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Asia Pacific Physics Conference (APPC)

Numerical Analysis of Quantum Phase Transitions with Dynamic Control of Anisotropy

Shinya Yasuda, Synge Todo

Published Mar 19, 2014• 0 citations
Publisher

Abstract

We present a method that optimizes the aspect ratio of a lattice during the Monte Carlo simulations and realizes the virtually isotropic lattice automatically. Based on the correlation lengths, the system linear lengths, including the imaginary-time direction for the quantum models, are tuned by the Robbins-Monro algorithm. The method enables us to compare directly the value of critical amplitude among different anisotropic models, and helps with identifying the universality class. We apply the present method to anisotropic and transverse field Ising models and two kinds of dimerized Heisenberg antiferromagnets, the staggered and columnar dimer models, and discuss the universality and the corrections to scaling.

DOI
10.7566/jpscp.1.012127

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We present a method that optimizes the aspect ratio of a lattice during the Monte Carlo simulations and realizes the virtually isotropic lattice automatically. Based on the correlation lengths, the system linear lengths, including the imaginary-time direction for the quantum models, are tuned by the Robbins-Monro algorithm. The method enables us to compare directly the value of critical amplitude among different anisotropic models, and helps with identifying the universality class. We apply the present method to anisotropic and transverse field Ising models and two kinds of dimerized Heisenberg antiferromagnets, the staggered and columnar dimer models, and discuss the universality and the corrections to scaling.
We present a method that optimizes the aspect ratio of a spatially anisotropic quantum lattice model during the quantum Monte Carlo simulation, and realizes the virtually isotropic lattice automatically. The anisotropy is removed by using the Robbins-Monro algorithm based on the correlation length in each direction. The method allows for comparing directly the value of the critical amplitude among different anisotropic models, and identifying the universality more precisely. We apply our method to the staggered dimer antiferromagnetic Heisenberg model and demonstrate that the apparent nonuniversal behavior is attributed mainly to the strong size correction of the effective aspect ratio due to the existence of the cubic interaction.
The quantum phase transition of the one-dimensional long-range transverse-field Ising model is explored by combining the quantum Monte Carlo method and stochastic parameter optimization, specifically achieved by tuning correlation ratios so that space and imaginary time are isotropic. In our simulations, the simulator automatically determines the parameters to sample from, even without prior knowledge of the critical point and universality class. The leading-order finite-size corrections are eliminated by comparing two systems with different sizes; this procedure is also performed automatically. Varying the decay exponent of the long-range interaction, σ, we investigate σ dependence of the dynamical exponent and the other critical exponents precisely in the mean-field, nonuniversal, and two-dimensional classical Ising universality regimes. We successfully obtained numerical evidence supporting σ=7/4 as the universality boundary between the latter two.
We formulate a cluster Monte Carlo method for the anisotropic limit of Ising models on (d+1)-dimensional lattices, which in effect, are equivalent with d-dimensional quantum transverse Ising models. Using this technique, we investigate the transverse Ising models on the square, triangular, Kagome, honeycomb, and simple-cubic lattices. The Monte Carlo data are analyzed by finite-size scaling. In each case we find, as expected, that the critical behavior fits well in the (d+1)-dimensional Ising universality class. For the transverse Ising model on the square lattice, we determine the Binder cumulant of the classical counterpart for a range of aspect ratios between the system sizes in the third or "classical" direction and that in the other two directions. Matching this universal function with the case of the isotropic Ising model yields the length ratio relating the isotropic Ising model with the anisotropic limit. The efficiency of the present algorithm is reflected by the precision of its results, which improves significantly on earlier analyses.
We have developed a unified finite-size scaling method for quantum phase transitions that requires no prior knowledge of the dynamical exponent zzz. During a quantum Monte Carlo simulation, the temperature is automatically tuned by the Robbins-Monro stochastic approximation method, being proportional to the lowest gap of the finite-size system. The dynamical exponent is estimated in a straightforward way from the system-size dependence of the temperature. As a demonstration of our novel method, the two-dimensional S=1/2S=1/2S=1/2 quantum XYXYXY model in uniform and staggered magnetic fields is investigated in the combination of the world-line quantum Monte Carlo worm algorithm. In the absence of a uniform magnetic field, we obtain the fully consistent result with the Lorentz invariance at the quantum critical point, z=1z=1z=1, i.e., the three-dimensional classical XYXYXY universality class. Under a finite uniform magnetic field, on the other hand, the dynamical exponent becomes two, and the mean-field universality with effective dimension (2+2)(2+2)(2+2) governs the quantum phase transition.
The S=1/2S=1/2S=1/2 and S=1S=1S=1 two-dimensional quantum Heisenberg antiferromagnets on the anisotropic dimerized square lattice are investigated by the quantum Monte Carlo method. By finite-size-scaling analyses on the correlation lengths, the ground-state phase diagram parametrized by strengths of the dimerization and of the spatial anisotropy is determined much more accurately than the previous works. It is confirmed that the quantum critical phenomena on the phase boundaries belong to the same universality class as that of the classical three-dimensional Heisenberg model. Furthermore, for S=1,S=1,S=1, we show that all the spin-gapped phases, such as the Haldane and dimer phases, are adiabatically connected in the extended-parameter space, though they are classified into different classes in terms of the string order parameter in the one-dimensional, i.e., the zero-interchain-coupling, case.
Motivated by the unexpected Monte Carlo results as well as the theoretical proposal of a large correction to scaling for the critical theory of the 2-d staggered-dimer spin-1/2 Heisenberg model on the square lattice, we study the phase transitions induced by dimerization of several dimerized quantum Heisenberg models with spatial anisotropy using first principles Monte Carlo method. Remarkably, while our Monte Carlo data for all the models considered here, including the herringbone- and ladder-dimer models on the square lattice, are compatible with the recently proposed scenario of an enhanced correction to scaling, we find it is likely that the enhanced correction to scaling manifests itself as amplification of the nonuniversal prefactors appeared in the scaling forms. In other words, our data are in consistence with the established numerical values for the critical exponents, including the confluent exponent, in the O(3) universality class. Convincing numerical evidence is provided to support this proposed scenario.