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

The Ising Model

Sergey G. Abaimov

Springer series in synergeticsJan 1, 20150 citations

Dynamical Process of a Bit-Width Reduced Ising Model With Simulated Annealing

Shuta Kikuchi, Nozomu Togawa, Shu Tanaka

Ising machines have attracted attention as fast and high-precision solvers for combinatorial optimization problems, which are often formulated as ground-state (lowest-energy) search problems of the Ising model. Since the bit-width of coefficients on Ising machines is limited, the Ising model must be transformed into a bit-width reduced (BWR) Ising model which reduces the coefficients. In a bit-width reduction method, which adds auxiliary spins, the ground state of the BWR Ising model is theoretically the same as the Ising model before bit-width reduction (original Ising model). However, the dynamical process of the BWR Ising model towards the ground state remains to be elucidated. Therefore, using simulated annealing (SA), we compared the dynamical processes of these models. The result shows that the dynamical process differs significantly by model. Analysis from the viewpoint of statistical mechanics found that the BWR Ising model has two characteristic properties: an effective temperature and a slow relaxation. It is suggested that these properties alter the effective temperature and spin flip probability in the BWR Ising model, contributing to the dynamical differences. Therefore, to obtain the same dynamical process as the original Ising model, we proposed SA parameters for the BWR Ising model. We demonstrated the proposed SA parameter using a square lattice Ising model, in which the interaction and magnetic field were set uniformly to the same positive values or randomly. As experimental evaluations, the dynamical process of the BWR Ising model became closer to that of the original Ising model.Ising machines have attracted attention as fast and high-precision solvers for combinatorial optimization problems, which are often formulated as ground-state (lowest-energy) search problems of the Ising model. Since the bit-width of coefficients on Ising machines is limited, the Ising model must be transformed into a bit-width reduced (BWR) Ising model which reduces the coefficients. In a bit-width reduction method, which adds auxiliary spins, the ground state of the BWR Ising model is theoretically the same as the Ising model before bit-width reduction (original Ising model). However, the dynamical process of the BWR Ising model towards the ground state remains to be elucidated. Therefore, using simulated annealing (SA), we compared the dynamical processes of these models. The result shows that the dynamical process differs significantly by model. Analysis from the viewpoint of statistical mechanics found that the BWR Ising model has two characteristic properties: an effective temperature and a slow relaxation. It is suggested that these properties alter the effective temperature and spin flip probability in the BWR Ising model, contributing to the dynamical differences. Therefore, to obtain the same dynamical process as the original Ising model, we proposed SA parameters for the BWR Ising model. We demonstrated the proposed SA parameter using a square lattice Ising model, in which the interaction and magnetic field were set uniformly to the same positive values or randomly. As experimental evaluations, the dynamical process of the BWR Ising model became closer to that of the original Ising model.

arXiv:2304.12796IEEE AccessJan 1, 20237 citations

A Bit-Width Reducing Method for Ising Models Guaranteeing the Ground-State Output

Yuta Yachi, Masashi Tawada, Nozomu Togawa

Ising machines are being developed as an efficient computing alternative for solving combinatorial optimization problems. Ising machines solve a combinatorial optimization problem by transforming it into a data structure called an Ising model. However, Ising machines have hardware limitations and hence the input Ising model coefficients must be limited to a certain range. Existing coefficient bit-width reducing methods do not guarantee that the output solution is optimum without adding too many extra spins. In this paper, we propose a new bit-width reducing method for Ising models. In this method, we first partition an original Ising model into several Ising models, each of which has the same graph topology, and the coefficient bit-width is reduced to be dealt with by the target Ising machine. Next, we obtain a ground-state solution for every partitioned Ising model. At that time, we can theoretically prove that, if all the partitioned Ising models have a common ground-state solution, it also gives a ground-state solution for the original non-bit-width-reduced Ising model. Experimental results demonstrate that the theorem is empirically true.Ising machines are being developed as an efficient computing alternative for solving combinatorial optimization problems. Ising machines solve a combinatorial optimization problem by transforming it into a data structure called an Ising model. However, Ising machines have hardware limitations and hence the input Ising model coefficients must be limited to a certain range. Existing coefficient bit-width reducing methods do not guarantee that the output solution is optimum without adding too many extra spins. In this paper, we propose a new bit-width reducing method for Ising models. In this method, we first partition an original Ising model into several Ising models, each of which has the same graph topology, and the coefficient bit-width is reduced to be dealt with by the target Ising machine. Next, we obtain a ground-state solution for every partitioned Ising model. At that time, we can theoretically prove that, if all the partitioned Ising models have a common ground-state solution, it also gives a ground-state solution for the original non-bit-width-reduced Ising model. Experimental results demonstrate that the theorem is empirically true.

IEEE International System-on-Chip Conference (SOCC)Sep 5, 20232 citations

The Ising Model

Benjamin A. Stickler, E. Schachinger

Springer eBooksJan 1, 20160 citations

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}