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arXiv:cond-mat/0308298 · Journal of the Physical Society of Japan

OrderNMonte Carlo Algorithm for Fermion Systems Coupled with Fluctuating Adiabatical Fields

Nobuo Furukawa, Yukitoshi Motome

Published Jun 15, 2004• 42 citations
arXivHTMLPDF

Abstract

An improved algorithm is proposed for Monte Carlo methods to study fermion systems interacting with adiabatical fields. To obtain a weight for each Monte Carlo sample with a fixed configuration of adiabatical fields, a series expansion using Chebyshev polynomials is applied. By introducing truncations of matrix operations in a systematic and controlled way, it is shown that the cpu time is reduced from O(N^3) to O(N) where N is the system size. Benchmark results show that the implementation of the algorithm makes it possible to perform systematic investigations of critical phenomena using system-size scalings even for an electronic model in three dimensions, within a realistic cpu timescale.

DOI
10.1143/jpsj.73.1482

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An improved algorithm is proposed for Monte Carlo methods to study fermion systems interacting with adiabatical fields. To obtain a weight for each Monte Carlo sample with a fixed configuration of adiabatical fields, a series expansion using Chebyshev polynomials is applied. By introducing truncations of matrix operations in a systematic and controlled way, it is shown that the cpu time is reduced from O(N^3) to O(N) where N is the system size. Benchmark results show that the implementation of the algorithm makes it possible to perform systematic investigations of critical phenomena using system-size scalings even for an electronic model in three dimensions, within a realistic cpu timescale.
A new Monte Carlo method is proposed for fermion systems interacting with classical degrees of freedom. To obtain a weight for each Monte Carlo sample with a fixed configuration of classical variables, the moment expansion of the density of states by Chebyshev polynomials is applied instead of the direct diagonalization of the fermion Hamiltonian. This reduces a cpu time to scale as O(Ndim2log⁡Ndim)O(N_{\rm dim}^{2} \log N_{\rm dim})O(Ndim2​logNdim​) compared to O(Ndim3)O(N_{\rm dim}^{3})O(Ndim3​) for the diagonalization in the conventional technique; NdimN_{\rm dim}Ndim​ is the dimension of the Hamiltonian. Another advantage of this method is that parallel computation with high efficiency is possible. These significantly save total cpu times of Monte Carlo calculations because the calculation of a Monte Carlo weight is the bottleneck part. The method is applied to the double-exchange model as an example. The benchmark results show that it is possible to make a systematic investigation using a system-size scaling even in three dimensions within a realistic cpu timescale.
A new algorithm for Monte Carlo calculation of the double exchange model is studied. The algorithm is commonly applicable to wide classes of strongly correlated electron systems which involve itinerant electrons coupled with thermodynamically fluctuating fields. Using moment expansions of the density of states with Chebyshev polynomials, the algorithm provides an efficient calculation on large size clusters, especially on parallel computers. Benchmark calculations are performed on Beowulf-type cluster systems with over 100 CPUs in parallel. PACS code: 02.70.Lq 71.10.Fd 75.10.-b 75.70.Pa
Microscopic models of classical degrees of freedom coupled to noninteracting fermions occur in many different contexts. Prominent examples from solid state physics are descriptions of colossal magnetoresistance manganites and diluted magnetic semiconductors, or auxiliary field methods for correlated electron systems. Monte Carlo simulations are vital for an understanding of such systems, but notorious for requiring the solution of the fermion problem with each change in the classical field configuration. We present an efficient, truncation-free O(N) method on the basis of Chebyshev expanded local Green functions, which allows us to simulate systems of unprecedented size N.
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.
We introduce regular series expansion for weakly and moderately correlated fermionic systems based on the fluctuating local field approach. The method relies on explicitly accounting for the leading fluctuating mode(s) and is therefore suitable for medium-sized lattices. It can also be used as a solver for the cluster approximations for infinite-size lattices. We introduce a classical fluctuating field coupled to a fermionic collective mode(s). In this way, fluctuations in selected modes are treated in a nonperturbative way. Other degrees of freedom are accounted for in the diagram expansion performed for each value of the fluctuating field. The method is benchmarked for the U/t=1U/t=1U/t=1 and U/t=2U/t=2U/t=2 Hubbard lattices at half filling. Results for susceptibility in the antiferromagnetic channel along with the spectral function are compared with the numerically exact quantum Monte Carlo data. Calculations up to the third order of the series expansion are performed and show a uniform convergence to the reference result for the susceptibility. This convergence is observed well below the dynamical mean-field theory (DMFT) Ne\'el temperature and makes our practically simple method applicable in a much wider temperature range than DMFT-based diagrammatic schemes.
We develop an efficient Monte Carlo algorithm, which accelerates slow Monte Carlo dynamics in quasi-one-dimensional Ising spin systems. The loop algorithm of the quantum Monte Carlo method is applied to the classical spin models with highly anisotropic exchange interactions. Both correlation time and real CPU time are reduced drastically. The algorithm is demonstrated in the layered triangular-lattice antiferromagnetic Ising model. We have obtained the relation between the transition temperature and the exchange interaction parameters, which modifies the result of the chain-mean-field theory.