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Journal of the Physical Society of Japan

New Quantum Monte Carlo Approach to Ground-State Phase Transitions in Quantum Spin Systems

Yoshihiko Nonomura

Published Jan 15, 1998• 20 citations
Publisher

Abstract

A new efficient scheme for the finite-size-scaling study of ground-state phase transitions in quantum spin systems is proposed on the basis of the quantum Monte Carlo method. That is, Trotter numbers and simulated temperatures are scaled in accordance with system sizes, and only one simulation is required for each system size. The transition point is estimated independently of the critical exponents using the non-equilibrium relaxation of the order parameter in a very large system. This scheme is confirmed by the estimation of the critical phenomenon of the one-dimensional S =1/2 asymmetric XY model.

DOI
10.1143/jpsj.67.5

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Journal of the Physical Society of Japan•Apr 1, 1989•47 citations
A new efficient scheme for the finite-size-scaling study of ground-state phase transitions in quantum spin systems is proposed on the basis of the quantum Monte Carlo method. That is, Trotter numbers and simulated temperatures are scaled in accordance with system sizes, and only one simulation is required for each system size. The transition point is estimated independently of the critical exponents using the non-equilibrium relaxation of the order parameter in a very large system. This scheme is confirmed by the estimation of the critical phenomenon of the one-dimensional S =1/2 asymmetric XY model.
We propose a new efficient scheme for the quantum Monte Carlo study of quantum critical phenomena in quantum spin systems. Rieger and Young's Trotter-number-dependent finite-size scaling in quantum spin systems and Ito et al 's evaluation of the transition point with the non-equilibrium relaxation in classical spin systems are combined and generalized. That is, only one Trotter number and one inverse temperature proportional to system sizes are taken for each system size, and the transition point of the transformed classical spin model is estimated as the point at which the order parameter shows power-law decay. The present scheme is confirmed by the determination of the critical phenomenon of the one-dimensional asymmetric XY model in the ground state. The estimates of the transition point and the critical exponents , and are in good agreement with the exact solutions. The dynamical critical exponent is also estimated as , which is consistent with that of the two-dimensional Ising model.
A generalization to the quantum case of a recently introduced algorithm [Tomita and Okabe, Phys. Rev. Lett. 86, 572 (2001)] for the determination of the critical temperature of classical spin models is proposed. We describe a simple method to automatically locate critical points in (Quantum) Monte Carlo simulations. The algorithm assumes the existence of a finite correlation length in at least one of the two phases surrounding the quantum critical point. We illustrate these ideas on the example of the critical interchain coupling for which coupled antiferromagnetic S=1S=1S=1 spin chains order at T=0.T=0.T=0. Finite-size scaling relations are used to determine the exponents, $\ensuremath{\nu}=0.72(2)$ and $\ensuremath{\eta}=0.038(3)$ in agreement with previous estimates.
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.
Ground-state phase transition of site-diluted Heisenberg antiferromagnets on a square lattice is studied. By using the continuous-time loop algorithm, we perform large-scale quantum Monte Carlo simulation on large systems at quite low temperatures. It is found that the critical concentration of magnetic sites is independent of the spin size S, and equal to the classical percolation threshold. However, the existence of quantum fluctuations makes the critical exponents deviate from those of the classical percolation transition. It is found that the transition is not universal, i.e., the critical exponents depend on the spin size S.
The scaling of the transition temperature into an ordered phase close to a quantum critical point as well as the order parameter fluctuations inside the quantum critical region provide valuable information about universal properties of the underlying quantum critical point. Here, we employ quantum Monte Carlo simulations to examine these relations in detail for two-dimensional quantum systems that exhibit a finite-temperature Ising-transition line in the vicinity of a quantum critical point that belongs to the universality class of either (i) the three-dimensional Ising model for the case of the quantum Ising model in a transverse magnetic field on the square lattice or (ii) the chiral Ising transition for the case of a half-filled system of spinless fermions on the honeycomb lattice with nearest-neighbor repulsion. While the first case allows large-scale simulations to assess the scaling predictions to a high precision in terms of the known values for the critical exponents at the quantum critical point, for the later case, we extract values of the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$, related to the order parameter fluctuations, which we discuss in relation to other recent estimates from ground-state quantum Monte Carlo calculations as well as analytical approaches.
We develop an interface approach to bulk phase transitions and ordered states by using Monte Garlo simulations, and apply it to antiferromagnetic q -state Potts models with q =3∼6 on the simple cubic lattice. A stiffness exponent a defined by Δ F ∼ L a , where Δ F is the interface free energy for a system of size L , is introduced as a measure of the stiffness of the ordered phase against an external stress. Applying finite-size scaling to Δ F and to the squared order parameter enable one to determine each of T c , ν, β and γ in order where T c is the critical temperature and the others are critical exponents. This approach also provides a means to study properties of the ordered state through a and interface profiles. In the q =3 and 4 models the estimated exponents ν, β, γ and a indicate new universality classes. The q =5 model shows a phase transition, while the q =6 model does not. The q =3∼5 models have non-integer values of a at low temperatures. The phase transitions in these models are confirmed to be due to entropy gains.