Critical Universality Classes Research Articles

Log-Normal Waiting Time Widths Characterize Dynamics

Many astronomical phenomena, including Fast Radio Bursts and Soft Gamma Repeaters, consist of brief distinct aperiodic events. The intervals between these events vary randomly, but there are periods of greater activity, with shorter mean intervals, and of lesser activity, with longer mean intervals. A single dimensionless parameter, the width of a log-normal function fitted to the distribution of waiting times between events, quantifies the variability of the activity. This parameter describes its dynamics in analogy to the critical exponents and universality classes of renormalization group theory. If the distribution of event strengths is a power law, the width of the log-normal fit is independent of the detection threshold and is a robust measure of the dynamics of the phenomenon.

Universality of stress-anisotropic and stress-isotropic jamming of frictionless spheres in three dimensions: Uniaxial versus isotropic compression.

We numerically study a three-dimensional system of athermal, overdamped, frictionless spheres, using a simplified model for a non-Brownian suspension. We compute the bulk viscosity under both uniaxial and isotropic compression as a means to address the question of whether stress-anisotropic and stress-isotropic jamming are in the same critical universality class. Carrying out a critical scaling analysis of the system pressure p, shear stress σ, and macroscopic friction μ=σ/p, as functions of particle packing fraction ϕ and compression rate ε[over ̇], we find good agreement for all critical parameters comparing the isotropic and anisotropic cases. In particular, we determine that the bulk viscosity diverges as p/ε[over ̇]∼(ϕ_{J}-ϕ)^{-β}, with β=3.36±0.09, as jamming is approached from below. We further demonstrate that the average contact number per particle Z can also be written in a scaling form as a function of ϕ and ε[over ̇]. Once again, we find good agreement between the uniaxial and isotropic cases. We compare our results to prior simulations and theoretical predictions.

Disorder in dissipation-induced topological states: Evidence for a different type of localization transition

The quest for nonequilibrium quantum phase transitions is often hampered by the tendency of driving and dissipation to give rise to an effective temperature, resulting in classical behavior. Could this be different when the dissipation is engineered to drive the system into a nontrivial quantum coherent steady state? In this work we shed light on this issue by studying the effect of disorder on recently-introduced dissipation-induced Chern topological states, and examining the eigenmodes of the Hermitian steady state density matrix or entanglement Hamiltonian. We find that, similarly to equilibrium, each Landau band has a single delocalized level near its center. However, using three different finite size scaling methods we show that the critical exponent $\nu$ describing the divergence of the localization length upon approaching the delocalized state is significantly different from equilibrium if disorder is introduced into the non-dissipative part of the dynamics. This indicates a different type of nonequilibrium quantum critical universality class accessible in cold-atom experiments.

Magnetocaloric effect near room temperature and critical behaviour of Fe doped MnCo0.7Fe0.3Ge

We report on the critical magnetic behaviour, universality class and magnetocaloric properties of Mn0.7Fe0.3Co0.7Fe0.3Ge. At room temperature, the alloy is found to crystallize in the hexagonal structure of Ni2In-type. Paramagnetic to ferromagnetic phase transition is found to occur at a Curie temperature of about 298 K. The saturation magnetization value at low temperatures is about 2.79 μB/f.u. Kouvel-Fisher and modified Arrott's Plot methods along with magnetocaloric method are implemented to determine the critical exponents which are found to be self-consistent. The critical exponents suggest short range magnetic interactions with 3D-Heisenberg universality class of the alloy as inferred from the relationJ(r)~1/r5.02. Around the Curie temperature, the maximum change in the magnetic entropy, under the influence of 90 kOe, is about −3.91 J kg−1 K−1.

Rapid Kerr imaging characterization of the magnetic properties of two-dimensional ferromagnetic Fe3GeTe2

Van der Waals (vdW) ferromagnetic materials have attracted considerable attention in the nanomaterial community, which could provide a unique platform to study magnetism at the nanoscale. Along this direction, many interesting results have been reported, including the electric field control of magnetism and topological spin textures. In this report, we present a rapid and spatially resolved imaging method to study the dimensionality-dependent magnetic properties of Fe3GeTe2 (FGT) nanoflakes. Our method is named as polar magneto-optical Kerr imaging microscopy magnetometry (p-MIMM), which is made possible by analyzing the intensity evolution of wide-field polar magneto-optical Kerr effect (MOKE) images that were collected by varying magnetic fields, thicknesses, and temperatures. In particular, spatially resolved MOKE hysteresis loops can be acquired in the FGT nanoflakes with a submicrometer resolution. By analyzing the evolution of the relative (saturated) MOKE intensity as a function of temperature, we further study the critical exponent and universality class and its dependence on the FGT nanoflake thickness. Combining the polar MOKE images with the calculated MOKE hysteresis loops, a detailed magnetic phase diagram summarizing an evolution of the stripe domain, single domain, and paramagnetic state is further validated. Our results suggest that the wide-field p-MIMM can be conveniently used for rapidly examining the magnetic properties of versatile vdW magnetic materials.

Dynamical scaling analysis on critical universality class for fully frustrated XY models in two dimensions

Critical properties of the fully frustrated XY models on square and triangular lattices are investigated by means of the nonequilibrium relaxation (NER) method. We examine the validity of the conclusion on the universality class of the chiral transition previously obtained by the NER method, in which that belongs to a non-Ising-type one. To clarify it, we analyze the NER of fluctuations for longer Monte Carlo steps on lager lattices than those performed previously. The calculation is made at the chiral transition temperatures estimated carefully by the use of recently improved dynamical scaling analysis. The result indicates that the asymptotic behavior of the time-dependent exponents, which should converge to the critical exponents, show the same tendency as those obtained previously in shorter times. We also apply the improved dynamical scaling analysis to the estimation of Kosterlitz-Thouless (KT) transitions and confirm the existence of the double transitions for the chiral and KT phases with more reliable estimations.

Non-classical critical exponents at Bose–Einstein condensation

We show that ideal Bose–Einstein condensation (BEC) in d = 3 dimensions is a non-classical critical second order phase transition with exponents , , , , and , obeying all the scaling equalities. These results are found with no approximations or assumptions. The previous exponents are a critical universality class on its own, different from the so-far accepted notion that BEC belongs to the spherical model universality class.

Critical Behavior and Universality Classes for an Algorithmic Phase Transition in Sparse Reconstruction

Recovery of an N-dimensional, K-sparse solution $${\mathbf {x}}$$ from an M-dimensional vector of measurements $${\mathbf {y}}$$ for multivariate linear regression can be accomplished by minimizing a suitably penalized least-mean-square cost $$||{\mathbf {y}}-{\mathbf {H}} {\mathbf {x}}||_2^2+\lambda V({\mathbf {x}})$$ . Here $${\mathbf {H}}$$ is a known matrix and $$V({\mathbf {x}})$$ is an algorithm-dependent sparsity-inducing penalty. For ‘random’ $${\mathbf {H}}$$ , in the limit $$\lambda \rightarrow 0$$ and $$M,N,K\rightarrow \infty $$ , keeping $$\rho =K/N$$ and $$\alpha =M/N$$ fixed, exact recovery is possible for $$\alpha $$ past a critical value $$\alpha _c = \alpha (\rho )$$ . Assuming $${\mathbf {x}}$$ has iid entries, the critical curve exhibits some universality, in that its shape does not depend on the distribution of $${\mathbf {x}}$$ . However, the algorithmic phase transition occurring at $$\alpha =\alpha _c$$ and associated universality classes remain ill-understood from a statistical physics perspective, i.e. in terms of scaling exponents near the critical curve. In this article, we analyze the mean-field equations for two algorithms, Basis Pursuit ( $$V({\mathbf {x}})=||{\mathbf {x}}||_{1} $$ ) and Elastic Net ( $$V({\mathbf {x}})= ||{\mathbf {x}}||_{1} + \tfrac{g}{2} ||{\mathbf {x}}||_{2}^2$$ ) and show that they belong to different universality classes in the sense of scaling exponents, with mean squared error (MSE) of the recovered vector scaling as $$\lambda ^\frac{4}{3}$$ and $$\lambda $$ respectively, for small $$\lambda $$ on the critical line. In the presence of additive noise, we find that, when $$\alpha >\alpha _c$$ , MSE is minimized at a non-zero value for $$\lambda $$ , whereas at $$\alpha =\alpha _c$$ , MSE always increases with $$\lambda $$ .

Critical phases in the raise and peel model

The raise and peel model (RPM) is a nonlocal stochastic model describing the space and time fluctuations of an evolving one dimensional interface. Its relevant parameter u is the ratio between the rates of local adsorption and nonlocal desorption processes (avalanches) The model at u = 1 is the first example of a conformally invariant stochastic model. For small values u < u0 the model is known to be noncritical, while for u > u0 it is critical. Although previous studies indicate that u0 = 1, a determination of u0 with a reasonable precision is still missing. By calculating numerically the structure function of the height profiles in the reciprocal space we confirm with good precision that indeed u0 = 1. We establish that at the conformal invariant point u = 1 the RPM has a roughening transition with dynamical and roughness critical exponents z = 1 and , respectively. For u > 1 the model is critical with a u-dependent dynamical critical exponent that tends towards zero as . However at 1/u = 0 the RPM is exactly mapped into the totally asymmetric exclusion problem. This last model is known to be noncritical (critical) for open (periodic) boundary conditions. Our numerical studies indicate that the RPM as , due to its nonlocal dynamical processes, has the same large-distance physics no matter what boundary condition we chose. For u > 1, our numerical analysis shows that in contrast to previous predictions, the region is composed of two distinct critical phases. For the height profiles are rough (), and for the height profiles are flat at large distances (). We also observed that in both critical phases (u > 1) the RPM at short length scales, has an effective behavior in the Kardar–Parisi–Zhang critical universality class, that is not the true behavior of the system at large length scales.

Effects of diffusion in competitive contact processes on bipartite lattices

We investigate the influence of particle diffusion in the two-dimension contact process (CP) with a competitive dynamics in bipartite sublattices, proposed in de Oliveira and Dickman (2011 Phys. Rev. E 84 011125). The particle creation depends on its first and second neighbors and the extinction increases according to the local density. In contrast to the standard CP model, mean-field theory and numerical simulations predict three stable phases: inactive (absorbing), active symmetric and active asymmetric, signed by distinct sublattice particle occupations. Our results from MFT and Monte Carlo simulations reveal that low diffusion rates do not destroy sublattice ordering, ensuring the maintenance of the asymmetric phase. On the other hand, for diffusion larger than a threshold value Dc, the sublattice ordering is suppressed and only the usual active (symmetric)-inactive transition is presented. We also show the critical behavior and universality classes are not affected by the diffusion.

Critical Behaviors and Universality Classes of Percolation Phase Transitions on Two-Dimensional Square Lattice* *Supported by the National Natural Science Foundation of China under Grant No. 11121403

We have investigated both site and bond percolation on two-dimensional lattice under the random rule and the product rule respectively. With the random rule, sites or bonds are added randomly into the lattice. From two candidates picked randomly, the site or bond with the smaller size product of two connected clusters is added when the product rule is taken. Not only the size of the largest cluster but also its size jump are studied to characterize the universality class of percolation. The finite-size scaling forms of giant cluster size and size jump are proposed and used to determine the critical exponents of percolation from Monte Carlo data. It is found that the critical exponents of both size and size jump in random site percolation are equal to that in random bond percolation. With the random rule, site and bond percolation belong to the same universality class. We obtain the critical exponents of the site percolation under the product rule, which are different from that of both random percolation and the bond percolation under the product rule. The universality class of site percolation differs different from that of bond percolation when the product rule is used.

Asymptotics of a Fredholm Determinant Corresponding to the First Bulk Critical Universality Class in Random Matrix Models

We study the determinant det(I−KPII) of an integrable Fredholm operator KPII acting on the interval (−s, s) whose kernel is constructed out of the Ψ-function associated with the Hastings–McLeod solution of the second Painleve equation. This Fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random Hermitian matrix chosen from the unitary ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density. Using the Riemann–Hilbert method, we evaluate the large s-asymptotics of det(I−KPII) .

Universality class of quantum criticality in the two-dimensional Hubbard model at intermediate temperatures (t2/U≪T≪t)

We show that the dilute Fermi gas quantum critical universality class quantitatively describes the Mott/metal crossover of the two-dimensional Hubbard model for temperatures somewhat less than (roughly half) the tunneling but much greater than (roughly twice) the superexchange energy. We calculate the observables expected to be universal near the transition --- density and compressibility --- with numerically exact determinantal quantum Monte Carlo. We find they are universal functions of the chemical potential. Despite arising from the strongly correlated regime of the Hubbard model, these functions are given by the weakly interacting, dilute Fermi gas model. These observables and their derivatives are the only expected universal static observables of this universality class, which we also confirm by verifying there is no scaling collapse of the kinetic energy, fraction of doubly occupied sites, and nearest neighbor spin correlations. Our work resolves the universality class of the intermediate temperature Mott/metal crossover, which had alternatively been proposed to be described by more exotic theories. However, in the presence of a Zeeman magnetic field, we find that interplay of spin with itinerant charge can lead to physics beyond the dilute Fermi gas universality class.

Nonequilibrium Relaxation Study of Effect of Randomness on Critical Universality Classes of Ferromagnetic Phase Transitions

The ferromagnetic critical exponents are estimated along the phase boundary for the gauge glass model in three dimensions. The nonequilibrium relaxation method is applied to estimate the transition temperature and critical exponents. Together with the previously obtained results for the ± J Ising model in two and three dimensions, universality classes are examined for these disordered models. The result reveals a direct numerical confirmation of the Harris criterion for typical models with positive, zero and negative values of α 0 , the exponent for the specific heat in the regular case. In addition, we show a nonuniversal behavior of the dynamical exponent z with respect to the randomness for these models irrespective of α 0 .

Free energy in a magnetic field and the universal scaling equation of state for the three-dimensional Ising model

We have substantially extended the high-temperature and low-magnetic-field (and the related low-temperature and high-magnetic-field) bivariate expansions of the free energy for the conventional three-dimensional Ising model and for a variety of other spin systems generally assumed to belong to the same critical universality class. In particular, we have also derived the analogous expansions for the Ising models with spin s=1,3/2,.. and for the lattice euclidean scalar field theory with quartic self-interaction, on the simple cubic and the body-centered cubic lattices. Our bivariate high-temperature expansions, which extend through K^24, enable us to compute, through the same order, all higher derivatives of the free energy with respect to the field, namely all higher susceptibilities. These data make more accurate checks possible, in critical conditions, both of the scaling and the universality properties with respect to the lattice and the interaction structure and also help to improve an approximate parametric representation of the critical equation of state for the three-dimensional Ising model universality class.

Universality of Ionic Criticality: Size- and Charge-Asymmetric Electrolytes

Grand-canonical simulations designed to resolve critical universality classes are reported for z:1 hard-core electrolyte models with diameter ratios lambda=a+/a- less than or approximately equal 6. For z=1 Ising-type behavior prevails. Unbiased estimates of Tc(lambda) are within 1% of previous (biased) estimates but the critical densities are approximately 5% lower. Ising character is also established for the 2:1 and 3:1 equisized models, along with critical amplitudes and improved Tc estimates. For z=3, however, strong finite-size effects reduce the confidence level although classical and O (n>or=3) criticality are excluded.

Superfluid-insulator transitions on the triangular lattice

We report on a phenomenological study of superfluid to Mott insulator transitions of bosons on the triangular lattice, focusing primarily on the interplay between Mott localization and geometrical charge frustration at 1/2-filling. A general dual vortex field theory is developed for arbitrary rational filling factors f, based on the appropriate projective symmetry group. At the simple non-frustrated density f=1/3, we uncover an example of a deconfined quantum critical point very similar to that found on the half-filled square lattice. Turning to f=1/2, the behavior is quite different. Here, we find that the low-energy action describing the Mott transition has an emergent nonabelian SU(2)\times U(1) symmetry, not present at the microscopic level. This large nonabelian symmetry is directly related to the frustration-induced quasi-degeneracy between many charge-ordered states not related by microscopic symmetries. Through this ``pseudospin'' SU(2)symmetry, the charged excitations in the insulator close to the Mott transition develop a skyrmion-like character. This leads to an understanding of the recently discovered supersolid phase of the triangular lattice XXZ model (cond-mat/0505258, cond-mat/0505257, cond-mat/0505298) as a ``partially melted'' Mott insulator. The latter picture naturally explains a number of puzzling numerical observations of the properties of this supersolid. Moreover, we predict that the nearby quantum phase transition from this supersolid to the Mott insulator is in the recently-discovered non-compact CP^1 critical universality class (PRB 70, 075104 (2004)). A description of a broad range of other Mott and supersolid states, and a diverse set of quantum critical points between them, is also provided.

Charge and density fluctuations lock horns: ionic criticality with power-law forces

How do charge and density fluctuations compete in ionic fluids near gas-liquid criticality when quantum mechanical effects play a role ? To gain some insight, long-range $\Phi^{{\mathcal{L}}}_{\pm \pm} / r^{d+\sigma}$ interactions (with $\sigma>0$), that encompass van der Waals forces (when $\sigma = d = 3$), have been incorporated in exactly soluble, $d$-dimensional 1:1 ionic spherical models with charges $\pm q_0$ and hard-core repulsions. In accord with previous work, when $d>\min \{\sigma, 2\}$ (and $q_0$ is not too large), the Coulomb interactions do not alter the ($q_0 = 0$) critical universality class that is characterized by density correlations at criticality decaying as $1/r^{d-2+\eta}$ with $\eta = \max \{0, 2-\sigma\}$. But screening is now algebraic, the charge-charge correlations decaying, in general, only as $1/r^{d+\sigma+4}$; thus $\sigma = 3$ faithfully mimics known \textit{non}critical $d=3$ quantal effects. But in the \textit{absence} of full ($+, -$) ion symmetry, density and charge fluctuations mix via a transparent mechanism: then the screening \textit{at criticality} is \textit{weaker} by a factor $r^{4-2\eta}$. Furthermore, the otherwise valid Stillinger-Lovett sum rule fails \textit{at} criticality whenever $\eta =0$ (as, e.g., when $\sigma>2$) although it remains valid if $\eta >0$ (as for $\sigma<2$ or in real $d \leq 3$ Ising-type systems).

The Lifshitz line of the disordered and microemulsion phase in an A/B/A–B three component homopolymer/diblock copolymer mixture

Thermal composition fluctuations were measured in a homopolymer blend dPB/PS (dPB/dPS) of critical composition mixed with different amounts of a symmetric diblock copolymer dPB–PS (PB and PS being polybutadiene and polystyrene, respectively) with small-angle neutron scattering (SANS). From thermal fluctuations the two-phase boundary, two critical universality classes and the Lifshitz line were derived. The bicontinuous microemulsion phase could be identified by the characteristic peak positions of samples prepared in bulk, block and film contrast. A non-monotonic Lifshitz line (LL) showing a dependence on temperature was found. LL is located in the disordered and bicontinuous microemulsion phase.

Critical properties of an aperiodic model for interacting polymers

We investigate the effects of aperiodic interactions on the critical behavior of an interacting two-polymer model on hierarchical lattices (equivalent to the Migadal-Kadanoff approximation for the model on Bravais lattices), via renormalization-group and tranfer-matrix calculations. The exact renormalization-group recursion relations always present a symmetric fixed point, associated with the critical behavior of the underlying uniform model. If the aperiodic interactions, defined by s ubstitution rules, lead to relevant geometric fluctuations, this fixed point becomes fully unstable, giving rise to novel attractors of different nature. We present an explicit example in which this new attractor is a two-cycle, with critical indices different from the uniform model. In case of the four-letter Rudin-Shapiro substitution rule, we find a surprising closed curve whose points are attractors of period two, associated with a marginal operator. Nevertheless, a scaling analysis indicates that this attractor may lead to a new critical universality class. In order to provide an independent confirmation of the scaling results, we turn to a direct thermodynamic calculation of the specific-heat exponent. The thermodynamic free energy is obtained from a transfer matrix formalism, which had been previously introduced for spin systems, and is now extended to the two-polymer model with aperiodic interactions.