A systematic and unifying method is developed and demonstrated for obtaining exact solutions of n-site (n even integer) Ising correlations on various planar lattices. The scheme, which is exceedingly more simple than using solely traditional Pfaffian techniques, embodies five mapping theorems in alliance with algebraic correlation identities. In the theoretical framework, the triangular Ising model plays an overarching role. In particular, considering a select 7-site cluster of the triangular Ising model, the knowledge of all its 11 even-number correlations defined upon this cluster (where only four of the correlations need to be actually calculated by Pfaffian procedures) is shown to be sufficient for determining exactly all honeycomb, decorated-honeycomb, and kagom\'e Ising model even-number correlations upon their correspondingly select 10-site, 19-site, and 9-site clusters, respectively. The relative ease and direct applicability of the present approach are highlighted not only by the resulting large numbers of n-site (n even-integer) correlation solutions (e.g., approximately 85 and 50 for the honeycomb and kagom\'e Ising models, respectively) and their large ${n}_{\mathrm{max}}$ values ( ${n}_{\mathrm{max}=8}$ ,10,18 for the kagom\'e, honeycomb, and decorated-honeycomb Ising models, respectively) but also by the realization that the exact solutions for Ising multisite correlations upon the kagom\'e lattice (one of the four regular lattices in two dimensions) are apparently the first to explicitly appear in the literature beyond its nearest-neighbor pair correlation (energy).A systematic and unifying method is developed and demonstrated for obtaining exact solutions of n-site (n even integer) Ising correlations on various planar lattices. The scheme, which is exceedingly more simple than using solely traditional Pfaffian techniques, embodies five mapping theorems in alliance with algebraic correlation identities. In the theoretical framework, the triangular Ising model plays an overarching role. In particular, considering a select 7-site cluster of the triangular Ising model, the knowledge of all its 11 even-number correlations defined upon this cluster (where only four of the correlations need to be actually calculated by Pfaffian procedures) is shown to be sufficient for determining exactly all honeycomb, decorated-honeycomb, and kagom\'e Ising model even-number correlations upon their correspondingly select 10-site, 19-site, and 9-site clusters, respectively. The relative ease and direct applicability of the present approach are highlighted not only by the resulting large numbers of n-site (n even-integer) correlation solutions (e.g., approximately 85 and 50 for the honeycomb and kagom\'e Ising models, respectively) and their large ${n}_{\mathrm{max}}$ values (${n}_{\mathrm{max}=8}$,10,18 for the kagom\'e, honeycomb, and decorated-honeycomb Ising models, respectively) but also by the realization that the exact solutions for Ising multisite correlations upon the kagom\'e lattice (one of the four regular lattices in two dimensions) are apparently the first to explicitly appear in the literature beyond its nearest-neighbor pair correlation (energy).

Physical review. B, Condensed matterApr 1, 198856 citations

Correlation function in Ising models

C. Ruge, P. Zhu, F. Wagner

arXiv:hep-lat/9403009Physica A Statistical Mechanics and its ApplicationsSep 1, 199442 citations

Ising ferromagnets in Ising-percolation square lattices, an example of Ising-Ising coupling

Jafar Cheraghalizadeh, M. N. Najafi

In this paper we consider the Ising model (called the dynamical Ising model with dynamical temperature $T$ ) on the Ising-correlated percolation lattice. To construct this lattice, another Ising model (which is called the quenched Ising model with quench artificial temperature ${T}_{q}$ ) is employed to model the correlations between imperfections. We argue that for each quenched temperature, there is a particular dynamical temperature ${T}_{c}({T}_{q})$ for which the dynamical Ising model becomes critical. We present some finite-size arguments (based on the moments of the magnetism and the energy as well as the spanning cluster probability) to extract the critical points and also show that they are compatible with the finite-size scaling of the singular point of the magnetic susceptibility. The model is thoroughly characterized in and out of these points. We find that the critical behaviors of the model change significantly with respect to the regular Ising model as well as the Ising model on the uncorrelated percolation lattice. It is shown that for the critical lattice, ${T}_{q}={T}_{c}^{\text{square}\phantom{\rule{4.pt}{0ex}}\text{Ising}\phantom{\rule{4.pt}{0ex}}\text{model}}\ensuremath{\approx}2.269$, the critical temperature for the dynamical Ising model in the thermodynamic limit is ${T}_{c}({T}_{q}={T}_{c}^{\text{square}\phantom{\rule{4.pt}{0ex}}\text{Ising}\phantom{\rule{4.pt}{0ex}}\text{model}})=1.94\ifmmode\pm\else\textpm\fi{}0.005$, and the fractal dimension of the exterior perimeter of geometrical spin clusters is ${D}_{f}^{T={T}_{c}({T}_{q}=2.269)}=1.408\ifmmode\pm\else\textpm\fi{}0.002$. Many quantities, such as the dynamical critical temperatures, all local and global critical exponents, and the fractal dimension of loops ${D}_{f}$ , scale with the quench temperature in a power-law fashion, with some critical exponents that are reported. Significantly we see that ${D}_{f}^{T={T}_{c}({T}_{q})}\ensuremath{-}{D}_{f}^{T={T}_{c}({T}_{q}=2.269)}\ensuremath{\sim}\frac{1}{\sqrt{\ensuremath{\zeta}({T}_{q})}}$ in which $\ensuremath{\zeta}({T}_{q})$ is the correlation length of the quenched Ising model at temperature ${T}_{q}$ .In this paper we consider the Ising model (called the dynamical Ising model with dynamical temperature $T$) on the Ising-correlated percolation lattice. To construct this lattice, another Ising model (which is called the quenched Ising model with quench artificial temperature ${T}_{q}$) is employed to model the correlations between imperfections. We argue that for each quenched temperature, there is a particular dynamical temperature ${T}_{c}({T}_{q})$ for which the dynamical Ising model becomes critical. We present some finite-size arguments (based on the moments of the magnetism and the energy as well as the spanning cluster probability) to extract the critical points and also show that they are compatible with the finite-size scaling of the singular point of the magnetic susceptibility. The model is thoroughly characterized in and out of these points. We find that the critical behaviors of the model change significantly with respect to the regular Ising model as well as the Ising model on the uncorrelated percolation lattice. It is shown that for the critical lattice, ${T}_{q}={T}_{c}^{\text{square}\phantom{\rule{4.pt}{0ex}}\text{Ising}\phantom{\rule{4.pt}{0ex}}\text{model}}\ensuremath{\approx}2.269$, the critical temperature for the dynamical Ising model in the thermodynamic limit is ${T}_{c}({T}_{q}={T}_{c}^{\text{square}\phantom{\rule{4.pt}{0ex}}\text{Ising}\phantom{\rule{4.pt}{0ex}}\text{model}})=1.94\ifmmode\pm\else\textpm\fi{}0.005$, and the fractal dimension of the exterior perimeter of geometrical spin clusters is ${D}_{f}^{T={T}_{c}({T}_{q}=2.269)}=1.408\ifmmode\pm\else\textpm\fi{}0.002$. Many quantities, such as the dynamical critical temperatures, all local and global critical exponents, and the fractal dimension of loops ${D}_{f}$, scale with the quench temperature in a power-law fashion, with some critical exponents that are reported. Significantly we see that ${D}_{f}^{T={T}_{c}({T}_{q})}\ensuremath{-}{D}_{f}^{T={T}_{c}({T}_{q}=2.269)}\ensuremath{\sim}\frac{1}{\sqrt{\ensuremath{\zeta}({T}_{q})}}$ in which $\ensuremath{\zeta}({T}_{q})$ is the correlation length of the quenched Ising model at temperature ${T}_{q}$.

Physical review. ENov 26, 20184 citations

CORRELATION LENGTHS IN ISING MODELS

Walter Selke, Nenad M. Svrakic, Paul J. Upton

WORLD SCIENTIFIC eBooksApr 1, 19921 citation

Efficient algorithm for computing correlation functions of the two-dimensional random-bond Ising model

Xintian Wu, Juan Zhao

Extending the recent work of the bond-propagation algorithm [Y. L. Loh and E. W. Carlson, Phys. Rev. Lett. 97, 227205 (2006)], we present an efficient algorithm for calculating the correlation functions between any two (or more) spins of Ising models in two dimensions. With this algorithm, the spontaneous magnetization, correlation length, and susceptibility can be calculated. The results for the usual Ising model with different size lattices are compared with the exact results. We also present the correlation functions for the $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model.Extending the recent work of the bond-propagation algorithm [Y. L. Loh and E. W. Carlson, Phys. Rev. Lett. 97, 227205 (2006)], we present an efficient algorithm for calculating the correlation functions between any two (or more) spins of Ising models in two dimensions. With this algorithm, the spontaneous magnetization, correlation length, and susceptibility can be calculated. The results for the usual Ising model with different size lattices are compared with the exact results. We also present the correlation functions for the $\ifmmode\pm\else\textpm\fi{}J$ random-bond Ising model.

Physical Review BSep 3, 20096 citations

Correlations in inhomogeneous Ising models

W. F. Wolff, J. Zittartz

The European Physical Journal BDec 1, 198228 citations

{n}_{\mathrm{max}}

values (

{n}_{\mathrm{max}=8}

,10,18 for the kagom\'e, honeycomb, and decorated-honeycomb Ising models, respectively) but also by the realization that the exact solutions for Ising multisite correlations upon the kagom\'e lattice (one of the four regular lattices in two dimensions) are apparently the first to explicitly appear in the literature beyond its nearest-neighbor pair correlation (energy).

In this paper we consider the Ising model (called the dynamical Ising model with dynamical temperature

T

) on the Ising-correlated percolation lattice. To construct this lattice, another Ising model (which is called the quenched Ising model with quench artificial temperature

{T}_{q}

) is employed to model the correlations between imperfections. We argue that for each quenched temperature, there is a particular dynamical temperature

{T}_{c}({T}_{q})

for which the dynamical Ising model becomes critical. We present some finite-size arguments (based on the moments of the magnetism and the energy as well as the spanning cluster probability) to extract the critical points and also show that they are compatible with the finite-size scaling of the singular point of the magnetic susceptibility. The model is thoroughly characterized in and out of these points. We find that the critical behaviors of the model change significantly with respect to the regular Ising model as well as the Ising model on the uncorrelated percolation lattice. It is shown that for the critical lattice, ${T}_{q}={T}_{c}^{\text{square}\phantom{\rule{4.pt}{0ex}}\text{Ising}\phantom{\rule{4.pt}{0ex}}\text{model}}\ensuremath{\approx}2.269$, the critical temperature for the dynamical Ising model in the thermodynamic limit is ${T}_{c}({T}_{q}={T}_{c}^{\text{square}\phantom{\rule{4.pt}{0ex}}\text{Ising}\phantom{\rule{4.pt}{0ex}}\text{model}})=1.94\ifmmode\pm\else\textpm\fi{}0.005$, and the fractal dimension of the exterior perimeter of geometrical spin clusters is ${D}_{f}^{T={T}_{c}({T}_{q}=2.269)}=1.408\ifmmode\pm\else\textpm\fi{}0.002$. Many quantities, such as the dynamical critical temperatures, all local and global critical exponents, and the fractal dimension of loops

{D}_{f}

, scale with the quench temperature in a power-law fashion, with some critical exponents that are reported. Significantly we see that ${D}_{f}^{T={T}_{c}({T}_{q})}\ensuremath{-}{D}_{f}^{T={T}_{c}({T}_{q}=2.269)}\ensuremath{\sim}\frac{1}{\sqrt{\ensuremath{\zeta}({T}_{q})}}$ in which $\ensuremath{\zeta}({T}_{q})$ is the correlation length of the quenched Ising model at temperature

{T}_{q}