Ising Exponents Research Articles

Neural-network quantum-state study of the long-range antiferromagnetic Ising chain.

We investigate quantum phase transitions in the transverse field Ising chain with algebraically decaying long-range (LR) antiferromagnetic interactions using the variational Monte Carlo method with the restricted Boltzmann machine employed as a trial wave function ansatz. First, we measure the critical exponents and the central charge through the finite-size scaling analysis, verifying the contrasting observations in the previous tensor network studies. The correlation function exponent and the central charge deviate from the short-range (SR) Ising values at a small decay exponent α_{LR}, while the other critical exponents examined are very close to the SR Ising exponents regardless of α_{LR} examined. However, in the further test of the critical Binder ratio, we find that the universal ratio of the SR limit does not hold for α_{LR}<2, implying a deviation in the criticality. On the other hand, we find evidence of the conformal invariance breakdown in the conformal field theory (CFT) test of the correlation function. The deviation from the CFT description becomes more pronounced as α_{LR} decreases, although a precise breakdown threshold is yet to be determined.

Phase diagram in cyclopentanone + water + secondary butyl alcohol and light scattering investigations near the LCST

We report light scattering investigations and study of phase diagram in the system cyclopentanone (CP) + water (W) + secondary butyl alcohol (sBA), a more promising system for exploring the Quadruple Critical Point (QCP) as compared to methyl ethyl ketone (MEK) + W + sBA. Unlike in MEK + W + sBA [A. P. Naseer and P. K. M. Unni, J. Mol. Liq., 296, 111,776 (2019)], the diameter of the tunnel D≡TU-TL decreases from 66.626 K at XCP≡Xketone = 0.1000 to 24.926 K at Xketone = 0.2500 without addition of a fourth component. Our studies indicate that addition of only 0.0018 weight fraction of KSCN (i.e., 5% of the concentration of KSCN required in case of the system MEK + W + sBA), is sufficient to split the tunnel-like portion into two paraboloids with an intervening one-phase region in between. The tunnel-like feature in the phase diagram of CP + W + sBA occurs for 0.1000≤XCP≤0.2000 and for XCP>0.2000, the tunnel is open towards higher water weight fractions, though the critical line of UCSTs and LCSTs is continuous. Light scattering investigations show Fisher renormalization of the susceptibility exponent γ for 0.1750≤Xketone≡XCP≤0.2500 with a near complete renormalization observed for Xketone≡XCP = 0.2500 and normal Ising exponents observed for the region 0.1750>Xketone≡XCP>0.2500. Similarly, Ising-like behavior is observed at both low and high XCP weight-fractions while a tendency for mean-field behavior is observed for Xketone = 0.2500. Both these features appear generic to ternary systems involving (ketone + water + secondary butyl alcohol). The correlation length amplitude of critical concentration fluctuation ξ0 starts from a value of 6.5Å for XCP = 0.15 and saturates at 8.5Å in the range 0.175≤XCP≤0.350.

Long-Range Influence of a Pump on a Critical Fluid.

A pump coupled to a conserved density generates long-range modulations, resulting from the non-equilibrium nature of the dynamics. We study how these modulations are modified at the critical point where the system exhibits intrinsic long-range correlations. To do so, we consider a pump in a diffusive fluid, which is known to generate a density profile in the form of an electric dipole potential and a current in the form of a dipolar field above the critical point. We demonstrate that while the current retains its form at the critical point, the density profile changes drastically. At criticality, in d<4 dimensions, the deviation of the density from the average is given by sgn[cos(θ)]|cos(θ)/r^{(d-1)}|^{1/δ} at large distance r from the pump and angle θ with respect to the pump's orientation. At short distances, there is a crossover to a cos(θ)/r^{d-3+η} profile. Here δ and η are Ising critical exponents. The effect of the local pump on the domain wall structure below the critical point is also considered.

Husimi-lattice solutions and the coherent-anomaly-method analysis for hard-square lattice gases.

Although lattice gases composed of particles preventing up to their kth nearest neighbors from being occupied (the kNN models) have been widely investigated in the literature, the location and the universality class of the fluid-columnar transition in the 2NN model on the square lattice are still a topic of debate. Here, we present grand-canonical solutions of this model on Husimi lattices built with diagonal square lattices, with 2L(L+1) sites, for L⩽7. The systematic sequence of mean-field solutions confirms the existence of a continuous transition in this system, and extrapolations of the critical chemical potential μ_{2,c}(L) and particle density ρ_{2,c}(L) to L→∞ yield estimates of these quantities in close agreement with previous results for the 2NN model on the square lattice. To confirm the reliability of this approach, we employ it also for the 1NN model, where very accurate estimates for the critical parameters μ_{1,c} and ρ_{1,c}-for the fluid-solid transition in this model on the square lattice-are found from extrapolations of data for L⩽6. The nonclassical critical exponents for these transitions are investigated through the coherent anomaly method (CAM), which in the 1NN case yields β and ν differing by at most 6% from the expected Ising exponents. For the 2NN model, the CAM analysis is somewhat inconclusive, because the exponents sensibly depend on the value of μ_{2,c} used to calculate them. Notwithstanding, our results suggest that β and ν are considerably larger than the Ashkin-Teller exponents reported in numerical studies of the 2NN system.

Semiconductor of spinons: from Ising band insulator to orthogonal band insulator

We use the ionic Hubbard model to study the effects of strong correlations on a two-dimensional semiconductor. The spectral gap in the limit where on-site interactions are zero is set by the staggered ionic potential, while in the strong interaction limit it is set by the Hubbard U. Combining mean field solutions of the slave spin and slave rotor methods, we propose two interesting gapped phases in between: (i) the insulating phase before the Mott phase can be viewed as gapping a non-Fermi liquid state of spinons by the staggered ionic potential. The quasi-particles of underlying spinons are orthogonal to physical electrons, giving rise to the ‘ARPES-dark’ state where the ARPES gap will be larger than the optical and thermal gap. (ii) The Ising insulator corresponding to ordered phase of the Ising variable is characterized by single-particle excitations whose dispersion is controlled by Ising-like temperature and field dependences. The temperature can be conveniently employed to drive a phase transition between these two insulating phases where Ising exponents become measurable by ARPES and cyclotron resonance. The rare earth monochalcogenide semiconductors where the magneto-resistance is anomalously large can be a candidate system for the Ising band insulator. We argue that the Ising and orthogonal insulating phases require strong enough ionic potential to survive the downward renormalization of the ionic potential caused by Hubbard U.

Critical behavior of the extended Hubbard model with bond dimerization

Exploiting the matrix-product-state based density-matrix renormalization group (DMRG) technique we study the one-dimensional extended (U-V) Hubbard model with explicit bond dimerization in the half-filled band sector. In particular we investigate the nature of the quantum phase transition, taking place with growing ratio V/U between the symmetry-protected-topological and charge-density-wave insulating states. The (weak-coupling) critical line of continuous Ising transitions with central charge c=1/2 terminates at a tricritical point belonging to the universality class of the dilute Ising model with c=7/10. We demonstrate that our DMRG data perfectly match with (tricritical) Ising exponents, e.g., for the order parameter β=1/8 (1/24) and correlation length ν=1 (5/9). Beyond the tricritical Ising point, in the strong-coupling regime, the quantum phase transition becomes first order.

Transfer-matrix study of a hard-square lattice gas with two kinds of particles and density anomaly

Using transfer matrix and finite-size scaling methods, we study the thermodynamic behavior of a lattice gas with two kinds of particles on the square lattice. Only excluded volume interactions are considered, so that the model is athermal. Large particles exclude the site they occupy and its four first neighbors, while small particles exclude only their site. Two thermodynamic phases are found: a disordered phase where large particles occupy both sublattices with the same probability and an ordered phase where one of the two sublattices is preferentially occupied by them. The transition between these phases is continuous at small concentrations of the small particles and discontinuous at larger concentrations, both transitions are separated by a tricritical point. Estimates of the central charge suggest that the critical line is in the Ising universality class, while the tricritical point has tricritical Ising (Blume-Emery-Griffiths) exponents. The isobaric curves of the total density as functions of the fugacity of small or large particles display a minimum in the disordered phase.

Erratum: Coexistence curves at liquid-liquid critical points: Ising exponents and extended scaling [Phys. Rev. A14, 1770 (1976)

Erratum: Coexistence curves at liquid-liquid critical points: Ising exponents and extended scaling [Phys. Rev. A<b>14</b>, 1770 (1976)

Compressible metamagnetic model: Renormalization-group approach

A layered compressible metamagnetic Ising model is studied within the renormalization-group approach. The ferro- and antiferromagnetic couplings depend linearly on volume, so shear forces are infinite, as in Domb's compressible ferromagnetic model [J. Chem. Phys. 25, 783 (1956)]. The Hamiltonian is expanded around the mean-field solution and the resulting functional contains both the biquadratic term, characteristic of compressibility, and the rigid metamagnetic characteristic cubic terms. However, a different contribution, related to the anisotropy of the spin-lattice coupling, arises. Mean-field behavior is reproduced only partially. However, differently from the infinite-shear compressible ferromagnetic case, some of the fixed points can be accessed within the physical domain of the variables, for some range of the interaction parameters. In this case, the continuous transition line at intermediate magnetic fields presents Fisher-renormalized critical exponents [Phys. Rev. 176, 257 (1968)]. The two tricritical points present Ising exponents (at low field) and Fisher-renormalized Gaussian exponents.

Effects of magnetoelastic coupling: critical behavior and structure deformation

We study a 3D crystal where each atom interacts with neighbors via elastic and magnetic interactions by Monte Carlo simulation. The distance dependence of both interactions is supposed to be the Lennard–Jones potential and the spins are of the Ising type. When the magnetic interaction strength is much smaller than the elastic one, the magnetic transition remains in the 3D Ising criticality. With larger magnetic interaction, the critical exponents get very close to those of the 3D XY universality and not far from Fisher renormalized Ising exponents. For strong magnetic interaction, we show that as the temperature increases the crystal is broken into ferromagnetic domains separated by domain walls consisting of contracted antiferromagnetic spin pairs.

Anisotropic ordering in a two-temperature lattice gas

We consider a two-dimensional lattice gas model with repulsive nearest- and next-nearest-neighbor interactions that evolves in time according to anisotropic Kawasaki dynamics. The hopping of particles along the principal directions is governed by two heat baths at different temperatures ${\mathrm{T}}_{\mathrm{x}}$ and ${\mathrm{T}}_{\mathrm{y}}$. The stationary states of this nonequilibrium model are studied using a simple mean-field theory and linear stability analysis. The results are improved by a generalized dynamical mean-field approximation. In the stable ordered state the particles form parallel chains which are oriented along the direction of the higher temperature. In the resulting phase diagram in the ${\mathrm{T}}_{\mathrm{x}}$-${\mathrm{T}}_{\mathrm{y}}$ plane the critical temperature curve shows a weak maximum as a function of the parallel temperature which is confirmed by Monte Carlo simulations. Finite-size scaling analysis suggests that the model leaves the equilibrium universality class of the x-y model with cubic anisotropy and is described by the Ising exponents.

Specific-heat critical behavior of the structural random-field system Dy(AsxV1-x)O4.

The specific-heat critical behaviors of the pure Ising transition in ${\mathrm{DyVO}}_{4}$ and the random-field Ising transition in Dy(${\mathrm{As}}_{\mathit{x}}$${\mathrm{V}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$)${\mathrm{O}}_{4}$ have been measured. The critical exponent \ensuremath{\alpha} and amplitude ratio ${\mathit{A}}^{+}$/${\mathit{A}}^{\mathrm{\ensuremath{-}}}$ for ${\mathrm{DyVO}}_{4}$, which has ${\mathit{T}}_{\mathit{c}}$=14.82 K, are in excellent agreement with the known pure d=3 Ising exponents. The random fields generated by the local strain fields in Dy(${\mathrm{As}}_{\mathit{x}}$${\mathrm{V}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$)${\mathrm{O}}_{4}$ depress the transitions to ${\mathit{T}}_{\mathit{c}}$=13.45 K and ${\mathit{T}}_{\mathit{c}}$=7.7 K for x=0.05 and x=0.15, respectively. With increasing x, the transition becomes rounded and severely depressed in size. The value of \ensuremath{\alpha} appears large and negative for x=0.15. As the random-field strength increases, i.e., the value of x gets larger, the amount of entropy change associated with the transition rapidly decreases. Where the missing entropy reappears is not adequately understood. \textcopyright{} 1996 The American Physical Society.

Critical behavior of a cationic-surfactant-water-salt system near and far from the Krafft temperature.

The cationic micellar system docecylammonium chloride +water+KCl has been studied by light scattering near the critical point for different concentrations of salt. For the lowest value of salt concentration, the critical point is rather close to the Krafft line, and therefore there is a critical mixture in equilibrium with two noncritical phases: vapor and pure solid surfactant. Both the correlation length and osmotic susceptibility data for the critical mixtures can be accurately described by the usual three-dimensional Ising exponents for all values of the concentration of KCl (${\mathit{c}}_{\mathrm{KCl}}$). This is in agreement with theoretical predictions that indicate that approaching a critical end point does not affect the critical exponents. Two off-critical mixtures have also been studied, and their behavior has been found to be compatible with the linear model equation of state.

Critical behavior of a cationic surfactant in an aqueous salt solution

The critical behavior of a cationic surfactant, butyl-tetradecyldimethylammonium bromide, in an aqueous 4M–NaBr solution was investigated concerning the isothermal osmotic compressibility and long-range correlation length by means of light scattering, coexistence curve, and turbidity. The critical exponents were obtained as γ′=1.39±0.04 and ν′=0.70±0.03 together with ξ0=(2.48±0.06) nm, and β′=0.375±0.01. The temperature dependence of turbidity showed a good accordance with the result of light scattering measurement. Negative α′ was obtained from the scaling relation. The critical exponents are in good agreement with the Fisher-renormalization prediction for the ternary system (almost fully renormalized Ising exponents).

Critical exponents and phase diagram of polymer blend solutions: polystyrene/ poly(methyl methacrylate)/d6-benzene system

Critical behaviour of blend solutions of polystyrene ( M w = 3.55 × 10 5)/poly(methyl methacrylate) ( M w = 3.27 × 10 5) in d 6-benzene was investigated by static light scattering. The phase diagram (spinodal, binodal and cloud points) was determined at a fixed total polymer concentration. The system had the lower critical solution temperature. The inverse isothermal osmotic compressibility ( ∂π ∂c 1 T ) and correlation length ζ of concentration fluctuations were determined as functions of the reduced temperature ϵ= | T s T−1 | near the stability limit, where T is the absolute temperature and T s is the spinodal temperature. Both ( ∂π ∂c 1 T ) and ξ were described by single exponents over the investigated temperature range. The critical exponents γ and ν for ( ∂π ∂c 1 T ) and ζ were 1.23 and 0.63, respectively. They were very close to the three-dimensional Ising exponents and obviously different from the mean-field or Fisher's renormalized Ising exponents, as theoretically predicted by Broseta et al.

Ising droplets in five dimensions

Monte Carlo simulations indicate that the size distribution of Coniglio-Klien droplets in the five-dimensional nearest-neighbor Ising model corresponds to mean field Ising exponents.

The susceptibility critical exponent for a nonaqueous ionic binary mixture near a consolute point

We report turbidity measurements of a nonaqueous ionic solution of triethyl n-hexylammonium triethyl n-hexylboride in diphenyl ether. A classical susceptibility critical exponent γ=1.01±0.01 is obtained over the reduced temperature range 10−4≤ t≤10−1. The best fits of the sample transmission had a standard deviation of 0.39% over this range. Ising and spherical model critical exponents are firmly excluded. The correlation length amplitude ξ0 from fitting is 1.0±0.2 nm which is much larger than values found in neutral fluids and some aqueous binary mixtures.

MONTE CARLO STUDIES OF A NEW MODEL DRIVEN DIFFUSIVE SYSTEM WITH REPULSIVE INTERACTIONS

We propose a new class of driven lattice gas with repulsive nearest-neighbor interactions. Particles are allowed to jump to empty next-nearest-neighbor (nnn) sites in addition to the standard nearest-neighbor moves. In contrast to previous model with repulsive interactions, the external driving field (E) acts only along the nnn directions and does not destroy ground state sublattice ordering. Extensive Monte Carlo simulations in two dimensions for small E are consistent with a line of continuous transitions with Ising exponents. First-order transitions are also found for larger E.

Description of critical dynamics by static geometry of clusters

Critical slowing-down of relaxation (exponent z ) is modelled by clusters, which grow and recede at the end of connected sequences of particles. Consequently the relaxation time of a cluster of size s is proportional to the number of back and forth steps needed to travel its typical connected length l s . This and an assumption on mutual screening of ends, lead to z =[(2.5 ϱ − 0.25)( γ + β )− β ] ν , where ϱ is a static geometric exponent in l s ∼ s ϱ . Ising exponents ϱ measured (elsewhere), with the help of simulation, give z ϱ,β,ν . In addition, a Flory-like approximation is derived here; it relates ϱ to β and ν analytically, giving z βν . In D = 2, 3 and 4 − ɛ , respectively, we findz β,ν = 2.14, 2.05 and 2 + ɛ 2 75.1, while z ϱ,β,ν = 2.17(4) and 2.04(4); both sets of values are in fair agreement with the literature.

Scaling of critical scattering in β-CuZn

We have investigated the temperature scaling of the critical scattering of the binary alloy Cu 0.5Zn 0.5 by neutron scattering using time-of-flight and triple axis spectrometers. The temperature behaviour of the short range order scattering is well described by power laws with Ising exponents for all T⩾ T c in accordance with recent theoretical predictions.