Estimates Of Critical Exponents Research Articles

Unveiling universal aspects of the cellular anatomy of the brain

Recent cellular-level volumetric brain reconstructions have revealed high levels of anatomic complexity. Determining which structural aspects of the brain to focus on, especially when comparing with computational models and other organisms, remains a major challenge. Here we quantify aspects of this complexity and show evidence that brain anatomy satisfies universal scaling laws, establishing the notion of structural criticality in the cellular structure of the brain. Our framework builds upon understanding of critical systems to provide clear guidance in selecting informative structural properties of cellular brain anatomy. As an illustration, we obtain estimates for critical exponents in the human, mouse and fruit fly brains and show that they are consistent between organisms, to the extent that data limitations allow. Such universal quantities are robust to many of the microscopic details of the cellular structures of individual brains, providing a key step towards generative computational models of the cellular structure of the brain, and also clarifying in which sense one animal may be a suitable anatomic model for another.

Critical behaviours of anisotropic XY ferromagnet in the presence of random field

The anisotropic XY ferromagnet has been studied by Monte Carlo simulation in a three dimensional simple cubic lattice. The increase in critical temperature (ferro-para transition) has been noticed with increasing the strength of anisotropy. The effects of random fields (both with full circular symmetry and in angular window) on the critical temperature are investigated systematically in the anisotropic XY ferromagnet in three dimensions. Reduction of the critical temperature of anisotropic XY ferromagnet has been observed in the presence of random field. The compensating field (the required amount of field which preserves the critical temperature for isotropic XY ferromagnet) has been studied as a function of the strength of anisotropy. The compensating field was found to depend linearly on the strength of anisotropy. We have also studied the effects of random field confined in the angular window and observed the reduction of the critical temperature with increase of the angular extension. The critical behaviours are formalised by the usual finite size analysis and the estimation of critical exponents for the susceptibility and the specific heat.

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.

Dynamical Scaling Analysis for ±J Ising Model in Three Dimensions

The ±J Ising model in three dimensions is investigated using dynamical scaling analysis for a nonequilibrium relaxation process. The spin-glass (SG) transition is analyzed under an assumed dynamical exponent z proposed for the scaling analysis of the SG transition. To estimate dynamical exponent z precisely, we introduce an extrapolation scheme for the asymptotic value of the logarithmic derivative of the dynamical Binder parameter. We obtain the phase diagram in the temperature–concentration (p–T) space, as well as the p-dependence of critical exponents γ, ν, and z. For the static exponents of the SG transition, the universality is strongly indicated numerically along the whole region of the paramagnetic (PM)–SG phase boundary. A remarkable crossover behavior is observed for estimations of critical exponents at the boundary close to the multicritical point (MCP) of PM–ferromagnetic (FM)–SG phases. In addition, the dynamical assumption for z resulting from the slow relaxation in the SG region disappears in the FM region, and the crossover between them occurs around the Nishimori Line.

Continued functions and critical exponents: tools for analytical continuation of divergent expressions in phase transition studies

Resummation methods using continued functions are implemented to converge divergent series appearing in perturbation problems related to continuous phase transitions in field theories. In some cases, better convergence properties are obtained using continued functions than diagonal Pade approximants, which are extensively used in literature. We check the reliability of critical exponent estimates derived previously in universality classes of O(n)-symmetric models (classical phase transitions) and Gross-Neveu-Yukawa models (quantum phase transitions) using new methods.

O(N) models with boundary interactions and their long range generalizations

We study the critical properties of scalar field theories in d + 1 dimensions with O(N) invariant interactions localized on a d-dimensional boundary. By a combination of large N and epsilon expansions, we provide evidence for the existence of non-trivial O(N) BCFTs in 1 < d < 4. Due to having free fields in the bulk, these models possess bulk higher-spin currents which are conserved up to terms localized on the boundary. We suggest that this should lead to a set of protected spinning operators on the boundary, and give evidence that their anomalous dimensions vanish. We also discuss the closely related long-range O(N ) models in d dimensions, and in particular study a weakly coupled description of the d = 1 long range O(N) model near the upper critical value of the long range parameter, which is given in terms of a non-local non-linear sigma model. By combining the known perturbative descriptions, we provide some estimates of critical exponents in d = 1.

Real-space renormalization of randomly vacated lattices: a renormalization group for jamming?

Jamming occurs in granular materials, as well as in emulsions, dense suspensions, and other amorphous, particulate systems. When the packing fraction $\phi$, defined as the ratio of particle volume to system volume, is increased past a critical value $\phi_c$, a liquid-solid phase transition occurs, and grains are no longer able to rearrange. Previous studies have shown evidence of spatial correlations that diverge near $\phi = \phi_c$, but there has been no explicit spatial renormalization group (RG) scheme that has captured this transition. Here, I present a candidate for such a scheme, using a block-spin-like transformation of a randomly vacated lattice of grains. I define a real-space RG transformation based on local mechanical stability. This model displays a critical packing fraction $\phi_c$ and gives estimates of critical exponents in two and three dimensions.

Bootstrapping MN and tetragonal CFTs in three dimensions

Conformal field theories (CFTs) with MN and tetragonal global symmetry in d=2+1d=2+1 dimensions are relevant for structural, antiferromagnetic and helimagnetic phase transitions. As a result, they have been studied in great detail with the \varepsilon=4-dε=4−d expansion and other field theory methods. The study of these theories with the nonperturbative numerical conformal bootstrap is initiated in this work. Bounds for operator dimensions are obtained and they are found to possess sharp kinks in the MN case, suggesting the existence of full-fledged CFTs. Based on the existence of a certain large-NN expansion in theories with MN symmetry, these are argued to be the CFTs predicted by the \varepsilonε expansion. In the tetragonal case no new kinks are found, consistently with the absence of such CFTs in the \varepsilonε expansion. Estimates for critical exponents are provided for a few cases describing phase transitions in actual physical systems. In two particular MN cases, corresponding to theories with global symmetry groups O(2)^2\rtimes S_2O(2)2⋊S2 and O(2)^3\rtimes S_3O(2)3⋊S3, a second kink is found. In the O(2)^2\rtimes S_2O(2)2⋊S2 case it is argued to be saturated by a CFT that belongs to a new universality class relevant for the structural phase transition of NbO_22 and paramagnetic-helimagnetic transitions of the rare-earth metals Ho and Dy. In the O(2)^3\rtimes S_3O(2)3⋊S3 case it is suggested that the CFT that saturates the second kink belongs to a new universality class relevant for the paramagnetic-antiferromagnetic phase transition of the rare-earth metal Nd.

Observation of nonscalar and logarithmic correlations in two- and three-dimensional percolation.

In bulk percolation, we exhibit operators that insert N clusters with any given symmetry under the symmetric group S_{N}. At the critical threshold, this leads to predictions that certain combinations of two-point correlation functions depend logarithmically on distance, without the usual power law. The behavior under rotations of certain amplitudes of correlators is also determined exactly. All these results hold in any dimension, 2≤d≤6. Moreover, in d=2 the critical exponents and universal logarithmic prefactors are obtained exactly. We test these predictions against extensive simulations of critical bond percolation in d=2 and 3, for all correlators up to N=4 (d=2) and N=3 (d=3), finding excellent agreement. In d=3 we further obtain precise numerical estimates for critical exponents and logarithmic prefactors.

Deconfined criticality from the QED3 -Gross-Neveu model at three loops

The QED$_3$-Gross-Neveu model is a (2+1)-dimensional U(1) gauge theory involving Dirac fermions and a critical real scalar field. This theory has recently been argued to represent a dual description of the deconfined quantum critical point between Neel and valence bond solid orders in frustrated quantum magnets. We study the critical behavior of the QED$_3$-Gross-Neveu model by means of an epsilon expansion around the upper critical space-time dimension of $D_c^+=4$ up to the three-loop order. Estimates for critical exponents in 2+1 dimensions are obtained by evaluating the different Pade approximants of their series expansion in epsilon. We find that these estimates, within the spread of the Pade approximants, satisfy a nontrivial scaling relation which follows from the emergent SO(5) symmetry implied by the duality conjecture. We also construct explicit evidence for the equivalence between the QED$_3$-Gross-Neveu model and a corresponding critical four-fermion gauge theory that was previously studied within the 1/N expansion in space-time dimensions 2<D<4.

A functional perspective on emergent supersymmetry

We investigate the emergence of mathcal{N} = 1 supersymmetry in the long-range behavior of three-dimensional parity-symmetric Yukawa systems. We discuss a renormalization approach that manifestly preserves supersymmetry whenever such symmetry is realized, and use it to prove that supersymmetry-breaking operators are irrelevant, thus proving that such operators are suppressed in the infrared. All our findings are illustrated with the aid of the ϵ-expansion and a functional variant of perturbation theory, but we provide numerical estimates of critical exponents that are based on the non-perturbative functional renormalization group.

Turbulent mixing of a critical fluid: The non-perturbative renormalization

Non-perturbative Renormalization Group (NPRG) technique is applied to a stochastical model of a non-conserved scalar order parameter near its critical point, subject to turbulent advection. The compressible advecting flow is modeled by a random Gaussian velocity field with zero mean and correlation function 〈υjυi〉∼(Pji⊥+αPji∥)/kd+ζ. Depending on the relations between the parameters ζ, α and the space dimensionality d, the model reveals several types of scaling regimes. Some of them are well known (model A of equilibrium critical dynamics and linear passive scalar field advected by a random turbulent flow), but there is a new nonequilibrium regime (universality class) associated with new nontrivial fixed points of the renormalization group equations. We have obtained the phase diagram (d, ζ) of possible scaling regimes in the system. The physical point d=3, ζ=4/3 corresponding to three-dimensional fully developed Kolmogorov's turbulence, where critical fluctuations are irrelevant, is stable for α≲2.26. Otherwise, in the case of “strong compressibility” α≳2.26, the critical fluctuations of the order parameter become relevant for three-dimensional turbulence. Estimations of critical exponents for each scaling regime are presented.

Minimally subtracted six-loop renormalization of O(n) -symmetric ϕ4 theory and critical exponents

We present the perturbative renormalization group functions of $O(n)$-symmetric $\phi^4$ theory in $4-2\varepsilon$ dimensions to the sixth loop order in the minimal subtraction scheme. In addition, we estimate diagrams without subdivergences up to 11 loops and compare these results with the asymptotic behaviour of the beta function. Furthermore, we perform a resummation to obtain estimates for critical exponents in three and two dimensions.

Corrections to finite-size scaling in the 3D Ising model based on nonperturbative approaches and Monte Carlo simulations

Corrections to scaling in the 3D Ising model are studied based on nonperturbative analytical arguments and Monte Carlo (MC) simulation data for different lattice sizes [Formula: see text]. Analytical arguments show the existence of corrections with the exponent [Formula: see text], the leading correction-to-scaling exponent being [Formula: see text]. A numerical estimation of [Formula: see text] from the susceptibility data within [Formula: see text] yields [Formula: see text], in agreement with this statement. We reconsider the MC estimation of [Formula: see text] from smaller lattice sizes, [Formula: see text], using different finite-size scaling methods, and show that these sizes are still too small, since no convergence to the same result is observed. In particular, estimates ranging from [Formula: see text] to [Formula: see text] are obtained, using MC data for thermodynamic average quantities, as well as for partition function zeros. However, a trend toward smaller [Formula: see text] values is observed in one of these cases in a refined estimation from extended data up to [Formula: see text]. We discuss the influence of [Formula: see text] on the estimation of critical exponents [Formula: see text] and [Formula: see text].

Molecular dynamics simulation of a binary mixture near the lower critical point.

2,6-lutidine molecules mix with water at high and low temperatures but in a wide intermediate temperature range a 2,6-lutidine/water mixture exhibits a miscibility gap. We constructed and validated an atomistic model for 2,6-lutidine and performed molecular dynamics simulations of 2,6-lutidine/water mixture at different temperatures. We determined the part of demixing curve with the lower critical point. The lower critical point extracted from our data is located close to the experimental one. The estimates for critical exponents obtained from our simulations are in a good agreement with the values corresponding to the 3D Ising universality class.

Functional renormalization group approach to the Yang-Lee edge singularity

We determine the scaling properties of the Yang-Lee edge singularity as described by a one-component scalar field theory with imaginary cubic coupling, using the nonperturbative functional renormalization group in $3 \leq d\leq 6$ Euclidean dimensions. We find very good agreement with high-temperature series data in $d = 3$ dimensions and compare our results to recent estimates of critical exponents obtained with the four-loop $\epsilon = 6-d$ expansion and the conformal bootstrap. The relevance of operator insertions at the corresponding fixed point of the RG $\beta$ functions is discussed and we estimate the error associated with $\mathcal{O}(\partial^4)$ truncations of the scale-dependent effective action.

Nematic order in a simple-cubic lattice-spin model with full-ranged dipolar interactions.

In a previous paper [Phys. Rev. E 90, 022506 (2014)PLEEE81539-375510.1103/PhysRevE.90.022506], we studied the thermodynamic and structural properties of a three-dimensional simple-cubic lattice model with dipolarlike interaction, truncated at nearest-neighbor separation, for which the existence of an ordering transition at finite temperature had been proven mathematically; here we extend our investigation, addressing the full-ranged counterpart of the model, for which the critical behavior had been investigated theoretically and experimentally. In addition, the existence of an ordering transition at finite temperature had been proven mathematically as well. Both models exhibited the same continuously degenerate ground-state configuration, possessing full orientational order with respect to a suitably defined staggered magnetization (polarization), but no nematic second-rank order; in both cases, thermal fluctuations remove the degeneracy, so that nematic order does set in at low but finite temperature via a mechanism of order by disorder. On the other hand, there were recognizable quantitative differences between the two models as for ground-state energy and critical exponent estimates; the latter were found to agree with early renormalization-group calculations and with experimental results.

Phase diagram and critical behavior of the antiferromagnetic Ising model in an external field

We study the critical properties of the antiferromagnetic spin-1/2 Ising model in an external field on the square lattice. Using tomographic entropic sampling, a flat-histogram simulation method, we estimate the number of configurations, , and related microcanonical averages in the energy-magnetization space, for system sizes L = 10–30. The critical line and exponents are calculated using finite-size scaling analysis in the temperature-external field plane. With these estimates in hand, we perform detailed studies of critical behavior using Metropolis sampling of larger systems (). These results are compared to several approximate theoretical methods. Our estimates of critical exponents and Binder’s reduced fourth cumulant along the critical line are in very good agreement with their respective literature values for the two-dimensional Ising universality class. We verify as well that the specific heat scales along the critical line, as expected for an Ising-like critical point.

Критические индексы и псевдо-$\epsilon$-разложение

Представлены псевдо-$\epsilon$-разложения ($\tau$-ряды) для критических индексов трехмерной $O(n)$-симметричной модели типа $\lambda\phi^4$, найденные на основе шестипетлевых ренормгрупповых разложений. Численные результаты приведены для физически интересных случаев $n = 1$, $n = 2$, $n = 3$ и $n = 0$, а также для $4 \le n \le 32$ с тем, чтобы выявить общие свойства полученных рядов. Псевдо-$\epsilon$-разложения индексов $\gamma$ и $\alpha$ имеют малые и быстро убывающие по модулю коэффициенты, так что вполне приемлемые численные оценки дает прямое суммирование $\tau$-рядов, а обращение к аппроксимантам Паде позволяет получить высокоточные результаты. Напротив, коэффициенты псевдо-$\epsilon$-разложения индекса поправки к скейлингу $\omega$ не имеют выраженной тенденции к убыванию при физических значениях $n$. Однако соответствующие ряды знакопеременны, и для получения надежных численных оценок здесь также оказывается достаточным использование простых аппроксимант Паде. Таким образом, технику псевдо-$\epsilon$-разложения можно рассматривать как своеобразный метод пересуммирования, превращающий расходящиеся ренормгрупповые ряды в разложения, удобные c вычислительной точки зрения.

The Ising Spin Glass in dimension four

The critical behaviors of the bimodal and Gaussian Ising spin glass (ISG) models in dimension four are studied through extensive numerical simulations, and from an analysis of high temperature series expansion (HTSE) data of Klein et al. (1991). The simulations include standard finite size scaling measurements, thermodynamic limit regime measurements, and analyses which provide estimates of critical exponents without any consideration of the critical temperature. The higher order HTSE series for the bimodal model provide accurate estimates of the critical temperature and critical exponents. These estimates are independent of and fully consistent with the simulation values. Comparisons between ISG models in dimension four show that the critical exponents and the critical constants for dimensionless observables depend on the form of the interaction distribution of the model.