Non-Hermitian Physics

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.

Spontaneous non-Hermiticity in the (2+1)-dimensional Gross-Neveu model

“…In the present paper, we show that in the simplest (2+1)-D massless Gross-Neveu (GN) model (for the first time, it was discussed in Ref. [10]) with four-fermion interaction (initially, its Lagrangian is Hermitian and PT -symmetric 1 ) the non-Hermiticity can also arise spontaneously. In contrast to the situation observed in the massless NJL model, in (2+1)-D massless GN model (i) both PT -and anti-PT -symmetric non-Hermitian ground states are allowed to be realised.…”

“…Conclusions. As a result, we see that at λ > λ crit = 2N π 2 , where λ ≡ ΛG(Λ) is the dimensionless coupling constant of the model (see in the last paragraph of the previous subsection III A), there might appear, on the same footing, three different phases in the framework of the Hermitian massless (2+1)-D GN model (1). One of them is characterized by dynamical appearance of the Hermitian mass term M H in the Lagrangian, i.e.…”

“…Morerover, it is symmetric with respect to space parity P, time reversal T and PT symmetries, which we now discuss in more detail within the framework of model (1). In (2+1) dimensions the space reflection, or parity, transformation P is defined by (t, x, y) P −→ (t, −x, y).…”

“…Generation of the M N H1 mass term. First, let us explore, using the CJT approach, the possibility of the dynamic appearance of a non-Hermitian and PT symmetric mass term M N H1 (13) in the original (2+1)-D GN model (1). In this case we should find the solution of the gap equation (25) in the form…”

“…For a long time, it was believed that to describe quantum systems it is necessary to use theories with Hermitian Hamiltonians (or Lagrangians), since in this case the energy spectrum is real. However, in recent decades, it has been understood that there are situations, especially in open physical systems interacting with the environment, which can be effectively considered in the framework of non-Hermitian Hamiltonians (see, e.g., in review [1]). Moreover, it was claimed that if non-Hermitian theories have in addition a space-time reflection symmetry PT , then its energy spectrum is real [2,3], i.e.…”