Mossel and Caux (2012) extended the thermodynamic Bethe ansatz (TBA) to generalized Gibbs ensembles (GGEs) and proved existence and uniqueness of the saddle-point solutions only for the repulsive Lieb-Liniger model with polynomial driving functions obeying simple coercivity conditions, explicitly leaving open the mathematical status of these generalized TBA equations for integrable models with string solutions, nested Bethe ansatz structures, attractive interactions, and for non-polynomial driving functions. The intervening physics literature has derived and used such equations for XXZ and sine-Gordon-type models, nested SU(N) chains, and overlap-based quench-action driving terms, but only formally or numerically; the only rigorous existence-uniqueness theorem for related equations (Hilfiker-Runkel, 2019) covers a disjoint ground-state class. The problem is to establish well-posedness - existence and uniqueness under explicit hypotheses, or provable ill-posedness (non-existence or non-uniqueness) - of the generalized TBA equations for these classes, thereby putting the equations that define generalized Gibbs macrostates in the models where they are actually used on a rigorous footing.

Background

A one-dimensional integrable quantum system possesses an infinite family of mutually commuting conserved charges Q^0,Q^1,Q^2=H^0,…\hat Q_0,\hat Q_1,\hat Q_2=\hat H_0,\dots (particle number, momentum, Hamiltonian, and higher charges), which strongly constrains its dynamics. The Bethe ansatz exactly diagonalizes such models: eigenstates are parametrized by a set of rapidities {λj}\{\lambda_j\}, and in the thermodynamic limit (volume L→∞L\to\infty at fixed particle density) sums over rapidities become continuous rapidity densities ρ(λ)\rho(\lambda). The thermodynamic Bethe ansatz (TBA) determines the equilibrium ρ(λ)\rho(\lambda) from a variational problem: the Yang-Yang functional combines the energy with the entropy of a Fermi-type gas of rapidities, and its saddle-point condition is a nonlinear integral equation for the pseudo-energy ε(λ)=ln⁡(ρh(λ)/ρ(λ))\varepsilon(\lambda)=\ln(\rho^{\rm h}(\lambda)/\rho(\lambda)), where ρh\rho^{\rm h} is the density of holes (unoccupied rapidity levels). For the repulsive Lieb-Liniger Bose gas (delta-function interacting bosons, coupling c>0c>0) it reads

ε(λ)+∫−∞∞dν2π 2c(λ−ν)2+c2 ln⁡ ⁣(1+e−ε(ν))=ε0(λ),\varepsilon(\lambda)+\int_{-\infty}^{\infty}\frac{d\nu}{2\pi}\,\frac{2c}{(\lambda-\nu)^2+c^2}\,\ln\!\left(1+e^{-\varepsilon(\nu)}\right)=\varepsilon_0(\lambda),

with driving term ε0(λ)=βλ2−βμ\varepsilon_0(\lambda)=\beta\lambda^{2}-\beta\mu at inverse temperature β\beta and chemical potential μ\mu, in units where the single-particle energy is λ2\lambda^{2} (in the rapidity normalization of Mossel and Caux used throughout); Yang and Yang proved that this equation has a unique solution obtainable by iteration, a result later recast in the language of large deviations by Dorlas, Lewis and Pule.

When an isolated integrable system is driven out of equilibrium by a quantum quench (a sudden change of a Hamiltonian parameter), it fails to thermalize in the usual sense: the conserved charges retain the memory of the initial condition. The stationary state is expected to be described by a generalized Gibbs ensemble (GGE), a density matrix exp⁡(−∑nβnQ^n)\exp(-\sum_n\beta_n\hat Q_n) whose generalized inverse temperatures {βn}\{\beta_n\} are fixed by the initial expectations of all charges. For the Lieb-Liniger gas the charge eigenvalues on a Bethe state are the power sums Qn=∑jλjnQ_n=\sum_j\lambda_j^n, so the GGE simply replaces the quadratic driving term of the thermal TBA by a polynomial ε0(λ)=∑nβnλn\varepsilon_0(\lambda)=\sum_n\beta_n\lambda^n. Mossel and Caux proved that the resulting generalized TBA (gTBA) equation still has a unique solution obtainable by iteration, provided ε0\varepsilon_0 is bounded from below and ε0(λ)→+∞\varepsilon_0(\lambda)\to+\infty as λ→±∞\lambda\to\pm\infty. They also argued that the gTBA is the only practical route to GGE averages in the interacting gas, since the higher charges are not normal-ordered (i.e. not arranged with all creation operators to the left of the annihilation operators) operators whose expectation values are directly computable.

Most integrable models lie outside this proof. In the gapped regime of the spin-1/2 XXZ Heisenberg chain (the anisotropic nearest-neighbour spin-1/2 chain) and in the sine-Gordon field theory (a relativistic integrable field theory in one spatial dimension whose excitations include towers of bound states called breathers), the Bethe roots organize into strings - regular arrays of complex rapidities interpreted as bound states - producing (typically infinitely many) coupled TBA equations. Models with several particle species or internal degrees of freedom require the nested Bethe ansatz, with several types of rapidities and matrix-valued scattering. In the attractive regime of the Lieb-Liniger model (c<0c<0) the spectrum itself is built from strings and the thermodynamic kernels become singular. Finally, the quench action - a variational formulation of post-quench dynamics in which the steady state is the saddle point of an overlap-based action - produces gTBA equations whose driving function is the extensive logarithm of the overlap between the initial state and Bethe eigenstates; this driving is generically not a polynomial, and in the paradigmatic BEC-to-Lieb-Liniger quench it has a logarithmic singularity at the origin.

The only rigorous existence and uniqueness theory for TBA-type equations beyond the Lieb-Liniger thermal case is due to Hilfiker and Runkel, who treated Y-systems - functional relations Yn(x+i)Yn(x−i)=∏m(1+Ym(x))GnmY_n(x+i)Y_n(x-i)=\prod_{m}(1+Y_m(x))^{G_{nm}} encoding the analytic structure of TBA - with a finite coupling matrix GG that is non-negative, irreducible, real-diagonalizable and of spectral radius below 2, under the assumption that the YY-functions have no zeros in a strip around the real axis. This covers ground-state problems of certain massive integrable quantum field theories, but not the finite-entropy (filling) equations of GGE saddle points, nor the string, nested, or attractive structures listed above. Whether the gTBA equations, as used throughout the literature on integrable quenches, are well-posed remains mathematically open.

Problem Statement

Mossel and Caux proved, for the repulsive Lieb-Liniger model (c>0c>0), that the generalized TBA saddle-point equation

ε(λ)+∫−∞∞dν2π 2c(λ−ν)2+c2 ln⁡ ⁣(1+e−ε(ν))=ε0(λ)\varepsilon(\lambda)+\int_{-\infty}^{\infty}\frac{d\nu}{2\pi}\,\frac{2c}{(\lambda-\nu)^2+c^2}\,\ln\!\left(1+e^{-\varepsilon(\nu)}\right)=\varepsilon_0(\lambda)

with polynomial driving function ε0(λ)=∑nβnλn\varepsilon_0(\lambda)=\sum_n\beta_n\lambda^n has a unique solution, obtainable by iteration, whenever ε0\varepsilon_0 is bounded from below and ε0(λ)→+∞\varepsilon_0(\lambda)\to+\infty as λ→±∞\lambda\to\pm\infty. The problem is to extend this well-posedness theory - or rigorously establish its failure - to the generalized TBA equations of integrable models outside this case, namely:

(a) models whose thermodynamic Bethe ansatz involves string structures, i.e. families of bound-state (complex-rapidity) Bethe roots, such as the gapped spin-1/2 XXZ Heisenberg chain or the sine-Gordon model with its breather towers, whose TBA consist of (generally infinitely many) coupled nonlinear integral equations;

(b) models solved by the nested Bethe ansatz, with several rapidity species and matrix-valued scattering;

(c) attractive regimes, e.g. the attractive Lieb-Liniger model c<0c<0, where the root structure is built from strings and the effective kernels and sources become singular;

(d) driving functions ε0\varepsilon_0 that are not polynomials - in particular functions with isolated singularities, which arise in practice from quench-action overlap data.

Scope of the four classes: (a)-(c) are feature-defined families of models; the models named as examples (introduced by "such as" or "e.g.") are illustrative members rather than exhaustive representatives, so settling a class requires a result governing every model with the stated feature. Class (d) is a property of the driving function rather than a model family, and it cuts across (a)-(c), so settling (d) requires the non-polynomial-driving result for each model class among (a)-(c) to which it is applied. The classes are not disjoint: the attractive regime (c) is itself a string case, and a non-polynomial driving function (d) can be imposed on any of (a)-(c).

A complete answer should, for each class treated, state the precise integral equations (kernels, source terms, filling/normalization conditions, function space), prove existence and uniqueness of the solution ε(λ)\varepsilon(\lambda) (equivalently of the rapidity density ρ(λ)\rho(\lambda)) under explicit hypotheses generalizing the bounded-below and coercivity conditions of the repulsive case, and characterize the convergence of the iterative scheme used in practice; alternatively, it should prove that for identified classes within (a)-(d) the equations are ill-posed - admitting no solution, or several inequivalent solutions - thereby exhibiting genuine non-uniqueness of the generalized Gibbs macrostate. The single scientific objective uniting these cases is a mathematically controlled generalized TBA: the saddle-point equations that define generalized Gibbs ensembles in the models where the field actually uses them. Partial progress - a clean well-posedness theorem for any one of the classes (a)-(d) with sharp hypotheses, or a rigorous ill-posedness result for a specific case - constitutes a genuine partial answer; the problem is fully resolved when the rigorous status (well-posed with explicit conditions, or provably ill-posed) is settled for all four classes.

References

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