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AIP conference proceedings

Gaussian Quantum Monte Carlo methods with symmetry projection

Fakher F. Assaad

Published Jan 1, 2006• 3 citations
Publisher

Abstract

Recently Corney and Drummond introduced the sign‐free Gaussian Quantum Monte Carlo method (GQMC) for the Hubbard model. We generalize their ideas to derive stochastic differential equations (SDEs) for the general electronic structure problem. We show that the method fails to reproduce the correct symmetries of the ground state leading to systematic errors at low temperatures. We propose to restore these symmetries a posteriori by projecting the resulting density matrix onto the symmetry sectors of the ground state. The so produced results are accurate and reliable, even for parameters where an auxiliary field QMC approach fails due to a severe sign problem. Possible sources for the systematic errors are discussed.

DOI
10.1063/1.2178044

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Recently Corney and Drummond introduced the sign‐free Gaussian Quantum Monte Carlo method (GQMC) for the Hubbard model. We generalize their ideas to derive stochastic differential equations (SDEs) for the general electronic structure problem. We show that the method fails to reproduce the correct symmetries of the ground state leading to systematic errors at low temperatures. We propose to restore these symmetries a posteriori by projecting the resulting density matrix onto the symmetry sectors of the ground state. The so produced results are accurate and reliable, even for parameters where an auxiliary field QMC approach fails due to a severe sign problem. Possible sources for the systematic errors are discussed.
Gaussian quantum Monte Carlo (GQMC) is a stochastic phase space method for fermions with positive weights. In the example of the Hubbard model close to half filling it fails to reproduce all the symmetries of the ground state leading to systematic errors at low temperatures. In a previous work [Phys. Rev. B 72, 224518 (2005)] we proposed to restore the broken symmetries by projecting the density matrix obtained from the simulation onto the ground state symmetry sector. For ground state properties, the accuracy of this method depends on a large overlap between the GQMC and exact density matrices. Thus, the method is not rigorously exact. We present the limits of the approach by a systematic study of the method for two and three leg Hubbard ladders for different fillings and on-site repulsion strengths. We show several indications that the systematic errors stem from nonvanishing boundary terms in the partial integration step in the derivation of the Fokker-Planck equation. Checking for spiking trajectories and slow decaying probability distributions provides an important test of the reliability of the results. Possible solutions to avoid boundary terms are discussed. Furthermore we compare results obtained from two different sampling methods: Reconfiguration of walkers and the Metropolis algorithm.
A sign-free Monte Carlo method for the Hubbard model has recently been proposed by Corney and Drummond. High-precision measurements on small clusters show that ground-state correlation functions are not correctly reproduced. We argue that the origin of this mismatch lies in the fact that the low-temperature density matrix does not have the symmetries of the Hamiltonian. Here we show that supplementing the algorithm with symmetry projection schemes provides reliable and accurate estimates of ground-state properties.
We examine Gaussian-basis Monte Carlo method (GBMC) introduced by Corney and Drummond. This method is based on an expansion of the density-matrix operator rho by means of the coherent Gaussian-type operator basis Lambda and does not suffer from the minus sign problem. The original method, however, often fails in reproducing the true ground state and causes systematic errors of calculated physical quantities because the samples are often trapped in some metastable or symmetry broken states. To overcome this difficulty, we combine the quantum-number projection scheme proposed by Assaad, Werner, Corboz, Gull and Troyer in conjunction with the importance sampling of the original GBMC method. This improvement allows us to carry out the importance sampling in the quantum-number-projected phase-space. Some comparisons with the previous quantum-number projection scheme indicate that, in our method, the convergence with the ground state is accelerated, which makes it possible to extend the applicability and widen the range of tractable parameters in the GBMC method. The present scheme offers an efficient practical way of computation for strongly correlated electron systems beyond the range of system sizes, interaction strengths and lattice structures tractable by other computational methods such as the quantum Monte Carlo method.
We show how symmetry properties can be used to greatly increase the accuracy and efficiency in auxiliary-field quantum Monte Carlo (AFQMC) calculations of electronic systems. With the Hubbard model as an example, we study symmetry preservation in two aspects of ground-state AFQMC calculations, the Hubbard-Stratonovich transformation and the form of the trial wave function. It is shown that significant improvement over state-of-the-art calculations can be achieved. In unconstrained calculations, the implementation of symmetry often leads to shorter convergence time and much smaller statistical errors and thereby a substantial reduction of the sign problem. Moreover, certain excited states become possible to calculate which are otherwise beyond reach. In calculations with constraints, the use of symmetry can reduce the systematic error from the constraint. It also allows release-constraint calculations, leading to essentially exact results in many cases. Detailed comparisons are made with exact diagonalization results. Accurate ground-state energies are then presented for larger system sizes in the two-dimensional repulsive Hubbard model.
We present in detail the recently developed multiconfigurational symmetrized projector quantum Monte Carlo method for excited states of the Hubbard model. We describe the implementation of the Monte Carlo method for a multiconfigurational trial wave function. We give a detailed discussion of issues related to the symmetry of the projection procedure that validates our Monte Carlo procedure for excited states. In this context we discuss various averaging procedures for the Green function and present an analysis of the errors incurred in these procedures. We study the ground-state energy and correlation functions of the one-dimensional Hubbard model at half-filling to confirm these analyses. We then study the energies and correlation functions of excited states of Hubbard chains. Hubbard rings away from half-filling are also studied and the pair binding energies for holes of 4n4n4n and 4n+24n+24n+2 systems are compared with the Bethe ansatz results of Fye, Martins, and Scalettar [Phys. Rev. B 42, 6809 (1990)]. Our study of the two-dimensional Hubbard model includes the $4\ifmmode\times\else\texttimes\fi{}2$ ladder and the $3\ifmmode\times\else\texttimes\fi{}4$ lattice with periodic boundary conditions. The $3\ifmmode\times\else\texttimes\fi{}4$ lattice is nonbipartite and amenable to exact diagonalization studies and is, therefore, a good candidate for checks on the method. We are able to reproduce accurately the energies of ground and excited states, both at and away from half-filling. We study the properties of the $4\ifmmode\times\else\texttimes\fi{}2$ Hubbard ladder with bond alternation as the correlation strength and filling are varied. The method reproduces the correlation functions accurately. We also examine the severity of sign problem for one- and two-dimensional systems.
Reliable simulations of correlated quantum systems, including high-temperature superconductors and frustrated magnets, are increasingly desired nowadays to further our understanding of essential features in such systems. Quantum Monte Carlo (QMC) is a unique numerically exact and intrinsically unbiased method to simulate interacting quantum many-body systems. More importantly, when QMC simulations are free from the notorious fermion sign problem, they can reliably simulate interacting quantum models with large system size and low temperature to reveal low-energy physics such as spontaneously broken symmetries and universal quantum critical behaviors. Here, we concisely review recent progress made in developing new sign-problem-free QMC algorithms, including those employing Majorana representation and those utilizing hot-spot physics. We also discuss applications of these novel sign-problem-free QMC algorithms in simulations of various interesting quantum many-body models. Finally, we discuss possible future directions of designing sign-problem-free QMC methods.