Physical review. B, Solid state

Hubbard model for a disordered linear chain: Probability distribution of exchange

George Theodorou

Published Sep 1, 1977• 43 citations

Abstract

The Hubbard model for a disordered linear chain with a half-filled band is studied. At low temperatures ($T\ensuremath{\lesssim}\frac{U}{4k}$) and sufficiently small transfer integrals, the Hamiltonian via a perturbation expansion reduces to that of a disordered one-dimensional Heisenberg antiferromagnet. It is found that the coupling constant $J$ has for $n\ensuremath{\gg}1$ the behavior $J=D{n}^{\ensuremath{\alpha}}{\ensuremath{\beta}}^{2n}$, where $D$, $\ensuremath{\alpha}$, and $\ensuremath{\beta}$ depend on the parameters of the Hamiltonian and $n$ is the number of intermediate sites between localized spins, and is a random variable. An expression for the probability distribution $P(J)$ of exchange integral $J$ is also obtained. For small $J$, $P(J)\ensuremath{\propto}\frac{1}{{J}^{1\ensuremath{-}c}}{|\mathrm{ln}(\frac{J}{D})|}^{\ensuremath{\alpha}c}$. That is, $P(J)$ has a singularity at the origin for $c<1$, where $c$ depends on the parameters of the Hamiltonian.

DOI: 10.1103/physrevb.16.2254

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Physical review. B, Solid state

Hubbard model for a disordered linear chain: Probability distribution of exchange

George Theodorou

Published Sep 1, 1977• 43 citations

Publisher

Abstract

The Hubbard model for a disordered linear chain with a half-filled band is studied. At low temperatures (

T\ensuremath{\lesssim}\frac{U}{4k}

) and sufficiently small transfer integrals, the Hamiltonian via a perturbation expansion reduces to that of a disordered one-dimensional Heisenberg antiferromagnet. It is found that the coupling constant

J

has for

n\ensuremath{\gg}1

the behavior

J=D{n}^{\ensuremath{\alpha}}{\ensuremath{\beta}}^{2n}

, where

D

\ensuremath{\alpha}

, and

\ensuremath{\beta}

depend on the parameters of the Hamiltonian and

n

is the number of intermediate sites between localized spins, and is a random variable. An expression for the probability distribution

P(J)

of exchange integral

J

is also obtained. For small

J

P(J)\ensuremath{\propto}\frac{1}{{J}^{1\ensuremath{-}c}}{|\mathrm{ln}(\frac{J}{D})|}^{\ensuremath{\alpha}c}

. That is,

P(J)

has a singularity at the origin for

c<1

, where

c

depends on the parameters of the Hamiltonian.

DOI: 10.1103/physrevb.16.2254

Related papers

Low-temperature conductivity in a narrow half-filled Hubbard chain

Jean-Pierre Gallinar

In this paper, the low-temperature (${k}_{B}T\ensuremath{\ll}U$) frequency-dependent $\ensuremath{\sigma}(\ensuremath{\omega})$ conductivity in a half-filled Hubbard chain is calculated to the lowest order in the parameter $\frac{t}{U}$ and in closed form to all orders in $\frac{t}{\ensuremath{\omega}}\ensuremath{-}U$. Here $t$ is the nearest-neighbor hopping integral and $U$ the intra-atomic Coulomb interaction. The problem is formulated in terms of the number of random walks of an electron and a hole which leave the spin configuration unchanged, at temperatures such that ${k}_{B}T\ensuremath{\ll}t\ensuremath{\ll}U$ (where ${k}_{B}$ is Boltzmann's constant). The conductivity then depends on spin configuration, and for both the antiferromagnetic (AF) ( $T=0$ K) ground state and the equiprobable or random arrangement ( $R$ ) (${k}_{B}T\ensuremath{\gg}\frac{{t}^{2}}{U}$) an exact expression is found. For the partially disordered regime ($T\ensuremath{\simeq}{T}_{N}$) the thermodynamical averaging is expressed in terms of a phenomenological spin-spin correlation length $l$ which interpolates between the AF and $R$ exact limits. This solution is such that for $l=\ensuremath{\infty}$ the AF line shape is recovered except at the center of the spectrum (i.e., at $\ensuremath{\omega}=U$), where the absorption always diverges logarithmically. For a ferromagnetic ( $l=0$ ) arrangement induced by a magnetic field, the absorption then vanishes. The line shapes obtained have sharp edges at $|\ensuremath{\omega}|=U\ifmmode\pm\else\textpm\fi{}4t$ for all temperatures. At the critical value of $4t=U$ the AF ground state is found to undergo a metallic transition. A closed-form analytical expression in terms of the three elliptical integrals is given for the absorption lines. Except for the AF case, which has a square-root edge singularity, the absorption is found to vanish linearly. Finally, a comparison is made with some recent calculations.In this paper, the low-temperature (${k}_{B}T\ensuremath{\ll}U$) frequency-dependent $\ensuremath{\sigma}(\ensuremath{\omega})$ conductivity in a half-filled Hubbard chain is calculated to the lowest order in the parameter $\frac{t}{U}$ and in closed form to all orders in $\frac{t}{\ensuremath{\omega}}\ensuremath{-}U$. Here $t$ is the nearest-neighbor hopping integral and $U$ the intra-atomic Coulomb interaction. The problem is formulated in terms of the number of random walks of an electron and a hole which leave the spin configuration unchanged, at temperatures such that ${k}_{B}T\ensuremath{\ll}t\ensuremath{\ll}U$ (where ${k}_{B}$ is Boltzmann's constant). The conductivity then depends on spin configuration, and for both the antiferromagnetic (AF) ($T=0$ K) ground state and the equiprobable or random arrangement ($R$) (${k}_{B}T\ensuremath{\gg}\frac{{t}^{2}}{U}$) an exact expression is found. For the partially disordered regime ($T\ensuremath{\simeq}{T}_{N}$) the thermodynamical averaging is expressed in terms of a phenomenological spin-spin correlation length $l$ which interpolates between the AF and $R$ exact limits. This solution is such that for $l=\ensuremath{\infty}$ the AF line shape is recovered except at the center of the spectrum (i.e., at $\ensuremath{\omega}=U$), where the absorption always diverges logarithmically. For a ferromagnetic ($l=0$) arrangement induced by a magnetic field, the absorption then vanishes. The line shapes obtained have sharp edges at $|\ensuremath{\omega}|=U\ifmmode\pm\else\textpm\fi{}4t$ for all temperatures. At the critical value of $4t=U$ the AF ground state is found to undergo a metallic transition. A closed-form analytical expression in terms of the three elliptical integrals is given for the absorption lines. Except for the AF case, which has a square-root edge singularity, the absorption is found to vanish linearly. Finally, a comparison is made with some recent calculations.

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Maciej Kozarzewski, Peter Prelovšek, Marcin Mierzejewski

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Paata Kakashvili, G. I. Japaridze

Jan 1, 20045 citations

Effective Hamiltonian for a half-filled Hubbard chain with alternating on-site interactions

Paata Kakashvili, George I Japaridze

We derive an effective spin Hamiltonian for the one-dimensional half-filled Alternating Hubbard model in the limit of strong on-site repulsion. We show that the effective Hamiltonian is a spin $S=1/2$ Heisenberg chain with asymmetric next-nearest-neighbor (nnn) exchange.We derive an effective spin Hamiltonian for the one-dimensional half-filled Alternating Hubbard model in the limit of strong on-site repulsion. We show that the effective Hamiltonian is a spin $S=1/2$ Heisenberg chain with asymmetric next-nearest-neighbor (nnn) exchange.

arXiv:cond-mat/0405048Journal of Physics Condensed MatterAug 3, 20048 citations

The Hubbard model for a disordered linear chain with a half-filled band is studied. At low temperatures (

T\ensuremath{\lesssim}\frac{U}{4k}

J

has for

n\ensuremath{\gg}1

the behavior

J=D{n}^{\ensuremath{\alpha}}{\ensuremath{\beta}}^{2n}

, where

D

\ensuremath{\alpha}

, and

\ensuremath{\beta}

depend on the parameters of the Hamiltonian and

n

is the number of intermediate sites between localized spins, and is a random variable. An expression for the probability distribution

P(J)

of exchange integral

J

is also obtained. For small

J

P(J)\ensuremath{\propto}\frac{1}{{J}^{1\ensuremath{-}c}}{|\mathrm{ln}(\frac{J}{D})|}^{\ensuremath{\alpha}c}

. That is,

P(J)

has a singularity at the origin for

c<1

, where

c

depends on the parameters of the Hamiltonian.

\frac{t}{U}

and in closed form to all orders in $\frac{t}{\ensuremath{\omega}}\ensuremath{-}U$. Here

t

is the nearest-neighbor hopping integral and

U

the intra-atomic Coulomb interaction. The problem is formulated in terms of the number of random walks of an electron and a hole which leave the spin configuration unchanged, at temperatures such that ${k}_{B}T\ensuremath{\ll}t\ensuremath{\ll}U$ (where

{k}_{B}

is Boltzmann's constant). The conductivity then depends on spin configuration, and for both the antiferromagnetic (AF) (

T=0

K) ground state and the equiprobable or random arrangement (

R

) (${k}_{B}T\ensuremath{\gg}\frac{{t}^{2}}{U}$) an exact expression is found. For the partially disordered regime ($T\ensuremath{\simeq}{T}_{N}$) the thermodynamical averaging is expressed in terms of a phenomenological spin-spin correlation length

l

which interpolates between the AF and

R

exact limits. This solution is such that for $l=\ensuremath{\infty}$ the AF line shape is recovered except at the center of the spectrum (i.e., at $\ensuremath{\omega}=U$), where the absorption always diverges logarithmically. For a ferromagnetic (

l=0

) arrangement induced by a magnetic field, the absorption then vanishes. The line shapes obtained have sharp edges at $|\ensuremath{\omega}|=U\ifmmode\pm\else\textpm\fi{}4t$ for all temperatures. At the critical value of

4t=U

the AF ground state is found to undergo a metallic transition. A closed-form analytical expression in terms of the three elliptical integrals is given for the absorption lines. Except for the AF case, which has a square-root edge singularity, the absorption is found to vanish linearly. Finally, a comparison is made with some recent calculations.

\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 derive an effective spin Hamiltonian for the one-dimensional half-filled Alternating Hubbard model in the limit of strong on-site repulsion. We show that the effective Hamiltonian is a spin

S=1/2

Heisenberg chain with asymmetric next-nearest-neighbor (nnn) exchange.