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

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.