Three-dimensional Ising Universality Class Research Articles

Machine learning phase transitions of the three-dimensional Ising universality class* *Supported by the National Natural Science Foundation of China (12275102) and the National Key Research and Development Program of China (2022YFA1604900)

Exploration of the QCD phase diagram and critical point is one of the main goals in current relativistic heavy-ion collisions. The QCD critical point is expected to belong to a three-dimensional (3D) Ising universality class. Machine learning techniques are found to be powerful in distinguishing different phases of matter and provide a new way to study the phase diagram. We investigate phase transitions in the 3D cubic Ising model using supervised learning methods. It is found that a 3D convolutional neural network can be trained to effectively predict physical quantities in different spin configurations. With a uniform neural network architecture, it can encode phases of matter and identify both second- and first-order phase transitions. The important features that discriminate different phases in the classification processes are investigated. These findings can help study and understand QCD phase transitions in relativistic heavy-ion collisions.

Thermal melting of discrete time crystals: A dynamical phase transition induced by thermal fluctuations

The stability of a discrete time crystal against thermal fluctuations has been studied numerically by solving a stochastic Landau-Lifshitz-Gilbert equation of a periodically driven classical system composed of interacting spins, each of which couples to a thermal bath. It is shown that in the thermodynamic limit, even though the long-range temporal crystalline order is stable at low temperature, it is melting above a critical temperature, at which the system experiences a nonequilibrium phase transition. The critical behaviors of the continuous phase transition have been systematically investigated, and it is shown that despite the genuine nonequilibrium feature of such a periodically driven system, its critical properties fall into the three-dimensional Ising universality class with a dynamical exponent ($z=2$) identical to that in the critical dynamics of a kinetic Ising model without driving.

Fixed point behavior of cumulants in the three-dimensional Ising universality class

High-order cumulants and factorial cumulants of conserved charges are suggested for the study of the critical dynamics in heavy-ion collision experiments. In this paper, using the parametric representation of the three-dimensional Ising model which is believed to belong to the same universality class as quantum chromo-dynamics, the temperature dependence of the second- to fourth-order (factorial) cumulants of the order parameter is studied. It is found that the values of the normalized cumulants are independent of the external magnetic field at the critical temperature, which results in a fixed point in the temperature dependence of the normalized cumulants. In finite-size systems simulated using the Monte Carlo method, this fixed point behavior still exists at temperatures near the critical. This fixed point behavior has also appeared in the temperature dependence of normalized factorial cumulants from at least the fourth order. With a mapping from the Ising model to QCD, the fixed point behavior is also found in the energy dependence of the normalized cumulants (or fourth-order factorial cumulants) along different freeze-out curves.

Restoring isotropy in a three-dimensional lattice model: The Ising universality class

We study a generalized Blume-Capel model on the simple cubic lattice. In addition to the nearest neighbor coupling there is a next to next to nearest neighbor coupling. In order to quantify spatial anisotropy, we determine the correlation length in the high temperature phase of the model for three different directions. It turns out that the spatial anisotropy depends very little on the dilution parameter $D$ of the model and is essentially determined by the ratio of the nearest neighbor and the next to next to nearest neighbor coupling. This ratio is tuned such that the leading contribution to the spatial anisotropy is eliminated. Next we perform a finite size scaling (FSS) study to tune $D$ such that also the leading correction to scaling is eliminated. Based on this FSS study, we determine the critical exponents $\nu=0.62998(5)$ and $\eta=0.036284(40)$, which are in nice agreement with the more accurate results obtained by using the conformal bootstrap method. Furthermore we provide accurate results for fixed point values of dimensionless quantities such as the Binder cumulant and for the critical couplings. These results provide the groundwork for broader studies of universal properties of the three-dimensional Ising universality class.

Cumulants and factorial cumulants in the three-dimensional Ising universality class

The high-order cumulants and factorial cumulants of conserved charges are suggested to study the critical dynamics in heavy ion collisions. In this paper, using parametric representation of the three-dimensional Ising model, density plots on the phase diagram and temperature dependence of the order parameter cumulants and factorial cumulants are studied and compared. In the vicinity of the critical point, cumulants and factorial cumulants can not be distinguished. Far away from the critical point, sign changes occur in the factorial cumulants comparing with the same order cumulants. The cause of sign changes is analyzed. These features may be used to measure the distance to the critical point.

Second critical point in two realistic models of water.

The hypothesis that water has a second critical point at deeply supercooled conditions was formulated to provide a thermodynamically consistent interpretation of numerous experimental observations. A large body of work has been devoted to verifying or falsifying this hypothesis, but no unambiguous experimental proof has yet been found. Here, we use histogram reweighting and large-system scattering calculations to investigate computationally two molecular models of water, TIP4P/2005 and TIP4P/Ice, widely regarded to be among the most accurate classical force fields for this substance. We show that both models have a metastable liquid-liquid critical point at deeply supercooled conditions and that this critical point is consistent with the three-dimensional Ising universality class.

Liquid-liquid critical point of water

Chemical Physics Despite a broad range of experimental observations that indirectly point out the possible existence of a liquid-liquid phase transition in deeply supercooled water, no unambiguous experiment has shown this yet. The challenges of performing experiments under such conditions are associated with the inevitable rapid crystallization of the metastable liquid state, and therefore computational simulations become a crucial alternative. Using two accurate classical models of water and long (more than 40 microseconds) isobaricisothermal molecular dynamics simulations, Debenedetti et al. provide strong computational evidence of the presence of a second metastable liquid-liquid critical point in the deeply supercooled region that has critical behavior consistent with the three-dimensional Ising universality class. Science this issue p. [289][1] [1]: /lookup/doi/10.1126/science.abb9796

High-resolution Monte Carlo study of the order-parameter distribution of the three-dimensional Ising model.

We apply extensive Monte Carlo simulations to study the probability distribution P(m) of the order parameter m for the simple cubic Ising model with periodic boundary condition at the transition point. Sampling is performed with the Wolff cluster flipping algorithm, and histogram reweighting together with finite-size scaling analyses are then used to extract a precise functional form for the probability distribution of the magnetization, P(m), in the thermodynamic limit. This form should serve as a benchmark for other models in the three-dimensional Ising universality class.

Dynamic critical exponent z of the three-dimensional Ising universality class: Monte Carlo simulations of the improved Blume-Capel model.

We study purely dissipative relaxational dynamics in the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model on the simple cubic lattice by using local algorithms. We perform a finite size scaling analysis of the integrated autocorrelation time of the magnetic susceptibility in equilibrium at the critical point. We obtain z=2.0245(15) for the dynamic critical exponent. As a complement, fully magnetized configurations are suddenly quenched to the critical temperature, giving consistent results for the dynamic critical exponent. Furthermore, our estimate of z is fully consistent with recent field theoretic results.

Probing QCD critical fluctuations from intermittency analysis in relativistic heavy-ion collisions

It is shown that intermittency, a self-similar correlation with respect to the size of the phase space volume, is sensitive to critical density fluctuations of baryon numbers in a system belonging to the three-dimensional (3D) Ising universality class. The relation between intermittency index and relative baryon density fluctuation is obtained. We thus suggest that measuring the intermittency in relativistic heavy-ion collisions could be used as a good probe of density fluctuations associated with the QCD critical phenomena. From recent preliminary results on neutron density fluctuations in central Au + Au collisions at sNN=7.7,11.5,19.6,27,39,62.4 and 200 GeV at RHIC/STAR, the collision energy dependence of intermittency index is extracted and shows a non-monotonic behavior with a peak at around 20 - 27 GeV, indicating that the strength of intermittency becomes the largest in this energy region. The transport UrQMD model without implementing critical physics cannot describe the observed behavior.

Two- and three-point functions at criticality: Monte Carlo simulations of the improved three-dimensional Blume-Capel model.

We compute two- and three-point functions at criticality for the three-dimensional Ising universality class. To this end, we simulate the improved Blume-Capel model at the critical temperature on lattices of a linear size up to L=1600. As a check, also simulations of the spin-1/2 Ising model are performed. We find f_{σσε}=1.051(1) and f_{εεε}=1.533(5) for operator product expansion coefficients. These results are consistent with but less precise than those recently obtained by using the bootstrap method. An important ingredient in our simulations is a variance reduced estimator of N-point functions. Finite size corrections vanish with L^{-Δ_{ε}}, where L is the linear size of the lattice and Δ_{ε} is the scaling dimension of the leading Z_{2}-even scalar ε.

Finite-size behaviour of generalized susceptibilities in the whole phase plane of the Potts model**Supported by Fund Project of National Natural Science Foundation of China (11647093, 11405088, 11521064), Fund Project of Sichuan Provincial Department of Education (16ZB0339), Fund Project of Chengdu Technological University (2016RC004) and the Major State Basic Research Development Program of China (2014CB845402)

We study the sign distribution of generalized magnetic susceptibilities in the temperature-external magnetic field plane using the three-dimensional three-state Potts model. We find that the sign of odd-order susceptibility is opposite in the symmetric (disorder) and broken (order) phases, but that of the even-order one remains positive when it is far away from the phase boundary. When the critical point is approached from the crossover side, negative fourth-order magnetic susceptibility is observable. It is also demonstrated that non-monotonic behavior occurs in the temperature dependence of the generalized susceptibilities of the energy. The finite-size scaling behavior of the specific heat in this model is mainly controlled by the critical exponent of the magnetic susceptibility in the three-dimensional Ising universality class.

Critical behavior in the presence of an order-parameter pinning field

Can one investigate more efficiently quantum critical behavior by employing a local order-parameter pinning field, which explicitly breaks the symmetry of the model under investigation? To answer this question, the authors consider the two-dimensional square-lattice bilayer quantum Heisenberg model using a world-line quantum Monte Carlo method. The pinning-field approach is found to accurately locate the quantum critical point over a wide range of pinning-field strengths. However, the identification of the quantum critical scaling behavior is found to be complicated by the fact that the pinning field introduces strong corrections to scaling. A renormalization group analysis exhibits important analogies to surface critical phenomena, and a crossover effect to an infinite pinning-field fixed point, which the authors study in detail by simulations of an improved classical lattice model in the three-dimensional Ising universality class. At the infinite pinning-field fixed point, the short-distance expansion of the order-parameter profile exhibits a new universal critical exponent, which characterizes the experimentally relevant critical adsorption on a line defect. This result also implies the presence of slowly decaying scaling corrections, which are analyzed here in detail.

Structure and dynamics of binary liquid mixtures near their continuous demixing transitions

The dynamic and static critical behavior of a family of binary Lennard-Jones liquid mixtures, close to their continuous demixing points (belonging to the so-called model H' dynamic universality class), are studied computationally by combining semi-grand canonical Monte Carlo simulations and large-scale molecular dynamics (MD) simulations, accelerated by graphic processing units (GPU). The symmetric binary liquid mixtures considered cover a variety of densities, a wide range of compressibilities, and various interactions between the unlike particles. The static quantities studied here encompass the bulk phase diagram (including both the binodal and the λ-line), the correlation length, and the concentration susceptibility, of the finite-sized systems above the bulk critical temperature Tc, the compressibility and the pressure at Tc. Concerning the collective transport properties, we focus on the Onsager coefficient and the shear viscosity. The critical power-law singularities of these quantities are analyzed in the mixed phase (above Tc) and non-universal critical amplitudes are extracted. Two universal amplitude ratios are calculated. The first one involves static amplitudes only and agrees well with the expectations for the three-dimensional Ising universality class. The second ratio includes also dynamic critical amplitudes and is related to the Einstein-Kawasaki relation for the interdiffusion constant. Precise estimates of this amplitude ratio are difficult to obtain from MD simulations, but within the error bars our results are compatible with theoretical predictions and experimental values for model H'. Evidence is reported for an inverse proportionality of the pressure and the isothermal compressibility at the demixing transition, upon varying either the number density or the repulsion strength between unlike particles.

Behavior and finite-size effects of the sixth order cumulant in the three-dimensional Ising universality class**Supported by Fund Project of Sichuan Provincial Department of Education (16ZB0339), Fund Project of Chengdu Technological University for Doctor (2016RC004), Major State Basic Research Development Program of China (2014CB845402) and National Natural Science Foundation of China (11405088, 11221504)

The high-order cumulants of conserved charges are suggested to be sensitive observables to search for the critical point of Quantum Chromodynamics (QCD). This has been calculated to the sixth order in experiments. Corresponding theoretical studies on the sixth order cumulant are necessary. Based on the universality of the critical behavior, we study the temperature dependence of the sixth order cumulant of the order parameter using the parametric representation of the three-dimensional Ising model, which is expected to be in the same universality class as QCD. The density plot of the sign of the sixth order cumulant is shown on the temperature and external magnetic field plane. We found that at non-zero external magnetic field, when the critical point is approached from the crossover side, the sixth order cumulant has a negative valley. The width of the negative valley narrows with decreasing external field. Qualitatively, the trend is similar to the result of Monte Carlo simulation on a finite-size system. Quantitatively, the temperature of the sign change is different. Through Monte Carlo simulation of the Ising model, we calculated the sixth order cumulant of different sizes of systems. We discuss the finite-size effects on the temperature at which the cumulant changes sign.

Local and long-range magnetic order of thespin−32systemCoSb2O6

The ${\mathrm{Co}}^{2+}$ ions of ${\mathrm{CoSb}}_{2}{\mathrm{O}}_{6}$ exhibit local magnetic order below $\ensuremath{\sim}80$ K followed by long-range antiferromagnetic order below ${T}_{N}=13.45$ K. Analysis of the magnetic susceptibility above ${T}_{N}$ using an Ising model of Co-Co dimers yields a magnetic exchange coupling ${J}_{\ensuremath{\parallel}}/{k}_{\mathrm{B}}=\ensuremath{-}10.603(6)$ K (${k}_{\mathrm{B}}$ is the Boltzmann constant). The transition at ${T}_{N}$ is accompanied by a spin gap in the heat capacity $[{\mathrm{\ensuremath{\Delta}}}_{2}/{k}_{\mathrm{B}}=33.92(9)$ K]. Highly anisotropic behavior of the thermal expansion is observed, and the influence of local magnetic order is evident. A critical exponent of $\ensuremath{\alpha}=0.103(3)$ is obtained from analysis of the critical behavior of the heat capacity and thermal expansion near ${T}_{N}$, similar to the value expected for the three-dimensional Ising universality class.

Emergent Coulombic criticality and Kibble-Zurek scaling in a topological magnet

When a classical system is driven through a continuous phase transition, its nonequilibrium response is universal and exhibits Kibble-Zurek scaling. We explore this dynamical scaling in the novel context of a three-dimensional topological magnet with fractionalized excitations, namely the liquid-gas transition of the emergent mobile magnetic monopoles in dipolar spin ice. Using field-mixing and finite-size scaling techniques, we place the critical point of the liquid-gas line in the three-dimensional Ising universality class. We then demonstrate Kibble-Zurek scaling for sweeps of the magnetic field through the critical point. Unusually slow microscopic time scales in spin ice offer a unique opportunity to detect this universal nonequilibrium physics in current experimental setups.

Condensation of Methane in the Metal-Organic Framework IRMOF-1: Evidence for Two Critical Points.

Extensive grand canonical Monte Carlo simulations in combination with successive umbrella sampling are used to investigate the condensation of methane in the nanoporous crystalline material IRMOF-1. Two different types of novel condensation transitions are found, each of them ending in a critical point: (i) a fluid-fluid transition at higher densities (the analog of the liquid-gas transition in the bulk) and (ii) a phase transition at low densities on the surface of the IRMOF-1 structure. The nature of these transitions is different from the usual capillary condensation in thin films and cylindrical pores where the coexisting phases are confined in one or two of the three spatial dimensions. In contrast to that, in IRMOF-1 the different phases can be described as bulk phases that are inhomogeneous due to the presence of the metal-organic framework. As a consequence, the condensation transitions in IRMOF-1 belong to the three-dimensional (3D) Ising universality class.

Critical behaviour of a 3D Ising-like system in the ρ^6 model approximation: Role of the correction for the potential averaging

The critical behaviour of systems belonging to the three-dimensional Ising universality class is studied theoretically using the collective variables (CV) method. The partition function of a one-component spin system is calculated by the integration over the layers of the CV phase space in the approximation of the non-Gaussian sextic distribution of order-parameter fluctuations (the \rho^6 model). A specific feature of the proposed calculation consists in making allowance for the dependence of the Fourier transform of the interaction potential on the wave vector. The inclusion of the correction for the potential averaging leads to a nonzero critical exponent of the correlation function \eta and the renormalization of the values of other critical exponents. The contributions from this correction to the recurrence relations for the \rho^6 model, fixed-point coordinates and elements of the renormalization-group linear transformation matrix are singled out. The expression for a small critical exponent \eta is obtained in a higher non-Gaussian approximation.

Thermodynamic Casimir effect: Universality and corrections to scaling

We study the thermodynamic Casimir force for films in the three-dimensional Ising universality class with symmetry breaking boundary conditions. We focus on the effect of corrections to scaling and probe numerically the universality of our results. In particular, we check the hypothesis that corrections are well described by an effective thickness ${L}_{0,\mathrm{eff}}={L}_{0}+c{({L}_{0}+{L}_{s})}^{1\ensuremath{-}\ensuremath{\omega}}+{L}_{s}$, where $c$ and ${L}_{s}$ are system specific parameters and $\ensuremath{\omega}\ensuremath{\approx}0.8$ is the exponent of the leading bulk correction. We simulate the improved Blume-Capel model and the spin-1/2 Ising model on the simple cubic lattice. First, we analyze the behavior of various quantities at the critical point. Taking into account corrections $\ensuremath{\propto}{L}_{0}^{\ensuremath{-}\ensuremath{\omega}}$ in the case of the Ising model, we find good consistency of results obtained from these two different models. In particular, we get from the analysis of our data for the Ising model for the difference of Casimir amplitudes ${\ensuremath{\Delta}}_{+\ensuremath{-}}\ensuremath{-}{\ensuremath{\Delta}}_{++}=3.200(5)$, which nicely compares with ${\ensuremath{\Delta}}_{+\ensuremath{-}}\ensuremath{-}{\ensuremath{\Delta}}_{++}=3.208(5)$ obtained by studying the improved Blume-Capel model. Next, we study the behavior of the thermodynamic Casimir force for large values of the scaling variable $x=t{({L}_{0}/{\ensuremath{\xi}}_{0})}^{1/\ensuremath{\nu}}$. It can be obtained up to an overall amplitude by expressing the partition function of the film in terms of eigenvalues and eigenstates of the transfer matrix and boundary states. Here, we demonstrate how this amplitude can be computed with high accuracy. Finally, we discuss our results for the scaling functions ${\ensuremath{\theta}}_{+\ensuremath{-}}$ and ${\ensuremath{\theta}}_{++}$ of the thermodynamic Casimir force for the whole range of the scaling variable. We conclude that our numerical results are in accordance with universality. Corrections to scaling are well approximated by an effective thickness.