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arXiv:0904.3100 · Physical Review Letters

Green-Function-Based Monte Carlo Method for Classical Fields Coupled to Fermions

Alexander Weiße

Published Apr 17, 2009• 20 citations
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Abstract

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.

DOI
10.1103/physrevlett.102.150604

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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.
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.
Models of noninteracting fermions coupled to auxiliary classical fields are relevant to the understanding of a wide variety of problems in many-body physics, e.g., the description of manganites, diluted magnetic semiconductors, or strongly interacting electrons on lattices. We present a flat-histogram Monte Carlo algorithm that simulates a statistical ensemble that allows one to directly acquire the partition function at all temperatures for such systems. The defining feature of the algorithm is that it utilizes the complete thermodynamic information from the full energy spectrum of noninteracting fermions available during sampling of the configuration space of the classical fields. We benchmark the method for the classical Ising and Potts models in two dimensions, as well as the Falicov-Kimball model describing itinerant electrons interacting with heavy ions.
Models of fermions interacting with classical degrees of freedom are applied to a large variety of systems in condensed matter physics. For this class of models, Weiße [Phys. Rev. Lett. 102 , 150604 (2009)] has recently proposed a very efficient numerical method, called O( N ) Green-Function-Based Monte Carlo (GFMC) method, where a kernel polynomial expansion technique is used to avoid the full numerical diagonalization of the fermion Hamiltonian matrix of size N , which usually costs O( N 3 ) computational complexity. Motivated by this background, in this paper we apply the GFMC method to the double exchange model in three spatial dimensions. We mainly focus on the implementation of GFMC method using both MPI on a CPU-based cluster and Nvidia's Compute Unified Device Architecture (CUDA) programming techniques on a GPU-based (Graphics Processing Unit based) cluster. The time complexity of the algorithm and the parallel implementation details on the clusters are discussed. We also show the performance scaling for increasing Hamiltonian matrix size and increasing number of nodes, respectively. The performance evaluation indicates that for a 32 3 Hamiltonian a single GPU shows higher performance equivalent to more than 30 CPU cores parallelized using MPI.
We tutorially review the determinantal Quantum Monte Carlo method for fermionic systems, using the Hubbard model as a case study. Starting with the basic ingredients of Monte Carlo simulations for classical systems, we introduce aspects such as importance sampling, sources of errors, and finite-size scaling analyses. We then set up the preliminary steps to prepare for the simulations, showing that they are actually carried out by sampling discrete Hubbard-Stratonovich auxiliary fields. In this process the Green's function emerges as a fundamental tool, since it is used in the updating process, and, at the same time, it is directly related to the quantities probing magnetic, charge, metallic, and superconducting behaviours. We also discuss the as yet unresolved "minus-sign problem", and two ways to stabilize the algorithm at low temperatures.
We tutorially review the determinantal Quantum Monte Carlo method for fermionic systems, using the Hubbard model as a case study. Starting with the basic ingredients of Monte Carlo simulations for classical systems, we introduce aspects such as importance sampling, sources of errors, and finite-size scaling analyses. We then set up the preliminary steps to prepare for the simulations, showing that they are actually carried out by sampling discrete Hubbard-Stratonovich auxiliary fields. In this process the Green's function emerges as a fundamental tool, since it is used in the updating process, and, at the same time, it is directly related to the quantities probing magnetic, charge, metallic, and superconducting behaviours. We also discuss the as yet unresolved ‘minus-sign problem', and two ways to stabilize the algorithm at low temperatures.
We develop a general numerical method to study the zero-temperature properties of strongly correlated electron models on large lattices. The technique, which resembles Green's-function Monte Carlo, projects the ground-state component from a trial wave function with no approximations. We use this method to determine the phase diagram of the two-dimensional $t\ensuremath{-}J$ model, using the Maxwell construction to investigate electronic phase separation. The shell effects of fermions on finite-sized periodic lattices are minimized by keeping the number of electrons fixed at a closed-shell configuration and varying the size of the lattice. Results obtained for various electron numbers corresponding to different closed shells indicate that the finite-size effects in our calculation are small. For any value of interaction strength, we find that there is always a value of the electron density above which the system can lower its energy by forming a two-component phase separated state. Our results are compared with other calculations on the $t\ensuremath{-}J$ model. We find that the most accurate results are consistent with phase separation at all interaction strengths.