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 Juttner, Klumper 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.
Highlights
The Hubbard model [1] arises as an approximate description of correlated electrons in narrow-band materials
The Hubbard model has attracted the attention of high energy physicists due to its multiple connections [2,3,4,5,6,7,8] with the integrable spin chains emerging in the context of N =4 Super Yang-Mills theory [9,10,11]
The purpose of this article is to add a little piece to the jigsaw, by recasting the Thermodynamic Bethe Ansatz equations of Takahashi as a nonlinear Riemann-Hilbert problem, reminiscent of the Quantum Spectral Curve formulation recently obtained for the study of anomalous dimensions in AdS/CFT [31]
Summary
The Hubbard model [1] arises as an approximate description of correlated electrons in narrow-band materials. The TBA was introduced in the AdS/CFT context [27,28,29] to overcome the so-called wrapping problem [30] affecting the Beisert-Staudacher equations This very complicated set of TBA equations was recast into the greatly simplified form of a nonlinear matrix Riemann-Hilbert problem: the Quantum Spectral Curve or Pμ-system [31, 32]. (1.14), (1.15) coincide with the infinite Trotter number limit of the exact Bethe Ansatz diagonalising the Quantum Transfer Matrix Since the latter equations are the starting point for the derivation of the NLIEs of [15], our analysis provides the missing link between the two different approaches to the Hubbard model thermodynamics. Appendix D contains a dictionary linking this work to the paper [15]
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