Revisiting the Fermion Sign Problem from the Structure of Lee-Yang Zeros. I. The Form of Partition Function for Indistinguishable Particles and Its Zeros at 0~K

TL;DR

This paper analyzes the distribution of Lee-Yang zeros of the partition function for indistinguishable particles at zero temperature, revealing insights into the fermion sign problem and phase transition phenomena via complex analysis.

Contribution

It introduces a polynomial reformulation of the partition function in the complex plane and characterizes the zeros' distribution, offering new understanding of the fermion sign problem at zero temperature.

Findings

Zeros are located at specific points , /2, /3, ..., /(N-1) at 0 K

Zeros disrupt analytic continuation of thermodynamic quantities

A zero at  induces an extra free energy term and can signal phase transition

Abstract

To simulate indistinguishable particles, recent studies of path-integral molecular dynamics formulated their partition function $Z$ as a recurrence relation involving a variable $\xi$, with $\xi=1$(-1) for bosons (fermions). Inspired by Lee-Yang phase transition theory, we extend $\xi$ into the complex plane and reformulate $Z$ as a polynomial in $\xi$. By analyzing the distribution of the partition function zeros, we gain insights into the analytical properties of indistinguishable particles, particularly regarding the fermion sign problem (FSP). We found that at 0~K, the partition function zeros for $N$-particles are located at $\xi=-1$, $-1/2$, $-1/3$, $\cdots$, $-1/(N-1)$. This distribution disrupts the analytic continuation of thermodynamic quantities, expressed as functions of $\xi$ and typically performed along $\xi=1\to-1$, whenever the paths intersect these zeros. Moreover, we highlight the zero at $\xi = -1$, which induces an extra term in the free energy of the fermionic systems compared to ones at other $\xi=e^{i\theta}$ values. If a path connects this zero to a bosonic system with identical potential energies, it brings a transition resembling a phase transition. These findings provide a fresh perspective on the successes and challenges of emerging FSP studies based on analytic continuation techniques.