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 $\xi =-1, -1/2, -1/3, ..., -1/(N-1)$. 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.

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• Bosons
• Fermions
• Lee-Yang & Fisher zeroes
• Path-integral methods