Physical review. B, Solid state

Two exactly soluble Ising systems with new boundary conditions

H. C. Bolton, C. H. Law

Published Jul 1, 1977• 2 citations

Abstract

Two Ising systems are investigated with the two extended-mean-field boundary conditions recently considered by Bolton and Johnson. The systems are $N$ spins on a line with boundary spins on either side and at each end. The exact solutions are obtained by the use of the modified-transfer-matrix method. A general formula for the critical temperature is found for each system. The critical temperature is found to take an asymptotic form similar to that obtained by Ferdinand and Fisher for the two-dimensional planar infinite Ising lattice. The thermodynamic limits of the results are discussed.

DOI: 10.1103/physrevb.16.502

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Monte Carlo studies on small Ising systems

H. C. Bolton, David W. R. Gruen

Physical review. B, Solid stateMay 1, 19775 citations

Series-expansion analysis of critical-temperature shifts of finite Ising lattices with "self-consistent" boundary conditions

H. C. Bolton, C. H. Law

Physical review. B, Condensed matterJan 1, 19791 citation

On Ising models

Hoede, C., Zandvliet, Henricus J.W.

Jun 1, 20060 citations

Exactly soluble Ising models with anisotropic interactions and arbitrary external magnetic field

Pedro L. Garrido, Joaquín Marro

Journal of Physics A Mathematical and GeneralMay 11, 19874 citations

Exact Solution Of The Ising Model On The 15 X 15 Square Lattice With Free Boundary Conditions

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Approximation method for spin-½ Ising models

S. Gartenhaus

Physical review. B, Condensed matterFeb 1, 198310 citations

Physical review. B, Solid state

Two exactly soluble Ising systems with new boundary conditions

H. C. Bolton, C. H. Law

Published Jul 1, 1977• 2 citations

Publisher

Abstract

Two Ising systems are investigated with the two extended-mean-field boundary conditions recently considered by Bolton and Johnson. The systems are

N

spins on a line with boundary spins on either side and at each end. The exact solutions are obtained by the use of the modified-transfer-matrix method. A general formula for the critical temperature is found for each system. The critical temperature is found to take an asymptotic form similar to that obtained by Ferdinand and Fisher for the two-dimensional planar infinite Ising lattice. The thermodynamic limits of the results are discussed.

DOI: 10.1103/physrevb.16.502

Related papers

Monte Carlo studies on small Ising systems

H. C. Bolton, David W. R. Gruen

Results are presented for the critical temperatures ${T}_{n}$ of small $n\ifmmode\times\else\texttimes\fi{}n$ Ising clusters using one of the extended mean-field boundary conditions recently introduced by Bolton and Johnson. The spin flips were simulated numerically. The values of ${T}_{n}$ were obtained by extrapolating to zero the long-range order below ${T}_{n}$ . The extrapolation to the infinite limit $n=\ensuremath{\infty}$ was performed using an expansion in powers of $\frac{1}{n}$ and the final result agrees to 2% with the Onsager value.Results are presented for the critical temperatures ${T}_{n}$ of small $n\ifmmode\times\else\texttimes\fi{}n$ Ising clusters using one of the extended mean-field boundary conditions recently introduced by Bolton and Johnson. The spin flips were simulated numerically. The values of ${T}_{n}$ were obtained by extrapolating to zero the long-range order below ${T}_{n}$. The extrapolation to the infinite limit $n=\ensuremath{\infty}$ was performed using an expansion in powers of $\frac{1}{n}$ and the final result agrees to 2% with the Onsager value.

Physical review. B, Solid stateMay 1, 19775 citations

Series-expansion analysis of critical-temperature shifts of finite Ising lattices with "self-consistent" boundary conditions

H. C. Bolton, C. H. Law

Finite Ising $N\ifmmode\times\else\texttimes\fi{}N$ ($N=1\ensuremath{-}5$) and $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ ($N=1\ensuremath{-}8$) systems with one of the three self-consistent extended mean-field boundary conditions all around and on both sides, respectively, are studied. The boundary conditions used are: ( $a$ ) extended mean-field boundary condition using average, abbreviated EA (in which each boundary spin takes some average value); ( $b$ ) extended mean-field boundary condition using probability, abbreviated EP (in which each boundary spin takes the values \ifmmode\pm\else\textpm\fi{}1 with certain probabilities) and ( $c$ ) extended Bethe-Peierls, EBP (in which each boundary spin is assumed to be acted on by some effective field). The self-consistent equations are then derived by letting the average magnetization of the internal spins of the system be equal to that of the boundary spins. The finite systems considered therefore represent some high-order systematic generalization of the familiar molecular-field and Bethe-Peierls approximations. The equations for the critical temperature are formulated graphically in terms of some high-temperature expansion series which are calculated to the ninth and tenth orders. Two of the series in the equations of $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ for $N\ensuremath{\rightarrow}\ensuremath{\infty}$ are directly related to the layer and local susceptibilities considered by Binder and Hohenberg for the Ising infinite half plane. Critical temperatures obtained numerically with the use of truncation and extrapolation techniques. Results on the systems with EA and EP indicate that the shifts of their critical temperatures can be described by a power law ${N}^{\ensuremath{-}\ensuremath{\lambda}}$ where $\ensuremath{\lambda}=\frac{1}{{\ensuremath{\gamma}}_{1}}$ and ${\ensuremath{\gamma}}_{1}(\ensuremath{\approx}\frac{11}{8})$ is the critical exponent of the layer susceptibility. This is different from the value $\ensuremath{\lambda}=1$ which is believed true for two-dimensional finite systems with periodic or free boundary conditions. No conclusive remark can be made to systems with EBP through they suggest some even higher value (\ensuremath{\ge} 2).Finite Ising $N\ifmmode\times\else\texttimes\fi{}N$ ($N=1\ensuremath{-}5$) and $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ ($N=1\ensuremath{-}8$) systems with one of the three self-consistent extended mean-field boundary conditions all around and on both sides, respectively, are studied. The boundary conditions used are: ($a$) extended mean-field boundary condition using average, abbreviated EA (in which each boundary spin takes some average value); ($b$) extended mean-field boundary condition using probability, abbreviated EP (in which each boundary spin takes the values \ifmmode\pm\else\textpm\fi{}1 with certain probabilities) and ($c$) extended Bethe-Peierls, EBP (in which each boundary spin is assumed to be acted on by some effective field). The self-consistent equations are then derived by letting the average magnetization of the internal spins of the system be equal to that of the boundary spins. The finite systems considered therefore represent some high-order systematic generalization of the familiar molecular-field and Bethe-Peierls approximations. The equations for the critical temperature are formulated graphically in terms of some high-temperature expansion series which are calculated to the ninth and tenth orders. Two of the series in the equations of $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ for $N\ensuremath{\rightarrow}\ensuremath{\infty}$ are directly related to the layer and local susceptibilities considered by Binder and Hohenberg for the Ising infinite half plane. Critical temperatures obtained numerically with the use of truncation and extrapolation techniques. Results on the systems with EA and EP indicate that the shifts of their critical temperatures can be described by a power law ${N}^{\ensuremath{-}\ensuremath{\lambda}}$ where $\ensuremath{\lambda}=\frac{1}{{\ensuremath{\gamma}}_{1}}$ and ${\ensuremath{\gamma}}_{1}(\ensuremath{\approx}\frac{11}{8})$ is the critical exponent of the layer susceptibility. This is different from the value $\ensuremath{\lambda}=1$ which is believed true for two-dimensional finite systems with periodic or free boundary conditions. No conclusive remark can be made to systems with EBP through they suggest some even higher value (\ensuremath{\ge} 2).

Physical review. B, Condensed matterJan 1, 19791 citation

On Ising models

Hoede, C., Zandvliet, Henricus J.W.

For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent $2-\sqrt{2} \approx 0.5858,$ comparable to the exponent $4/7 \approx 0.5714$ found from renormalization arguments.For various Ising models two approaches are discussed, one is that of simulating lattices, also called gauging on exact equations, the other is that of calculating analytical expressions for the boundary free energy of Ising lattices. The first approach allows to conjecture a solution for some Ising models, that have sofar not been solved, once some exact partial result for the problem is known. The second approach aims at furnishing such a partial result in the form of a condition for the critical temperature. An example of such a result was recently given for the 2D Ising square lattice with nearest and next-nearest-neighbor interactions. The critical line that separates the ordered (ferromagnetic) phase from the disordered (paramagnetic) phase showed good agreement in the moderate and strong nearest neighbor coupling limit with several results obtained by Monte Carlo, transfer matrix and series expansion results. We extend the discussion of the critical line, finding an excellent fit, now also in other points, like the Padé point, as well as cusp behavior at the Onsager point where the lattice decouples into two 2D square lattices with only nearest-neighbor interaction. Combination of this result with a geometrical argument in the simulation approach leads to a critical exponent $2-\sqrt{2} \approx 0.5858,$ comparable to the exponent $4/7 \approx 0.5714$ found from renormalization arguments.

Jun 1, 20060 citations

Exactly soluble Ising models with anisotropic interactions and arbitrary external magnetic field

Pedro L. Garrido, Joaquín Marro

The authors consider several Ising models with anisotropic interactions which can be solved exactly in the thermodynamic limit, for instance using a transfer matrix method. These include a two-dimensional system where each spin interacts directly with the two nearest neighbours along one of the principal axes of the lattice and has a coherent-field coupling with the spins along the second principal direction and a multidimensional system with coherent-field couplings along any of the principal axes.The authors consider several Ising models with anisotropic interactions which can be solved exactly in the thermodynamic limit, for instance using a transfer matrix method. These include a two-dimensional system where each spin interacts directly with the two nearest neighbours along one of the principal axes of the lattice and has a coherent-field coupling with the spins along the second principal direction and a multidimensional system with coherent-field couplings along any of the principal axes.

Journal of Physics A Mathematical and GeneralMay 11, 19874 citations

Exact Solution Of The Ising Model On The 15 X 15 Square Lattice With Free Boundary Conditions

Seung‐Yeon Kim

The square-lattice Ising model is the simplest system showing phase transitions (the transition between the paramagnetic phase and the ferromagnetic phase and the transition between the paramagnetic phase and the antiferromagnetic phase) and critical phenomena at finite temperatures. The exact solution of the squarelattice Ising model with free boundary conditions is not known for systems of arbitrary size. For the first time, the exact solution of the Ising model on the square lattice with free boundary conditions is obtained after classifying all ) spin configurations with the microcanonical transfer matrix. Also, the phase transitions and critical phenomena of the square-lattice Ising model are discussed using the exact solution on the square lattice with free boundary conditions.The square-lattice Ising model is the simplest system showing phase transitions (the transition between the paramagnetic phase and the ferromagnetic phase and the transition between the paramagnetic phase and the antiferromagnetic phase) and critical phenomena at finite temperatures. The exact solution of the squarelattice Ising model with free boundary conditions is not known for systems of arbitrary size. For the first time, the exact solution of the Ising model on the square lattice with free boundary conditions is obtained after classifying all ) spin configurations with the microcanonical transfer matrix. Also, the phase transitions and critical phenomena of the square-lattice Ising model are discussed using the exact solution on the square lattice with free boundary conditions.

World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer EngineeringNov 20, 20110 citations

Approximation method for spin-½ Ising models

S. Gartenhaus

An approximation for calculating the thermodynamic properties of spin-\textonehalf{} Ising models is described. The method is based on the formulation of Kramers and Wannier for the square lattice, and involves expressing the free energy in terms of the limiting maximum eigenvalue of a matrix sequence which represents the contribution from a single spin of the lattice. The application of this formulation to various two- and three-dimensional lattices shows that many of these are characterized by an identically structured matrix sequence with exactly two nonvanishing elements in each row and column. To explore the potential of using this similarity and its simplicity for analyzing Ising models we consider the two-dimensional square lattice in an external field and approximate the doubly infinite lattice by a semi-infinite one consisting of $n (\ensuremath{\le}10)$ infinitely long rows. Calculations are carried out and compared with (1) Onsager's formula for the free energy, (2) the Onsanger-Yang formula for the spontaneous magnetization, and (3) values calculated from series. Except for the region around the critical point where the correlation length exceeds $n$ lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.An approximation for calculating the thermodynamic properties of spin-\textonehalf{} Ising models is described. The method is based on the formulation of Kramers and Wannier for the square lattice, and involves expressing the free energy in terms of the limiting maximum eigenvalue of a matrix sequence which represents the contribution from a single spin of the lattice. The application of this formulation to various two- and three-dimensional lattices shows that many of these are characterized by an identically structured matrix sequence with exactly two nonvanishing elements in each row and column. To explore the potential of using this similarity and its simplicity for analyzing Ising models we consider the two-dimensional square lattice in an external field and approximate the doubly infinite lattice by a semi-infinite one consisting of $n (\ensuremath{\le}10)$ infinitely long rows. Calculations are carried out and compared with (1) Onsager's formula for the free energy, (2) the Onsanger-Yang formula for the spontaneous magnetization, and (3) values calculated from series. Except for the region around the critical point where the correlation length exceeds $n$ lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.

Physical review. B, Condensed matterFeb 1, 198310 citations

Two Ising systems are investigated with the two extended-mean-field boundary conditions recently considered by Bolton and Johnson. The systems are

N

Results are presented for the critical temperatures

{T}_{n}

of small $n\ifmmode\times\else\texttimes\fi{}n$ Ising clusters using one of the extended mean-field boundary conditions recently introduced by Bolton and Johnson. The spin flips were simulated numerically. The values of

{T}_{n}

were obtained by extrapolating to zero the long-range order below

{T}_{n}

. The extrapolation to the infinite limit $n=\ensuremath{\infty}$ was performed using an expansion in powers of

\frac{1}{n}

and the final result agrees to 2% with the Onsager value.

a

) extended mean-field boundary condition using average, abbreviated EA (in which each boundary spin takes some average value); (

b

) extended mean-field boundary condition using probability, abbreviated EP (in which each boundary spin takes the values \ifmmode\pm\else\textpm\fi{}1 with certain probabilities) and (

c

) extended Bethe-Peierls, EBP (in which each boundary spin is assumed to be acted on by some effective field). The self-consistent equations are then derived by letting the average magnetization of the internal spins of the system be equal to that of the boundary spins. The finite systems considered therefore represent some high-order systematic generalization of the familiar molecular-field and Bethe-Peierls approximations. The equations for the critical temperature are formulated graphically in terms of some high-temperature expansion series which are calculated to the ninth and tenth orders. Two of the series in the equations of $N\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$ for $N\ensuremath{\rightarrow}\ensuremath{\infty}$ are directly related to the layer and local susceptibilities considered by Binder and Hohenberg for the Ising infinite half plane. Critical temperatures obtained numerically with the use of truncation and extrapolation techniques. Results on the systems with EA and EP indicate that the shifts of their critical temperatures can be described by a power law ${N}^{\ensuremath{-}\ensuremath{\lambda}}$ where $\ensuremath{\lambda}=\frac{1}{{\ensuremath{\gamma}}_{1}}$ and ${\ensuremath{\gamma}}_{1}(\ensuremath{\approx}\frac{11}{8})$ is the critical exponent of the layer susceptibility. This is different from the value $\ensuremath{\lambda}=1$ which is believed true for two-dimensional finite systems with periodic or free boundary conditions. No conclusive remark can be made to systems with EBP through they suggest some even higher value (\ensuremath{\ge} 2).

2-\sqrt{2} \approx 0.5858,

comparable to the exponent

4/7 \approx 0.5714

found from renormalization arguments.

n

lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.