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Physical review. B, Condensed matter

Thermodynamic Bethe ansatz equations of one-dimensional Hubbard model and high-temperature expansion

Minoru Takahashi, Masahiro Shiroishi

Published Apr 4, 2002• 27 citations
Publisher

Abstract

A numerical method to calculate thermodynmic Bethe ansatz equations is proposed based on Newton's method. Thermodynamic quantities of the one-dimensional Hubbard model are numerically calculated and compared with high-temperature expansion and numerical results of the quantum transfer matrix method by J\"uttner, Kl\"umper, and Suzuki. The coincidence is surprisingly good. We obtain the high-temperature expansion of the grand potential up to ${\ensuremath{\beta}}^{6}.$

DOI
10.1103/physrevb.65.165104

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A numerical method to calculate thermodynmic Bethe ansatz equations is proposed based on Newton's method. Thermodynamic quantities of the one-dimensional Hubbard model are numerically calculated and compared with high-temperature expansion and numerical results of the quantum transfer matrix method by J"uttner, Kl"umper, and Suzuki. The coincidence is surprisingly good. We obtain the high-temperature expansion of the grand potential up to \ensuremathβ6.{\ensuremath{\beta}}^{6}.\ensuremathβ6.
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 β6β^6β6.
The ``string'' structure of one-dimensional Hubbard model eigenstates is studied numerically and is used to suggest the validity of thermodynamic Bethe-ansatz equations (TBAE). We furthermore solve TBAE in a strong-coupling expansion series form and obtain the thermodynamic potential which agrees with the known high-temperature expansion series. We also calculate various thermodynamic quantities using our solution and provide some additional features of the strongly correlated one-dimensional Hubbard model.
We investigate the grand potential of the one-dimensional Hubbard model in the high-temperature limit, calculating the coefficients of the high-temperature expansion $(\ensuremath{\beta}$-expansion) of this function up to order ${\ensuremath{\beta}}^{4}$ by an alternative method. The results derived are analytical and do not involve any perturbation expansion in the hopping constant, being valid for arbitrary density of electrons in the one-dimensional model. In the half-filled case, we compare our analytical results for the specific heat and the magnetic susceptibility, in the high-temperature limit, with the ones obtained by Beni et al. [Phys. Rev. B 8, 3329 (1973)] and Takahashi's integral equations, showing that the latter result does not take into account the complete energy spectrum of the one-dimensional Hubbard model. The exact integral solution by J\"uttner et al. [Nucl. Phys. B 522, 471 (1998)] is applied to the determination of the range of validity of our expansion in $\ensuremath{\beta}$ in the half-filled case, for several different values of U.
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.
We present symbolic algorithms designed to investigate the perturbative expansions of the d-Hubbard model. These methods are part of recent developments in many-body perturbation theory that aim to reformulate Feynman diagrams in terms of divided differences. The direct application of this technique to the one-dimensional Hubbard model yields the coefficients of the grand potential up to the sixth order, expressed in terms of both the interacting potential (U expansions) and high-temperature expansions (β expansions). A key feature of this approach is the ability to merge the β expansions to any desired order. To verify our analytical results, we compare the derived magnetic susceptibility with the exact solution of the quantum transfer matrix method in the half-filled case.
The phase diagram and thermodynamic properties of the Hubbard model in one dimension are calculated numerically within the generalized self-consistent approach over a wide range of temperature T, magnetic field h, interaction strength U/t and electron concentration n. The temperature variation of the transverse magnetization, double occupied sites and chemical potential at various n, U/t and h is also analyzed. The wave vector q for the magnetic order parameter in the vicinity of the half-filling in the metallic phase shows a crossover from incommensurate magnetic phase (0<q<π) into the state with strong antiferromagnetic correlations (q=π) as temperature is increased. Overall our numerical results in the approximate theory at finite temperatures are in good quantitative and qualitative agreement with the Bethe-ansatz results.