Critical behavior of a three-dimensional random-bond Ising model using finite-time scaling with extensive Monte Carlo renormalization-group method

Abstract

We have investigated the critical behavior of a three-dimensional random-bond Ising model for a series of the disorder strength by a finite-time scaling combining with Monte Carlo renormalization-group method in the presence of a linearly varying temperature. The method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures. The static exponents obtained agree well with most existing results, verify both the hyperscaling and the Rushbrooke scaling laws and their combined scaling law, which in turn validate their asymptotic nature, and corroborate the universality of the relevant random fixed point with respect to the forms of disorder. The dynamic critical exponent z is estimated to 2.114(51), which is compatible with those obtained from experiments and renormalization-group analyses. The exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point. They also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover. Our results demonstrate the effectiveness of the finite-time scaling method.

Microcrack connectivity in rocks: a real-space renormalization group approach for 3D anisotropic bond percolation

“…Therefore, on the phase boundaries which consist of fixed points, the probabilities are unchanged after iteration, that is bond percolation and the renormalized bond probabilities and the values of critical exponents can be determined in this way. Furthermore, Xiong and his co-workers [24] developed this method to investigate the critical behavior of a 3D random-bond Ising model. However, the Monte Carlo renormalization group method has not been applied in 3D anisotropic percolation and the results of this method are discrete points in the parameter space, through which a phase diagram is dicult to draw.…”

Phase transitions and critical phenomena in a three-dimensional site-diluted Potts model

“…The dependences of m, w, V n and C on L are plotted according to expressions (8)(9)(10)(11) for calculation the critical exponents of a, b, g and n. Analysis of data, made using the nonlinear least-square method, allows to define the values as follows b=n ¼ 0:691ð24Þ, g=n ¼ 1:521ð30Þ, 1=n ¼ 1:342ð13Þ, a=n ¼ À0:187ð11Þ. Then, exponents a¼ À0.139(11), b¼0.514(24) and g¼1.133(30) are derived by means of values of n, obtained for 3D diluted Potts model with q¼4.…”

“…This criterion is satisfied only by the systems whose effective Hamiltonian is isomorphous relative to the Ising model in the critical point. Recently, the critical properties of disordered Ising model were studied well enough [3,[5][6][7][8][9][10]. Unlike 3D site-diluted Ising model, the three-dimensional 3-and 4-state Potts models with quenched non-magnetic impurities is still at issue.…”

Complete scaling makes differences along the critical isobar: Molecular dynamics simulations of Lennard-Jones fluid with finite-time scaling

“…FTS has been applied effectively both to classical [57][58][59][60][61][62] and quantum phase transitions in lattice models [63][64][65]. Analogue to FSS that circumvents the problem that ξ is longer than the lattice size L, FTS overcomes the critical slowing down of a divergent t eq by devising a controllable time scale t dr .…”