Physical review. B, Condensed matter

Exact elementary excitations with a complex wave vector in the one-dimensional Hubbard model

S. T. Chui

Published Apr 15, 1980• 8 citations

Abstract

Computer calculations of finite Hubbard chains indicate two peaks for the specific heat for a large Coulomb interaction $U$ for band fillings less than 1/2. Even though the origin of this is physically obvious, no elementary excitations with a gap energy $U$ have been explicitly demonstrated from the exact solution of the Hubbard model. We here demonstrate that there exists elementary excitation with a complex wave vector $k$ for the $N\ensuremath{-}1$ spin-up case. These excitations with complex wave vectors have never been discussed previously. These results suggest similar excitations for the case with more general spin arrangements and may account for the specific-heat results.

DOI: 10.1103/physrevb.21.3269

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Excitation spectrum of the one-dimensional Hubbard model

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

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Physical review. B, Condensed matter

Exact elementary excitations with a complex wave vector in the one-dimensional Hubbard model

S. T. Chui

Published Apr 15, 1980• 8 citations

Publisher

Abstract

Computer calculations of finite Hubbard chains indicate two peaks for the specific heat for a large Coulomb interaction

U

for band fillings less than 1/2. Even though the origin of this is physically obvious, no elementary excitations with a gap energy

U

have been explicitly demonstrated from the exact solution of the Hubbard model. We here demonstrate that there exists elementary excitation with a complex wave vector

k

for the

N\ensuremath{-}1

spin-up case. These excitations with complex wave vectors have never been discussed previously. These results suggest similar excitations for the case with more general spin arrangements and may account for the specific-heat results.

DOI: 10.1103/physrevb.21.3269

Related papers

Excitation spectrum of the one-dimensional Hubbard model

Cornelius F. Coll

We have extended calculations of the zero-temperature excitation spectrum of the one-dimensional Hubbard model to the case where the number of electrons is less than the number of sites in the chain. The results are computed as a function of the ratio $\frac{U}{t}$ , where $U$ represents the on-site Coulomb repulsion and $t$ is the transfer integral, assumed to be nonzero only for nearest neighbors. Exact calculations are made for the energy and momentum of excitations having single-particle character. Unlike the situation for the half-filled band, we find no gap in the excitation spectrum. We have also considered excitations of the spin-wave type. These are shown to vary linearly with momentum for small momentum. The group velocity for small momentum is found to be inversely proportional to the magnetic susceptibility.We have extended calculations of the zero-temperature excitation spectrum of the one-dimensional Hubbard model to the case where the number of electrons is less than the number of sites in the chain. The results are computed as a function of the ratio $\frac{U}{t}$, where $U$ represents the on-site Coulomb repulsion and $t$ is the transfer integral, assumed to be nonzero only for nearest neighbors. Exact calculations are made for the energy and momentum of excitations having single-particle character. Unlike the situation for the half-filled band, we find no gap in the excitation spectrum. We have also considered excitations of the spin-wave type. These are shown to vary linearly with momentum for small momentum. The group velocity for small momentum is found to be inversely proportional to the magnetic susceptibility.

Physical review. B, Solid stateMar 1, 1974166 citations

Spin and charge excitations of the degenerate Hubbard model in one dimension

P. Schlottmann

The degenerate Hubbard chain with N internal degrees of freedom is integrable if site occupations of more than two electrons are excluded. The ground state of the model has several unusual properties; e.g., the magnetic susceptibility has logarithmic singularities as the field H\ensuremath{\rightarrow}0 and also the specific-heat coefficient is singular. An extensive discussion of ground-state properties as a function of U and the band filling is presented. The elemental charge and spin excitations are derived for arbitrary U and band filling. If, on average, there is exactly one electron per site as a function of U, a qualitative change in the low-energy charge excitations is found at a critical value ${\mathit{U}}_{\mathit{c}}$ . This change is probably associated with a Mott metal-insulator transition ( ${\mathit{U}}_{\mathit{c}}$ =0 for N=2). The Fermi velocity is finite for U ${\mathit{U}}_{\mathit{c}}$ but vanishes for U> ${\mathit{U}}_{\mathit{c}}$ (for one electron per site). The spin-wave velocity is inversely proportional to the magnetic susceptibility for all U and band fillings. There are N-1 elemental spin-wave branches. For arbitrary band filling there are then N-1 soft modes at different finite momenta, all related to the Fermi surface. For exactly one electron per site, the spin-excitation spectrum can be mapped onto the one of the SU(N) Heisenberg chain. The implications of the hole states on superconductivity are also discussed.The degenerate Hubbard chain with N internal degrees of freedom is integrable if site occupations of more than two electrons are excluded. The ground state of the model has several unusual properties; e.g., the magnetic susceptibility has logarithmic singularities as the field H\ensuremath{\rightarrow}0 and also the specific-heat coefficient is singular. An extensive discussion of ground-state properties as a function of U and the band filling is presented. The elemental charge and spin excitations are derived for arbitrary U and band filling. If, on average, there is exactly one electron per site as a function of U, a qualitative change in the low-energy charge excitations is found at a critical value ${\mathit{U}}_{\mathit{c}}$. This change is probably associated with a Mott metal-insulator transition (${\mathit{U}}_{\mathit{c}}$=0 for N=2). The Fermi velocity is finite for U${\mathit{U}}_{\mathit{c}}$ but vanishes for U>${\mathit{U}}_{\mathit{c}}$ (for one electron per site). The spin-wave velocity is inversely proportional to the magnetic susceptibility for all U and band fillings. There are N-1 elemental spin-wave branches. For arbitrary band filling there are then N-1 soft modes at different finite momenta, all related to the Fermi surface. For exactly one electron per site, the spin-excitation spectrum can be mapped onto the one of the SU(N) Heisenberg chain. The implications of the hole states on superconductivity are also discussed.

Physical review. B, Condensed matterFeb 1, 199132 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.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 stateMar 1, 1972201 citations

Finite-temperature properties of the Hubbard chain with bond-charge interaction

Fabrizio Dolcini, Arianna Montorsi

We investigate the one-dimensional (1D) Hubbard model with an additional bond-charge interaction, recently considered in the description of compounds that exhibit strong 1D features above the temperature of ordered phases. The partition function of the model is exactly calculated for a value of the bond-charge coupling; the behavior of the specific heat and spin susceptibility as a function of temperature is derived at arbitrary filling, and particularly discussed across the occurring metal-insulator transition. The results show that the bond-charge terms weaken the spin excitations of the system.We investigate the one-dimensional (1D) Hubbard model with an additional bond-charge interaction, recently considered in the description of compounds that exhibit strong 1D features above the temperature of ordered phases. The partition function of the model is exactly calculated for a value of the bond-charge coupling; the behavior of the specific heat and spin susceptibility as a function of temperature is derived at arbitrary filling, and particularly discussed across the occurring metal-insulator transition. The results show that the bond-charge terms weaken the spin excitations of the system.

arXiv:cond-mat/0201391Physical review. B, Condensed matterAug 15, 200219 citations

Thermodynamic Quantities of the One-Dimensional Hubbard Model at Finite Temperatures

Tatsuya Usuki, Norio Kawakami, Ayao Okiji

The thermodynamic quantities of the 1D Hubbard model are investigated exactly by means of the Bethe Ansatz solution at finite temperatures. For an arbitrary electron concentration, the magnetic susceptibility, the charge susceptibility and the specific heat are calculated numerically as a function of temperature. On the basis of the obtained results, the characteristic behavior, which appears in the charge excitation near half-filling, is discussed, and is understood to be closely related to the gap formation of the Mott-type insulator.The thermodynamic quantities of the 1D Hubbard model are investigated exactly by means of the Bethe Ansatz solution at finite temperatures. For an arbitrary electron concentration, the magnetic susceptibility, the charge susceptibility and the specific heat are calculated numerically as a function of temperature. On the basis of the obtained results, the characteristic behavior, which appears in the charge excitation near half-filling, is discussed, and is understood to be closely related to the gap formation of the Mott-type insulator.

Journal of the Physical Society of JapanApr 15, 199062 citations

Pair theory of the Hubbard Hamiltonian

H. Barentzen

A detailed account is given of a recently developed pair theory of the Hubbard Hamiltonian for the case of a half-filled band. The resulting pair Hamiltonian is an exact reformulation of the original Hubbard Hamiltonian in terms of pair operators which satisfy elementary boson commutation relations. The Hamiltonian takes the familiar form of a sum of a quadratic part representing independent-pair energies and a quartic part representing pair interactions. The matrix elements entering the pair Hamiltonian are the pair energies and products of pair wave functions, which are formally given as the solutions of a certain eigenvalue equation. By the solving of the eigenvalue problem, explicit analytical expressions are obtained for both pair energies and wave functions for an arbitrary number of dimensions. For the case $M=1$ , where $M$ is the number of spin-down electrons, pair interactions do not enter since there is only one electron-hole pair, and the exact energies of the Hubbard Hamiltonian are obtained. This will be explicitly verified for the one-dimensional case; here we demonstrate that the pair energies agree with the exact energies found by other authors for this case.A detailed account is given of a recently developed pair theory of the Hubbard Hamiltonian for the case of a half-filled band. The resulting pair Hamiltonian is an exact reformulation of the original Hubbard Hamiltonian in terms of pair operators which satisfy elementary boson commutation relations. The Hamiltonian takes the familiar form of a sum of a quadratic part representing independent-pair energies and a quartic part representing pair interactions. The matrix elements entering the pair Hamiltonian are the pair energies and products of pair wave functions, which are formally given as the solutions of a certain eigenvalue equation. By the solving of the eigenvalue problem, explicit analytical expressions are obtained for both pair energies and wave functions for an arbitrary number of dimensions. For the case $M=1$, where $M$ is the number of spin-down electrons, pair interactions do not enter since there is only one electron-hole pair, and the exact energies of the Hubbard Hamiltonian are obtained. This will be explicitly verified for the one-dimensional case; here we demonstrate that the pair energies agree with the exact energies found by other authors for this case.

Physical review. B, Condensed matterOct 15, 19837 citations

Computer calculations of finite Hubbard chains indicate two peaks for the specific heat for a large Coulomb interaction

U

for band fillings less than 1/2. Even though the origin of this is physically obvious, no elementary excitations with a gap energy

U

have been explicitly demonstrated from the exact solution of the Hubbard model. We here demonstrate that there exists elementary excitation with a complex wave vector

k

for the

N\ensuremath{-}1

\frac{U}{t}

, where

U

represents the on-site Coulomb repulsion and

t

is the transfer integral, assumed to be nonzero only for nearest neighbors. Exact calculations are made for the energy and momentum of excitations having single-particle character. Unlike the situation for the half-filled band, we find no gap in the excitation spectrum. We have also considered excitations of the spin-wave type. These are shown to vary linearly with momentum for small momentum. The group velocity for small momentum is found to be inversely proportional to the magnetic susceptibility.

{\mathit{U}}_{\mathit{c}}

. This change is probably associated with a Mott metal-insulator transition (

{\mathit{U}}_{\mathit{c}}

=0 for N=2). The Fermi velocity is finite for U

{\mathit{U}}_{\mathit{c}}

but vanishes for U>

{\mathit{U}}_{\mathit{c}}

(for one electron per site). The spin-wave velocity is inversely proportional to the magnetic susceptibility for all U and band fillings. There are N-1 elemental spin-wave branches. For arbitrary band filling there are then N-1 soft modes at different finite momenta, all related to the Fermi surface. For exactly one electron per site, the spin-excitation spectrum can be mapped onto the one of the SU(N) Heisenberg chain. The implications of the hole states on superconductivity are also discussed.

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.

M=1

, where

M

is the number of spin-down electrons, pair interactions do not enter since there is only one electron-hole pair, and the exact energies of the Hubbard Hamiltonian are obtained. This will be explicitly verified for the one-dimensional case; here we demonstrate that the pair energies agree with the exact energies found by other authors for this case.