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Physical Review E

Green’s function Monte Carlo method with exact imaginary-time propagation

Kevin E. Schmidt, Parhat Niyaz, Amanda L. Vaught, Michael A. Lee

Published Jan 1, 2005• 44 citations
Publisher

Abstract

We present a general formulation of the Green's function Monte Carlo method in imaginary-time quantum Monte Carlo which employs exact propagators. This algorithm has no time-step errors and is obtained by minimal modifications of the time-independent Green's function Monte Carlo method. We describe how the method can be applied to the many-body Schrödinger equation, lattice Hamiltonians, and simple field theories. Our modification of the Green's function Monte Carlo algorithm is applied to the ground state of liquid 4He. We calculate the zero-temperature imaginary-time diffusion constant and relate that to the effective mass of a mass-four "impurity" atom in liquid 4He.

DOI
10.1103/physreve.71.016707

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We present a general formulation of the Green's function Monte Carlo method in imaginary-time quantum Monte Carlo which employs exact propagators. This algorithm has no time-step errors and is obtained by minimal modifications of the time-independent Green's function Monte Carlo method. We describe how the method can be applied to the many-body Schrödinger equation, lattice Hamiltonians, and simple field theories. Our modification of the Green's function Monte Carlo algorithm is applied to the ground state of liquid 4He. We calculate the zero-temperature imaginary-time diffusion constant and relate that to the effective mass of a mass-four "impurity" atom in liquid 4He.
A novel scheme to solve the quantum eigenvalue problem through the imaginary-time Green function Monte Carlo method is presented. This method is applicable not only to the ground states, but also to the excited states of generic system. We demonstrate the actual utility of the method with the numerical examples on three simple systems. Typeset using REVTEX 1 The stochastic quantization has been successfully applied in recent years to the various problems of physics ranging from quantum chromo dynamics to the plasmon oscillation [1-3]. While it can handle very complex systems which is usually beyond the approach either by the direct diagonalization or by the perturbation, it has been only applicable to the calculation of the ground state properties of the system. In this paper, we propose a new method based on the imaginary-time Green function Monte Carlo approach, which is extendable to the excited states of Hamiltonian system. We have primarily in mind the non-relativistic quantum few body problem such as encountered in nuclear physics [4-6].
Abstract In this paper Kalos proposed what is now known as the Green’s function quantum Monte Carlo (GFQMC) method for solving the Schrödinger equation. Kalos made the “critical” observation that the time-independent Schrödinger equation could be transformed into an integral equation for which the Green’s function is known if the boundary conditions correspond to the wavefunction vanishing at infinity. In a 3n-dimensional space R the integral form is (modified here) an equation which may be simulated by a random walk process in which particles at positions R’ are altered in weight by the V / E term and moved to new positions R obtained by sampling the Green’s function G(R’,R). As for simple diffusion, repetition leads to a distribution of particles corresponding to the lowest-energy wavefunction from which the energy E may be determined. Since time is not involved, there is no time-step error associated with the procedure. The term V / E must remain positive if mixed signs for particle weights are to be avoideda. Kalos examined the utility of the method in determining the boson ground-state energies for several three- and four-body nuclear problems involving pairwise-additive interactions with square, Gaussian, and exponential wells. The errors in the computed binding energies were estimated to be about 1%.
We present a numerically stable Quantum Monte Carlo algorithm to calculate zero-temperature imaginary-time Green functions $ G(\vec{r}, τ) $ for Hubbard type models. We illustrate the efficiency of the algorithm by calculating the on-site Green function $ G(\vec{r}=0, τ) $ on 4×44 \times 44×4 to 12×1212 \times 1212×12 lattices for the two-dimensional half-filled repulsive Hubbard model at U/t=4U/t = 4U/t=4. By fitting the tail of $ G(\vec{r}=0, τ) $ at long imaginary time to the form e−τΔce^{-τΔ_c}e−τΔc​, we obtain a precise estimate of the charge gap: Δc=0.67±0.02Δ_c = 0.67 \pm 0.02Δc​=0.67±0.02 in units of the hopping matrix element. We argue that the algorithm provides a powerful tool to study the metal-insulator transition from the insulator side.
Abstract This paper describes a different Q11C method for solving the Schrödinger equation, one unlike variational, diffusion, and Green’s function methods. It can be applied to many-body systems to produce solutions for ground states. For systems of fermions, antisymmetrization can be imposed to overcome the problem of node location, but the instability of the method is a serious shortcoming, so that the method as described is limited to treating relatively simple systems. Nevertheless, this auxiliary field quantum 1,fonte Carlo (AFMC) method was demonstrated to be successful for test cases reported in this paper, and in time has been improved to allow treatment of realistic systemsa. The method makes use of the imaginary-time propagation of an initial wavefunction toward the ground-state function. It requires use of a trial wavefunction which is the product of single-particle orbitals for bosons or an antisymmetrized product for fermions. A Hubbard-Stratonovich transformation of the propagator changes the many-body problem to a problem of a collection of single particles interacting with a fluctuating (auxiliary) field which replaces the many-body interactions. The authors developed an integral expression for the energy which, after discretization, could be evaluated by a Metropolis Monte Carlo integration forward in time. The algorithm was tested for one-dimensional systems of particles: 6 to 20 bosons with varied interactions, 2 to 6 neutron-proton pairs, others with repulsive potentials. Importance sampling was found to be critical and computational requirements to be large. For spin systems on a lattice the AFMC method was considered feasible, but for general multidimensional systems the method appeared to be limited by its computation requirements.
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