Dirac theory and Hermitian interaction terms

Bhabani Prasad Mandal, Saurabh Gupta

We consider a couple of examples to study the pseudo-Hermitian interaction in relativistic quantum mechanics. Rasbha interaction, commonly used to study the spin Hall effect, is considered with imaginary coupling. The corresponding Dirac Hamiltonian is shown to be parity pseudo-Hermitian. In the other example we consider parity pseudo-Hermitian scalar interaction with arbitrary parameter in Dirac theory. In both cases we show that the energy spectrum is real and all the other features of nonrelativistic pseudo-Hermitian formulation are present. Using the spectral method, the positive definite metric operator (η) has been calculated explicitly for both the models to ensure positive definite norms for the state vectors.

Mustafa Sarisaman, Murat Taş, Enes Talha Kırca

In this study, motivated by recent advancements in non-Hermitian physics, we explore new characteristics of Dirac semimetals (DSMs) using the spectral singularities by means of scattering techniques, with the goal of uncovering additional unique properties. To achieve this, we investigate how the axion texture of a DSM affects its topological properties by analyzing its interaction with electromagnetic waves. We examine the transverse electric (TE) mode configuration, where the magneto-electric effect induces a dichroic property in these materials. This behavior is particularly interesting and commonly seen in potential DSM candidates. Consequently, we report for the first time that a dichroic DSM generates 12 unique topological laser types. We discover that surface currents are generated by topological terms on the surface of the DSM slab. Furthermore, we examine how the θ term associated with axions in topological materials contributes to these topological properties. Our study reveals distinct topological role of the θ term more clearly than ever before. Our results confirm that the topological properties of DSMs with a single Dirac cone remain stable under external influences and that a topologically robust DSM laser can be developed accordingly

Gordon Walter Semenoff

Carl M. Bender

The average quantum physicist on the street would say that a quantum-mechanical Hamiltonian must be Dirac Hermitian (invariant under combined matrix transposition and complex conjugation) in order to guarantee that the energy eigenvalues are real and that time evolution is unitary. However, the Hamiltonian H = p 2 + ix 3 , which is obviously not Dirac Hermitian, has a positive real discrete spectrum and generates unitary time evolution, and thus it defines a fully consistent and physical quantum theory. Evidently, the axiom of Dirac Hermiticity is too restrictive. While H = p 2 + ix 3 is not Dirac Hermitian, it is PT symmetric; that is, invariant under combined parity P (space reflection) and time reversal T . The quantum mechanics defined by a PT -symmetric Hamiltonian is a complex generalization of ordinary quantum mechanics. When quantum mechanics is extended into the complex domain, new kinds of theories having strange and remarkable properties emerge. In the past few years, some of these properties have been verified in laboratory experiments. A particularly interesting PT-symmetric Hamiltonian is H = p 2 − x 4 , which contains an upside-down potential. This potential is discussed in detail, and it is explained in intuitive as well as in rigorous terms why the energy levels of this potential are real, positive, and discrete. Applications of PT -symmetry in quantum field theory are also discussed.

R. Gerritsma, Gerhard Kirchmair, F. Zähringer +3 more