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Quantum Monte Carlo Method without Hubbard-Stratonovich Transformation for Many-body Systems

J. Wang, Weiwei Pan, D. Y. Sun

Published Oct 17, 2021• 0 citations
OpenAlex

Abstract

By precisely writing down the matrix element of the local Boltzmann operator ($e^(-{\tau}h$), where h is the paired operator formed by two interacting terms conjugate to each other), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. Our formula presents remarkable outcomes such that: 1) the Hubbard-Stratonovich transformation is not necessary, and the auxiliary field is no longer needed, which can improve the computational efficiency. 2) The new formula improves the accuracy of the Suzuki-Trotter decomposition. Our method also provides a potential theoretical framework for analyzing and solving the fermion sign problem. As an example, we studied the one-dimensional half-filled Hubbard model at a finite temperature. The obtained results were in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm could be used in various studies.

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By precisely writing down the matrix element of the local Boltzmann operator (e(−τhe^(-{\tau}he(−τh), where h is the paired operator formed by two interacting terms conjugate to each other), we have proposed a new path integral formulation for quantum field theory and developed a corresponding Monte Carlo algorithm. Our formula presents remarkable outcomes such that: 1) the Hubbard-Stratonovich transformation is not necessary, and the auxiliary field is no longer needed, which can improve the computational efficiency. 2) The new formula improves the accuracy of the Suzuki-Trotter decomposition. Our method also provides a potential theoretical framework for analyzing and solving the fermion sign problem. As an example, we studied the one-dimensional half-filled Hubbard model at a finite temperature. The obtained results were in excellent agreement with the known solutions. The new formula and Monte Carlo algorithm could be used in various studies.
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.
By precisely writing down the matrix element of the local Boltzmann operator ([Formula: see text], where [Formula: see text] 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 the current formula, the Hubbard-Stratonovich transformation is not necessary, accordingly the determinant calculation is not needed, 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-principles calculations. 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 could be applied to various studies in future.
By precisely writing down the matrix element of the local Boltzmann operator, 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.
Ground-state properties of the Hubbard model on a two-dimensional square lattice are studied by the auxiliary-field quantum Monte Carlo method. Accurate results for energy, double occupancy, effective hopping, magnetization, and momentum distribution are calculated for interaction strengths of U/tU/tU/t from 2 to 8, for a range of densities including half-filling and n=0.3,0.5,0.6,0.75n=0.3,0.5,0.6,\phantom{\rule{0.16em}{0ex}}0.75n=0.3,0.5,0.6,0.75, and 0.8750.8750.875. At half-filling, the results are numerically exact. Away from half-filling, the constrained path Monte Carlo method is employed to control the sign problem. Our results are obtained with several advances in the computational algorithm, which are described in detail. We discuss the advantages of generalized Hartree-Fock trial wave functions and its connection to pairing wave functions, as well as the interplay with different forms of Hubbard-Stratonovich decompositions. We study the use of different twist angle sets when applying the twist averaged boundary conditions. We propose the use of quasirandom sequences, which improves the convergence to the thermodynamic limit over pseudorandom and other sequences. With it and a careful finite size scaling analysis, we are able to obtain accurate values of ground-state properties in the thermodynamic limit. Detailed results for finite-sized systems up to $16\ifmmode\times\else\texttimes\fi{}16$ are also provided for benchmark purposes.
The fermion sign problem, the biggest obstacle in quantum Monte Carlo calculations, is completely solved in this paper. Here, we find a strategy, in which the contribution from those negative-weighted paths is thoroughly cancelled or replaced by some positive-weighted paths. The crucial point lies on the Feynman path integral formula proposed in our group, which allows us to deeply analyze the Boltzmann weight of each path. Through mathematical proof, we demonstrate that physical quantities can be exactly calculated within a specific kind of paths, which have positive the Boltzmann weight. With this finding, a new Monte Carlo method is proposed, in which the fermion sign problem is absent. As an example, the current method is applied to the two-dimensional Hubbard model, and the results do manifest the correctness.
We present a unified theory of the variational Monte Carlo (VMC) and determinant quantum Monte Carlo (DQMC) methods using a novel density matrix formulation of VMC. We introduce an efficient algorithm for VMC to compute correlation functions and expectation values based on the auxiliary field Hirsch-Hubbard-Stratonovic transformation. We show that this new approach to VMC converges significantly faster than its traditional implementations. Furthermore, we generalize the Trotter-Suzuki decomposition to finite imaginary time steps τ∼O(1)τ\sim O(1)τ∼O(1) and develop a variational quantum Monte Carlo (VQMC) method accordingly, which is more accurate than VMC and can incorporate quantum fluctuations more efficiently. The two extreme limits of the VQMC method, namely infinitesimal and infinite imaginary time steps, correspond to the DQMC and VMC techniques, respectively. We demonstrate that our VQMC allows us to access lower temperatures in comparison with the conventional DQMC before the sign problem comes into play. We finally show that our VQMC can also enhance the accuracy of the projector Monte Carlo methods by providing better and less biased candidates for the trial wave functions, requiring shorter projection times for a given accuracy and alleviating the sign problem further.