Abstract:

Recent years have witnessed groundbreaking developments in non-Hermitian systems, which transcend the framework of Hermitian operators in conventional quantum mechanics to reveal new physical laws inherent in complex eigenvalues and non-Hermitian symmetries. Unlike Hermitian systems, non-Hermitian systems achieve unified description of dynamical evolution through the real-imaginary dual structure of complex eigenvalues, manifesting unique phenomena including exceptional points, non-orthogonal eigenstates, and the non-Hermitian skin effect (NHSE). These distinctive properties originate from the non-Hermitian nature of their Hamiltonians, such as pseudo-Hermiticity constraining real spectra, the separation of algebraic-geometric multiplicities in Jordan block structures, and the generalized Brillouin zone theory reconstructing bulk-boundary correspondence. Prototypical models like the HatanoNelson model and non-Hermitian Su-Schrieffer-Heeger (SSH) model demonstrate nonreciprocal hopping and topological responses in complex energy spectra, providing paradigmatic platforms for understanding NHSE and energy band singularities. In symmetry and topological classification, non-Hermitian systems extend the traditional Altland-Zirnbauer tenfold classification to the 38-fold Bernard-LeClair symmetry classes, encompassing pseudo-Hermiticity, chiral symmetry, and combined conjugation-transposition operations. Topological classification progresses through dual approaches: K-theory mapping to Hermitian frameworks and homotopy theory analyzing deformation characteristics of complex band manifolds. Future challenges involve developing universal theories for high-dimensional systems, elucidating crystalline symmetry impacts on classification, and implementing experimental platforms, ultimately advancing applications in open quantum systems, nonequilibrium physics, and novel device engineering.

Key words:  non-Hermitian systems, pseudo-Hermiticity, Hatano-Nelson model, nonHermitian SSH model, symmetry, topological phase