Non-Hermitian physics

A review is given on the foundations and applications of non-Hermitian\nclassical and quantum physics. First, key theorems and central concepts in\nnon-Hermitian linear algebra, including Jordan normal form, biorthogonality,\nexceptional points, pseudo-Hermiticity and parity-time symmetry, are delineated\nin a pedagogical and mathematically coherent manner. Building on these, we\nprovide an overview of how diverse classical systems, ranging from photonics,\nmechanics, electrical circuits, acoustics to active matter, can be used to\nsimulate non-Hermitian wave physics. In particular, we discuss rich and unique\nphenomena found therein, such as unidirectional invisibility, enhanced\nsensitivity, topological energy transfer, coherent perfect absorption,\nsingle-mode lasing, and robust biological transport. We then explain in detail\nhow non-Hermitian operators emerge as an effective description of open quantum\nsystems on the basis of the Feshbach projection approach and the quantum\ntrajectory approach. We discuss their applications to physical systems relevant\nto a variety of fields, including atomic, molecular and optical physics,\nmesoscopic physics, and nuclear physics with emphasis on prominent\nphenomena/subjects in quantum regimes, such as quantum resonances,\nsuperradiance, continuous quantum Zeno effect, quantum critical phenomena,\nDirac spectra in quantum chromodynamics, and nonunitary conformal field\ntheories. Finally, we introduce the notion of band topology in complex spectra\nof non-Hermitian systems and present their classifications by providing the\nproof, firstly given by this review in a complete manner, as well as a number\nof instructive examples. Other topics related to non-Hermitian physics,\nincluding nonreciprocal transport, speed limits, nonunitary quantum walk, are\nalso reviewed.\n