Two-dimensional Ising Ferromagnet Research Articles

Dynamical critical behavior of the two-dimensional three-state Potts model.

We investigate the dynamical critical behavior of the two-dimensional three-state Potts model with single spin-flip dynamics in equilibrium. We focus on the mean-squared deviation of the magnetization M (MSD_{M}) as a function of time, as well as on the autocorrelation function of M. Our simulations reveal the existence of two crossover behaviors at times τ_{1}∼L^{z_{1}} and τ_{2}∼L^{z_{2}}, separating three dynamical regimes. MSD_{M} appears to shift from ordinary diffusion in the first regime, to anomalous diffusion in the second, and finally to be constant in the third regime. The magnetization autocorrelation function, on the other hand, is found to fluctuate between exponential decay, stretched-exponential decay, and then again exponential decay along these three regimes. This behavior is in agreement with the one reported recently for the two-dimensional Ising ferromagnet [Phys. Rev. E 108, 034118 (2023)2470-004510.1103/PhysRevE.108.034118], indicating that the existence of two dynamic critical exponents is not a peculiarity of the Ising model itself. A comparison of both MSD_{M} and the magnetization's autocorrelation function suggests that within our numerical accuracy the exponents z_{1} and z_{2} are shared between the Ising and three-state Potts models at least for the particular case of single spin-flip dynamics studied here, even though their equilibrium universality classes are clearly distinct. Continuity of MSD_{M} requires that α(z_{2}-z_{1})=γ/ν-z_{1}, in which α is the anomalous exponent in the intermediate regime. Since the ratio γ/ν is not shared between the two models, it follows that α is not shared either, an aspect well verified in our simulations. Finally, we also discuss the relevance of our main findings using another useful observable, namely the line magnetization M_{l}.

Critical dynamical behavior of the Ising model.

We investigate the dynamical critical behavior of the two- and three-dimensional Ising models with Glauber dynamics in equilibrium. In contrast to the usual standing, we focus on the mean-squared deviation of the magnetization M,MSD_{M}, as a function of time, as well as on the autocorrelation function of M. These two functions are distinct but closely related. We find that MSD_{M} features a first crossover at time τ_{1}∼L^{z_{1}}, from ordinary diffusion with MSD_{M}∼t, to anomalous diffusion with MSD_{M}∼t^{α}. Purely on numerical grounds, we obtain the values z_{1}=0.45(5) and α=0.752(5) for the two-dimensional Ising ferromagnet. Related to this, the magnetization autocorrelation function crosses over from an exponential decay to a stretched-exponential decay. At later times, we find a second crossover at time τ_{2}∼L^{z_{2}}. Here, MSD_{M} saturates to its late-time value ∼L^{2+γ/ν}, while the autocorrelation function crosses over from stretched-exponential decay to simple exponential one. We also confirm numerically the value z_{2}=2.1665(12), earlier reported as the single dynamic exponent. Continuity of MSD_{M} requires that α(z_{2}-z_{1})=γ/ν-z_{1}. We speculate that z_{1}=1/2 and α=3/4, values that indeed lead to the expected z_{2}=13/6 result. A complementary analysis for the three-dimensional Ising model provides the estimates z_{1}=1.35(2),α=0.90(2), and z_{2}=2.032(3). While z_{2} has attracted significant attention in the literature, we argue that for all practical purposes z_{1} is more important, as it determines the number of statistically independent measurements during a long simulation.

Effects of geometry, boundary condition and dynamical rules on the magnetic relaxation of Ising ferromagnet

We have studied the magnetic relaxation behavior of a two-dimensional Ising ferromagnet by Monte Carlo simulation. Our primary goal is to investigate the effects of the system’s geometry (area preserving), boundary conditions and dynamical rules on the relaxation behavior. The Glauber and Metropolis dynamical rules have been employed. The systems with periodic and open boundary conditions are studied. The major findings are the exponential relaxation and the dependence of relaxation time ([Formula: see text]) on the aspect ratio [Formula: see text] (length over breadth having fixed area). A power law dependence ([Formula: see text]) has been observed for larger values of aspect ratio ([Formula: see text]). The exponent ([Formula: see text]) has been found to depend linearly ([Formula: see text]) on the system’s temperature ([Formula: see text]). The transient behaviors of the spin-flip density have been investigated for both surface and bulk/core. The size dependencies of saturated spin-flip density significantly differ for the surface and the bulk/core. Both the saturated bulk/core and saturated surface spin-flip density was found to follow the logarithmic dependence [Formula: see text] with the system size. The faster relaxation was observed for open boundary condition with any kind (Metropolis/Glauber) of dynamical rule. Similarly, Metropolis algorithm yields faster relaxation for any kind (open/periodic) of boundary condition.

Construction of Long-Range Magnetic Sequences on Different Surfaces of BaTiO3.

Using the first-principles spin-density-functional theory calculations, we studied the origin of ferromagnetism from non-magnetic ferroelectric barium titanate (BaTiO3 ) and found out vacancies in different surface can successfully contribute to the origin of ferromagnetism. Accurately, our findings demonstrate that both O and Ti vacancies induce ferromagnetism on the (001) and (010) surfaces of BaTiO3 , and the optimal Ti-O bond length can control the vacancy-induced spin density that is delocalized or concentrated in the real space outside the vacancy, and it helps to enhance our understanding on the long-range magnetic order induced by the vacancy. In addition, intrinsic magnetism is shown on the defect-free (110) surface, and the structure is found to be a near-ideal two-dimensional Ising ferromagnet with large magnetocrystalline anisotropy, and it supplies the platform for studying basic spin behavior of BaTiO3 and more according materials.

Weighted averages in population annealing: Analysis and general framework.

Population annealing is a powerful sequential Monte Carlo algorithm designed to study the equilibrium behavior of general systems in statistical physics through massive parallelism. In addition to the remarkable scaling capabilities of the method, it allows for measurements to be enhanced by weighted averaging [J. Machta, Phys. Rev. E 82, 026704 (2010)1539-375510.1103/PhysRevE.82.026704], admitting to reduce both systematic and statistical errors based on independently repeated simulations. We give a self-contained introduction to population annealing with weighted averaging, generalize the method to a wide range of observables such as the specific heat and magnetic susceptibility and rigorously prove that the resulting estimators for finite systems are asymptotically unbiased for essentially arbitrary target distributions. Numerical results based on more than 10^{7} independent population annealing runs of the two-dimensional Ising ferromagnet and the Edwards-Anderson Ising spin glass are presented in depth. In the latter case, we also discuss efficient ways of measuring spin overlaps in population annealing simulations.

Phase diagram of the dipolar Ising ferromagnet on a kagome lattice

We study the field--temperature phase diagram of the two-dimensional dipolar Ising ferromagnet on a kagome lattice with a specific ratio between the exchange and dipolar constants, $\delta = 1$. Using the stochastic cutoff (SCO) $O(N)$ Monte Carlo method, we calculated order parameters for stripe and bubble phases and other thermodynamical quantities. We find two kinds of stripe phases at low fields, where the arrangement of the branch spins neighboring the stripe frame varies, and two bubble phases at high fields, in which three-spin domains (bubbles) form a regular triangular lattice but the triangular array of bubbles changes on a kagome lattice. We also find that with increasing the field, there exist a disordered phase between the stripe and bubble phases and between the two bubble phases. We discuss the details of the features of these phases and phase transitions.

Asymptotic states of Ising ferromagnets with long-range interactions.

It is known that, after a quench to zero temperature (T=0), two-dimensional (d=2) Ising ferromagnets with short-range interactions do not always relax to the ordered state. They can also fall in infinitely long-lived striped metastable states with a finite probability. In this paper, we study how the abundance of striped states is affected by long-range interactions. We investigate the relaxation of d=2 Ising ferromagnets with power-law interactions by means of Monte Carlo simulations at both T=0 and T≠0. For T=0 and the finite system size, the striped metastable states are suppressed by long-range interactions. In the thermodynamic limit, their occurrence probabilities are consistent with the short-range case. For T≠0, the final state is always ordered. Further, the equilibration occurs at earlier times with an increase in the strength of the interactions.

Universality Class of the Nonequilibrium Phase Transition in Two-Dimensional Ising Ferromagnet Driven by Propagating Magnetic Field Wave

The finite size analysis of the nonequilibrium phase transition, in two dimensional Ising ferromagnet driven by plane propagating magneticwave, is studied by Monte Carlo simulation. It is observed that the system undergoes a nonequilibrium dynamicphase transition from a high temperature dynamically symmetric ({\it propagating}) phase to a low temperature dynamically symmetry-broken ({\it pinned}) phase as the system is cooled below the transition temperature. This transition temperature is determined precisely by studying the fourth order Binder Cumulant of the dynamic order parameter as a function of temperature for different system sizes ($L$). From the finite size analysis of dynamic order parameter ($Q_L \sim L^{-{{\beta} \over {\nu}}}$) and the dynamic susceptibility ($\chi^{Q}_{L} \sim L^{{\gamma} \over {\nu}}$), we have estimated the critical exponents $\beta/\nu=0.146\pm0.025$ and $\gamma/\nu=1.869\pm0.135$ (measured from the data read at the critical temperature obtained from Binder cumulant), and $\gamma/\nu=1.746\pm0.017$ (measured from the peak positions of dynamic susceptibility). Our results indicate that such driven Ising ferromagnet belongs to the same universality class of the two-dimensional equilibrium Ising ferromagnet (where $\beta/\nu=1/8$ and $\gamma/\nu=7/4$), within the limits of statistical errors.

Temperature-field phase diagram of the two-dimensional dipolar Ising ferromagnet

We study field-induced phase transitions in the two-dimensional dipolar Ising ferromagnet with a specific ratio between the exchange and dipolar constants, $\delta=1$, which exhibits a stripe-ordered phase with the width of one lattice unit at low temperatures without magnetic field. By using a mean-field (MF) approximation and a Monte Caro (MC) method with the stochastic-cutoff algorithm, which is an $O(N)$ simulation method, we show the temperature-field phase diagrams. In the MC study the orientational order and the structure factor are evaluated. Second-order transition points are determined by a finite-size-scaling analysis and first-order transition points are identified by the analysis of the energy histogram. Although both the MF and MC phase diagrams consist of wide regions of several stripe-ordered phases and narrow regions between them characterized by complicated stripe patterns, they show qualitative and quantitative differences in possible phases and phase boundaries. In the MF phase diagram, three main stripe-ordered phases exhibit a nesting structure, while in the MC phase diagram, two main stripe-ordered phases are located separately, which causes a characteristic field-induced reentrant transition of the orientational order.

Raman fingerprint of two terahertz spin wave branches in a two-dimensional honeycomb Ising ferromagnet

Two-dimensional (2D) magnetism has been long sought-after and only very recently realized in atomic crystals of magnetic van der Waals materials. So far, a comprehensive understanding of the magnetic excitations in such 2D magnets remains missing. Here we report polarized micro-Raman spectroscopy studies on a 2D honeycomb ferromagnet CrI3. We show the definitive evidence of two sets of zero-momentum spin waves at frequencies of 2.28 terahertz (THz) and 3.75 THz, respectively, that are three orders of magnitude higher than those of conventional ferromagnets. By tracking the thickness dependence of both spin waves, we reveal that both are surface spin waves with lifetimes an order of magnitude longer than their temporal periods. Our results of two branches of high-frequency, long-lived surface spin waves in 2D CrI3 demonstrate intriguing spin dynamics and intricate interplay with fluctuations in the 2D limit, thus opening up opportunities for ultrafast spintronics incorporating 2D magnets.

Self-averaging in the random two-dimensional Ising ferromagnet.

We study sample-to-sample fluctuations in a critical two-dimensional Ising model with quenched random ferromagnetic couplings. Using replica calculations in the renormalization group framework we derive explicit expressions for the probability distribution function of the critical internal energy and for the specific heat fluctuations. It is shown that the disorder distribution of internal energies is Gaussian, and the typical sample-to-sample fluctuations as well as the average value scale with the system size L like ∼Llnln(L). In contrast, the specific heat is shown to be self-averaging with a distribution function that tends to a δ peak in the thermodynamic limit L→∞. While previously a lack of self-averaging was found for the free energy, we here obtain results for quantities that are directly measurable in simulations, and implications for measurements in the actual lattice system are discussed.

Analysis of the phase transition in the two-dimensional Ising ferromagnet using a Lempel-Ziv string-parsing scheme and black-box data-compression utilities.

In this work we consider information-theoretic observables to analyze short symbolic sequences, comprising time series that represent the orientation of a single spin in a two-dimensional (2D) Ising ferromagnet on a square lattice of size L(2)=128(2) for different system temperatures T. The latter were chosen from an interval enclosing the critical point T(c) of the model. At small temperatures the sequences are thus very regular; at high temperatures they are maximally random. In the vicinity of the critical point, nontrivial, long-range correlations appear. Here we implement estimators for the entropy rate, excess entropy (i.e., "complexity"), and multi-information. First, we implement a Lempel-Ziv string-parsing scheme, providing seemingly elaborate entropy rate and multi-information estimates and an approximate estimator for the excess entropy. Furthermore, we apply easy-to-use black-box data-compression utilities, providing approximate estimators only. For comparison and to yield results for benchmarking purposes, we implement the information-theoretic observables also based on the well-established M-block Shannon entropy, which is more tedious to apply compared to the first two "algorithmic" entropy estimation procedures. To test how well one can exploit the potential of such data-compression techniques, we aim at detecting the critical point of the 2D Ising ferromagnet. Among the above observables, the multi-information, which is known to exhibit an isolated peak at the critical point, is very easy to replicate by means of both efficient algorithmic entropy estimation procedures. Finally, we assess how good the various algorithmic entropy estimates compare to the more conventional block entropy estimates and illustrate a simple modification that yields enhanced results.

Erratum: Glassy Chimeras could be blind to quantum speedup: Designing better benchmarks for quantum annealing machines [Phys. Rev. X4, 021008 (2014)

Erratum to Phys. Rev. X 4, 021008 (2014): The critical exponent associated with the ferromagnetic susceptibility was computed incorrectly. Furthermore, Ising ferromagnets on the Chimera topology have the same universality class as two-dimensional Ising ferromagnets.

Critical wetting in the two-dimensional Ising ferromagnet confined between inhomogeneous walls

We present a numerical study of the critical wetting behavior of an Ising magnet confined between two walls, separated by a distance L, where short-range inhomogeneous surface magnetic fields act. So, samples are assumed to have a size L × M, L being the width and M the length, respectively. By considering surface fields varying spatially with a given wavelength or period (λ), H 1(x,λ) with 1 ≤ x ≤ M, we found that the wetting temperature is given by the exact result of Abraham [D.B. Abraham, Phys. Rev. Lett. 44, 1165 (1980)] provided that an effective field given by the spacial average value $$\left( {H_{eff} \equiv \tfrac{1} {\lambda }\int_0^\lambda {H_1 (x,\lambda )dx > 0} } \right)$$ is considered. The above results hold in the low wavelength regime, while for λ → ∞ and a bivaluated surface field (i.e., H max for x ≤ M/ 2, and δ H max for x>M/ 2, with 0 <δ< 1), one observes two almost independent wetting transitions, both being compatible with Abraham’s exact results corresponding to H max and δ H max, respectively. On the other hand, for H 1(x,λ) ≠ 0 but H eff = 0 bulk standard critical behavior results is observed.

Communication: On nucleation statistics in small systems.

Non-stationary random formation of stable nuclei from a small metastable system is considered. Distribution of waiting times to observe the first nucleus is examined, and it is shown that the steady-state nucleation rate is given by inverse of the standard deviation, which is independent of the post-critical size n where the nucleus is detected. The mean time, on the other hand, is n-sensitive and contains additional information on transient nucleation and growth effects. The method is applied to Monte Carlo data on nucleation in a cold two-dimensional Ising ferromagnet with Metropolis dynamics, where nucleation rates obtained earlier from low-temperature cluster expansions can provide a strict independent test.

Phase transition in a two-dimensional Ising ferromagnet based on the generalized zero-temperature Glauber dynamics

At zero temperature, based on the Ising model, the phase transition in a two-dimensional square lattice is studied using the generalized zero-temperature Glauber dynamics. Using Monte Carlo (MC) renormalization group methods, the static critical exponents and the dynamic exponent are studied; the type of phase transition is found to be of the first order.

Influence of nonuniform surface magnetic fields in wetting transitions in a confined two-dimensional Ising ferromagnet

Wetting transitions are studied in the two-dimensional Ising ferromagnet confined between walls where competitive surface fields act. In our finite samples of size L×M, the walls are separated by a distance L, M being the length of the sample. The surface fields are taken to be short-range and nonuniform, i.e., of the form H(1),δH(1),H(1),δH(1),..., where the parameter -1≤δ≤1 allows us to control the nonuniformity of the fields. By performing Monte Carlo simulations we found that those competitive surface fields lead to the occurrence of an interface between magnetic domains of different orientation that runs parallel to the walls. In finite samples, such an interface undergoes a localization-delocalization transition, which is the precursor of a true wetting transition that takes place in the thermodynamic limit. By exactly working out the ground state (T=0), we found that besides the standard nonwet and wet phases, a surface antiferromagnetic-like state emerges for δ<-1/3 and large fields (H(1)>3), H(1)(tr)/J=3, δ(tr)=-1/3,T=0, being a triple point where three phases coexist. By means of Monte Carlo simulations it is shown that these features of the phase diagram remain at higher temperatures; e.g., we examined in detail the case T=0.7×T(cb). Furthermore, we also recorded phase diagrams for fixed values of δ, i.e., plots of the critical field at the wetting transition (H(1w)) versus T showing, on the one hand, that the exact results of Abraham [Abraham, Phys. Rev. Lett. 44, 1165 (1980)] for δ=1 are recovered, and on the other hand, that extrapolations to T→0 are consistent with our exact results. Based on our numerical results we conjectured that the exact result for the phase diagram worked out by Abraham can be extended for the case of nonuniform fields. In fact, by considering a nonuniform surface field of some period λ, with λ<<M, e.g., [H(1)(x,λ)>0], one can obtain the effective field H(eff) at a λ coarse-grained level given by H(eff)=1/λ∑(x=1)(λ)H(1)(x,λ). Then we conjectured that the exact solution for the phase diagram is now given by H(eff)/J=F(T), where F(T) is a function of the temperature T that straightforwardly follows from Abraham's solution. The conjecture was exhaustively tested by means of computer simulations. Furthermore, it is found that for δ≠1 the nonwet phase becomes enlarged, at the expense of the wet one, i.e., a phenomenon that we call "surface nonuniformity-induced nonwetting," similar to the already known case of "roughness-induced nonwetting."

Nature versus nurture: Predictability in low-temperature Ising dynamics

Consider a dynamical many-body system with a random initial state subsequently evolving through stochastic dynamics. What is the relative importance of the initial state ("nature") versus the realization of the stochastic dynamics ("nurture") in predicting the final state? We examined this question for the two-dimensional Ising ferromagnet following an initial deep quench from T=∞ to T=0. We performed Monte Carlo studies on the overlap between "identical twins" raised in independent dynamical environments, up to size L=500. Our results suggest an overlap decaying with time as t(-θ)(h) with θ(h)=0.22 ± 0.02; the same exponent holds for a quench to low but nonzero temperature. This "heritability exponent" may equal the persistence exponent for the two-dimensional Ising ferromagnet, but the two differ more generally.

Limiting shapes in two-dimensional Ising ferromagnets

We consider an Ising model on a square lattice with ferromagnetic spin-spin interactions spanning beyond nearest neighbors. Starting from initial states with a single unbounded interface separating ordered phases, we investigate the evolution of the interface subject to zero-temperature spin-flip dynamics. We consider an interface which is initially (i) the boundary of the quadrant or (ii) the boundary of a semi-infinite bar. In the former case the interface recedes from its original location in a self-similar diffusive manner. After a rescaling by √[t], the shape of the interface becomes more and more deterministic; we determine this limiting shape analytically and verify our predictions numerically. The semi-infinite bar acquires a stationary shape resembling a finger, and this finger translates along its axis. We compute the limiting shape and the velocity of the Ising finger.

A Genuine Two‐Dimensional Ising Ferromagnet with Magnetically Driven Re‐entrant Transition

BaFe2(PO4)2 consists of honeycomb layers containing high-spin d6 Fe2+ ions. It is the first genuine two-dimensional oxide Ising ferromagnet (FM) with uniaxial magnetism parallel to the crystallographic three-fold axis. BaFe2(PO4)2 undergoes a re-entrant structural transition between 170 K and its FM critical temperature (65.5 K) owing to competition between the uniaxial magnetism and Jahn–Teller instability.