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 $O(N_{\rm dim}^{2} \log N_{\rm dim})$ compared to $O(N_{\rm dim}^{3})$ for the diagonalization in the conventional technique; $N_{\rm dim}$ 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 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 $O(N_{\rm dim}^{2} \log N_{\rm dim})$ compared to $O(N_{\rm dim}^{3})$ for the diagonalization in the conventional technique; $N_{\rm dim}$ 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.

arXiv:cond-mat/0007407Journal of the Physical Society of JapanDec 15, 199972 citations

Scattering theory on the lattice and with a Monte Carlo method

H. Kröger, Kevin J. M. Moriarty, JEAN POTVIN

We present an alternative time-dependent method of calculating the S matrix in quantum systems governed by a Hamiltonian. In the first step one constructs a new Hamiltonian that describes the physics of scattering at energy E with a reduced number of degrees of freedom. Its matrix elements are computed with a Monte Carlo projector method. In the second step the scattering matrix is computed algebraically via diagonalization and exponentiation of the new Hamiltonian. Although we have in mind applications in many-body systems and quantum field theory, the method should be applicable and useful in such diverse areas as atomic and molecular physics, nuclear physics, high-energy physics and solid-state physics. As an illustration of the method, we compute s-wave scattering of two nucleons in a nonrelativistic potential model (Yamaguchi potential), for which the S matrix is known exactly.We present an alternative time-dependent method of calculating the S matrix in quantum systems governed by a Hamiltonian. In the first step one constructs a new Hamiltonian that describes the physics of scattering at energy E with a reduced number of degrees of freedom. Its matrix elements are computed with a Monte Carlo projector method. In the second step the scattering matrix is computed algebraically via diagonalization and exponentiation of the new Hamiltonian. Although we have in mind applications in many-body systems and quantum field theory, the method should be applicable and useful in such diverse areas as atomic and molecular physics, nuclear physics, high-energy physics and solid-state physics. As an illustration of the method, we compute s-wave scattering of two nucleons in a nonrelativistic potential model (Yamaguchi potential), for which the S matrix is known exactly.

Physical Review ASep 1, 19904 citations

On a full Monte Carlo approach to quantum mechanics

Jean Michel Sellier, Ivan Dimov

Physica A Statistical Mechanics and its ApplicationsJul 17, 201610 citations

A study of parallelizing O(N) Green-function-based Monte Carlo method for many fermions coupled with classical degrees of freedom

Shixun Zhang, Shinichi Yamagia, Seiji Yunoki

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.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.

arXiv:1303.6016Journal of Physics Conference SeriesAug 12, 20133 citations

Monte Carlo Simulations in Quantum Mechanics

Andreas Wipf

Lecture notes in physicsJan 1, 20210 citations

O(N_{\rm dim}^{2} \log N_{\rm dim})

compared to

O(N_{\rm dim}^{3})

for the diagonalization in the conventional technique;

N_{\rm dim}

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.