arXiv:cond-mat/9806294 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Fisher zeros of theQ-state Potts model in the complex temperature plane for nonzero external magnetic field

Seung‐Yeon Kim, Richard J. Creswick

Published Dec 1, 1998• 24 citations

Abstract

The microcanonical transfer matrix is used to study the distribution of the Fisher zeros of the $Q>2$ Potts models in the complex temperature plane with nonzero external magnetic field ${H}_{q}.$ Unlike the Ising model for ${H}_{q}\ensuremath{\ne}0,$ which has only a nonphysical critical point (the Fisher edge singularity), the $Q>2$ Potts models have physical critical points for ${H}_{q}<0$ as well as the Fisher edge singularities for ${H}_{q}>0.$ For ${H}_{q}<0,$ the crossover of the Fisher zeros of the Q-state Potts model into those of the $(Q\ensuremath{-}1)$-state Potts model is discussed, and the critical line of the three-state Potts ferromagnet is determined. For ${H}_{q}>0,$ we investigate the edge singularity for finite lattices and compare our results with the high-field, low-temperature series expansion of Enting. For $3<~Q<~6,$ we find that the specific heat, magnetization, susceptibility, and the density of zeros diverge at the Fisher edge singularity with exponents ${\ensuremath{\alpha}}_{e},$ ${\ensuremath{\beta}}_{e},$ and ${\ensuremath{\gamma}}_{e},$ which satisfy the scaling law ${\ensuremath{\alpha}}_{e}+2{\ensuremath{\beta}}_{e}+{\ensuremath{\gamma}}_{e}=2.$

DOI: 10.1103/physreve.58.7006

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Fisher zeros of the Ising antiferromagnet in an arbitrary nonzero magnetic field

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Density of states, Potts zeros, and Fisher zeros of theQ-state Potts model for continuousQ

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arXiv:cond-mat/9806294 · Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics

Fisher zeros of theQ-state Potts model in the complex temperature plane for nonzero external magnetic field

Seung‐Yeon Kim, Richard J. Creswick

Published Dec 1, 1998• 24 citations

arXiv HTML PDF

Abstract

The microcanonical transfer matrix is used to study the distribution of the Fisher zeros of the

Q>2

Potts models in the complex temperature plane with nonzero external magnetic field

{H}_{q}.

Unlike the Ising model for

{H}_{q}\ensuremath{\ne}0,

which has only a nonphysical critical point (the Fisher edge singularity), the

Q>2

Potts models have physical critical points for

{H}_{q}<0

as well as the Fisher edge singularities for

{H}_{q}>0.

For

{H}_{q}<0,

the crossover of the Fisher zeros of the Q-state Potts model into those of the

(Q\ensuremath{-}1)

-state Potts model is discussed, and the critical line of the three-state Potts ferromagnet is determined. For

{H}_{q}>0,

we investigate the edge singularity for finite lattices and compare our results with the high-field, low-temperature series expansion of Enting. For

3<~Q<~6,

we find that the specific heat, magnetization, susceptibility, and the density of zeros diverge at the Fisher edge singularity with exponents

{\ensuremath{\alpha}}_{e},

{\ensuremath{\beta}}_{e},

and

{\ensuremath{\gamma}}_{e},

which satisfy the scaling law

{\ensuremath{\alpha}}_{e}+2{\ensuremath{\beta}}_{e}+{\ensuremath{\gamma}}_{e}=2.

DOI: 10.1103/physreve.58.7006

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Fisher zeros of the Ising antiferromagnet in an arbitrary nonzero magnetic field

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From the exact partition functions of the Ising model on L x L square lattices (up to L = 14) in an arbitrary nonzero external magnetic field H, the precise distributions of the Fisher zeros in the complex temperature (a = e(2betaJ)) plane are obtained. Our results indicate that in the limit L-->infinity the Fisher zeros for H not = 0 do cut the positive real axis at the antiferromagnetic critical point ac(H); these values are compared with the results of closed-form approximations for the antiferromagnetic critical line. From the Fisher zeros the thermal scaling exponent yt = 1 is also obtained along the critical line, indicating the logarithmic singularity of the specific heat for the antiferromagnetic Ising model even in a strong uniform magnetic field.From the exact partition functions of the Ising model on L x L square lattices (up to L = 14) in an arbitrary nonzero external magnetic field H, the precise distributions of the Fisher zeros in the complex temperature (a = e(2betaJ)) plane are obtained. Our results indicate that in the limit L-->infinity the Fisher zeros for H not = 0 do cut the positive real axis at the antiferromagnetic critical point ac(H); these values are compared with the results of closed-form approximations for the antiferromagnetic critical line. From the Fisher zeros the thermal scaling exponent yt = 1 is also obtained along the critical line, indicating the logarithmic singularity of the specific heat for the antiferromagnetic Ising model even in a strong uniform magnetic field.

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Partition function zeros of the honeycomb-lattice Ising antiferromagnet in the complex magnetic-field plane

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Density of the Fisher zeros for the three-state and four-state Potts models

Seung‐Yeon Kim

The numbers of states up to $L=12$ for the three-state Potts model and up to $L=10$ for the four-state Potts model on $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with self-dual boundary condition are enumerated using the microcanonical transfer matrix exploiting the permutation symmetry of the model. From these numbers of states, the densities of the Fisher zeros $g(\ensuremath{\theta})$ of the partition function in the complex $u=(1+(Q\ensuremath{-}1){e}^{\ensuremath{-}\ensuremath{\beta}J})∕\sqrt{Q}$ plane are determined for the zeros on the unit circle ${u}_{0}={e}^{i\ensuremath{\theta}}$. For small $\ensuremath{\theta}$ the density of zeros obeys the finite-size scaling, allowing us to estimate the order parameter and the thermal exponent.The numbers of states up to $L=12$ for the three-state Potts model and up to $L=10$ for the four-state Potts model on $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with self-dual boundary condition are enumerated using the microcanonical transfer matrix exploiting the permutation symmetry of the model. From these numbers of states, the densities of the Fisher zeros $g(\ensuremath{\theta})$ of the partition function in the complex $u=(1+(Q\ensuremath{-}1){e}^{\ensuremath{-}\ensuremath{\beta}J})∕\sqrt{Q}$ plane are determined for the zeros on the unit circle ${u}_{0}={e}^{i\ensuremath{\theta}}$. For small $\ensuremath{\theta}$ the density of zeros obeys the finite-size scaling, allowing us to estimate the order parameter and the thermal exponent.

Physical Review EJul 8, 200411 citations

The microcanonical transfer matrix is used to study the distribution of the Fisher zeros of the

Q>2

Potts models in the complex temperature plane with nonzero external magnetic field

{H}_{q}.

Unlike the Ising model for

{H}_{q}\ensuremath{\ne}0,

which has only a nonphysical critical point (the Fisher edge singularity), the

Q>2

Potts models have physical critical points for

{H}_{q}<0

as well as the Fisher edge singularities for

{H}_{q}>0.

For

{H}_{q}<0,

the crossover of the Fisher zeros of the Q-state Potts model into those of the

(Q\ensuremath{-}1)

-state Potts model is discussed, and the critical line of the three-state Potts ferromagnet is determined. For

{H}_{q}>0,

we investigate the edge singularity for finite lattices and compare our results with the high-field, low-temperature series expansion of Enting. For

3<~Q<~6,

we find that the specific heat, magnetization, susceptibility, and the density of zeros diverge at the Fisher edge singularity with exponents

{\ensuremath{\alpha}}_{e},

{\ensuremath{\beta}}_{e},

and

{\ensuremath{\gamma}}_{e},

which satisfy the scaling law

{\ensuremath{\alpha}}_{e}+2{\ensuremath{\beta}}_{e}+{\ensuremath{\gamma}}_{e}=2.

The microcanonical transfer matrix is used to study the distribution of Yang-Lee zeros of the

Q

-state Potts model in the complex magnetic field ( ${x\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}e}^{\ensuremath{\beta}h}$) plane for the first time. Finite size scaling suggests that at (and below) the critical temperature the zeros lie close to, but not on, the unit circle with the two exceptions of the critical point

x\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1

(

h\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0

) itself and the zeros in the limit

T\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0

The microcanonical transfer matrix is used to evaluate the exact integer values for the number of states $\ensuremath{\Omega}(E,M)$, as a function of energy

E

and magnetization

M

, of the Ising model in an external magnetic field

H

on $L\ifmmode\times\else\texttimes\fi{}2L$ honeycomb lattices

(L=6,8,\dots{},14)

and (due to memory limitations) the exact real values for $\overline{\ensuremath{\Omega}}(a,M)={\ensuremath{\sum}}_{E}\ensuremath{\Omega}(E,M){a}^{E}$

(L=16,\dots{},22)

, where $a={e}^{2\ensuremath{\beta}J}$. Given $\ensuremath{\Omega}(E,M)$ or $\overline{\ensuremath{\Omega}}(a,M)$, the exact partition functions $Z(a,x)={\ensuremath{\sum}}_{E,M}\ensuremath{\Omega}(E,M){a}^{E}{x}^{M}={\ensuremath{\sum}}_{M}\overline{\ensuremath{\Omega}}(a,M){x}^{M}$, where $x={e}^{\ensuremath{-}2\ensuremath{\beta}H}$, are obtained. From the exact partition functions, the precise distributions of the partition function zeros in the complex

x

plane of the honeycomb-lattice Ising antiferromagnet in a uniform external magnetic field, where the famous circle theorem for the partition function zeros of the Ising ferromagnet is not applicable, are determined. In addition, the critical points

{x}_{cp}(a)

and the magnetic scaling exponents

{y}_{h}(a)

of the honeycomb-lattice Ising antiferromagnet in a uniform magnetic field are estimated using its partition function zeros in the complex

x

plane. Our results for the critical points are also compared to those of the two different closed-form approximations.

The properties of the partition function zeros in the complex temperature plane (Fisher zeros) and in the complex

Q

plane (Potts zeros) are investigated for the

Q

-state Potts model in an arbitrary nonzero external magnetic field

H_q

, using the exact partition function of the one-dimensional model.

The numbers of states up to

L=12

for the three-state Potts model and up to

L=10

for the four-state Potts model on $L\ifmmode\times\else\texttimes\fi{}L$ square lattice with self-dual boundary condition are enumerated using the microcanonical transfer matrix exploiting the permutation symmetry of the model. From these numbers of states, the densities of the Fisher zeros $g(\ensuremath{\theta})$ of the partition function in the complex $u=(1+(Q\ensuremath{-}1){e}^{\ensuremath{-}\ensuremath{\beta}J})∕\sqrt{Q}$ plane are determined for the zeros on the unit circle ${u}_{0}={e}^{i\ensuremath{\theta}}$. For small $\ensuremath{\theta}$ the density of zeros obeys the finite-size scaling, allowing us to estimate the order parameter and the thermal exponent.