons that we recall and/or improve below. Before this, we briefly recall the definition of the operators of interest as Friedrichs extensions of non-negative quadratic forms. We conclude the section with standard definitions for the spectral data we are interested in, and by recalling elements of Girardeau's exact solution for the impenetrable 1D Bose gas [START_REF] Girardeau | Relationship between systems of impenetrable bosons and fermions in one dimension[END_REF][START_REF] Minguzzi | Strongly interacting trapped one-dimensional quantum gases: Exact solution[END_REF][START_REF] Mistakidis | Cold atoms in low dimensions -a laboratory for quantum dynamics[END_REF].

2.1. Domain of the Hamiltonians. We start by defining the domains on which the operators 2D and 1D will be defined hereafter. Throughout the paper, we use Friedrichs extensions to define the Hamiltonians in a weak (distributional) sense. For a more general discussion on that topic, readers can refer to, for example, [31, Chapters 2 and 3] or [START_REF] Reed | Methods of Modern Mathematical Physics. I. Functional analysis[END_REF][START_REF]Methods of Modern Mathematical Physics. II. Fourier analysis, self-adjointness[END_REF].

2.1.1. 2D model. We first discuss 2D . As an operator on 2 sym (ℝ 2 ), it can clearly be defined on the subspace

 2D ∶=  ∞ (ℝ 2 ∖ 2D ) ∩ 2 sym (ℝ 2 ), where 2D represents the diagonals in ℝ 2 , i.e. 2D ∶= ( 1 , … , ) ∈ ℝ 2 ∶ = for some ≠ and  ∞ (ℝ 2 ∖ 2D
) is the set of smooth functions with compact support contained in ℝ 2 ∖ 2D . Our operator is symmetric and corresponds to the quadratic form  2D defined by

 2D ( ) ∶= ⟨ | 2D ⟩ 2 = ∑ =1 ∫ ℝ 2 | | 2 + ( )| | 2 , ∈  2D .
Since the above is non-negative, i.e.  2D ( ) ≥ 0 for all ∈  2D , it is closable and we denote its closure by ( 2D ,  ( 2D )), where  ( 2D ) is the completion of

 2D under the norm √  2D (⋅) + ‖⋅‖ 2
and is a dense subspace of

 ( 2D ) = ∈ 2 sym (ℝ 2 ) ∶ ∀ , , | | ∈ 2 (ℝ 2 )
is independent of [36, Theorem 5].

2.1.2. 1D model. Our approach to 1D is similar. It can be defined naturally on the dense subspace  1D ∶=  ∞ (ℝ ∖ 1D ) ∩ 2 sym (ℝ ) that is a nice enough subspace of symmetric functions vanishing on diagonals. Clearly, ( 1D ,  1D ) is symmetric and its quadratic form is

 1D ( ) ∶= ⟨ | 1D ⟩ 2 = ∑ =1 ∫ ℝ | | | | | | 2 + | | 2 | | 2 , ∈  1D ,
which is also bounded below by 0. Then its closure is defined on  ( 1D ), the completion of  1D under the norm

√  1D (⋅) + ‖⋅‖ 2 i.e.  ( 1D ) = ∈ 1 0 (ℝ ∖ 1D ) ∩ 2 sym (ℝ ) ∶ ∀ , ∈ 2 ℝ , where 1 0 (ℝ ∖ 1D ) is the completion of  ∞ (ℝ ∖ 1D
) under the natural 1 -norm. The Friedrichs extension of ( 1D ,  1D ) has domain ( 1D ) contained in  ( 1D ). According to the trace theorem [15, Section 5.5], there exists a unique bounded linear operator

∶ 1 (ℝ ) → ⨁ 1≤ < ≤ 2 ( 1 , … , ) ∈ ℝ ∶ = (2.1) such that = ( | = ) < if ∈ 1 (ℝ ) ∩ (ℝ ).
A function in 1 (ℝ ) has zero trace on the diagonals 1D , i.e. = 0, if and only if it is in 1 0 (ℝ ∖ 1D ).

Hardy inequalities for 2D

anyons. An important ingredient of our proof below is that the magnetic-gauge picture 2D Hamiltonian defined above controls the inverse square distance between any pair of particles. Such Hardy-type inequalities were investigated in [START_REF] Hoffmann-Ostenhof | Many-particle Hardy Inequalities[END_REF][START_REF] Solovej | Hardy and Lieb-Thirring inequalities for anyons[END_REF][START_REF]Local exclusion principle for identical particles obeying intermediate and fractional statistics[END_REF][START_REF] Larson | Exclusion bounds for extended anyons[END_REF][START_REF] Qvarfordt | Exchange and exclusion in the non-abelian anyon gas[END_REF], and we recall some of the known results here. We first consider the case where two particles out of are isolated from the others. A Hardy inequality for any ≠ 0 follows from [START_REF] Solovej | Hardy and Lieb-Thirring inequalities for anyons[END_REF] in this case. We next deal with the general case to obtain an inequality for any ≠ 0 by combining the isolated case with estimates from [START_REF] Hoffmann-Ostenhof | Many-particle Hardy Inequalities[END_REF].

Theorem 2.1 (Isolated two-anyon Hardy).

Let be an open subset of ℝ 2( -1) , and let ∈ (0, ∞] be a constant. Then

( ; , 3 , … , ) ∶= ⟂ | | 2 + ∑ =3 + 2( -) ⟂ | | + 2( -) | | 2 + -2( -) ⟂ | | -2( -) | | 2
is antipodal-antisymmetric with respect to . By definition of , we have

| | < 2 | | -| | , ∀ ≥ 3. Hence, ∧ = 0 on ∖{ } and ∫ Γ ⋅ d = 2
for any simple loop Γ in ∖{ } enclosing { }. Applying [34, Lemma 2] to , we obtain Theorem 2.4 (Many-anyon Hardy for ∈ (0, 2)).

Let ∈ 2 sym (ℝ 2 ) such that ∀ , | | ∈ 2 (ℝ 2
), then we have

∑ =1 ∫ ℝ 2 | | | | 2 ≥ 2 ( -1) 2 + 3( -2) ∑ 1≤ < ≤ ∫ ℝ 2 | | 2 | -| 2 .
with as in (2.3).

Observe that the constant in the above scales as -2 for large , which is worst than the optimal -1 obtained in [START_REF] Solovej | Hardy and Lieb-Thirring inequalities for anyons[END_REF]Theorem 4] for special values of . We shall work at a fixed in the sequel.

Proof. With the same notation as in Theorem 2.1, we let be ∞ and let be ℝ 2( -1) . It is clear that

ℝ 2 ∖ ∞ = ⋃ =3 , where ∶= ( 1 , … , ) ∈ ℝ 2 ∶ | | 1 -2 | | ≥ | | 1 + 2 -2 | | . Hence, for ( 1 , … , ) ∈ , we have | | 1 -2 | | ≥ max | | 1 -| | , | | 2 -| | , and then 2 ≤ 3 | | 1 -2 | | 2 ,
where 2 is defined in Lemma 2.2. Therefore, with the help of the estimate (2.5), it turns out that

∑ =3 ∫ ℝ 2 | | 1 | | 2 + | | 2 | | 2 + | | | | 2 ≥ ∑ =3 3 ∫ ℝ 2 | | 2 2 ≥ ∑ =3 3 ∫ | | 2 2 ≥ ∑ =3 ∫ | | 2 | 1 -2 | 2 ≥ ∫ ℝ 2 ∖ ∞ | | 2 | 1 -2 | 2 . (2.6) Combining (2.
2) with (2.6), we obtain

-2 + -1 ∫ ℝ 2 | | 1 | | 2 + | | 2 | | 2 + ∑ =3 ∫ ℝ 2 | | | | 2 ≥ ∫ ℝ 2 | | 2 | 1 -2 | 2 . (2.7)
Since is symmetric, the estimate (2.7) above implies

-2 + -1 ∫ ℝ 2 | Corollary 2.5 ( 1 -

regularity).

Consider the quadratic form domain of the harmonic oscillator

1 (ℝ 2 ) ∶= ∈ 1 (ℝ 2 ) ∶ ∀ , | | ∈ 2 (ℝ 2 ) .
Then the domain of the quadratic form of 2D ,  ( 2D ), is contained in 1 (ℝ 2 ).

Proof. Let be in  ( 2D ), then ∈ 2 sym (ℝ 2 ) and | | ∈ 2 (ℝ 2 ) for all . Using the Cauchy-Schwarz inequality and applying Theorem 2.4 to , we have

∫ ℝ 2 | | | | | | 2 ≤ ∑ 1≤ ≤ , ≠ ∫ ℝ 2 | | 2 | -| 2 < ∞, which implies that | | ∈ 2 (ℝ 2 ). Since | | is also in 2 (ℝ 2 ), | | | | | | = | | | i - | | | is clearly in 2 (ℝ 2
) for all , which concludes the proof.

Eigenvalues and eigenfunctions.

2.3.1. 1D model. The Tonks-Girardeau gas is exactly soluble via Bose/Fermi mapping [START_REF] Girardeau | Relationship between systems of impenetrable bosons and fermions in one dimension[END_REF][START_REF] Minguzzi | Strongly interacting trapped one-dimensional quantum gases: Exact solution[END_REF][START_REF] Mistakidis | Cold atoms in low dimensions -a laboratory for quantum dynamics[END_REF]. This means that there is a one-to-one correspondence between its eigenfunctions and those of the free Fermi gas (Slater determinants). Any eigenfunction 1D of 1D is of the form

1D ( 1 , … , ) = ∏ < sgn( -) det , ( ( ))
,

where 1 , … , are eigenfunctions of the one-particle harmonic oscillator

-+ 2 .
The corresponding eigenvalue 1D is the sum of the one-particle eigenvalues corresponding to 1 , … , . With this correspondence, we know that the eigenfunctions of 1D are all of the form

1D ( 1 , … , ) = ( 1 , … , ) ∏ 1≤ < ≤ | | | - | | | -( 2 1 +⋯+ 2 )∕2
for some polynomial , and that they form an orthonormal basis of 2 (ℝ ).

2D model.

As regards the eigenfunctions of 2D , we prove that

Proposition 2.6 (Diagonalization of 2D

).

There exists an orthonormal basis (Ψ 2D ) ∈ℕ * of 2 sym (ℝ 2 ), and a non-decreasing divergent sequence ( 2D ) ∈ℕ * such that 2D Ψ 2D = 2D Ψ 2D .

Proof. Since the quadratic form  2D is non-negative,

2D + 1 ∶ ( 2D ) → 2 sym (ℝ 2
) is invertible and

( 2D + 1) -1 ∶ 2 sym (ℝ 2 ) → ( 2D ) ⊂ 2 sym (ℝ 2
) is bounded. With the help of Corollary 2.5 above and the fact that 1 (ℝ 2 ) is compactly embedded in 2 (ℝ 2 ), we deduce that ( 2D + 1) -1 is a compact operator. Applying the spectral theorem [31, Chapters 4 and 5], there exists an orthonormal basis (Ψ 2D ) ∈ℕ * of 2 sym (ℝ 2 ), and a decreasing vanishing sequence ( ) ∈ℕ * such that

( 2D + 1) -1 Ψ 2D = Ψ 2D .
This implies that such Ψ 2D 's are all in ( 2D ) and

2D Ψ 2D = ( -1 -1)Ψ 2D ,
which concludes the proof.

We shall characterize eigenvalues by standard min-max formulae [31, Section 5.5]: Theorem 2.7 (Courant-Fischer min-max formulae). Let ( , ( )) be a bounded-below self-adjoint operator, and let (,  ( )) be its corresponding closed quadratic form. Then

∶= inf subspace ⊂( ) dim = sup Ψ∈ , ‖Ψ‖=1 ⟨Ψ| Ψ⟩ = inf subspace ⊂ ( ) dim = sup Ψ∈ , ‖Ψ‖=1 (Ψ)
is the -th eigenvalue of counted with multiplicity or is inf ( ), the bottom of the essential spectrum of .

Thanks to the diagonalization property of 2D and 1D , we know that they have empty essential spectrum. Hence, their eigenvalues can be characterized by the min-max formulae above.

ENERGY UPPER BOUNDS

In this section, we bound the 2D eigenvalues from above by constructing appropriate trial states: Theorem 3.1 (Upper bounds for 2D eigenvalues). Let 2D and 1D be the -th eigenvalues of 2D and 1D respectively (counting the multiplicity), and let be the ground energy of HO . Then we have the following inequality:

2D ≤ + 1D .
Proof. According to the min-max formulae in Theorem 2.7, the energy 2D can be written as 2D = inf linearly independent

Ψ 1 ,…,Ψ ∈ ( 2D ) sup ‖Ψ‖ 2 (ℝ 2 ) =1 Ψ∈span{Ψ 1 ,…,Ψ }  2D (Ψ). (3.1)
We hence obtain an upper bound for 2D by constructing trial states of the form

Ψ( 1 , … , ) = ( 1 , … , ) ( 1 , … , ) -i ( 1 ,…, ) (3.2)
with ∈  ( 1D ), and as in (1.4) and (1.6) respectively. Notice that is smooth away from the -direction diagonals, i.e. is well-defined and differentiable on

Λ 0 ∶= ( 1 , … , ) ∈ ℝ 2 , | -| > 0, ∀1 ≤ < ≤ . ( 3.3)
One can check that has a gap on the -direction diagonals, hence, fails to be in 1 loc (ℝ 2 ), but is however in 1 loc (Λ 0 ). A direct calculation based on (1.8) yields

-i = -i Ψ i = -i Ψ i + Ψ i = -i + Ψ i = Ψ i (3.4)
in the sense of distributions in  ′ (Λ 0 ). Hence Ψ ∈ 2 (Λ 0 ). But ∈  ( 1D ) vanishes on the diagonals and thus Ψ is continuous across the components of Λ 0 despite the jumps in its phase. It follows that (3.4) also holds around the singularity of , i.e. can be extended to  ′ (ℝ 2 ). Hence, Ψ is in 2 (ℝ 2 ), and then the trial state Ψ is indeed in  ( 2D ) with the energy relation:

 2D (Ψ) = ∑ =1 ∫ ℝ 2 | | | Ψ | | | 2 + ( )|Ψ| 2 = ∑ =1 ∫ ℝ 2 | + 1 2 | | 2 | | 2 | | 2 =  1D ( ) + . ( 3.5)
Take now 1D 1 , … , 1D to be orthonormal eigenfunctions for the 1D eigenvalues 1D 1 , … , 1D respectively. The corresponding trial states as in (3.2) are clearly orthonormal in 2 (ℝ 2 ). Using (3.1) and (3.5), we obtain

2D ≤ sup Ψ= -i , ∈span{ 1D 1 ,…, 1D }  2D (Ψ), = + sup ∈span{ 1D 1 ,…, 1D }  1D ( ) = + 1D ,
where the suprema are over 2 -normalized functions. This concludes the proof of Theorem 3.1.

ENERGY LOWER BOUNDS

In this section, we will conclude the proof of Theorem 1.1 by proving the lower bounds matching the upper bounds of Theorem 3.1.

Theorem 4.1 (Lower bounds for 2D eigenvalues).

Let 2D and 1D be the -th eigenvalues of 2D and 1D respectively (counting the multiplicity), and let be the ground energy of HO . Then

lim inf →0 2D - ≥ 1D .
Let Ψ 2D be an eigenfunction of the 2D Hamiltonian, corresponding to the eigenvalue 2D . Recall that defined in (1.4) is strictly positive, so that we can rewrite Ψ 2D as

Ψ 2D ( 1 , … , ) = Φ ( 1 , … , ) ( 1 , … , ).
Then we can express the energy 2D in terms of Φ and as follows: With the same notation as above, the following identity holds:

To make this idea more rigorous, we express some properties of { , } in terms of the weighted Sobolev spaces

2 ℝ 2 ∶= 2 ℝ 2 ; 2 1 ( 1 , … , )d 1 ⋯ d , 1 Λ 0 ∶= 1 Λ 0 ; 2 1 ( 1 , … , )d 1 ⋯ d . Proposition 4.3 (Convergence of { , } ).
After possibly extracting a subsequence, , → 0, strongly in 2 (Ω ) for any bounded subset Ω ⊂ ℝ 2 and weakly in 1 Λ 0 with Λ 0 as in (3.3). Furthermore, the limit 0, ( 1 , … , ) = 0, ( 1 , … , ) ∈ 1 (ℝ ∖ 1D ) has no dependence on the -coordinates, and it satisfies the following property:

lim inf →0 2D - ≥ ∑ =1 ∫ ℝ ∖ 1D | | | 0, | | | 2 + ∫ ℝ | | 2 | | 0, | | 2 . (4.8)
Proof. From (4.7) we deduce that { , } is bounded in 1 (Λ 0 ). According to the Banach-Alaoglu Theorem, we can extract a (not relabeled) subsequence converging weakly to some function 0, in the Hilbert space 1 Λ 0 . Because of the coefficient -1 multiplying the -derivatives in (4.7), we deduce that the limit 0, only depends on the -coordinates. Then we have

lim inf →0 ‖ ‖ , ‖ ‖ 2 ( Λ 0) ≥ ‖ ‖ 0, ‖ ‖ 2 ( Λ 0) = ‖ ‖ 0, ‖ ‖ 2 (ℝ ∖ 1D ) = ‖ ‖ 0, ‖ ‖ 2 (ℝ ) , lim inf →0 ‖ ‖ ‖ , ‖ ‖ ‖ 2 ( Λ 0) ≥ ‖ ‖ ‖ 0, ‖ ‖ ‖ 2 ( Λ 0) = ‖ ‖ ‖ 0, ‖ ‖ ‖ 2 (ℝ ∖ 1D ) , ∀ . ( 4.9)
Using Sobolev embeddings, ( , ) converges strongly to 0, in the weighted Sobolev space 2 ( ) ∶= 2 ; 2 1 ( 1 , … , )d 1 ⋯ d for any bounded set ⊂ Λ 0 , equivalently for any bounded set ⊂ ℝ 2 . After passing to a further subsequence, ( , ) also converges to 0, almost everywhere on . By a diagonal argument, we can choose a subsequence such that ( , ) converges to 0, almost everywhere on ℝ 2 . Then, with the help of Fatou's lemma, we have weakly in ( 2 (Λ 0 )) 2 , we obtain that such convergence is in fact strong, which concludes the proof of Lemma 5.1.

lim inf →0 ∫ ℝ 2 | | 2 | | , | | 2 2 1 ≥ ∫ ℝ 2 | | 2 | | 0, | | 2 2 1 = ∫ ℝ | | 2 | | 0, | | 2 , ∀ . ( 4
Proof of Theorem 1.2. Using a diagonal extraction, we can construct a sequence { , } which converges to some 0, in the sense of Proposition 4.3 for all . We know from Proposition 4.11 that these limit 0, are eigenfunctions corresponding to the 1D eigenvalues 1D respectively, and from now on we denote them by 1D . Similarly to the proof of Proposition 4.11 above, we can deduce that { 1D } is an orthonormal set in 2 (ℝ ), hence, it is an orthonormal basis.

Recall the notation in the proof of Proposition 4.3, we have

‖ ‖ ‖ Ψ 2D -1D ‖ ‖ ‖ 2 (ℝ 2 ) = ‖ ‖ ‖ -i -1D ‖ ‖ ‖ 2 (ℝ 2 ) ≤ ‖ ‖ ‖ -1D -i ‖ ‖ ‖ 2 (ℝ 2 ) + ‖ ‖ ‖ 1D -i -1 ‖ ‖ ‖ 2 (ℝ 2 ) ≤ ‖ ‖ ‖ , -1D ‖ ‖ ‖ 2 (ℝ 2 ) + ‖ ‖ ‖ ‖ 1D -i √ -1 ‖ ‖ ‖ ‖ 2 (ℝ 2
) .

The first term ‖ , -1D ‖ 2 (ℝ 2 ) vanishes as goes to 0 because of the strong convergence of Lemma 4.10. It is clear that 1D ( -i √ -1) is dominated by 2 1D , so using the dominated convergence theorem, the second term ‖ 1D ( -i √ -1)‖ 2 (ℝ 2 ) also vanishes as goes to 0. Hence,

Ψ 2D -1D → 0 strongly in 2 (ℝ 2 ).
Recall the definition of in (1.5). We can rewrite it with the same notation as in the proof of Proposition 4.3:

( 1 , … , ) = ∫ ℝ , ( 1 , … , ) 2 1 ( 1 , … , )d 1 ⋯ d .
Meanwhile, we can rewrite the eigenfunction 1D in the similar form:

1D ( 1 , … , ) = ∫ ℝ 1D ( 1 , … , ) 2 1 ( 1 , … , )d 1 ⋯ d .
Similarly to their derivatives, we then obtain

‖ ‖ ‖ -1D ‖ ‖ ‖ 1 (ℝ ∖ 1D ) ≤ ‖ ‖ ‖ , -1D ‖ ‖ ‖ 1 (Λ 0 ) . (5.2)
Since { , } converges strongly to 1D in 1 (Λ 0 ) by Lemma 5.1 above, the control (5.2) implies → 1D strongly in 1 (ℝ ∖ 1D ), which concludes the proof of Theorem 1.2.

ECOLE NORMALE SUPÉRIEURE DE LYON & CNRS, UMPA (UMR 5669)

Email address: nicolas.rougerie@ens-lyon.fr ECOLE NORMALE SUPÉRIEURE DE LYON, UMPA (UMR 5669) Email address: qiyun.yang@ens-lyon.fr

Proposition 4 . 2 (

 42 Energy decoupling).

  2 sym (ℝ 2 ) [31, Theorem 3.10]. For simplicity, we denote by  2D the closure in the sequel. Then there exists a unique self-adjoint extension of 2D , called Friedrichs extension, with extended domain ( 2D ) contained in  ( 2D ) [31, Corollary 3.17]. One can check that

  With the help of Proposition 4.9, the limit 0, is indeed in  ( 1D ).According to the proof of Proposition 4.3 and Lemma 4.11, the right-hand side of (5.1) is bounded from below by 1D , which implies that

	.10) on | Moreover, Inequal-Combining (4.6), (4.9) and (4.10), we obtain (4.8) and conclude the proof. Applying the diamagnetic inequality to Φ , we obtain Let ⊂ ℝ -1 be defined in Lemma 4.8, then integrating over (-√ ∕2, √ ∕2) × both sides of the identity (4.26), we have ity (4.8) implies lim inf →0 2D -≥  1D ( 0, ). (4.29) Combining with the upper bounds in Theorem 3.1, we deduce  1D ( 0, ) ≤ 1D . (4.30) →0 ‖ ‖ , ‖ ‖( 2 (Λ 0 )) 2 = ‖ ‖ ‖ 0, ‖ ‖ ‖( 2 (Λ 0 )) 2 . ( lim Combining it with the fact proved in Proposition 4.3 that

1 , … , )d 1 ⋯ d ≤ (4.36) , →

1D

d ≥ 2 ∫ Ω( ,

,…, ) | | 2 | | 2 d , (2.4)

+ | | | | 2 + ∑ 1≤ ≤ , ≠ , ∫ ℝ 2 | | | | 2 ≥ ∫ ℝ 2 | | 2 | -| 2 , ≠ .Summing over all possible pairs, we conclude the proof of Theorem 2.4.One application of the above is to clarify the quadratic domain of the 2D magnetic-gauge operator: it must be contained in the usual 1 space.

√∕2, √ ∕2 ) ×

where

Integrating the estimate (2.4) over on both sides gives

Using the fact that

we finally get the isolated Hardy inequality (2.2).

To estimate the integral on the complementary domain of the isolated two-anyon domain defined in Theorem 2.1, we need the following lemma, which follows immediately from [START_REF] Hoffmann-Ostenhof | Many-particle Hardy Inequalities[END_REF]Lemma 3.6]: Lemma 2.2 (Three-particle Hardy). Let ∈ 1 (ℝ 2 ), then for any integer ∈ [3, ], we have

where

Remark 2.3 (Three-particle Hardy for anyons).

With the help of the diamagnetic inequality from [START_REF]Local exclusion and Lieb-Thirring inequalities for intermediate and fractional statistics[END_REF]Lemma 4], | | is in 1 (ℝ 2 ) provided that ∈ 2 sym (ℝ 2 ) and ∀ , | | ∈ 2 (ℝ 2 ). Applying Lemma 2.2 above to such | | and using the diamagnetic inequality, yields

(2.5)

The above gives a control on three-particle encounters that we may combine with Theorem 2.1 to prove a many-anyons Hardy inequality on the whole configuration space with fewer restrictions on than in [START_REF] Solovej | Hardy and Lieb-Thirring inequalities for anyons[END_REF]Theorem 4]. where (Ψ) ∶= i Ψ Ψ -Ψ Ψ . , and we complete the proof by summing over .

Returning to (3.4), we have

Then Proposition 4.2 becomes

where ( 1 , … , ) ∶= Φ ( 1 , … , ) i ( 1 ,…, ) . We next rescale the tightly-confined space variables and denote

Then Equation (4.4) becomes

where 1 is defined by setting = 1 in (1.4).

Applying the upper bounds in Theorem 3.1 to Equation (4.6), yields

where the right-hand side does not depend on . Intuitively, it follows from the above that , will have less and less dependence on the -coordinates as goes to 0. In the limit, , will depend only on the -coordinates and the left-hand side of (4.7) will reduce to the energy of , under the 1D Hamiltonian 1D . Asymptotically the left-hand side of (4.7) will be larger than the corresponding eigenvalue 1D (concluding the proof) if we can prove that , converges to a function of the -coordinates only, vanishing on the diagonals.

Then Proposition 4.2 implies

Note that the limit 0, in Proposition 4.3 may not be unique, i.e. it depends on the extracted subsequence. We will later recover some form of uniqueness by proving that these limits are all eigenfunctions of 1D corresponding to the eigenvalues 1D . Hence, the limit 0, is unique if and only if the energy level 1D is non-degenerate.

In order to finish the proof for the target lower bounds, it principally remains to prove that the right-hand side of Inequation (4.8) is bounded from below by 1D , i.e.

The inequality of (4.12) follows from the next series of lemmas. The first four aim at proving that the limit 0, is in the quadratic form domain of 1D , i.e. 0, ∈  ( 1D ). The main issue is to prove that the limit 0, vanishes on diagonals (Proposition 4.9). The last two lemmas are to check that the limit 0, is indeed an eigenfunction of 1D corresponding to the eigenvalue 1D .

Lemma 4.5 (Being 1 accross diagonals).

Consider ∈ 2 sym (ℝ ) such that ∈ 1 (ℝ ∖ 1D ). Then is in 1 (ℝ ). Proof. We have that is in 1 outside of an interface, and continuous across the interface, because of the symmetry constraint. That is globally in 1 is then a classical result.

It suffices to show that [14, Theorem 3 in Section 5.8.2], for any unit ∈ ℝ and ℎ ∈ (0, 1), there exists some constant independent of ℎ and such that

Since is symmetric on ℝ , it is equivalent to prove that, under the same notations,

Fix ℎ ∈ (0, 1) and a unit . Let

. There exists a permutation on {1, … , } such that

Then we have

Under this construction, +ℎ( ) ( ) is actually in Λ for almost every ∈ Λ, that is, except for a zero measure set in ℝ , + ℎ( ) ( ) and are both in Λ. Since a zero-measure set does not matter in our proof, we will ignore it from now on. On the other hand, the domain Λ is clearly convex, and is in 1 (Λ). So we can apply the Newton-Leibniz theorem [16, Section 4.9.2] to on Λ as follows:

One can check that |ℎ( )| ≤ ℎ. Thus, the identity (4.13) above leads to

It is not hard to verify that for each and ℎ, the function

for each permutation on {1, … , }, where

With this observation, we can integrate over Λ on both sides of (4.14) and obtain

, which concludes the proof of Proposition 4.5.

From Proposition 4.3, we know that the limit 0, ∈ 1 (ℝ ∖ 1D ), and that with the help of the upper bounds in Theorem 3.1, 0, ∈ 2 (ℝ ) for all . Since Ψ 2D , and are all symmetric, = Ψ 2D -1 i is clearly symmetric, and so are , and its limit 0, . With the help of Proposition 4.5, we obtain that the limit 0, ∈ 1 (ℝ ). Therefore, to prove that 0, ∈  ( 1D ), it remains to prove that 0, has zero trace on 1D , i.e. 0, ∈ 1 0 (ℝ ∖ 1D ). Since we already know that 0, is symmetric, it suffices to prove that the limit 0, vanishes on the diagonal { 1 = 2 }. Lemma 4.6 (Estimate near { 1 = 2 }). Consider 1 defined in (1.4) for = 1 and , defined in (4.5). Then we have

for some constant ′ > 0 independent of , where

Proof. Thanks to the upper bounds in Theorem 3.1 and the energy decoupling in Proposition 4.2, there is a constant independent of such that

where Ψ 2D = Φ with the same notation as above. Consider the region where is small, i.e. for > 0, consider

and observe that in the region Ω (recall the definition of in (1.3)),

then we can take out the -direction ground state from the integral in (4.15) as follows:

(4.17)

We define an open set in ℝ 2( -1) ,

One can check that is contained in Ω . Applying the isolated Hardy inequality in Theorem 2.1 to the function Φ and the domain , we obtain that

Combining (4.16), (4.17) and (4.18), we deduce

which after rescaling in the -direction, ↦ √ , yields

for some constant ′ independent of . One can check that × is contained in ̃ when = √ , which concludes the proof of Lemma 4.6.

We shall use a simple consequence of the Sobolev embedding:

Then, to prove { | | 2 } is bounded in ( ), it suffices to prove { } is bounded in ( ). Using Hölder's inequality, yields

for ∈ [1, min{2, ∕( -2)}] and for such that

which shows { } is bounded in ( ) because of the Sobelev embeddings mentioned above. Therefore, it follows from (4.20) that { } is bounded in ( ), which concludes the proof.

In the following, we would like to concentrate on the properties of , in the -direction, so we consider the function , on ℝ defined by Let be a bounded open subset in ℝ -1 . For a constant ≥ 0, we define

Then when ≥ , we have

for some constants > 0 and ∈ (0, 1) both independent of .

Proof. Using the Cauchy-Schwarz inequality, we have

where

is a constant independent of and is defined in Lemma 4.6. And this implies that { , } is bounded in 2 (ℝ ):

On the other hand, we can control the derivative of , as follows:

(Cauchy-Schwarz inequality)

With the help of (4.11), the estimate in (4.24) becomes

Using Hölder's inequality next, it turns out that

where = 1--1 ∈ (0, 1) and | | is the Lebesgue measure of in ℝ -1 . Since {| , | 2 } is bounded in 1, ( ), the estimate in (4.27) implies

(4.28) for some constant ′ > 0 independent of .

On the other hand, if

for some constant ′′ > 0 independent of , which together with the controls in (4.28) shows that

for some constant > 0 independent of , concluding the proof.

Proposition 4.9 (Vanishing on diagonals).

The limit 0, defined in Proposition 4.3 vanishes on the diagonals 1D .

Proof. Notice that | 0, | can be rewritten as

where is defined in Lemma 4.6 and 1 is defined in (4.23). Let , be defined in (4.21). Then for each bounded subset ⊂ ℝ , using the Cauchy-Schwarz inequality, we have

where Ω is a bounded subset in ℝ 2 such that Ω contains × . Hence,

). Meanwhile, we know that { , } is bounded in 1 (ℝ ) from the proof of Lemma 4.8. Then according to the Banach-Alaoglu Theorem, { , } also converges weakly to 1 | 0, | in 1 (ℝ ) after passing to a subsequence. Since the trace operator defined in (2.1) is continuous,

weakly in 2 (ℝ -1 ). Therefore, with the estimate in Lemma 4.8, for each > 0 independent of , we have

which implies that 0, ( 2 , 2 , 3 , … , ) equals to 0 in 2 ( ) for any bounded open subset in ℝ -1 and for any constant > 0 independent of . Letting finally → 0 after → 0 it follows that 0, ( 2 , 2 , 3 , … , ) equals to 0 in 2 (ℝ -1 ), i.e. 0, vanishes on the diagonal { 1 = 2 }, which concludes the proof of Proposition 4.9.

We next need the following strong 2 -compactness:

With the same notation as in Proposition 4.3, we have , → 0, strongly in 2 (ℝ 2 ). Hence, the limit 0, is normalized in 2 (ℝ ).

Proof. Applying the diamagnetic inequality to the 2D eigenfunction Ψ 2D , we obtain

We next decompose | | Ψ 2D | | into its orthogonal projection on and a remainder:

where

By definition, we can deduce and , are orthogonal on -direction:

) and then (4.31) above yields

, (4.33) where in the last inequality we also use the fact that the second eigenvalue of -2 + -2 2 is equal to 3 . Combining the above estimate (4.33) with the upper bounds in Theorem 3.1, we can deduce that { , } converges strongly to 0 in 2 (ℝ 2 ) as goes to 0. After rescaling the -coordinates, ↦ √ , the identity (4.32) becomes

where , is the same one as in (4.5) and

We then note that

→ 0. With the same notation as in Proposition 4.3, it follows that

.

Since {| , |} is bounded in 1 (ℝ 2 ) by Proposition 4.4, we obtain that { , } is bounded in 1 (ℝ ). Again using the Banach-Alaoglu Theorem, { , } also converges weakly to | 0, | in 1 (ℝ ) after passing to a subsequence, i.e. , -| 0, | ⇀ 0 in 1 (ℝ ). Referring to [START_REF] Lewin | Describing lack of compactness in Sobolev spaces[END_REF]Lemma 14], for a positive sequence { } going to infinity, again after passing to a subsequence, { , -| 0, |} satisfies the following local estimate:

On the other hand, the orthogonality of and in the -direction leads to

for some constant ̃ independent of . It follows that

which together with the weak convergence

Lemma 4.11 (Limiting eigenfunctions).

The limit 0, defined in Proposition 4.3 is an eigenfunction of 1D corresponding to the eigenvalue 1D .

Proof. Let { , } converge to 0, in the sense of the one in Proposition 4.3. Then after passing to a subsequence, { , } converges to some 0, in the same sense for all ≤ . Proposition 4.10 tells us that { , } converges strongly to 0, in 2 (ℝ 2 ). Therefore, for ≠ ≤ , we have

i.e. { 0, } =1 is an orthonormal set in 2 (ℝ ).

We now prove the lemma by induction. Firstly, for = 1, it is clear that  1D ( 0,1 ) ≥ 1D 1 , and with the upper bounds in (4.30) we obtain that 0,1 is the eigenfunction corresponding to 1D 1 . Suppose that for = -1, the set of limit { 0, } -1 =1 is a set of eigenfunctions corresponding to the eigenvalues { 1D } -1 =1 respectively. According to the identity (4.38), we have 0, ⟂ 2 (ℝ ) { 0, } -1 =1 , which implies  1D ( 0, ) ≥ 1D . With the help of the upper bounds in (4.30), this gives  1D ( 0, ) = 1D .

Then we can deduce that 0, is an eigenfunction corresponding to 1D . Thanks to Lemma 4.11,Inequality (4.29) gives

which concludes the proof of Theorem 4.1.

RELATION BETWEEN EIGENFUNCTIONS

In this section, we conclude the proof for the relation between eigenfunctions stated in Theorem 1.2.

Similarly to Lemma 4.10, we also have the corresponding strong 1 -compactness: With the same notation as in Proposition 4.3, we have , → 0, strongly in 1 (Λ 0 ).

Proof. Since we have proved the corresponding strong 2 -compactness in Lemma 4.10, it remains to prove that , → 0, strongly in ( 2 (Λ 0 )) 2 .

Recalling the estimate (4.7), it follows that , → 0 = 0, strongly in 2 (Λ 0 ) for all , and that (5.1)


Bastien Belzunces

Sophie Hoyau

Jérôme Cuny

Fabienne Bessac
email: fabienne.bessac@purpan.fr

Desorption Mechanism of a Pesticide from an Hydrated Calcium Montmorillonite Unraveled by Molecular Dynamics Simulations

Fenhexamid, N-(2,3-dichloro-4-hydroxyphenyl)-1-methylcyclohexanecarboxamide, a fungicide, is authorized in the European Union to be used in agriculture. Active molecule behavior in the environment and their transfer to environmental components (soil, water and air) are primordial topics to study to limit environmental pollution. Following a step by step approach, we studied fenhexamid. In previous works, we worked on the fenhexamid pesticide isolated, then on its complexes with one or two Ca 2+ and N a + , the most abundant cations into soils, in the gas phase. We also reported theoretical calculations about fenhexamid in interaction with a clay surface: a Camontmorillonite. In the present paper, potentials of mean force describing overall fenhexamid desorption from a Camontmorillonite surface in presence of water were 1 computed from molecular dynamics simulations combined with umbrella sampling. Several mechanisms were envisaged. The results allow to quantitatively estimate the free energy of desorption and to provide a detailed insight into the hydration sphere of Ca 2+ during the desorption process.

Introduction

The development of water resources is a major issue regarding the impacts of climate change, population ows (tourism, urban sprawl), epidemics, etc. All uses of water are concerned: drinking water supply and sanitation, agricultural and industrial uses. However, these water resources are often contaminated with many organic compounds. In this context, studying the behavior of organic contaminants such as pesticides (PEST) in water and the interactions between these contaminants and natural inorganic substrates such as clay is of paramount importance. The process of pesticide desorption from clay is not easy to monitor experimentally, while computational chemistry can provide signicant input into soil and water contaminant behavior. Several pesticides (atrazine, metamitron, fenhexamid...) have already been studied in our group. The global project is carried out using a step by step approach. The pesticide is considered: (i) isolated and (ii) in interaction with cations [START_REF] Mignon | Ab initio molecular dynamics study of the hydration of Li + , Na + and K + in a montmorillonite model. Inuence of isomorphic substitution[END_REF] ;

(iii) in interaction with soil surfaces [START_REF] Belzunces | Theoretical study of the atrazine pesticide interaction with pyrophyllite and Ca2+ -montmorillonite clay surfaces[END_REF][START_REF] Belzunces | Interaction of Metamitron and Fenhexamid with Ca2+-Montmorillonite Clay Surfaces: A Density Functional Theory Molecular Dynamics Study[END_REF] ; (iv) all the previous systems are currently studied in the presence of water. The present work concerns more particularly the fenhexamid pesticide. Smectite clay minerals are abundant in soils and have very interesting properties as high swelling behavior, high surface area, large negative charge... Due to those properties, smectites are used as catalysts, sorbents and barriers for contaminants (radioactive waste repositories)... Montmorillonite family belongs to smectite type and shares basic structure with pyrophyllite, an aluminum phyllosilicate of formula: Al 2 (Si 4 O 10 )(OH) 2 organized in layers. Temperature ramp experiments measuring the weight loss of water from smectite type of clay have shown at least two kinds of interlayer water in smectites 69 . A rst type of water is readily removable through drying or heating, a second kind of water is more strongly bound. Water lost below 150 • C is assigned to bulk and weakly bound water. The highest peak temperatures for magnesium, calcium, strontium, barium, and lithium clay are found to correlate with the hydration energies of the cations and are related to the interaction of the clay with the cation. About six water molecules per cation was found, at the rst step of hydration for divalent cations, and about three water molecules for lithium and sodium cations. In addition, in montmorillonite (Mont), IR experiments have shown that spectral absorptions due to water molecules are inuenced by both the interlayer cation and the moisture content 911 . First, adsorbed water on Mont solvates the cationic counterions.

Trends in the sorbed amount of water, swelling properties, and the enthalpy of hydration have led to deduce that the principal mechanism underlying the observed variations is the hydration of the cationic counterions.

In dierent theoretical works on pyrophyllite 12 and montmorillonite 1316 , the water contents were varied in order to get insight into swelling mechanisms and into the cation hydration state when the moisture content increases. In particular, Holmboe group studied cation hydration in the interlayer of NaMont and CaMont going from dry clay (0W), up to the tenth-layer hydrate called 10W using CLAYFF force eld Molecular Dynamics (MD) simulations (see Table 1) [START_REF] Holmboe | Molecular Dynamics Simulations of Water and Sodium Diusion in Smectite Interlayer Nanopores as a Function of Pore Size and Temperature[END_REF][START_REF] Yang | Atomistic simulations of cation hydration in sodium and calcium montmorillonite nanopores[END_REF] . For CaMont, at 1W in the basal spacing d 001 of about 13.7 Å, Ca 2+ are close to the Mont surface and hydrated by 7 water molecules which are not spherically distributed around Ca 2+ . From 2W to higher hydration states, Ca 2+ are surrounded by 8 water molecules in their rst hydration sphere (HS) but only from the 3W hydrate state, Ca 2+ are hydrated with an evenly distributed spherical rst coordination sphere. At the 5W hydration state, the basal spacing d 001 is equal to ≈ 24.8 Å for both NaMont and CaMont. [START_REF] Katz | Calcium ion coordination: A comparison with that of Beryllium, Magnesium and Zinc[END_REF][START_REF] Ikeda | Hydration properties of magnesium and calcium ions from constrained rst principles molecular dynamics[END_REF] . Moreover, the rst HS was found to be long-lived at room temperature (300 K). Half of the initial solvating water molecules are still in the HS after 1 ns of simulation time (Car-Parrinello MD within the NVT ensemble using the hybrid DFT functional B3LYP). In contrast, at higher temperature, 500 K, frequent exchanges are observed between HS. At 900 K, the rate of exchanges is raised to a great extent compared to 300 K. In average, the residence time is estimated to be between 0 and 6.4 ps [START_REF] Adeagbo | Ca2+ solvation as a function of p, T, and pH from ab initio simulation[END_REF] .

Water adsorption on Mont surface is greatly inuenced by the cationic counterions just like organic molecule adsorption on Mont is greatly inuenced by the cationic counterions and the water loading of Mont interlayer space. Due to Mont adsorption properties, this clay family has been considered as highly ecient materials for the removal of contaminants such as pesticides from wastewater. Mont combines several strengths: it is inexpensive, environmentally friendly, abundant and ecient to adsorb a wide range of contaminant types [START_REF] Zhu | Adsorbents based on montmorillonite for contaminant removal from water: A review[END_REF] .

Theoretical groups have previously been working on adsorption of small organic molecules on Mont. Among them, [START_REF] Clausen | Adsorption of Low-Molecular-Weight Molecules on a Dry Clay Surface: An Ab Initio Study[END_REF] ; sorption of the anionic form of the 2,4D pesticide (2,4dichlorophenoxyacetic acid) on dry and hydrated surfaces of a Camontmorillonite has been investigated using ab initio molecular dynamics simulation at room temperature [START_REF] Tunega | Formation of 2,4D complexes on montmorillonites an ab initio molecular dynamics study[END_REF] ; carbon dioxide intercalation in hydrated Namontmorillonite has also been under investigations [START_REF] Myshakin | Molecular Dynamics Simulations of Carbon Dioxide Intercalation in Hydrated Na-Montmorillonite[END_REF] .

To the authors' knowledge, one other work in scientic literature reports the combination of MD and umbrella sampling to compute the desorption of organic molecules, paracetamol and carbamazepine, from a watered clay surface, Nasaponite. This study was made in the force eld framework using LAMMPS software. [START_REF] Corbin | Adsorption of Pharmaceuticals onto Smectite Clay Minerals: A Combined Experimental and Theoretical Study[END_REF] Moreover, prior DFT researches using umbrella sampling to understand adsorption and desorption mechanisms on surfaces and bulk systems have been conducted, for instance: (i) the exploration of the free-energy prole of dimethylamine at the airwater surface 33 ; (ii) the degradation of diisopropyl methylphosphonate on γ -Al 2 O 3 surface [START_REF] Biswas | High-Temperature Decomposition of Diisopropyl Methylphosphonate on Alumina: Mechanistic Predictions from Ab Initio Molecular Dynamics[END_REF] . The present work is more ambitious and allows the quantitative estimation of the free energy of desorption of the fenhexamid (Fen, see Figure 1) from a Ca Mont clay surface in presence of water within the DFT framework. Fen is a fungicide for preventive use against Botrytis cinerea, the agent of gray rot. After a brief presentation of no substitution in T sheets;

Ca 2+ compensating cations in the interlayer.

X-ray data from Wardle et al. 35 on pyrophyllite (2:1 dioctahedral smectite with neither substitution, nor interlayer compensating cation) was our starting point to build Mont model.

Unit cell parameters (a, b, c, α, β and γ) optimized at PBE-D2 level were in very good agreement with experimental data (see previous works [START_REF] Belzunces | Theoretical study of the atrazine pesticide interaction with pyrophyllite and Ca2+ -montmorillonite clay surfaces[END_REF][START_REF] Belzunces | Interaction of Metamitron and Fenhexamid with Ca2+-Montmorillonite Clay Surfaces: A Density Functional Theory Molecular Dynamics Study[END_REF] ). The triclinic computational cell consisted of six pyrophyllite unit cell (3a×2b×c) in order to accommodate fenhexamid pesticide in the interlayer. Four Al 3+ were substituted by Mg 2+ in O sheets, and the resultant negative charge was compensated by two Ca 2+ cations in the interlayer. In a previous work, we showed that Fen interacts with only one cation [START_REF] Belzunces | Interaction of Metamitron and Fenhexamid with Ca2+-Montmorillonite Clay Surfaces: A Density Functional Theory Molecular Dynamics Study[END_REF] . In the following, the Ca 2+ in interaction with the pesticide is noted Ca1 while Ca2 designates the Ca 2+ , which does not interact with the pesticide. In the following, computational cell parameters were set to: 15.500 Å×

CPMD parameters

Car-Parrinello molecular dynamics simulations [START_REF] Car | Unied Approach for Molecular Dynamics and Density-Functional Theory[END_REF] were performed under periodic boundary conditions using the Perdew-Burke-Ernzerhof (PBE) 3741 functional as implemented in the CPMD package 42 . The D2 Grimme's dispersion correction 4345 has been added (PBE-D2).

Valence-core interactions have been described using Troullier-Martins norm-conserving pseu-dopotentials [START_REF] Hamann | Norm-Conserving Pseudopotentials[END_REF][START_REF] Troullier | Ecient pseudopotentials for plane-wave calculations[END_REF] in combination with a plane-wave basis set. The cut-o energy for plane waves is 80 Ry and the Monkhorst-Pack method [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF] was used to sample the Brillouin zone at the Γ point only within the triclinic computational cell. Hydrogen atoms were replaced by deuterium atoms. The ctitious mass of the electrons was µ = 500 a.u. and the time step 4.5 a.u.. Hydrated FenCaMont starting point originates from a 12 ps molecular dynamics (MD) trajectory performed within the NVT ensemble with a Langevin thermostat [START_REF] Ceriotti | Equation with Colored Noise for Constant-Temperature Molecular Dynamics Simulations[END_REF] at 350 K 5052 . 132 water molecules are present in the interlayer space measuring around ≈18 Å in all the trajectories of this work. These conditions corresponds to the 5W hydrate state qualied by Holmboe and coworkers at 298 K (see Table 1) [START_REF] Holmboe | Molecular Dynamics Simulations of Water and Sodium Diusion in Smectite Interlayer Nanopores as a Function of Pore Size and Temperature[END_REF][START_REF] Yang | Atomistic simulations of cation hydration in sodium and calcium montmorillonite nanopores[END_REF] . To model the desorption of Fen from the hydrated surface of Mont, the Umbrella Sampling method 5355 was implemented.

Eleven biased Car-Parrinello molecular dynamics simulations were conducted to move the system under study from one thermodynamically stable state to another following a reaction coordinate noted ξ (see Table 2). PLUMED, an open-source, community-developed library was used to bias the trajectories 5658 . The simulations lasted between 11 and 25 ps. An equilibration time of 4 ps was removed from the analysis for each simulation. With the eleven trajectories, a total duration of 177 ps was simulated.

Fen desorption from watered CaMont has been quantied by deriving the potential of mean force (PMF). The PMF characterizes the change in the free energy of the system due to the changes in its conguration 5962 . The free energy F of a system in thermodynamic equilibrium in an NVT statistical ensemble is dened as:

F = -k B T lnZ ( 1
)
where Z is the canonical partition function and k B is the Boltzmann constant. For two interacting species, the PMF, W (ξ), is a function of the reaction coordinate ξ. The chosen reaction coordinate ξ is the distance between the geometric center of Fen and the clay surface, i.e., the plane containing the basal oxygen, Ob. The whole set of Ob atoms forms the Mont surface, noted surf hereafter. Increasing ξ should make possible to switch from Fen adsorbed on Mont to Fen desorbed. The PMF is the free energy prole of the system as a function of the separation distance ξ between CaMont and Fen:

W (ξ) = -k B T ln(g(ξ)) (2)
where g(ξ) is the distribution function of distances for the pair (Fen;CaMont). Each biased dynamic simulation made possible to sample the conguration space around a value ξ i of ξ by introducing into the Hamiltonian of the system for each simulation an harmonic potential V centered on the ξ i value:

V = 1 2 k i (ξ -ξ i ) 2 (3)
The harmonic constant k i is chosen to allow for an adequate sampling of the potential energy surface (PES). Table 2 gives the k i values as well as the spacing of the umbrella windows, which are the two parameters to control the sampling of the PES.

Table 2: Parameters of the 11 biased MD trajectories: i is the window index, ξ i is the value of the reaction coordinate around which the harmonic potential oscillated, k i is the associated harmonic constant and t i the duration of the simulation (ps). The free energy surface was reconstructed from the distributions of ξ resulting from each biased MD simulation using the WHAM self-consistent method (weighted histogram approximation method) [START_REF] Kumar | The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method[END_REF][START_REF] Alan | WHAM: the weighted histogram analysis method[END_REF] . In addition, before plotting our free energy curves corresponding to the Fen desorption from the surface, we checked if our sampling was sucient. Indeed, we plotted the distribution functions of the reaction coordinate ξ and we were careful about their overlaps. These distributions are centered on the values of ξ around which the harmonic potentials of the umbrella sampling oscillate. We put Figure S1(i) in supplementary material that represents the plot of these dierent distributions as a function of ξ.

i ξ i (Å) k i (kcal.mol -1 .Å -2 ) t i (

Results and discussion

When a pesticide is desorbed from the CaMont surface, several mechanisms can be envisaged. First, the pesticide is desorbed and Ca 2+ stays close to the surface (Figure 3(b)).

Second, the pesticide and the interlayer cation are desorbed together (Figure 3(c)). In a last mechanism, we can envisage that PEST-Ca 2+ complex moves away from the surface andnally, dissociates (Figure 3(d)). In the following, the simulated desorption prole is described and the Casurf distances are analyzed to determine, which is the desorption mechanism.

Then, the hydration state of both cations helps to better understand the reaction path.

Moreover, correlation between positions and hydration states of Ca 2+ is highlighted. for the minimums or wells and M1 and M2 for the maximums along the reaction coordinate ξ.

Description of the desorption prole and Casurf distances

We demonstrate in the Supporting Information that those maximums are transition states.

For all the trajectories, OeOe and OeOb radial distribution functions (RDF) were plotted to conrm the liquid phase of water all along the desorption. On Figure S2, the OeOe and OeOb RDF are given for each singularity:

OeOe distances between the oxygen atoms of water molecules For each singular- ity, the proles are almost stackable. In addition, the rst peak of each distribution corresponds to a distance of ≈ 2.8 Å, which is characteristic of liquid water;

OeOb distances between the oxygen atoms of the water molecules Oe and the oxygen atoms of the surface Ob Similarly, for each singularity, the distributions are comparable, which indicates a similar hydration of the surface all along the desorption. The rst peak at ≈ 3.0 Å indicates that the water molecules are not in direct contact with the surface of the montmorillonite.

During the process, the global water structure does not seem to be aected by the desorption of Fen from Mont and the inevitable induced reorganizations.

In order to follow the movements of the cations during the desorption, we evaluated the distance distribution functions of both Ca 2+ to the surface, denoted Ca1surf and Ca2 surf thereafter (see Figure S3). Within P1, Ca1 is close to the surface and the Ca1surf distribution has a double peak with two maximums at ≈ 1.1 Å and ≈ 1.6 Å (see Figure S3(a)). Ca2 is further from Mont with three preferred positions ≈ 2.7, 4.1 and 5.2 Å. For P2, the Ca1surf distribution displays a single peak centered at 2.3 Å, while the distribution of Ca2surf has a peak centered at 2.5 Å. For P3, the distributions share common features with P1. The Ca1surf distribution has only one maximum at 1.4 Å, as for Ca2, two preferred positions ≈ 2.5 and 4.0 Å exist. These observations show that the simulated mechanism is the one shown in Figure 3(b). In P1 and P2, Ca1 interacts with Fen whereas, in P3, the FenCa1 interaction is broken and Ca1 comes back closer to the surface.

Hydration state of the cations

In the analysis of the desorption free energy prole of Fen from the montmorillonite surface, the hydration states of both Ca 2+ are important to explain the process. Thus, the arrangement of both Ca 2+ rst HS is investigated in the following.

On Figure S4, the CaOe RDF have been extracted from the trajectories and sorted by singularities of the free energy prole: i) for Ca1Oe, all the proles are similar, with a rst peak centered on 2.3 Å and the Ca1Oe RDF rst minimum equals zero at 3.0 Å; ii) for Ca2Oe, all the curves are also similar, the rst peak is more intense but also centered on 2.3 Å and the Ca2Oe RDF rst minimum also equals zero at 3.0 Å. Thus, a water molecule has been counted in the rst HS of a calcium cation when d Ca-Oe ≤ 3.0 Å.

Then, to better understand the shape of the distributions of Figure S3, we sorted the structures according to the number of water molecules in the rst HS (see Figures 5 for wells and 6 for maximums). For both cations Ca1 and Ca2, we were able to deconvolute the peaks of Casurf distributions and to draw the following information:

Ca1 is hydrated by 1 to 4 water molecules: 1 to 2 for the structures in P1, 2 to 3 for M1 and P2 and 2 to 4 (with a majority of 3 water molecules around Ca1) for M2 and P3. Moreover on average, Ca1surf distances are similar for P1 and P3 (≈ 1.4 Å). For M1, P2 and M2, this average distance is longer, ≈ 2.4 Å, because Ca1 interacts with Fen, which moves away from the surface. For the majority of these structures, three water molecules solvate Ca1.

Ca2 is solvated by 5 to 8 water molecules: 5 to 8 for P1, 5 for M1, 5 (to 6) for P2 and M2, and 5 to 7 for P3. In M1 and P2, Ca2 is localized at an average distance of about 2.5 Å from Mont with an inner rst HS, thus including the surface in a square-based bi-pyramid structure. This kind of structures with 5 water molecules in the rst HS is predominant in M2 (Ca2surf distance ≈ 2.5 Å) but Ca2 can also be solvated by 6

water molecules (Ca2surf distance ≈ 3.9 Å). In P1 and P3, more hydration states are accessible to Ca2, while Ca1 is located near the surface (≈ 1.4 Å). In P1 and P3, Ca2 displays dierent HS: a square-based pyramid (5 H 2 O), an octahedron (6 H 2 O), and a distorted square antiprism (8 H 2 O) (both the latest are among the most stable of the rst HS of Ca 2+ according to the literature). In the interlayer space of an hydrated montmorillonite system, Holmboe and coworkers showed that the 5W hydration state allows full hydration of Ca 2+ cations [START_REF] Holmboe | Molecular Dynamics Simulations of Water and Sodium Diusion in Smectite Interlayer Nanopores as a Function of Pore Size and Temperature[END_REF][START_REF] Yang | Atomistic simulations of cation hydration in sodium and calcium montmorillonite nanopores[END_REF] . For Holmboe at 300 K, full hydration for and not yet by water molecule screening. Thus, when Ca1 moves away from the surface, Ca2

Correlation between positions and hydration states of both cations

has to do the same (B3 and B4). Thus, Ca1Ca2 distances become longer in average (see Figure 10) than in dry Mont and Ca2 rst HS grows to 6 till 8 water molecules. In B2, Ca1 solvated by one water molecule stays close to Mont and Ca2 is further from the surface. In this case, Ca1Ca2 distances are shorter than in dry Mont, which is probably possible thanks to the screening of Ca2 by the higher number of water molecules into its rst HS (6 to 8 but mostly 7). Experimental evidence allows to list two dierent types of interaction of water with the cations of the clay: i) water molecules bound to the cations attached to the clay surface (inner HS); ii) water molecules bound to fully hydrated cations detached from the surface (outer HS). Snapshot P1 belongs to B1 and presents 5 water molecules around Ca2 and a Ca2surf distance of 2.80 Å (see Figure 9). Thus, the Ca2 inner HS is a square-based bi-pyramid structures formed by 5 water molecules and the surface. Ca1 stays attached to the surface and to Fen by the monodentate complexation site Om. Thus, only one water molecule enter the space between Fen and the surface near Ca1 to form its rst HS.

On Figure 7 water molecules and HS Ca2 contains mostly 5 water molecules.

On Figure 9, snapshot M2 belonging to B'2 shows a structure for which, both Ca 2+ cations are in a 6 apices HS (see Figure 9). For Ca2, 6 water molecules are at the apices of a squarebased bi-pyramid. For Ca1, 4 water molecules, Om and the surface constitute the apices of a square-based bi-pyramid again. Ca2 is located at the center of an outer HS whereas Ca1 is at the center of an inner HS.

Alternative desorption mechanisms

In the previously presented desorption prole, the interaction between Ca1 and Fen is lost in P3 in agreement with the mechanism shown on Figure 3 Å). The interaction between Ca1 and the surface is lost but the monodentate interaction between Fen and Ca1 via Om is preserved. Thus, ξ is adapted to deal with the desorption of FenCa1 complex from the surface (see Figure 3(c)). We decided to take over from a B'3 structure and with the same reaction coordinate ξ to launch 6 new biased MD trajectories (see windows 9 to 14 on Table 3 and overlapping distributions of the umbrella sampling simulations on Figure S1(ii)). Therefore, we obtained a new end of prole for ξ > 5.8 Å that we superimposed to the rst in Figure 11. This new prole corresponds to the desorption of the FenCa1 complex from the surface (Figure 3(c)). After M2, the new prole diers from the previous one with a minimum P3c at ξ = 6.23 Å and a maximum M3c for ξ = 6.74 Å.

The new prole is energetically disfavored with respect to the previous one. The bottoms of the wells P3 and P3c are ≈ 2 kcal/mol apart in free energy and oset by 0.3 Å. In P3c, the Table 3: Parameters of the 14 biased MD trajectories: i is the window index, ξ i is the value of the reaction coordinate around which the harmonic potential oscillated, k i is the associated harmonic constant and t i the duration of the simulation (ps). In bold, the characteristic of the 6 trajectories that dier from the rst energy prole are given. This work shows that once one pathway is taken the other is no longer accessible except by crossing a free energy barrier of more than 5 kcal/mol (see Figure 13). By pursuing (c) pathway beyond M3c, the FenCa1 complex could evolve towards dissociation to increase the hydration energy of each entity: Ca1 and Fen. This pathway is evoked in Figure 3 as (d) mechanism. However, the (b) desorption mechanism will remain preferred.

i ξ i (Å) k i (kcal.mol -1 .Å -2 ) t i (ps) 1

Figure 1 :

 1 Figure 1: Structural formula of the fenhexamid, C 14 H 17 Cl 2 N O 2 .

Figure 2 :

 2 Figure 2: Montmorillonite representation. Mont contains 4 substitutions of Al 3+ (ligth blue) by M g 2+ (yellow) in the O sheet and 2 Ca 2+ cations (magenta) compensate the net negative charge carried by the clay layer and are situated above substitutions (ditrigonal ring of basal oxygens, Ob). The whole set of Ob atoms forms Mont surface, noted surf. Si ions in the T sheet are in dark blue. δ designates the interlayer spacing and d 001 , the basal spacing.

Figure 4

 4 Figure 4 displays the calculated desorption prole of a fenhexamid molecule from the montmorillonite surface. The prole presents three free energy wells separated by two barriers of 2.5 and 5.0 kcal.mol -1 . The singularities of the free energy prole are noted P1, P2 and P3

Figure 3 :

 3 Figure 3: Mechanism propositions for the desorption of a pesticide (PEST) from a clay surface (MONT): (a) Denition of the Ca1 and Ca2 cations. Ca1 is the Ca 2+ interacting with PEST and Ca2 does not interact with PEST; (b) PEST alone is desorbed from MONT, Ca1 remains attached to MONT; (c) PESTCa1 complex is desorbed from MONT; (d) PEST and Ca1 have a concerted departure from MONT but the interaction between PEST and Ca1 is not retained during the desorption.

Figure 4 :

 4 Figure 4: Free energy prole of fenhexamid desorption from the montmorillonite surface as a function of ξ, the distance from the geometric center of fenhexamid to surf, the surface containing the oxygen atoms Ob (see Figure 2). Every purple cross represents a structure visited during the 11 MD trajectories. The continuous green line gives the free energy prole of the desorption after smoothing. P1, P2, P3 are the minimums or energetic wells of this prole. M1 and M2 are the maximums. The zones delimited by the red horizontal lines enclose the structures considered as belonging to the minimums and to the maximums. Numbers on the top of the graph are the ξ values for each singularity.

Ca 2+ includes 8 Figure 5 :Figure 6 :

 856 Figure 5: Ca1surf and Ca2surf normalized distributions for each of the wells P1, P2 and P3 of the free energy prole. In each well, the structures were sorted according to the number of water molecules in the rst HS (d Oe-Ca ≤ 3.0 Å) of Ca1 or Ca2.

Figures 7 and 8

 8 Figures 7 and 8 report the density of structures regarding to Ca1surf and Ca2-surf values visited for the minimums (Figure7) and the maximums (Figure8) of the free energy desorption prole. Moreover, Figure9reports four snapshots of structures along the free energy prole of desorption of Fen from Mont: one for each minimum and one for the M2 maximum, which diers from other singularities. The number of water molecules in the rst HS of Ca1 and Ca2 is given as well as the Ca1Ca2, Ca1surf, Ca2surf and Ca1Om distances (Om is the oxygen atom of the Fen carbonyl group).

Figure 7 :

 7 Figure 7: Density of structures for the three wells P1, P2 and P3 of Fen desorption from the montmorillonite, at given Ca1surf and Ca2surf distances. Distances are in Å, the logarithm (log) of the density of structures is represented (scale from -4 to 0). In round boxes, the numbers of water molecules in the rst HS around Ca1 (blue) and Ca2 (red) are also given (in bold: the most abundant; in dashed circles: rst HS including the surface, square-based bi-pyramid structures formed by 5 water molecules and the surface). In square boxes, distances corresponding to the maximums of density of structures are put forward.

Figure 8 :

 8 Figure 8: Density of structures for both minimums M1 and M2 of Fen desorption from the montmorillonite, at given Ca1surf and Ca2surf distances. Distances are in Å, the logarithm (log) of the density of structures is represented (scale from -5 to 0). In round boxes, the numbers of water molecules in the rst HS around Ca1 (blue) and Ca2 (red) are also given (in bold: the most abundant; in dashed circles: rst HS including the surface, square-based bi-pyramid structures formed by 5 water molecules and the surface). In square boxes, distances corresponding to the maximums of density of structures are put forward.

Figure 9 :

 9 Figure 9: Snapshots taken in the 3 minimums, P1, P2 and P3, and in the M2 maximum of the free energy prole of desorption of Fen from Mont. For clarity, only the Ob oxygen atoms are represented for the Mont surface. For the same reason, only the water molecules involved into the rst HS of Ca1 and Ca2 are represented (d Oe-Ca ≤ 3.0 Å). HS Ca1 and HS Ca2 are the numbers of water molecules in the rst HS of Ca1 and Ca2, respectively. Ca1Ca2, Ca1surf, Ca2surf and Ca1Om distances are given in Å. Om is the oxygen atom of the Fen carbonyl group.

4 Å

 4 P2, the density of structures shows only one basin with an high density of structures corresponding to Ca1surf ≈ 2.3 Å and Ca2surf ≈ 2.5 Å. Ca1 is solvated by two but predominantly three water molecules and Ca2 by mostly 5. Localization of the density goes along with Ca1Ca2 distances around 10 Å comparable with dry Mont. Compared to P1, Ca1 positions are further from the surface and Ca1 are at almost equal distances to Mont than Ca2. For the second minimum, the snapshot P2 shows again 5 water molecules around Ca2 and an even shorter Ca2surf distance than in P1 (2.66 Å, see Figure9). The Ca2 inner HS is a square-based bi-pyramid structure formed by 5 water molecules and the surface. Here, Ca1 is still attached to the surface and to Fen (Ca1Om = 2.25 Å). As Fen moves away from the surface from P1 to P2, a second and a third water molecules enter the space between Fen and the surface near Ca1. Its HS contains 3 water molecules.For P3, Ca1 is closer to the surface, while the interaction between Ca1 and the oxygen atom of Fen carbonyl group is lost. The structures present Ca1surf distances around 1.corresponding to mostly three water molecules in the rst HS. These water molecules insert the space between Fen and Ca1. Again as in P2, Ca1Ca2 distances are in average roughly equal to the CaCa distance in dry Mont. If the rst HS of Ca2 counts 5 water molecules, thus Ca2surf is about 2.5 Å, while if the rst HS of Ca2 counts 6 to 7 water molecules, thus Ca2surf is longer being around 4.0 Å. In the P3 minimum, while Fen moves away from the surface,the space between Fen and the surface can host additional water molecules. In snapshot P3, we chose to show on Figure9, a structure for which, Ca2 has 6 apices and Ca1 has 5. For Ca2, 6 water molecules are at the apices of a square-based bi-pyramid. For Ca1, 4 water molecules and the surface constitute the ve apices of a square-based pyramid. Ca2 is located at the center of an outer HS whereas Ca1 is at the center of an inner HS.On Figure8, for M1 and M2 maximums, the density of structures according to both distances Ca1surf and Ca2surf are given. For M1 transition state (see FigureS6(a)), the density is localized around Ca1surf = 2.4 Å and Ca2-surf = 2.6 Å. HS Ca1 contains mostly 3 water molecules and HS Ca2 5. This graph is quite similar to P2 graph on Figure7. Thus M1 resembles the well, which is downstream and could be qualied of a late connecting maximum. On the other hand, M2 graph connecting P2 and P3 looks like the P2 well, which is upstream, for the basin around Ca1surf = 2.5 Å and Ca2surf = 2.5 Å (noted B'1). As well as for P2, in B'1, HS Ca1 is mostly 3 and HS Ca2 = 5. However, new basins are visited: B'2. Ca1surf = 2.5 Å and Ca2surf = 3.9 Å, HS Ca1 contains two to four but mostly 3 water molecules and HS Ca2 contains mostly 6 water molecules; B'3. Ca1surf = 3.3 Å and Ca2surf = 2.5 Å, HS Ca1 contains three to ve but mostly 3

  (b). To conrm this mechanism is preferred or not, an alternative mechanism has been studied. When the chosen reaction coordinate ξ increases beyond 5.7 Å (M2), all the structures end up in well P3 with Ca1-surf distances around 1.4 Å. Thus, one question arises: is it possible that Fen desorbed from Mont along with Ca1 (Figure 3(c))? If so, is this mechanism energetically accessible? For the structures of B'3 (Figure 8 M2), Ca1 is moving away from the surface (Ca1surf = 3.3

Figure 11 :

 11 Figure 11: Free energy prole of fenhexamid desorption from the montmorillonite surface as a function of ξ, the distance from the geometric center of fenhexamid to surf, the surface containing the oxygen atoms Ob (see Figure 2). Every purple or green cross represents a structure visited during the 11 (b) and 14 (c) MD simulations. The continuous green line gives the free energy prole of the desorption after smoothing. P1, P2, P3 and P3c are the minimums or energetic wells of this prole. M1, M2 and M3c are the maximums. The zones delimited by the red horizontal lines enclose the structures considered as belonging to the minimums and to the maximums. Numbers on the top of the graph are the ξ values for each singularity.

Figure 12 :

 12 Figure 12: Ca1surf and Ca2surf normalized distributions for the P3c well and the M3c maximum of the free energy prole. In P3c and M3c, the structures were sorted according to the number of water molecules in the rst HS (d Oe-Ca ≤ 3.0 Å) of Ca1 or Ca2.

Figure 13 :

 13 Figure13: Free energy prole of Ca1 moving away from the surface. ξ ′ is the distance between Ca1 and surf, the surface containing the oxygen atoms Ob (see Figure2). Every purple cross represents a structure visited during the MD simulations. The continuous green line gives the free energy prole after smoothing. PI and PII are the minimums or energetic wells of this prole. MI is the maximum. The zones delimited by the red horizontal lines enclose the structures considered as belonging to the minimums and to the maximums. Numbers on the top of the graph are the ξ ′ values for each singularity.

Table 1 :

 1 Water contents and basal spacing, d 001 , for dierent hydrate states of Mont. NW: Nth-layer hydrate state and n: number of water molecules per unit cell per layer-hydrate[START_REF] Holmboe | Molecular Dynamics Simulations of Water and Sodium Diusion in Smectite Interlayer Nanopores as a Function of Pore Size and Temperature[END_REF][START_REF] Yang | Atomistic simulations of cation hydration in sodium and calcium montmorillonite nanopores[END_REF] .The geometrical structure and energetics of hydrated Ca 2+ have also been studied in vacuum by theoretical calculations or via EXAFS neutron diraction experiments[START_REF] Badyal | Understanding the Eects of Concentration on the Solvation Structure of Ca2+ in Aqueous Solution. II: Insights into Longer Range Order from Neutron Diraction Isotope Substitution[END_REF] . Coordination numbers of 6, 7 and 8 are common for the rst HS of Ca 2+ . For hydrated structures of the form Ca(H 2 O) 2+ i .jH 2 O, i water molecules are in the rst coordination sphere and j water molecules are in the second coordination sphere (hydrogen bonded to water molecules in the rst sphere). The clusters Ca(H 2 O) 2+ 4 .2H 2 O and Ca(H 2 O) 2+ 5 .1H 2 O were signicantly higher in relative energy than Ca(H 2 O) 2+ 6 (> 8 kcal.mol -1 ) 20,21 . On the contrary,

	NW	0W	1W	2W	3W	4W	5W	10W
	d 001 (Å) 9.5-10.1 12.0-12.7 15.0-15.7 18.0-19.0 21.4-22.0 ≈24.8 ≈40.7
	n	0	5	10	15	20	25	50
				3

  Mignon et al. have studied DNA bases adsorption energies on dry NaMont[START_REF] Mignon | Theoretical Study of the Adsorption of RNA/DNA Bases on the External Surfaces of Na+ -Montmorillonite[END_REF][START_REF] Mignon | Theoretical study of the adsorption of DNA bases on the acidic external surface of montmorillonite[END_REF] , and cytosine adsorption on hydrated NaMont 26 with ab initio MD simulations;

	investigated the adsorption of single molecules of volatiles, such as water, ethanol, ethyl ac-
	etate, pyridine, toluene, and n-octane, on the dry surface of a smectite clay (NaMont) using
	DFT MD
	Szczerba et al. have performed force eld molecular dynamics simulations in order to study
	the interactions of ethylene glycol with watered smectite (CaMont) 27 ; Aggarwal et al. have
	used force eld quantum mechanics to estimate relative energies for the adsorption of non-
	ionic organic solutes in clay mineral-water-solute systems (KMont) 28 ; Clausen et al. have

  PI and PII, and one maximum, MI. PI bottom of the well is less than 1 kcal/mol lower in free energy than PII. The barrier from PI is about 6 kcal/mol, which is consistent with the exclusivity of the paths. In FigureS5, Ca1surf and Ca2surf normalized distributions for each of the PI, PII and MI singularities are presented. For the rst HS of Ca1, PI is in accordance with the P3 well of path (b). However, for all structures in PI, HS Ca1 = 3 because the Ca1surf distance is in between 1.225 and 1.485 Å excluding HS Ca1 = 2 and HS Ca1 = 4 structures. Similarly, PII is consistent with P3c (path(c)) with mostly 4 water molecules in the Ca1 rst HS. Going from PI to PII via MI, an increase in HS Ca1 is observed from 3 to 5. Ca2, the non interacting cation, has an HS of mostly 6 or 7

	2 3 4 5 6 7 8 prole shows two minimums, water molecules.	4.41 4.70 4.89 5.05 5.55 5.58 5.65 5.78	14.3 28.6 114.3 14.3 57.1 114.3 114.3 57.1	20 20 13 20 12 13 16 25

Acknowledgement

This work was granted access to the HPC resources of CALMIP supercomputing center under the allocation 2018-[P1222] and of CINES under the allocation 2018-[cpq7002] made by GENCI.

Ca1surf and Ca2surf distances increase with the number of water molecules into their rst HS. One may wonder why no structure corresponding to the couple Ca1surf ≈ 1.6 Å and

Supporting Information Available

The following les are available free of charge.