Lattice Ising Model Research Articles

Dynamic magnetic characteristics of the kinetic Ising model under the influence of randomness.

In this paper, we propose to solve the issues of long-range or next-neighbor interactions by introducing randomness. This approach is applied to the square lattice Ising model. The Monte Carlo method with the Metropolis algorithm is utilized to calculate the critical temperature T_{C}^{*} under equilibrium thermodynamic phase transition conditions and to investigate the characterization of randomness in terms of magnetization. In order to further characterize the effect of this randomness on the magnetic system, clustering coefficients C_{p} are introduced. Furthermore, we investigate a number of dynamic magnetic behaviors, including dynamic hysteresis behaviors and metamagnetic anomalies. The results indicate that noise has the effect of destabilizing the system and promoting the dynamic phase transition. When the system is subjected to noise, the effect of this noise can be mitigated by the addition of a time-oscillating magnetic field. Finally, the evolution of anomalous metamagnetic fluctuations under the influence of white noise is examined. The relationship between the bias field corresponding to the peak of the curve h_{b}^{peak} and the noise parameter σ satisfies the exponential growth equation, which is consistent with other results.

Operator product expansion for radial lattice quantization of 3D ϕ4 theory

At its critical point, the three-dimensional lattice Ising model is described by a conformal field theory (CFT), the 3D Ising CFT. Instead of carrying out simulations on Euclidean lattices, we use the quantum finite elements method to implement radially quantized critical ϕ4 theory on simplicial lattices approaching R×S2. Computing the four-point function of identical scalars, we demonstrate the power of radial quantization by the accurate determination of the scaling dimensions Δε and ΔT as well as ratios of the operator product expansion coefficients fσσε and fσσT of the first spin-0 and spin-2 primary operators ε and T of the 3D Ising CFT. Published by the American Physical Society 2024

A SPIN-1 ISING MODEL INVESTIGATION OF THE MAGNETIC SYSTEM IS CARRIED OUT WITHIN THE CONTEXT OF GENERALIZED STATISTICAL MECHANICS

In this study magnetization has been investigated with the help of Ising model in the frame of non-extensive statistical mechanics where a behavior of interacting elementary moments ensemble is taken into consideration. To examine the physical systems with three states and two order parameters, researchers employ the spin-1 single lattice Ising model or three-state systems. Within this model, various thermodynamic characteristics of phenomena like ferromagnetism in binary alloys, liquid mixtures, liquid-crystal mixtures, freezing, magnetic order, phase transformations, semi-stable and unstable states, ordered and disordered transitions have been investigated for three distinct forms associated with q < 1, q = 1, and q > 1. In this context, q represents the non-extensivity parameter of Tsallis statistics.

Bootstrap, Markov Chain Monte Carlo, and LP/SDP hierarchy for the lattice Ising model

Bootstrap is an idea that imposing consistency conditions on a physical system may lead to rigorous and nontrivial statements about its physical observables. In this work, we discuss the bootstrap problem for the invariant measure of the stochastic Ising model defined as a Markov chain where probability bounds and invariance equations are imposed. It is described by a linear programming (LP) hierarchy whose asymptotic convergence is shown by explicitly constructing the invariant measure from the convergent sequence of moments. We also discuss the relation between the LP hierarchy for the invariant measure and a recently introduced semidefinite programming (SDP) hierarchy for the Gibbs measure of the statistical Ising model based on reflection positivity and spin-flip equations.

Calculation of critical exponents on fractal lattice Ising model by higher-order tensor renormalization group method.

The critical behavior of the Ising model on a fractal lattice, which has the Hausdorff dimension log_{4}12≈1.792, is investigated using a modified higher-order tensor renormalization group algorithm supplemented with automatic differentiation to compute relevant derivatives efficiently and accurately. The complete set of critical exponents characteristic of a second-order phase transition was obtained. Correlations near the critical temperature were analyzed through two impurity tensors inserted into the system, which allowed us to obtain the correlation lengths and calculate the critical exponent ν. The critical exponent α was found to be negative, consistent with the observation that the specific heat does not diverge at the critical temperature. The extracted exponents satisfy the known relations given by various scaling assumptions within reasonable accuracy. Perhaps most interestingly, the hyperscaling relation, which contains the spatial dimension, is satisfied very well, assuming the Hausdorff dimension takes the place of the spatial dimension. Moreover, using automatic differentiation, we have extracted four critical exponents (α, β, γ,andδ) globally by differentiating the free energy. Surprisingly, the global exponents differ from those obtained locally by the technique of the impurity tensors; however, the scaling relations remain satisfied even in the case of the global exponents.

The real space renormalization group treatment of the FCSC Ising lattice

A large-scale renormalization group study of the Ising model for the square, honeycomb, triangular, simple cubic, and body-centered (BC) cubic lattices has been performed recently by us. We complement those studies with the structurally more complicated face-centered cubic lattice Ising model. The results obtained from the real space renormalization group (RSRG) treatment of the face-centered simple cubic (FCSC) Ising lattice have been presented in this work. The difficulty due to its non-self-dual decimation transformation property and high numbers of nearest neighbors in the treatment is overcome with some relevant approximations. The approximation is based on keeping only the pairwise interactions in the decimated lattice which is apparently in the form of a tetragonal structure. Within this approximation, the renormalized coupling strength is related to the coupling constant of the original lattice. The critical coupling strength for the decimated tetragonal structure is calculated as 0.0905.

Statistical mechanics approach to the holographic renormalization group: Bethe lattice Ising model and p-adic AdS/CFT

Abstract The Bethe lattice Ising model—a classical model of statistical mechanics for the phase transition—provides a novel and intuitive understanding of the prototypical relationship between tensor networks and the anti-de Sitter (AdS)/conformal field theory (CFT) correspondence. After analytically formulating a holographic renormalization group for the Bethe lattice model, we demonstrate the underlying mechanism and the exact scaling dimensions for the power-law decay of boundary-spin correlations by introducing the relation between the lattice network and an effective Poincaré metric on a unit disk. We compare the Bethe lattice model in the high-temperature region with a scalar field in AdS2, and then discuss its more direct connection to the p-adic AdS/CFT. In addition, we find that the phase transition in the interior induces a crossover behavior of boundary-spin correlations, depending on the depth of the corresponding correlation path.

Topological dualities in the Ising model

We relate two classical dualities in low-dimensional quantum field theory: Kramers-Wannier duality of the Ising and related lattice models in $2$ dimensions, with electromagnetic duality for finite gauge theories in $3$ dimensions. The relation is mediated by the notion of boundary field theory: Ising models are boundary theories for pure gauge theory in one dimension higher. Thus the Ising order/disorder operators are endpoints of Wilson/'t Hooft defects of gauge theory. Symmetry breaking on low-energy states reflects the multiplicity of topological boundary states. In the process we describe lattice theories as (extended) topological field theories with boundaries and domain walls. This allows us to generalize the duality to non-abelian groups; finite, semi-simple Hopf algebras; and, in a different direction, to finite homotopy theories in arbitrary dimension.

Lifted directed-worm algorithm.

Nonreversible Markov chains can outperform reversible chains in the Markov chain Monte Carlo method. Lifting is a versatile approach to introducing net stochastic flow in state space and constructing a nonreversible Markov chain. We present here an application of the lifting technique to the directed-worm algorithm. The transition probability of the worm update is optimized using the geometric allocation approach; the worm backscattering probability is minimized, and the stochastic flow breaking the detailed balance is maximized. We demonstrate the performance improvement over the previous worm and cluster algorithms for the four-dimensional hypercubic lattice Ising model. The sampling efficiency of the present algorithm is approximately 80, 5, and 1.7 times as high as those of the standard worm algorithm, the Wolff cluster algorithm, and the previous lifted worm algorithm, respectively. We estimate the dynamic critical exponent of the hypercubic lattice Ising model to be z≈0 in the worm and the Wolff cluster updates. The lifted version of the directed-worm algorithm can be applied to a variety of quantum systems as well as classical systems.

Strong simulation of tracking single photons with which-way-detectors in linear optics

Which-way-detectors (WWDs) are path-entangled detectors characterizing mutual exclusivity between path information and interference visibility in wave-particle duality experiments. We show surprisingly that WWDs allow to utilize single photons distinguishable in time domain to realize linear optical circuits where tracking their paths is exponentially hard for strong simulation analogous to rectangular lattice based Ising models. Distinguishable photons have scalability advantages of generation and detection compared with indistinguishable photons by promising both theoretical and experimental improvements in linear optical computing including boson sampling. We calculate strong simulation complexities by using variable elimination (VE) method for undirected graphs related to tensor network contraction for quantum circuits and recursive Feynman path-integral (RFPI) method to reduce space complexity. Two designs include either a single photon touring m times or m single photons propagating sequentially through an optical circuit composed of n beam splitters and phase shifters entangled with n WWDs. VE method for tracking results in undirected graphs matching with (2 m − 1) × (n + 1) and m × (n + 1) lattice Ising models with computational complexities of and in time and and in space for single and multi-photon based designs, respectively. We exploit RFPI method for m ≫ n to reduce space complexities to polynomial levels with respect to n and log m. Probability amplitude of specific cases of multi-photon design is represented in terms of Ising partition function with purely imaginary weights to characterize sampling complexity. Open issues about sampling complexity and experimental implementation of multi-WWD set-ups are discussed.

Analytic average magnetization expression for the body-centered cubic Ising lattice

Recently we have performed large-scale average magnetization studies of the Ising model for the square, honeycomb, triangular, and simple cubic lattice. We want to complement those studies with the structurally more complicated body-centered cubic lattice Ising model in this paper. We have relevantly calculated the order parameter or average magnetization expression for the body-centered cubic Ising lattice based on the recently developed interrelation, given by $\langle\sigma_{0, i}\rangle= \langle\tanh[K(\sigma_{1,i}+\sigma_{2,i}+\dots +\sigma_{z,i})+H]\rangle$. We will use the same conjectures for the odd spins correlation functions as in our previous works. The mathematical forms of the odd spin correlation functions will be eventually elucidated by some physical discussions. We have seen that the obtained average magnetization expression with the numerical data for the body-centered cubic lattice Ising model is in good agreement.

Ising model on a 2D additive small-world network

In this article, we have employed Monte Carlo simulations to study the Ising model on a two-dimensional additive small-world network (A-SWN). The system model consists of a LxL square lattice where each site of the lattice is occupied for a spin variable that interacts with the nearest neighbor and has a certain probability p of being additionally connected at random to one of its farther neighbors. The system is in contact with a heat bath at a given temperature T and it is simulated by one-spin flip according to the Metropolis prescription. We have calculated the thermodynamic quantities of the system, such as, the magnetization per spin m, magnetic susceptibility chi, and the reduced fourth-order Binder cumulant U as a function of T for several values of lattice size L and additive probability p. We also have constructed the phase diagram for the equilibrium states of the model in the plane T versus p showing the existence of a continuous transition line between the ferromagnetic F and paramagnetic P phases. Using the finite-size scaling (FSS) theory, we have obtained the critical exponents for the system, where varying the parameter p, we have observed a change in the critical behavior from the regular square lattice Ising model to A-SWN.

The location of the Fisher zeros and estimates of y T = 1/ν are found for the Baxter–Wu model

The Fisher zeros of the Baxter–Wu model are examined for the first time and for two series of finite-sized systems, with ‘spherical’ boundary conditions, their location is found to be extremely simple. They lie on the unit circle in the complex sinh[2βJ 3] plane. This is the same location as the Fisher zeros of the square lattice Ising model with nearest neighbour interactions and Brascamp–Kunz boundary conditions. The Baxter–Wu model is an Ising model with three-site interactions, J 3, on the triangle lattice. From the leading Fisher zeros, using finite-size scaling, accurate estimates of the critical exponent 1/ν are obtained and emphasis is placed on using different variables such as exp[−2βJ 3], exp[−4βJ 3], and sinh[2βJ 3] to enhance the accuracy of estimates. Furthermore, using the imaginary parts of the leading zeros versus the real part of the leading zeros, yields different results. This is similar to results of Janke and Kenna for the nearest neighbour, Ising model on the square lattice and extends this behaviour to a multisite interaction system in a different universality class than the pair-interaction cases.

Magnetic susceptibility of the square lattice Ising model

In this work, we obtained an analytical relation for the susceptibility of the square lattice Ising model. Our investigation is based on an average magnetization interrelation which was recently obtained by us. To proceed further, we have to make a mathematical conjecture about the three-site correlation function appearing in the average magnetization interrelation. We presented the conjectured mathematical form of the three spin correlation function with the relation, \(\langle \sigma _{1}\sigma _{2}\sigma _{3}\rangle =a(K,H)\langle \sigma \rangle +[1-a(K,H)]\langle \sigma \rangle ^{(1+\beta ^{-1})}\). Here, \(\beta\) denotes the critical exponent for the average magnetization and a(K, H) is a function whose behavior will be described around the critical point with an arbitrary constant. To elucidate the relevance of the method, we have first calculated the susceptibility of the 1D chain as an example, and the obtained susceptibility expression for the 1D chain is equivalent to the result of the susceptibility obtained by the conventional method. Applying the same method, we obtained the values of the magnetic critical exponent \(\gamma\) of the square lattice Ising model. The values of \(\gamma\) are obtained as \(\gamma =1.72\) for \(T\!>\!T_{c}\), and \(\gamma =0.91\) for \(T\!<\!T_{c}\).

Relevant spontaneous magnetization relations for the triangular and the cubic lattice Ising model

The spontaneous magnetization relations for the 2D triangular and the 3D cubic lattices of the Ising model are derived by a new tractable and easily calculable mathematical method. The result obtained for the triangular lattice is compared with the already available result to test and investigate the relevance of the new mathematical method. From this comparison, it is seen that the agreement of our result is almost the same or almost equivalent to the previously obtained exact result. The new approach is, then, applied to the long-standing 3D cubic lattice, and the corresponding expression for the spontaneous magnetism is derived. The relation obtained is compared with the already existing numerical results for the 3D lattice. The essence of the method used in this paper is based on exploiting the main characteristic of the order parameter of a second order phase transition which provides a more direct physical insight into the calculation of the spontaneous magnetization of the Ising model.

Asymptotics of Bordered Toeplitz Determinants and Next-to-Diagonal Ising Correlations

We prove the analogue of the strong Szegő limit theorem for a large class of bordered Toeplitz determinants. In particular, by applying our results to the formula of Au-Yang and Perk (Physica A 144:44–104, 1987) for the next-to-diagonal correlations \(\langle \sigma _{0,0}\sigma _{N-1,N} \rangle \) in the square lattice Ising model, we rigorously justify that the next-to-diagonal long-range order is the same as the diagonal and horizontal ones in the low temperature regime. We also confirm the leading and subleading terms in an asymptotic formula of Cheng and Wu (Phys Rev 164:719–735, 1967) for \(\langle \sigma _{0,0}\sigma _{M,N} \rangle \) when \(M=N\) and \(M=N-1\), thereby establishing the anisotropy-dependence of the subleading term in the asymptotics of the next-to-diagonal correlations. We use Riemann-Hilbert and operator theory techniques, independently and in parallel, to prove these results.

Critical Phenomena and Phase Transitions in Large Lattices within Monte-Carlo Based Non-perturbative Approaches

Critical phenomena and Goldstone mode effects in spin models with the O(n) rotational symmetry are considered. Starting with Goldstone mode singularities in the XY and O(4) models, we briefly review various theoretical concepts, as well as state-of-the-art Monte Carlo simulation results. They support recent results of the GFD (grouping of Feynman diagrams) theory, stating that these singularities are described by certain nontrivial exponents, which differ from those predicted earlier by perturbative treatments. Furthermore, we present the recent Monte Carlo simulation results of the three-dimensional Ising model for lattices with linear sizes up to L = 1536, which are very large as compared to L ≤ 128 usually used in the finite-size scaling analysis. These results are obtained, using a parallel OpenMP implementation of the Wolff single-cluster algorithm. The finite-size scaling analysis of the critical exponent η, assuming the usually accepted correction-to-scaling exponent ω ≈ 0.8, shows that η is likely to be somewhat larger than the value 0.0335 ± 0.0025 of the perturbative renormalization group (RG) theory. Moreover, we have found that the actual data can be well described by different critical exponents: η = ω =1/8 and ν = 2/3, found within the GFD theory.

The square lattice Ising model on the rectangle III: Hankel and Toeplitz determinants

Based on the results obtained in (Hucht 2017 J. Phys. A: Math. Theor. 50 065201), we show that the partition function of the anisotropic square lattice Ising model on the L × M rectangle, with open boundary conditions in both directions, is given by the determinant of an M/2 × M/2 Hankel matrix, that equivalently can be written as the Pfaffian of a skew-symmetric M × M Toeplitz matrix. The M − 1 independent matrix elements of both matrices are Fourier coefficients of a certain symbol function, which is given by the ratio of two characteristic polynomials. These polynomials are associated to the different directions of the system, encode the respective boundary conditions, and are directly related through the symmetry of the considered Ising model under exchange of the two directions. The results can be generalized to other boundary conditions and are well suited for the analysis of finite-size scaling functions in the critical scaling limit using Szegő’s theorem.

Density functional for the lattice gas from fundamental measure theory.

We construct a density functional for the lattice gas or Ising model on square and cubic lattices based on lattice fundamental measure theory. To treat the nearest-neighbor attractions between the lattice gas particles, the model is mapped to a multicomponent model of hard particles with additional lattice polymers where effective attractions between particles arise from the depletion effect. The lattice polymers are further treated via the introduction of polymer clusters (labelled by the numbers of polymer they contain) such that the model becomes a multicomponent model of particles and polymer clusters with nonadditive hard interactions. The density functional for this nonadditive hard model is constructed with lattice fundamental measure theory. The resulting bulk phase diagram recovers the Bethe-Peierls approximation and planar interface tensions show a considerable improvement compared to the standard mean-field functional and are close to simulation results in three dimensions. We demonstrate the existence of planar interface solutions at chemical potentials away from coexistence when the equimolar interface position is constrained to arbitrary real values.

A universal framework for metropolis Monte Carlo simulation of magnetic Curie temperature

Recent research of two-dimensional magnetism intrigues rapidly growing attention and broadens the prospects for utility in nano-devices. However, understanding the new magnetic phenomena and the behavior of magnetic centers is still one of the challenges. The Heisenberg model with the Metropolis Monte Carlo method provides an accurate description with continuous degrees of freedom while effective and universal spin update algorithms remain highly desirable. In this study, we propose algorithms for the magnetization switching in the classical Heisenberg model based on the concept of Euler angles and quaternion, which update the spins simply by a rotation matrix and convert to sphere and Cartesian coordinates in a very convenient way. The proposed methods are fully tested and validated by comparisons with the benchmarks of both the two-dimensional square lattice Ising model and the three-dimensional cubic lattice Heisenberg model. As an application example of the two-dimensional ferromagnetic material of CrI3, the simulated Curie temperature is about 42 K, which is in good agreement with the experimental value of 45 K. The update algorithms together with other configuration schemes are compiled into an easy-to-operate program named SEU-mtc, aiming to execute post-processing analysis of the spin microstates and greatly improve the efficiency of Curie temperature simulations based on ab initio methods.