A Riemann-Hilbert formulation for the finite temperature Hubbard model

TL;DR

This paper reformulates the finite temperature Hubbard model's thermodynamic equations into a Riemann-Hilbert problem, simplifying their solution and linking different integrability approaches.

Contribution

It introduces a Riemann-Hilbert formulation for the Hubbard model's TBA equations, enabling easier numerical solutions and connecting to existing methods like Quantum Transfer Matrix.

Findings

Equivalence of TBA equations to a Riemann-Hilbert problem.

Reduction to three coupled nonlinear integral equations.

Facilitates numerical analysis of the Hubbard model.

Abstract

Inspired by recent results in the context of AdS/CFT integrability, we reconsider the Thermodynamic Bethe Ansatz equations describing the 1D fermionic Hubbard model at finite temperature. We prove that the infinite set of TBA equations are equivalent to a simple nonlinear Riemann-Hilbert problem for a finite number of unknown functions. The latter can be transformed into a set of three coupled nonlinear integral equations defined over a finite support, which can be easily solved numerically. We discuss the emergence of an exact Bethe Ansatz and the link between the TBA approach and the results by J\"uttner, Kl\"umper and Suzuki based on the Quantum Transfer Matrix method. We also comment on the analytic continuation mechanism leading to excited states and on the mirror equations describing the finite-size Hubbard model with twisted boundary conditions.