Monte Carlo Eigenvalue Methods in Quantum Mechanics and Statistical Mechanics
M. P. Nightingale (Department of Physics, University of Rhode Island,, Kingston, RI), C. J. Umrigar (Cornell Theory Center, Laboratory of Atomic, and Solid State Physics, Cornell University, Ithaca, NY)
TL;DR
This paper reviews Monte Carlo eigenvalue methods used in quantum and statistical mechanics, emphasizing their mathematical similarities and the importance of optimized trial states for reducing errors.
Contribution
It provides a unified overview of Monte Carlo eigenvalue techniques across quantum and statistical mechanics, highlighting their common stochastic foundation and error reduction strategies.
Findings
Monte Carlo methods are stochastic implementations of the power method.
Optimized trial states significantly reduce Monte Carlo estimation errors.
Methods are applicable across diverse problems in quantum and statistical mechanics.
Abstract
In this review we discuss, from a unified point of view, a variety of Monte Carlo methods used to solve eigenvalue problems in statistical mechanics and quantum mechanics. Although the applications of these methods differ widely, the underlying mathematics is quite similar in that they are stochastic implementations of the power method. In all cases, optimized trial states can be used to reduce the errors of Monte Carlo estimates.
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Taxonomy
TopicsTheoretical and Computational Physics