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arXiv:1501.04651 · Journal of High Energy Physics

A Riemann-Hilbert formulation for the finite temperature Hubbard model

Andrea Cavaglià, Martina Cornagliotto, Massimo Mattelliano, Roberto Tateo

Published Jun 1, 2015• 23 citations

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 Jüttner, Klümper 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.

DOI: 10.1007/jhep06(2015)015

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•Jan 1, 2005•5 citations

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 Jüttner, Klümper 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.

New numerical method to calculate thermodynmic Bethe ansatz equations is proposed based on Newton's method. Thermodynamic quantities of one-dimensional Hubbard model is numerically calculated and compared with high temperature expansion and numerical results of quantum transfer matrix method by Jüttner, Klümper and Suzuki. The coincidence is surprisingly good. We get high-temperature expansion of grand potential up to

β^6

.

The excited state thermodynamic Bethe ansatz (TBA) equations for the spinless Fermion model are presented by the quantum transfer matrix (QTM) approach. We introduce a more general family called T-functions and explore functional relations among them (T-system) and their certain combinations (Y-system). {}From their analytical property, we derive a closed set of non-linear integral equations which characterize the correlation length of $$ at any finite temperatures. Solving these equations numerically, we explicitly determine the correlation length, which coincides with earlier results with high accuracy.

The thermodynamic Bethe ansatz (TBA) and the excited state TBA equations for an integrable spin chain related to the Lie superalgebra osp (1|2) are proposed by the quantum transfer matrix (QTM) method. We introduce the fusion hierarchy of the QTM and derive the functional relations among them (T-system) and their certain combinations (Y-system). Their analytical property leads to the nonlinear integral equations which describe the free energy and the correlation length at any finite temperatures. With regard to the free energy, they coincide with the TBA equation based on the string hypothesis.

Abstract We construct an integrable Hubbard model with impurity site containing spin and charge degrees of freedom. The Bethe ansatz equations for the Hamiltonian are derived and two alternative sets of equations for the thermodynamical properties. For this study, the thermodynamical Bethe ansatz (TBA) and the quantum transfer matrix (QTM) approach are used. The latter approach allows for a consistent treatment by use of a finite set of non-linear integral equations. In both cases, TBA and QTM, the contribution of the impurity to the thermodynamical potential is given by integral expressions.

Abstract. In this work, we present a proof of the existence of real and ordered solutions to the generalized Bethe Ansatz equations for the one dimensional Hubbard model on a finite lattice, with periodic boundary conditions. The existence of a continuous set of solutions extending from any U> 0 to U = ∞ is also shown. This continuity property, when combined with the proof that the wavefunction obtained with the generalized Bethe Ansatz is normalizable, is relevant to the question of whether or not the solution gives us the ground state of the finite system, as suggested by Lieb and Wu. Lastly, for the absolute ground state at half-filling, we show that the solution converges to a distribution in the thermodynamic limit. This limit distribution satisfies the integral equations that led to the well known solution of the 1D Hubbard model. PACS numbers: 71.10.Fd, 71.10.Hf, 71.30.+h. 1.