The stationary state reached after the interaction quench of the Lieb-Liniger Bose gas from a non-interacting condensate is exactly known at the quench-action saddle point, yet it resists the generalized Gibbs ensemble (GGE) description that organizes other exactly solved quenches: its rapidity distribution decays algebraically, ρ(λ)∼λ−4\rho(λ)\simλ^{-4}, so the even local conserved charges Q2n=∑jλj2nQ_{2n}=\sum_j λ_j^{2n} diverge for n>1n>1 (the divergence of Q4Q_4 signals infinite energy fluctuations), and no explicit Gibbs-form density matrix with computable weights has been constructed. Consequently the dynamical correlation functions of this non-thermal equilibrium state (NTES) — which display a characteristic negative-energy dispersion branch originating from the algebraic tail — are governed by no known generalized detailed-balance relation, unlike thermal states whose line shapes are fixed by the Boltzmann factor. The problem is to construct the missing equilibrium description: an explicit GGE-type ensemble built from local, quasi-local, or suitably generalized conserved quantities with explicitly computable weights that reproduces the exact saddle point including its algebraic tail, and to derive the generalized detailed-balance relation governing the line shape of dynamical correlators in this non-thermal equilibrium state. (The BEC-to-BCS quench of the attractive Gaudin-Yang model shares the algebraic-tail feature, and a similar lower dispersion branch is conjectured there.)

Background

The model and the quench. The Lieb-Liniger model describes NN bosons on a ring of length LL interacting through repulsive point contacts of strength cc. It is integrable: it possesses an infinite family of mutually commuting conserved charges, and the Bethe ansatz (the exact solution scheme for such one-dimensional systems) labels every eigenstate by NN quasi-momenta, called rapidities, {λj}\{λ_j\}. The eigenvalues of the local conserved charges on such a state are Qn=∑jλjnQ_n=\sum_j λ_j^n, so Q2Q_2 is (up to a factor) the energy and Q0Q_0 the particle number. A quantum quench is the sudden change of a Hamiltonian parameter — here the interaction cc: the system is prepared in the ground state at c=0c=0, a Bose-Einstein condensate (BEC) in which all bosons occupy the zero-momentum state, and then evolves unitarily at finite cc.

Relaxation and the generalized Gibbs ensemble. In a generic (non-integrable) system, late-time local observables after a quench are predicted by a thermal Gibbs ensemble at the energy injected by the quench. In integrable systems the infinite conserved charges block thermalization: local observables equilibrate instead to a generalized Gibbs ensemble (GGE), a density matrix of the form Z−1exp⁡(−∑nμnQn)\mathcal{Z}^{-1}\exp(-\sum_n μ_n Q_n) whose weights μnμ_n are fixed by the initial state's charge expectation values. For many exactly solved quenches the GGE, combined with the thermodynamic Bethe ansatz (TBA — the nonlinear integral equations governing the filling function ϑ(λ)\vartheta(λ), the occupation fraction of rapidity configurations, and the rapidity or root density ρ(λ)\rho(λ)), provides a compact equilibrium description from which observables can be computed.

The obstruction for the BEC quench. For the BEC interaction quench of the Lieb-Liniger model the late-time stationary state — a non-thermal equilibrium state (NTES) — is in fact known exactly, but not through a GGE. The quench action method, a thermodynamically exact variational procedure built from the overlaps of the initial BEC with the Bethe eigenstates, locates the representative saddle-point eigenstate and yields a generalized TBA equation, first derived by De Nardis, Wouters, Brockmann and Caux,

ϵ(λ)=ln⁡ ⁣[λ2c2(λ2c2+14)]−h′−12πK∗ln⁡ ⁣[1+e−ϵ](λ),\epsilon(λ)=\ln\!\left[\frac{λ^2}{c^2}\left(\frac{λ^2}{c^2}+\frac{1}{4}\right)\right]-h'-\frac{1}{2π}K*\ln\!\left[1+e^{-\epsilon}\right](λ),

with kernel K(x,y)=2c/(c2+(x−y)2)K(x,y)=2c/(c^2+(x-y)^2) and a Lagrange multiplier h′h' fixing the density. Its solution has an algebraic tail, ρ(λ)∼λ−4\rho(λ)\simλ^{-4} for any finite cc (crossing to ∼λ−2\simλ^{-2} in the Tonks-Girardeau limit c→∞c\to∞ of impenetrable bosons), in contrast to the exponential decay of any thermal state. The tail makes the expectation values of Q2nQ_{2n} diverge for n>1n>1, so the GGE weights cannot be determined; moreover the BEC–Bethe-state overlap functional contains log⁡λ\logλ contributions with no expansion in powers of the charge eigenvalues. The stationary state is therefore exactly computable, but it has no known representation as an explicit equilibrium ensemble. Kormos, Shashi, Chou, Caux and Imambekov, who first identified the tail, proposed a regularized GGE built from the charges of a qq-boson lattice regularization of the continuum model; it reproduces the tail asymptotics only through Padé-type approximants, not as an exact construction.

Dynamical correlations and detailed balance. The one-body dynamical correlation function g1(k,ω)g_1(k,ω) is the Fourier transform of the field-field correlator ⟨Ψ†(x,t)Ψ(0,0)⟩⟨Ψ^†(x,t)Ψ(0,0)⟩ and is probed by momentum-resolved spectroscopy; inserting a complete set of eigenstates rewrites it as a sum over intermediate Bethe states weighted by squared matrix elements (form factors). In a thermal equilibrium state (TES) the line shape of such dynamical correlators is governed by the detailed-balance relation (DBR): spectral weights at excitation energies ±ω±ω are related by the Boltzmann factor e−βωe^{-βω}, which confines the weight mainly to positive energies. For GGE-type states a generalized DBR is expected; De Nardis and Panfil derived one for the density structure factor in the small-momentum limit, valid for any equilibrium state of GGE form. Chen and Cheng computed g1(k,ω)g_1(k,ω) in the BEC-quench NTES exactly, by combining the quench action with a systematic classification of the intermediate states, and found a qualitatively distinct spectral distribution: a second, lower dispersion branch at negative energies, whose microscopic origin is the algebraic tail — a small number of exterior particles at large rapidity that move essentially freely. A similar structure is expected in other quenches with algebraic stationary tails, notably the BEC-to-BCS quench of the attractive Gaudin-Yang model (two-component fermions with contact attraction) solved by Rylands, Calabrese and Bertini, where the rapidity distributions also decay as 1/k41/k^4. What remains missing is the compact equilibrium description of the BEC-quench NTES: an explicit ensemble, and the generalized detailed-balance relation that would play the role the Boltzmann factor plays for thermal states.

Problem Statement

Consider the interaction quench of the Lieb-Liniger model in which the non-interacting condensate ground state (a Bose-Einstein condensate at coupling c=0c=0) is released into a finite repulsive coupling cc. The late-time stationary state (the NTES) is exactly known at the quench-action saddle point: its rapidity density ρ(λ)\rho(λ) solves the generalized TBA equation above and decays algebraically, ρ(λ)∼λ−4\rho(λ)\simλ^{-4} (crossing to ∼λ−2\simλ^{-2} in the Tonks-Girardeau limit). Because of this tail, the even local conserved charges Q2n=∑jλj2nQ_{2n}=\sum_jλ_j^{2n} have divergent expectation values for n>1n>1 (the divergence of Q4Q_4 signals infinite energy fluctuations), and the BEC–Bethe-state overlap functional contains log⁡λ\logλ terms with no expansion in the charge eigenvalues, so the standard GGE logic — writing the state as Z−1exp⁡(−∑nμnQn)\mathcal{Z}^{-1}\exp(-\sum_nμ_nQ_n) with weights fixed by the initial state — is inapplicable. The problem is to supply the missing equilibrium description of this state, in two linked parts that jointly constitute one objective:

(1) Construct an explicit generalized Gibbs description of the post-BEC-quench stationary state: an explicit set of conserved quantities (local, quasi-local, or appropriately generalized, with their operator or eigenvalue-density form specified), defined independently of the saddle-point solution they are to reproduce and with finite expectation values on the stationary state, together with explicitly computable weights determined from the pre-quench (BEC) state, whose Gibbs density matrix reproduces the exact quench-action saddle point in the thermodynamic limit — the root density and filling function solving the generalized TBA equation, including the λ−4λ^{-4} tail, and the expectation values of all local operators with finite moments. A construction that takes the conserved quantity to be the generalized-TBA driving term itself (the logarithmic term ln⁡[λ2/c2(λ2/c2+1/4)]−h′\ln[λ^2/c^2(λ^2/c^2+1/4)]-h' of the equation above) or a generic functional of the stationary root density, with unit weight, does not satisfy this requirement.

(2) Derive the generalized detailed-balance relation for such states: the non-thermal analogue of the thermal detailed-balance relation, i.e. a compact relation determining how the spectral weight of dynamical correlation functions is distributed between positive and negative excitation energies at each momentum. For the one-body dynamical correlation function g1(k,ω)g_1(k,ω) in the BEC-quench NTES it must reproduce the two-branch structure — finite spectral weight on both the lower (negative-energy) and upper (positive-energy) branches, with the lower-branch weight exceeding the upper-branch weight — and should be expressible through the same data as the ensemble of part (1).

The two parts are connected: the explicit ensemble of part (1) is the natural domain of validity of, and the route to, the generalized detailed balance of part (2). Fitting a parametric expression to the already-computed g1(k,ω)g_1(k,ω) without deriving it from an equilibrium ensemble does not answer the problem.

References

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