Quantum-Classical Correspondence of Non-Hermitian Symmetry Breaking

TL;DR

This paper establishes a universal quantum-classical correspondence for non-Hermitian symmetry breaking, linking spectral transitions to semiclassical orbit symmetries using complex path integrals and a generalized Gutzwiller trace formula.

Contribution

It introduces a novel theoretical framework connecting real and complex energy levels to classical orbit symmetries in non-Hermitian systems, unifying various spectral phenomena.

Findings

Real energy levels correspond to symmetry-preserving orbits.

Complex conjugate levels arise from symmetry-breaking pairs of orbits.

Exceptional points are inherently quantum, not classical phenomena.

Abstract

Real-to-complex spectral transitions and the associated spontaneous symmetry breaking of eigenstates are central to non-Hermitian physics, yet a comprehensive and universal theory that precisely describes the underlying physical mechanisms for each individual state remains elusive. Here, we resolve the mystery by employing the complex path integral formalism and developing a generalized Gutzwiller trace formula. These methodologies enable us to establish a universal quantum-classical correspondence that precisely links the real or complex nature of individual energy levels to the symmetry properties of their corresponding semiclassical orbits. Specifically, in systems with a general $\eta$-pseudo-Hermitian symmetry, real energy levels are quantized along periodic orbits that preserve the corresponding classical $S_\eta$ symmetry. In contrast, complex conjugate energy levels arise from semiclassical orbits that individually break the $S_\eta$ symmetry but together form $S_\eta$-symmetric pairs. This framework provides a unified explanation for the spectral behaviors in various continuous non-Hermitian models and for the $\mathcal{PT}$ transition in two-level systems. Besides, we demonstrate that the exceptional point is inherently a quantum phenomenon, as it cannot be described by a single classical orbit. Our work uncovers the physical mechanism of non-Hermitian symmetry breaking and introduces a new perspective with broad implications for the control and application of non-Hermitian phenomena.