Fractional Path Integral Monte Carlo

TL;DR

This paper introduces a novel Fractional Path Integral Monte Carlo method to study fractional Schrödinger equations, revealing how fractional derivatives influence particle delocalization and potential condensation phenomena.

Contribution

It develops a new Monte Carlo algorithm for fractional Schrödinger equations, deriving an analytic fractional density matrix and applying it to both free particles and helium simulations.

Findings

Fractional Laplacian enhances particle delocalization.

Fractional Hamiltonians may exhibit atypical condensation.

Algorithm enables study of complex fractional potentials.

Abstract

Fractional derivatives are nonlocal differential operators of real order that often appear in models of anomalous diffusion and a variety of nonlocal phenomena. Recently, a version of the Schr\"odinger Equation containing a fractional Laplacian has been proposed. In this work, we develop a Fractional Path Integral Monte Carlo algorithm that can be used to study the finite temperature behavior of the time-independent Fractional Schr\"odinger Equation for a variety of potentials. In so doing, we derive an analytic form for the finite temperature fractional free particle density matrix and demonstrate how it can be sampled to acquire new sets of particle positions. We employ this algorithm to simulate both the free particle and $^{4}$He (Aziz) Hamiltonians. We find that the fractional Laplacian strongly encourages particle delocalization, even in the presence of interactions, suggesting that fractional Hamiltonians may manifest atypical forms of condensation. Our work opens the door to studying fractional Hamiltonians with arbitrarily complex potentials that escape analytical solutions.