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Physical review. B, Condensed matter

Multifractal and critical properties of the Ising model

Wojciech Jeżewski

Published May 1, 1998• 6 citations
Publisher

Abstract

Multifractal properties of the two-dimensional Ising model are studied by means of the $f(\ensuremath{\alpha})$ spectra of singularities of probability measures supported by energy spectra. These measures are determined through calculations of energy degeneracies for finite-size systems up to $12\ifmmode\times\else\texttimes\fi{}12$ spins in cases of square and triangular lattices and up to $22\ifmmode\times\else\texttimes\fi{}22$ spins in the case of the hexagonal lattice. The calculations are performed in an exact manner using the transfer-matrix method. It is argued that, in the thermodynamic limit, the scaling exponent ${\ensuremath{\alpha}}_{\mathrm{max}}$ associated with the most probable energy of the system takes at the critical temperature a minimum value and, consequently, it is argued that a given system reveals a phase transition at a finite temperature if ${\ensuremath{\alpha}}_{\mathrm{max}}$ possesses a minimum at the finite temperature. It is also shown that, in the thermodynamic limit, the spectrum $f(\ensuremath{\alpha})$ shrinks when the critical temperature is approached.

DOI
10.1103/physrevb.57.10240

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Multifractal properties of the two-dimensional Ising model are studied by means of the f(\ensuremathα)f(\ensuremath{\alpha})f(\ensuremathα) spectra of singularities of probability measures supported by energy spectra. These measures are determined through calculations of energy degeneracies for finite-size systems up to 12\ifmmode×\else\texttimes\fi1212\ifmmode\times\else\texttimes\fi{}1212\ifmmode×\else\texttimes\fi12 spins in cases of square and triangular lattices and up to 22\ifmmode×\else\texttimes\fi2222\ifmmode\times\else\texttimes\fi{}2222\ifmmode×\else\texttimes\fi22 spins in the case of the hexagonal lattice. The calculations are performed in an exact manner using the transfer-matrix method. It is argued that, in the thermodynamic limit, the scaling exponent \ensuremathαmax{\ensuremath{\alpha}}_{\mathrm{max}}\ensuremathαmax​ associated with the most probable energy of the system takes at the critical temperature a minimum value and, consequently, it is argued that a given system reveals a phase transition at a finite temperature if \ensuremathαmax{\ensuremath{\alpha}}_{\mathrm{max}}\ensuremathαmax​ possesses a minimum at the finite temperature. It is also shown that, in the thermodynamic limit, the spectrum f(\ensuremathα)f(\ensuremath{\alpha})f(\ensuremathα) shrinks when the critical temperature is approached.
We compute the finite-size corrections to the free energy, internal energy and specific heat of the critical two-dimensional spin-1/2 Ising model on a triangular and hexagonal lattices wrapped on a torus. We find the general form of the finite-size corrections to these quantities, as well as explicit formulas for the first coefficients of each expansion. We analyze the implications of these findings on the renormalization-group description of the model. Key Words: Ising model; finite-size scaling; corrections to scaling; renormalization group; irrelevant operators; scaling functions. It is well-known that phase transitions in statistical-mechanical systems can occur only in the infinite-volume limit. In any finite system, all thermodynamic quantities (such as the magnetic susceptibility and the specific heat) are analytic functions of all parameters (such as the temperature and the magnetic field); but near a critical
The corrections to finite-size scaling in the critical two-point correlation function G(r) of 2D Ising model on a square lattice have been studied numerically by means of exact transfer-matrix algorithms. The systems have been considered, including up to 800 spins. The calculation of G(r) at a distance r equal to the half of the system size L shows the existence of an amplitude correction proportional to 1/L^2. A nontrivial correction ~1/L^0.25 of a very small magnitude also has been detected, as it can be expected from our recently developed GFD (grouping of Feynman diagrams) theory. Monte Carlo simulations of the squared magnetization of 3D Ising model have been performed by Wolff's algorithm in the range of the reduced temperatures t =< 0.000086 and system sizes L =< 410. The effective critical exponent beta_eff(t) tends to increase above the currently accepted numerical values. The critical coupling K_c=0.22165386(51) has been extracted from the Binder cumulant data within 96 =< L =< 384. The critical exponent 1/nu, estimated from the finite-size scaling of the derivatives of the Binder cumulant, tends to decrease slightly below the RG value 1.587 for the largest system sizes. The finite-size scaling of accurately simulated maximal values of the specific heat C_v in 3D Ising model confirms a logarithmic rather than power-like critical singularity of C_v.
The spectrum of the transfer matrix of a semi-infinite two-dimensional Isin model with marginally inhomogeneous couplings has been determined exactly in the finite-size-scaling limit at the bulk critical point. Depending on the local temperature at the boundary the surface phase transition of this system is either of first order or of second order with nonuniversal critical exponents. For a second-order transition, the excitation energies of the transfer matrix are inversely proportional to the linear size (N) of the system, but the levels are not equidistantly spaced. On the other hand, for a first-order transition the lowest gap vanishes algegraically, but faster than 1/N.
The dynamics of Ising spins on a two-dimensional fractal lattice has been studied by Monte Carlo simulations. Both the limits $L>>\ensuremath{\xi}$ and $L<<\ensuremath{\xi}$ were studied where LLL is the size of the sample and $\ensuremath{\xi}$ the correlation length. Exact transfer matrix calculations were also carried out to check that the correct equilibrium values were reached. In both limits the predicted breakdown of conventional dynamic scaling was obtained. Using finite-size scaling $\ensuremath{\tau}\ensuremath{\sim}{L}^{z}$ for $L<<\ensuremath{\xi}$, an effective temperature-dependent critical exponent zzz was obtained consistent with the recent prediction $z\ensuremath{\propto}\frac{1}{T}$ for small TTT.
We use transfer-matrix and finite-size scaling methods to investigate the location and properties of the multicritical point of two-dimensional Ising spin glasses on square, triangular, and honeycomb lattices with both binary and Gaussian disorder distributions. For square and triangular lattices with binary disorder, the estimated position of the multicritical point is in numerical agreement with recent conjectures regarding its exact location. For the remaining four cases, our results indicate disagreement with the respective versions of the conjecture, though by very small amounts, never exceeding 0.2%. Our results for (i) the correlation-length exponent $\ensuremath{\nu}$ governing the ferroparamagnetic transition, (ii) the critical domain-wall energy amplitude $\ensuremath{\eta}$, (iii) the conformal anomaly ccc, (iv) the finite-size susceptibility exponent $\ensuremath{\gamma}/\ensuremath{\nu}$, and (v) the set of multifractal exponents ${{\ensuremath{\eta}}_{k}}$ associated to the moments of the probability distribution of spin-spin correlation functions at the multicritical point are consistent with universality as regards lattice structure and disorder distribution and in good agreement with existing estimates.
Multifractal detrended fluctuation analysis has emerged as a standard tool for characterizing scale invariance in complex systems, yet its application to discrete spin models is frequently marred by reports of "spurious multifractality" that contradict established theory. In this work, we resolve this controversy by establishing a rigorous protocol for the analysis of discrete lattice snapshots. Using the two-dimensional Ising model as a benchmark, we demonstrate that the previously reported broad singularity spectra are finite-size artifacts dominated by lattice discreteness effects in the negative moment regime (q<0). By restricting the analysis to positive moments and performing a systematic finite-size scaling (FSS) analysis, we show that the spectral width collapses to zero (Δα→0) in the thermodynamic limit. The method accurately recovers the monofractal exponent of the Ising universality class (α≈H≈0.875), consistent with conformal field theory. Applying this FSS protocol to the random bond Ising model, we reveal that the broad multifractal dispersion often attributed to quenched disorder also flattens systematically with increasing system size. These results demonstrate that apparent multifractality in finite discrete lattices is predominantly driven by preasymptotic discreteness artifacts, underscoring the universal necessity of FSS to avoid false positives in statistical mechanics.