The energy density of an Ising half plane lattice

M. Assis, Barry M. McCoy

Published Jan 1, 2010• 4 citations

Abstract

We compute the energy density at arbitrary temperature of the half-plane Ising lattice with a boundary magnetic field Hb at a distance M rows from the boundary and compare limiting cases of the exact expression with recent calculations at T = Tc done by means of discrete complex analysis methods.

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The energy density of an Ising half-plane lattice

M Assis, Barry M. McCoy

We compute the energy density at arbitrary temperature of the half-plane Ising lattice with a boundary magnetic field H b at a distance M rows from the boundary and compare limiting cases of the exact expression with recent calculations at T = T c done by means of discrete complex analysis methods.

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On properties of the Ising model for complex energy/temperature and magnetic field

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Partition function of finite Ising model (spin=1/2) was exactly obtained by a computer for three dimensional even lattice 4×4×2=32 with periodic boundary condition in the presence of a magnetic field. Specific heat was calculated as a function of temperature. The zeros of the partition functions of the complex temperature plane and magnetic field plane were investigated.

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Theory of Toeplitz Determinants and the Spin Correlations of the Two-Dimensional Ising Model. IV

Barry M. McCoy, Tai Tsun Wu

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A Connection Between Complex-Temperature Properties of the 1D and 2D Spin s Ising Model

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Jan 1, 199416 citations

Phase Boundary of an Ising Antiferromagnet

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The phase boundary in the magnetic field–temperature ( H - T ) plane of an Ising antiferromagnet in a simple quadratic lattice is obtained by studying the zeros of the partition function in the complex temperature plane. The critical line in the weak field region is given by T c ( H )/ T c (0)=1-α×( m H / J ) 2 , and the value of α is estimated from our numerical results and compared with previous determinations.

Journal of the Physical Society of JapanDec 15, 19890 citations

The energy density of an Ising half plane lattice

M. Assis, Barry M. McCoy

Published Jan 1, 2010• 4 citations

OpenAlex

Abstract

Related papers

The energy density of an Ising half-plane lattice

M Assis, Barry M. McCoy

We compute the energy density at arbitrary temperature of the half-plane Ising lattice with a boundary magnetic field H b at a distance M rows from the boundary and compare limiting cases of the exact expression with recent calculations at T = T c done by means of discrete complex analysis methods.

arXiv:1011.5463Journal of Physics A Mathematical and TheoreticalFeb 10, 20112 citations

On properties of the Ising model for complex energy/temperature and magnetic field

V. Matveev, Robert E. Shrock

We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable $u=e^{-4K}$ , where $K=J/(k_BT)$ , for nonzero external magnetic field, $H$ . Exact results are given for the phase diagram in the $u$ plane for the model in one dimension and on infinite-length quasi-one-dimensional strips. In the case of real $h=H/(k_BT)$ , these results provide new insights into features of our earlier study of this case. We also consider complex $h=H/(k_BT)$ and $μ=e^{-2h}$ . Calculations of complex- $u$ zeros of the partition function on sections of the square lattice are presented. For the case of imaginary $h$ , i.e., $μ=e^{iθ}$ , we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the $u$ plane. We find that in this case, the phase boundary ${\cal B}_u$ contains a real line segment extending through part of the physical ferromagnetic interval $0 \le u \le 1$ , with a right-hand endpoint $u_{rhe}$ at the temperature for which the Yang-Lee edge singularity occurs at $μ=e^{\pm iθ}$ . Conformal field theory arguments are used to relate the singularities at $u_{rhe}$ and the Yang-Lee edge.We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable $u=e^{-4K}$, where $K=J/(k_BT)$, for nonzero external magnetic field, $H$. Exact results are given for the phase diagram in the $u$ plane for the model in one dimension and on infinite-length quasi-one-dimensional strips. In the case of real $h=H/(k_BT)$, these results provide new insights into features of our earlier study of this case. We also consider complex $h=H/(k_BT)$ and $μ=e^{-2h}$. Calculations of complex-$u$ zeros of the partition function on sections of the square lattice are presented. For the case of imaginary $h$, i.e., $μ=e^{iθ}$, we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the $u$ plane. We find that in this case, the phase boundary ${\cal B}_u$ contains a real line segment extending through part of the physical ferromagnetic interval $0 \le u \le 1$, with a right-hand endpoint $u_{rhe}$ at the temperature for which the Yang-Lee edge singularity occurs at $μ=e^{\pm iθ}$. Conformal field theory arguments are used to relate the singularities at $u_{rhe}$ and the Yang-Lee edge.

arXiv:0711.4639Journal of Physics A Mathematical and TheoreticalMar 17, 200823 citations

Statistical Mechanics of Finite Ising Model

Chikao Kawabata, Masaki Takeuchi, Yukihiko Karaki

Journal of the Physical Society of JapanMar 1, 19771 citation

Theory of Toeplitz Determinants and the Spin Correlations of the Two-Dimensional Ising Model. IV

Barry M. McCoy, Tai Tsun Wu

We consider the rectangular Ising model on a half-plane of infinite extent and study some of the consequences connected with the presence of the boundary. Only the spins on the boundary row are allowed to interact with a magnetic field $\mathfrak{H}$ . The method of Pfaffians is employed to obtain exact expressions for the partition function. It is found that the free energy is the sum of two terms, one of which is independent of $\mathfrak{H}$ and proportional to the total number of lattice sites, while the other depends on $\mathfrak{H}$ and is proportional to the number of lattice sites on the boundary. This separation makes it possible to define various thermodynamic quantities associated with the boundary. In particular, the boundary magnetization is shown to be discontinuous, in the ferromagnetic case, at zero magnetic field for temperatures below the bulk critical temperature ${T}_{c}$ . This discontinuity, which is the spontaneous boundary magnetization, goes to zero as ${(1\ensuremath{-}\frac{T}{{T}_{c}})}^{\frac{1}{2}}$ as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$. For $T={T}_{c}$ , the discontinuity is of course absent, and the boundary magnetization behaves as $\ensuremath{-}\mathfrak{H}\mathrm{ln}\mathfrak{H}$ for small $\mathfrak{H}$ . The boundary susceptibility at zero magnetic field in the ferromagnetic case exhibits a logarithmic singularity at $T={T}_{c}$ , both above and below transition. An interesting feature is that the ferromagnetic boundary magnetization, although discontinuous for $T<{T}_{c}$ , may be analytically continued beyond the point $\mathfrak{H}=0$ . We interpret this as a hystersis phenomenon which we study in detail by computing the probability distribution function for the average boundary spin. The correlation function for two spins, both on the boundary row, is also obtained exactly and its asymptotic behavior is given. Finally, we derive an expression for the magnetization in any row and explicitly evaluate it for the second row, i.e., the row next to the boundary.We consider the rectangular Ising model on a half-plane of infinite extent and study some of the consequences connected with the presence of the boundary. Only the spins on the boundary row are allowed to interact with a magnetic field $\mathfrak{H}$. The method of Pfaffians is employed to obtain exact expressions for the partition function. It is found that the free energy is the sum of two terms, one of which is independent of $\mathfrak{H}$ and proportional to the total number of lattice sites, while the other depends on $\mathfrak{H}$ and is proportional to the number of lattice sites on the boundary. This separation makes it possible to define various thermodynamic quantities associated with the boundary. In particular, the boundary magnetization is shown to be discontinuous, in the ferromagnetic case, at zero magnetic field for temperatures below the bulk critical temperature ${T}_{c}$. This discontinuity, which is the spontaneous boundary magnetization, goes to zero as ${(1\ensuremath{-}\frac{T}{{T}_{c}})}^{\frac{1}{2}}$ as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$. For $T={T}_{c}$, the discontinuity is of course absent, and the boundary magnetization behaves as $\ensuremath{-}\mathfrak{H}\mathrm{ln}\mathfrak{H}$ for small $\mathfrak{H}$. The boundary susceptibility at zero magnetic field in the ferromagnetic case exhibits a logarithmic singularity at $T={T}_{c}$, both above and below transition. An interesting feature is that the ferromagnetic boundary magnetization, although discontinuous for $T<{T}_{c}$, may be analytically continued beyond the point $\mathfrak{H}=0$. We interpret this as a hystersis phenomenon which we study in detail by computing the probability distribution function for the average boundary spin. The correlation function for two spins, both on the boundary row, is also obtained exactly and its asymptotic behavior is given. Finally, we derive an expression for the magnetization in any row and explicitly evaluate it for the second row, i.e., the row next to the boundary.

Physical ReviewOct 10, 1967195 citations

A Connection Between Complex-Temperature Properties of the 1D and 2D Spin s Ising Model

Victor Matveev, Robert E. Shrock

Although the physical properties of the 2D and 1D Ising models are quite different, we point out an interesting connection between their complex-temperature phase diagrams. We carry out an exact determination of the complex-temperature phase diagram for the 1D Ising model for arbitrary spin s and show that in the u_s=e^{-K/s^2} plane (i) it consists of N_{c,1D}=4s^2 infinite regions separated by an equal number of boundary curves where the free energy is non-analytic; (ii) these curves extend from the origin to complex infinity, and in both limits are oriented along the angles \\theta_n = (1+2n)\\pi/(4s^2), for n=0,..., 4s^2-1; (iii) of these curves, there are N_{c,NE,1D}=N_{c,NW,1D}=[s^2] in the first and second (NE and NW) quadrants; and (iv) there is a boundary curve (line) along the negative real u_s axis if and only if s is half-integral. We note a close relation between these results and the number of arcs of zeros protruding into the FM phase in our recent calculation of partition function zeros for the 2D spin s Ising model.Although the physical properties of the 2D and 1D Ising models are quite different, we point out an interesting connection between their complex-temperature phase diagrams. We carry out an exact determination of the complex-temperature phase diagram for the 1D Ising model for arbitrary spin s and show that in the u_s=e^{-K/s^2} plane (i) it consists of N_{c,1D}=4s^2 infinite regions separated by an equal number of boundary curves where the free energy is non-analytic; (ii) these curves extend from the origin to complex infinity, and in both limits are oriented along the angles \\theta_n = (1+2n)\\pi/(4s^2), for n=0,..., 4s^2-1; (iii) of these curves, there are N_{c,NE,1D}=N_{c,NW,1D}=[s^2] in the first and second (NE and NW) quadrants; and (iv) there is a boundary curve (line) along the negative real u_s axis if and only if s is half-integral. We note a close relation between these results and the number of arcs of zeros protruding into the FM phase in our recent calculation of partition function zeros for the 2D spin s Ising model.

Jan 1, 199416 citations

Phase Boundary of an Ising Antiferromagnet

S. Dasgupta

Journal of the Physical Society of JapanDec 15, 19890 citations

We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable

u=e^{-4K}

, where

K=J/(k_BT)

, for nonzero external magnetic field,

H

. Exact results are given for the phase diagram in the

u

plane for the model in one dimension and on infinite-length quasi-one-dimensional strips. In the case of real

h=H/(k_BT)

, these results provide new insights into features of our earlier study of this case. We also consider complex

h=H/(k_BT)

and

μ=e^{-2h}

. Calculations of complex-

u

zeros of the partition function on sections of the square lattice are presented. For the case of imaginary

h

, i.e.,

μ=e^{iθ}

, we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the

u

plane. We find that in this case, the phase boundary

{\cal B}_u

contains a real line segment extending through part of the physical ferromagnetic interval

0 \le u \le 1

, with a right-hand endpoint

u_{rhe}

at the temperature for which the Yang-Lee edge singularity occurs at

μ=e^{\pm iθ}

. Conformal field theory arguments are used to relate the singularities at

u_{rhe}

and the Yang-Lee edge.

\mathfrak{H}

. The method of Pfaffians is employed to obtain exact expressions for the partition function. It is found that the free energy is the sum of two terms, one of which is independent of

\mathfrak{H}

and proportional to the total number of lattice sites, while the other depends on

\mathfrak{H}

and is proportional to the number of lattice sites on the boundary. This separation makes it possible to define various thermodynamic quantities associated with the boundary. In particular, the boundary magnetization is shown to be discontinuous, in the ferromagnetic case, at zero magnetic field for temperatures below the bulk critical temperature

{T}_{c}

. This discontinuity, which is the spontaneous boundary magnetization, goes to zero as ${(1\ensuremath{-}\frac{T}{{T}_{c}})}^{\frac{1}{2}}$ as $T\ensuremath{\rightarrow}{T}_{c}\ensuremath{-}$. For

T={T}_{c}

, the discontinuity is of course absent, and the boundary magnetization behaves as $\ensuremath{-}\mathfrak{H}\mathrm{ln}\mathfrak{H}$ for small

\mathfrak{H}

. The boundary susceptibility at zero magnetic field in the ferromagnetic case exhibits a logarithmic singularity at

T={T}_{c}

, both above and below transition. An interesting feature is that the ferromagnetic boundary magnetization, although discontinuous for

T<{T}_{c}

, may be analytically continued beyond the point

\mathfrak{H}=0

. We interpret this as a hystersis phenomenon which we study in detail by computing the probability distribution function for the average boundary spin. The correlation function for two spins, both on the boundary row, is also obtained exactly and its asymptotic behavior is given. Finally, we derive an expression for the magnetization in any row and explicitly evaluate it for the second row, i.e., the row next to the boundary.