O. Vazquez-Candanedo, J. C. Hernandez-Herrejon, F. M. Izrailev, D. N. Christodoulides

We investigate the properties of PT-symmetric tight-binding models by considering both bounded and unbounded models. For the bounded case, we obtain closed form expressions for the corresponding energy spectra and we analyze the structure of eigenstates as well as their dependence on the gain/loss contrast parameter. For unbounded PT-lattices, we explore their scattering properties through the development of analytical models. Based on our approach we identify a mechanism that is responsible to the emergence of localized states that are entirely due to the presence of gain and loss. The derived expressions for the transmission and reflection coefficients allow one to better understand the role of PT-symmetry in energy transport problems occurring in such PT-symmetric tight-binding settings. Our analytical results are further exemplified via pertinent examples.

http://arxiv.org/abs/1309.6708

Mesoscale and Nanoscale Physics (cond-mat.mes-hall)

A. Grod, S. Kuzhel

Non-self-adjoint Schrodinger operators A which correspond to non-symmetric zero-range potentials are investigated. For a given A, the description of non-real eigenvalues, spectral singularities and exceptional points are obtained; the possibility of interpretation of A as a self-adjoint operator in a Krein space is studied, the problem of similarity of A to a self-adjoint operator in a Hilbert space is solved.

http://arxiv.org/abs/1309.5482

Mathematical Physics (math-ph); Spectral Theory (math.SP); Quantum Physics (quant-ph)

Carlos A. Margalli, J. David Vergara

The quantization of higher order time derivative theories including interactions is unclear. In this paper in order to solve this problem, we propose to consider a complex version of the higher order derivative theory and map this theory to a real first order theory. To achieve this relationship, the higher order derivative formulation must be complex since there is not a real canonical transformation from this theory to a real first order theory with stable interactions. In this manner, we work with a non-Hermitian higher order time derivative theory. To quantize this complex theory, we introduce reality conditions that allow us to map the complex higher order theory to a real one, and we show that the resulting theory is regularizable and renormalizable for a class of interactions.

http://arxiv.org/abs/1309.2928

High Energy Physics – Theory (hep-th)

Chao Hang, Dmitry A. Zezyulin, Vladimir V. Konotop, Guoxiang Huang

We propose a scheme of creating a tunable highly nonlinear defect in a one-dimensional photonic crystal. The defect consists of an atomic cell filled in with two isotopes of three-level atoms. The probe-field refractive index of the defect can be made parity-time (PT) symmetric, which is achieved by proper combination of a control field and of Stark shifts induced by a far-off-resonance field. In the PT-symmetric system families of stable nonlinear defect modes can be formed by the probe field.

http://arxiv.org/abs/1309.2839

Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)

Jianke Yang

For the one-dimensional nonlinear Schroedinger equations with parity-time (PT) symmetric potentials, it is shown that when a real symmetric potential is perturbed by weak PT-symmetric perturbations, continuous families of asymmetric solitary waves in the real potential are destroyed. It is also shown that in the same model with a general PT-symmetric potential, symmetry breaking of PT-symmetric solitary waves do not occur. Based on these findings, it is conjectured that one-dimensional PT-symmetric potentials cannot support continuous families of non-PT-symmetric solitary waves.

http://arxiv.org/abs/1309.1652

Pattern Formation and Solitons (nlin.PS); Optics (physics.optics)

Francisco M. Fernández, Javier Garcia

We analyse several non-Hermitian Hamiltonians with antiunitary symmetry from the point of view of their point-group symmetry. We show that PT-symmetric Hamiltonians with point-group symmetry \(C_{2v}\) exhibit complex eigenvalues for all values of a potential parameter. In such cases the PT phase transition takes place at the trivial Hermitian limit.

http://arxiv.org/abs/1309.0808

Quantum Physics (quant-ph)

V.N.Rodionov

The modified Dirac equations for the massive particles with the replacement of the physical mass \(m\) with the help of the relation \(m\rightarrow m_1 + \gamma_5 m_2\) are investigated. It is shown that for a free fermion theory with a \(\gamma_5\) mass term, the finiteness of the mass spectrum at the value \(m_{max}= {m_1}^2/2m_2\) takes place. In this case the region of the unbroken \(\cal PT\)-symmetry may be expressed by means of the simple restriction of the physical mass \(m\leq m_{max}\). Furthermore, we have that the areas of unbroken \(\cal PT\)-symmetry \(m_1\geq m_2\geq 0\), which guarantees the reality values of the physical mass \(m\), consists of three different parametric subregions: i) \(0\leq m_2 < m_1/\sqrt{2}\), \,\,ii) \(m_2=m_1/\sqrt{2}=m_{max},\) \,\,(iii) \(m_1/\sqrt{2}< m_2 \leq m_1\). It is vary important, that only the first subregion (i) defined mass values \(m_1,m_2,\) which correspond to the description of traditional particles in the modified models, because this area contain the possibility transform the modified model to the ordinary Dirac theory. The second condition (ii) is defined the “maximon” – the particle with maximal mass \(m=m_{max}\). In the case (iii) we have to do with the unusual or “exotic” particles for description of which Hamiltonians and equations of motion have no a Hermitian limit. The formulated criterions may be used as a major test in the process of the division of considered models into ordinary and “exotic fermion theories”.

http://arxiv.org/abs/1309.0231

High Energy Physics – Theory (hep-th); High Energy Physics – Phenomenology (hep-ph); Mathematical Physics (math-ph); Quantum Physics (quant-ph)