## Abstract

-symmetric quantum mechanics (PTQM) has become a hot area of research and investigation. Since its beginnings in 1998, there have been over 1000 published papers and more than 15 international conferences entirely devoted to this research topic. Originally, PTQM was studied at a highly mathematical level and the techniques of complex variables, asymptotics, differential equations and perturbation theory were used to understand the subtleties associated with the analytic continuation of eigenvalue problems. However, as experiments on -symmetric physical systems have been performed, a simple and beautiful physical picture has emerged, and a -symmetric system can be understood as one that has a balanced loss and gain. Furthermore, the phase transition can now be understood intuitively without resorting to sophisticated mathe- matics. Research on PTQM is following two different paths: at a fundamental level, physicists are attempting to understand the underlying mathematical structure of these theories with the long-range objective of applying the techniques of PTQM to understanding some of the outstanding problems in physics today, such as the nature of the Higgs particle, the properties of dark matter, the matter–antimatter asymmetry in the universe, neutrino oscillations and the cosmological constant; at an applied level, new kinds of -synthetic materials are being developed, and the phase transition is being observed in many physical contexts, such as lasers, optical wave guides, microwave cavities, superconducting wires and electronic circuits. The purpose of this Theme Issue is to acquaint the reader with the latest developments in PTQM. The articles in this volume are written in the style of mini-reviews and address diverse areas of the emerging and exciting new area of -symmetric quantum mechanics.

## 1. Brief introduction to -symmetric quantum theory

-symmetric Hamiltonians were discovered and explored in detail by Bender & Boettcher [1] (see also Bender [2]). Such Hamiltonians often exhibit two parametric regions, a region of *unbroken* symmetry in which the eigenvalues are *all real* and a region of *broken* symmetry in which some of the eigenvalues are real, and the remaining eigenvalues are complex. [Here, represents parity (space reflection) and represents time reversal.] A phase transition occurs at the boundary between these two regions. The class of -symmetric Hamiltonians considered by Bender and Boettcher [1,2] is
1.1
where *ε* is a real parameter. These Hamiltonians are not Dirac–Hermitian except at *ε*=0. (*H* is *Dirac–Hermitian* if *H*=*H*^{†}, where *H*^{†} is the transpose and complex conjugate of *H*.) However, they are symmetric because under space reflection *x* changes sign and under time reversal *i* changes sign. For the Hamiltonians (1.1), the region of unbroken symmetry is *ε*≥0 and the region of broken symmetry is −1<*ε*<0. At *ε*=1 and at *ε*=2 we obtain
1.2
and, surprisingly, the eigenvalues of these Hamiltonians are all real, positive and discrete. However, at , for example, we obtain *H*=*p*^{2}+*x*^{2}(*ix*)^{−1/2}, which has an infinite number of complex eigenvalues and only a finite number of real eigenvalues (figure 1). A rigorous proof that the eigenvalues are real when *ε*≥0 is given by Dorey *et al.* [3–5]. In contrast, if a quantum-mechanical Hamiltonian is Dirac–Hermitian, then its energy eigenvalues are always real. Thus, a Dirac–Hermitian Hamiltonian cannot exhibit a phase transition where its eigenvalues go from being real to being complex.
-symmetric systems are interesting because they are intermediate between open systems (systems in contact with an external environment) and closed (isolated) systems. An open system typically suffers loss to or gain from its environment and thus cannot be in equilibrium. A -symmetric system is special because, while it is in contact with the environment, the loss and gain are precisely balanced. The loss and gain leads may be widely separated, but, if the parameters of the system are adjusted to support a sufficiently rapid internal circulation, the system can be in equilibrium and thus mimic a closed system. A -symmetric system in equilibrium (that is, one for which the system has an unbroken PT symmetry) typically exhibits Rabi oscillations between its modes. However, if the parameters of the system are varied to weaken sufficiently the internal circulation, the Rabi oscillations cease. This is the signal that the system has undergone a transition to a broken -symmetric phase and is no longer in equilibrium. A system in a broken -symmetric phase mimics an open system.

The phase transition has been observed in laboratory experiments on a remarkably diverse variety of physical -symmetric systems including superconducting wires [6,7], optical wave-guides [8–10], atomic diffusion [11], microwave cavities [12], NMR [13], lasers [14–16], electronic circuits [17], mechanical oscillators [18], photonic lattices [19–21], -symmetric graphene [22] and atomic beams [23].

## 2. Note added in proof

We note with sadness the very recent passing of Boris Samsonov, who made many contributions to the theoretical development of quantum mechanics.

## Footnotes

One contribution of 17 to a Theme Issue ‘𝒫𝒯 quantum mechanics’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.