Three-dimensional Ising Model Research Articles

Harvesting physical predictions from asymptotically safe quantum field theories

Asymptotic safety is a powerful mechanism for obtaining a consistent and predictive quantum field theory beyond the realm of perturbation theory. It hinges on an interacting fixed point of the Wilsonian renormalization group flow which controls the microscopic dynamics. Connecting the fixed point to observations requires constructing the set of effective actions compatible with this microscopic dynamics. Technically, this information is stored in the UV-critical surface of the fixed point. In this work, we describe a novel approach for extracting this information based on analytical and pseudospectral methods. Our construction is illustrated at the level of the three-dimensional Ising model and easily generalizes to any asymptotically safe quantum field theory. It also constitutes an important step toward setting up a well-founded swampland program within the gravitational asymptotic safety program. Published by the American Physical Society 2024

In-depth investigation of phase transition phenomena in network models derived from lattice models

Lattice models exhibit significant potential in investigating phase transitions, yet they encounter numerous computational challenges. To address these issues, this study introduces a Monte Carlo-based approach that transforms lattice models into a network model with intricate inter-node correlations. This framework enables a profound analysis of Ising, JQ, and XY models. By decomposing the network into a maximum entropy component and a conservative component, under the constraint of detailed balance, this work derives an estimation formula for the temperature-dependent magnetic induction in Ising models. Notably, the critical exponent β in the Ising model aligns well with the established results, and the predicted phase transition point in the three-dimensional Ising model exhibits a mere 0.7% deviation from numerical simulations.

Long range order for three-dimensional random field Ising model throughout the entire low temperature regime

Long range order for three-dimensional random field Ising model throughout the entire low temperature regime

3D-Ising-type magnetic interactions stabilized by the extremely large uniaxial magnetocrystalline anisotropy in layered ferromagnetic Cr2Te3

We investigate the magnetocrystalline anisotropy, critical behavior, and magnetocaloric effect in ferromagnetic-layered Cr2Te3. We have studied the critical behavior around the Curie temperature (TC) using various techniques, including the modified Arrott plot (MAP), the Kouvel-Fisher method (KF), and critical isothermal analysis (CI). The derived critical exponents β = 0.353(4) and γ = 1.213(5) fall in between the three-dimensional (3D) Ising and 3D Heisenberg type models, suggesting complex magnetic interactions by not falling into any single universality class. On the other hand, the renormalization group theory, employing the experimentally obtained critical exponents, suggests 3D-Ising-type magnetic interactions decaying with distance as J(r) = r−4.89. We also observe an extremely large uniaxial magnetocrystalline anisotropy energy (MAE) of Ku = 2065 kJ/m3, the highest ever found in any CrxTey based systems, originating from the noncollinear ferromagnetic ground state as predicted from the first-principles calculations. The self-consistent renormalization theory (SCR) suggests Cr2Te3 to be an out-of-plane itinerant ferromagnet. Further, a maximum entropy change of -ΔSMmax≈ 2.08 J/kg − K is estimated around TC for the fields applied parallel to the c-axis.

QCD critical point, Lee-Yang edge singularities, and Padé resummations

I analyze the trajectory of the Lee-Yang edge singularities of the QCD equation of state in the complex baryon chemical potential (μB) plane for different values of the temperature by using the recent lattice results for the Taylor expansion coefficients up to eighth order in μB and various resummation techniques that blend in Padé expansions and conformal maps. By extrapolating from this information, I estimate for the location of the QCD critical point: Tc≈100 MeV, μc≈580 MeV. I also estimate the crossover slope at the critical point to be α1≈9∘ and further constrain the nonuniversal mapping parameters between the three-dimensional Ising model and QCD equations of state. Published by the American Physical Society 2024

Operator product expansion for radial lattice quantization of 3D ϕ4 theory

At its critical point, the three-dimensional lattice Ising model is described by a conformal field theory (CFT), the 3D Ising CFT. Instead of carrying out simulations on Euclidean lattices, we use the quantum finite elements method to implement radially quantized critical ϕ4 theory on simplicial lattices approaching R×S2. Computing the four-point function of identical scalars, we demonstrate the power of radial quantization by the accurate determination of the scaling dimensions Δε and ΔT as well as ratios of the operator product expansion coefficients fσσε and fσσT of the first spin-0 and spin-2 primary operators ε and T of the 3D Ising CFT. Published by the American Physical Society 2024

Duality transformations and the entanglement entropy of gauge theories

The study of entanglement in gauge theories is expected to provide insights into many fundamental phenomena, including confinement. However, calculations of quantities related to entanglement in gauge theories are limited by ambiguities that stem from the non-factorizability of the Hilbert space. In this work we study lattice gauge theories that admit a dual description in terms of spin models, for which the replica trick and Rényi entropies are well defined. In the first part of this work, we explicitly perform the duality transformation in a replica geometry, deriving the structure of a replica space for a gauge theory. Then, in the second part, we calculate, by means of Monte Carlo simulations, the entropic c-function of the ℤ2 gauge theory in three spacetime dimensions, exploiting its dual description in terms of the three-dimensional Ising model.

On nonintegrability of three-dimensional Ising model

On nonintegrability of three-dimensional Ising model

Critical behavior in the ferromagnet MgMn6Sn6 single crystal with HfFe6Ge6-type structure

A thorough investigation was carried out to analyze the magnetic critical properties of single crystals belonging to the HfFe6Ge6-type structure. MgMn6Sn6 exhibit a ferromagnetic transition below TC ≈ 300 K. The magnetization measurements were analyzed around the phase transition temperature of this ferromagnetic material for two different orientations: H//ab and H//c. Using Kouvel‐Fisher method, we can determine critical exponents β, γ, and δ in this study. To evaluate accuracy of these values, the scaling equation m=f±(h) and Widom scaling law δ=1+γ/β were applied. Critical exponents γ = 1.191 ± 0.007 at TC = 300.92 K, β = 0.326 ± 0.003 at TC = 300.96 K, and δ = 4.653 with TC ≈ 300 K indicate that the three-dimensional Ising model (β = 0.325, γ = 1.24, and δ = 4.82) is the better model for H//ab. The mean‐field and 3D‐Heisenberg models were the best models below TC and above TC for H//c, respectively. The MgMn6Sn6 single crystal exhibits a notable anisotropic transition in its magnetic phase according to all experimental findings. The presence of an anisotropic state with short-range order in the ferromagnetic system are revealed by analyzing electron spin resonance (ESR) spectroscopy. Our research provides experimental proof of a holistic comprehension regarding the magnetic phase-transition characteristics demonstrated by MgMn6Sn6.

Uncertainty Analysis for Ferromagnetic-Paramagnetic Phase Transition Behavior of Magnetic Materials

This study aims to model the phase transition of ferromagnetic materials using two- (2D) and three-dimensional (3D) Ising models, incorporating long-range magnetic spin-to-spin interactions and the influence of an external magnetic field. The 2D Ising model is investigated mainly for a 4×4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4\ imes 4$$\\end{document} domain, while its extension to larger domains is also explored. For the 3D Ising model, the selection of representative volume element is discussed in terms of free energy. An uncertainty quantification formulation is integrated into the Ising model to capture the uncertainties related to the external magnetic field and evaluate their effects on the phase transition from a ferromagnetic to a paramagnetic state. Despite the common use and known effectiveness of the Ising model, there is still no universally accepted exact solution for the 3D domains, making it an ongoing research focus. The research also acknowledges the complexities and limitations arising from experimental methods, despite the insights they provide on the microscopic dynamics of ferromagnetic materials during phase transitions.

Ising string beyond the Nambu-Goto action

A major result of the effective string theory (EST) description of confinement is the so-called “low-energy universality”, which states that the first few terms of the large distance expansion of any EST are universal and coincide with those of the Nambu-Goto action. Going beyond this approximation is one of the most interesting open problems in the EST. In the higher-order terms beyond Nambu-Goto several important pieces of physical information are encoded, which could improve our understanding of the physical mechanisms behind confinement and of the physical degrees of freedoms which originate the EST. In this paper we evaluate numerically the first two of these corrections in the case of the three-dimensional gauge Ising model. The first of them turns out to be negative; γ3=−0.00048(4), similar (but not equal) to the one recently measured in the SU(2) Yang-Mills theory in three dimensions and compatible with the bootstrap bound γ3≥−1768. Published by the American Physical Society 2024

Heterogeneous nucleation in the random field Ising model.

We investigate the nucleation dynamics of the three-dimensional random field Ising model under an external field. We use umbrella sampling to compute the free-energy cost of a critical nucleus and use forward flux sampling for the direct estimation of nucleation rates. For moderate to strong disorder, our results indicate that the size of the nucleating cluster is not a good reaction coordinate, contrary to the pure Ising model. We rectify this problem by introducing a coordinate that also accounts for the location of the nucleus. Using the free energy barrier to predict the nucleation rate, we find reasonable agreement, although deviations become stronger as disorder increases. We attribute this effect to cluster shape fluctuations. We also discuss finite-size effects on the nucleation rate.

Magnetic properties and critical behaviors of the nodal-line semimetal candidate ErIn3

The AuCu3-type intermetallic compounds ReIn3 (Re = a rare earth ion) with type-IV magnetic space groups are predicted to show topologically nontrivial electronic states. Here, we grow ErIn3 single crystals, and study their magnetic properties and critical behaviors by means of the magnetic susceptibility, and magnetization isotherm measurements. Combining a detailed analysis of the magnetic susceptibility and isothermal magnetization, we find that this compound harbors a complicated magnetic phase diagram, and its magnetic moment arrangement appears not to simply follow the fashion as observed in the isostructural counterpart GdIn3 (it adopts a conventional type-C magnetic structure that belongs to type-IV magnetic space groups). A careful study of the magnetic properties around the antiferromagnetic (AFM)-paramagnetic transition yields the critical exponents β = 0.309 (0.297), γ = 1.117 (1.038), and δ = 4.617 (4.454), indicating that the tricritical mean field model or the three-dimensional Ising model works for ErIn3’s magnetic behaviors and the presence of a long-range AFM interaction therein. Besides, the exchange interaction distance J(r) ∼ r −4.665 as well confirms a long-range magnetic coupling in ErIn3. Our results offer the clues that the magnetic structure varies from one member of ReIn3 family to another, and to confirm their electronic features in the AFM phases further experimental and theoretical studies are still desired.

Study of magnetic properties and magnetocrystalline anisotropy of CoPb3 alloys by Ab initio calculations and Monte Carlo simulations for three-dimensional Ising, XY and Heisenberg models

The electronic and magnetic properties of CoPb3 compound have been calculated using the frst-principles density functional calculations and Monte Carlo simulation of Ising, XY and Heisenberg models. This compound exhibit metallic character. The total magnetic moments of CoPb3 using GGA and GGA + U approximations are 1.53874 and 2.1487 μB/f.u, respectively. The magnetic anisotropy shows that the easy axis is along the axis [100]. CoPb3 has perpendicular magnetic anisotropy (PMA) at T = 0 K. The critical temperature follows the trend Ising model > XY model > Heisenberg model using the MCS method. The isothermal magnetic entropy change was calculated. The maximum magnetic entropy changes are 0.24 J K−1 kg−1, 8.69 J K−1 kg−1 and 1.16 J K−1 kg−1 for Ising, XY and Heisenberg models respectively.

Critical dynamical behavior of the Ising model.

We investigate the dynamical critical behavior of the two- and three-dimensional Ising models with Glauber dynamics in equilibrium. In contrast to the usual standing, we focus on the mean-squared deviation of the magnetization M,MSD_{M}, as a function of time, as well as on the autocorrelation function of M. These two functions are distinct but closely related. We find that MSD_{M} features a first crossover at time τ_{1}∼L^{z_{1}}, from ordinary diffusion with MSD_{M}∼t, to anomalous diffusion with MSD_{M}∼t^{α}. Purely on numerical grounds, we obtain the values z_{1}=0.45(5) and α=0.752(5) for the two-dimensional Ising ferromagnet. Related to this, the magnetization autocorrelation function crosses over from an exponential decay to a stretched-exponential decay. At later times, we find a second crossover at time τ_{2}∼L^{z_{2}}. Here, MSD_{M} saturates to its late-time value ∼L^{2+γ/ν}, while the autocorrelation function crosses over from stretched-exponential decay to simple exponential one. We also confirm numerically the value z_{2}=2.1665(12), earlier reported as the single dynamic exponent. Continuity of MSD_{M} requires that α(z_{2}-z_{1})=γ/ν-z_{1}. We speculate that z_{1}=1/2 and α=3/4, values that indeed lead to the expected z_{2}=13/6 result. A complementary analysis for the three-dimensional Ising model provides the estimates z_{1}=1.35(2),α=0.90(2), and z_{2}=2.032(3). While z_{2} has attracted significant attention in the literature, we argue that for all practical purposes z_{1} is more important, as it determines the number of statistically independent measurements during a long simulation.

Mapping out the thermodynamic stability of a QCD equation of state with a critical point using active learning

The Beam Energy Scan Theory (BEST) collaboration's equation of state (EoS) incorporates a three-dimensional Ising model critical point into the quantum chromodynamics (QCD) equation of state from lattice simulations. However, it contains four free parameters related to the size and location of the critical region in the QCD phase diagram. Certain combinations of the free parameters lead to acausal or unstable realizations of the EoS that should not be considered. In this work, we use an active learning framework to rule out pathological EoS efficiently. We find that checking stability and causality for a small portion of the parameters' range is sufficient to construct algorithms that perform with $>96$% accuracy across the entire parameter space. Though in this work we focus on a specific case, our approach can be generalized to any EoS containing a parameter space-class correspondence.

Efficient calculation of three-dimensional tensor networks

We have proposed an efficient algorithm to calculate physical quantities in the translational invariant three-dimensional tensor networks, which is particularly relevant to the study of the three-dimensional classical statistical models and the ($2+1$)-dimensional quantum lattice models. In the context of a classical model, we determine the partition function by solving the dominant eigenvalue problem of the transfer matrix, whose left and right dominant eigenvectors are represented by two projected entangled simplex states. These two projected entangled simplex states are not Hermitian conjugate to each other but are appropriately arranged so that their inner product can be computed much more efficiently than in the usual prescription. For the three-dimensional Ising model, the calculated internal energy and spontaneous magnetization agree with the published results in the literature. The possible improvement and extension to other models are also discussed.

Topological correlations in three-dimensional classical Ising models: An exact solution with a continuous phase transition

We study a three-dimensional (3D) classical Ising model that is exactly solvable when some coupling constants take certain imaginary values. The solution combines and generalizes the Onsager-Kaufman solution [L. Onsager, Phys. Rev. 65, 117 (1944); B. Kaufman, Phys. Rev. 76, 1232 (1949)] of the 2D Ising model and the solution of Kitaev's honeycomb model [A. Kitaev, Ann. Phys, 321, 2 (2006)], leading to a three-parameter phase diagram with a third-order phase transition between two distinct phases. Interestingly, the phases of this model are distinguished by topological features: the expectation value of a certain family of loop observables depend only on the topology of the loop (whether the loop is contractible), and are quantized at rational values that differ in the two phases. We show that a related exactly solvable 3D classical statistical model with real coupling constants also shows the topological features of one of these phases. Furthermore, even in the model with complex parameters, the partition function has some physical relevance, as it can be interpreted as the transition amplitude of a quantum dynamical process and may shed light on dynamical quantum phase transitions.

Comment on Zhang, D. Exact Solution for Three-Dimensional Ising Model. Symmetry 2021, 13, 1837

We show that Zhang Degang’s claimed solution of the three-dimensional Ising model has fatal irreparable errors.

Reply to Perk, J.H.H. Comment on “Zhang, D. Exact Solution for Three-Dimensional Ising Model. Symmetry 2021, 13, 1837”

In this reply, I point out that the comment by Jacques H. H. Perk [...]