Physical review. B, Condensed matter

Hierarchical quantum Ising model

Zhifang Lin, Ruibao Tao

Published Jun 1, 1990• 2 citations

Abstract

A quantum Ising chain with both the exchange couplings and the transverse fields arranged in a hierarchical way is considered. Exact analytical results for the critical line and dispersion relation of the low-energy excitations are obtained. It is shown that only when ${\mathit{R}}_{1}$=${\mathit{R}}_{2}$>${\mathit{R}}_{\mathit{c}}$, where ${\mathit{R}}_{1}$ and ${\mathit{R}}_{2}$ are the hierarchical parameters for the exchange couplings and the transverse fields, respectively, does the system preserve a logarithmic singularity in the specific heat.

DOI: 10.1103/physrevb.41.11597

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Physical review. B, Condensed matter

Hierarchical quantum Ising model

Zhifang Lin, Ruibao Tao

Published Jun 1, 1990• 2 citations

Publisher

Abstract

{\mathit{R}}_{1}

{\mathit{R}}_{2}

{\mathit{R}}_{\mathit{c}}

, where

{\mathit{R}}_{1}

and

{\mathit{R}}_{2}

are the hierarchical parameters for the exchange couplings and the transverse fields, respectively, does the system preserve a logarithmic singularity in the specific heat.

DOI: 10.1103/physrevb.41.11597

Related papers

Phase transition and scaling properties of the energy spectrum for a hierarchical quantum Ising model

Zhifang Lin, Ruibao Tao

We study a one-dimensional quantum Ising model where the exchange couplings and the transverse fields are hierarchically distributed. Exact analytical results for the critical line, energy gap, and dispersion relation of the low-energy excitations are obtained. It is shown that only in the case of ${\mathit{R}}_{1}$ = ${\mathit{R}}_{2}$ =R> ${\mathit{R}}_{\mathit{c}}$ , where ${\mathit{R}}_{1}$ and ${\mathit{R}}_{2}$ are the hierarchical parameters for the exchange couplings and the transverse fields, does the system preserve the logarithmic singularity in the specific heat. Using renormalization-group techniques and an iteration map for the transfer matrix, we have presented the analytical results for the scaling properties for the energy spectra of the models with ${\mathit{R}}_{1}$ = ${\mathit{R}}_{2}$ and ${\mathit{R}}_{2}$ =1. From knowledge of the scaling properties of the energy spectra, we have exhibited a close relationship between the disappearance of the logarithmic singularity in the specific heat and the anomalous scaling behavior at the lower band edges.We study a one-dimensional quantum Ising model where the exchange couplings and the transverse fields are hierarchically distributed. Exact analytical results for the critical line, energy gap, and dispersion relation of the low-energy excitations are obtained. It is shown that only in the case of ${\mathit{R}}_{1}$=${\mathit{R}}_{2}$=R>${\mathit{R}}_{\mathit{c}}$, where ${\mathit{R}}_{1}$ and ${\mathit{R}}_{2}$ are the hierarchical parameters for the exchange couplings and the transverse fields, does the system preserve the logarithmic singularity in the specific heat. Using renormalization-group techniques and an iteration map for the transfer matrix, we have presented the analytical results for the scaling properties for the energy spectra of the models with ${\mathit{R}}_{1}$=${\mathit{R}}_{2}$ and ${\mathit{R}}_{2}$=1. From knowledge of the scaling properties of the energy spectra, we have exhibited a close relationship between the disappearance of the logarithmic singularity in the specific heat and the anomalous scaling behavior at the lower band edges.

Physical review. B, Condensed matterJul 1, 19900 citations

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{\mathit{R}}_{1}

{\mathit{R}}_{2}

{\mathit{R}}_{\mathit{c}}

, where

{\mathit{R}}_{1}

and

{\mathit{R}}_{2}

are the hierarchical parameters for the exchange couplings and the transverse fields, respectively, does the system preserve a logarithmic singularity in the specific heat.

{\mathit{R}}_{1}

{\mathit{R}}_{2}

=R>

{\mathit{R}}_{\mathit{c}}

, where

{\mathit{R}}_{1}

and

{\mathit{R}}_{2}

are the hierarchical parameters for the exchange couplings and the transverse fields, does the system preserve the logarithmic singularity in the specific heat. Using renormalization-group techniques and an iteration map for the transfer matrix, we have presented the analytical results for the scaling properties for the energy spectra of the models with

{\mathit{R}}_{1}

{\mathit{R}}_{2}

and

{\mathit{R}}_{2}

=1. From knowledge of the scaling properties of the energy spectra, we have exhibited a close relationship between the disappearance of the logarithmic singularity in the specific heat and the anomalous scaling behavior at the lower band edges.