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

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 $\xi =-1$, $-1\mathrm{/}2$, $-1\mathrm{/}3$, $\cdots$, $-1\mathrm{/}(N-1)$. 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.