Ising Criticality Research Articles

Critical short-time behavior of majority-vote model on scale-free networks.

We discuss the short-time behavior of the majority vote dynamics on scale-free networks at the critical threshold. A heterogeneous mean-field theory on the critical short-time behavior of the majority-vote model on scale-free networks is introduced. In addition, the heterogeneous mean-field predictions are compared with extensive Monte Carlo simulations of the short-time dependencies of the order parameter and the susceptibility. Closed expressions of the dynamical exponent z and the time correlation exponent ν_{∥} are obtained. The short-time scaling is compatible with a nonuniversal critical behavior for 5/2<γ<7/2. However, for γ≥7/2, we have the mean-field Ising criticality with additional logarithmic corrections for γ=7/2, the same as the stationary scaling.

Random Field Ising Model Criticality in a Complex Binary Liquid System.

While Ising criticality in classical liquids has been firmly established both theoretically and experimentally, much less is known about criticality in liquids in which the growth of the correlation length is frustrated by finite-size effects. A theoretical approach for dealing with this issue is the random-field Ising model (RFIM). While experimental critical-exponent values have been reported for magnetic samples (here, we consider γ, ν and η), little experimental information is available for critical fluctuations in corresponding liquid systems. In this paper, we present a study on a binary liquid consisting of 3-methyl pyridine and heavy water in a very light-weight porous gel. We find that the experimental results are in agreement with the theoretical predictions from the RFIM.

Quantum Simulation of the Tricritical Ising Model in Tunable Josephson Junction Ladders.

Modern hybrid superconductor-semiconductor Josephson junction arrays are a promising platform for analog quantum simulations. Their controllable and nonsinusoidal energy-phase relation opens the path to implement nontrivial interactions and study the emergence of exotic quantum phase transitions. Here, we propose the analysis of an array of hybrid Josephson junctions defining a two-leg ladder geometry for the quantum simulation of the tricritical Ising phase transition. This transition provides the paradigmatic example of minimal conformal models beyond Ising criticality and its excitations are intimately related to Fibonacci non-Abelian anyons and topological order in two dimensions. We study this superconducting system and its thermodynamic phases based on bosonization and matrix-product-state techniques. Its effective continuous description in terms of a three-frequency sine-Gordon quantum field theory suggests the presence of the targeted tricritical point and the numerical simulations confirm this picture. Our results indicate which experimental observables can be adopted in realistic devices to probe the physics and the phase transitions of the model. Additionally, our proposal provides a useful one-dimensional building block to design exotic topological order in two-dimensional scalable Josephson junction arrays.

Dimensional reduction of Kitaev spin liquid at quantum criticality

We investigate the fate of the Kitaev spin liquid (KSL) under the influence of an external magnetic field h in the [001] direction and upon tuning bond anisotropy of the Kitaev coupling Kz keeping Kx=Ky=K. Guided by density matrix renormalization group, exact diagonalization, and with insights from parton mean-field theory, we uncover a field-induced gapless-to-gapless Lifshitz transition from the nodal KSL to an intermediate gapless phase. The intermediate phase sandwiched between hc1 and hc2, which persists for a wide range of anisotropy Kz/K&gt;0, is composed of weakly coupled one-dimensional quantum critical chains. This intermediate phase is a dimensional crossover, which asymptotically leads to the one-dimensional quantum Ising criticality characterized by the (1+1)D conformal field theory as the field reaches the phase transition at hc2. Beyond hc2 the system enters a partially polarized phase describable as effectively decoupled bosonic chains in which spin waves propagate along the one-dimensional zigzag direction. Our findings provide a comprehensive phase diagram and offer insights into the unusual physics of dimensional reduction generated by a uniform magnetic field in an otherwise two-dimensional quantum spin liquid. Published by the American Physical Society 2024

Machine learning renormalization group for statistical physics

We develop a machine-learning renormalization group (MLRG) algorithm to explore and analyze many-body lattice models in statistical physics. Using the representation learning capability of generative modeling, MLRG automatically learns the optimal renormalization group (RG) transformations from self-generated spin configurations and formulates RG equations without human supervision. The algorithm does not focus on simulating any particular lattice model but broadly explores all possible models compatible with the internal and lattice symmetries given the on-site symmetry representation. It can uncover the RG monotone that governs the RG flow, assuming a strong form of the c-theorem. This enables several downstream tasks, including unsupervised classification of phases, automatic location of phase transitions or critical points, controlled estimation of critical exponents, and operator scaling dimensions. We demonstrate the MLRG method in two-dimensional lattice models with Ising symmetry and show that the algorithm correctly identifies and characterizes the Ising criticality.

Operator Product Expansion Coefficients of the 3D Ising Criticality via Quantum Fuzzy Spheres.

Conformal field theory (CFT) plays a crucial role in the study of various critical phenomena. While much attention has been paid to the critical exponents of different universalities, which correspond to the conformal dimensions of CFT primary fields, other important and intricate data such as operator product expansion (OPE) coefficients governing the fusion of two primary fields, have remained largely unexplored, especially in dimensions higher than 2D (or equivalently, 1+1D). Motivated by the recently proposed fuzzy sphere regularization, we investigate the operator content of 3D Ising criticality from a microscopic perspective. We first outline the procedure for extracting OPE coefficients on the fuzzy sphere and then compute 13 OPE coefficients of low-lying CFT primary fields. Our results are highly accurate and in agreement with the numerical conformal bootstrap data of 3D Ising CFT. Moreover, we were able to obtain 4 OPE coefficients, including f_{T_{μν}T_{ρη}ε}, which were previously unknown, thus demonstrating the superior capabilities of our scheme. Expanding the horizon of the fuzzy sphere regularization from the state perspective to the operator perspective opens up new avenues for exploring a wealth of new physics.

Quantum spin liquids bootstrapped from Ising criticality in Rydberg arrays

Arrays of Rydberg atoms constitute a highly tunable, strongly interacting venue for the pursuit of exotic states of matter. We develop a new strategy for accessing a family of fractionalized phases known as quantum spin liquids in two-dimensional Rydberg arrays. We specifically use effective field theory methods to study arrays assembled from Rydberg chains tuned to an Ising phase transition that famously hosts emergent fermions propagating within each chain. This highly entangled starting point allows us to naturally access spin liquids familiar from Kitaev's honeycomb model, albeit from an entirely different framework. In particular, we argue that finite-range repulsive Rydberg interactions, which frustrate nearby symmetry-breaking orders, can enable coherent propagation of emergent fermions between the chains in which they were born. Delocalization of emergent fermions across the full two-dimensional Rydberg array yields a gapless Z2 spin liquid with a single massless Dirac cone. Here, the Rydberg occupation numbers exhibit universal power-law correlations that provide a straightforward experimental diagnostic of this phase. We further show that explicitly breaking symmetries perturbs the gapless spin liquid into gapped, topologically ordered descendants: Breaking lattice symmetries generates toric-code topological order, whereas introducing chirality generates non-Abelian Ising topological order. In the toric-code phase, we analytically construct microscopic incarnations of non-Abelian defects, which can be created and transported by dynamically controlling the atom positions in the array. Our work suggests that appropriately tuned Rydberg arrays provide a cold-atoms counterpart of solid-state 'Kitaev materials' and, more generally, spotlights a new angle for pursuing experimental platforms for Abelian and non-Abelian fractionalization.

Giant Viscoelasticity near Mott Criticality in PbCrO_{3} with Large Lattice Anomalies.

Coupling of charge and lattice degrees of freedom in materials can produce intriguing electronic phenomena, such as conventional superconductivity where the electrons are mediated by lattice for creating supercurrent. The Mott transition, which is a source for many fascinating emergent behaviors, is originally thought to be driven solely by correlated electrons with an Ising criticality. Recent studies on the known Mott systems have shown that the lattice degree of freedom is also at play, giving rise to either Landau or unconventional criticality. However, the underlying coupling mechanism of charge and lattice degrees of freedom around the Mott critical end point remains elusive, leading to difficulties in understanding the associated Mott physics. Here, we report a study of Mott transition in cubic PbCrO_{3} by measuring the lattice parameter, using high-pressure x-ray diffraction techniques. The Mott criticality in this material is revealed with large lattice anomalies, which is governed by giant viscoelasticity that presumably results from a combination of lattice elasticity and electron viscosity. Because of the viscoelastic effect, the lattice of this material behaves peculiarly near the critical end point, inconsistent with any existing university classes. We argue that the viscoelasticity may play as a hidden degree of freedom behind the Mott criticality.

Study of the Transport Properties of the Critical Binary Mixture Triethylamine − Water with an Ionic Impurity

The binary liquid mixture of triethylamine + water (TEA–W) has a lower consolute point at a critical composition of 32.27 mass. % triethylamine. The shear viscosity (η) and the electrical conductivity (σ) in the single phase region of this system with added (K+, Cl–) ions at various concentrations are measured in the vicinity and far from the critical temperature TC. For the pure system without KCl salt, the viscosity measurements yield an enhancement, as expected, for the Ising criticality with a crossover to a regular behavior. Shear viscosity data are consistent with a power-law divergence η = η0(Qζ0)zt–y predicted by the mode-coupling and dynamic renormalization group theories. In the temperature range ∆T = TC – T &lt; 2 ºC, the electrical conductivity (σ) exhibits a monotonous deviation from the Vogel–Fulcher–Tammann (VFT) behavior. This anomaly is described by a power law t1 –α, where t is the reduced temperature (T –TC)/TC, and α is the critical exponent of the specific heat anomaly at constant pressure. For the electrolyte mixtures, the obtained critical exponent values are in the range of those expected by the theoretical calculations for the Ising 3D universality class. By combining the viscosity and the electrical conductivity data, the value of the computed Walden product has been determined, and the salt dissociation degrees, as well as the Debye screening length, have been estimated.

Microscopic characterization of Ising conformal field theory in Rydberg chains

Rydberg chains provide an appealing platform for probing conformal field theories (CFTs) that capture universal behavior in a myriad of physical settings. Focusing on a Rydberg chain at the Ising transition separating charge density wave and disordered phases, we establish a detailed link between microscopics and low-energy physics emerging at criticality. We first construct lattice incarnations of primary fields in the underlying Ising CFT including chiral fermions -- a nontrivial task given that the Rydberg chain Hamiltonian does not admit an exact fermionization. With this dictionary in hand, we compute correlations of microscopic Rydberg operators, paying special attention to finite, open chains of immediate experimental relevance. We further develop a method to quantify how second-neighbor Rydberg interactions tune the sign and strength of four-fermion couplings in the Ising CFT. Finally, we determine how the Ising fields evolve when four-fermion couplings drive an instability to Ising tricriticality. Our results pave the way to a thorough experimental characterization of Ising criticality in Rydberg arrays, and can inform the design of novel higher-dimensional phases based on coupled critical chains.

Liquid–Liquid Phase Behavior of Diesel/Biofuel Blends: Ternary Mixtures of n-Hexadecane, 2,2,4,4,6,8,8-Heptamethylnonane, and Ethanol and the Binary Subsystems

Mixtures of n-hexadecane and 2,2,4,4,6,8,8-heptamethylnonane are common diesel fuel surrogates. This paper describes a systematic investigation of the liquid–liquid phase behavior of the ternary mixtures of the normal alkane n-hexadecane and its isomer 2,2,4,4,6,8,8-heptamethylnonane (both CnH2n+2; n = 16) with ethanol as a model system of surrogate fuels. By applying the cloud point method in the temperature range of T = 240–330 K at ambient pressure (p = 1013 mbar), the complete liquid–liquid phase diagram is obtained. The binary mixtures of the alcohol with both isomeric alkanes show partial miscibility with upper critical solution points. The behavior of the ternary mixtures is examined by investigating the quasi-binary subsystems of mixtures of the alcohol with mixtures of the two isomers for various fixed mole ratios. By changing the ratio of the different isomers, the region of the phase diagram of the alcohol mixture with the n-alkane is gradually shifted toward lower temperatures when adding the branched isomer toward the phase diagram of the binary mixture of the alcohol with the isomeric compound. The general shape of the phase bodies of the pseudobinaries, predefined by the ratio of the isomeric components, shows only a slight change. For the numerical analysis of the measured phase behavior of the pseudobinary mixtures, concepts are used that are applied for binary solutions presuming Ising criticality. A master curve is obtained when applying the corresponding state principle on the binary and the pseudobinary mixtures. Superposition of the different results allows for a comprehensive description of the liquid–liquid phase behavior, and it estimates the loci of critical line and tie lines. The limits of the range of validity of the descriptions are determined by the appearance of solid phases, which are also reported here. This study is expected to establish a basis for optimizing blends of petrochemical fuels and fuels from renewable sources.

Emergence of a finite-size-scaling function in the supervised learning of the Ising phase transition

We investigate the connection between the supervised learning of the binary phase classification in the ferromagnetic Ising model and the standard finite-size-scaling theory of the second-order phase transition. Proposing a minimal one-free-parameter neural network model, we analytically formulate the supervised learning problem for the canonical ensemble being used as a training data set. We show that just one free parameter is capable enough to describe the data-driven emergence of the universal finite-size-scaling function in the network output that is observed in a large neural network, theoretically validating its critical point prediction for unseen test data from different underlying lattices yet in the same universality class of the Ising criticality. We also numerically demonstrate the interpretation with the proposed one-parameter model by providing an example of finding a critical point with the learning of the Landau mean-field free energy being applied to the real data set from the uncorrelated random scale-free graph with a large degree exponent.

Glide symmetry breaking and Ising criticality in the quasi-1D magnet CoNb2O6

We construct a microscopic spin-exchange Hamiltonian for the quasi-one-dimensional (1D) Ising magnet [Formula: see text] that captures detailed and hitherto-unexplained aspects of its dynamic spin structure factor. We perform a symmetry analysis that recalls that an individual Ising chain in this material is buckled, with two sites in each unit cell related by a glide symmetry. Combining this with numerical simulations benchmarked against neutron scattering experiments, we argue that the single-chain Hamiltonian contains a staggered spin-exchange term. We further argue that the transverse-field-tuned quantum critical point in [Formula: see text] corresponds to breaking this glide symmetry, rather than an on-site Ising symmetry as previously believed. This gives a unified microscopic explanation of the dispersion of confined states in the ordered phase and quasiparticle breakdown in the polarized phase at high transverse field.

Role of many-body correlation in slow dynamics of glass-forming liquids: intrinsic or perturbative

Upon cooling or densification, a supercooled liquid shows drastic slowing down toward its glass-transition point. The physical mechanism behind this slow glassy dynamics has been a matter of discussion for a long time, but there has still been no consensus on its origin. Recently, we have found that for systems mainly interacting with steric repulsions, glassy structural order (or, angular order) generally develops upon cooling and its correlation length, , grows as ( is the bare correlation length, T is the temperature, T0 is the hypothetical ideal glass transition, and d is the spatial dimensionality). This ordering is difficult to detect by two-body density correlation since it is a consequence of sterically-induced (entropically-driven) many-body correlation that lowers local free energy. Interestingly, the power-law growth of with the exponent of 2/d is reminiscent of the Ising criticality. We also find that the structural relaxation time diverges as (: the microscopic relaxation time, K is a fragility index, is the Boltzmann constant), suggesting that glass transition is a consequence of Ising-like criticality with growing activation energy. Unlike ordinary critical phenomena, the activation energy of particle motion increases in proportion to the root of the correlation volume of , implying that the particle motion is strongly correlated in that volume. This relation indicates that the impact of spatial fluctuations of the order parameter on slow dynamics is not perturbative but intrinsic. Although we need further study to confirm our claim, we hope that the discussion in this article would provide a good starting point for further consideration of the physical nature of glass transition.

Revisiting the Liquid-Liquid Phase Behavior of n-Alkanes and Ethanol.

Mixtures of alkanes and ethanol are important in many areas, for example, as fuel blends. This paper describes new experimental data obtained for the liquid-liquid equilibrium phase behavior of normal alkanes (n-alkanes; CnH2n+2; 9 ≤ n ≤ 24) with ethanol. The results were obtained by applying the cloud point method in a temperature range of T = 230-423 K at ambient pressure. All systems are partially miscible with an upper critical solution point. The two phase regions of the phase diagrams show no indication of any obvious optical irregularities, like birefringence, coloring, formation of schlieren, or remarkable turbidity, except critical opalescence. With increasing length of the molecular chain of the n-alkanes, the (liquid-liquid) critical point is shifted to higher temperatures and higher ethanol content. The data are analyzed numerically implying Ising criticality. The nonsymmetric shape of the phase body is considered in different approaches for describing the diameter by presuming (a) the validity of the rectilinear diameter rule, (b) a nonlinear diameter predicted in the theory of complete scaling, and (c) combining both concepts. The numerical analysis yields the critical temperature, the critical composition, the width, and the diameter of the phase diagrams. The results are compared with literature data sets from similar mixtures; these data are also evaluated in terms of the models applied here. Phase diagrams of 13 different sets of mixtures are measured and analyzed to extract general aspects of the behavior of the normal alkane-ethanol mixtures. A simple Flory-Huggins-like approach allows a semiquantitative description of the experimental results of the critical temperatures. Therefore, it confirms the picture of molecular ordering within the solutions.

Critical Scaling Behaviors of Entanglement Spectra**Supported by the start-up funding from Westlake University, and the National Natural Science Foundation of China under Grant No 11974288.

We investigate the evolution of entanglement spectra under a global quantum quench from a short-range correlated state to the quantum critical point. Motivated by the conformal mapping, we find that the dynamical entanglement spectra demonstrate distinct finite-size scaling behaviors from the static case. As a prototypical example, we compute real-time dynamics of the entanglement spectra of a one-dimensional transverse-field Ising chain. Numerical simulation confirms that the entanglement spectra scale with the subsystem size l as ∼l−1 for the dynamical equilibrium state, much faster than ∝ ln−1 l for the critical ground state. In particular, as a byproduct, the entanglement spectra at the long time limit faithfully gives universal tower structure of underlying Ising criticality, which shows the emergence of operator-state correspondence in the quantum dynamics.

Effect of Isomerism on the Liquid–Liquid Phase Behavior of Mixtures of 1-Alkyl-3-methylimidazolium Bis((trifluoromethyl)sulfonyl)amide Ionic Liquids with Heptanol

Systematic investigations on the liquid–liquid phase behavior of binary mixtures of ionic liquids (IL) of the type 1-alkyl-3-methylimidazolium bis((trifluoromethyl)sulfonyl)amide (CxmimNTf2, x = 2, 4, 6, 8, and 10) with n-heptanol and its branched isomers 3-ethyl-3-pentanol and 2,4-dimethyl-3-pentanol have been performed. The phase diagrams were determined at atmospheric pressure in the temperature range (T) of 263–448 K by applying the cloud point method. The phase diagrams are analyzed in terms of well-established numerical concepts based on Ising criticality. The results are discussed, and trends and explanations for the shifts of the range of solubility are given. The systems show partial miscibility with upper critical solution temperatures. Comparing the solutions in branched alcohols with those in the linear isomer, the critical solution points are found to be remarkably shifted toward higher temperatures and lower IL contents. The trend becomes more pronounced for ILs with shorter alkyl side chains where also a change of the shape of the phase diagrams is observed.

Ground-state phases and quantum criticality of a one-dimensional Peierls model with spin-dependent sign-alternating potentials

We consider a one-dimensional commensurate Peierls insulator in the presence of spin-dependent sign-alternating potentials. In a continuum description, the latter supply the fermions with spin-dependent "relativistic" masses $m_{\uparrow,\downarrow}$. The ground-state phase diagram describes three gapped phases: the CDW and SDW-like band insulator phases sandwiched by a "mixed" phase in which the CDW and SDW superstructures coexist with a nonzero spontaneous dimerization (SD). The critical lines separating the massive phases belong to the Ising universality class. The Ising criticality is accompanied by the Kohn anomaly in the renormalized phonon spectrum. We derive a Ginzburg criterion which specifies a narrow region around the critical point where quantum fluctuations play a dominant role, rendering the adiabatic (or mean-field) approximation inapplicable. A full account of quantum effects is achieved in the anti-adiabatic limit where the effective low-energy theory represents a massive version the N=4 Gross-Neveu model. Using Abelian bosonization we demonstrate that the description of the SD phase, including its critical boundaries, is well approximated by a sum of two effective double-frequency sine-Gordon (DSG) models subject to self-consistency conditions that couple the charge and spin sectors. Using the well-known critical properties of the DSG model we obtain the singular parts of the dimerization order parameter and staggered charge and spin susceptibilities near the Ising critical lines. We show that, in the anti-adiabatic limit, on the line $m_{\downarrow} = 0$ there exists of a Berezinskii-Kosterlitz-Thouless critical point separating a Luttinger-liquid gapless phase from the spontaneously dimerized one. We also discuss topological excitations of the model carrying fractional charge and spin. \pacs{71.10.Pm; 71.27.+a; 71.45.Lr; 75.40.Kb}

Critical phase behavior in multi-component fluid mixtures: Complete scaling analysis.

We analyze the critical gas-liquid phase behavior of arbitrary fluid mixtures in their coexistence region. We focus on the setting relevant for polydisperse colloids, where the overall density and composition of the system are being controlled, in addition to temperature. Our analysis uses the complete scaling formalism and thus includes pressure mixing effects in the mapping from thermodynamic fields to the effective fields of 3D Ising criticality. Because of fractionation, where mixture components are distributed unevenly across coexisting phases, the critical behavior is remarkably rich. We give scaling laws for a number of important loci in the phase diagram. These include the cloud and shadow curves, which characterise the onset of phase coexistence, a more general set of curves defined by fixing the fractional volumes of the coexisting phases to arbitrary values, and conventional coexistence curves of the densities of coexisting phases for fixed overall density. We identify suitable observables (distinct from the Yang-Yang anomalies discussed in the literature) for detecting pressure mixing effects. Our analytical predictions are checked against numerics using a set of mapping parameters fitted to simulation data for a polydisperse Lennard-Jones fluid, allowing us to highlight crossovers where pressure mixing becomes relevant close to the critical point.

Spin-liquid behaviour and the interplay between Pokrovsky-Talapov and Ising criticality in the distorted, triangular-lattice, dipolar Ising antiferromagnet

We study the triangular-lattice Ising model with dipolar interactions, inspired by its realisation in artificial arrays of nanomagnets. We show that a classical spin-liquid forms at intermediate temperatures, and that its behaviour can be tuned by temperature and/or a small lattice distortion between a string Luttinger liquid and a domain-wall-network state. At low temperature there is a transition into a magnetically ordered phase, which can be first-order or continous with a crossover in the critical behaviour between Pokrovsky-Talapov and 2D-Ising universality. When the Pokrovsky-Talapov criticality dominates, the transition is essentially of the Kasteleyn type.