2D Ising System Research Articles

Dynamic Phase Transition in 2D Ising Systems: Effect of Anisotropy and Defects.

We investigate the dynamic phase transition in two-dimensional Ising models whose equilibrium characteristics are influenced by either anisotropic interactions or quenched defects. The presence of anisotropy reduces the dynamical critical temperature, leading to the expected result that the critical temperature approaches zero in the full-anisotropy limit. We show that a comprehensive understanding of the dynamic behavior of systems with quenched defects requires a generalized definition of the dynamic order parameter. By doing so, we demonstrate that the inclusion of quenched defects lowers the dynamic critical temperature as well, with a linear trend across the range of defect fractions considered. We also explore if and how it is possible to predict the dynamic behavior of specific magnetic systems with quenched randomness. Various geometric quantities, such as a defect potential index, the defect dipole moment, and the properties of the defect Delaunay triangulation, prove useful for this purpose.

Experimental critical dynamics of 3-methyl pyridine/D2O mixtures without and with antagonistic salt

We observed criticality in the structure and dynamics of a three-dimensional (3D) and two-dimensional (2D) Ising system consisting of 3-methyl pyridine $\text{(3MP)}/{\mathrm{D}}_{2}\mathrm{O}$ without and with antagonistic salt. We could describe both dynamic criticalities by the Kawasaki crossover function. The dynamic critical exponent was $z=0.063\ifmmode\pm\else\textpm\fi{}0.020$ and $0.005\ifmmode\pm\else\textpm\fi{}0.019$ for three and two dimensions, which confirms earlier observations in the 3D case and confirms expectations in the 2D case. The amplitudes of the critical dynamics are governed by the bare viscosities experimentally, and by the coefficient $R$ theoretically [the latter is proportional to ${(4\ensuremath{-}d)}^{\ensuremath{-}1}$ with the dimensionality $d$]. This finding is in accordance with the lubrication effect [N. Gov, A. G. Zilman, and S. Safran, Phys. Rev. E 70, 011104 (2004)], which is also connected to lamellar systems of the Brazovskii criticality [M. Gvaramia et al., Colloid Polym. Sci. 297, 1507 (2019)]. This lubrication effect is tightly connected to a laminar flow enforced by the domain structure, and it also holds for our 2D Ising system. The experimental techniques employed were small-angle neutron and x-rays scattering for the static criticality, and dynamic light scattering and neutron spin echo spectroscopy for the critical dynamics. Furthermore, the criticality of the viscosities was measured.

The high-Q static scattering of 3-methyl pyridine/D2O mixtures without and with antagonistic salt

Here we focus on the high-Q small-angle neutron scattering where we observed deviations from an ideal power law Q−2. From theory, this deviation Δ arises from the critical correlation-function exponent ηd in d dimensions. The investigated systems were 3-methyl pyridine/D2O without and with antagonistitc salt. They display the critical behavior of a 3d and 2d Ising system, respectively. In the first case, the value of Δ matches the ideal value η3 well, but in the latter case the value of Δ is affected by the two confined dimensions and the third dimension.

Nonextensive statistical field theory

We introduce a field-theoretic approach for describing the critical behavior of nonextensive systems, systems displaying global correlations among their degrees of freedom, encoded by the nonextensive parameter q. As some applications, we report, to our knowledge, the first analytical computation of both universal static and dynamic q-dependent nonextensive critical exponents for O(N) vector models, valid for all loop orders and |q−1|<1. Then emerges the new nonextensive O(N)q universality class. Both exponents ηq and zq reduce to those when q=1 for N→∞. We employ six independent methods which furnish identical results. Particularly, the results for nonextensive 2d Ising systems, exact within the referred approximation, agree with that obtained from computer simulations, within the margin of error, as better as q is closer to 1. We argue that the present approach can be applied to all models described by extensive statistical field theory as well. The results show an interplay between global correlations and fluctuations.

Permutation-Loewner entropy analysis for 2-dimensional Ising system interface

The structure of the 2-dimensional (2D) Ising system interface has been investigated in various theoretical statistical physics studies. Although the relationships between the interface geometry and physical properties of the system have already been indicated, in this study, we consider their alternative expressions using the chordal Loewner evolution and permutation entropy method. The Loewner driving forces corresponding to the Ising interfaces for below and at the critical temperature Tc were numerically calculated and analyzed by examining their pattern frequency, entropy, and distance to white noise in the theoretical scheme of the permutation entropy method. Using this permutation-Loewner entropy (PLE) method, we found an exponential function-type relation between the PLE and mean energy of the Ising system, which provides a connection between the theory of Loewner evolution and 2D statistical mechanical systems.

Crystal field effect on the magnetic field dependence cooling exponent of 3D simple cubic spin-7/2 Ising model: A Monte Carlo study

Based on the Monte Carlo simulation, the magneto caloric effect of a spin-7/2 3D-Ising system with pure ferromagnetic interaction was studied. The effective magnetic field h dependent entropy change ΔS was examined and result shows that ΔS follows a quadratic polynomial of hn where n is the cooling exponent. In the high field region the quadratic term h2n becomes comparable with the linear term hn. The quadratic term is strongly suppressed by the increment of the crystal field coefficient D. The ratio of the coefficient of the linear and quadratic term B/C is nearly 250 for D = 4 J (J is the exchange coefficient) and the quadratic term becomes negligibly small compared with the linear term. Furthermore in this ferromagnetic Ising system n continuously decreases from 0.71 to 0.62 for the rise of effective crystal field coefficient De from 0 to 4. This happens due to the enhancement of the + 7/2 spin from 70% to 87% for the increment of the same amount of De. It turns the system from spin-σ to spin-1/2 3D-Ising in nature.

Crystal field effect on the magneto caloric properties of the mixed spin-7/2 and spin-3/2 3D-ising ferrimagnet: A Monte Carlo simulation

It has been observed by the Monte Carlo simulation that the Magneto Caloric effect (MCE) of the ferrimagnetic mixed spin-7/2 and spin-3/2 3D Ising system is influenced by the change of crystal field coefficients. The peak value of the magnetic entropy change ΔSP has a second order polynomial dependency on the reduced crystal field coefficient De. This relation is only valid for the low magnetic field and low De values. The inverse of the coefficients of the polynomials has a linear dependency with the 4/3 power of the magnetic field. Full width half maximum (FWHM) of temperature dependence ΔS curve increases with the increment of De. The relative cooling power (RCP) increases with De and at De = 4 a large percentage change of ΔRCP % (~55%) is observed for the reduced field h = 5.

Behavior of information flow near criticality.

Recent experiments have indicated that many biological systems self-organize near their critical point, which hints at a common design principle. While it has been suggested that information transmission is optimized near the critical point, it remains unclear how information transmission depends on the dynamics of the input signal, the distance over which the information needs to be transmitted, and the distance to the critical point. Here we employ stochastic simulations of a driven two-dimensional Ising system and study the instantaneous mutual information and the information transmission rate between a driven input spin and an output spin. The instantaneous mutual information varies nonmonotonically with the temperature but increases monotonically with the correlation time of the input signal. In contrast, there exists not only an optimal temperature but also an optimal finite input correlation time that maximizes the information transmission rate. This global optimum arises from a fundamental trade-off between the need to maximize the frequency of independent input messages, the necessity to respond fast to changes in the input, and the need to respond reliably to these changes. The optimal temperature lies above the critical point but moves toward it as the distance between the input and output spin is increased.

Different universality classes of isostructural UTX compounds (T=Rh,Co,Co0.98Ru0.02;X=Ga,Al)

Magnetization isotherms of the $5f$-electron ferromagnets URhGa, UCoGa, and $\mathrm{U}{\mathrm{Co}}_{0.98}{\mathrm{Ru}}_{0.02}\mathrm{Al}$ were measured at temperatures in the vicinity of their Curie temperature to investigate the critical behavior near the ferromagnetic phase transition. These compounds adopt the layered hexagonal ZrNiAl-type structure and exhibit huge uniaxial magnetocrystalline anisotropy. The critical \ensuremath{\beta}, \ensuremath{\gamma}, and \ensuremath{\delta} exponents were determined by analyzing Arrott-Noakes plots, Kouvel-Fisher plots, critical isotherms, scaling theory, and Widom scaling relations. The values obtained for URhGa and UCoGa can be explained by the results of the renormalization group theory for a two-dimensional (2D) Ising system with long-range interactions similar to URhAl reported by other investigators. On the other hand, the critical exponents determined for $\mathrm{U}{\mathrm{Co}}_{0.98}{\mathrm{Ru}}_{0.02}\mathrm{Al}$ are characteristic of a three-dimensional (3D) Ising ferromagnet with short-range interactions suggested in previous studies also for the itinerant $5f$-electron paramagnet UCoAl situated near a ferromagnetic transition. The change from the 2D to the 3D Ising system is related to the gradual delocalization of $5f$ electrons in the series of the URhGa, URhAl, and UCoGa to $\mathrm{U}{\mathrm{Co}}_{0.98}{\mathrm{Ru}}_{0.02}\mathrm{Al}$ and UCoAl compounds and appears close to the strongly itinerant nonmagnetic limit. This indicates possible new phenomena that may be induced by the change of dimensionality in the vicinity of the quantum critical point.

Loewner driving force of the interface in the 2-dimensional Ising system as a chaotic dynamical system.

The interfaces in the 2-dimensional (2D) ferromagnetic Ising system below and at the critical temperature Tc were numerically analyzed in the framework of discrete Loewner evolution. We numerically calculated Loewner driving forces corresponding to the interfaces in the 2D Ising system and analyzed them using nonlinear time series analyses. We found that the dynamics of the Loewner driving forces showed chaotic properties wherein their intermittency, sensitivity to initial condition, and autocorrelation change depending on the temperature T of the system. It is notable that while the Loewner driving forces have deterministic properties, they have Gaussian-type probability distributions whose variance increases as T→Tc, indicating that they are examples of the Gaussian chaos. Thus, the obtained Loewner driving forces can be considered a chaotic dynamical system whose bifurcation is dominated by the temperature of the Ising system. This perspective for the dynamical system was discussed in relation to the extension and/or generalization of the stochastic Loewner evolution.

Near Degeneracy of Magnetic Phases in Two-Dimensional Chromium Telluride with Enhanced Perpendicular Magnetic Anisotropy.

The discovery of atomically thin van der Waals magnets (e.g., CrI3 and Cr2Ge2Te6) has triggered a renaissance in the study of two-dimensional (2D) magnetism. Most of the 2D magnetic compounds discovered so far host only one single magnetic phase unless the system is at a phase boundary. In this work, we report the near degeneracy of magnetic phases in ultrathin chromium telluride (Cr2Te3) layers with strong perpendicular magnetic anisotropy highly desired for stabilizing 2D magnetic order. Single-crystalline Cr2Te3 nanoplates with a trigonal structure (space group P3̅1c) were grown by chemical vapor deposition. The bulk magnetization measurements suggest a ferromagnetic (FM) order with an enhanced perpendicular magnetic anisotropy, as evidenced by a coercive field as large as ∼14 kOe when the field is applied perpendicular to the basal plane of the thin nanoplates. Magneto-optical Kerr effect studies confirm the intrinsic ferromagnetism and characterize the magnetic ordering temperature of individual nanoplates. First-principles density functional theory calculations suggest the near degeneracy of magnetic orderings with a continuously varying canting from the c-axis FM due to their comparable energy scales, explaining the zero-field kink observed in the magnetic hysteresis loops. Our work highlights Cr2Te3 as a promising 2D Ising system to study magnetic phase coexistence and switches for ultracompact information storage and processing.

Distinguishing the Intrinsic Antiferromagnetism in Polycrystalline LiCoPO4 and LiMnPO4 Olivines.

We report a detailed investigation of the long-range magnetic ordering in polycrystalline samples of LiCoPO4 and LiMnPO4, which belong to a series of well-known olivine cathode materials LiMPO4 (M = Mn, Fe, Co, Ni). Samples were prepared by hydrothermal and solid state methods. The magnetic susceptibility is found to be strongly field-dependent, impacting the antiferromagnetic transition temperature and the bifurcation of the FC and ZFC curves. We discuss the role synthesis conditions have on impurity formation and particle size. We report neutron powder diffraction data for the samples prepared by solid state methods. Based upon representational analysis of the observed reflections, we affirm the magnetic structure Pnma' for LiCoPO4 and Pn'm'a' for LiMnPO4. The refined magnetic moments from these models are 3.28(4) μB for LiCoPO4 and 4.28(3) μB for LiMnPO4. We also study the onset of magnetic ordering in each sample and affirm that the ordering temperature is 22 K for LiCoPO4 and 34 K for LiMnPO4. The critical parameters describing those transitions are βc = 0.21(4) (LiCoPO4) and βc = 0.31(3) (LiMnPO4). These values are characteristic of a 3D Ising system for LiMnPO4 and intermediate behavior between a 2D and 3D Ising system for LiCoPO4. We compare these observations with other reports proposing lower magnetic symmetry.

Magnetocaloric effect in the 2D dilute Ising system

We consider the magnetocaloric effect (MCE) for the 2D Ising system with a fixed concentration of non-magnetic mobile charged impurities. Pseudospin formalism and highperformance Monte Carlo simulations are used to describe the system and, in particular, to obtain the temperature dependences of the MCE parameters for our system. The effect of the impurityimpurity interaction strength on the concentration dependence of the MCE parameters is discussed, as well as the possibility of using the MCE effect to reveal a hidden frustration in the system.

Spin Seebeck Effect from Antiferromagnetic Magnons and Critical Spin Fluctuations in Epitaxial FeF_{2} Films.

We report a longitudinal spin Seebeck effect (SSE) study in epitaxially grown FeF_{2}(110) antiferromagnetic (AFM) thin films with strong uniaxial anisotropy over the temperature range of 3.8-250K. Both the magnetic-field and temperature-dependent SSE signals below the Néel temperature (T_{N}=70 K) of the FeF_{2} films are consistent with a theoretical model based on the excitations of AFM magnons without any net induced static magnetic moment. In addition to the characteristic low-temperature SSE peak associated with the AFM magnons, there is another SSE peak at T_{N} which extends well into the paramagnetic phase. All the SSE data taken at different magnetic fields up to 12T near and above the critical point T_{N} follow the critical scaling law very well with the critical exponents for magnetic susceptibility of 3D Ising systems, which suggests that the AFM spin correlation is responsible for the observed SSE near T_{N}.

Phase Transition in Frustrated Magnetic Thin Film-Physics at Phase Boundaries.

In this review, we outline some principal theoretical knowledge of the properties of frustrated spin systems and magnetic thin films. The two points we would like to emphasize: (i) the physics in low dimensions where exact solutions can be obtained; (ii) the physics at phase boundaries where interesting phenomena can occur due to competing interactions of the two phases around the boundary. This competition causes a frustration. We will concentrate our attention on magnetic thin films and phenomena occurring near the boundary of two phases of different symmetries. Two-dimensional (2D) systems are in fact the limiting case of thin films with a monolayer. Naturally, we will treat this case at the beginning. We begin by defining the frustration and giving examples of frustrated 2D Ising systems that we can exactly solve by transforming them into vertex models. We will show that these simple systems already contain most of the striking features of frustrated systems such as the high degeneracy of the ground state (GS), many phases in the GS phase diagram in the space of interaction parameters, the reentrance occurring near the boundaries of these phases, the disorder lines in the paramagnetic phase, and the partial disorder coexisting with the order at equilibrium. Thin films are then presented with different aspects: surface elementary excitations (surface spin waves), surface phase transition, and criticality. Several examples are shown and discussed. New results on skyrmions in thin films and superlattices are also displayed. By the examples presented in this review we show that the frustration when combined with the surface effect in low dimensions gives rise to striking phenomena observed in particular near the phase boundaries.

Clifford algebra approach of 3D Ising model

We develop a Clifford algebra approach for 3D Ising model. By utilizing some mathematical facts of the direct product of matrices and their trace, we expand the dimension of the transfer matrices V of the 3D Ising system by adding unit matrices I (with compensation of a factor) and adjusting their sequence, which do not change the trace of the transfer matrices V (Theorem I: Trace Invariance Theorem). It allows us to perform a linearization process on sub-transfer-matrices (Theorem II: Linearization Theorem). It is found that locally for each site j, the internal factor Wj in the transfer matrices can be treated as a boundary factor, which can be dealt with by a procedure similar to the Onsager-Kaufman approach for the boundary factor U in the 2D Ising model. This linearization process splits each sub-transfer matrix into 2n sub-spaces (and the whole system into 2nl sub-spaces). Furthermore, a local transformation is employed on each of the sub-transfer matrices (Theorem III: Local Transformation Theorem). The local transformation trivializes the non-trivial topological structure, while it generalizes the topological phases on the eigenvectors. This is induced by a gauge transformation in the Ising gauge lattice that is dual to the original 3D Ising model. The non-commutation of operators during the processes of linearization and local transformation can be dealt with to be commutative in the framework of the Jordan-von Neumann-Wigner procedure, in which the multiplication in Jordan algebras is applied instead of the usual matrix multiplication AB (Theorem IV: Commutation Theorem).

Spectral density and calculation of free energy

We examine a spectral density, which characterizes the degeneracy of each energy, for 2D and 3D Ising models. We compare two approximations for the spectral density, the polynomial approximation (Dixon et al., 2005) and the nv-approximation suggested by us in Kryzhanovsky and Litinskii (2017). Our analysis shows that while the free energy is almost independent of the approximation method, its derivatives depend on the approximation, and they differ significantly near the critical temperature. We show that for the polynomial approximation, the ground state of the system is reached at a finite temperature, in contradiction with fundamental physical principles. For the 2D Ising system, the nv-approximation is also incorrect. However, it works rather well when the dimension of the system is equal to three or higher.

Critical behavior of magnetization in URhAl: Quasi-two-dimensional Ising system with long-range interactions

The critical behavior of dc magnetization in the uranium ferromagnet URhAl with the hexagonal ZrNiAl-type crystal structure has been studied around the ferromagnetic transition temperature T_C. The critical exponent beta for the temperature dependence of the spontaneous magnetization below T_C, gamma for the magnetic susceptibility, and delta for the magnetic isotherm at T_C have been obtained with a modified Arrott plot, a Kouvel-Fisher plot, the critical isotherm analysis and the scaling analysis. We have determined the critical exponents as beta = 0.287 +- 0.005, gamma = 1.47 +- 0.02, and delta = 6.08 +- 0.04 by the scaling analysis and the critical isotherm analysis. These critical exponents satisfy the Widom scaling law delta=1+gamma/beta. URhAl has strong uniaxial magnetic anisotropy, similar to its isostructural UCoAl that has been regarded as a three-dimensional (3D) Ising system in previous studies. However, the universality class of the critical phenomenon in URhAl does not belong to the 3D Ising model (beta = 0.325, gamma = 1.241, and delta = 4.82) with short-range exchange interactions between magnetic moments. The determined exponents can be explained with the results of the renormalization group approach for a two-dimensional (2D) Ising system coupled with long-range interactions decaying as J(r)~r^-(d+sigma) with sigma = 1.44. We suggest that the strong hybridization between the uranium 5f and rhodium 4d electrons in the U-Rh_I layer in the hexagonal crystal structure is a source of the low dimensional magnetic property. The present result is contrary to current understandings of the physical properties in a series of isostructural UTX uranium ferromagnets (T: transition metals, X: p-block elements) based on the 3D Ising model.

Viscosity of a net-baryon fluid near the QCD critical point

In the dynamics of the QCD critical point, the net-baryon fluid, linked to the slow component of the order parameter, relaxes to a 3d Ising system in equilibrium. An analytical study of shear and bulk viscosity, with constraints imposed by the dynamics of the critical net-baryon fluid, the universality property and the requirements of a class of strong coupling theories, is performed in the neighbourhood of the critical point. It is found that the shear viscosity of the net-baryon fluid is restricted in the domain $1.6\leq 4\pi\frac{\eta}{s}\leq 3.7$ for $T_c < T \leq 2T_c$ whereas the bulk viscosity is small, $4\pi\frac{\zeta}{s} < 0.05$ (for $T>1.27 T_c$) but rising towards the singularity at $T=T_c$.

A quantum simulation approach for a three-dimensional Ising spin model—Comparison to mean field theory

We use a recently developed quantum simulation approach to study the properties of a three-dimensional Ising model consisting of S = 1/2 quantum spins localized at the sites of a simple cubic lattice. We assume nearest-neighbor interaction between spins with an exchange interaction that can be either ferromagnetic or antiferromagnetic. It is found that the computational method quickly converges towards the expected equilibrium spin configurations. The resulting spontaneous magnetization curves corresponding to the two types of magnetic interactions under consideration were found to be almost identical to the ones obtained via quantum mean field theory at all temperatures. The derived total energies, total free energies, magnetic entropies and specific heats per mole of spins show no sizeable differences from known theoretical values. Furthermore, the results of the simulations for two different 3D Ising systems containing 4×4×4 and 20×20×20 spins localized at the sites of a simple cubic lattice were found to be almost identical to each other. This finding suggests that the self-consistent algorithm approach of the current simulation method allows one to obtain the physical bulk properties of a large magnetic system by relying on simulations of a much smaller spin system sample. Therefore, the method presently considered appears to be not only very accurate as gauged by comparison to mean field theory, but also able to greatly increase the speed of simulations.