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Journal of Physics Conference Series

Generalized Quantum Monte Carlo Algorithm in Eigen Basis

V. А. Kashurnikov, Andrey V. Krasavin, Yaroslav Zhumagulov, Petr F. Kartsev

Published Nov 1, 2019• 0 citations
Publisher

Abstract

Abstract New generalized continuous-time quantum Monte Carlo algorithm is developed. The algorithm is absolutely universal in the sense that it allows to calculate models specified in an arbitrary basis (in the basis of occupation numbers; in the momentum representation; in the basis of the eigenfunctions of small clusters into which the entire system is divided, or in any other basis): the algorithm requires only the specification of the matrix elements between the basis functions defined in a convenient basis. The implemented algorithm is applied to one-dimensional Hubbard model.

DOI
10.1088/1742-6596/1389/1/012080

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Abstract New generalized continuous-time quantum Monte Carlo algorithm is developed. The algorithm is absolutely universal in the sense that it allows to calculate models specified in an arbitrary basis (in the basis of occupation numbers; in the momentum representation; in the basis of the eigenfunctions of small clusters into which the entire system is divided, or in any other basis): the algorithm requires only the specification of the matrix elements between the basis functions defined in a convenient basis. The implemented algorithm is applied to one-dimensional Hubbard model.
We propose a quantum Monte Carlo algorithm capable of simulating the Bose-Hubbard model on arbitrary graphs, obviating the need for devising lattice-specific updates for different input graphs. We show that with our method, which is based on the recently introduced permutation matrix representation quantum Monte Carlo [Gupta, Albash, and Hen, J. Stat. Mech. (2020) 073105], the problem of adapting the simulation to a given geometry amounts to generating a cycle basis for the graph on which the model is defined, a procedure that can be carried out efficiently and in an automated manner. To showcase the versatility of our approach, we provide simulation results for Bose-Hubbard models defined on two-dimensional lattices as well as on a number of random graphs.
We present a numerically exact continuous-time quantum Monte Carlo algorithm for fermions with a general interaction nonlocal in space-time. The new determinantal grand-canonical scheme is based on a stochastic series expansion for the partition function in the interaction representation. The method is particularly applicable for multiband, time-dependent correlations since it does not invoke the Hubbard-Stratonovich transformation. The test calculations for exactly solvable models, as well results for the Green function and for the time-dependent susceptibility of the multiband supersymmetric model with a spin-flip interaction are discussed.
Abstract By precisely writing down the matrix element of the local Boltzmann operator (e-th, where h is the Hermitian conjugate pairs of off-diagonal operators), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. With current formula, the Hubbard–Stratonovich transformation is not necessary, and is not based on the determinant approach, which can improve the computational efficiency. The results show that, the simulation time has the square-law scaling with system sizes, which is comparable with the usual first-principle calculation. The current formula also improves the accuracy of the Suzuki–Trotter decomposition. As an example, we have studied the one-dimensional half-filled Hubbard model at finite temperature. The obtained results are in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm can be used in various studies.
Continuous-time determinantal algorithm is proposed for the quantum Monte Carlo simulation of the interacting fermions. The scheme does not invoke Hubbard-Stratonovich transformation. The fermionic action is divided into two parts. One of them contains the interaction and certain additional terms; another one is purely Gaussian. The first part is considered as a perturbation. Terms of the series expansion for the partition function are generated in a random walk process. The sign problem and the complexity of the algorithm are analyzed. We argue that the scheme should be useful particularly for the systems with non-local interaction.