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.
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.
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.
Lee–Yang theory clearly demonstrates where the phase transition of many-body systems occurs and the asymptotic behavior near the phase transition using the partition function under complex parameters. The complex parameters make the direct investigation of Lee–Yang theory in practical systems challenging. Here we construct a non-Hermitian quantum system that can correspond to the one-dimensional Ising model with imaginary parameters through the equality of partition functions. By adjusting the non-Hermitian parameter, we successfully obtain the partition function under different imaginary magnetic fields and observe the Lee–Yang zeros. We also observe the critical behavior of free energy in vicinity of Lee–Yang zero that is consistent with theoretical prediction. Our work provides a protocol to study Lee–Yang zeros of the one-dimensional Ising model using a single-qubit non-Hermitian system.
We investigate the Lee--Yang zeros (LYZ) of inhomogeneous anisotropic XY spin chains with Dzyaloshinskii--Moriya (DM) interactions in the complex transverse-field plane, focusing on their fundamental connection to quantum phase transitions. We systematically study uniform chains, period-2 and period-3 modulated chains, and Fibonacci quasiperiodic chains of lengths 5 and 8. As the DM coupling strength
D increases, the LYZ exhibit qualitatively distinct topological evolutions on the complex plane: the complex zeros of the uniform chain collapse toward the real axis; periodic chains feature either bifurcation of closed zero contours before all zeros become real or a single re-emergence of complex zeros; quasiperiodic chains exhibit repeated annihilation and revival of complex zeros. Analytical derivations demonstrate that this diverse behavior originates from DM-induced shifts of folded bands and the modulation of zero positions by anisotropic pairing at particle--hole band crossings. Our results establish a direct correspondence between LYZ topology and band deformation: isolated contact points of zeros with the real axis correspond to discrete quantum critical fields, while continuous real-zero intervals directly identify gapless chiral phases. Accordingly, beyond locating phase boundaries, LYZ can distinguish characteristic phases and serve as an intuitive probe for band folding and phase diagram restructuring.
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.
Coupling spin models to complex external fields can give rise to interesting phenomena like zeroes of the partition function (Lee-Yang zeroes, edge singularities) or oscillating propagators. Unfortunately, it usually also leads to a severe sign problem that can be overcome only in special cases; if the partition function has zeroes, the sign problem is even representation-independent at these points. In this study, we couple the N-state Potts model in different ways to a complex external magnetic field and discuss the above mentioned phenomena and their relations based on analytic calculations (1D) and results obtained using a modified cluster algorithm (general D) that in many cases either cures or at least drastically reduces the sign-problem induced by the complex external field.