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Physical Review Letters

Monte Carlo Evaluation of Functional Integrals Using Coherent-State Slater Determinants

Y. Avishai, J. Richert

Published Apr 18, 1983• 9 citations
Publisher

Abstract

The feasibility of using Monte Carlo methods in finite fermion systems has been investigated. The basic tool is a (recently suggested) functional integral representation of $Z=\mathrm{tr}[\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}II)]$ using real coherent states of Slater determinants. Conditions for using the discretized ($N$-point lattice) version as a starting point for Monte Carlo integrations are studied. The limit of large $\ensuremath{\beta}$ is achieved only with large $\ensuremath{\surd}N$. For a two-fermion system numerical calculations of the ground-state energy agree farily well with the exact value.

DOI
10.1103/physrevlett.50.1175

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Comment on "Monte Carlo Evaluation of Functional Integrals Using Coherent-State Slater Determinants"

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In a recent letter [1], Avishai and Richert proposed a Monte Carlo evaluation of many-fermion functional integrals using real coherent-state Slater determinants.

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The feasibility of using Monte Carlo methods in finite fermion systems has been investigated. The basic tool is a (recently suggested) functional integral representation of Z=tr[exp(\ensuremath−\ensuremathβII)]Z=\mathrm{tr}[\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}II)]Z=tr[exp(\ensuremath−\ensuremathβII)] using real coherent states of Slater determinants. Conditions for using the discretized (NNN-point lattice) version as a starting point for Monte Carlo integrations are studied. The limit of large \ensuremathβ\ensuremath{\beta}\ensuremathβ is achieved only with large \ensuremath√N\ensuremath{\surd}N\ensuremath√N. For a two-fermion system numerical calculations of the ground-state energy agree farily well with the exact value.
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.
We develop a general numerical method to study the zero-temperature properties of strongly correlated electron models on large lattices. The technique, which resembles Green's-function Monte Carlo, projects the ground-state component from a trial wave function with no approximations. We use this method to determine the phase diagram of the two-dimensional $t\ensuremath{-}J$ model, using the Maxwell construction to investigate electronic phase separation. The shell effects of fermions on finite-sized periodic lattices are minimized by keeping the number of electrons fixed at a closed-shell configuration and varying the size of the lattice. Results obtained for various electron numbers corresponding to different closed shells indicate that the finite-size effects in our calculation are small. For any value of interaction strength, we find that there is always a value of the electron density above which the system can lower its energy by forming a two-component phase separated state. Our results are compared with other calculations on the $t\ensuremath{-}J$ model. We find that the most accurate results are consistent with phase separation at all interaction strengths.
Fermion quantum Monte Carlo (QMC) methods that work in a general basis space may be more effective for some problems than the traditional diffusion QMC method. Projection of the ground state energy is achieved by random walks in the space of Slater determinants whose one‐particle orbitals are expressed in terms of the chosen basis set. A complication is that the determinants will in general acquire complex phases. This a consequence of ground state Monte Carlo projection using the Hubbard‐Stratonovich transformation of the two‐body interaction. To control the resulting “sign” decay, we describe a method we have recently introduced for the propagation of phaseless determinants. The approximation relies on importance sampling with a trial wave function. The approximation has features in common with diffusion MC fixed node and lattice‐model constrained path methods. Using a planewave basis and non‐local pseudopotentials, we apply the method to Si atom, dimer, and 2, 16, 54 atom (216 electrons) bulk supercells. Single Slater determinant wave functions from density functional theory calculations were used as |ΨT〉 with no additional optimization. The calculated binding energy of Si2 and cohesive energy of bulk Si are in excellent agreement with experiments and are comparable to the best existing theoretical results.
We propose a new quantum Monte Carlo algorithm to compute fermion ground-state properties. The ground state is projected from an initial wave function by a branching random walk in an over-complete basis space of Slater determinants. By constraining the determinants according to a trial wave function $|{\ensuremath{\Psi}}_{T}〉$, we remove the exponential decay of signal-to-noise ratio characteristic of the sign problem. The method is variational and is exact if $|{\ensuremath{\Psi}}_{T}〉$ is exact. We report results on the two-dimensional Hubbard model up to size $16\ifmmode\times\else\texttimes\fi{}16$, for various electron fillings and interaction strengths.
An algorithm is developed for determining the exact ground state properties of quantum many-body systems which is equally applicable to bosons and fermions. The Schroedinger eigenvalue equation for the ground state energy is recast into the form of a many-dimensional integral through the use of the Hubbard-Stratonovitch representation of the imaginary time many- body evolution operator. The resulting functional integral is then evaluated stochastically. The algorithm is tested for an exactly soluble boson system and is then extended to include fermions and repulsive potentials. Importance sampling is crucial to the success of the method, particularly for more complex systems. Improved computational efficiency is attained by performing the calculations in momentum space.