The Yang–Yang equation is the nonlinear integral equation whose unique solution determines the thermodynamics of the repulsive Lieb–Liniger Bose gas. Kozlowski (Lett. Math. Phys. 104, 55–74 (2014)) proved that this solution admits an all-order low-temperature asymptotic expansion whose leading term is the zero-temperature dressed energy, the proof resting on the positivity of the Lieb kernel K(λ)=2c/(λ2+c2)K(\lambda)=2c/(\lambda^2+c^2). The paper's conclusion poses the problems of extending the method to the analogous nonlinear integral equations governing the thermodynamics of other integrable models — the XXZ spin-1/2 chain and the Hubbard model being the canonical examples — and of treating rigorously the large-volume (1/L1/L) corrections to ground-state and low-lying excitation energies. Later work by the same school rigorously analysed the quantum-transfer-matrix nonlinear integral equation of the massless XXZ chain (0≤Δ<10\le\Delta<1, generic anisotropy), its application to the free energy being conditional on two conjectures; this leaves open the massive and negative-anisotropy regimes, the Hubbard model, the conjecture-free free-energy asymptotics, and — entirely — the rigorous 1/L1/L corrections.

Background

The Lieb–Liniger model describes one-dimensional bosons interacting through the repulsive two-body delta potential c δ(x)c\,\delta(x) with c>0c>0. Its exact spectrum follows from the Bethe ansatz (a reduction of the many-body eigenvalue problem to algebraic equations for the particles' rapidities), and in the thermodynamic limit the free energy is expressed through the unique solution ε(λ)\varepsilon(\lambda) of the nonlinear integral equation introduced by Yang and Yang (1969) — the Yang–Yang equation

ε(λ)=λ2−h−T2π∫RK(λ−μ) ln⁡ ⁣(1+e−ε(μ)/T) dμ,K(λ)=2cλ2+c2,\varepsilon(\lambda)=\lambda^2-h-\frac{T}{2\pi}\int_{\mathbb{R}}K(\lambda-\mu)\,\ln\!\big(1+\mathrm{e}^{-\varepsilon(\mu)/T}\big)\,\mathrm{d}\mu,\qquad K(\lambda)=\frac{2c}{\lambda^2+c^2},

where hh is the chemical potential and TT the temperature. As T→0T\to0, ε\varepsilon converges to the dressed energy ε0\varepsilon_0 — the energy of a particle-like excitation once interactions with the Fermi sea are included — which vanishes at the edges ±q\pm q of the Fermi zone. The original derivation was formal; a fully rigorous derivation of the thermodynamics via large deviations was given by Dorlas, Lewis and Pulé (1989).

Kozlowski (2014) proved that the unique solution of the Yang–Yang equation admits a complete asymptotic expansion in even powers of TT, with remainders uniform in a complex strip around the real axis, and simultaneously established the unique solvability of the coupled system (ε0,q)(\varepsilon_0,q) governing the zero-temperature limit. Earlier low-temperature calculations for thermodynamic Bethe ansatz (TBA) type equations (e.g. Bortz 2007) provided consistent algorithms but no existence proof. The engine of the 2014 proof is the positivity of the Lieb kernel K(λ)=2c/(λ2+c2)K(\lambda)=2c/(\lambda^2+c^2): it yields invertibility of the associated Fredholm integral operator (the integral operator with kernel KK) on L2L^2 and bounds that remain stable as T→0T\to0.

The same TBA structure recurs across one-dimensional integrable models, but the kernels lose positivity. For the XXZ spin-1/2 chain — the anisotropic Heisenberg chain with anisotropy Δ=cos⁡ζ\Delta=\cos\zeta — the finite-temperature thermodynamics is described either by the hierarchies of coupled nonlinear integral equations of Gaudin and Takahashi, or, in the quantum transfer matrix (QTM) approach (an auxiliary transfer matrix arising from a Trotter decomposition of the partition function), by a single nonlinear integral equation of Destri–de Vega type whose kernel has poles at λ=±iζ\lambda=\pm\mathrm{i}\zeta off the real axis; in the massive regime Δ>1\Delta>1, strings (bound-state patterns of Bethe roots, i.e. of the rapidities solving the Bethe equations) appear. The one-dimensional Hubbard model is governed by Takahashi's infinite coupled set of such equations, with separate charge and spin Fermi zones. None of the positivity structure of the Lieb kernel survives verbatim, so the extension of the 2014 theorem to these lattice-model analogues of the Yang–Yang equation is a genuinely open mathematical problem.

A second, closely related direction concerns finite size. At zero temperature on a chain of LL sites, conformal field theory (the continuum field theory describing the critical points of lattice models) predicts E0(L)=e∞L−πvccent/(6L)+o(1/L)E_0(L)=e_\infty L-\pi v c_{\rm cent}/(6L)+o(1/L), where vv is the Fermi velocity and ccentc_{\rm cent} the central charge (the number characterizing the universality class of the critical theory), together with analogous 1/L1/L corrections for low-lying excited states. Bethe-ansatz derivations of these corrections (de Vega–Woynarovich 1985; Destri–de Vega 1995; the recent distribution approach of Granet–Jacobsen–Saleur 2018) remain heuristic, and the same 2014 conclusion named their rigorous treatment as open. The common objective of both directions is to bring the thermodynamics of integrable models beyond the repulsive Bose gas under the same rigorous control.

Problem Statement

Extend the rigorous low-temperature theory of the Yang–Yang equation, established for the repulsive Lieb–Liniger gas by Kozlowski (2014), to the nonlinear integral equations governing the thermodynamics of further Bethe-ansatz-integrable models, and make the large-volume behaviour of these models rigorous. The source conclusion poses these as two independent expectations, and a rigorous resolution of either direction, in the following precise senses, counts as a solution of this statement.

For the low-temperature direction: for at least one model class not yet treated — for instance the massive antiferromagnetic (Δ>1\Delta>1) or negative-anisotropy (−1<Δ<0-1<\Delta<0) regimes of the XXZ spin-1/2 chain, the one-dimensional Hubbard model (Takahashi's coupled equations or their quantum-transfer-matrix analogue), or another integrable model whose thermodynamics is governed by an analogous Yang–Yang-type nonlinear integral equation — establish:

(1) for one and the same model class: (i) existence and uniqueness of the solution of the model's thermodynamic nonlinear integral equation(s) at low temperature; (ii) an asymptotic expansion in powers of TT to all orders whose leading term is the T=0T=0 dressed energy, with remainder bounds uniform on explicitly specified domains — the direct analogue of the 2014 main theorem, including the solvability of the zero-temperature dressed-energy–Fermi-boundary system; and (iii) in the same rigorous sense, the low-temperature asymptotics of the per-site free energy without unresolved conjectures (in the quantum transfer matrix setting, eigenvalue dominance and commutativity of the thermodynamic and Trotter limits are so far only conjectured at low temperature). All three items (i)–(iii) are jointly required, for that same model class, for direction (1) to count as resolved.

For the large-volume direction, independently of item (1), establish:

(2) for a Bethe-ansatz-integrable model not yet rigorously treated — for instance the XXZ spin-1/2 chain away from the free-fermion point, or the one-dimensional Hubbard model at non-zero UU — the leading O(1/L)O(1/L) corrections to the ground-state energy and to the energies of the low-lying excited states at T=0T=0, with explicit coefficients (for instance the central-charge term −πvccent/(6L)-\pi v c_{\rm cent}/(6L)) derived rather than assumed. The low-lying excited states counted here are those whose 1/L1/L coefficients are governed by the same conformal field theory as the ground state. Direction (2) is resolved only if the derived coefficients are not already rigorously established, for that model and regime, in the cited literature.

The two directions jointly form the programme of rigorous low-temperature and finite-size control of Yang–Yang-type thermodynamics beyond the positive-kernel Bose gas; because the source poses them as separate expectations, neither direction is conditioned on the other. Completing the already-advanced massless XXZ case (0≤Δ<10\le\Delta<1, generic ζ\zeta) by removing its two conjectures equally counts as a solution. A submission must state precisely which model, parameter regime, and statements it covers, so that a domain expert can judge the increment over the published partial results.

References

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