Completing the Riemann-Hilbert formulation of the finite-temperature one-dimensional Hubbard model: low temperatures, strong coupling, and excited thermodynamic branches

The one-dimensional fermionic Hubbard model at finite temperature admits two classical exact descriptions: Takahashi's thermodynamic Bethe ansatz (TBA), an infinite tower of coupled integral equations, and the quantum transfer matrix (QTM) nonlinear integral equations (NLIEs) of Jüttner, Klümper and Suzuki. Cavaglià, Cornagliotto, Mattelliano and Tateo (JHEP 06 (2015) 015) recast the TBA as a nonlinear Riemann-Hilbert problem for four P-functions on a two-sheeted Riemann surface, reduced it to three coupled NLIEs on a finite support, and solved them numerically in a high-temperature window under the assumption that the ground-state P-functions carry no zeros on the first sheet. That assumption is known to fail at low temperature and for excited states, where Bethe-root zeros migrate to the first sheet, and the iteration empirically breaks down below a threshold temperature; the excited thermodynamic branches that encode finite-temperature correlation lengths were left unsystematized. The open problem is to construct the modified resolvent parametrization and NLIEs that remain valid with first-sheet zeros, characterize the breakdown boundary in the temperature-coupling-field-filling parameter space, classify the excited free-energy branches and connect their finite-temperature branching structure to the zero-temperature phase diagram including the Mott-insulating phase, and validate the extended equations against the independent TBA/QTM results.

Background

The one-dimensional Hubbard model describes spin-1/2 fermions hopping between neighboring lattice sites with a repulsive on-site interaction U (written u = U/4 in the conventions of the 2015 paper); it is the paradigmatic exactly solvable model of strongly correlated electrons. In one dimension its spectrum was found by Lieb and Wu through the Bethe ansatz, an infinite set of algebraic equations for charge momenta and spin rapidities. At finite temperature T, chemical potential $\mu$ and magnetic field B, the thermodynamics in the thermodynamic limit follows from the thermodynamic Bethe ansatz (TBA) of Takahashi (1972): an infinite set of coupled nonlinear integral equations for the so-called dressed energies (interaction-renormalized excitation energies) of the charge rapidities and of families of complex spin rapidities called strings, whose regular arrangement in the complex plane is assumed by the string hypothesis. A second, equivalent route rewrites the quantum partition function as a two-dimensional classical model on an $L \times \bar{N}$ lattice via a Suzuki-Trotter decomposition (a path-integral slicing whose transverse size N-bar, the Trotter number, is sent to infinity with the transfer-matrix parameter fixed at $1/(T\bar{N})$, denoted $u$ in the plain typeface of the 2015 paper and distinct from the interaction coupling $u = U/4$); the column-to-column transfer matrix of this classical model is the quantum transfer matrix (QTM), and the free energy is the largest QTM eigenvalue. Jüttner, Klümper and Suzuki (1998) reduced the QTM Bethe equations to three coupled nonlinear integral equations (NLIEs) of Klümper-Batchelor-Pearce and Destri-de Vega type. The two routes were known to agree numerically, but no direct analytic bridge between the TBA and QTM equations existed for the Hubbard model. Inspired by AdS/CFT integrability, where the complicated TBA/Y-system (functional relations among transfer-matrix eigenvalues) of planar N=4 super-Yang-Mills theory was compressed into the Quantum Spectral Curve (QSC), a nonlinear Riemann-Hilbert problem for a finite set of Q-functions, Cavaglià, Cornagliotto, Mattelliano and Tateo (2015) proved that Takahashi's TBA is equivalent to a closed system of functional relations for four functions $P^{H/V}_\pm(z)$ (with auxiliary functions $F^{ab}$), analytic on a two-sheeted Riemann surface in the spectral variable z (parametrizing momenta through the Hubbard dispersion), with prescribed large-z asymptotics; the Y-functions of the TBA are ratios of T-functions built from the P's. Parametrizing the P-functions through two density functions $\rho_H$, $\rho_V$ by resolvents (Cauchy transforms of the densities along the cut) reduces the problem to three coupled NLIEs on the finite interval z in (-1,1), which a Chebyshev-discretized iteration solves in seconds. The Bethe roots of an exact Bethe ansatz, which is precisely the infinite-Trotter-number limit of the QTM Bethe equations, appear as zeros of the P-functions; the ground state at high temperature has all of them on the second sheet. The catch: the uniqueness of the resolvent parametrization is proved under the assumption that the P-functions have no zeros on the first sheet, and the numerical checks were confined to the region |B| < 1, |\mu| < 1, 0 < u < 2, T $\geq$ 1, while the iteration breaks down below a threshold temperature that increases with u, consistent with the second-to-first-sheet zero flow established in the QTM analysis. Excited thermodynamic branches of the free energy matter physically because their gaps above the ground branch are the inverse correlation lengths of local operators at finite temperature (in QTM language, spacings between the leading eigenvalues). Before 2015 only isolated such branches had been computed, all through the QTM: the spin correlation length at half filling by Tsunetsugu, free energy and correlation lengths by Klümper and Bariev, and the one-particle Green's function at half filling by Umeno, Shiroishi and Klümper. The general mechanism of excited branches, established in the 1990s TBA literature, is analytic continuation in the external parameters: zeros driven across an integration contour contribute residue terms to the equations, and genuine branch points of the free energy occur only when a pair of zeros pinches the contour from opposite sides. In the free-fermion limit u = 0 of the Hubbard model the branch points can be written in closed form and coincide with the boundaries of the four zero-temperature phases; at u > 0 the T = 0 phase diagram gains a fifth, Mott-insulating phase (interaction-driven charge gap at half filling), and how its presence reshapes the finite-temperature branch structure is unknown.

Problem Statement

Within the Riemann-Hilbert/P-function reformulation of Takahashi's finite-temperature TBA for the one-dimensional Hubbard model (coupling u, temperature T, magnetic field B, chemical potential $\mu$), the ground state is described by three coupled nonlinear integral equations on the finite support z in (-1,1) whose derivation assumes that the P-functions $P^{H/V}_\pm$ carry no zeros on the first sheet of their two-sheeted Riemann surface; this holds in the verified region |B| < 1, |\mu| < 1, 0 < u < 2, T $\geq$ 1, while the iterative scheme empirically breaks down below a threshold temperature that increases with u, consistently with the known flow of Bethe-root zeros from the second to the first sheet at low temperature, and even the ground state is expected to carry first-sheet zeros near T = 0. The problem is to complete this formulation, which is a question about the analytic structure of this reformulation and not about the computability of Hubbard thermodynamics at low temperature (already available through the older quantum-transfer-matrix NLIEs). Specifically: (i) construct the modified resolvent parametrization and the corresponding finite-support NLIEs that remain valid when finitely many P-function zeros lie on the first sheet, and characterize the boundary in the (T, u, B, $\mu$) space at which the original no-first-sheet-zero equations and their iteration cease to be valid; (ii) turn the analytic-continuation mechanism into a systematic treatment of excited thermodynamic branches of the free energy: the explicit residue modifications of the equations produced by energy-carrying Bethe roots crossing the integration contours, a classification of the resulting excited branches (which control finite-temperature correlation lengths of local operators), and the location of the genuine branch points where pairs of zeros pinch the contour, generalizing the closed-form free-fermion loci $\lambda_B\lambda_\mu^{-1} e^{\pm 2/T} = -1$ and $\lambda_B\lambda_\mu e^{\pm 2/T} = -1$ (fugacities $\lambda_B = e^{B/T}$, $\lambda_\mu = e^{\mu/T}$), with the finite-temperature branching structure related to the zero-temperature phase diagram including the fifth, Mott-insulating phase present at u > 0; (iii) demonstrate the extended equations numerically at low temperature and for representative excited states, with agreement in the overlap regions with the independent TBA and quantum-transfer-matrix results and correct limiting behaviour (u $\to$ 0 free fermions, T $\to$ 0 Luttinger-liquid and Mott asymptotics). A resolution consists of all the components listed above under (i), (ii) and (iii): the modified equations together with their derivation and the characterization of the validity boundary, the residue modifications and the classification of excited branches including the branch-point loci, the generalization of the closed-form free-fermion loci and the connection of the branching structure to the zero-temperature phase diagram, and reproducible numerics establishing the stated agreements and the limiting behaviours.

References

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