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arXiv:cond-mat/0509149 · Physical Review B

Symmetry projection schemes for Gaussian Monte Carlo methods

Fakher F. Assaad, Philipp Werner, Philippe Corboz, Emanuel Gull, Matthias Troyer

Published Dec 30, 2005• 30 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.72.224518

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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.
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.
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.
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 benchmark constrained-path Monte Carlo (CPMC) on the triangular-lattice Hubbard model for several fillings and UUU values and show that symmetry-adapted trial wave functions substantially improve quantitative accuracy. Away from half-filling, simple free-electron-based trials that preserve the ground state symmetry yield energy deviations ≲1%\lesssim 1\%≲1% from exact diagonalization and density matrix renormalization group results. At half-filling, strong frustration in the intermediate to large UUU regimes necessitates symmetry-projected trials to reach comparable accuracy, where both free-electron and symmetry-broken Hartree-Fock trials incur substantial constraint bias. Since the computational cost of CPMC with symmetry projection scales polynomially with system size, our results motivate its use as a practical route for studying competing ground states in strongly correlated, frustrated systems.
We describe and discuss a recently proposed quantum Monte Carlo algorithm to compute the ground-state properties of various systems of interacting fermions. In this method, the ground state is projected from an initial wave function by a branching random walk in an overcomplete basis of Slater determinants. By constraining the determinants according to a trial wave function |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉, we remove the exponential decay of signal-to-noise ratio characteristic of the sign problem. The method is variational and is exact if |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉 is exact. We illustrate the method by describing in detail its implementation for the two-dimensional one-band Hubbard model. We show results for lattice sizes up to 16\ifmmode\times\else\texttimes\fi{}16 and for various electron fillings and interaction strengths. With simple single-determinant wave functions as |${\mathrm{\ensuremath{\psi}}}_{\mathrm{T}}$〉, the method yields accurate (often to within a few percent) estimates of the ground-state energy as well as correlation functions, such as those for electron pairing. We conclude by discussing possible extensions of the algorithm.