Monte Carlo Evaluation of Feynman Path Integrals in Imaginary Time.

Leonard Frank Register, Michael A. Stroscio, Michael A. Littlejohn

Виктор Павлович Маслов, Alexander Mikhailovich Chebotarev

Christopher C. Gerry, J. Kiefer

A numerical procedure for evaluating Feynman’s path integral in imaginary time without using the Monte Carlo ‘‘heat bath’’ method is discussed. The method is essentially iterative matrix multiplication and bears a close resemblance to the Feynman procedure of summing all paths by time slicing and then integrating. The procedure is used to determine ground-state energy eigenvalues for the harmonic oscillator, double-well anharmonic oscillator, and the Morse oscillator.

William Hugh Baird

Soohaeng Yoo Willow, So Hirata

A new, alternative set of interpretation rules of Feynman-Goldstone diagrams for many-body perturbation theory is proposed, which translates diagrams into algebraic expressions suitable for direct Monte Carlo integrations. A vertex of a diagram is associated with a Coulomb interaction (rather than a two-electron integral) and an edge with the trace of a Green's function in real space and imaginary time. With these, 12 diagrams of third-order many-body perturbation (MP3) theory are converted into 20-dimensional integrals, which are then evaluated by a Monte Carlo method. It uses redundant walkers for convergence acceleration and a weight function for importance sampling in conjunction with the Metropolis algorithm. The resulting Monte Carlo MP3 method has low-rank polynomial size dependence of the operation cost, a negligible memory cost, and a naturally parallel computational kernel, while reproducing the correct correlation energies of small molecules within a few mEh after 10(6) Monte Carlo steps.

Åsmund Larsen, F. Ravndal

The statistical mechanics of a quantum harmonic oscillator can easily be formulated in terms of Feynman path integrals in imaginary time. By straightforward integrations, the partition function, the internal energy, and correlation functions are found. When time is discretized, the same results can be obtained directly using Monte Carlo methods on a computer. In the zero temperature limit, the usual quantum mechanical results are recovered.