Emptiness Formation Probability in 1D Bose Liquids
EEmptiness Formation Probability in 1D Bose Liquids
Hsiu-Chung Yeh
School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA
Alex Kamenev
School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA andWilliam I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, Minnesota 55455, USA
We study emptiness formation probability (EFP) in interacting 1D Bose liquids. That is theprobability that a snapshot of its ground state reveals exactly zero number of particles within theinterval | x | < R . For a weakly interacting liquid there is parametrically wide regime n − < R < ξ (here n is the average density and ξ is the healing length), where EFP exhibits a non-trivial crossoverfrom the Poisson to the Gaussian behavior. We employ the instanton technique [A. Abanov, 2004]to study quantitative details of these regime and compare it with previously reported limited cases. I. INTRODUCTION
Recent precision measurements of particle number fluc-tuations in ultra cold quantum gases [1–3] have revivedinterest [4–7] in large deviations statistics in many-bodysystems. Emptiness formation probability (EFP) is prob-ably the most iconic and widely studied measure of suchlarge deviations. It plays a special role in the theory ofBethe Ansatz [8] integrable models [9–12] and is a testbed for development of non-perturbative techniques, suchas the instanton calculus [13]. The EFP, P E F P ( R ), is theprobability that no particles are found within the spaceinterval ( − R, R ) in the ground state of a one-dimensional(1D) many-body system with the average density n . P E F P ( R ) = N (cid:89) i =1 (cid:90) | x i |≥ R dx i | Ψ g ( x , x , ..., x N ) | , (1)where Ψ g ( x , x , ..., x N ) is the normalized ground statewave function of N-particle system. Even in integrablemodels, where Ψ g is known via Bethe Ansatz, calculationof the multiple integral over the restricted interval is stilla formidable task. A similar idea was first discuss inrandom matrix theory (RMT) [14], where the probabilitythat no eigenvalues fall within a certain interval of energyspectrum for different ensembles was studied [15].For integrable systems, the problem is often formu-lated in terms of spin-1 / l consecutive spin tobe “up” in the ground state of the chain. Via Jordan-Wigner transformation, such formulation is equivalent tothe absence of quasiparticles on l consecutive sites [16].In these cases EFP is found to be expressed in terms ofFredholm determinants [9, 16–19]. Even though EFP canbe related to known mathematical constructions, extract-ing its asymptotic behavior is still extremely challenging.The exact answers are known so far only in a handful ofisolated points in the parameter space [16, 20].This makes EFP an attractive playground for devel-opment of approximate asymptotic techniques. Moststudies have been focusing on the regime nR (cid:29) ● ● ● ● ● ● ● ● ● ● ● ● ●●●●●●●●●●●●●● Slope: 2Slope: 1 ��� ��� � � ������������ � / ξ � FIG. 1. Function f ( R/ξ ), Eq. (4), for a weakly repulsivebosons in log-log scale. The numerical results show a crossoverfor the exponent of EFP from linear (green dashed line) toquadratic (blue dashed line) and the red dashed line is fittedwith first few points. P E F P ( R ) (cid:28)
1. In this limit the problem may be studiedwithin the semiclassical instanton approximation, whereln P E F P ( R ) is associated with (twice) the classical ac-tion along a certain dynamical trajectory of the Euler-Lagrange equations [13]. Such classical problem needs tobe solved with the boundary conditions imposed on both“past” and “future” boundaries, which makes it not aneasy task, neither analytically nor numerically. Similarstructures are known in the theory of rare events in clas-sical stochastic systems [22–25].In this work we focus on EFP in the repulsive Lieb-Liniger (LL) model [26], of spinless bosons with the re-pulsive delta-potential in 1D. The ground (and excited)states of the model may be written through the Betheansatz [26] and its thermodynamic characteristics areknown exactly in terms of the microscopic parameters[26]. In particular, one may find the sound velocity v s and thus define the healing (or correlation) length as ξ = ( mv s ) − , (2)where m is a mass of bosonic particles. In the limitof impenetrable interactions (the Tonks-Girardeau limit a r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov [27]), ξ = ( πn ) − , and the model is equivalent to the freefermions. Their (squared) ground state wave functioncoincides with the joint probability distribution of eigen-values in the circular unitary random matrix ensemble[28]. The exact answer for the free fermion EFP is thusknown from RMT [29, 30] − ln P E F P ( R ) = 12 R ξ + 14 ln( R/ξ ) + O (1) . (3)Within the instanton approach the leading term here wasderived by A. Abanov [31] through a beautiful applica-tion of the complex valued functions theory. The onlytreatment away from the Tonks-Girardeau case, we arefamiliar with, is Ref. [32], which conjectured EFP in thelimit ξ (cid:28) R , see our discussion below.Our particular focus here is on the opposite limit ofthe weakly interacting bosons. A defining feature of thisregime is that the mean distance between the particles ismuch shorter than the correlation length, n − (cid:28) ξ . Asa result, there is a wide range n − < R < ξ , which wasnot previously discussed in the literature.Our main finding is that, through the entire range n − (cid:28) R , the logarithm of EFP may be expressed as: − ln P E F P = nξ × f (cid:18) Rξ (cid:19) , (4)where f ( r ) is a universal function, as long as n − (cid:28) ξ ,plotted in Fig. 1. Its asymptotic limits are: f ( r ) ≈ . r + 1 . r + O ( r ); r (cid:28) . (5)The leading term here is consistent with P E F P ≈ e − Rn ,which is the Poisson probability of finding the interval 2 R empty of independent (i.e. non-interacting) randomlyplaced particles with the mean density n . Indeed, thelimit R (cid:28) ξ is reached in the non-interacting case (i.e. ξ → ∞ ). The latter is characterized by the uniformground state | Ψ g | = L − N , where L = N/n is the systemsize. From Eq. (1): P E F P = (cid:0) L − RL (cid:1) N N →∞ → e − Rn .The other limit is: f ( r ) ≈ . r + 0 . r + O (ln r ); r (cid:29) . (6)Now the leading term corresponds to the GaussianEFP, P E F P ≈ exp {− . R n/ξ } . The Gaussian large- R asymptotic of the zero-temperature EFP may be ar-gued on the very general ground [31]. The specific co-efficient, found here for the weakly interacting limit, isnew. It is at odds with the conjecture of Ref. [32], − ln P E F P = 4(
R/ξ ) , which is parametrically inconsis-tent with our scaling, Eq. (4).The linear in r term in Eq. (6) is consistent with beingzero. Indeed, in all cases with short-range interactions,where exact results are available [16, 20], such term isindeed absent. We believe that this is a generic featureof short-range interacting system and provide a pertur-bative argument to that effect in section III. Curiously, the Calogero-Sutherland model with the inverse squarelong-range interactions exhibits a non-zero O ( r ) term(i.e. ∼ R term in the large R asymptotic of − ln P EFP )[14, 33]. Our numerical accuracy is not sufficient to es-tablish a coefficient of ln R term in Eq. (6).The paper is organized as follows. In Section II weformulate an instanton approach for calculation of EFPfor weakly interacting bosons. Numerical solution of cor-responding Euler-Lagrange equations, discussion of thelimiting cases and comparison with other works may befound in Section III. Appendix A is devoted to the freefermion limit as a test-drive of our numerical procedure. II. INSTANTON CALCULUS FOR WEAKLYINTERACTING BOSONS
Here we adopt the hydrodynamic instanton approachto emptiness formation, developed by A. Abanov [31, 34–37]. It is justified in the macroscopic emptiness regime, n − (cid:28) R , where EFP is exponentially small. It is thusexpected to be given by an optimal evolution trajectoryin the space of the system’s hydrodynamic degrees offreedom. In our case the latter are the local particledensity, ρ ( x, t ), and the local current, j ( x, t ). The twoare rigidly related by the continuity equation, ∂ t ρ + ∂ x j = 0 . (7)The classical action, that yields proper hydrodynamicequations as its extremal conditions, is given by [27] S [ ρ, j ] = (cid:90) (cid:90) dxdt (cid:104) mj ρ − V ( ρ ) (cid:105) ; (8) V ( ρ ) = c ρ − n ) + ( ∂ x ρ ) mρ , (9)where m is the mass of a particle. The Lagrangian inEq. (8) consists of the kinetic energy of the current alongwith the potential energy (equation of state) V ( ρ ). Forweakly interacting Bose liquid the latter is quadratic indensity deviations from its equilibrium value, n , with theinteraction parameter, c . The correlation length is givenby ξ = 1 /mv s = ( mnc ) − / . It satisfies the weak in-teraction criterion, n − (cid:28) ξ , as long as γ ≡ mc/n (cid:28) ρ and j , under the con-tinuity constraint, Eq. (7), yields classical Euler equationof the hydrodynamic flow (with the quantum pressurecontribution) [39]. Solutions of this equation do not leadto the formation of emptiness. The reason is that theemptiness is a large quantum fluctuation (similar to tun-neling), which is located in a classically forbidden regionof the phase space. The instanton approach is based onthe realization that the quantum transition amplitudeis given by the path integral (cid:82) D ρ D je iS [ ρ,j ] δ ( ∂ t ρ + ∂ x j ),with proper boundary conditions. The integration con-tours over the field variables may be then deformed intothe complex plane to pass through a classically forbiddenstationary configuration that reaches the required empti-ness. The probability of such rare event is P ∝ | e iS i nst | ,where the classical action along the instanton trajectory, S i nst , acquires a (positive) imaginary part.Before proceeding with the analytical continuation tothe complex plane, it is convenient to pass from a La-grangian formalism, Eq. (7), to the Hamiltonian one.To this end we introduce a new auxiliary field ∂ x θ ( x, t )and perform the Hubbard-Stratonovich transformationfor the kinetic energy term ∼ mj / (2 ρ ) in e iS [ ρ,j ] . Thisbrings terms − ρ ( ∂ x θ ) / (2 m ) + j∂ x θ to the action. Onemay then integrate by parts the last term (assuming peri-odic boundary conditions in the x direction) and employthe continuity relation to find: S [ ρ, θ ] = (cid:90) (cid:90) dxdt (cid:104) θ∂ t ρ − ρ ( ∂ x θ ) m − V ( ρ ) (cid:105) , (10)where we neglected the factor (cid:112) det[ ρ ] from the Hubbard-Stratonovich transformation, since it goes beyond the ac-curacy of the instanton approach. Notice that the fields ρ and θ are not subject to any constraints and play therole of the canonical pair.We are now on the position to perform the analyticalcontinuation. Following the standard treatment of tun-neling, it is achieved by the Wick rotation to imaginarytime t → − iτ . The resulting equations of motions maybe solved with real ρ and purely imaginary θ (the in-tegration contour in θ is deformed to pass through animaginary saddle point). It is convenient thus to redefine θ → iθ such that the saddle point solutions for both ρ and θ are real functions (in imaginary time), while thenew θ integration runs along the imaginary axis. Thecorresponding Eucledian action acquires the Hamilton-ain form S [ ρ, θ ] = i (cid:90) (cid:90) dxdτ [ θ∂ τ ρ − H ( ρ, θ )]; (11) H ( ρ, θ ) = ρ ( ∂ x θ ) m − c ρ − n ) − ( ∂ x ρ ) mρ . (12)Notice that the potential V ( ρ ) enters the effective Hamil-tonian, H ( ρ, θ ), with the “wrong” sign, mirroring the in-verted potential in the tunneling problem.The equations of motion, that follow from the action(11), are not the most convenient for the numerical solu-tion. To facilitate the latter, we found useful to performthe canonical transformation ( ρ, θ ) → ( Q, P ) to the newpair of the conjugated fields Q ( x, τ ) = (cid:112) ρ ( x, τ ) e − θ ( x,τ ) and P ( x, τ ) = (cid:112) ρ ( x, τ ) e θ ( x,τ ) , or conversely ρ = P Q and θ = ln( P/Q ). Substituting these into Eq. (11), one finds for the action S [ Q, P ] = i (cid:90) (cid:90) dxdτ [ P ∂ τ Q − H ( Q, P )]+ i (cid:90) dx P Q ln PQ (cid:12)(cid:12)(cid:12) τ = τ f τ = τ i , (13) H ( Q, P ) = − ∂ x P ∂ x Q m − c ( P Q − n ) , (14)where τ i ( f ) are initial(final) times of the optimal trajec-tory, discussed below.Variables Q, P may be considered as an analytical con-tinuation of the real-time degrees of freedom Q ↔ Ψ and P ↔ ¯Ψ. The first line of Eq. (13) is nothing but theanalytical continuation of the Gross-Pitaevskii (GP) ac-tion [40], ∼ | ∂ x Ψ | / m + c ( | Ψ | − n ) /
2. However, wouldwe start directly from the GP action, we would miss theboundary term, the second line in Eq. (13). This bound-ary term [41], i (cid:82) dxρ θ (cid:12)(cid:12) τ = τ f τ = τ i , does not alter the equationsof motion, but contributes to the instanton action. Itscontribution appears to be of the paramount importancein the regime n − < R < ξ . To the best of our knowledge,it was first introduced in the context of classical stochas-tic systems by Krapivsky, Meerson, and Sasorov [24], butwas not discussed so far in the quantum context.It is convenient to pass to dimensionless coordinatesand fields: x → ξx , τ → τ / ( nc ), P → √ nP , Q → √ nQ .In terms of them the Euclidean action takes the form S = inξ (cid:18)(cid:90) (cid:90) dxdτ (cid:20) P ∂ τ Q + ∂ x P ∂ x Q P Q − (cid:21) + 12 (cid:90) dx P Q ln PQ (cid:12)(cid:12)(cid:12) τ = τ i τ = τ f (cid:19) . (15)The corresponding equations of motion acquire the uni-versal parameter-free form: ∂ τ Q = 12 ∂ x Q − ( P Q − Q, (16) ∂ τ P = − ∂ x P + ( P Q − P. (17)These partial differential equations are known asAblowitz-Kaup-Newell-Segur (AKNS) system [42], whichis integrable with the inverse scattering method. Re-markably, exactly these equations appear in the studiesof rare events in Kardar-Parisi-Zhang classical stochasticequation [25, 43, 44].We can now specify the boundary conditions, appro-priate for the emptiness formation problem. We are look-ing for a transition amplitude from a uniform state at adistant past, τ i = −∞ , to a state with the emptiness,i.e. zero density for | x | < R , at the observation time, τ f = 0. This leads to the conditions: ρ ( x, −∞ ) = n and ρ ( | x | < R,
0) = 0. Outside of the interval x ∈ ( − R, R )at the observation time τ f = 0, the density is notfixed and is to be integrated out in the boundary term i (cid:82) dxρ θ (cid:12)(cid:12) τ = τ f . This fixes θ ( | x | > R,
0) = 0. In terms ofthe dimensionless coordinates and fields
Q, P , these readas:
P Q ( x, −∞ ) = 1; (18) P ( x,
0) = (cid:110) , | x | < R/ξ,Q ( x, , | x | > R/ξ. (19)The zero density constraint within the emptiness interval ρ = QP = 0, may be enforced by either P = 0, or Q = 0.This choice is arbitrary, since Q and P are interchange-able by a canonical transformation.The program now is as follows: one needs to solvethe stationary field equations (16) and (17), subject tothe boundary conditions (18) and (19). The resultinginstanton trajectory is to be substituted into the action(15) (including the boundary term (!)), resulting in theinstanton action S i nst ( R ). The semiclassical transitionamplitude is then given e iS i nst ( R ) , resulting finally in theEFP of the form − ln P E F P ( R ) = 2 (cid:61) S i nst ( R ) . (20)One notices then that the Eqs. (16), (17) are free fromany parameters, while the boundary conditions (18), (19)depend on the single parameter, R/ξ . The form of theaction (15) immediately implies the result, Eq. (4), where f ( R/ξ ) is twice the value of the double integral plus theboundary term, within the large round brackets on theright hand side of Eq. (15), evaluated along the optimaltrajectory.
III. RESULTS AND DISCUSSION
The equations of motion (16), (17) are of the AKNStype and thus are, in principle, integrable. However, theboundary conditions (18), (19) are not the initial valueproblem, which could be treated with the inverse scatter-ing approach. Although a lot is known about solutions ofEqs. (16), (17) (see, eg., discussion of their multi-solitonconfigurations in Ref. [25])), we were not able to find theiranalytical treatment, suitable for EFP setup, formulatedabove. We thus resorted to a numerical approach.We use Chernykh-Stepanov algorithm [23, 45] to solveequations of motion iteratively. The algorithm takes theadvantage of the diffusive character of Eq. (16) in the forward time and Eq. (17) in the backward time. The twoequations are successively evolved Q -forward, followedby P -backward in time to converge to desired solutions.The diffusive character of the equations provides stabilityfor such iteration scheme, making the ( Q, P ) variablesadvantageous over the ( ρ, θ ) pair. The results are stillpresented in terms of the more physically intuitive ( ρ, θ )degrees of freedom.At the initial backward-propagating step, we put Q ( x, τ ) = 1 and P ( x,
0) = θ ( | x | − R ), here θ ( x ) is theHeaviside step function. Then P ( x, τ ) is determined frombackward evolution of Eq. (17) up to a large negativetime τ = − T . Next we update the initial condition for Q from Q ( x, − T ) P ( x, − T ) = 1, cf. Eq. (18), and evolveEq. (16) forward in time up to τ = 0, with P ( x, τ ) foundin the first step. This way we obtain new Q ( x, τ ), whichwe use to update initial conditions for P at τ = 0, ac-cording to Eq. (19) and evolve P backward in time again, etc . We then evaluate the action (15), and check that itsvalue does not depend on the choice of the large negativeinitial time, − T .The evolution of density and (imaginary) phase areshown in Figs. 2 and 3 for R/ξ = 1 and 20. The corre-sponding f ( R/ξ ) is presented in Fig. 1. Its numerical fitsin the regimes
R/ξ (cid:28)
R/ξ (cid:29)
R/ξ (cid:28) ξ → ∞ is equivalent to c → Q and P become pure diffusion and anti-diffusion, while the dy-namical part of the action (15) is (cid:82)(cid:82) dxdτ P [ ∂ τ Q − ∂ x Q ],which is nullified on the equation of motion. The onlycontribution to the action is thus the boundary term (cid:82) dxP Q ln PQ (cid:12)(cid:12) τ i = −∞ = (cid:82) dxρ θ (cid:12)(cid:12) τ i = −∞ = − (cid:82) dx θ ( x, −∞ )(the final time, τ f = 0, does not contribute either in viewof Eq. (19)). It numerical evaluation gives 1 . R (cid:28) ξ .In the opposite R/ξ (cid:29)
1, it is useful to look at theaction (11) and rescale variables in an alterantive way: x → Rx , τ → Rmξτ , ρ → nρ and θ → ( R/ξ ) θ . TheEuclidean action takes the form S = inξ (cid:90)(cid:90) dxdτ (cid:104) R ξ (cid:18) θ∂ τ ρ − ρ ( ∂ x θ ) ρ − (cid:19) + ( ∂ x ρ ) ρ (cid:105) . (21)The first line here is the leading term, ∝ ( R/ξ ) , whichis given by the hydrodynamic action without quantumpressure. It corresponds to the leading Gaussian termin EFP, Eq. (6). One notices the absence of the linear,in R/ξ , term, consistent with our numerical finding. Thequantum pressure correction (the second line in Eq. (21),seems to be of the order O (1). This may be mislead-ing, since an attempt to treat the quantum pressure as aperturbative correction, seems to lead to logarithmicallydivergent integrals. This is probably the reason why theleading correction to the Gaussian result is of the orderof O (ln R/ξ ), cf. Eq. (6). However this type of termsexceeds the accuracy of the instanton approximation.The message from Eq. (21) is that the Gaussian partof EFP in the limit R (cid:29) ξ , can be found without thequantum pressure. This is in agreement with the successof such hydrodynamic theory [31] to obtain exact resultsvis-a-vis the Gaussian limit. Most notable case is the freefermion Tonks-Girardeau limit, cf. Eq. (3). We have nu-merically explored this known limit, see Appendix A, asa test-drive of our numerical procedure. We have foundcoefficient 0 . / τ = - � τ = - � τ = - ��� τ = - ��� τ = � - � - � - � � � � ������������������� � / ξ ρ � τ = - � τ = - � τ = - ��� τ = - ��� τ = - ���� - � - � - � � � � � - � - � - � - �� � / ξ θ τ = - � τ = - � τ = - ��� τ = - ��� τ = - ���� - � - � - � � � � � - �� - ����� � / ξ � FIG. 2. Time evolution of the density ρ ( x, τ ), imaginaryphase θ ( x, τ ) and velocity v ( x, τ ) for weakly interactingbosons with R/ξ = 1. The density evolves from the uni-form value, ρ = n , at large negative τ towards the emptinessof size 2 R at τ = 0. At τ → | x | < R and thus is not shown. The velocity v isrelated to spatial gradient of phase v = ∂ x θ/m. We conclude with a brief comparison with some previ-ously published results on EFP. The only analytic work,we know of, on EFP in 1D interacting boson model isa conjecture by Its, Korepin and Waldron [32]. In theweakly interacting limit, the leading term at large R isclaimed to be − ln P E F P = 4(
R/ξ ) . This is in a para-metric disagreement with our main result (4). On theother hand, calculations based on the bosonisation pro-cedure [31] are in a parametric agreement with Eq. (4). τ = - �� τ = - �� τ = - �� τ = - � τ = � - �� - �� - �� � �� �� �������� � / ξ ρ � τ = - �� τ = - �� τ = - �� τ = - � τ = - � - �� - �� - �� � �� �� �� - �� - �� - �� - �� - �� - �� � / ξ θ τ = - �� τ = - �� τ = - �� τ = - � τ = - � - �� - �� - �� � �� �� �� - � - ���� � / ξ � FIG. 3. Same as Fig. 2 for
R/ξ = 20.
Bosonization only allows for a treatment of a small sup-pression of density, rather than the emptiness. If onearbitrarily takes such “small” supression all the way tozero density, its probability is consistent with Eq. (4).There is also a number of results on EFP in antiferro-magnetic spin-1/2 XXZ chain with the Hamiltonian H = ∞ (cid:88) j = −∞ (cid:2) S xj S xj +1 + S yj S yj +1 + ∆ S zj S zj +1 (cid:3) , (22)where ∆ is the anisotropy in the z -direction. It isproposed in Ref. [46] that EFP in the gapless regime, − < ∆ ≤
1, is − ln P E F P ∼ Al + B ln l, (23)where A and B are constants depending on ∆ and l is thenumber of consecutive spin-polarized sites. An explicitexpression for the coefficient A was found to be A = ln (cid:104) Γ (1 / π √ π (cid:105) − (cid:90) ∞ dtt sinh ( tν )e − t cosh(2 tν ) sinh( t ) , (24)where parameter ν is defined through cos( πν ) = ∆. Thecorrespondence with the weakly interacting bosons maybe established for ∆ (cid:38) −
1, where the Luttinger parame-ter K = 1 / [2(1 − ν )] (cid:29)
1. Defining the correlation length(in lattice units) as ξ = K/ ( πn ), where the correspond-ing bosonic density is n = 1 / ξ (cid:29) − ln P E F P = n ξ l , (25)which is in parametric agreement with our result (4).To conclude, we have developed the instanton ap-proach that is capable to describe a complete crossover ofEFP from the Poisson to the Gaussian regime in the widerange of parameters, n − < R < ξ , available in weaklyinteracting bosonic 1D systems. Such systems are nowroutinely realized in cold atom experiments, where EFPmay be measured. IV. ACKNOWLEDGMENTS
We are indebted to A. Abanov, D. Gangardt and B.Meerson for valuable discussions. This work was sup-ported by NSF grant DMR-1608238.
Appendix A: Free Fermions Limit
In the free fermion limit the hydrodynamic potentialis given by V ( ρ ) = 2 (cid:90) πρπn dk π k m − µ ( ρ − n ) = π ( ρ − n ρ + 2 n )6 m , (A1)where µ = ( πn ) / m is the chemical potential. We sub-stitute it in the hydrodynamic action (11) to find S = i (cid:90) (cid:90) dxdτ (cid:104) θ∂ τ ρ − ρ ( ∂ x θ ) V ( ρ ) + ( ∂ x ρ ) ρ (cid:105) , (A2)where we kept the quantum pressure term from theweakly interacting case, since, as explained in SectionIII, it does not contribute in the large R limit anyways.We now proceed to the Q, P variables as above and thenmake them dimensionless, using ξ = 1 /πn appropriatefor the free fermions. The resulting equations of motionare ∂ τ Q = 12 ∂ x Q −
12 ( P Q − Q, (A3) ∂ τ P = − ∂ x P + 12 ( P Q − P, (A4) τ = - �� τ = - �� τ = - � τ = - � τ = � τ = - �� τ = - �� τ = - � τ = - � τ = � - �� - �� - �� � �� �� �������� � / ξ ρ � τ = - �� τ = - �� τ = - �� τ = - �� τ = - �� τ = - �� τ = - �� τ = - �� τ = - �� τ = - �� - �� - �� - �� � �� �� �� - �� - �� - � - � - � - �� � / ξ θ τ = - �� τ = - �� τ = - �� τ = - �� τ = - �� τ = - �� τ = - �� τ = - �� τ = - �� τ = - �� - �� - �� - �� � �� �� �� - ��� - ������������ � / ξ � FIG. 4. The upper panel is the time evolution of the density ρ for free fermions with R/ξ = 20. The middle and lowerone is the time evolution of phase θ and velocity v . The solidlines are numerical solutions of Eqs. (A3), (A4), using thealgorithm outlined in Section III, and dashed lines are theanalytical solutions of Ref. [31]. with the same boundary condition (18), (19) and themodified action S = inξ (cid:18)(cid:90) (cid:90) dxdτ (cid:104) P ∂ τ Q + ∂ x P ∂ x Q , (A5)+ ( P Q − P Q + 2) (cid:105) + 12 (cid:90) dx P Q ln PQ (cid:12)(cid:12)(cid:12) τ =0 τ = −∞ (cid:19) . The instanton solution is shown in Fig. 4, where we com-pare it to the analytical solution (without quantum pres-sure) of Ref. [31]. The corresponding optimal action isshown in Fig. 5. Its best fit is given − ln P E F P = 0 . R/ξ ) + O (ln R/ξ ) , (A6)where we used relation ξ = 1 /πn . This is in a very goodagreement with exact result for the free fermions, Eq. (3),[29, 30]. ● ● ● ● ● ● �� �� �� �� �� �� �� ���������������� � / ξ � � ���� ��� FIG. 5. 2 (cid:61) S i nst /nR vs. R/ξ for free fermions. The bluedashed line is a fit 0 . πR/ξ .[1] J. Esteve, J.-B. Trebbia, T. Schumm, A. Aspect, C. I.Westbrook, and I. Bouchoule, Physical review letters , 130403 (2006).[2] J. Armijo, T. Jacqmin, K. Kheruntsyan, and I. Bou-choule, Physical review letters , 230402 (2010).[3] T. Jacqmin, J. Armijo, T. Berrada, K. V. Kheruntsyan,and I. Bouchoule, Physical review letters , 230405(2011).[4] A. Del Campo, Physical Review A , 012113 (2011).[5] M. Pons, D. Sokolovski, and A. Del Campo, PhysicalReview A , 022107 (2012).[6] A. del Campo, New Journal of Physics , 015014 (2016).[7] M. Arzamasovs and D. M. Gangardt, Physical reviewletters , 120401 (2019).[8] H. Bethe, Zeitschrift f¨ur Physik , 205 (1931).[9] V. E. Korepin, A. G. Izergin, F. H. Essler, and D. B.Uglov, Physics Letters A , 182 (1994).[10] V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum inverse scattering method and correlation func-tions , Vol. 3 (Cambridge university press, 1997).[11] J. De Gier and V. Korepin, Journal of Physics A: Math-ematical and General , 8135 (2001).[12] H. E. Boos, V. E. Korepin, and F. A. Smirnov, NuclearPhysics B , 417 (2003).[13] H. Kleinert, Path integrals in quantum mechanics, statis-tics, polymer physics, and financial markets (World sci-entific, 2009).[14] M. L. Mehta,
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