2D Ising Model Research Articles

Resampling schemes in population annealing — numerical results

Population annealing (PA) is a population-based algorithm that is designed for equilibrium simulations of thermodynamic systems with a rough free energy landscape. It is known to be more efficient in doing so than standard Markov chain Monte Carlo alone. The algorithm has a number of parameters that can be fine-tuned to improve performance. While there is some theoretical and numerical work regarding most of these parameters, there appears to be a gap in the literature concerning the role of resampling in PA. Here, we present a numerical comparison of a number of resampling schemes for PA simulations of the 2D Ising model.

From steady-state TASEP model with open boundaries to 1D Ising model at negative fugacity

We expose a series of exact mappings between particular cases of four statistical physics models: (i) equilibrium 1D lattice gas with nearest-neighbor repulsion, (ii) (1 + 1)D combinatorial heap of pieces, (iii) directed random walks on a half-plane, and (iv) 1D totally asymmetric simple exclusion process (TASEP). In particular, we show that generating function of a 1D steady-state TASEP with open boundaries can be interpreted as a quotient of partition functions of 1D hard-core lattice gases with one adsorbing lattice site and negative fugacity. This result is based on the combination of a representation of a steady-state TASEP configurations in terms of (1 + 1)D heaps of pieces (HP) and a theorem of X Viennot which projects the partition function of (1 + 1)D HP onto that of a single layer of pieces, which in this case is a 1D hard-core lattice gas.

Mixed correlator dispersive CFT sum rules

Conformal field theory (CFT) dispersion relations reconstruct correlators in terms of their double discontinuity. When applied to the crossing equation, such dispersive transforms lead to sum rules that suppress the double-twist sector of the spectrum and enjoy positivity properties at large twist. In this paper, we construct dispersive CFT functionals for correlators of unequal scalar operators in position- and Mellin-space. We then evaluate these functionals in the Regge limit to construct mixed correlator holographic CFT functionals which probe scalar particle scattering in Anti-de Sitter spacetime. Finally, we test properties of these dispersive sum rules when applied to the 3D Ising model, and we use truncated sum rules to find approximate solutions to the crossing equation.

Critical temperature of one-dimensional Ising model with long-range interaction revisited

We present a generalized expression for the transfer matrix of finite and infinite one-dimensional spin chains within a magnetic field with spin pair interaction J/rp, where r∈{1,2,3,…,nv} is the distance between two spins, nv is the number of nearest neighbors reached by the interaction, and 1≤p≤2. Using this transfer matrix, we calculate the partition function, the Helmholtz free energy, and the specific heat for both finite and infinite ferromagnetic 1D Ising models within a zero external magnetic field. We focus on the temperature Tmax where the specific heat reaches its maximum, needed to compute J/(kBTmax) numerically for every value of nv∈{1,2,3,…,30}, which we interpolate and then extrapolate up to the critical temperature as a function of p, by using a novel inter-extrapolation function. We use two different procedures to reach the infinite spin chain with an infinite interaction range: increasing the chain size as well as the interaction range by the same amount and increasing the interaction range for the infinite chain. As we expected, both extrapolations lead to the same critical temperature within their uncertainties by two different concurrent curves. Critical temperatures fall within the upper and lower bounds reported in the literature, showing a better coincidence with many existing approximations for p close to 1 than the p values near 2. It is worth mentioning that the well-known cases for nearest (original Ising model) and next-nearest neighbor interactions are recovered doing nv=1 and nv=2, respectively.

Models of Ni- and Co-ion occupation in LiNi0.5Co0.5PO4 orthophosphate and its magnetic structure

The neutron diffraction, magnetic and heat capacity measurements have been carried out to study the polycrystalline sample LiNi0.5Co0.5PO4 prepared by the glycerol-nitrate synthesis method. Models of Ni- and Co-ion occupation the 4c octahedral position in a crystal structure LiNi0.5Co0.5PO4 are calculated for a paramagnetic state. The best model is the Ni- and Co-ions occupy the 4c site in Pnma patent space-group in sequence Ni–Co–Ni–Co or Co–Ni–Co–Ni. It is shown that nickel ions form ab planes alternating with the planes of cobalt ions in the direction of the c crystallographic axis. At 7 K, an average magnetic moment of 3d-ions is equal to 1.90 (9) μ B. The moments are ordering antiferromagnetically and parallel to the bc plane decreasing to zero at 15 K. In the high-spin state a temperature dependence of the Ni2+/Co2+ ion-magnetic moment is well described within the 2D Ising model with order parameter β = 0.198 and Néel temperature T N = 14.1 (1) K, obtained from heat capacity data. This temperature agrees well with T cr = 14.3 (2) K, determined with magnetic measurement. Maybe the short-range magnetic order exists in LiNi0.5Co0.5PO4 over temperature region (14–16) K, that is confirmed by the maximum on a temperature dependence of the magnetization at 16.1 (5) K.

RETRACTED: Structural, magnetic properties, critical behaviors and magnetic entropy changes of La0.7-x Gd x Ca0.3MnO3 (x=0, 0.05, 0.1) manganites

Abstract Structural, magnetic properties, critical behaviors and magnetic entropy changes of La0.7-x Gd x Ca0.3MnO3 (x=0, 0.05, 0.1) polycrystalline manganites have been investigated. The X-ray diffraction characterization shows that all the samples can be well indexed on an orthorhombic structure with Pnma space group. Magnetic measurements show that polycrystalline samples sequentially display the characteristics of cluster spin glass states, ferromagnetic states, ferromagnetic paramagnetic coexistence states and pure paramagnetic states with increasing temperature. The Curie temperature (Tc) increases with increasing doping concentration x, and the ferromagnetic paramagnetic transform into a second-order phase transitions near the Tc. Critical behaviors have been studied through the modified Arrott plots and the Kouvel-Fisher method. The critical exponents of polycrystalline samples are determined to be close to the critical exponents of the tricritical mean field model (x=0), 3D Ising model (x=0.05) and 3D Heisenberg model (x=0.1), indicating that their ferromagnetic coupling may be the result of the short-range interactions between spins in this system. The maximum magnetic entropy changes reach values of 3.99 J/(kg·K), 2.81 J/(kg·K) and 4.20 J/(kg·K) at the magnetic field of 7T, and the relative cooling power (RCP) values of La0.7-x Gd x Ca0.3MnO3 (x=0, 0.05, 0.1) are 478.8 J/kg, 431.1 J/kg and 536.98 J/kg respectively.

Topological Quantum Statistical Mechanics and Topological Quantum Field Theories

The Ising model describes a many-body interacting spin (or particle) system, which can be utilized to imitate the fundamental forces of nature. Although it is the simplest many-body interacting system of spins (or particles) with Z2 symmetry, the phenomena revealed in Ising systems may afford us lessons for other types of interactions in nature. In this work, we first focus on the mathematical structure of the three-dimensional (3D) Ising model. In the Clifford algebraic representation, many internal factors exist in the transfer matrices of the 3D Ising model, which are ascribed to the topology of the 3D space and the many-body interactions of spins. They result in the nonlocality, the nontrivial topological structure, as well as the long-range entanglement between spins in the 3D Ising model. We review briefly the exact solution of the ferromagnetic 3D Ising model at the zero magnetic field, which was derived in our previous work. Then, the framework of topological quantum statistical mechanics is established, with respect to the mathematical aspects (topology, algebra, and geometry) and physical features (the contribution of topology to physics, Jordan–von Neumann–Wigner framework, time average, ensemble average, and quantum mechanical average). This is accomplished by generalizations of our findings and observations in the 3D Ising models. Finally, the results are generalized to topological quantum field theories, in consideration of relationships between quantum statistical mechanics and quantum field theories. It is found that these theories must be set up within the Jordan–von Neumann–Wigner framework, and the ergodic hypothesis is violated at the finite temperature. It is necessary to account the time average of the ensemble average and the quantum mechanical average in the topological quantum statistical mechanics and to introduce the parameter space of complex time (and complex temperature) in the topological quantum field theories. We find that a topological phase transition occurs near the infinite temperature (or the zero temperature) in models in the topological quantum statistical mechanics and the topological quantum field theories, which visualizes a symmetrical breaking of time inverse symmetry.

Ising-like Critical Behavior of Vortex Lattices in an Active Fluid.

Turbulent vortex structures emerging in bacterial active fluids can be organized into regular vortex lattices by weak geometrical constraints such as obstacles. Here we show, using a continuum-theoretical approach, that the formation and destruction of these patterns exhibit features of a continuous second-order equilibrium phase transition, including long-range correlations, divergent susceptibility, and critical slowing down. The emerging vorticity field can be mapped onto a two-dimensional (2D) Ising model with antiferromagnetic nearest-neighbor interactions by coarse graining. The resulting effective temperature is found to be proportional to the strength of the nonlinear advection in the continuum model.

Solving Conformal Field Theories with Artificial Intelligence

In this Letter, we deploy for the first time reinforcement-learning algorithms in the context of the conformal-bootstrap program to obtain numerical solutions of conformal field theories (CFTs). As an illustration, we use a soft actor-critic algorithm and find approximate solutions to the truncated crossing equations of two-dimensional CFTs, successfully identifying well-known theories like the 2D Ising model and the 2D CFT of a compactified scalar. Our methods can perform efficient high-dimensional searches that can be used to study arbitrary (unitary or nonunitary) CFTs in any spacetime dimension.

Magnetic behavior of ordered DO3-type Fe3Al Heusler alloy

Fe3Al alloy is synthesized in a vacuum arc-melting furnace under an argon atmosphere. The prepared ingot is annealed in a vacuum furnace at 800 °C for 1hr, further at 500 °C for 48 hrs to facilitate the re-crystallization. The alloys are crystallized in a cubic DO3 structure. The magnetic properties are derived from M−H, and M−T measurements and the thermal properties like transition temperature (Tc), heat flow, and heat capacity (Cp) are measured from DSC. We have observed the magnetic transition at a temperature of 925°K with an applied magnetic field of 5000 Oe. The saturation magnetization (Ms) is found to be 188 emu/g, and remanence magnetization (Mr) of 0.78 emu/g. From M−T and DSC measurements it appears that the samples have ferromagnetic ordering with second-order phase transition. The critical exponent is derived by applying mean-field, 3D-Heisenberg, 3D-Ising, and tricritical models.

Shaken dynamics on the 3d cubic lattice

On the space of ±1 spin configurations on the 3d square lattice, we consider the shaken dynamics, a parallel Markovian dynamics that can be interpreted in terms of Probabilistic Cellular Automata. The transition probabilities are defined in terms of pair ferromagnetic Ising-type Hamiltonians with nearest neighbor interaction J, depending on an additional parameter q, measuring the tendency of the system to remain locally in the same state. Odd times and even times have different transition probabilities. We compute the stationary measure of the shaken dynamics and we investigate its relation with the Gibbs measure for the 3d Ising model. It turns out that the two parameters J and q tune the geometry of the underlying lattice. We conjecture the existence of unique line of critical points in J−q plane. By a judicious use of perturbative methods we delimit the region where such curve must lie and we perform numerical simulation to determine it. Our method allows us to find in a unified way the critical values of J for Ising model with first neighbors interaction, defined on a whole class of lattices, intermediate between the two-dimensional hexagonal and the three-dimensional cubic one, such as, for example, the tetrahedral lattice. Finally we estimate the critical exponents of the magnetic susceptibility and show that our model captures a dimensional transition in the geometry of the system at q=0.

Restricted Boltzmann machine representation for the groundstate and excited states of Kitaev Honeycomb model

In this work, the capability of restricted Boltzmann machines (RBMs) to find solutions for the Kitaev honeycomb model with periodic boundary conditions is investigated. The measured groundstate energy of the system is compared and, for small lattice sizes (e.g. with 18 spinors), shown to agree with the analytically derived value of the energy up to a deviation of . Moreover, the wave-functions we find have overlap with the exact ground state wave-functions. Furthermore, the possibility of realizing anyons in the RBM is discussed and an algorithm is given to build these anyonic excitations and braid them for possible future applications in quantum computation. Using the correspondence between topological field theories in (2 + 1)d and 2d conformal field theories, we propose an identification between our RBM states with the Moore-Read state and conformal blocks of the 2d Ising model.

The dynamic critical exponent z for 2d and 3d Ising models from five-loop ε expansion

We calculate the dynamical critical exponent z for 2d and 3d Ising universality classes by means of minimally subtracted five-loop ε expansion obtained for the one-component model A. This breakthrough turns out to be possible through the successful adaptation of the Sector Decomposition technique to the problems of critical dynamics. The obtained fifth perturbative order accompanied by the use of advanced resummation techniques for asymptotic series allows us to find highly accurate numerical estimates of z: for two- and three-dimensional cases we obtain 2.14(2) and 2.0235(8) respectively. The numbers found are in good agreement with recent results obtained using different approaches.

Global exploration of phase behavior in frustrated Ising models using unsupervised learning techniques

We apply a set of machine-learning (ML) techniques for the global exploration of the phase diagrams of two frustrated 2D Ising models with competing interactions. Based on raw Monte Carlo spin configurations generated for random system parameters, we apply principal-component analysis (PCA) and auto-encoders to achieve dimensionality reduction, followed by clustering using the DBSCAN method and a support-vector machine classifier to construct the transition lines between the distinct phases in both models. The results are in very good agreement with available exact solutions, with the auto-encoders leading to quantitatively superior estimates, even for a data set containing only 1400 spin configurations. In addition, the results suggest the existence of a relationship between the structure of the optimized auto-encoder latent space and physical characteristics of both systems. This indicates that the employed approach can be useful in perceiving fundamental properties of physical systems in situations where a priori theoretical insight is unavailable.

Post-spontaneous-symmetry-breaking power-laws after a very strong earthquake: Indication for the preparation of a new strong earthquake or not?

It has recently been found that the evolution of the preparation of a strong earthquake (EQ), as it is monitored through fracture-induced electromagnetic emissions (EME) in the MHz band, presents striking similarity with the evolution of a thermal system as temperature drops, since distinct steps of the evolution of the phenomenon of spontaneous symmetry breaking (SSB) can be identified. Here, the study of fracture-induced EME in the MHz band in analogy to thermal systems is extended to the phase of local fracture structures that follow after the SSB (and the occurrence of the main EQ). By comparing fracture-induced MHz EME associated with the strongest EQs (MW=6.9) that occurred in Greece during the last twenty years with the 3D Ising model, a way to distinguish whether a possible identification of post-SSB power-laws immediately after a very strong EQ is a sign for the preparation of a new strong EQ or not is provided. In the suggested approach, the time series analysis method known as the method of critical fluctuations is used, enhanced by the autocorrelation function.

A Method of the Riemann–Hilbert Problem for Zhang’s Conjecture 2 in a Ferromagnetic 3D Ising Model: Topological Phases

A method of the Riemann–Hilbert problem is employed for Zhang’s conjecture 2 proposed in Philo. Mag. 87 (2007) 5309 for a ferromagnetic three-dimensional (3D) Ising model in a zero external magnetic field. In this work, we first prove that the 3D Ising model in the zero external magnetic field can be mapped to either a (3 + 1)-dimensional ((3 + 1)D) Ising spin lattice or a trivialized topological structure in the (3 + 1)D or four-dimensional (4D) space (Theorem 1). Following the procedures of realizing the representation of knots on the Riemann surface and formulating the Riemann–Hilbert problem in our preceding paper [O. Suzuki and Z.D. Zhang, Mathematics 9 (2021) 776], we introduce vertex operators of knot types and a flat vector bundle for the ferromagnetic 3D Ising model (Theorems 2 and 3). By applying the monoidal transforms to trivialize the knots/links in a 4D Riemann manifold and obtain new trivial knots, we proceed to renormalize the ferromagnetic 3D Ising model in the zero external magnetic field by use of the derivation of Gauss–Bonnet–Chern formula (Theorem 4). The ferromagnetic 3D Ising model with nontrivial topological structures can be realized as a trivial model on a nontrivial topological manifold. The topological phases generalized on wavevectors are determined by the Gauss–Bonnet–Chern formula, in consideration of the mathematical structure of the 3D Ising model. Hence we prove the Zhang’s conjecture 2 (main theorem). Finally, we utilize the ferromagnetic 3D Ising model as a platform for describing a sensible interplay between the physical properties of many-body interacting systems, algebra, topology, and geometry.

Non-invertible topological defects in 4-dimensional $\mathbb {Z}_2$ pure lattice gauge theory

Abstract We explore topological defects in the 4D pure $\mathbb {Z}_2$ lattice gauge theory. This theory has 1-form $\mathbb {Z}_{2}$ center symmetry as well as Kramers–Wannier–Wegner (KWW) duality. We construct the KWW duality topological defects in a similar way to those constructed by Aasen et al. [J. Phys. A 49, 354001 (2016)] for the 2D Ising model. These duality defects turn out to be non-invertible. We also construct 1-form $\mathbb {Z}_{2}$ symmetry defects as well as the junctions between the KWW duality defects and 1-form $\mathbb {Z}_{2}$ center symmetry defects. The crossing relations between these defects are derived. The expectation values of some configurations of these topological defects are calculated by using these crossing relations.

Phase-ordering kinetics in the Allen-Cahn (Model A) class: Universal aspects elucidated by electrically induced transition in liquid crystals.

The two-dimensional (2D) Ising model is the statistical physics textbook example for phase transitions and their kinetics. Quenched through the Curie point with Glauber rates, the late-time description of the ferromagnetic domain coarsening finds its place at the scalar sector of the Allen-Cahn (or Model A) class, which encompasses phase-ordering kinetics endowed with a nonconserved order parameter. Resisting exact results sought for theoreticians since Lifshitz's first account in 1962, the central quantities of 2D Model A-most scaling exponents and correlation functions-remain known up to approximate theories whose disparate outcomes urge experimental assessment. Here we perform such assessment based on a comprehensive study of the coarsening of 2D twisted nematic liquid crystals whose kinetics is induced by a superfast electrical switching from a spatiotemporally chaotic (disordered) state to a two-phase concurrent, equilibrium one. Tracking the dynamics via optical microscopy, we first show the sharp evidence of well-established Model A aspects, such as the dynamic exponent z=2 and the dynamic scaling hypothesis, to then move forward. We confirm the Bray-Humayun theory for Porod's regime describing intradomain length scales of the two-point spatial correlators and show that their nontrivial decay beyond the Porod's scale can be captured in a free-from-parameter fashion by Gaussian theories, namely the Ohta-Jasnow-Kawasaki (OJK) and Mazenko theories. Regarding time-related statistics, we corroborate the aging hypothesis in Model A systems, which includes the collapse of two-time correlators into a master curve whose format is, actually, best accounted for by a solution of the local scaling invariance theory: the same solution that fits the 2D nonconserved Ising model correlator along with the Fisher-Huse conjecture. We also suggest the true value for the local persistence exponent in Model A class, in disfavor of the exact outcome for the diffusion and OJK equations. Finally, we observe a fractal morphology for persistence clusters and extract their universal dimension. Given its accuracy and possibilities, this experimental setup may work as a prototype to address further universality issues in the realm of nonequilibrium systems.

Connecting Complex Electronic Pattern Formation to Critical Exponents

Scanning probes reveal complex, inhomogeneous patterns on the surface of many condensed matter systems. In some cases, the patterns form self-similar, fractal geometric clusters. In this paper, we advance the theory of criticality as it pertains to those geometric clusters (defined as connected sets of nearest-neighbor aligned spins) in the context of Ising models. We show how data from surface probes can be used to distinguish whether electronic patterns observed at the surface of a material are confined to the surface, or whether the patterns originate in the bulk. Whereas thermodynamic critical exponents are derived from the behavior of Fortuin–Kasteleyn (FK) clusters, critical exponents can be similarly defined for geometric clusters. We find that these geometric critical exponents are not only distinct numerically from the thermodynamic and uncorrelated percolation exponents, but that they separately satisfy scaling relations at the critical fixed points discussed in the text. We furthermore find that the two-dimensional (2D) cross-sections of geometric clusters in the three-dimensional (3D) Ising model display critical scaling behavior at the bulk phase transition temperature. In particular, we show that when considered on a 2D slice of a 3D system, the pair connectivity function familiar from percolation theory displays more robust critical behavior than the spin-spin correlation function, and we calculate the corresponding critical exponent. We discuss the implications of these two distinct length scales in Ising models. We also calculate the pair connectivity exponent in the clean 2D case. These results extend the theory of geometric criticality in the clean Ising universality classes, and facilitate the broad application of geometric cluster analysis techniques to maximize the information that can be extracted from scanning image probe data in condensed matter systems.

Work, Entropy Production, and Thermodynamics of Information under Protocol Constraints

In many real-world situations, there are constraints on the ways in which a physical system can be manipulated. We investigate the entropy production (EP) and extractable work involved in bringing a system from some initial distribution $p$ to some final distribution $p'$, given that the set of master equations available to the driving protocol obeys some constraints. We first derive general bounds on EP and extractable work, as well as a decomposition of the nonequilibrium free energy into an "accessible free energy" (which can be extracted as work, given a set of constraints) and an "inaccessible free energy" (which must be dissipated as EP). In a similar vein, we consider the thermodynamics of information in the presence of constraints, and decompose the information acquired in a measurement into "accessible" and "inaccessible" components. This decomposition allows us to consider the thermodynamic efficiency of different measurements of the same system, given a set of constraints. We use our framework to analyze protocols subject to symmetry, modularity, and coarse-grained constraints, and consider various examples including the Szilard box, the 2D Ising model, and a multi-particle flashing ratchet.