Physical review. B, Condensed matter

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

Chi-Ning Chen, Chin‐Kun Hu

Published May 1, 1991• 5 citations

Abstract

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.

DOI: 10.1103/physrevb.43.11519

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Physical review. B, Condensed matter

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

Chi-Ning Chen, Chin‐Kun Hu

Published May 1, 1991• 5 citations

Publisher

Abstract

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.

DOI: 10.1103/physrevb.43.11519

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

Percolation renormalization-group approach to theq-state Potts model

Chin‐Kun Hu, Chi-Ning Chen

Based on the connection between the q-state Potts model (QPM) and the q-state bond-correlated percolation model (QBCPM), we have proposed percolation renormalization-group (PRG) methods to calculate the free energy, the critical point, and critical exponents for the QPM. Our methods are free from inconsistency in Larsson's method. We have carried out the ${\ensuremath{\lambda}}_{1}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\lambda}}_{1}$ to ${\ensuremath{\lambda}}_{2}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\lambda}}_{2}$ renormalization-group (RG) transformations in our methods for several values of original cell sizes ${\ensuremath{\lambda}}_{1}$ and final cell sizes ${\ensuremath{\lambda}}_{2}$ with various boundary conditions. We find that such RG transformations with large ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$ and periodic boundary condition usually give accurate physical quantities for the QPM. The advantages and generalization of our approach are also discussed.Based on the connection between the q-state Potts model (QPM) and the q-state bond-correlated percolation model (QBCPM), we have proposed percolation renormalization-group (PRG) methods to calculate the free energy, the critical point, and critical exponents for the QPM. Our methods are free from inconsistency in Larsson's method. We have carried out the ${\ensuremath{\lambda}}_{1}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\lambda}}_{1}$ to ${\ensuremath{\lambda}}_{2}$\ifmmode\times\else\texttimes\fi{}${\ensuremath{\lambda}}_{2}$ renormalization-group (RG) transformations in our methods for several values of original cell sizes ${\ensuremath{\lambda}}_{1}$ and final cell sizes ${\ensuremath{\lambda}}_{2}$ with various boundary conditions. We find that such RG transformations with large ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$ and periodic boundary condition usually give accurate physical quantities for the QPM. The advantages and generalization of our approach are also discussed.

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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 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.

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Physical review. B, Condensed matterJul 1, 19891 citation

Monte Carlo study of the Potts model on the square and the simple cubic lattices

Chin‐Kun Hu, Kit-Sing Mak

It has been shown that the partition function of the q-state Potts model (QPM) is the generating function of a q-state bond-correlated percolation model (QBCPM) such that the spontaneous magnetization M, the magnetic susceptibility \ensuremath{\chi}, the internal energy U, and the specific heat ${C}_{h}$ for the QPM are related to the percolation probability P, the mean cluster size S, the average number of occupied bond p\ifmmode\bar\else\textasciimacron\fi{}, and the fluctuations of occupied bonds F for the QBCPM. We use the Monte Carlo simulation method of Swendsen and Wang and the multiple labeling technique of Hoshen and Kopelman to calculate P, S, p\ifmmode\bar\else\textasciimacron\fi{}, and F for the QBCPM on the square and the simple cubic lattice. Our calculations give accurate critical points for the QPM.It has been shown that the partition function of the q-state Potts model (QPM) is the generating function of a q-state bond-correlated percolation model (QBCPM) such that the spontaneous magnetization M, the magnetic susceptibility \ensuremath{\chi}, the internal energy U, and the specific heat ${C}_{h}$ for the QPM are related to the percolation probability P, the mean cluster size S, the average number of occupied bond p\ifmmode\bar\else\textasciimacron\fi{}, and the fluctuations of occupied bonds F for the QBCPM. We use the Monte Carlo simulation method of Swendsen and Wang and the multiple labeling technique of Hoshen and Kopelman to calculate P, S, p\ifmmode\bar\else\textasciimacron\fi{}, and F for the QBCPM on the square and the simple cubic lattice. Our calculations give accurate critical points for the QPM.

Physical review. B, Condensed matterSep 1, 198929 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

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 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 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.

{C}_{h}

for the QPM are related to the percolation probability P, the mean cluster size S, the average number of occupied bond p\ifmmode\bar\else\textasciimacron\fi{}, and the fluctuations of occupied bonds F for the QBCPM. We use the Monte Carlo simulation method of Swendsen and Wang and the multiple labeling technique of Hoshen and Kopelman to calculate P, S, p\ifmmode\bar\else\textasciimacron\fi{}, and F for the QBCPM on the square and the simple cubic lattice. Our calculations give accurate critical points for the QPM.

{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.