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Journal of Physics Conference Series

Accelerated path integral calculations for many-body systems

Antun Balaž, Ivana Vidanović, Aleksandar Bogojević

Published Aug 1, 2008• 0 citations
Publisher

Abstract

We present a generalization of a recently developed method for accelerated Monte Carlo calculation of path integrals of generic many particle systems. Using this method and the derived hierarchy of efiective actions we calculate the energy spectra of a two particle model with quartic interaction for several values of coupling and demonstrate agreement with analytical results governing the increase in efficiency of the new method.

DOI
10.1088/1742-6596/128/1/012048

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We present a generalization of a recently developed method for accelerated Monte Carlo calculation of path integrals of generic many particle systems. Using this method and the derived hierarchy of efiective actions we calculate the energy spectra of a two particle model with quartic interaction for several values of coupling and demonstrate agreement with analytical results governing the increase in efficiency of the new method.
A recently developed method systematically improved the convergence of generic path integrals for transition amplitudes, partition functions, expectation values and energy spectra. This was achieved by analytically constructing a hierarchy of discretized effective actions indexed by a level number p and converging to the continuum limit as 1/N p . Here we apply the above general method to numerical calculations using Metropolis Monte Carlo simulations of energy expectation values and energy spectra. We analyze and compare the ensuing increase in efficiency of several orders of magnitude.
By recursively solving the underlying Schrödinger equation, we set up an efficient systematic approach for deriving analytic expressions for discretized effective actions. With this, we obtain discrete short-time propagators for both one and many particles in arbitrary dimension to orders that have not been accessible before. They can be used to substantially speed up numerical Monte Carlo calculations of path integrals, as well as for setting up an alternative analytical approximation scheme for energy spectra, density of states, and other statistical properties of quantum systems.
We develop an approach of calculating the many-body path integral based on the linked cluster expansion method. First, we derive a linked cluster expansion and we give the diagrammatic rules for calculating the free energy and the pair distribution function g(r)g(r)g(r) as a systematic power-series expansion in the particle density. We also present a structured Pad\'e approximation scheme in the momentum space to determine g(r)g(r)g(r). The calculated g(r)g(r)g(r) for distinguishable particles interacting with Lennard-Jones and hard-sphere potential in various attempted schemes of approximation of the diagrammatic series compares very well with the results of path-integral Monte Carlo simulation. Our method is applicable to a wide range of problems of current general interest and may be extended to the case of identical particles and, in particular, to the case of the many-fermion problem.
We formulate the quasielastic response of a non-relativistic many-body system at zero temperature in terms of ground state density matrix elements and real time path integrals that embody the final state interactions. While the former provide the weight for a conventional Monte Carlo calculation, the latter require a more sophisticated treatment. We argue that the recently developed Stationary Phase Monte Carlo technique can be used to study the approach to "Y-scaling." We perform calculations for a particle in a potential well in one and three dimensions and compare them to the exact results available for these models. We then derive an eikonal approximation to the Path Integrals. This method is suitably generalized to treat strongly repulsive interactions, and allows comparison to Silver's theory of final state interactions in a straightforward way. We also give an exact prescription to calculate the scaling limit for potentials comprising a hard core. Finally, we study the approach to scaling in a model 4He nucleus, and find good agreement with experimental data.
We present in detail a formulation of the shell model as a path integral and Monte Carlo techniques for its evaluation. The formulation, which linearizes the two-body interaction by an auxiliary field, is quite general, both in the form of the effective ``one-body'' Hamiltonian and in the choice of ensemble. In particular, we derive formulas for the use of general (beyond monopole) pairing operators, as well as a novel extraction of the canonical (fixed-particle-number) ensemble via an activity expansion. We discuss the advantages and disadvantages of the various formulations and ensembles and give several illustrative examples. We also discuss and illustrate calculation of the imaginary-time response function and the extraction, by maximum entropy methods, of the corresponding strength function. Finally, we discuss the ``sign problem'' generic to fermion Monte Carlo calculations, and prove that a wide class of interactions are free of this limitation.
We propose a new method to calculate ground state position dependent observables in quantum many-body systems. The method, which we call the path-integral diffusion Monte Carlo (PI-DMC) method, is essentially a combination of path-integral Monte Carlo (PIMC) and diffusion Monte Carlo (DMC) methods. The distribution resulting from a DMC simulation is further propagated in imaginary time by PIMC sampling. Tests of the new method for simple cases such as the harmonic oscillator, a double well, and a set of ten coupled harmonic oscillators show that the results for several observables converge rapidly to the exact result.