Fermion sign problem and the structure of Lee-Yang zeros. II. Finite temperature results for a model system without interactions
Beyond the analysis of the Lee-Yang (LY) zero of $\xi$ at $0$ K presented by our previous work [He et. al. Phys. Rev. E 113, 24115 (2026)], it is important but intricate to understand how these zeros evolve with temperature ($T$). Here, we use an analytically solvable noninteracting one-dimensional particle-on-a-ring model to address this. We determine the trajectories of these zeros and analyze how their evolution with $T$ reshapes the analytic structure of the partition function. In particular, the zero originating from $\xi=-1$ at $T=0$ remains close to $-1$ at low $T$, where it governs the sign factor and strongly constrains continuation along the real $\xi$ axis. This explains why both direct extrapolation and implicit schemes such as contour-based fitting can fail in the low-$T$ regime, even at high fitting order, while becoming reasonable again once the relevant zeros move away at higher $T$s. Furthermore, based on the polynomial structure of the partition function, we propose a new fitting strategy for low-$T$ fermionic properties. The key is to first obtain reliable high-$T$ fermionic properties by continuing sign-problem-free data in $\xi\in[0,1]$ to $\xi=-1$, and then extend this information toward lower $T$ through $T$-fitting of the $\xi$-independent remainder $\phi(\beta)=Z_{\text{F}}$. These results provide a solvable benchmark for diagnosing the validity of analytic continuation and suggest a possible route toward treating more realistic interacting fermionic systems.