Ising Ferromagnet Research Articles

What will be the Euclidean dimension of an Ising ferromagnetic cubic shell?

The equilibrium and nonequilibrium properties of an Ising ferromagnetic cubic shell have been extensively studied by Monte Carlo simulation using Metropolis single spin flip algorithm. Although, geometrically the Euclidean dimension of the cubical shell is three, interestingly, the Ising ferromagnetic cubic shell undergoes ferromagnetic phase transition at a temperature which is very close to that for two-dimensional Ising ferromagnet. Surprisingly, the Ising ferromagnetic cubic shell shows a strange (neither exponential nor stretched exponential) kind of relaxation behaviour, instead of exponential relaxation as usually observed in the two dimensional Ising ferromagnet. The metastable lifetime of a ferromagnetic Ising cubical shell is studied as a function of the applied magnetic field. Here also, the cubic shell behaves more likely a two-dimensional object as found from statistical analysis and comparison with Becker-Döring prediction of classical nucleation theory.

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}.

Superconcentration for minimal surfaces in first passage percolation and disordered Ising ferromagnets

We consider the standard first passage percolation model on Zd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Z}}^ d$$\\end{document} with a distribution G taking two values 0<a<b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<a<b$$\\end{document}. We study the maximal flow through the cylinder [0,n]d-1×[0,hn]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[0,n]^ {d-1}\ imes [0,hn]$$\\end{document} between its top and bottom as well as its associated minimal surface(s). We prove that the variance of the maximal flow is superconcentrated, i.e. in O(nd-1logn)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(\\frac{n^{d-1}}{\\log n})$$\\end{document}, for h≥h0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h\\ge h_0$$\\end{document} (for a large enough constant h0=h0(a,b)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_0=h_0(a,b)$$\\end{document}). Equivalently, we obtain that the ground state energy of a disordered Ising ferromagnet in a cylinder [0,n]d-1×[0,hn]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[0,n]^ {d-1}\ imes [0,hn]$$\\end{document} is superconcentrated when opposite boundary conditions are applied at the top and bottom faces and for a large enough constant h≥h0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h\\ge h_0$$\\end{document} (which depends on the law of the coupling constants). Our proof is inspired by the proof of Benjamini–Kalai–Schramm (Ann Probab 31:1970–1978, 2003). Yet, one major difficulty in this setting is to control the influence of the edges since the averaging trick used in Benjamini et al. (Ann Probab 31:1970–1978, 2003) fails for surfaces. Of independent interest, we prove that minimal surfaces (in the present discrete setting) cannot have long thin chimneys.

Enhanced magnetocaloric effect in melt-spun Mn[formula omitted]Fe[formula omitted]Ge3 (x = 0 and 1) ribbons with anisotropy induced spin-dimensionality crossover

We report the structural, magnetic, magnetocaloric and ferromagnetic resonance properties of melt-spun Mn5−xFexGe3 (x = 0 and 1) ribbons. The melt-spun Mn5Ge3 ribbons exhibit hexagonal structure with preferential growth along c-axis while Fe doping retains the structure but changes preferential growth in in-plane direction. Fe doping enhances the ferromagnetic to paramagnetic phase transition temperature of Mn5Ge3 ribbons. Elaborate isothermal magnetization data analyses in the critical region yield critical exponents β= 0.344(1), γ= 1.284(1) and δ= 4.70(1) with critical temperature TC= 303.64(1) K for x = 0 while β= 0.389(1), γ= 1.334(1) and δ= 4.441(1) with TC= 330.84(1) K for x = 1. Further, the magnetization data in the vicinity of TC satisfies the magnetic scaling equations of state characteristic of a second-order phase transition. The magnetocaloric performance is reflected through maximum entropy change −ΔSmmax(T) ∼ 5.0(2.9) J kg−1 K−1; refrigerant capacity RC ∼ 505(302) J kg−1 and temperature-averaged entropy TEC(10 K) ∼ 4.92(2.28) J kg−1 K−1 at an applied field of 5 T for x = 0(1) ribbons, respectively. −ΔSmmax(T) embracing the critical region is found to be well-described by the critical exponents that are consistent with those determined by the modified Arrott plots and Kouvel–Fisher analyses. Renormalization group calculations further confirm that Mn5Ge3 and Mn4FeGe3 systems behave as 3D Ising and 3D Heisenberg ferromagnets in the critical region wherein short-range extended magnetic interactions decay as J(r) ≈ r−4.995 and r−4.895 with intermoment distance, respectively. Ferromagnetic resonance study reveals that with the introduction of Fe, the magnetocrystaline anisotropy (Ku∼1.4×106 erg/cm3) of Mn5Ge3 gets suppressed so much so that the system exhibits a crossover from 3D Ising to 3D Heisenberg universality class in the critical region.

Exploring the equilibrium and dynamic phase transition properties of the Ising ferromagnet on a decorated triangular lattice.

We study the equilibrium and dynamic phase transition properties of a two-dimensional Ising model on a decorated triangular lattice under the influence of a time-dependent magnetic field composed of a periodic square wave part plus a time-independent bias term. Using Monte Carlo simulations with a standard Metropolis algorithm, we determine the equilibrium critical behavior in zero field. At a fixed temperature corresponding to the multidroplet regime, we locate the relaxation time and the dynamic critical half period at which a dynamic phase transition takes place between ferromagnetic and paramagnetic states. Benefiting from finite-size scaling theory, we estimate the dynamic critical exponent ratios for the dynamic order parameter and its scaled variance, respectively. The response function of the average energy is found to follow a logarithmic scaling as a function of lattice size. At the critical half period and in the vicinity of a small bias field regime, the average of the dynamic order parameter obeys a scaling relation with a dynamic scaling exponent which is very close to the equilibrium critical isotherm value. Finally, in the slow critical dynamics regime, investigation of metamagnetic fluctuations in the presence of bias field reveals a symmetric double-peak behavior for the scaled variance contours of the dynamic order parameter and average energy. Our results strongly resemble those previously reported for kinetic Ising models.

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.

Evolution of the inverse magnetocaloric effect for random-field to spin-glass crossover in SmCaCoMnO6−SmMnO3 nanocomposite

We observe an enhancement of inverse magnetocaloric effect (MCE) in $\mathrm{Sm}\mathrm{Ca}\mathrm{Co}\mathrm{Mn}{\mathrm{O}}_{6}$ (94%) and $\mathrm{Sm}\mathrm{Mn}{\mathrm{O}}_{3}$ (6%) nanocomposite due to uniform external field-induced random field (RF) to spin glass (SG) crossover. RF is generally observed in diluted antiferromagnet and dilute dipole-coupled Ising ferromagnet systems. Under the application of magnetic-field ($H$), such systems may show a crossover from RF to SG at low temperatures ($T$). Here, we report a similar type of crossover but in ferrimagnetic $\mathrm{Sm}\mathrm{Ca}\mathrm{Co}\mathrm{Mn}{\mathrm{O}}_{6}$ and antiferromagnetic $\mathrm{Sm}\mathrm{Mn}{\mathrm{O}}_{3}$ nanocomposite. A model of site dilution of the ferrimagnetic phase by the embedded antiferromagnetic regions is proposed. In this composite system, the RF appears at $T&lt;35$ K, which shows a crossover to SG at $T&lt;15$ K under the applied field. The enhanced inverse MCE (with an entropy change, $\ensuremath{-}\mathrm{\ensuremath{\Delta}}{S}_{M}\ensuremath{\sim}\ensuremath{-}0.95$ J ${\mathrm{Kg}}^{\ensuremath{-}1}\phantom{\rule{0.16em}{0ex}}{\mathrm{K}}^{\ensuremath{-}1}$ for a change in $H\ensuremath{\sim}70$ kOe at $T\ensuremath{\sim}7.5$ K) due to the SG state is the consequence of field-induced RF-SG crossover giving highly $H$-dependent $\ensuremath{-}\mathrm{\ensuremath{\Delta}}{S}_{M}$ in the SG regime, a requisite for large MCE.

Anomalous Hall metal and fractional Chern insulator in twisted transition metal dichalcogenides

We predict robust Ising ferromagnetism driven by Coulomb interaction in the metallic phase of twisted transition metal dichalcogenides homobilayers for a range of small twist angles. Due to the presence of spin-valley locking and Chern band, the completely spin polarized state -- a half metal -- has a spin gap and exhibits anomalous Hall effect. We also find that near a magic angle where the Chern band is predicted to be exceptionally flat, the anomalous Hall metal at $1/3$ filling may become unstable at low temperature to a $\sqrt{3}\times\sqrt{3}$ charge density wave, or a fractional Chern insulator.

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.

A closure for the master equation starting from the dynamic cavity method

We consider classical spin systems evolving in continuous time with interactions given by a locally tree-like graph. Several approximate analysis methods have earlier been reported based on the idea of Belief Propagation / cavity method. We introduce a new such method which can be derived in a more systematic manner using the theory of Random Point Processes. Within this approach, the master equation governing the system’s dynamics is closed via a set of differential equations for the auxiliary cavity probabilities. The numerical results improve on the earlier versions of the closure on several important classes of problems. We re-visit here the cases of the Ising ferromagnet and the Viana–Bray spin-glass model.

Dynamic phase transitions on the kagome Ising ferromagnet.

We perform extensive Monte Carlo simulations to investigate the dynamic phase transition properties of the two-dimensional kinetic Ising model on the kagome lattice in the presence of square-wave oscillating magnetic field. Through detailed finite-size scaling analysis, we study universality aspects of the nonequilibrium phase transition. Obtained critical exponents indicate that the two-dimensional kagome-lattice kinetic Ising model belongs to the same universality class with the corresponding Ising model in equilibrium. Moreover, dynamic critical exponent of the local moves used in simulations is determined with high precision. Our numerical results are compatible with the previous ones on kinetic Ising models.

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.

Neutron depolarization due to ferromagnetism and spin freezing in CePd1−xRhx

We report neutron depolarization measurements of the suppression of long-range ferromagnetism and the emergence of magnetic irreversibilities and spin freezing in CePd$_{1-x}$Rh$_x$ around $x^*\approx0.6$. Tracking the temperature versus field history of the neutron depolarization, we find clear signatures of long-range Ising ferromagnetism below a Curie temperature $T_{\rm C}$ for $x=0.4$ and a spin freezing of tiny ferromagnetic clusters below a freezing temperature $T_{\rm F1}$ for $x>x^*$. Under zero-field-cooling/field-heating and for $x>x^*$ a reentrant temperature dependence of the neutron depolarization between $T_{\rm F2}<T_{\rm F1}$ and $T_{\rm F1}$ is microscopically consistent with a thermally activated growth of the cluster size. The evolution of the depolarization as well as the reentrant temperature dependence as a function of Rh content are consistent with the formation of a Kondo-cluster glass below $T_{\rm F1}$ adjacent to a ferromagnetic quantum phase transition at $x^*$.

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.

Uniaxial ferromagnetism in the kagome metal TbV6Sn6

The synthesis and characterization of the vanadium-based kagome metal TbV${_6}$Sn${_6}$ is presented. X-ray measurements confirm this material forms with the same crystal structure type as the recently investigated kagome metals GdV$_6$Sn$_6$ and YV$_6$Sn$_6$, with space group symmetry P6/mmm. A signature of a phase transition at 4.1K is observed in heat capacity, resistivity, and magnetic susceptibility measurements, and both resistivity and magnetization measurements exhibit hysteresis in magnetic field. Furthermore, a strikingly large anisotropy in the magnetic susceptibility was observed, with the c-axis susceptibility nearly 100 times the ab plane susceptibility at 2K. This is highly suggestive of uniaxial ferromagnetism, and the large size of 9.4$\mu_b$/f.u. indicates the Tb$^{3+}$ $4f$ electronic moments cooperatively align perpendicular to the V kagome lattice plane. The entropy at the phase transition is nearly Rln(2), indicating that the CEF ground state of the Tb$^{3+}$ ion is a doublet, and therefore the sublattice of $4f$ electrons in this material can be shown to map at low temperatures to the Ising model in a D$_{6h}$ symmetry environment. Hall measurements at temperatures from 300K to 1.7K can be described by two-band carrier transport at temperatures below around 150K, with a large increase in both hole and electron mobilities, similar to YV$_6$Sn$_6$, and an anomalous Hall effect is seen below the ordering temperature. Angle-resolved photoemission measurements above the magnetic ordering temperature reveal typical kagome dispersions. Our study presents TbV${_6}$Sn${_6}$ as an ideal system to study the interplay between Ising ferromagnetism and non-trivial electronic states emerging from a kagome lattice.

High Curie temperature and large perpendicular magnetic anisotropy in two-dimensional half metallic OsI3 monolayer with quantum anomalous Hall effect

The 5d transition heavy elements with large spin-orbit coupling (SOC) effect can induce a large magnetic anisotropy energy (MAE) in magnetic materials. Meanwhile, the asymmetric occupation of 5d orbital electrons in transition metal trihalides can also trigger interesting orbital ordering. Here, the electronic structure and magnetic properties of two-dimensional (2D) OsI3 monolayer were investigated by first-principles calculations. The OsI3 monolayer has two orbital orderings with low spin states, where the half-metallic state is an Ising ferromagnet with a Curie temperature (TC) of 208 K and a perpendicular magnetic anisotropy (PMA) of −45.8 meV, but the Mott insulator state has an in-plane magnetic anisotropy (IMA) of 13.7 meV. The half-metallic state included non-self-consistent SOC calculations opens a band gap of 118.3 meV, showing a quantum anomalous Hall effect (QAHE) with a Chern number (C) of −1. A topologically trivial band gap opened in self-consistent SOC calculations, where QAHE of C = −2 can be realized by shifting Fermi surface. The in-plane biaxial strain can significantly tailor the MAE of the half-metallic state, yielding a PMA of −33.0 and −50.7 meV at a compressive strain of 6% and −2%. Meanwhile, TC rises to 275 K at a tensile strain of 6%. The transition between Mott insulator and half-metallic states can also be regulated by strain. These findings suggest that OsI3 monolayer is promising for 2D magnetism and spintronics.

Critical properties of the Ising ferromagnet VI[formula omitted] monolayer

Motivated by the experimental observation of two-dimensional van der Waals ferromagnets, we investigated the critical properties of Ising ferromagnet VI3 monolayer using the real-space renormalization group method. It is worth noting that a block-spin construction we choose contains an even number of sites. Thereupon, we present a scheme that combines the majority rule and central rule to determine the block-spin values. The critical temperature TC is calculated to be 46.8 K under the first-order cumulant-expansion approximation, which is in excellent agreement with the recent numerical result of TC = 46 K using the Metropolis Monte Carlo simulations based on the classical Heisenberg model, and is slightly lower than the experimental value TC = 50 K of the corresponding bulk VI3. Furthermore, the critical exponents β and ν are estimated, and compared with the mean-field results and the exact solutions, respectively. Our numerical result β≅0.324 suggests that the magnetic phase transition of the VI3 monolayer appears to lie between the two-dimensional Ising universality class (β=0.125) and three-dimensional Heisenberg universality class (β=0.365). The theoretical results in this paper may stimulate further experimental studies of the VI3 monolayer.

On the neural network flow of spin configurations

We study the so-called neural network flow of spin configurations in the 2-d Ising ferromagnet. This flow is generated by successive reconstructions of spin configurations, obtained by an artificial neural network like a restricted Boltzmann machine or an autoencoder. It was reported recently that this flow may have a fixed point at the critical temperature of the system, and even allow the computation of critical exponents. Here we focus on the flow produced by a fully-connected autoencoder, and we refute the claim that this flow converges to the critical point of the system by directly measuring physical observables, and showing that the flow strongly depends on the network hyperparameters. We explore the network metric, the reconstruction error, and we relate it to the so called intrinsic dimension of data, to shed light on the origin and properties of the flow.

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.