Phase Transition In Ising Model Research Articles

Anomalous Dynamical Phase Transitions of the Ising Model Studied by the Nonequilibrium Ensemble Method: Freezing-by-Heating and Permanent Liquid.

We investigate the nonequilibrium, dynamical phase transition behavior of the one-dimensional Ising model, using the trajectory ensemble approach in the context of the large deviation theory. We introduce a double-biased ensemble, named the s, g-ensemble, based on nonequilibrium steady-state trajectories. The ensemble invokes the time-integrated, trajectory energy as an order parameter, coupled to its conjugate g-field in addition to the dynamical activity and its conjugate field s in the trajectory space. Using the dynamical free energy obtained from the large deviation formalism, we explore the rich behaviors of the dynamical phase transition of the 1D Ising model in the (s, g, T) parameter space, with T being temperature. Among other features, we discover that novel, anomalous dynamical phase transitions are possible due to the decoupling between the dynamical activity and trajectory energy under specific conditions. In particular, we observe that the system exhibits a freezing-by-heating phenomenon as the dynamical activity decreases with temperature under a specific condition. We also find a permanent liquid phase when the equilibrium temperature and the nonequilibrium g-field are exactly balanced against each other. Our results provide a useful tool for exploring the dynamical phase transition phenomena to be investigated in various systems.

Incremental learning of phase transition in Ising model: Preprocessing, finite-size scaling and critical exponents

We investigated an unsupervised machine learning to recognize the phase transition in Ising model by using principal component analysis (PCA) with a preprocessing of raw data, and we realized the large-scale calculations and finite-size scaling analysis based on the incremental PCA. It was shown that the two different phases are clearly and reasonably recognized by the k-means clustering of the first and second principal components. Taking the normalized first principal component as the order parameter, we calculated the critical properties near the phase transition point by using the finite-size scaling method. Our results of the critical temperature and critical exponents are consistent with the classical values, which indicates that the first principal component is capable of catching main features of the phase transition. The present scheme provides an alternative way with emphasis on data preprocessing and incremental PCA for the unsupervised machine learning of the phase transition in Ising model and its related spin models.

How a complex Zeeman field drives quantum phase transitions in the Ising model: From the perspective of geometric phase and symmetry

Inspired by the realizations of imaginary magnetic field in a non-Hermitian atomic systems by laser assisted spin-selective dissipations (Lapp et al., 2019; Li et al., 2019), we investigate the quantum phase transitions of Ising model driven by the constant and staggered complex Zeeman field and find that the quantum phase transition is due to the competition of antiferromagnetic phase and ferromagnetic phase. With the Jordan–Wigner transformation, the model is transformed to a lattice model. By discussing the geometric phase and symmetry of the equivalent model, we can understand the Lee–Yang zeros in a new perspective. The studies indicate the complex parameter significantly changes the topological structure of the complex energy spectrum and the parity–time PT symmetry of the Hamiltonian. Due to the chiral symmetry of the model, the chiral indices are used as topological numbers to distinguish different topological phases. We find the quantum phase transition occurs when the coupling |J| is equal to complex Zeeman field |h| whether in the constant or staggered case.

Image Preparation for Machine-Learning Analysis of Multiple-Size Spin Systems

A new idea of preparing data is proposed for application of machine-learning methods to analysis of phase transitions of Ising model. Presenting spin configurations with different sizes in single-s...

Discrete truncated Wigner approach to dynamical phase transitions in Ising models after a quantum quench

By means of the discrete truncated Wigner approximation we study dynamical phase transitions arising in the steady state of transverse-field Ising models after a quantum quench. Starting from a fully polarized ferromagnetic initial condition these transitions separate a phase with nonvanishing magnetization along the ordering direction from a symmetric phase upon increasing the transverse field. We consider two paradigmatic cases, a one-dimensional long-range model with power-law interactions $\propto 1/r^{\alpha}$ decaying algebraically as a function of distance $r$ and a two-dimensional system with short-range nearest-neighbour interactions. In the former case we identify dynamical phase transitions for $\alpha \lesssim 2$ and we extract the critical exponents from a data collapse of the steady state magnetization for up to 1200 lattice sites. We find identical exponents for $\alpha \lesssim 0.5$, suggesting that the dynamical transitions in this regime fall into the same universality class as the nonergodic mean-field limit. The two-dimensional Ising model is believed to be thermalizing, which we also confirm using exact diagonalization for small system sizes. Thus, the dynamical transition is expected to correspond to the thermal phase transition, which is consistent with our data upon comparing to equilibrium quantum Monte-Carlo simulations. We further test the accuracy of the discrete truncated Wigner approximation by comparing against numerically exact methods such as exact diagonalization, tensor network as well as artificial neural network states and we find good quantitative agreement on the accessible time scales. Finally, our work provides an additional contribution to the understanding of the range and the limitations of qualitative and quantitative applicability of the discrete truncated Wigner approximation.

Abnormal Calcium Leak from Cardiac Sarcoplasmic Reticulum: New Insights Offered by Statistical Physics

Heart muscle contraction is activated by a synchronized systolic Ca release from sarcoplasmic reticulum (SR) via Ca sparks. In disease, Ca sparks fail to terminate, causing a diastolic Ca leak that decreases contraction amplitude and increases the risk of arrhythmia. The mechanisms and treatment of the abnormal Ca leak remain unclear. We have recently shown that spark termination emerges as collective behavior (synchronized closings) of Ca release channels (RyRs) that is identical to synchronization of spin orientation in ferromagnets described by a phase transition in Ising model. We employed the Ising approach to investigate and classify mechanisms of spark termination failure and Ca leak. Methods and Results: The key parameters determining whether spark termination succeeds or fails are SR Ca and RyR opening/closing rates. They define analogues of magnetic field h and the inverse temperature (β) in Ising model. Sparks terminate via a phase transition known as “h polarity reversal” and the leak emerges when h fails to change its sign. This happens when the SR depletes insufficiently and RyR openings remained partially synchronized via Ca-induced-Ca-release, generating long-lasting sparks. Leak can also occur via β, known as Onsager's “order-disorder” transition. This happens at low SR Ca, reducing RyR current and RyRs interactions, resulting in independent RyR openings. The disorder leak is distinguished from synchronized leak by larger Peierls contour lengths reflecting degree of disorder. Abnormal leak results from either a probability imbalance during synchronized firing of RyRs or from disordered RyR firing. Each leak type requires different and balanced treatment to shift RyR operation towards normal spark termination.

Critical Limit Principle of Maximum Entropy and Reinterpretation of Scaling Laws

The critical limit principle of maximum entropy is put forward, it’s a sufficient condition to obtain accurate critical points, and ensure that the new phase system is still in the maximum entropy state. Two representations for the phase transition of Ising models are found; the universal formula of critical points is explained by thermodynamics. From the point of view of fractal geometry and the correspondence between symmetry and conservation, the scaling laws are reinterpreted. The self consistence equations for the universal class are set up, by which and the scaling laws higher accurate exponents to date are obtained. The temperature where the self similar transformation disappears is calculated.

Thermodynamic, critical properties and phase transitions of the Ising model on a square lattice with competing interactions

Abstract The thermodynamic and critical properties, and phase transitions of two-dimensional Ising model on a square lattice with competing interactions are investigated by the Monte Carlo method. Estimations are made for the magnitude relations of the next-nearest-neighbor and nearest-neighbor exchange interactions r = J 2 / J 1 in the value ranges of 0.1≤ r ≤1.0. The anomalies of thermodynamic observables are shown to be present in this model on the interval 0.45≤ r ≤0.5. The phase diagram for the dependence of the critical temperature on a value of next-nearest neighbor interaction is plotted. A phase transition for all values in the interval 0.45≤ r ≤0.5 is shown to be a second order. Our data show that the temperature of the heat capacity maximum at r =0.5 tends to a finite value. The static critical exponents of the heat capacity α , susceptibility γ , order parameter β , correlation length ν , and the Fisher exponent η are calculated by means of the finite-size scaling theory. It is found that the change in next-nearest neighbor interaction value in the range 0.7≤ r ≤1.0 leads to nonuniversal critical behavior.

Phase transitions in Ising models on directed networks.

We examine Ising models with heat-bath dynamics on directed networks. Our simulations show that Ising models on directed triangular and simple cubic lattices undergo a phase transition that most likely belongs to the Ising universality class. On the directed square lattice the model remains paramagnetic at any positive temperature as already reported in some previous studies. We also examine random directed graphs and show that contrary to undirected ones, percolation of directed bonds does not guarantee ferromagnetic ordering. Only above a certain threshold can a random directed graph support finite-temperature ferromagnetic ordering. Such behavior is found also for out-homogeneous random graphs, but in this case the analysis of magnetic and percolative properties can be done exactly. Directed random graphs also differ from undirected ones with respect to zero-temperature freezing. Only at low connectivity do they remain trapped in a disordered configuration. Above a certain threshold, however, the zero-temperature dynamics quickly drives the model toward a broken symmetry (magnetized) state. Only above this threshold, which is almost twice as large as the percolation threshold, do we expect the Ising model to have a positive critical temperature. With a very good accuracy, the behavior on directed random graphs is reproduced within a certain approximate scheme.

Frustrations and phase transitions in the Ising model on square lattice

The phase transitions in two-dimensional antiferromagnetic Ising model are studied on a square lattice by taking the interactions of next-nearest neighbors into account. The model is investigated on basis of the replica Monte Carlo algorithm and the histogrammic analysis of data. The diagram of the critical temperature dependence on an interaction value of next-nearest neighbors is plotted. The studied model reveals the phase transition of second order and the frustration points depending on relative interaction values and magnetic field. A effect, heat capacity splitting near the frustration points, is found.

Secondary Phase Transition of Ising Model

Lttice-spin phonons are considered, which make the heat capacity at the critical temperature satisfy experimental observations better. There is a BEC phase transition in an Ising model attributable to the lattice-spin phonons. We proved that the spin-wave theory only is available after BEC transition, and the magnons have the same characteristics as the lattice-spin phonons’, resulting from quantum effect. Energy-level overlap effect at ultra-low temperature is found. A prediction of BEC phase transition in a crystal is put forward as our theory generalization.

Ising Model Phase Transition Calculation for Ferro-Paramagnetic Lattice

The position of the phase transition in the two dimensional Ising model were determined byusing Monte Carlo simulation in a quadratic for area of variable length with external magnetic fieldswitched off (B = 0). The magnetization (M) per site (µ), magnetic susceptibility (x) of aferromagnetic and paramagnetic materials were calculated as a function of temperature T for(20×20,40×40,60×60), (80×80,120×120,200×200) spin lattice interactions. Nearest neighborinteraction is assumed (i.e. each spin has 4 neighbors); uses periodic boundary conditions. The Curietemperature (Tc = 2.27 J/kB ) is determined by measuring the magnetic susceptibility at which theferromagnetic and paramagnetic undergoes a phase change from order to disorder. There is thus aphase transition defined by the Curie temperature. The Monte Carlo method were used to check theseresults and to confirm the phase transition. The data are analyzed using the Curie-Weiss law whichcontains the Curie temperature as a parameter.

Ising Model Phase Transition Calculation for Ferro-Paramagnetic Lattice

The position of the phase transition in the two dimensional Ising model were determined byusing Monte Carlo simulation in a quadratic for area of variable length with external magnetic fieldswitched off (B = 0). The magnetization (M) per site (µ), magnetic susceptibility (x) of aferromagnetic and paramagnetic materials were calculated as a function of temperature T for(20×20,40×40,60×60), (80×80,120×120,200×200) spin lattice interactions. Nearest neighborinteraction is assumed (i.e. each spin has 4 neighbors); uses periodic boundary conditions. The Curietemperature (Tc = 2.27 J/kB ) is determined by measuring the magnetic susceptibility at which theferromagnetic and paramagnetic undergoes a phase change from order to disorder. There is thus aphase transition defined by the Curie temperature. The Monte Carlo method were used to check theseresults and to confirm the phase transition. The data are analyzed using the Curie-Weiss law whichcontains the Curie temperature as a parameter.

The Ising model and critical behavior of transport in binary composite media

We present a general theory for critical behavior of transport in binary composite media. The theory holds for lattice and continuum percolation models in both the static case with real parameters and the quasi–static case (frequency dependent) with complex parameters. Through a direct, analytic correspondence between the magnetization of the Ising model and the effective parameter problem of two phase random media, we show that the critical exponents of the transport coefficients satisfy the standard scaling relations for phase transitions in statistical mechanics. Our work also shows that delta components form in the underlying spectral measures at the spectral endpoints precisely at the percolation threshold pc and at 1 − pc. This is analogous to the Lee-Yang-Ruelle characterization of the Ising model phase transition, and identifies these transport transitions with the collapse of spectral gaps in these measures.

A mean field Ising model for cortical rotation in amphibian one-cell stage embryos

We propose a new physical mechanism of cortical rotation generation in one-cell embryos of amphibians based on a phase transition in the ensemble of microtubules localized to the cortical region of the cell interior. Microtubules, protein polymers formed from tubulin heterodimers, are highly negatively charged, which results in strong electrostatic interactions over tens of nanometers, even in the presence of counterions that partially screen electrostatic interactions. A simplified model that offers a plausible representation of these effects is based on the Ising Hamiltonian, which has been robustly applied to explain a wide range of order–disorder transitions in physics, chemistry and other sciences. An Ising model phase transition, especially with the supercooperative flow alignment effect of global rotation of the cortex, provides an alternative to models of cortical rotation based on microtubule polymerization or motor molecules. Insofar as there is any reality to the concept that microtubules are involved in consciousness, we propose that cortical rotation in the one-cell embryo is a better place to look for the purported microtubule entanglement or coherence properties than the adult brain.

Fast Phylogeny Reconstruction Through Learning of Ancestral Sequences

Given natural limitations on the length DNA sequences, designing phylogenetic reconstruction methods which are reliable under limited information is a crucial endeavor. There have been two approaches to this problem: reconstructing partial but reliable information about the tree (Mossel in IEEE Comput. Biol. Bioinform. 4:108–116, 2007; Daskalakis et al. in SIAM J. Discrete Math. 25:872–893, 2011; Daskalakis et al. in Proc. of RECOMB 2006, pp. 281–295, 2006; Gronau et al. in Proc. of the 19th Annual SODA 2008, pp. 379–388, 2008), and reaching “deeper” in the tree through reconstruction of ancestral sequences. In the latter category, Daskalakis et al. (Proc. of the 38th Annual STOC, pp. 159–168, 2006) settled an important conjecture of M. Steel (My favourite conjecture. Preprint, 2001), showing that, under the CFN model of evolution, all trees on n leaves with edge lengths bounded by the Ising model phase transition can be recovered with high probability from genomes of length O(logn) with a polynomial time algorithm. Their methods had a running time of O(n 10). Here we enhance our methods from Daskalakis et al. (Proc. of RECOMB 2006, pp. 281–295, 2006) with the learning of ancestral sequences and provide an algorithm for reconstructing a sub-forest of the tree which is reliable given available data, without requiring a-priori known bounds on the edge lengths of the tree. Our methods are based on an intuitive minimum spanning tree approach and run in O(n 3) time. For the case of full reconstruction of trees with edges under the phase transition, we maintain the same asymptotic sequence length requirements as in Daskalakis et al. (Proc. of the 38th Annual STOC, pp. 159–168, 2006), despite the considerably faster running time.

Sequences of phase transitions in Ising models on correlated networks

Using the generic example of Ising spins on scale-free networks, we demonstrate that degree-degree correlations can induce a large number of thermodynamically stable states in networks that otherwise exhibit only the two completely ordered states. The additional stable states are related to the layered network structure. As one increases the temperature, a cascade of first-order phase transitions is found, at which some layers of the network become disordered, while others remain ordered. Negative degree-degree correlations are found to stabilize ordered layers against thermal fluctuations. Positively correlated networks can exhibit an infinite number of ground states and phase transitions, while in negatively correlated networks both numbers are finite.

First- and second-order phase transitions in Ising models on small-world networks: Simulations and comparison with an effective field theory

We perform simulations of random Ising models defined over small-world networks and we check the validity and the level of approximation of a recently proposed effective field theory. Simulations confirm a rich scenario with the presence of multicritical points with first- or second-order phase transitions. In particular, for second-order phase transitions, independent of the dimension d0 of the underlying lattice, the exact predictions of the theory in the paramagnetic regions, such as the location of critical surfaces and correlation functions, are verified. Quite interestingly, we verify that the Edwards-Anderson model with d0=2 is not thermodynamically stable under graph noise.

Critical Binder cumulant in a two-dimensional anisotropic Ising model with competing interactions

The Binder cumulant at the phase transition of Ising models on square lattices with ferromagnetic couplings between nearest neighbors and with competing antiferromagnetic couplings between next-nearest neighbors, along only one diagonal, is determined using Monte Carlo techniques. In the phase diagram a disorder line occurs separating regions with monotonically decaying and with oscillatory spin-spin correlations. Findings on the variation of the critical cumulant with the ratio of the two interaction strengths are compared to related recent results based on renormalization-group calculations.

Phase transition of a one-dimensional Ising model with distance-dependent connections

The critical behavior of the Ising model on a one-dimensional network, which has long-range connections at distances l>1 with the probability theta(l) approximately l(-m), is studied by using Monte Carlo simulations. Through analyzing the Ising model on networks with different m values, this paper discusses the impact of the global correlation, which decays with the increase of m, on the phase transition of the Ising model. Adding the analysis of the finite-size scaling of the order parameter [M], it is observed that in the whole range of 0<m<2, a finite-temperature transition exists, and the critical exponents show consistence with mean-field values, which indicates a mean-field nature of the phase transition.