We report on large-scale Wang-Landau Monte Carlo simulations of the critical behavior of two spin models in two- (2d) and three-dimensions (3d), namely the 2d random-bond Ising model and the pure 3d Blume-Capel model at zero crystal-field coupling. The numerical data we obtain and the relevant finite-size scaling analysis provide clear answers regarding the universality aspects of both models. In particular, for the random-bond case of the 2d Ising model the theoretically predicted strong universality's hypothesis is verified, whereas for the second-order regime of the Blume-Capel model, the expected $d=3$ Ising universality is verified. Our study is facilitated by the combined use of the Wang-Landau algorithm and the critical energy subspace scheme, indicating that the proposed scheme is able to provide accurate results on the critical behavior of complex spin systems.
Transverse field effect on thermodynamic properties of the spin-3/2 Blume-Capel model on rectangular lattice in which the interactions in perpendicular directions differ in signs is studied within the mean field approximation. Phase diagrams in the (transverse field, temperature) plane are constructed for various values of single-ion anisotropy.
The phase diagram of the spin-3/2 Blume-Capel model in two dimensions is explored by conventional finite-size scaling, conformal invariance and Monte Carlo simulations. The model in its $τ$-continuum Hamiltonian version is also considered and compared with others spin-3/2 quantum chains. Our results indicate that differently from the standard spin-1 Blume-Capel model there is no multicritical point along the order-disorder transition line. This is in qualitative agreement with mean field prediction but in disagreement with previous approximate renormalization group calculations. We also presen…
In this work we investigate the Blume-Capel model with infinite-range ferromagnetic interactions and under the influence of a quenched disorder - a random crystal field. For a suitable choice of the random crystal field the model displays a wealth of multicritical behavior, continuous and first-order transition lines, as well as re-entrant behavior. The resulting phase diagrams show a variety of topologies as a function of the disorder parameter $\textit{p}$. A comparison with recent results on the Blume-Capel model in random crystal field is discussed.
This investigation employs a square lattice and Glauber dynamics methodology to probe the effects of diluteness in the crystal-field interaction in a Blume-Capel Ising system under an oscillating magnetic-field. Fourteen different phase diagrams have been observed in temperature-magnetic-field space as the concentration of the crystal-field interactions is varied. Besides, a comparison is given with the results of the pure spin-1 Ising systems.
The first order phase transition of the three-dimensional Blume Capel are investigated using cooling algorithm which improved from Creutz Cellular Automaton for the $D/J=2.9$ parameter value in the first order phase transition region. The analysis of the data using the finite-size effect and the histogram technique indicate that the magnetic susceptibility maxima and the specific heat maxima increase with the system volume ($L^{d}$) at $% D/J=2.9$.
In this letter we study the local persistence of the two--dimensional Blume-- Capel Model by extension of the concept of Glauber dynamics. We verify that for any value of the ratio $α=D/J$ between anisotropy $D$ and exchange $J$ the persistence shows a power law behavior. In particular for $α<0$ we find a persistence exponent $θ_{l}=0.2096(13)$, \textit{i.e.} in the Ising universality class. For $α>0$ ($α\neq 1$) we observe the occurrence of blocking.
The global persistence exponent $θ_g$ is calculated for the two-dimensional Blume-Capel model following a quench to the critical point from both disordered states and such with small initial magnetizations. Estimates are obtained for the nonequilibrium critical dynamics on the critical line and at the tricritical point. Ising-like universality is observed along the critical line and a different value $θ_g =1.080(4)$ is found at the tricritical point.
In this work we investigate the Blume-Capel model with infinite-range ferromagnetic interactions and under the influence of a quenched disorder - a random crystal field. For a suitable choice of the random crystal field the model displays a wealth of multicritical behavior, continuous and first-order transition lines, as well as re-entrant behavior. The resulting phase diagrams show a variety of topologies as a function of the disorder parameter $\textit{p}$. A comparison with recent results on the Blume-Capel model in random crystal field is discussed.
In this paper we study the short-time behavior of the Blume-Capel model at the tricritical point as well as along the second order critical line. Dynamic and static exponents are estimated by exploring scaling relations for the magnetization and its moments at early stage of the dynamic evolution. Our estimates for the dynamic exponents, at the tricritical point, are $z= 2.215(2)$ and $θ= -0.53(2)$.
We report on numerical simulations of the two-dimensional Blume-Capel ferromagnet embedded in the triangular lattice. The model is studied in both its first- and second-order phase transition regime for several values of the crystal field via a sophisticated two-stage numerical strategy using the Wang-Landau algorithm. Using classical finite-size scaling techniques we estimate with high accuracy phase-transition temperatures, thermal, and magnetic critical exponents and we give an approximation of the phase diagram of the model.
We study the Blume-Capel model on the square lattice. To allow for wetting and interfacial adsorption, the spins on opposite boundaries are fixed in two different states, "+1" and "-1", with reduced couplings at one of the boundaries. Using mainly Monte Carlo techniques, of Metropolis and Wang-Landau type, phase diagrams showing bulk and wetting transitions are determined. The role of the non-boundary state, "0", adsorbed preferably at the interface between "-1" and "+1" rich regions, is elucidated.
We study the zero temperature steady state of the random field Blume Capel model with spin-flip Glauber dynamics on a random regular graph. The magnetization m as a function of the external field H is observed to have double hysteresis loops with a return point memory. We also solve the model on a Bethe lattice in the approximation that the spin relaxation dynamics is abelian and find good agreement between simulations on random regular graphs and Bethe lattice calculations for negative values of H.
The trilayered Blume-Capel ($S=1$) magnet with nearest neighbour intralayer ferromagnetic and nearest neighbour interlayer antiferromagnetic interaction is studied by Monte Carlo simulation. Depending on the relative interaction strength and the value of anisotropy the critical temperature (where all the sublattice magnetisations and consequently the total magnetisation vanishes) and the compensation temperature (where the total magnetisation vanishes for a special combination of nonzero sublattice magnetisations) are estimated. The comprehensive phase diagrams with lines of critical temperatu…
We investigate the critical properties of the spin-3/2 Blume-Capel model in two dimensions on a random lattice with quenched connectivity disorder. The disordered system is simulated by applying the cluster hybrid Monte Carlo update algorithm and re-weighting techniques. We calculate the critical temperature as well as the critical point exponents $γ/ν$, $β/ν$, $α/ν$, and $ν$. We find that, contrary of what happens to the spin-1/2 case, this random system does not belong to the same universality class as the regular two-dimensional ferromagnetic model.
A dilute Ising ferromagnet is considered in this study using renormalization group techniques with a hierarchical lattice. A series of phase diagrams have been produced that probe the effects of varying the temperature and concentration of nonmagnetic impurities. Each phase diagram corresponds to a different strength for the internal coupling coefficients on our lattice. Phases have been interpreted and critical exponents calculated for the higher order transitions.
Low-temperature magnetization processes and magnetocaloric properties of a geometrically frustrated spin-1 Blume-Capel model on a triangular lattice are studied by Monte Carlo simulations. The model is found to display qualitatively different behavior depending on the sign of the single-ion anisotropy D. For positive values of D we observe two magnetization plateaus, similar to the spin-1/2 Ising antiferromagnet, and negative isothermal entropy changes for any field intensity. For a range of small negative values of D there are four magnetization plateaus and the entropy changes can be either …
Using the approach based on the Bogoliubov inequality for free energy, we have studied the magnetic properties of the spin-3/2 Blume-Capel quantum model in the presence of the random transverse single-ion anisotropy (RTSIA). By analyzis of the phase diagrams in $τ\times φ_{x} $, $m \times φ_{x}$ and $m \times τ$ plans, we have obtained information about how the presence of anisotropy affects the properties of the system. We have shown that the RTSIA is responsible for changing the topology of the phase diagrams, compared to the pure model, leading to the appearance tricritical point (TCP).
We broaden the study of the statistical physics of the spin-$S$ Blume-Capel model with ferromagnetic mean-field interactions $J$ in competition with short-range antiferromagnetic interactions $K$ in a linear chain in the thermodynamic limit. This work describes the critical behavior of the model when the $S$ takes a half integer and an integer value. In both cases the phase diagrams exhibit new ferromagnetic phases (for certain values of $K$) enclosed by branches emerging from the first-order frontiers of the pure ferromagnetic model. For finite temperatures the complex topologies were obtaine…
We study purely dissipative relaxational dynamics in the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model on the simple cubic lattice by using local algorithms. We perform a finite size scaling analysis of the integrated autocorrelation time of the magnetic susceptibility in equilibrium at the critical point. We obtain $z=2.0245(15)$ for the dynamic critical exponent. As a complement, fully magnetized configurations are suddenly quenched to the critical temperature, giving consistent results for the dynamic critical exponent. Furthermore, our …
We use a self-consistent Ornstein-Zernike approximation to study the Blume-Capel ferromagnet on three-dimensional lattices. The correlation functions and the thermodynamics are obtained from the solution of two coupled partial differential equations. The theory provides a comprehensive and accurate description of the phase diagram in all regions, including the wing boundaries in non-zero magnetic field. In particular, the coordinates of the tricritical point are in very good agreement with the best estimates from simulation or series expansion. Numerical and analytical analysis strongly sugges…
We applied a mean-field approach associated to Monte Carlo simulations in order to study the spin-1 ferromagnetic Blume-Capel model in the square and the linear lattice. This new technique, which we call MFT-MC, determines the molecular field as the magnetization response of a Monte Carlo simulation. The resulting phase diagram is qualitatively correct, in contrast to effective-field approximations, in which the first-order line is not perpendicular to the anisotropy axis at low temperatures. Thermodynamic quantities, as the entropy and the specific heat curves can be obtained so as to analyze…
We carry out a numerical and analytic analysis of the Yang-Lee zeros of the 1D Blume-Capel model with periodic boundary conditions and its generalization on Feynman diagrams for which we include sums over all connected and non-connected rings for a given number of spins. In both cases, for a specific range of the parameters, the zeros originally on the unit circle are shown to departure from it as we increase the temperature beyond some limit. The curve of zeros can bifurcate and become two disjoint arcs as in the 2D case. We also show that in the thermodynamic limit the zeros of both Blume-Ca…
The spin-3/2 Blume-Capel model on a rectangular lattice with the ferromagnetic bilinear short-range interaction $K^{\rm F}=K(1+x)$ in one direction and the anti-ferromagnetic one $K^{\rm AF}=K(-1+x)$ in the perpendicular direction under a longitudinal magnetic field and in the presence of a single-ion anisotropy is investigated within the mean field approximation. The quantity $x$ characterizes coupling between the interactions in the perpendicular directions and its absolute value does not exceed unity. Basing on the study of temperature dependencies of sublattice magnetizations the phase dia…
We consider the Blume-Capel model with the quantum tunneling between the excited states. We find a magnetically ordered phase transition induced by quantum fluctuation in a model. The model has no phase transition in the corresponding classical case. Usually, quantum fluctuation breaks ordered phase as in the case of the transverse field Ising model. However, in present case, an ordered phase is induced by quantum fluctuation. Moreover, we find a phase transition between a quantum paramagnetic phase and a classical diamagnetic phase at zero temperature. We study the properties of the phase tra…
The phase-diagram of the two-dimensional Blume-Capel model with a random crystal field is investigated within the framework of a real-space renormalization group approximation. Our results suggest that, for any amount of randomness, the model exhibits a line of Ising-like continuous transitions, as in the pure model, but no first-order transition. At zero temperature the transition is also continuous, but not in the same universality class as the Ising model. In this limit, the attractor (in the renormalization group sense) is the percolation fixed point of the site diluted spin-1/2 Ising mode…
We present an extensive Monte Carlo investigation of the metastable lifetime through the reversal of the magnetization of spin-$s$ Ising and Blume-Capel models, where $s=\{1/2, 1, 3/2, 2, 5/2, 3, 7/2\}$. The mean metastable lifetime (or reversal time) is studied as a function of the applied magnetic field and for both models is found to obey the Becker-Doring theory, as was initially developed for the case of $s=1/2$ Ising ferromagnet within the classical nucleation theory. Moreover, the decay of metastable volume fraction nicely follows the Avrami's law for all values of $s$ and for both mode…
The dynamic magnetic behaviors of the spin-1 Blume-Capel Ising bilayer system (BCIS) are studied in an oscillating external magnetic field on a two-layer square lattice by utilizing the mean field theory based on Glauber-type stochastic dynamics (DMFT). The dynamic equations describing the time-dependencies of the average magnetizations are obtained with the Master equation. The dynamic phases in this system are found by solving these dynamic equations. The temperature dependence of the dynamic order parameters is examined to characterize the nature (continuous or discontinuous) of the phase t…
We study the thermodynamic properties of the three-dimensional Blume-Capel model on the simple cubic lattice by means of computer simulations. In particular, we implement a parallelized variant of the multicanonical approach and perform simulations by keeping a constant temperature and crossing the phase boundary along the crystal-field axis. We obtain numerical data for several temperatures in both the first- and second-order regime of the model. Finite-size scaling analyses provide us with transition points and the dimensional scaling behavior in the numerically demanding first-order regime,…
Critical phenomena and universality behavior of ferromagnetic thin films described by a spin-1 Blume-Capel Hamiltonian has been examined for various thickness values ranging from 3 to 40 layers. Using effective field theory, we have found that crystal field interactions significantly affects the critical value of surface to bulk ratio of exchange interactions $R_{c}$ at which the critical temperature becomes independent of film thickness $L$. Moreover, we have extracted the shift exponent $λ$ from computed data. Based on the results, we have shown that in the presence of surface exchange enhan…
We study the transient behavior of damage propagation in the two-dimensional spin-$1$ Blume-Capel model using Monte Carlo simulations with Metropolis dynamics. We find that, for a particular region in the second-order transition regime of the crystal field--temperature phase diagram of the model, the average Hamming distance decreases exponentially with time in the weakly damaged system. Additionally, its rate of decay appears to depend linearly on a number of Hamiltonian parameters, namely the crystal field, temperature, applied magnetic field, but also on the amount of damage. Finally, a com…
Systems of particles in a confining potential exhibit a spatially dependent density which fundamentally alters the nature of phase transitions that occur. A specific instance of this situation, which is being extensively explored currently, concerns the properties of ultra-cold, optically trapped atoms. Of interest is how the superfluid-insulator transition is modified by the inhomogeneity, and, indeed, the extent to which a sharp transition survives at all. This paper explores a classical analog of these systems, the Blume-Capel model with a spatially varying single ion anisotropy and/or temp…
We review the pertinent features of the phase diagram of the zero-field Blume-Capel model, focusing on the aspects of transition order, finite-size scaling and universality. In particular, we employ a range of Monte Carlo simulation methods to study the 2D spin-1 Blume-Capel model on the square lattice to investigate the behavior in the vicinity of the first-order and second-order regimes of the ferromagnet-paramagnet phase boundary, respectively. To achieve high-precision results, we utilize a combination of (i) a parallel version of the multicanonical algorithm and (ii) a hybrid updating sch…
We investigate by means of Monte Carlo simulations the dynamic phase transition of the two-dimensional kinetic Blume-Capel model under a periodically oscillating magnetic field in the presence of a quenched random crystal-field coupling. We analyze the universality principles of this dynamic transition for various values of the crystal-field coupling at the originally second-order regime of the corresponding equilibrium phase diagram of the model. A detailed finite-size scaling analysis indicates that the observed nonequilibrium phase transition belongs to the universality class of the equilib…
The Blume-Capel model with infinite-range interactions presents analytical solutions in both canonical and microcanonical ensembles and therefore, its phase diagram is known in both ensembles. This model exhibits nonequivalent solutions and the microcanonical thermodynamical features present peculiar behaviors like nonconcave entropy, negative specific heat, and a jump in the thermodynamical temperature. Examples of nonequivalent ensembles are in general related to systems with long-range interactions that undergo canonical first-order phase transitions. Recently, the extended gaussian ensembl…
The wetting transition of the Blume-Capel model is studied by a finite-size scaling analysis of $L \times M$ lattices where competing boundary fields $\pm H_1$ act on the first row or last row of the $L$ rows in the strip, respectively. We show that using the appropriate anisotropic version of finite size scaling, critical wetting in $d=2$ is equivalent to a "bulk" critical phenomenon with exponents $α=-1$, $β=0$, and $γ=3$. These concepts are also verified for the Ising model. For the Blume-Capel model it is found that the field strength $H_{1c} (T)$ where critical wetting occurs goes to zero…
We consider the two-dimensional Blume-Capel model with zero chemical potential and small magnetic field evolving on a large but finite torus. We obtain sharp estimates for the transition time, we characterize the set of critical configurations, and we prove the metastable behavior of the dynamics as the temperature vanishes.
We employ numerical simulations and finite-size scaling techniques to investigate the properties of the dynamic phase transition that is encountered in the Blume-Capel model subjected to a periodically oscillating magnetic field. We mainly focus on the study of the two-dimensional system for various values of the crystal-field coupling in the second-order transition regime. Our results indicate that the present non-equilibrium phase transition belongs to the universality class of the equilibrium Ising model and allow us to construct a dynamic phase diagram, in analogy to the equilibrium case, …
A new approximating technique is developed so as to study the quantum ferromagnetic spin-1 Blume-Capel model in the presence of a transverse crystal field in the square lattice. Our proposal consists of approaching the spin system by considering islands of finite clusters whose frontiers are surrounded by non-interacting spins that are treated by the effective-field theory. The resulting phase diagram is qualitatively correct, in contrast to most effective-field treatments, in which the first-order line exhibits spurious behavior by not being perpendicular to the anisotropy axis at low tempera…
We investigate the phase diagram for the spin-$3/2$ ferromagnetic Blume-Capel model in a transverse crystal field using the standard mean-field approximation within the framework of Bogoliubov inequality for free energy. We draw a very rich phase diagram with first- and second-order transition lines; tricritical and tetracritical points; critical endpoint of order 2 and double critical endpoint. Additionally, the behaviour of magnetisation as a function of temperature over a wide range of values of both longitudinal and transverse crystal fields is also analysed. To the best of our knowledge, …
We have revisited the mean-field treatment for the Blume-Capel model under the presence of a discrete random magnetic field as introduced by Kaufman and Kanner. The magnetic field ($H$) versus temperature ($T$) phase diagrams for given values of the crystal field $D$ were recovered in accordance to Kaufman and Kanner original work. However, our main goal in the present work was to investigate the distinct structures of the crystal field versus temperature phase diagrams as the random magnetic field is varied because similar models have presented reentrant phenomenon due to randomness. Followin…
We analyze, within a mean-field approach, the stationary states of the kinetic spin-3/2 Blume-Capel model by the Glauber-type stochastic dynamics and subject to a time-dependent oscillating external magnetic field. The dynamic phase transition points are obtained by investigating the behavior of the dynamic magnetization as a function of temperature and as well as calculating the Liapunov exponent. Phase diagrams are constructed in the temperature and crystal-field interaction plane. We find five fundamental types of phase diagrams for the different values of the reduced magnetic field amplitu…
In this paper we solve the Blume-Capel model on a complete graph in the presence of random crystal field with a distribution, $P(Δ_i) =pδ(Δ_i-Δ)+(1-p) δ(Δ_i+Δ)$, using large deviation techniques. We find that the first order transition of the pure system is destroyed for $0.046<p<0.954$ for all values of the crystal field, $Δ$. The system has a line of continuous transition for this range of $p$ from $-\infty <Δ<\infty$. For values of $p$ outside this interval, the phase diagram of the system is similar to the pure model, with a tricritical point separating the line of first order and continuo…
In this work, the relaxation process of the spin-3/2 Blume-Capel model with quenched random crystal field on a two dimensional square lattice has been investigated by a method which combines the statistical equilibrium theory and the thermodynamic of linear irreversible processes. Generalized force and flux have been defined in irreversible thermodynamics limit. The kinetic equation for the magnetization has been obtained by using linear response theory. In order to understand effect of the first-order phase transition on the relaxation process, low temperature regions have been analyzed. And …
We investigate the first-order transition in the spin-1 two-dimensional Blume-Capel model in square lattices by revisiting the transfer-matrix method. With large strip widths increased up to the size of 18 sites, we construct the detailed phase coexistence curve which shows excellent quantitative agreement with the recent advanced Monte Carlo results. In the deep first-order area, we observe the exponential system-size scaling of the spectral gap of the transfer matrix from which linearly increasing interfacial tension is deduced with decreasing temperature. We find that the first-order signat…
We study the critical behavior of a frustrated Blume-Capel (BC) antiferromagnet on a triangular lattice by Monte Carlo simulations. For a reduced single-ion anisotropy strength $-1.47 \lesssim d < 0$ we find two phase transitions. The low-temperature phase is characterized by the antiferromagnetic long-range ordering (LRO) on two sublattices with the third one remaining in a non-magnetic state. At higher temperatures there is a critical region of the Berezinskii-Kosterlitz-Thouless (BKT) type with a power-law decaying spin-correlation function. For $-1.5 \le d \lesssim -1.47$, there is only on…
The renormalization group (RG) equation in the self-consistent local potential approximation (SC-LPA) suggested earlier for the description of continuous phase transitions in lattice models of the Landau-Ginzburg type has been applied to the solution of the spin-1 Blume-Capel model on the simple cubic lattice. The calculated transition temperatures of both continuous and the first-order phase transitions (FOPTs) in zero external field have been found to be in excellent agreement with the best available estimates. It has been argued that the SC-LPA RG equation may give more accurate and complet…
We investigate the tricritical scaling behavior of the two-dimensional spin-$1$ Blume-Capel model using the Wang-Landau method measuring the joint density of states for lattice sizes up to $48\times 48$ sites. The first-order transition curve is systematically determined employing the method of field mixing in conjunction with finite-size scaling, showing a significant deviation from the previous data points. Deep in the first-order area of the phase diagram, we also find that the specific heat exhibits a double-peak structure of the Schottky-like anomaly appearing with the transition peak. At…
We investigate the mixed-spin Blume-Capel model with spin-1/2 and spin-$S$ ($S=1$, $2$, and $3$) on the simple cubic and body-centered cubic lattices with single-ion-splitting crystal-field ($Δ$) by using the Metropolis and the Wang-Landau Monte Carlo methods. By numerical simulations, we prove that the tricritical point is spin-independent for both lattices. The positions of the tricritical point in the phase diagram are determined as ($Δ_t/J=2.978(1)$; $k_B T_t/J=0.439(1)$) and ($Δ_t/J=3.949(1)$; $k_B T_t/J=0.854(1)$) for the simple cubic and the body-centered cubic lattices, respectively. A…
The local properties of the spin one ferromagnetic Blume-Capel model defined on hierarchical lattices with dimension two and three are obtained by a numerical recursion procedure and studied as functions of the temperature and the reduced crystal-field parameter. The magnetization and the density of sites in the configuration S=0 state are carefully investigated at low temperature in the region of the phase diagram that presents the phenomenon of phase reentrance. Both order parameters undergo transitions from the ferromagnetic to the ordered paramagnetic phase with abrupt discontinuities that…
A spin-1 Blume-Capel model with dilute and random crystal fields is examined for honeycomb and square lattices by introducing an effective-field approximation that takes into account the correlations between different spins that emerge when expanding the identities. For dilute crystal fields, we have given a detailed exploration of the global phase diagrams of the system in $k_{B}T_{c}/J-D/J$ plane with the second and first order transitions, as well as tricritical points. We have also investigated the effect of the random crystal field distribution characterized by two crystal field parameter…
We study the Blume-Capel universality class in $d=\frac{10}{3}-ε$ dimensions. The RG flow is extracted by looking at poles in fractional dimension of three loop diagrams using $\overline{\rm MS}$. The theory is the only nontrivial universality class which admits an expansion to three dimensions with $ε=\frac{1}{3}<1$. We compute the relevant scaling exponents and estimate some of the OPE coefficients to the leading order. Our findings agree with and complement CFT results. Finally we discuss a family of nonunitary multicritical models which includes the Lee-Yang and Blume-Capel classes as spec…
The static critical exponents of the three dimensional Blume-Capel model which has a tricritical point at}$D/J=2.82${\small value are estimated for the standard and the cooling algorithms which improved from Creutz Cellular Automaton. The analysis of the data using the finite-size scaling and power law relations reproduce their well-established values in the}$D/J<3${\small and }$D/J<2.8${\small parameter region at standard and cooling algorithm, respectively. For the cooling algorithm at}$D/J=2.8$% {\small value of single-ion anisotropy parameter, the static critical exponents are estimated as…
The dynamical responses of Blume-Capel (S=1) ferromagnet to the plane propagating (with fixed frequency and wavelength) and standing magnetic field waves are studied sepa- rately in two dimensions by extensive Monte Carlo simulation. Depending on the values of temperature, amplitude of the propagating magnetic field and the strength of anisotropy, two different dynamical phases are observed. For a fixed value of anisotropy and the amplitude of the propagating magnetic field, the system undergoes a dynamical phase transition from a driven spin wave propagating phase to a pinned or spin frozen s…
