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 $ξ$, with $ξ=1$(-1) for bosons (fermions). Inspired by Lee-Yang phase transition theory, we extend $ξ$ into the complex plane and reformulate $Z$ as a polynomial in $ξ$. 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 $ξ=-1$, $-1/2$, $-1/3$, $\cdots$, $-1/(N-1)$. This distribution disrupts the analytic continuation of thermodynamic quantities, expressed as functions of $ξ$ and typically performed along $ξ=1\to-1$, whenever the paths intersect these zeros. Moreover, we highlight the zero at $ξ= -1$, which induces an extra term in the free energy of the fermionic systems compared to ones at other $ξ=e^{iθ}$ 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.