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Physical Review Letters

Fixed-Node Quantum Monte Carlo Method for Lattice Fermions

H. J. M. van Bemmel, D. F. B. ten Haaf, Wim van Saarloos, J. M. J. van Leeuwen, Guozhong An

Published Apr 11, 1994• 91 citations
Publisher

Abstract

We give a new prescription for performing random walks in configuration space for lattice fermion problems. Imposing a suitable condition for the wave function on nodal boundaries in configuration space enables us to devise a generalization of the fixed-node quantum Monte Carlo method, as it has been developed for continuum problems. It does not suffer from the sign problem and provides upper bounds for the energy of different candidates for the ground state. We present new results for the Hubbard model off half filling as a demonstration of the method.

DOI
10.1103/physrevlett.72.2442

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We give a new prescription for performing random walks in configuration space for lattice fermion problems. Imposing a suitable condition for the wave function on nodal boundaries in configuration space enables us to devise a generalization of the fixed-node quantum Monte Carlo method, as it has been developed for continuum problems. It does not suffer from the sign problem and provides upper bounds for the energy of different candidates for the ground state. We present new results for the Hubbard model off half filling as a demonstration of the method.
The fixed-node Monte Carlo method is extended to lattice fermion models. This method replaces the problem of finding the ground state of a many-fermion system by an effective eigenvalue problem of finding the lowest-energy wave function in a given region of the configuration space. It has previously only been applied to fermions moving in continuous space. The discreteness of the configuration space causes the algorithm to differ from that of the continuum. The method is tested against a known limiting case where exact results are available for comparison. Good agreement is found. Using the fixed-node Monte Carlo method, we study the domain-wall phase in the ground state of the two-dimensional Hubbard model. The existence of a domain-wall phase which has domains of antiferromagnetic phases separated by walls of holes is recently suggested by inhomogeneous Hartree-Fock (HF) and variational Monte Carlo (VMC) calculations. Large improvements of the energies are found. The domain walls are broader than those obtained by the HF and VMC calculations.
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.
A diffusion Monte Carlo algorithm is introduced that can determine the correct nodal structure of the wave function of a few-fermion system and its ground-state energy without an uncontrolled bias. This is achieved by confining signed random walkers to the points of a uniform infinite spatial grid, allowing them to meet and annihilate one another to establish the nodal structure without the fixed-node approximation. An imaginary-time propagator is derived rigorously from a discretized Hamiltonian, governing a non-Gaussian, sign-flipping, branching, and mutually annihilating random walk of particles. The accuracy of the resulting stochastic representations of a fermion wave function is limited only by the grid and imaginary-time resolutions and can be improved in a controlled manner. The method is tested for a series of model problems including fermions in a harmonic trap as well as the He atom in its singlet or triplet ground state. For the latter case, the energies approach from above with increasing grid resolution and converge within 0.015E_{h} of the exact basis-set-limit value for the grid spacing of 0.08 a.u. with a statistical uncertainty of 10^{-5}E_{h} without an importance sampling or Jastrow factor.
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 provide an extension to lattice systems of the reptation quantum Monte Carlo algorithm, originally devised for continuous Hamiltonians. For systems affected by the sign problem, a method to systematically improve upon the so-called fixed-node approximation is also proposed. The generality of the method, which also takes advantage of a canonical worm algorithm scheme to measure off-diagonal observables, makes it applicable to a vast variety of quantum systems and eases the study of their ground-state and excited-state properties. As a case study, we investigate the quantum dynamics of the one-dimensional Heisenberg model and we provide accurate estimates of the ground-state energy of the two-dimensional fermionic Hubbard model.