2D Ising Model Research Articles

Formation of magnetization plateaus in the 3D Ising model with the long-range RKKY interaction: application to rare-earth tetraborides

The standard Metropolis algorithm and the parallel tempering method are used to study the influence of the long-range RKKY interaction on the formation of magnetization plateaus in the 3D Ising model with magnetically coupled Shastry-Sutherland layers. It is shown that depending on the Fermi momentum k F the model exhibits the rich spectrum of magnetic solutions which is demonstrated by the appearance of the large number of magnetization plateaus on the magnetization curves. In particular, the following set of individual fractional magnetization plateaus with macroscopic stability regions is observed: m ∕m s = 1/8, 1/7, 1/6, 1/5, 2/9, 1/4, 1/3, 3/8, 2/5, 5/12, 3/7, 1/2, 5/9 and 2/3. In the regime of k F ~ 2π ∕1.24, corresponding to the real situation in TmB 4 , we have confirmed the existence of the magnetization plateau with m ∕m s = 1∕8, the nature of which was intensively discussed in recent experimental works. Since the change of k F can be induced by doping, our results provide predictions of the complete sequences of magnetization plateaus that could appear in tetraboride solid solutions. And finally, the current results are confronted with our previous ones obtained for the two-dimensional Ising model with the RKKY interaction in order to reveal the role of system dimension on the formation of magnetization plateaus.

Geometry of bounded critical phenomena

The quest for a satisfactory understanding of systems at criticality in dimensions d > 2 is a major field of research. We devise here a geometric description of bounded systems at criticality in any dimension d. This is achieved by altering the flat metric with a space dependent scale factor γ(x), x belonging to a bounded domain Ω. γ(x) is chosen in order to have a scalar curvature to be constant and matching the one of the hyperbolic space, the proper notion of curvature being—as called in the mathematics literature—the fractional Q-curvature. The equation for γ(x) is found to be the fractional Yamabe equation (to be solved in Ω) that, in absence of anomalous dimension, reduces to the usual Yamabe equation in the same domain. From the scale factor γ(x) we obtain novel predictions for the scaling form of one-point order parameter correlation functions. A (necessary) virtue of the proposed approach is that it encodes and allows to naturally retrieve the purely geometric content of two-dimensional boundary conformal field theory. From the critical magnetization profile in presence of boundaries one can extract the scaling dimension of the order parameter, Δ ϕ . For the 3D Ising model we find Δ ϕ = 0.518 142(8) which favorably compares (at the fifth decimal place) with the state-of-the-art estimate. A nontrivial prediction is the structure of two-point spin-spin correlators at criticality. They should depend on the fractional Q-hyperbolic distance calculated from the metric, in turn depending only on the shape of the bounded domain and on Δ ϕ . Numerical simulations of the 3D Ising model on a slab geometry are found to be in agreement with such predictions.

Ising model as Wilson-Majorana fermions

We show the equivalence of the 2D Ising model to standard free Euclidean lattice fermions of the Wilson Majorana type. The equality of the loop representations for the partition functions of both systems is established exactly for finite lattices with well-defined boundary conditions. The honeycomb lattice is particularly simple in this context and therefore discussed first and only then followed by the more familiar square lattice case.

Information geometry in quantum field theory: lessons from simple examples

Motivated by the increasing connections between information theory and high-energy physics, particularly in the context of the AdS/CFT correspondence, we explore the information geometry associated to a variety of simple systems. By studying their Fisher metrics, we derive some general lessons that may have important implications for the application of information geometry in holography. We begin by demonstrating that the symmetries of the physical theory under study play a strong role in the resulting geometry, and that the appearance of an AdS metric is a relatively general feature. We then investigate what information the Fisher metric retains about the physics of the underlying theory by studying the geometry for both the classical 2d Ising model and the corresponding 1d free fermion theory, and find that the curvature diverges precisely at the phase transition on both sides. We discuss the differences that result from placing a metric on the space of theories vs.~states, using the example of coherent free fermion states. We compare the latter to the metric on the space of coherent free boson states and show that in both cases the metric is determined by the symmetries of the corresponding density matrix. We also clarify some misconceptions in the literature pertaining to different notions of flatness associated to metric and non-metric connections, with implications for how one interprets the curvature of the geometry. Our results indicate that in general, caution is needed when connecting the AdS geometry arising from certain models with the AdS/CFT correspondence, and seek to provide a useful collection of guidelines for future progress in this exciting area.

Exact three spin correlation function relations for the square and the honeycomb Ising lattices

In this work, the order parameter and the two-site correlation functions are expressed properly using the decimation transformation process in the presence of an external field so that their applications lead to some significant physical results. Indeed, their applications produce or reproduce some relevant and important results which were included in cumbersome mathematics in the previous studies, if not in a form impossible to understand. The average magnetization or the order parameter <σ> is expressed as <σ0,i>=<tanh[κ(σ1,i+σ2,i+⋯+σz,i)+H]>. Here, κ is the coupling strength, z is the number of nearest neighbors. σ0,i denotes the central spin at the ith site, while σl,i, l=1,2,⋯,z are the nearest neighbor spins around the central spin. H is the normalized external magnetic field. We show that the application of this relation to the 1D Ising model reproduces readily the previously obtained exact results in the absence of an external field. Furthermore, the three-site correlation functions of square and honeycomb lattices of the form < σ1σ2σ3 > are analytically obtained. One finds that the three-site correlation functions are equal to f(κ) < σ > . Here f(κ) depends on the lattice types and is an analytic function of coupling constant. This result indicates that the critical properties of three-site correlation functions of those lattices are the same as the corresponding order parameters < σ > of those lattices. This will mean that the uniqueness of the average magnetization as an order parameter is questionable. The application of the two site correlation function relation obtained in this paper produces a relevant exact relation between four site correlation function and two site correlation functions for square lattice. It also produces a relation between correlation functions of honeycomb lattice. Making use of this relation leads to the exact calculation of the constant A appearing in the proposed correlation function relation in the scaling theory. It is also indicated that the correlation length ξ(κ) can be obtained exactly in the realm of scaling theory from the obtained correlation function relation for the honeycomb lattice. In addition, the average magnetization relation obtained in this paper leads easily to the result of the conventional mean field critical coupling strength which, as a first approximation, is equal to κc=1/z. It is also shown that systematic improvements of the values of the critical coupling strengths are possible with mean field treatments in this picture.

Cross derivative of the Gibbs free energy: A universal and efficient method for phase transitions in classical spin models

With an auxiliary weak external magnetic field, we reexamine the fundamental thermodynamic function, Gibbs free energy F(T, h), to study the phase transitions in the classical spin lattice models. A cross derivative, i.e. the second-order partial derivative of F(T, h) with respect to both temperature and field, is calculated to precisely locate the critical temperature, which also reveals the nature of a transition. The strategy is efficient and universal, as exemplified by the 5-state clock model, 2-dimensional (2D) and 3D Ising models, and the XY model, no matter a transition is trivial or exotic with complex excitations. More importantly, other conjugate pairs could also be integrated into a similar cross derivative if necessary, which would greatly enrich our vision and means to investigate phase transitions both theoretically and experimentally.

Electronic structure and magnetism of MTe2 (M = Ti, V, Cr, Mn, Fe, Co and Ni) monolayers

We study the electronic structure and magnetism of monolayer 3d transition-metal ditellurides MTe2 (M = Ti, V, Cr, Mn, Fe, Co and Ni) in trigonal prismatic H- and/or octahedral T-phase by means of the first-principles calculations. The results show that H-VTe2, T-VTe2, H-FeTe2 and T-MnTe2 monolayers exhibit intrinsic ferromagnetism, and the others have no ferromagnetism. The exchange splitting of V, Mn and Fe 3d orbitals is responsible for ferromagnetism. The exchange constant and the Curie temperature are estimated by using 2D Ising model and mean field theory. Among the four ferromagnetic monolayers, the H-VTe2 monolayer has the largest exchange constant and the corresponding Curie temperature is near room temperature. Calculations also show that the T-VTe2, H-VTe2 and H-FeTe2 monolayers have in-plane easy magnetization direction, while the easy direction of the T-MnTe2 monolayer is perpendicular to the layer. Moreover, we analyze the relative exchange strength of the four ferromagnetic monolayers based on the competition between the through-bond ferromagnetic interaction and the through-space antiferromagnetic interaction according to the Goodenough–Kanamori rules. The magnetocrystalline anisotropy is explained qualitatively based on the second-order perturbation theory from the spin-orbit coupling between 3d orbitals of M atoms.

A computational study of the magnetocaloric effect in the LaCr2Si2C compound

We have systematically established and predicted the magnetic behavior and magnetocaloric effect in LaCr2Si2C. In the present work, we have employed the Monte Carlo methods in the framework of the (3D) Ising model. Ferromagnetic order of LaCr2Si2C is observed below the Curie temperature, which is 18.5 K. From the variation of the magnetism and Binder’s fourth-order cumulate of the energy, we deduced that the magnetic phase transition is a second-order (FM) to paramagnetic (PM) transition above Tc. The impacts of both the exchange coupling and the temperature on the hysteresis cycle are treated. Other parameters have also been computed and established as the magnetic entropy change (ΔSM), which reaches 1.234 J/kg K in a field change of 5 T. The relative cooling power (RCP) attains 13.03 J/kg under an applied field of 5 T.

Influence of Interplane Interactions on Formation of Magnetization Plateaus in Generalized 3D Ising Models with Magnetically Coupled Shastry-Sutherland Layers

The combination of the standard Metropolis algorithm and the parallel tempering method is used to examine the influence of interplane interactions on the formation of magnetization plateaus in two types of generalized Ising models with magnetically coupled Shastry-Sutherland layers. In the first model, we consider the first (J1) and second (J2) intraplane and first (I1) and second (I2) interplane nearest neighbour (NN) interactions, while in the second one, we consider all intraplane and interplane interactions between spins inside an sphere of radius r < 6, and the interaction between two spins at Ri and Rj lattice sites is modeled by a simple one-parametric formula with exponentially decaying amplitudes $J_{\text {ij}} \sim e^{-\alpha |\textbf {R}_{\textbf {i}}-\textbf {R}_{\textbf {j}}|}$ . It is shown that the NN interplane interaction I1 does not change qualitatively the magnetic phase diagram of the model found for J1 and J2 (I1 = 0), while the next NN interaction I2 influences strongly both the stability regions of magnetic phases found for J1 and J2 (I1 = 0, I2 = 0) as well as the formation of new phases. In particular, we have found that the ferromagnetic interaction I2 suppresses the stability regions of magnetic plateau phases found for nonzero J1 and J2, while the antiferromagnetic I2 interaction forms, in addition to the standard 1/3 and 1/2 plateaus, the following set of new macroscopic magnetization plateaus: m/ms = 1/6,1/4,2/3, and 3/4. Practically the same plateaus are formed by the second model in the limit of strongly decaying spin interactions ( $\alpha \sim 4$ ), while in the opposite limit (α 1/2 and m/ms < 1/3 disappear and the magnetization curves become continuous in these regions.

Information flow in finite flocks

We explore information flow in finite active matter flocks by simulating the canonical Vicsek model and estimating the flow of information as a function of noise (the variability in the extent to which each animal aligns with its neighbours). We show that the global transfer entropy for finite flocks not only fails to peak near the phase transition, as demonstrated for the canonical 2D Ising model, but remains constant from the transition throughout the entire ordered regime to very low noise values. This provides a foundation for future study regarding information flow in more complex models and real-world flocking data.

QCD equation of state matched to lattice data and exhibiting a critical point singularity

We construct a family of equations of state for QCD in the temperature range 30 MeV $\leq T\leq$ 800 MeV and in the chemical potential range $0\leq \mu_B \leq$ 450 MeV. These equations of state match available lattice QCD results up to $\mathcal{O}(\mu_B^4)$ and in each of them we place a critical point in the 3D Ising model universality class. The position of this critical point can be chosen in the range of chemical potentials covered by the second Beam Energy Scan at RHIC. We discuss possible choices for the free parameters, which arise from mapping the Ising model onto QCD. Our results for the pressure, entropy density, baryon density, energy density and speed of sound can be used as inputs in the hydrodynamical simulations of the fireball created in heavy ion collisions. We also show our result for the second cumulant of the baryon number in thermal equilibrium, displaying its divergence at the critical point. In the future, comparisons between RHIC data and the output of the hydrodynamic simulations, including calculations of fluctuation observables, built upon the model equations of state that we have constructed may be used to locate the critical point in the QCD phase diagram, if there is one to be found.

3D-Ising critical behavior in antiperovskite-type ferromagneticlike Mn3GaN

In this work, a systematic investigation on magnetic critical behavior is performed for the first time on an antiperovskite-type Mn3GaN, which is prepared by intentionally modifying stoichiometry. According to the XRD results, the antiperovskite structure is well preserved, even though all lattice parameters shrink upon reducing Ga and N content down to 60%. The sample exhibits a ferromagneticlike feature with a Curie temperature (TC) of 394 K rather than frustrated behavior in stoichiometric Mn3GaN. Most importantly, the modified Arrott plots, Kouvel–Fisher plots, as well as critical isotherm method self-consistently co-confirm the critical exponents of β = 0.33, γ = 1.23, and δ = 4.7, unambiguously indicating that the critical behavior follows the 3D-Ising model around TC.

Magnetic Interactions and Magnetocaloric Effect in (La0.5Pr0.5)0.6Ba0.4MnO3: Effect of A-Site Codoping

We report on the structural, magnetic, and magnetocaloric properties of polycrystalline (La0.5Pr0.5)0.6Ba0.4MnO3 (LPBMO), which was fabricated by a conventional solid-state reaction method. LPBMO undergoes a second-order paramagnetic to ferromagnetic (PM-FM) phase transition around a Curie temperature TC ~ 277 K. The maximum magnetic entropy change is ~ 3.22 J kg-1 K-1 for a magnetic field change of 5 T. Based on the modified-Arrott plot and the iterative Kouvel Fisher methods, a set of critical exponents (beta = 0.514, gamma = 1.164, and delta = 3.265) are determined. These values are close to those expected for the mean field model with long range interactions below TC and for the 3D-Ising model with short range interactions above TC. Effect of A-site co-doping (La,Pr) on the magnetic and magnetocaloric properties of LPBMO is analyzed and discussed.

Critical Behavior in Ga-Doped Manganites La0.65Bi0.05Sr0.3Mn1 – xGaxO3 (x = 0 and 0.06)

In this paper, we present a detailed study of the critical behavior around paramagnetic–ferromagnetic phase transition in the nominal composition La0.65Bi0.05Sr0.3Mn1 – xGaxO3 (x = 0 and 0.06). Our powder specimens were synthesized using the conventional solid-state reaction method and analyzed by X-ray diffraction and magnetic measurements. X-ray diffraction analysis at room temperature shows that our specimens La0.65Bi0.05Sr0.3Mn1 – xGaxO3 (x = 0 and 0.06) adopt a rhombohedral structure with R-3C space group. Based on Banerjee criterion, the positive slopes of the Arrott plot confirm the second order magnetic phase transition of our samples. The critical exponent values are calculated by distinct techniques like modified Arrott plot (MAP), Kouvel–Fisher method (KF) and critical isotherm (CI). The determined critical exponents values are close to the 3D-Ising model. Furthermore, the substitution of Mn by Ga leads to the decrease of the critical exponent β and at the same time to the reduction of the Curie temperature TC. The validity of the determined exponent values is authenticated using the scaling hypothesis, in which the M(T, e) curves integrate in two universal curves below and above TC.

Computational complexity of spin-glass three-dimensional (3D) Ising model

In this work, the computational complexity of a spin-glass three-dimensional (3D) Ising model (for the lattice size N = lmn, where l, m, n are the numbers of lattice points along three crystallographic directions) is studied. We prove that an absolute minimum core (AMC) model consisting of a spin-glass 2D Ising model interacting with its nearest neighboring plane, has its computational complexity O(2mn). Any algorithms to make the model smaller (or simpler) than the AMC model will cut the basic element of the spin-glass 3D Ising model and lost many important information of the original model. Therefore, the computational complexity of the spin-glass 3D Ising model cannot be reduced to be less than O(2mn) by any algorithms, which is in subexponential time, superpolynomial.

Predicting Quantum Many-Body Dynamics with Transferable Neural Networks**Supported by the National Natural Science Foundation of China under Grant Nos 11874431 and 11804181, the National Key R&D Program of China under Grant No 2018YFA0306800, and the Guangdong Science and Technology Innovation Youth Talent Program under Grant Nos 2016TQ03X688 and 2018YFA0306504, and the Research Fund Program of the State Key Laboratory of

Advanced machine learning (ML) approaches such as transfer learning have seldom been applied to approximate quantum many-body systems. Here we demonstrate that a simple recurrent unit (SRU) based efficient and transferable sequence learning framework is capable of learning and accurately predicting the time evolution of the one-dimensional (1D) Ising model with simultaneous transverse and parallel magnetic fields, as quantitatively corroborated by relative entropy measurements between the predicted and exact state distributions. At a cost of constant computational complexity, a larger many-body state evolution is predicted in an autoregressive way from just one initial state, without any guidance or knowledge of any Hamiltonian. Our work paves the way for future applications of advanced ML methods in quantum many-body dynamics with knowledge only from a smaller system.

Critical phenomenon of the layered chiral helimagnetic YbNi3Al9

Two-dimensional layered YbNi3Al9 exhibits a chiral helimagnetic ground state, which is a candidate for the field-modulated chiral magnetic soliton. In this work, the magnetism and critical phenomenon of YbNi3Al9 are investigated. As H ⊥ c, a magnetic step with loop can be observed in the field-dependent magnetization, which may be corresponding to the possible chiral magnetic soliton phase transition. Based on the analysis of isothermal magnetization around TC, the critical exponents are obtained as β = 0.1370(2), γ = 1.7955(4), and δ = 14.1043(7), which fulfill the Widom scaling relation and Rushbrooke’s law. Moreover, the obtained critical exponents are testified by the modified Arrott plot and scaling hypothesis. The critical exponents of YbNi3Al9 are close to the theoretically prediction of 2D-Ising model with the spatial-dimensionality n = 2 and spin-dimensionality d = 1, indicating one-dimensional magnetic coupling in the two-dimensional framework. Based on universality scaling, we construct the detailed H–T phase diagram around the phase transition with H ⊥ c, which reveals that the field-induced magnetic transition for H ⊥ c is of the first-order type.

Ferromagnetic frustration in ternary nitride ZnFe3N.

As a new antiperovskite nitride, ZnFe3N was synthesized and characterized by almost completely substituting iron atoms at corner positions of γ'-Fe4N. The magnetic interactions of the system with the space group Pm3[combining macron]m are fully investigated. The critical behavior was investigated based on the measured magnetic data around the ferromagnetic phase transition temperature. In this work, the values of critical exponents (β, γ and δ) were obtained systematically using the Kouvel-Fisher method in the critical region. The Widom scaling law (δ = 1 + γβ-1) and the scaling equation (m = f±(h)) were used to reveal the reliability of these values. The values of the critical exponents (β = 0.325, γ = 1.228, and δ = 4.778) are different from those predicted by the three-dimensional (3D) Heisenberg model and mean-field model, and are very close to those of the 3D-Ising model. Combined with ESR analysis, the spin clusters induced by changes in chemical bonds are considered to be the cause for the existence of an anisotropic short-range ordered state in this ferromagnetic system.

Two-dimensional super-roughening in the three-dimensional Ising model.

We present a random-interface representation of the three-dimensional (3D) Ising model based on thermal fluctuations of a uniquely defined geometric spin cluster in the 3D model and its 2D cross section. Extensive simulations have been carried out to measure the global interfacial width as a function of temperature for different lattice sizes which is shown to signal the criticality of the model at T_{c} by forming a size-independent cusp in 3D, along with an emergent super-roughening at its 2D cross section. We find that the super-rough state is accompanied by an intrinsic anomalous scaling behavior in the local properties characterized by a set of geometric exponents which are the same as those for a pure 2D Ising model.

Critical behavior near the ferromagnetic–paramagnetic phase transition in La0.5Pr0.3Ba0.2Mn1−xTixO3 (x = 0.0 and 0.1)

The $${\text{La}}_{0.5 } { \Pr }_{0.3} {\text{Ba}}_{0.2} {\text{Mn}}_{1 - x} {\text{Ti}}_{x} {\text{O}}_{3}$$ (x = 0.0 and 0.1) compounds were elaborated by the solid-state reaction method. The X-ray diffraction characterization has indicated that these samples crystallized in the rhombohedral symmetry with $$R\bar{3}C$$ space group. The magnetic results which we obtain indicated that both samples show a paramagnetic–ferromagnetic transition with decreasing Curie temperature ($$T_{\text{C}}$$) from 170 K (x = 0.0) to 110 K (x = 0.1). The zero-field cold curve for x = 0.1 compound exhibit a clear cusp at low temperature (T = 35 K) that is generally related to a spin-glass or a cluster-glass state. Furthermore, the critical exponents’ values were founded by different techniques. Therefore, the average values of the critical exponent and the critical temperature obtained by the various methods are ($$\beta_{\text{moy}}$$ = 0.296; $$\gamma_{\text{moy}}$$ = 1.126; $$\delta_{\text{moy}}$$ = 4.875; $$T_{\text{C}}$$ = 177.63) and ($$\beta_{\text{moy}}$$ = 0.26; $$\gamma_{\text{moy}}$$ = 1.081; $$\delta_{\text{moy}}$$ = 5.155; $$T_{\text{C}}$$ = 109.99 K) for x = 0.0 and 0.1, respectively. The obtained critical exponents are in agreement with those forecasted out by the 3D-Ising model and the tricritical mean-field model for x = 0.0 and 0.1, respectively.