Two-dimensional Ising System Research Articles

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.

Compensation and Critical Behavior of Mixed Spin Ising Model with Four-Spin Interactions in a Crystal Field

We study the thermodynamical properties of two-dimensional mixed spin Ising system consisting of spin-[Formula: see text] and spin-1 with four-spin interactions in crystal field interactions on a square lattice within the framework of the finite cluster approximation based on single-site cluster theory. In this paper, the phase diagrams show qualitatively and quantitatively interesting features. In fact, the system exhibits a variety of interesting features resulting from the introduction of the four-spin interactions and the crystal field interactions. Finally, using the same method, we study the behavior of the total magnetization of the Ising model with four-spin interactions in the crystal field to show the existence of compensation points and point intervals for certain values of [Formula: see text].

Itinerant fermions on dilute frustrated Ising lattices

We consider itinerant spinless fermions as moving defects in a dilute two-dimensional frustrated Ising system where they occupy site vacancies. Fermions interact via local spin fluctuations and we analyze coupled self-consistent mean-field equations of the Green functions after expressing the spin and fermion operators in terms of Grassmann variables. The specific heat and effective mass are analyzed with the solutions satisfying the symmetry imposed by the coupling layout. At low temperature, we find that these solutions induce stripes along the lines of couplings with the same sign, and that a low fermion density yields a small effective mass.

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.

Kinetics of domain growth in Ising systems with bond disorder at regularly selected sites

We study the structure, phase behaviour and growth laws in Ising ferromagnets and binary mixtures with bond disorder introduced at regularly selected sites for a critical quench. The results presented here are from extensive Monte Carlo simulations on two-dimensional Ising systems. Domain growth in a ferromagnet is modelled by using nonconserved spin-flip (Glauber) kinetics, and phase separation in a binary (AB) mixture is modelled by conserved spin-exchange (Kawasaki) kinetics. In both cases, we observed that the domain growth law is consistent with the respective power-law growth with the variable growth exponent that depends on the number of disordered sites. The nonconserved Ising system follows dynamical scaling for all the number of disordered sites studied here; however, a small deviation is noticed for $$r\rightarrow 0$$ at a large number of disordered sites. Whereas, for the conserved case, dynamical scaling deviates from the master curve after a reasonable number of disordered lattice sites; we notice the formation of lamellar evolution morphology at a higher number of disordered sites at later times.

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.

Critical behavior of quasi-two-dimensional semiconducting ferromagnet Cr2Ge2Te6

The critical properties of the single-crystalline semiconducting ferromagnet CrGeTe$_3$ were investigated by bulk dc magnetization around the paramagnetic to ferromagnetic phase transition. Critical exponents $\beta = 0.200\pm0.003$ with critical temperature $T_c = 62.65\pm0.07$ K and $\gamma = 1.28\pm0.03$ with $T_c = 62.75\pm0.06$ K are obtained by the Kouvel-Fisher method whereas $\delta = 7.96\pm0.01$ is obtained by the critical isotherm analysis at $T_c = 62.7$ K. These critical exponents obey the Widom scaling relation $\delta = 1+\gamma/\beta$, indicating self-consistency of the obtained values. With these critical exponents the isotherm $M(H)$ curves below and above the critical temperatures collapse into two independent universal branches, obeying the single scaling equation $m = f_\pm(h)$, where $m$ and $h$ are renormalized magnetization and field, respectively. The determined exponents match well with those calculated from the results of renormalization group approach for a two-dimensional Ising system coupled with long-range interaction between spins decaying as $J(r)\approx r^{-(d+\sigma)}$ with $\sigma=1.52$.

Two-dimensional percolation transition in two atomic layers of Fe on W(110): Direct measurement of a static percolation critical exponent in a two-dimensional Ising system

When the coverage of the second atomic layer of Fe in an Fe/W(110) ultrathin film reaches a critical value, the system moves suddenly from a frustrated magnetic state without long-range order to an in-plane ferromagnetic state with long-range order, and displays many features of a percolation transition. Measurements of the magnetic susceptibility as the films are grown at 255 K show power law scaling that is limited by noise at low deposition, and by the dynamics of the paramagnetic, frustrated state at high deposition. Because the measurements represent a system driven by a finite field oscillating at a finite frequency, it is demonstated that the threshold deposition for percolation is bounded by the depositions where the real and imaginary components of the susceptibility have maxima. Fitting for the critical exponent of the static susceptibility at these bounds gives a bounded value for $\gamma_p=2.39\pm0.04$, in agreement with theory.

Dynamic Magnetic Hysteresis Properties in a Two-Dimensional Mixed Ising System Designed with Integer and Half-Integer Spins

Dynamic magnetic hysteresis properties in a two-dimensional mixed Ising system designed with integer and half-integer spins, namely spin-1 and spin-5/2, are investigated comprehensively on the basis of the mean-field approximation and the Glauber-type stochastic dynamics, namely dynamic mean-field approximation (DMFA). In particular, we examine the effect of Hamiltonian parameters on dynamic hysteresis properties. As a result of this examination, some characteristic hysteresis behaviors are obtained, such as the existence of single, double and triple hysteresis. Moreover, we study to obtain coercive field and remanent magnetization. Our results are compared with some experimental and theoretical works to find a qualitatively good agreement.

Droplets pinned at chemically inhomogenous substrates: A simulation study of the two-dimensional Ising case.

As a simplified model of a liquid nanostripe adsorbed on a chemically structured substrate surface, a two-dimensional Ising system with two boundaries at which surface fields act is studied. At the upper boundary, the surface field is uniformly negative, while at the lower boundary (a distance L apart), the surface field is negative only outside a range of extension b, where a positive surface stabilizes a droplet of the phase with positive magnetization for temperatures T exceeding the critical temperature T_{w} of the wetting transition of this model. We investigate the local order parameter profiles across the droplet, both in the directions parallel and perpendicular to the substrate, varying both b and T. Also, precursor effects to droplet formation as T approaches T_{w} from below are studied. In accord with theoretical predictions, for T>T_{w} the droplet is found to have the shape of a semiellipse, where the width (distance of the interface from the substrate) scale is proportional to b (b^{1/2}). So, the area of the droplet is proportional to b^{3/2}, and the temperature dependence of the corresponding prefactor, which also involves the interfacial stiffness, is studied.

A new look on the two-dimensional Ising model: thermal artificial spins

We present a direct experimental investigation of the thermal ordering in an artificial analogue of an asymmetric two-dimensional Ising system composed of a rectangular array of nano-fabricated magnetostatically interacting islands. During fabrication and below a critical thickness of the magnetic material the islands are thermally fluctuating and thus the system is able to explore its phase space. Above the critical thickness the islands freeze-in resulting in an arrested thermalized state for the array. Determining the magnetic state we demonstrate a genuine artificial two-dimensional Ising system which can be analyzed in the context of nearest neighbor interactions.

Dynamic hysteresis features in a two-dimensional mixed Ising system

The dynamic hysteresis features in a two-dimensional mixed spin (1,3/2) Ising system are studied by using the within the effective-field theory with correlations based on Glauber-type stochastic. The dynamic phase transition temperatures and dynamic hysteresis curves are obtained for both the ferromagnetic and antiferromagnetic interactions. It is observed that the dynamic hysteresis loop areas increase when the reduced temperatures increase, and the dynamic hysteresis loops disappear at certain reduced temperatures. The thermal behaviors of the coercivity and remanent magnetizations are also investigated. The results are compared with some theoretical and experimental works and found in a qualitatively good agreement.

High-precision work distributions for extreme nonequilibrium processes in large systems

The distributions of work for strongly nonequilibrium processes are studied using a very general form of a large-deviation approach, which allows one to study distributions down to extremely small probabilities of almost arbitrary quantities of interest for equilibrium, nonequilibrium stationary, and even nonstationary processes. The method is applied to quickly vary the external field in a wide range B = 3 ↔ 0 for a critical (T = 2.269) two-dimensional Ising system of size L × L = 128 × 128. To obtain free-energy differences from the work distributions, they must be studied in ranges where the probabilities are as small as 10^{-240}, which is not possible using direct simulation approaches. By comparison with the exact free energies, which are available for this model for the zero-field case, one sees that the present approach allows one to obtain the free energy with a very high relative precision of 10^{-4}. This works well also for a nonzero field, i.e., for a case where standard umbrella-sampling methods are not efficient to calculate free energies. Furthermore, for the present case it is verified that the resulting distributions of work for forward and backward processes fulfill Crooks theorem with high precision. Finally, the free energy for the Ising magnet as a function of the field strength is obtained.

Nonequilibrium magnetic properties in a two-dimensional kinetic mixed Ising system within the effective-field theory and Glauber-type stochastic dynamics approach

Nonequilibrium magnetic properties in a two-dimensional kinetic mixed spin-2 and spin-5/2 Ising system in the presence of a time-varying (sinusoidal) magnetic field are studied within the effective-field theory (EFT) with correlations. The time evolution of the system is described by using Glauber-type stochastic dynamics. The dynamic EFT equations are derived by employing the Glauber transition rates for two interpenetrating square lattices. We investigate the time dependence of the magnetizations for different interaction parameter values in order to find the phases in the system. We also study the thermal behavior of the dynamic magnetizations, the hysteresis loop area, and dynamic correlation. The dynamic phase diagrams are presented in the reduced magnetic field amplitude and reduced temperature plane and we observe that the system exhibits dynamic tricritical and reentrant behaviors. Moreover, the system also displays a double critical end point (B), a zero-temperature critical point (Z), a critical end point (E), and a triple point (TP). We also performed a comparison with the mean-field prediction in order to point out the effects of correlations and found that some of the dynamic first-order phase lines, which are artifacts of the mean-field approach, disappeared.

Small-network approximations for geometrically frustrated Ising systems

The study of frustrated spin systems often requires time-consuming numerical simulations. As the simplest approach, the classical Ising model is often used to investigate the thermodynamic behavior of such systems. Exploiting the small correlation lengths in frustrated Ising systems, we develop a method for obtaining first approximations to the energetic properties of frustrated two-dimensional Ising systems using small networks of less than 30 spins. These small networks allow much faster numerical simulations, and more importantly, analytical evaluations of their properties are numerically tractable. We choose Ising systems on the triangular lattice, the kagome lattice, and the triangular kagome lattice as prototype systems and find small systems that can serve as good approximations to these prototype systems. Through comparisons between the properties of extended models and small systems, we develop a set of criteria for constructing small networks to approximate general infinite two-dimensional frustrated Ising systems. This method of using small networks provides a different and efficient way to obtain a first approximation to the properties of frustrated spin systems.

Neutron diffraction study of the magnetic and structural phase transitions inBaFe2As2

We present the results of an investigation of both the magnetic and structural phase transitions in a high quality single crystalline sample of the undoped, iron pnictide compound BaFe$_2$As$_2$. Both phase transitions are characterized via neutron diffraction measurements which reveal simultaneous, continuous magnetic and structural orderings with no evidence of hysteresis, consistent with a single second order phase transition. The onset of long-range antiferromagnetic order can be described by a simple power law dependence $\phi(T)^2\propto(1-\frac{T}{T_N})^{2\beta}$ with $\beta=0.103\pm0.018$; a value near the $\beta=0.125$ expected for a two-dimensional Ising system. Biquadratic coupling between the structural and magnetic order parameters is also inferred along with evidence of three-dimensional critical scattering in this system.

Conjectures on the exact solution of three-dimensional (3D) simple orthorhombic Ising lattices

We report conjectures on the three-dimensional (3D) Ising model of simple orthorhombic lattices, together with details of calculations for a putative exact solution. Two conjectures, an additional rotation in the fourth curled-up dimension and weight factors on the eigenvectors, are proposed to serve as a boundary condition to deal with the topologic problem of the 3D Ising model. The partition function of the 3D simple orthorhombic Ising model is evaluated by spinor analysis, employing these conjectures. Based on the validity of the conjectures, the critical temperature of the simple orthorhombic Ising lattices could be determined by the relation of KK* = KK′ + KK″ + K′K″ or sinh 2K · sinh 2(K′ + K″ + (K′K″/K)) = 1. For a simple cubic Ising lattice, the critical point is putatively determined to locate exactly at the golden ratio , as derived from K* = 3K or sinh 2K · sinh 6K = 1. If the conjectures would be true, the specific heat of the simple orthorhombic Ising system would show a logarithmic singularity at the critical point of the phase transition. The spontaneous magnetization of the simple orthorhombic Ising ferromagnet is derived explicitly by the perturbation procedure, following the conjectures. The spin correlation functions are discussed on the terms of the Pfaffians, by defining the effective skew-symmetric matrix Aeff. The true range κx of the correlation and the susceptibility of the simple orthorhombic Ising system are determined by procedures similar to those used for the two-dimensional Ising system. The putative critical exponents derived explicitly for the simple orthorhombic Ising lattices are α = 0, β = 3/8, γ = 5/4, δ = 13/3, η = 1/8 and ν = 2/3, showing the universality behaviour and satisfying the scaling laws. The cooperative phenomena near the critical point are studied and the results based on the conjectures are compared with those of approximation methods and experimental findings. The putative solutions have been judged by several criteria. The deviations of the approximation results and the experimental data from the solutions are interpreted. Based on the solution, it is found that the 3D-to-2D crossover phenomenon differs with the 2D-to-1D crossover phenomenon and there is a gradual crossover of the exponents from the 3D to the 2D values. Special attention is also paid to the extra energy caused by the introduction of the fourth curled-up dimension and the states at/near infinite temperature revealed by the weight factors of the eigenvectors. The physics beyond the conjectures and the existence of the extra dimension are discussed. The present work is not only significant for statistical and condensed matter physics, but also fill the gap between the quantum field theory, cosmology theory, high-energy particle physics, graph theory and computer science.

Monte Carlo simulations for two-dimensional Ising system far from equilibrium

With a dynamic Monte Carlo simulation, which is free of critical slowing down, the critical behavior of two-dimensional Ising model at non-equilibrium states is investigated. We focus on the two-time autocorrelation function A(t, t′) to identify the dynamic exponent z in two different evolution stages, quenched from a high temperature state and a completely ordered state to the critical state at TC, respectively. By using the heat-bath algorithm, the dynamic critical exponent is estimated as z ≈ 2.16. As a result, the universality of the scaling behavior is verified numerically, and the exponent λC is determined complete by λC = β/v for the evolutions from an initial completely ordered state.

Properties of the interface in the confined Ising magnet with competing surface fields

A two-dimensional magnetic Ising system confined in an L × D geometry ( L ⪡ D ) in the presence of competing magnetic fields ( h) acting at opposite walls along the D -direction, exhibits an interface between domains of different orientation that run parallel to the walls. In the limit L → ∞ , this interface undergoes a wetting transition that occurs at the critical curve T w ( h ) , so that for T < T w ( h ) such an interface is bound to the walls, while for T w ( h ) ⩽ T < T cb the interface is freely fluctuating around the center of the film, where T cb is the bulk critical temperature. By considering both short- and long-range magnetic fields acting at the walls, we study the divergence of the (equilibrated) average position of the interface when approaching the wetting critical point. Furthermore, starting from a monodomain structure with the interface bound to one wall, we also study the dynamics of the interface unbinding.

Logarithmic corrections to correlation decay in two-dimensional random-bond Ising systems

The statistics of critical spin-spin correlation functions in Ising systems with nonfrustrated disorder are investigated on a strip geometry, via numerical transfer-matrix techniques. Conformal invariance concepts are used, in order to test for logarithmic corrections to pure power-law decay against distance. Fits of our data to conformal-invariance expressions, specific to logarithmic corrections to correlations on strips, give results with the correct sign, for the moments of order n = 0-4 of the correlation-function distribution. We find an interval of disorder strength along which corrections to pure-system behavior can be decomposed into the product of a known n -dependent factor and an approximately n -independent one, in accordance with predictions. A phenomenological fitting procedure is proposed, which takes partial account of subdominant terms of correlation-function decay on strips. In the low-disorder limit, it gives results in fairly good agreement with theoretical predictions, provided that an additional assumption is made.