Equilibration of a Tonks-Girardeau gas following a trap release
EEquilibration of a Tonks-Girardeau gas following a trap release
Mario Collura, Spyros Sotiriadis, and Pasquale Calabrese Dipartimento di Fisica dell’Universit`a di Pisa and INFN, 56127 Pisa, Italy
We study the non-equilibrium dynamics of a Tonks-Girardeau gas released from a parabolic trap to a circle.We present the exact analytic solution of the many body dynamics and prove that, for large times and in a prop-erly defined thermodynamic limit, the reduced density matrix of any finite subsystem converges to a generalizedGibbs ensemble. The equilibration mechanism is expected to be the same for all one-dimensional systems.
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The non-equilibrium dynamics of isolated many bodyquantum systems is currently in a golden age mainly due tothe experiments on trapped ultra-cold atomic gases [1–8] inwhich it is possible to measure the unitary non-equilibriumevolution without any significant coupling to the environment.A key question is whether the system relaxes to a stationarystate, and if it does, how to characterize from first principles itsphysical properties at late times. It is commonly believed that,depending on the integrability of the Hamiltonian governingthe time evolution, the behavior of local observables either canbe described by an effective thermal distribution or by a gen-eralized Gibbs ensemble (GGE), for non-integrable and inte-grable systems respectively (see e.g. [9] for a review). Whilethis scenario is corroborated by many investigations [10–29],a few studies [30–35] suggest that the behavior could be morecomplicated and in particular can depend on the initial state.In a global quantum quench, the initial condition is theground state of a translationally invariant Hamiltonian whichdiffers from the one governing the evolution by an experimen-tally tunable parameter such as a magnetic field. A differ-ent initial condition can be experimentally achieved [7, 8] byconsidering the non-equilibrium dynamics of a gas releasedfrom a parabolic trapping potential. It has been shown exper-imentally that the spreading of correlations is ballistic for anintegrable system and diffusive for a non-integrable one [8].However, when the gas expands in full space, for infinite timethe gas clearly reaches zero density (see e.g. [35–40] for atheoretical analysis) and it is rather confusing to distinguishthermal and GGE states. To circumvent this, Caux and Konik[41] have recently developed a new approach based on inte-grability to study the release of the Lieb-Liniger Bose gas [42]from a parabolic trap not in free space but on a closed circle(as sketched in Fig. 1), so that the gas has finite density. Ithas been numerically shown that the time averaged correla-tion functions are described by a GGE, apart from finite sizeeffects [41]. A preliminary analysis for non-integrable modelshas also been presented [43]. However, while this approachpermits effectively to calculate time-averaged quantities forrelatively large systems (the maximum number of particles is N = 56 [41]), the study of the time evolution is possible butmuch harder and it is difficult to establish whether (and inwhich sense) an infinite time limit exists.In order to overcome these limitations, we present herea full analytic solution of this non-equilibrium dynamics in FIG. 1: Left: sketch of the trap release dynamic in a circle. Right:Color plot of the numerical calculated density evolution for N =10 , , ∞ (from left to right) at N/L = 1 / and ωN = 5 . the limit of strong coupling, i.e. in the celebrated Tonks-Girardeau regime [44]. We will show that, in a properly de-fined thermodynamic (TD) limit, the reduced density matrixof any finite subsystem converges for long times to the GGEone. This implies that any measurable local observable willconverge to the GGE predictions. The model and quench protocol . We consider a one-dimensional Bose gas with delta pairwise interaction and inan external parabolic potential with Hamiltonian H = − N (cid:88) j =1 ∂ ∂x j + N (cid:88) j =1 ω x j + c (cid:88) i (cid:54) = j δ ( x i − x j ) , (1)where c > is the coupling constant (we set (cid:126) = m = 1 ).The translationally invariant Lieb-Liniger model is obtainedfor ω = 0 and on a circle of length L with periodic boundaryconditions (PBC). While Ref. [41] covers numerically arbi-trary c , to make an analytic progress we consider the strong-coupling limit of impenetrable bosons c → ∞ , correspondingalso to the low density n ≡ N/L (cid:28) regime for any c [42].For a trap release, the initial state is a Tonks Girardeau gasconfined by a parabolic potential, i.e. the ground state of Eq.(1) for a fixed ω . Following [44], the many body wave func-tion for the ground state of N impenetrable bosons is Ψ B ( x , · · · , x N ) = (cid:89) i
L = 400L = 800L = 1600TD Limit n ( x ,t ) n = 1/2, ω N = 5t/L x/L = 1/4(b)
L = 400L = 800L = 1600TD Limit n ( x ,t ) n = 1/2, ω N = 5t/L x/L = 3/8(c) -0.4 -0.2 0 0.2 0.400.20.40.60.81 t/L = 0t/L = 1/4t/L = 1/2t/L = 1TD Limit n ( x ,t ) n = 1/2, ω N = 5x/L(d) L = 1600
FIG. 2: (a,b,c) Time evolution of the density n ( x, t ) for different x/L and sizes. Dashed red lines indicate the equilibration value N/L at infinite time. (d) Density profile for L = 1600 at different rescaledtimes t/L . Symbols are the exact dynamics for finite N , while fullblack lines are the TD limit. with Φ j ( x,
0) = χ j ( x ) and H F the single particle free Hamil-tonian with PBC). These time evolved wave functions can becalculated from Eqs. (4) and (5) obtaining Φ j ( x, t ) = + ∞ (cid:88) p = −∞ Φ ∞ j ( x + pL, t ) , (6)where Φ ∞ j ( x, t ) = (1 − iωt ) j e − i tω x ω t (1 + iωt ) j +12 χ j (cid:16) x √ ω t (cid:17) , (7)is the time evolved eigenfunction in infinite space whichagrees with the result in [36]. The boson-fermion mappingremains valid for the time-dependent problem [45]. Time evolution of the density profile.
We start the analysisof the many body problem from the density profile n ( x, t ) = (cid:80) j | Φ j ( x, t ) | which shows clearly how a non-zero stationaryvalue can be achieved in a trap release experiment. From Eq.(7) we have for arbitrary time, N, L, ωn ( x, t ) = 1 √ ω t ∞ (cid:88) p,q = −∞ e i ω t ω t [( x + pL ) − ( x + qL ) ] × N − (cid:88) j =0 χ j (cid:18) x + pL √ ω t (cid:19) χ j (cid:18) x + qL √ ω t (cid:19) , (8)which, in the TD limit, because of the strongly oscillatingphase factor, reduces to the diagonal part p = q : n ( x, t ) = 1 √ ω t ∞ (cid:88) p = −∞ N − (cid:88) j =0 (cid:12)(cid:12)(cid:12)(cid:12) χ j (cid:16) x + pL √ ω t (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) , (9)and can be rewritten in terms of the TD limit of the particledensity at initial time n ( x ) = ( √ N ω − ω x ) /π as n ( x, t ) = 1 √ ω t ∞ (cid:88) p = −∞ n (cid:18) x + pL √ ω t (cid:19) , (10)In Figs. 1 and 2 we show the numerically calculated time de-pendent density for finite but large N which perfectly agreeswith the above TD prediction for any time. The two-point fermionic correlator C ( x, y ; t ) ≡(cid:104) Ψ † ( x, t )Ψ( y, t ) (cid:105) is given by C ( x, y ; t ) = N − (cid:88) j =0 Φ ∗ j ( x, t )Φ j ( y, t ) . (11)The numerical determination of this correlation function forfinite N is reported in Fig. 3 showing the approach to theinfinite time limit [46] C ( x, y ; t → ∞ ) = 2 n J [ √ ωN ( x − y )] √ ωN ( x − y ) , (12)with J ( z ) the Bessel function. The reduced density matrix and the GGE.
For a closed sys-tem evolving under Hamiltonian dynamics, the existence of astationary state may seem paradoxical because the whole sys-tem is always in a pure state and cannot be described by amixed state at infinite time. This ‘paradox’ is solved in thereduced density matrix formalism: given a space interval A ,the reduced density matrix is ρ A ( t ) = Tr B ρ ( t ) where B isthe complement of A and ρ ( t ) = | Ψ( t ) (cid:105)(cid:104) Ψ( t ) | is the densitymatrix of the whole system. With some abuse of language,we say that a system becomes stationary if, after the TD limitis taken for the whole system, the limit ρ A, ∞ = lim t →∞ ρ A ( t ) exists for any finite A [17]. Furthermore we say that a systemis described by a statistical ensemble with density matrix ρ E if the reduced density matrix ρ A,E ≡ Tr B ρ E equals ρ A, ∞ .For a gas of free fermions, by means of Wick theorem, anyobservable can be obtained from the two-point correlator. Theconstruction of ρ A in terms of C ( x, y ) in continuous space hasbeen detailed in [47] (generalizing the lattice approach [48]).As a crucial point, the non-local transformation mapping theTonks-Girardeau gas to free fermions is local within any given compact subspace, i.e. the bosonic degrees of freedom within A can be written only in terms of fermions in A . This is anal-ogous to lattice models such as the Ising chain [16–18]. Thus,if for finite x, y , C ( x, y, t → ∞ ) is described by a statisticalensemble, also ρ A will be and consequently the expectationvalue of any observable local within A . -20 -10 0 10 20-0.200.20.40.60.81 L = 400L = 800L = 1600Equil. R e [ C ( x , ;t ) ] n = 1/2, ω N = 5x(a) t/L = 0 -20 -10 0 10 20-0.100.10.20.30.40.50.6
L = 400L = 800L = 1600 R e [ C ( x , ;t ) ] n = 1/2, ω N = 5x(b) t/L = 1 -20 -10 0 10 20-0.100.10.20.30.40.50.6
L = 400L = 800L = 1600GGE R e [ C ( x , ;t ) ] n = 1/2, ω N = 5x(c) t/L = 2 -20 -10 0 10 20-0.100.10.20.30.40.50.6
L = 400L = 800L = 1600GGE R e [ C ( x , ;t ) ] n = 1/2, ω N = 5x(d) t/L = 4
FIG. 3: Snapshots of the correlation
Re[ C ( x, t )] at differentrescaled times t/L and sizes. For t/L = 0 the full line is the initialcorrelation in the TD limit, i.e. C ( x,
0) = sin[ √ ωNx ] /πx validfor x (cid:28) L . The full line for t/L = 2 , is the stationary value in Eq.(12). As time increases, two symmetric peaks are expelled from thecentral region. The inset in (d) shows the evolution for fixed x = 5 and L = 1600 : after the moving peak has been expelled, the corre-lation is damped in time and converges to the GGE. Because of integrability, it is natural to expect that Eq. (12)should be described by a GGE ρ GGE = Z − e − (cid:80) λ i I i , (13)with { I i } a complete set of local integrals of motion and λ i Lagrange multipliers fixed by the conditions (cid:104) Ψ | I i | Ψ (cid:105) =Tr[ ρ GGE I i ] , with | Ψ (cid:105) the many body initial state. However,for free fermions, one can work with the momentum occupa-tion modes ˆ n k = c † k c k which are non-local integrals of mo-tion, but can be written as linear combinations of local inte-grals of motion [28]. In the TD limit, the initial values of ˆ n k are (cid:104) Ψ | ˆ n k | Ψ (cid:105) = N − (cid:88) j =0 | A k,j | (cid:39) L (cid:114) Nω (cid:114) − k ωN , (14)and zero if the argument of the square root is negative. Inthe GGE we have Tr[ ρ GGE ˆ n k ] = ( e λ k + 1) − and equatingthe two, the λ k are derived. It is now straightforward to showthat C ( x, y ) in the GGE equals the infinite time limit of traprelease in Eq. (12) [46]. This shows that all stationary quanti-ties of the released gas are described by a GGE. Furthermore,in Ref. [19] it has been shown that all non-equal time station-ary properties are always determined by the same ensemble Ground State ω N = 8 ω N = 16 ω N = 32 S ( k ) n = 1 k/2 π FIG. 4: The GGE structure factor S ( k ) as function of k/ k F ( k F = πn ) for different initial trap potentials ωN compared withthe ground-state one (dashed line). describing the static quantities, and so, even in our case, theyare encoded solely in the GGE. The structure factor in the GGE.
The structure factor S ( k ) is the Fourier transform of the density-density correlation (cid:104) ˆ n ( x, t )ˆ n (0 , t ) (cid:105) . In any ensemble which is diagonal in theFourier modes, in the TD limit the structure factor can be writ-ten in terms of occupation modes n k as − S ( k ) = LN (cid:90) dq π n q n k − q = 4 √ nπ √ ωN f (cid:16) k √ ωN (cid:17) , (15)where the rhs is obtained using the GGE n k given in Eq. (14).Here f ( x ) = [(4 + x ) E (1 − /x ) − K (1 − /x )] | x | / if | x | < and zero otherwise where E ( z ) and K ( z ) are stan-dard elliptic functions and f (0) = 4 / . S ( k ) turns out to bean even function of k and monotonic for k > . The plotof S ( k ) for different initial trapping potentials is reported inFig. 4. S ( k ) resembles the one found numerically in [41] forthe Lieb-Liniger gas. Because of the trap release constraint √ N ω > √ n , we have S ( k ) > S (0) ≥ − / π =0 . . . . . This calculation shows how easy it is to ob-tain GGE predictions without solving the full non-equilibriumdynamics. The bosonic two-point function or one-body density matrix C B ( x, y ; t ) ≡ (cid:104) ˆΦ † ( x, t ) ˆΦ( y, t ) (cid:105) (with ˆΦ( y, t ) bosonic anni-hilation operator) is a non-trivial quantity whose calculationpresents difficulties also in thermal equilibrium [49]. How-ever, using the approach in [50], the computation is easy forlarge time and in the TD limit obtaining [46] C B ( x, y ; t → ∞ ) = C ( x, y ; t → ∞ ) e − n | x − y | , (16)with C ( x, y ; t → ∞ ) the fermion correlator in Eq. (12). Forsmall distances, C B ( x, y ; t → ∞ ) shows a singular behaviorof the form | x − y | which is different from its thermal coun-terpart | x − y | [49]. This behavior is strictly valid only inthe TD limit because for any finite N , at very small distances C B ( x, y ; t → ∞ ) crosses over to | x − y | as expected fromgeneral arguments [49]. This finite N crossover is numeri-cally demonstrated in [46]. Consequently, the momentum dis-tribution function has a large momentum tail of the form k − which crosses over to the standard k − for even larger k . Thislarge-momentum crossover should be a measurable signatureof the GGE. Trap to trap release . The case of a Tonks-Girardeau gasreleased not in a periodic system but in a larger harmonic traphas been solved by Minguzzi and Gangardt [36] who showedthat the system oscillates forever without relaxation. How-ever, even in this case, it is simple to see that the time aver-aged two-point correlations (and hence by Wick theorem anyobservable) are still described by a GGE.
Conclusions . In this letter we solved analytically the nonequilibrium dynamics of a Tonks Girardeau gas following atrap release to a periodic geometry as in Fig. 1. We prove thatfor long time and in the TD limit, any finite subsystem be-comes stationary and its behavior is described by a GGE. Thisprovides the first analytic proof of a GGE for an inhomoge-neous initial state. We stress that the mechanism responsiblefor the equilibration is very different from the one in a globalquantum quench since in the trap release it is due to the in-terference of the particles going around the circle many times.This equilibration mechanism is expected to be the same forany one dimensional gas released into a circle.Apart from the per se experimental interest [7, 8], these re-sults represent a first step towards a complete analytical un-derstanding of the famous quantum Newton cradle [2] at leastin the Tonks Girardeau limit.
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S. Petrov, V. Gritsev, S. Manz,S.Hofferberth, T. Schumm, E. Demler, and J. Schmiedmayer,Phys. Rev. A , 033604 (2009); M. Kormos, A. Shashi, Y.-Z.Chou, and A. Imambekov, arXiv:1204.3889. SUPPLEMENTARY MATERIALThe two-point fermionic correlation function
Plugging the one-particle time-evolved wavefunctions [Eqs. (6) and (7) in the main text] in the definition of the fermioniccorrelator C ( x, y ; t ) = (cid:80) Φ ∗ j ( x, t )Φ j ( y, t ) , we have C ( x, y ; t ) = 1 √ ω t ∞ (cid:88) p,q = −∞ exp (cid:26) i ω t ω t ) [( x + pL ) − ( y + qL ) ] (cid:27) × N − (cid:88) j =0 χ j (cid:18) x + pL √ ω t (cid:19) χ j (cid:18) y + qL √ ω t (cid:19) . (S1)Due to the oscillating phase factor, in the TD limit, the leading behavior of Eq. (S1) is given only by the diagonal part p = q , sothat C ( x, y ; t ) (cid:39) e i ω t ( x − y ω t √ ω t ∞ (cid:88) p = −∞ exp (cid:26) i ω t ( x − y ) pL ω t (cid:27) N − (cid:88) j =0 χ j (cid:18) x + pL √ ω t (cid:19) χ j (cid:18) y + pL √ ω t (cid:19) , (S2)that in the regime t (cid:29) ω − gives C ( x, y ; t ) (cid:39) ωt ∞ (cid:88) p = −∞ e i ( x − y ) pL/t N − (cid:88) j =0 χ j (cid:18) x + pLωt (cid:19) χ j (cid:18) y + pLωt (cid:19) (S3) = 1 √ ωt ∞ (cid:88) p = −∞ e i ( x − y ) pL/t N − (cid:88) j =0 ˜ χ j (cid:18) x + pL √ ωt (cid:19) ˜ χ j (cid:18) y + pL √ ωt (cid:19) . (S4)where we defined ˜ χ j ( x ) as the eigenfunctions of a harmonic oscillator with ω = 1 (i.e. ˜ χ j ( x ) = χ j ( x ) | ω =1 ).In the large time limit t/L → ∞ (but with t/L → ), the sum over p can be replaced by an integral ∞ (cid:88) p = −∞ γF ( γp ) −→ (cid:90) ∞−∞ dzF ( z ) , (S5)where γ = L/ ( √ ωt ) → , allowing to rewrite C ( x, y ; t ) as C ( x, y ; t → ∞ ) = 1 L (cid:90) ∞−∞ dz e i √ ω ( x − y ) z N − (cid:88) j =0 ˜ χ j (cid:16) γ xL + z (cid:17) ˜ χ j (cid:16) γ yL + z (cid:17) (S6) = 1 L (cid:90) ∞−∞ dz e i √ ω ( x − y ) z N − (cid:88) j =0 | ˜ χ j ( z ) | , (S7)where in the second line we used once again γ → . Now using lim N →∞ N − (cid:88) j =0 | ˜ χ j ( x ) | = √ N − x π ≡ ˜ n ( x ) , (S8)we have in the TD limit C ( x, y ; t → ∞ ) = 1 L (cid:90) ∞−∞ dk e i √ ω ( x − y ) k ˜ n ( k ) = (cid:90) ∞−∞ dk π e i ( x − y ) k (cid:104) Ψ | ˆ n k | Ψ (cid:105) = 2 n J (cid:104) √ ωN ( x − y ) (cid:105) √ ωN ( x − y ) , (S9)which is Eq. (12) in the main text. It is also clear that this is nothing but the Fourier transform of (cid:104) Ψ | ˆ n k | Ψ (cid:105) obtained in Eq.(14) in the main text from the GGE, thus showing that infinite time limit and GGE results for C ( x, y ) are equal. The two-point bosonic correlation function
The impenetrable bosons field operators ˆΦ( x ) are related to the fermionic ones ˆΨ( x ) by the Jordan-Wigner transformation ˆΨ( x ) = exp (cid:26) iπ (cid:90) x dz ˆΨ † ( z ) ˆΨ( z ) (cid:27) ˆΦ( x ) , ˆΨ † ( x ) = ˆΦ † ( x ) exp (cid:26) − iπ (cid:90) x dz ˆΨ † ( z ) ˆΨ( z ) (cid:27) . Consequently, the equal time two-point bosonic correlation function C B ( x, y ; t ) with y > x is C B ( x, y ; t ) ≡ (cid:104) ˆΦ † ( x ) ˆΦ( y ) (cid:105) = (cid:28) ˆΨ † ( x ) exp (cid:26) − iπ (cid:90) yx dz ˆΨ † ( z ) ˆΨ( z ) (cid:27) ˆΨ( y ) (cid:29) . (S10)Expanding the exponential and using Wick theorem one has C B ( x, y ; t ) = ∞ (cid:88) n =0 ( − iπ ) n n ! (cid:90) yx dz · · · (cid:90) yx dz n (cid:104) ˆΨ † ( x ) ˆΨ † ( z ) ˆΨ( z ) · · · ˆΨ † ( z n ) ˆΨ( z n ) ˆΨ( y ) (cid:105) (S11) = ∞ (cid:88) n =0 ( − n n ! (cid:90) yx dz · · · (cid:90) yx dz n det ij C ( x i , y j ; t ) , where the indices i, j run from to n , and we fixed x i = y i ≡ z i , ∀ i > , and x ≡ x, y ≡ y . The previous equation is aFredholm’s minor of the first order.In order to evaluate these correlators it is more convenient to introduce the N × N overlap matrix A ( x, y ; t ) with elements A ij ( x, y ; t ) ≡ (cid:90) yx dz Φ ∗ i ( z, t )Φ j ( z, t ) , i, j ∈ [0 , . . . , N − . (S12)in terms of which one has [50] C B ( x, y ; t ) = N − (cid:88) i,j =0 Φ ∗ i ( x, t ) B ij ( x, y ; t )Φ j ( y, t ) , (S13)where the N × N matrix B ( x, y ; t ) is B ( x, y ; t ) ≡ det[ P ]( P − ) T , with P ( x, y ; t ) ≡ I − y − x ) A ( x, y ; t ) , (S14)and I is the N × N identity matrix. The large-time limit.
Eq. (S13) is a good starting point to evaluate analytically the large-time limit of the bosonic correlationfunction. Indeed, the one-particle time-evolved functions are given in Eqs. (6) and (7) of the main text, and proceeding as in Eq.(S1) we can write in the TD and large-time limits Φ ∗ a ( z, t )Φ b ( z, t ) as Φ ∗ a ( z, t )Φ b ( z, t ) (cid:39) i a − b ωt ∞ (cid:88) p = −∞ χ ∗ a (cid:18) z + pLωt (cid:19) χ b (cid:18) z + pLωt (cid:19) (cid:39) i a − b L (cid:90) ∞−∞ dxχ ∗ a (cid:16) zωt + x (cid:17) χ b (cid:16) zωt + x (cid:17) = 1 L δ ab , (S15)where in the last equality we used the orthonormality of the eigenfunctions χ a ( x ) . Consequently, the large-time behavior of the A and P matrices is A ab ≡ (cid:90) yx dz Φ ∗ a ( z, t )Φ b ( z, t ) = y − xL δ ab , P ( x, y ; t → ∞ ) = (cid:18) − | x − y | L (cid:19) I . (S16)Clearly the above equations are valid as long as the rhs’ are finite, i.e. when | x − y | /L ∼ O (1) . For | x − y | (cid:28) L differentapproaches must be used, as e.g. expanding the determinant in Eq. (S11).From Eq. (S16), the B matrix is B ( x, y ; t → ∞ ) = (cid:18) − | x − y | L (cid:19) N − I , and lim N →∞ B ( x, y ; t → ∞ ) = I e − n | x − y | , (S17)where the large N limit has been taken keeping, as usual, n = N/L constant. Substituting Eq. (S17) in Eq. (S13), we finallyhave Eq. (16) in the main text, i.e. C B ( x, y ; t → ∞ ) = C ( x, y ; t → ∞ )e − n | x − y | . (S18) ω N = 100GGE, ω N = 200GGE, ω N = 400asymp. C B ( x , ;t → ∞ ) n = 0.5 x FIG. 5: Exact bosonic correlation function C B ( x, y ; t → ∞ ) calculated by discretizing the Fredholm’s minor in Eq. (S11). For large enough x , the data always agree with the prediction in Eq. (S18) (full lines), while for smaller x , the data approach it only for large enough ωN . Theinset shows a zoom for very small x , for which the asymptotic | x | behavior (dashed line) crosses over to a standard (non-singular) quadraticform. As anticipated in the main text, Eq. (S18) is valid only in the TD limit in the regime | x − y | /L ∼ O (1) . For | x − y | (cid:28) L ,the correlation function C B ( x, y ; t → ∞ ) crosses over to the standard singular behavior | x − y | , as expected from generalarguments. In order to show the correctness of this statement, we calculate numerically C B ( x, t → ∞ ) by discretizing theFredholm’s minor in Eq. (S11) as explained in Ref. [51] and using as input the GGE fermion correlation in Eq. (12) of the maintext. In Fig. 5, we report the numerically calculated C B ( x, t → ∞ ) as function of x for different values of ωN (we recall n = N/L is constant). It is clear that increasing N , the numerical data approach the asymptotic result in Eq. (S18). However, ifwe zoom in the region of very small distances, as done in the inset of Fig. 5, the | x − y | singularity is absent, as expected, andthe main singularity is of the form | x − y | while the leading behavior is non-singular ( x − y )2