Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/9910043 · Physical review. B, Condensed matter

Analytical results of the one-dimensional Hubbard model in the high-temperature limit

I. C. Charret, E. V. Corrêa Silva, S. M. de Souza, Onofre Rojas, M. T. Thomaz, A. T. Costa

Published Oct 29, 2001• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.64.195127

Related papers

One-Dimensional Hubbard Model in High Temperatures Through Many-Body Perturbation Theory

Mohamed Amine Tag, Abdelkader Hafdallah

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.

Acta Physica Polonica A•Feb 1, 2025•2 citations

Series expansion at high-temperature of the 2D Hubbard model via divided difference many-body perturbation theory

Chafia Bendjeroudib, Mohamed Amine Tag

Abstract We investigated the two-dimensional Hubbard model using many-body perturbation theory, focusing on the systematic computation of thermodynamic properties. The contributions of generalized Hugenholtz (Feynman) diagrams were evaluated symbolically and reformulated using divided differences, enabling efficient and precise calculations. We derived the coefficients of the grand potential expansion up to the eighth order in both the interaction potential (U-expansion) and the inverse temperature (β-expansion). Additionally, we presented results for magnetic susceptibility and heat capacity at half-filling, providing critical insights into the thermodynamic behavior of the model in this regime.

Physica Scripta•Apr 24, 2025•1 citation

Analytical Results for the Grand-Canonical Partition Function for Unidimensional Hubbard Model up to Order $β^5$

I. C. Charret, E. V. Correa Silva, S. M. de Souza +1 more

We calculate the exact analytical coefficients of the $β$ expansion of the grand-canonical partition function of the unidimensional Hubbard model up to order $β^5$, using an alternative method, based on properties of the Grassmann algebra. The results derived are non-perturbative and no restrictions on the set of parameters that characterize the model are required. By applying this method we obtain analytical results for the thermodynamical quantities, in the high-temeprature limit, for arbitrary density of electrons in the unidimensional chain.

arXiv:cond-mat/9705241•May 23, 1997•0 citations

Analytical results for the high-temperature expansion of the one-dimensional s=1XXZ model

Onofre Rojas, E. V. Corrêa Silva, W. A. Moura-Melo +2 more

Recently Rojas et al. [J. Math. Phys. 43, 1390 (2002)] obtained a closed analytical expression for the coefficients of arbitrary order in the cumulant expansion of a one-dimensional periodic chain model with nearest-neighbor interaction and spatial translation invariance; that approach can be applied equally well to both nonintegrable and exact integrable models. Here, we obtain the exact analytic expressions for the six lowest-order terms of the high-temperature expansion of the Helmholtz free energy per site of the nonintegrable one-dimensional spin-$1$ $\mathrm{XXZ}$ Heisenberg model. Our analytical results for the specific heat and the static magnetic susceptibility are compared, up to order ${\ensuremath{\beta}}^{6}$ for the ferromagnetic and antiferromagnetic phases, with the respective numerical results of a periodic chain with ten sites $(N=10)$ and numerical results in the literature [Bl\"ote, Physica B 79, 427 (1975)]. A very good agreement is obtained for both phases in the high-temperature regime.

Physical review. B, Condensed matter•Mar 31, 2003•4 citations

Thermodynamic Properties of the One-Dimensional Half-Filled-Band Hubbard Model

Hiroyuki Shiba, Philip A. Pincus

In order to investigate the thermal properties (specific heat, magnetic susceptibility, entropy, internal energy, and some correlation functions) of the one-dimensional half-filled-band Hubbard model, we have studied linear chains and rings containing two to six atoms, by performing machine calculations. Supplementing the low-temperature behavior obtained from the exact solution for the infinite chain by Lieb and Wu, our results should be suggestive of the properties of the infinite system throughout the entire temperature domain. It is shown that when the ratio of the correlation energy $U$ to the total width $\ensuremath{\Delta}$ of the band of single-particle excitations is larger than 1, the specific heat has two peaks. The high-temperature peak arises from the gradual metal-insulator transition (or the gradual formation of local moments), while the low-temperature peak is associated with the antiferromagnetic short-range ordering. When $\frac{U}{\ensuremath{\Delta}}$ becomes small, the two peaks merge into one. This picture is consistent with all other thermal properties, including correlation functions. The high-temperature properties are compared with the results predicted by Hubbard's approximate theory based on the truncation of the equations of motion of the Green's functions.

Physical review. B, Solid state•Mar 1, 1972•201 citations

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

Minoru Takahashi, Masahiro Shiroishi

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 ${\ensuremath{\beta}}^{6}.$

Physical review. B, Condensed matter•Apr 4, 2002•27 citations
We investigate the grand potential of the one-dimensional Hubbard model in the high-temperature limit, calculating the coefficients of the high-temperature expansion (\ensuremathβ(\ensuremath{\beta}(\ensuremathβ-expansion) of this function up to order \ensuremathβ4{\ensuremath{\beta}}^{4}\ensuremathβ4 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 \ensuremathβ\ensuremath{\beta}\ensuremathβ in the half-filled case, for several different values of U.
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.
Abstract We investigated the two-dimensional Hubbard model using many-body perturbation theory, focusing on the systematic computation of thermodynamic properties. The contributions of generalized Hugenholtz (Feynman) diagrams were evaluated symbolically and reformulated using divided differences, enabling efficient and precise calculations. We derived the coefficients of the grand potential expansion up to the eighth order in both the interaction potential (U-expansion) and the inverse temperature (β-expansion). Additionally, we presented results for magnetic susceptibility and heat capacity at half-filling, providing critical insights into the thermodynamic behavior of the model in this regime.
Analytical Results for the Grand-Canonical Partition Function for Unidimensional Hubbard Model up to Order β5β^5β5
We calculate the exact analytical coefficients of the βββ expansion of the grand-canonical partition function of the unidimensional Hubbard model up to order β5β^5β5, using an alternative method, based on properties of the Grassmann algebra. The results derived are non-perturbative and no restrictions on the set of parameters that characterize the model are required. By applying this method we obtain analytical results for the thermodynamical quantities, in the high-temeprature limit, for arbitrary density of electrons in the unidimensional chain.
Recently Rojas et al. [J. Math. Phys. 43, 1390 (2002)] obtained a closed analytical expression for the coefficients of arbitrary order in the cumulant expansion of a one-dimensional periodic chain model with nearest-neighbor interaction and spatial translation invariance; that approach can be applied equally well to both nonintegrable and exact integrable models. Here, we obtain the exact analytic expressions for the six lowest-order terms of the high-temperature expansion of the Helmholtz free energy per site of the nonintegrable one-dimensional spin-111 XXZ\mathrm{XXZ}XXZ Heisenberg model. Our analytical results for the specific heat and the static magnetic susceptibility are compared, up to order ${\ensuremath{\beta}}^{6}$ for the ferromagnetic and antiferromagnetic phases, with the respective numerical results of a periodic chain with ten sites (N=10)(N=10)(N=10) and numerical results in the literature [Bl\"ote, Physica B 79, 427 (1975)]. A very good agreement is obtained for both phases in the high-temperature regime.
In order to investigate the thermal properties (specific heat, magnetic susceptibility, entropy, internal energy, and some correlation functions) of the one-dimensional half-filled-band Hubbard model, we have studied linear chains and rings containing two to six atoms, by performing machine calculations. Supplementing the low-temperature behavior obtained from the exact solution for the infinite chain by Lieb and Wu, our results should be suggestive of the properties of the infinite system throughout the entire temperature domain. It is shown that when the ratio of the correlation energy UUU to the total width $\ensuremath{\Delta}$ of the band of single-particle excitations is larger than 1, the specific heat has two peaks. The high-temperature peak arises from the gradual metal-insulator transition (or the gradual formation of local moments), while the low-temperature peak is associated with the antiferromagnetic short-range ordering. When $\frac{U}{\ensuremath{\Delta}}$ becomes small, the two peaks merge into one. This picture is consistent with all other thermal properties, including correlation functions. The high-temperature properties are compared with the results predicted by Hubbard's approximate theory based on the truncation of the equations of motion of the Green's functions.