Fermion sign problem and the structure of Lee-Yang zeros: The form of the 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 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, ..., -1/(N-1). This distribution disrupts the analytic continuation of thermodynamic quantities, expressed as functions of ξ and typically performed along ξ=1→-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.