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.
Beyond the analysis of the Lee-Yang (LY) zero of
ξ 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
ξ=−1 at
T=0 remains close to
−1 at low
T, where it governs the sign factor and strongly constrains continuation along the real
ξ 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
Ts. 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
ξ∈[0,1] to
ξ=−1, and then extend this information toward lower
T through
T-fitting of the
ξ-independent remainder
ϕ(β)=ZF. 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.
Abstract To understand the distribution of the Yang–Lee zeros in quantum integrable field theories we analyse the simplest of these systems given by the 2D Yang–Lee model. The grand-canonical partition function of this quantum field theory, as a function of the fugacity z and the inverse temperature β , can be computed in terms of the thermodynamics Bethe Ansatz based on its exact S -matrix. We extract the Yang–Lee zeros in the complex plane by using a sequence of polynomials of increasing order N in z which converges to the grand-canonical partition function. We show that these zeros are distributed along curves which are approximate circles as it is also the case of the zeros for purely free theories. There is though an important difference between the interactive theory and the free theories, for the radius of the zeros in the interactive theory goes continuously to zero in the high-temperature limit β → 0 while in the free theories it remains close to 1 even for small values of β , jumping to 0 only at β = 0 .
By generalizing the recently developed path integral molecular dynamics for identical bosons and fermions, we consider the finite-temperature thermodynamic properties of fictitious identical particles with a real parameter ξ interpolating continuously between bosons (ξ = 1) and fermions (ξ = -1). Through general analysis and numerical experiments, we find that the average energy may have good analytical properties as a function of this real parameter ξ, which provides the chance to calculate the thermodynamical properties of identical fermions by extrapolation with a simple polynomial function after accurately calculating the thermodynamic properties of the fictitious particles for ξ ≥ 0. Using several examples, it is shown that our method can efficiently give accurate energy values for finite-temperature fermionic systems. Our work provides a chance to circumvent the fermion sign problem for some quantum systems.
The exchange antisymmetry between identical fermions gives rise to the well known fermion sign problem, in the form of large cancellation between positive and negative contribution to the partition function, making any simulation methods which directly sample this partition function exponentially difficult to converge. In this work, we employ path integral molecular dynamics (PIMD) and build upon the recently discovered fictitious particle model to investigate the fermion sign problem further. We consider the validity and invalidity condition for the method of parametrized path integral formulation of the partition function and extrapolation to circumvent the fermion sign problem. For the valid region of our method, our simulation shows that we may give accurate prediction of the energy for large fermion systems, which is much beyond the capability of the direct sampling in the traditional method. In particular, we find and verify a simple universal relation for high temperature noninteracting particles or strongly repulsive interacting particles at low temperatures.
Abstract The Chandler–Wolynes isomorphism gives a particularly transparent result for a two-level quantum system: after imaginary-time discretization, its equilibrium partition function becomes that of a one-dimensional ferromagnetic Ising chain, with tunneling events represented as Ising domain walls and all classical weights positive (Chandler 1987 Introduction to Modern Statistical Mechanics (Oxford University Press)). For general multistate quantum systems, any finite-dimensional Hamiltonian still admits an imaginary-time transfer-matrix representation as a one-dimensional chain with multivalued variables. However, this chain is a positive classical ensemble only when the off-diagonal couplings can be made real and non-positive. For three or more states, closed loops in the graph of diabatic states can carry gauge-invariant phases, which obstruct the removal of negative or complex path weights. We identify this obstruction as a loop-phase sign problem. Motivated by recent Lee–Yang-zero approaches to the fermion sign problem (He et al 2026 Phys. Rev. E 113 , 024115), we introduce loop fugacities associated with these fluxes and show that the associated Lee–Yang zeros diagnose the instability of analytic continuation from a sign-free point to a physical point with nontrivial loop phases.
Lee-Yang (LY) zeros play a fundamental role in the formulation of statistical physics in terms of (grand) partition functions, and assume theoretical significance for the phenomenon of phase transitions. In this paper, motivated by recent progress in cold Rydberg atom experiments, we explore the LY zeros in classical Rydberg blockade models. We find that the distribution of zeros of partition functions for these models in one dimension (1d) can be obtained analytically. We prove that all the LY zeros are real and negative for such models with arbitrary blockade radii. Therefore, no phase transitions happen in 1d classical Rydberg chains. We investigate how the zeros redistribute as one interpolates between different blockade radii. We also discuss possible experimental measurements of these zeros.