# Path Integral Monte Carlo Simulation of Degenerate Electrons: Permutation-Cycle Properties

## TL;DR

This paper analyzes permutation-cycle properties in path integral Monte Carlo simulations of degenerate electrons, focusing on finite-size effects and introducing a correlation function to better understand exchange cycles, with implications for improving fermionic PIMC methods.

## Contribution

It provides a detailed analysis of permutation-cycle behaviors in fermionic PIMC, including the development of a correlation function and insights into finite-size effects in electron systems.

## Key findings

- Exchange-cycle frequencies do not follow simple exponential laws due to finite size.
- Finite-size effects significantly influence permutation-cycle properties.
- The permutation-cycle correlation function offers new insights into cycle joint probabilities.

## Abstract

Being motivated by the surge of fermionic quantum Monte Carlo simulations at finite temperature, we present a detailed analysis of the permutation-cycle properties of path integral Monte Carlo (PIMC) simulations of degenerate electrons. Particular emphasis is put onto the uniform electron gas in the warm dense matter regime. We carry out PIMC simulations of up to $N=100$ electrons and investigate exchange-cycle frequencies, which are found not to follow any simple exponential law even in the case of ideal fermions due to the finite size of the simulation box. Moreover, we introduce a permutation-cycle correlation function, which allows us to analyse the joint probability to simultaneously find cycles of different lengths within a single configuration. Again, we find that finite-size effects predominate the observed behaviour. Finally, we briefly consider an inhomogeneous system, namely electrons in a $2D$ harmonic trap. We expect our results to be of interest for the further development of fermionic PIMC methods, in particular to alleviate the notorious fermion sign problem.