Quantum-Classical Correspondence of Non-Hermitian Symmetry Breaking

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.