Finite-time scaling via linear driving: application to the two-dimensional Potts model

Phases fluctuations and anomalous finite-time scaling in an externally applied field on finite-sized lattices

We study in detail the dynamic scaling of the three-dimensional Ising model under cooling on finite-sized lattices subject to an externally applied field whose magnitude fixes a scaled variable pertinent to it. Three different protocols, protocols A, B, and C, in which the field is applied either only below or only above the critical point besides during the whole process, respectively, are investigated. Anomalous finite-time scaling (FTS) are found in protocols B and C on a large lattice only and in protocol A when extrinsic self-similarity is broken. However, these anomalous scalings in cooling can be rectified unexpectedly by the recently found breaking-of-extrinsic-self-similarity exponent of FTS in zero-field heating, except in the case of protocol B in which a reduced zero-field cooling exponent is additionally required. This provides a distinctive source to the breaking-of-extrinsic-self-similarity exponents and thus confirms their validity. The different scaling behaviors of the three different protocols also shows that the so-called phases fluctuations at and above the critical point are more important than those below it. The previously found qualitative difference between zero-field heating and zero-field cooling is essentially the symmetry of the system states, namely whether they are ordered or disordered. We also confirm that there exists a revised FTS regime-the regime in which both the lattice sizes and driving rates are indispensable-in between the two end regimes of FTS and finite-size scaling even in the presence of the field. Its crossover to the end FTS regime is responsible for the anomalous scalings of the large lattice size in protocols B and C. However, our results show that the revised FTS is never needed for the usual order parameter in which no absolute value is taken. In addition, the end FTS regime in cooling is found to exhibit a special feature different from that in heating. Our results demonstrate that new exponents are needed for scaling in the whole driven process even in the case in which an external field is applied.

Pseudo-Yang-Lee Edge Singularity Critical Behavior in a Non-Hermitian Ising Model

The quantum phase transition of a one-dimensional transverse field Ising model in an imaginary longitudinal field is studied. A new order parameter M is introduced to describe the critical behaviors in the Yang-Lee edge singularity (YLES). The M does not diverge at the YLES point, a behavior different from other usual parameters. We term this unusual critical behavior around YLES as the pseudo-YLES. To investigate the static and driven dynamics of M, the (1+1) dimensional ferromagnetic-paramagnetic phase transition ((1+1) D FPPT) critical region, (0+1) D YLES critical region and the (1+1) D YLES critical region of the model are selected. Our numerical study shows that the (1+1) D FPPT scaling theory, the (0+1) D YLES scaling theory and (1+1) D YLES scaling theory are applicable to describe the critical behaviors of M, demonstrating that M could be a good indicator to detect the phase transition around YLES. Since M has finite value around YLES, it is expected that M could be quantitatively measured in experiments.

Linear relaxation in large two-dimensional Ising models

Critical dynamics in two-dimension Ising lattices up to 2048×2048 is simulated on field-programmable-gate-array- based computing devices. Linear relaxation times are measured from extremely long Monte Carlo simulations. The longest simulation has 7.1×10(16) spin updates, which would take over 37 years to simulate on a general purpose computer. The linear relaxation time of the Ising lattices is found to follow the dynamic scaling law for correlation lengths as long as 2048. The dynamic exponent z of the system is found to be 2.179(12), which is consistent with previous studies of Ising lattices with shorter correlation lengths. It is also found that Monte Carlo simulations of critical dynamics in Ising lattices larger than 512×512 are very sensitive to the statistical correlations between pseudorandom numbers, making it even more difficult to study such large systems.

Natural time analysis of critical phenomena

A quantity exists by which one can identify the approach of a dynamical system to the state of criticality, which is hard to identify otherwise. This quantity is the variance κ(1)(≡<χ(2)> - <χ>(2)) of natural time χ, where <f(χ)> = Σp(k)f(χ(k)) and p(k) is the normalized energy released during the kth event of which the natural time is defined as χ(k) = k/N and N stands for the total number of events. Then we show that κ(1) becomes equal to 0.070 at the critical state for a variety of dynamical systems. This holds for criticality models such as 2D Ising and the Bak-Tang-Wiesenfeld sandpile, which is the standard example of self-organized criticality. This condition of κ(1) = 0.070 holds for experimental results of critical phenomena such as growth of rice piles, seismic electric signals, and the subsequent seismicity before the associated main shock.

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

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.