Global solution of the infinite thermodynamic Bethe-ansatz string hierarchy of the one-dimensional attractive Hubbard model

The exact finite-temperature thermodynamics of the one-dimensional (1D) attractive Hubbard model is formally encoded in an infinite hierarchy of coupled nonlinear thermodynamic-Bethe-ansatz (TBA) integral equations that link real quasimomenta to infinitely many bound-state strings: spin $\Lambda$ strings and composite $k$ – $\Lambda$ strings of every length. In 2018 Cheng, Yu, Batchelor and Guan derived the closed equation of state, scaling functions, additivity rules and Wilson-ratio diagnostics of the FFLO-like phase only under a controlled truncation — strong attraction, low density, essentially length-1 strings — stating that the whole set of TBA equations admits no analytical or numerical solution as a whole. No work in the literature to date solves the full hierarchy for the attractive lattice model: later studies treat the repulsive regime, use mean-field or finite-size numerics, address the continuum Yang–Gaudin gas, or compute finite-particle dynamics without the string hypothesis. The problem therefore remains open: obtain the exact global thermodynamics of the attractive 1D Hubbard model from the complete string hierarchy — in closed form or as an exact convergent numerical solution with controlled string-length truncation — at arbitrary coupling, density, polarization and temperature, together with the quantitative role of longer $k$ – $\Lambda$ strings and the fate of the additivity rules and Wilson-ratio diagnostics outside the dilute strong-coupling regime.

Background

The one-dimensional Hubbard model describes spin-1/2 fermions hopping on a lattice with an on-site interaction $u$ (attractive for $u<0$ ); in the conventions of the source paper $H=-\sum_{j,a}(c^\dagger_{j,a}c_{j+1,a}+c^\dagger_{j+1,a}c_{j,a})+u\sum_j(1-2n_{j,\uparrow})(1-2n_{j,\downarrow})$ , with chemical potential $\mu$ and magnetic field $B$ coupling to the density and spin. It is a paradigm of exactly solvable quantum many-body physics: Lieb and Wu solved it by the nested Bethe ansatz, in which every eigenstate is labeled by quasimomenta $k$ (charge degrees of freedom) and spin rapidities $\Lambda$ . At finite temperature the thermodynamics follows from the thermodynamic Bethe ansatz (TBA), which converts the Bethe-ansatz quantization conditions into coupled nonlinear integral equations for dressed energies — the excitation energies of each species after interactions are folded in — whose solution yields the free energy and hence the full equation of state.

For the attractive regime, Takahashi's 1972 construction of these equations rests on the string hypothesis: the complex rapidities organize into bound-state families called strings — spin $\Lambda$ strings (a string of $n$ bound spin rapidities, a magnon-like spin bound state) and, characteristic of attraction, $k$ – $\Lambda$ strings (a bound state of $2m$ fermions, $m$ spin-up and $m$ spin-down, whose quasimomenta and rapidities form a complex string; the length-1 $k$ – $\Lambda$ string is a Cooper pair). The resulting TBA is an infinite hierarchy: one dressed energy $\epsilon'_n(\Lambda)$ per string length $n$ for each family, all coupled to the real- $k$ dressed energy $\epsilon_u(k)$ through convolution kernels. The free energy per site is obtained only after this whole coupled system is solved, so the exact thermodynamics at general parameters is tantamount to solving the hierarchy.

In 2018 Cheng, Yu, Batchelor and Guan used this framework to derive the universal thermodynamics of the attractive model at strong attraction ( $|u|\gg 1$ ) and low density ( $n_c\ll 1$ ), where only the real $k$ , the length-1 spin wave and the length-1 $k$ – $\Lambda$ string (the bound pair) contribute. In the partially polarized phase — a one-dimensional analogue of the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) paired state of a magnetized superconductor, with pairing at finite momentum set by the Fermi-surface mismatch — they obtained a closed equation of state from which follow effective chemical potentials of two free fluids (bound pairs and unpaired fermions), additivity rules (the compressibility and susceptibility decompose as sums of two ideal-gas contributions), dimensionless Wilson-ratio diagnostics (ratios such as $\chi T/C_v$ that stay constant across a phase and so map out phase boundaries), and quantum-critical scaling functions with dynamic exponent $z=2$ and correlation-length exponent $\nu=1/2$ .

The authors emphasize that these closed forms exist only because of the controlled truncation. In their words, the analysis of the coupled nonlinear integral equations 'provides a formidable challenge': 'it is challenging to solve infinitely many coupled nonlinear integral TBA equations, i.e., the desired analytical or numerical solution is not achievable by solving the whole set of TBA equations', an obstacle they say prevents a microscopic understanding of the Cooper-pairing mechanism and the many-body phenomena of the model. Outside the dilute strong-attraction corner — at weak or moderate attraction, at higher density up to half filling, or at temperatures where longer $k$ – $\Lambda$ strings ( $n\ge 2$ ) and higher spin strings are thermally populated — no exact global equation of state is known.

Problem Statement

Solve the complete infinite thermodynamic Bethe-ansatz string hierarchy of the one-dimensional attractive Hubbard model and obtain from it the exact global finite-temperature thermodynamics.

Concretely: determine the free energy per site $f(\mu,B,T,u)$ — with chemical potential $\mu$ , magnetic field $B$ , temperature $T$ and attraction strength $u<0$ — together with the density, magnetization, effective chemical potentials of the bound-pair and unpaired-fermion fluids (defined at general parameters as the dressed-energy thresholds of the bound-pair (length-1 $k$ – $\Lambda$ string) and unpaired-fermion (real- $k$ ) sectors, which reduce in the dilute strong-attraction limit to the two-fluid quantities of the Background), compressibility, spin susceptibility and specific heat, as the solution of the full Takahashi hierarchy retaining all length- $n$ spin $\Lambda$ strings and all length- $n$ $k$ – $\Lambda$ strings, valid at arbitrary coupling down to weak attraction, arbitrary density up to half filling (densities above half filling follow by the particle-hole symmetry of the model), arbitrary spin polarization, and arbitrary temperature. Any admissible solution must make quantitative the contributions of the $n\ge 2$ strings that the 2018 strong-coupling treatment discarded, and must settle whether the additivity rules, the Wilson-ratio constants and the quantum-critical scaling functions (there with $z=2$ , $\nu=1/2$ ) survive outside the dilute strong-attraction regime, including how the finite-temperature crossovers of the FFLO-like phase are modified at general coupling and density.

An admissible solution takes one of two forms: (a) a closed-form solution of the hierarchy — an explicit global equation of state or finite-temperature generating function from which the above observables follow by differentiation; or (b) an exact convergent numerical scheme for the full hierarchy with controlled truncation error in the string length, demonstrated across the whole $(\mu,B,T,u)$ parameter space including the regimes outside $|u|\gg 1$ and $n_c\ll 1$ . Either form must reproduce, as limiting cases, the strong-coupling/low-density equation of state, component chemical potentials, scaling functions and additivity rules of Cheng, Yu, Batchelor and Guan (2018); the exact zero-temperature phase boundaries (including the Wiener-Hopf results for the critical fields); and the Shiba-transform relation to the repulsive-model free energy.

References

Song Cheng, Yi-Cong Yu, M. T. Batchelor, and Xi-Wen Guan, "Universal thermodynamics of the one-dimensional attractive Hubbard model", Phys. Rev. B 97, 125145 (2018). DOI: 10.1103/PhysRevB.97.125145; arXiv:1708.07784; https://doi.org/10.1103/PhysRevB.97.125145 (source paper)
Minoru Takahashi, "One-Dimensional Hubbard Model at Finite Temperature", Prog. Theor. Phys. 47, 69 (1972). DOI: 10.1143/ptp.47.69; https://doi.org/10.1143/ptp.47.69
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