Physical review. B, Condensed matter

Geometrical factor and thermal properties of a sublattice dilute Potts model

Chin‐Kun Hu

Published Dec 1, 1985• 14 citations

Abstract

Based on the connection between a sublattice dilute q-state Potts model (SDQPM) and a correlated percolation model, we define a nonpercolating geometrical factor ${g}_{f}$ and a percolating geometrical factor ${g}_{p}$, which depend only on q and geometrical properties of the system. The thermal properties of the SDQPM may be derived simply from ${g}_{f}$ and ${g}_{p}$. The theory is confirmed by an exact calculation for one-dimensional SDQPM and may be extended to other interacting systems.

DOI: 10.1103/physrevb.32.7325

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Geometrical factor and thermal properties of the Potts model

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Percolation, finite-size scaling, and the thermal scaling power for the Potts model

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Physical review. B, Condensed matterJul 1, 19891 citation

Thermal Phase Transition of the Dilute s -State Potts and n -Vector Models at the Percolation Threshold

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Physical review. B, Condensed matter

Geometrical factor and thermal properties of a sublattice dilute Potts model

Chin‐Kun Hu

Published Dec 1, 1985• 14 citations

Publisher

Abstract

Based on the connection between a sublattice dilute q-state Potts model (SDQPM) and a correlated percolation model, we define a nonpercolating geometrical factor

{g}_{f}

and a percolating geometrical factor

{g}_{p}

, which depend only on q and geometrical properties of the system. The thermal properties of the SDQPM may be derived simply from

{g}_{f}

and

{g}_{p}

. The theory is confirmed by an exact calculation for one-dimensional SDQPM and may be extended to other interacting systems.

DOI: 10.1103/physrevb.32.7325

Related papers

Geometrical factor and thermal properties of the Potts model

Chin‐Kun Hu

Based on the connection between the q-state Potts model (QPM) and a q-state bond-correlated percolation model (QBCPM), the author defines a non-percolating geometrical factor g f and percolating geometrical factor g p which depend only on q and geometrical properties of the system. The thermal properties of the QPM could be derived simply from g f and g p . The idea is confirmed by an exact calculation for one-dimensional QPM and may be extended to other Ising-like spin models. The formulation also provides a geometrical meaning for the finite-size scaling and broadening at first-order phase transitions of the spin model.Based on the connection between the q-state Potts model (QPM) and a q-state bond-correlated percolation model (QBCPM), the author defines a non-percolating geometrical factor g f and percolating geometrical factor g p which depend only on q and geometrical properties of the system. The thermal properties of the QPM could be derived simply from g f and g p . The idea is confirmed by an exact calculation for one-dimensional QPM and may be extended to other Ising-like spin models. The formulation also provides a geometrical meaning for the finite-size scaling and broadening at first-order phase transitions of the spin model.

Journal of Physics A Mathematical and GeneralOct 21, 198616 citations

Fast algorithm to calculate exact geometrical factors for theq-state Potts model

Chi-Ning Chen, Chin‐Kun Hu

Based on the connection between the q-state Potts model (QPM) and a q-state bond-correlated percolation model, Hu has defined the percolating geometrical factor ${\mathit{N}}_{\mathit{p}}$ and the nonpercolating geometrical factors ${\mathit{N}}_{\mathit{f}}$ . From ${\mathit{N}}_{\mathit{p}}$ nd ${\mathit{N}}_{\mathit{f}}$ , we may obtain the thermal properties of the QPM and construct a percolation renormalization-group method to study the QPM. In this paper we present a fast algorithm to calculate exact ${\mathit{N}}_{\mathit{p}}$ and ${\mathit{N}}_{\mathit{f}}$ for the QPM on finite cells.Based on the connection between the q-state Potts model (QPM) and a q-state bond-correlated percolation model, Hu has defined the percolating geometrical factor ${\mathit{N}}_{\mathit{p}}$ and the nonpercolating geometrical factors ${\mathit{N}}_{\mathit{f}}$. From ${\mathit{N}}_{\mathit{p}}$ nd ${\mathit{N}}_{\mathit{f}}$, we may obtain the thermal properties of the QPM and construct a percolation renormalization-group method to study the QPM. In this paper we present a fast algorithm to calculate exact ${\mathit{N}}_{\mathit{p}}$ and ${\mathit{N}}_{\mathit{f}}$ for the QPM on finite cells.

Physical review. B, Condensed matterMay 1, 19915 citations

Percolation, finite-size scaling, and the thermal scaling power for the Potts model

Chin‐Kun Hu, Chi-Ning Chen

Based on the connection between the q-state Potts model (QPM) and q-state bond-correlated percolation model, a simple geometrical method to calculate the thermal scaling power ${y}_{t}$ for the QPM is presented. The method may be extended to other Ising-type models and hard-core particle models.Based on the connection between the q-state Potts model (QPM) and q-state bond-correlated percolation model, a simple geometrical method to calculate the thermal scaling power ${y}_{t}$ for the QPM is presented. The method may be extended to other Ising-type models and hard-core particle models.

Physical review. B, Condensed matterJul 1, 19891 citation

Thermal Phase Transition of the Dilute s -State Potts and n -Vector Models at the Percolation Threshold

Antonio Coniglio A

A general theory is given for the quenched dilute $s$ -state Potts and $n$ -vector models in any dimension $d$ . It is shown that for $T\ensuremath{\rightarrow}0$ at the percolation threshold the Potts thermal exponent ${\ensuremath{\nu}}_{T}$ equals the percolation exponent ${\ensuremath{\nu}}_{p}$, implying a crossover exponent $\ensuremath{\varphi}=1$, for any $s$ and $d$ . For the $n$ -vector model ( $n>1$ ), ${\ensuremath{\nu}}_{T}=\frac{{\ensuremath{\nu}}_{p}}{{\ensuremath{\zeta}}_{R}}$, where ${\ensuremath{\zeta}}_{R}$ is a resistivity critical exponent. Agreement with recent experiments for two-dimensional dilute Ising and Heisenberg systems is excellent.A general theory is given for the quenched dilute $s$-state Potts and $n$-vector models in any dimension $d$. It is shown that for $T\ensuremath{\rightarrow}0$ at the percolation threshold the Potts thermal exponent ${\ensuremath{\nu}}_{T}$ equals the percolation exponent ${\ensuremath{\nu}}_{p}$, implying a crossover exponent $\ensuremath{\varphi}=1$, for any $s$ and $d$. For the $n$-vector model ($n>1$), ${\ensuremath{\nu}}_{T}=\frac{{\ensuremath{\nu}}_{p}}{{\ensuremath{\zeta}}_{R}}$, where ${\ensuremath{\zeta}}_{R}$ is a resistivity critical exponent. Agreement with recent experiments for two-dimensional dilute Ising and Heisenberg systems is excellent.

Physical Review LettersJan 26, 1981365 citations

Cluster-size distribution and the magnetic property of a Potts model

Chin‐Kun Hu

Based on the connection between a q-state Potts model (QPM) and a q-state bond-correlated percolation model (QBCPM), we propose that in the calculation of the free energy and a physical quantity, e.g., the magnetic susceptibility, we need only to retain the terms of the most probable cluster-size distribution (MPCSD) defined in the text. For the one-dimensional model, the MPCSD may be calculated exactly. The free energy and the magnetic susceptibility determined by such MPCSD are the same as those calculated exactly by the transfer-matrix method. For the QPM on the lattice with dimensions d\ensuremath{\ge}2, the above assumption about the MPCSD implies that the magnetic susceptibility of the QPM is proportional to the mean cluster sizes of the QBCPM for both T> ${T}_{c}$ and T ${T}_{c}$ . The hypothesis on the MPCSD may be extended to other spin models. The implication of this work on the theory of the scaling laws of the critical exponents is also discussed.Based on the connection between a q-state Potts model (QPM) and a q-state bond-correlated percolation model (QBCPM), we propose that in the calculation of the free energy and a physical quantity, e.g., the magnetic susceptibility, we need only to retain the terms of the most probable cluster-size distribution (MPCSD) defined in the text. For the one-dimensional model, the MPCSD may be calculated exactly. The free energy and the magnetic susceptibility determined by such MPCSD are the same as those calculated exactly by the transfer-matrix method. For the QPM on the lattice with dimensions d\ensuremath{\ge}2, the above assumption about the MPCSD implies that the magnetic susceptibility of the QPM is proportional to the mean cluster sizes of the QBCPM for both T>${T}_{c}$ and T${T}_{c}$. The hypothesis on the MPCSD may be extended to other spin models. The implication of this work on the theory of the scaling laws of the critical exponents is also discussed.

Physical review. B, Condensed matterNov 1, 198613 citations

New percolation structures in dilute antiferromagnetic Potts models

Henry Phillip Fried, Michael Schick C

We introduce a new class of site-percolation models and study them by Monte Carlo methods. These percolation models are defined as the zero-temperature limit of a quench-diluted spin model whose symmetries are those of the antiferromagnetic three-state Potts model on a triangular lattice. Many of these percolation models exhibit a sequence of two percolation transitions. A new phase, excluded in the absence of dilution, is found at intermediate spin concentrations. This phase, which we call the chiral phase, exhibits the symmetries of only a subgroup of the full symmetry group of the Potts model. The critical properties of the transitions found in these percolation models have been studied. The results indicate that, for this entire class of percolation models, the observed transitions are all contained in the universality class of the geometric percolation (q=1 Potts) model. From this result we conclude that the breaking of symmetry, which is relevant for thermally driven transitions, is not relevant for percolation transitions.We introduce a new class of site-percolation models and study them by Monte Carlo methods. These percolation models are defined as the zero-temperature limit of a quench-diluted spin model whose symmetries are those of the antiferromagnetic three-state Potts model on a triangular lattice. Many of these percolation models exhibit a sequence of two percolation transitions. A new phase, excluded in the absence of dilution, is found at intermediate spin concentrations. This phase, which we call the chiral phase, exhibits the symmetries of only a subgroup of the full symmetry group of the Potts model. The critical properties of the transitions found in these percolation models have been studied. The results indicate that, for this entire class of percolation models, the observed transitions are all contained in the universality class of the geometric percolation (q=1 Potts) model. From this result we conclude that the breaking of symmetry, which is relevant for thermally driven transitions, is not relevant for percolation transitions.

Physical review. B, Condensed matterMar 1, 19908 citations

Based on the connection between a sublattice dilute q-state Potts model (SDQPM) and a correlated percolation model, we define a nonpercolating geometrical factor

{g}_{f}

and a percolating geometrical factor

{g}_{p}

, which depend only on q and geometrical properties of the system. The thermal properties of the SDQPM may be derived simply from

{g}_{f}

and

{g}_{p}

. The theory is confirmed by an exact calculation for one-dimensional SDQPM and may be extended to other interacting systems.

Based on the connection between the q-state Potts model (QPM) and a q-state bond-correlated percolation model, Hu has defined the percolating geometrical factor

{\mathit{N}}_{\mathit{p}}

and the nonpercolating geometrical factors

{\mathit{N}}_{\mathit{f}}

. From

{\mathit{N}}_{\mathit{p}}

{\mathit{N}}_{\mathit{f}}

, we may obtain the thermal properties of the QPM and construct a percolation renormalization-group method to study the QPM. In this paper we present a fast algorithm to calculate exact

{\mathit{N}}_{\mathit{p}}

and

{\mathit{N}}_{\mathit{f}}

for the QPM on finite cells.

Based on the connection between the q-state Potts model (QPM) and q-state bond-correlated percolation model, a simple geometrical method to calculate the thermal scaling power

{y}_{t}

for the QPM is presented. The method may be extended to other Ising-type models and hard-core particle models.

A general theory is given for the quenched dilute

s

-state Potts and

n

-vector models in any dimension

d

. It is shown that for $T\ensuremath{\rightarrow}0$ at the percolation threshold the Potts thermal exponent ${\ensuremath{\nu}}_{T}$ equals the percolation exponent ${\ensuremath{\nu}}_{p}$, implying a crossover exponent $\ensuremath{\varphi}=1$, for any

s

and

d

. For the

n

-vector model (

n>1

), ${\ensuremath{\nu}}_{T}=\frac{{\ensuremath{\nu}}_{p}}{{\ensuremath{\zeta}}_{R}}$, where ${\ensuremath{\zeta}}_{R}$ is a resistivity critical exponent. Agreement with recent experiments for two-dimensional dilute Ising and Heisenberg systems is excellent.

{T}_{c}

and T

{T}_{c}

. The hypothesis on the MPCSD may be extended to other spin models. The implication of this work on the theory of the scaling laws of the critical exponents is also discussed.