Physical review. B, Condensed matter

Gap problem for the Hubbard chain at finite temperatures: Solution via the Bethe-ansatz method

Arkady L. Kholodenko, Adolph L. Beyerlein

Published Jul 1, 1987• 1 citation

Abstract

The gap equation at zero temperature for the half-filled Hubbard chain obtained by Lieb and Wu is rederived with use of the formalism of Yang and Yang and then the results are extended to the case of finite temperatures. For the values of the coupling constant c in the range 0<c\ensuremath{\le}1.5 the gap \ensuremath{\Delta}(T) exhibits BCS-like behavior \ensuremath{\Delta}(T)=${\ensuremath{\Delta}}_{0}$(1-T/${T}_{c}$${)}^{1/2}$ while for the larger values of c, the data are well approximated by the equation \ensuremath{\Delta}(T)=${\ensuremath{\Delta}}_{0}$(1-T/${T}_{c}$) with ${\ensuremath{\Delta}}_{0}$ being the zero-temperature result of Lieb and Wu for the gap and ${T}_{c}$ some critical transition temperature.

DOI: 10.1103/physrevb.36.409

Related papers

The half-filled Hubbard chain in the Composite Operator Method: A comparison with Bethe Ansatz

M. M Sánchez, A Avella, F Mancini

arXiv:cond-mat/9807368Europhysics Letters (EPL)Nov 1, 19989 citations

Application of the finite-cell-scaling method to the one-dimensional Hubbard model

B. Fourcade, G. Spronken

Physical review. B, Condensed matterMay 1, 198413 citations

From a Hubbard dimer to a Hubbard model in d dimensions: a path integral solution

Arkady L. Kholodenko

Journal of Physics Condensed MatterJan 9, 19892 citations

Existence of Solutions to the Bethe Ansatz Equations for the 1D Hubbard Model: Finite Lattice and Thermodynamic Limit

Pedro S. Goldbaum

Jan 1, 20055 citations

Central charge for the half-filled Hubbard chain

Arkady L. Kholodenko

Zero-temperature finite-size Bethe ansatz calculations of the central charge for the Hubbard chain are compared with the finite-temperature Bethe ansatz calculations of the same charge.

Journal of Physics A Mathematical and GeneralDec 1, 19872 citations

The half-filled Hubbard chain in the Composite Operator Method: A comparison with Bethe Ansatz

M. M. Sánchez, A. Avella, F. Mancini

Jan 1, 19988 citations

Physical review. B, Condensed matter

Gap problem for the Hubbard chain at finite temperatures: Solution via the Bethe-ansatz method

Arkady L. Kholodenko, Adolph L. Beyerlein

Published Jul 1, 1987• 1 citation

Publisher

Abstract

{\ensuremath{\Delta}}_{0}

(1-T/

{T}_{c}$${)}^{1/2}

while for the larger values of c, the data are well approximated by the equation \ensuremath{\Delta}(T)=

{\ensuremath{\Delta}}_{0}

(1-T/

{T}_{c}

) with

{\ensuremath{\Delta}}_{0}

being the zero-temperature result of Lieb and Wu for the gap and

{T}_{c}

some critical transition temperature.

DOI: 10.1103/physrevb.36.409

Related papers

The half-filled Hubbard chain in the Composite Operator Method: A comparison with Bethe Ansatz

M. M Sánchez, A Avella, F Mancini

The one-dimensional Hubbard model at half-filling is studied in the framework of the Composite Operator Method using a static approximation. A solution characterized by strong antiferromagnetic correlations and a gap for any non-zero on-site interaction U is found. The corresponding ground-state energy, double occupancy and specific heat are in excellent agreement with those obtained within the Bethe Ansatz. These results show that the Composite Operator Method is an appropriate framework for the half-filled Hubbard chain and can be applied to evaluate properties, like the correlation functions, which cannot be obtained by means of the Bethe Ansatz, except for some limiting cases.The one-dimensional Hubbard model at half-filling is studied in the framework of the Composite Operator Method using a static approximation. A solution characterized by strong antiferromagnetic correlations and a gap for any non-zero on-site interaction U is found. The corresponding ground-state energy, double occupancy and specific heat are in excellent agreement with those obtained within the Bethe Ansatz. These results show that the Composite Operator Method is an appropriate framework for the half-filled Hubbard chain and can be applied to evaluate properties, like the correlation functions, which cannot be obtained by means of the Bethe Ansatz, except for some limiting cases.

arXiv:cond-mat/9807368Europhysics Letters (EPL)Nov 1, 19989 citations

Application of the finite-cell-scaling method to the one-dimensional Hubbard model

B. Fourcade, G. Spronken

The finite-cell-scaling method is applied to the one-dimensional Hubbard model. In the half-filled-band case and at zero temperature, the numerical results obtained for the ground-state energy differ from the exact solution by approximately 1%. For the range $1<\frac{U}{t}<5$ , where $U$ is the Coulomb interaction and $t$ is the transfer integral, the value of the normalized gap $\frac{\ensuremath{\Delta}}{t}$ agrees well with the exact result while it differs significantly for $0.2<\frac{U}{t}<0.9$ . For the latter range, we show that the gap varies as ${(\frac{U}{t})}^{\frac{1}{2}}\mathrm{exp}[{(a+\frac{\mathrm{bU}}{t})}^{\ensuremath{-}1}]$, where $a$ and $b$ are negative constants which depend on the size of the system, instead of ${(\frac{U}{t})}^{\frac{1}{2}}\mathrm{exp}(\frac{\ensuremath{-}2\ensuremath{\pi}t}{U})$ as in the exact solution.The finite-cell-scaling method is applied to the one-dimensional Hubbard model. In the half-filled-band case and at zero temperature, the numerical results obtained for the ground-state energy differ from the exact solution by approximately 1%. For the range $1<\frac{U}{t}<5$, where $U$ is the Coulomb interaction and $t$ is the transfer integral, the value of the normalized gap $\frac{\ensuremath{\Delta}}{t}$ agrees well with the exact result while it differs significantly for $0.2<\frac{U}{t}<0.9$. For the latter range, we show that the gap varies as ${(\frac{U}{t})}^{\frac{1}{2}}\mathrm{exp}[{(a+\frac{\mathrm{bU}}{t})}^{\ensuremath{-}1}]$, where $a$ and $b$ are negative constants which depend on the size of the system, instead of ${(\frac{U}{t})}^{\frac{1}{2}}\mathrm{exp}(\frac{\ensuremath{-}2\ensuremath{\pi}t}{U})$ as in the exact solution.

Physical review. B, Condensed matterMay 1, 198413 citations

From a Hubbard dimer to a Hubbard model in d dimensions: a path integral solution

Arkady L. Kholodenko

A new analytical method is proposed for solving the Hubbard model in d-dimensions. The method permits, in principle, the study of the Hubbard model at any temperature and any band filling. The accuracy of the method is tested against known Bethe ansatz results of Lieb and Wu (1968), and against known weak- and strong-coupling expansion results. The one-dimensional path integral solution obtained for the ground-state (half-filled band case) is in excellent qualitative agreement with Bethe ansatz results for all couplings and exactly reproduces weak- and strong-coupling expansion results.A new analytical method is proposed for solving the Hubbard model in d-dimensions. The method permits, in principle, the study of the Hubbard model at any temperature and any band filling. The accuracy of the method is tested against known Bethe ansatz results of Lieb and Wu (1968), and against known weak- and strong-coupling expansion results. The one-dimensional path integral solution obtained for the ground-state (half-filled band case) is in excellent qualitative agreement with Bethe ansatz results for all couplings and exactly reproduces weak- and strong-coupling expansion results.

Journal of Physics Condensed MatterJan 9, 19892 citations

Existence of Solutions to the Bethe Ansatz Equations for the 1D Hubbard Model: Finite Lattice and Thermodynamic Limit

Pedro S. Goldbaum

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.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.

Jan 1, 20055 citations

Central charge for the half-filled Hubbard chain

Arkady L. Kholodenko

Zero-temperature finite-size Bethe ansatz calculations of the central charge for the Hubbard chain are compared with the finite-temperature Bethe ansatz calculations of the same charge.

Journal of Physics A Mathematical and GeneralDec 1, 19872 citations

The half-filled Hubbard chain in the Composite Operator Method: A comparison with Bethe Ansatz

M. M. Sánchez, A. Avella, F. Mancini

PACS. 71.10.−w – Theories and models of many electron systems. PACS. 71.10.Fd – Lattice fermion models. PACS. 71.15.Th – Other methods. Abstract. – The one-dimensional Hubbard model at half-filling is studied in the framework of the Composite Operator Method using a static approximation. A solution characterized by strong antiferromagnetic correlations and a gap for any nonzero on-site interaction U is found. The corresponding ground-state energy, double occupancy and specific heat are in excellent agreement with those obtained within the Bethe Ansatz. These results show that the Composite Operator Method is an appropriate framework for the half-filled Hubbard chain and can be applied to evaluate properties, like the correlation functions, which cannot be obtained by means of the Bethe Ansatz, except for some limiting cases. Introduction. – New materials whose physics is dominated by electron correlations in narrow energy bands are actually a challenge for solid state physicists. The treatment of such systems is not trivial due to the competition between itinerant and localized behaviour of the electrons in these bands. The Hubbard Hamiltonian [1] is regarded as the simplest modelPACS. 71.10.−w – Theories and models of many electron systems. PACS. 71.10.Fd – Lattice fermion models. PACS. 71.15.Th – Other methods. Abstract. – The one-dimensional Hubbard model at half-filling is studied in the framework of the Composite Operator Method using a static approximation. A solution characterized by strong antiferromagnetic correlations and a gap for any nonzero on-site interaction U is found. The corresponding ground-state energy, double occupancy and specific heat are in excellent agreement with those obtained within the Bethe Ansatz. These results show that the Composite Operator Method is an appropriate framework for the half-filled Hubbard chain and can be applied to evaluate properties, like the correlation functions, which cannot be obtained by means of the Bethe Ansatz, except for some limiting cases. Introduction. – New materials whose physics is dominated by electron correlations in narrow energy bands are actually a challenge for solid state physicists. The treatment of such systems is not trivial due to the competition between itinerant and localized behaviour of the electrons in these bands. The Hubbard Hamiltonian [1] is regarded as the simplest model

Jan 1, 19988 citations

{\ensuremath{\Delta}}_{0}

(1-T/

{T}_{c}$${)}^{1/2}

while for the larger values of c, the data are well approximated by the equation \ensuremath{\Delta}(T)=

{\ensuremath{\Delta}}_{0}

(1-T/

{T}_{c}

) with

{\ensuremath{\Delta}}_{0}

being the zero-temperature result of Lieb and Wu for the gap and

{T}_{c}

some critical transition temperature.

1<\frac{U}{t}<5

, where

U

is the Coulomb interaction and

t

is the transfer integral, the value of the normalized gap $\frac{\ensuremath{\Delta}}{t}$ agrees well with the exact result while it differs significantly for

0.2<\frac{U}{t}<0.9

. For the latter range, we show that the gap varies as ${(\frac{U}{t})}^{\frac{1}{2}}\mathrm{exp}[{(a+\frac{\mathrm{bU}}{t})}^{\ensuremath{-}1}]$, where

a

and

b

are negative constants which depend on the size of the system, instead of ${(\frac{U}{t})}^{\frac{1}{2}}\mathrm{exp}(\frac{\ensuremath{-}2\ensuremath{\pi}t}{U})$ as in the exact solution.