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arXiv:cond-mat/0603402 · International Journal of Modern Physics B

MONTE CARLO SIMULATION OF BOSON LATTICES

VESA APAJA, Olav F. Syljuåsen

Published Dec 15, 2006• 1 citation
arXivHTMLPDF

Abstract

Boson lattices are theoretically well described by the Hubbard model. The basic model and its variants can be effectively simulated using Monte Carlo techniques. We describe two newly developed approaches, the Stochastic Series Expansion (SSE) with directed loop updates and continuous–time Diffusion Monte Carlo (CTDMC). SSE is a formulation of the finite temperature partition function as a stochastic sampling over product terms. Directed loops is a general framework to implement this stochastic sampling in a non–local fashion while maintaining detailed balance. CTDMC is well suited to finding exact ground–state properties, applicable to any lattice model not suffering from the sign problem; for a lattice model the evolution of the wave function can be performed in continuous time without any time discretization error. Both the directed loop algorithm and the CTDMC are important recent advances in development of computational methods. Here we present results for a Hubbard model for anti–ferromagnetic spin–1 bosons in one dimensions, and show evidence for a dimerized ground state in the lowest Mott lobe.

DOI
10.1142/s0217979206036168

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Boson lattices are theoretically well described by the Hubbard model. The basic model and its variants can be effectively simulated using Monte Carlo techniques. We describe two newly developed approaches, the Stochastic Series Expansion (SSE) with directed loop updates and continuous–time Diffusion Monte Carlo (CTDMC). SSE is a formulation of the finite temperature partition function as a stochastic sampling over product terms. Directed loops is a general framework to implement this stochastic sampling in a non–local fashion while maintaining detailed balance. CTDMC is well suited to finding exact ground–state properties, applicable to any lattice model not suffering from the sign problem; for a lattice model the evolution of the wave function can be performed in continuous time without any time discretization error. Both the directed loop algorithm and the CTDMC are important recent advances in development of computational methods. Here we present results for a Hubbard model for anti–ferromagnetic spin–1 bosons in one dimensions, and show evidence for a dimerized ground state in the lowest Mott lobe.
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.
The directed-loop quantum Monte Carlo method is generalized to the case of retarded interactions. Using the path integral, fermion-boson or spin-boson models are mapped to actions with retarded interactions by analytically integrating out the bosons. This yields an exact algorithm that combines the highly efficient loop updates available in the stochastic series expansion representation with the advantages of avoiding a direct sampling of the bosons. The application to electron-phonon models reveals that the method overcomes the previously detrimental issues of long autocorrelation times and exponentially decreasing acceptance rates. For example, the resulting dramatic speedup allows us to investigate the Peierls quantum phase transition on chains of up to 1282 sites.
A review of the loop algorithm , its generalizations, and its relation to some other Monte Carlo techniques is given. The loop algorithm is a quantum Monte Carlo procedure that employs non-local changes of worldline configurations, determined by local stochastic decisions. It is based on a formulation of quantum models of any dimension in an extended ensemble of worldlines and graphs, and is related to Swendsen-Wang algorithms. It can be represented directly on an operator level, both with a continuous imaginary time path integral and with the stochastic series expansion. It overcomes many of the difficulties of traditional worldline simulations. Autocorrelations are reduced by orders of magnitude. Grand-canonical ensembles, off-diagonal operators, and variance reduced estimators are accessible. In some cases, infinite systems can be simulated. For a restricted class of models, the fermion sign problem can be overcome. Transverse magnetic fields are handled efficiently, in contrast to strong diagonal fields. The method has been applied successfully to a variety of models for spin and charge degrees of freedom, including Heisenberg and XYZ spin models, hard-core bosons, Hubbard and t - J -models. Owing to the improved efficiency, precise calculations of asymptotic behaviour and of quantum critical exponents have been possible.
On the basis of a Feynman–Kac-type formula involving Poisson stochastic processes, a Monte Carlo algorithm has recently been introduced, which describes exactly the real- or imaginary-time evolution of many-body lattice quantum systems. We extend this algorithm to the exact simulation of time-dependent correlation functions. The techniques generally employed in Monte Carlo simulations to control fluctuations, namely reconfigurations and importance sampling, are adapted to the present algorithm and their validity is rigorously proved. We complete the analysis by several examples for the hard-core boson Hubbard model and for the Heisenberg model.
We introduce the concept of directed loops in stochastic series expansion and path-integral quantum Monte Carlo methods. Using the detailed balance rules for directed loops, we show that it is possible to smoothly connect generally applicable simulation schemes (in which it is necessary to include backtracking processes in the loop construction) to more restricted loop algorithms that can be constructed only for a limited range of Hamiltonians (where backtracking can be avoided). The "algorithmic discontinuities" between general and special points (or regions) in parameter space can hence be eliminated. As a specific example, we consider the anisotropic S=1/2 Heisenberg antiferromagnet in an external magnetic field. We show that directed-loop simulations are very efficient for the full range of magnetic fields (zero to the saturation point) and anisotropies. In particular, for weak fields and anisotropies, the autocorrelations are significantly reduced relative to those of previous approaches. The back-tracking probability vanishes continuously as the isotropic Heisenberg point is approached. For the XY model, we show that back tracking can be avoided for all fields extending up to the saturation field. The method is hence particularly efficient in this case. We use directed-loop simulations to study the magnetization process in the two-dimensional Heisenberg model at very low temperatures. For LxL lattices with L up to 64, we utilize the step structure in the magnetization curve to extract gaps between different spin sectors. Finite-size scaling of the gaps gives an accurate estimate of the transverse susceptibility in the thermodynamic limit: chi( perpendicular )=0.0659+/-0.0002.