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arXiv:cond-mat/0310465 · Physical Review B

Stochastic cluster series expansion for quantum spin systems

Kim Louis, Claudius Gros

Published Sep 30, 2004• 40 citations
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Abstract

In this paper we develop the stochastic cluster series expansion (SCSE) which is a (cluster) variant of the stochastic series expansion (SSE) method. For certain systems with longer-range interactions the SCSE is considerably more efficient than the standard implementation of the SSE, at low temperatures. As an application of this method we calculated the $T=0$ conductance for a linear Heisenberg chain with a (diagonal) next-nearest-neighbor interaction.

DOI
10.1103/physrevb.70.100410

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In this paper we develop the stochastic cluster series expansion (SCSE) which is a (cluster) variant of the stochastic series expansion (SSE) method. For certain systems with longer-range interactions the SCSE is considerably more efficient than the standard implementation of the SSE, at low temperatures. As an application of this method we calculated the T=0T=0T=0 conductance for a linear Heisenberg chain with a (diagonal) next-nearest-neighbor interaction.
Recently, the stochastic series expansion (SSE) has been proposed as a powerful Monte Carlo method, which allows simulations at low T for quantum-spin systems. We show that the SSE allows us to compute the magnetic conductance for various one-dimensional spin systems without further approximations. We consider various modifications of the anisotropic Heisenberg chain. We recover the Kane-Fisher scaling for one impurity in a Luttinger liquid and study the influence of noninteracting leads for the conductance of an interacting system.
The Stochastic Series Expansion (SSE) technique is a quantum Monte Carlo method that is especially efficient for many quantum spin systems and boson models. It was the first generic method free from the discretization errors affecting previous path integral based approaches. These lecture notes give a brief overview of the SSE method and its applications. In the introductory section, the representation of quantum statistical mechanics by the power series expansion of e−βH{\rm e}^{-βH}e−βH will be compared with path integrals in discrete and continuous imaginary time. Extensions of the SSE approach to ground state projection and quantum annealing in imaginary time will also be briefly discussed. The later sections introduce efficient sampling schemes (loop and cluster updates) that have been developed for many classes of models. A summary of generic forms of estimators for important observables are also given. Applications are discussed in the last section.
The stochastic series expansion (SSE) algorithm is one of the most powerful quantum Monte Carlo methods and has been extensively applied to the study of quantum many body systems. Its efficiency is particularly enhanced with a deterministic loop update scheme in the study of the S=1/2 quantum spin systems that preserve SU(2) spin rotational symmetry. Once the symmetry is broken, such as by an external field, a directed loop method is typically required, resulting in a significant reduction in efficiency. Inspired by the SSE approach developed for the quantum Ising model, we introduce a deterministic loop SSE method that is particularly suited for antiferromagnetic systems under a staggered magnetic field. This method enables separate investigations of longitudinal and transverse modes in magnetically ordered phases arising from spontaneous symmetry breaking. We benchmark the performance of our algorithm against the standard directed loop approach applied to the antiferromagnetic Heisenberg chain and demonstrate that our method substantially reduces CPU time per Monte Carlo step, thereby can outperform the directed loop algorithm in efficiency.
A quantum implementation of the Stochastic Series Expansion (SSE) Monte Carlo method is proposed, and it is shown that quantum SSE offers significant advantages over classical implementations of SSE. In particular, for problems where classical SSE encounters the sign problem, the cost of implementing a Monte Carlo iteration scales only linearly with system size in quantum SSE, while it may scale exponentially with system size in classical SSE. In cases where classical SSE can be efficiently implemented, quantum SSE still offers an advantage by allowing for more general observables to be measured.
We propose a universal stochastic series expansion (SSE) method for the simulation of the Heisenberg model with arbitrary spin and the Bose-Hubbard model with interaction. We report the calculations involving soft-core bosons with interaction by the SSE method. Moreover, we develop a simple procedure for increased efficiency of the algorithm. From calculation of integrated autocorrelation times we conclude that the method is efficient for both models and essentially eliminates the critical slowing down problem.
We investigate the Stochastic Series Expansion (SSE) technique, guided by Sandvik's algorithm of the quantum Monte Carlo simulation for spin-1/2 antiferromagnetic Heisenberg linear chain. The impact of the Monte Carlo steps, lattice sizes, and various temperatures on the variability and fidelity of simulation outcomes are studied. The findings show that the Monte Carlo steps where measurements are made (Msteps), and lattice size are crucial for stabilizing the results. Our results show that for larger lattice sizes, such as L=1024, achieving reliable outcomes necessitates extensive Monte Carlo simulations, specifically. The convergence in the properties is achieved with a high number of Monte Carlo steps starting from 7×10^4. Strong fluctuations are calculated between the results for each magnetic property and the existence of negative and positive in the specific heat at low temperatures.