Non-Hermitian Physics

TL;DR

The article surveys the foundations of non-Hermitian physics across classical and quantum realms, foregrounding the mathematical structure (Jordan form, biorthogonality, exceptional points, PT symmetry, pseudo-Hermiticity) that underpins open-system dynamics. It then maps these concepts onto a wide array of platforms—from photonics and circuits to biology and quantum many-body systems—showing how effective non-Hermitian operators capture resonance, decay, and transport phenomena. A major emphasis is placed on the topology of complex spectra, introducing generalized band topology notions (point/line gaps) and invariants in non-Hermitian settings, including higher-order EPs and spectral singularities. The review also connects non-Hermitian dynamics to quantum trajectories, Feshbach projections, and beyond-Markovian regimes, highlighting both foundational insights and practical implications for sensing, energy transfer, and controllable dissipation in engineered systems.

Abstract

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

Figures (49)

• Figure 1: Example of the Jordan normal form of a non-Hermitian matrix, where for eigenvalue $\lambda$ its geometric multiplicity $m^{\rm g}=4$ is smaller than the algebraic multiplicity $m^{\rm a}=7$ due to the existence of size-$2$ and size-$3$ Jordan blocks. Equation \ref{['jnfrel']} is satisfied as $1+1+2+3=7$.
• Figure 2: Inverse mean Petermann factor $1/\bar{K}_N$ versus the deviation $\bar{\gamma}=(\gamma-\gamma_\mathrm{EP}){N}$ of the gain/loss parameter $\gamma$ from its value at the exceptional point $\gamma_\mathrm{EP}$ multiplied by the length of disordered dimer chains ${N}$ZMC10. The Petermann factor diverges at the EP as $\bar{K}\propto\bar{\gamma}^{-1}$. The upper inset shows the distribution of the Petermann factors around the EP. The lower inset demonstrates the scaling $\bar{K}\propto \bar{\gamma}^{-1}$ in the vicinity of the EP, where the dashed line shows the slope with the exponent $-1$. Adapted from Ref. ZMC10. Copyright © 2010 by the American Physical Society.
• Figure 3: Real part of the eigenspectrum of the PT-symmetric non-Hermitian Hamiltonian $H_{a}=p^2-(ix)^a$ as a function of $a\in{\mathbb R}$. For $a\geq 2$, the PT symmetry is unbroken and eigenvalues are all real and positive. For $a< 2$, the PT symmetry is broken and an infinite number of complex conjugate pairs of eigenvalues appear in excited states. Note that only real parts of eigenvalues are plotted in this figure. Adapted from Ref. BCM98. Copyright © 1998 by the American Physical Society.
• Figure 4: Schematic illustration of the relation between the non-Hermitian Hamiltonian $H_g$ in Eq. \ref{['korff1']} whose spectrum is real for $g\leq1$ and the Hermitian Hamiltonian $h_g$ in Eq. \ref{['korff2']} that has the same spectrum CK08. In the non-Hermitian model $H_g$, all the terms are local including imaginary magnetic fields $\pm ig/2$ at the boundaries. After a similarity transformation $h_g=O^{-1}H_gO$, the Hermitian model acquires long-range hopping terms, which become increasingly long-ranged as $g$ approaches 1 at which the real-to-complex spectral transition occurs.
• Figure 5: (a) Schematic figure illustrating a microscopic realization of the generalized sine-Gordon model \ref{['sG']}, where a one-dimensional strongly correlated system is subject to a dissipative periodic potential $V(x)=\cos(x)+i\gamma\sin(x)$. (b) Low-energy many-body spectrum of the lattice model for the generalized sine-Gordon model. Real parts of eigenvalues are plotted against a non-Hermitian term $\gamma$. With increasing $\gamma$, the exceptional points (EPs) occur at finite $\gamma$ and the corresponding eigenvalues coalesce with the square-root scaling (inset). The crossing point between red and blue eigenvalues represents the Berezinskii-Kosterlitz-Thouless transition point YA17nc. See Sec. \ref{['Sec:QMBP']} for further discussions.

Theorems & Definitions (36)

• Example 2.1
• Theorem 2.1: Jordan normal form
• Theorem 2.2: Weyl's perturbation theorem
• Theorem 2.3: Stability of singular value spectra
• Theorem 2.4: Stability against general perturbations
• Example 2.2
• Theorem 2.5: Spectral shift
• Example 2.3
• Theorem 2.6: Pseudo Hermiticity
• Example 2.4