Gabriel Álvarez, Luis Martinez Alonso, Elena Medina
This paper deals with the determination of the S-curves in the theory of non-hermitian orthogonal polynomials with respect to exponential weights along suitable paths in the complex plane. It is known that the corresponding complex equilibrium potential can be written as a combination of Abelian integrals on a suitable Riemann surface whose branch points can be taken as the main parameters of the problem. Equations for these branch points can be written in terms of periods of Abelian differentials and are known in several equivalent forms. We select one of these forms and use a combination of analytic an numerical methods to investigate the phase structure of asymptotic zero densities of orthogonal polynomials and of asymptotic eigenvalue densities of random matrix models. As an application we give a complete description of the phases and critical processes of the standard cubic model.
http://arxiv.org/abs/1305.3028
Mathematical Physics (math-ph)
Ananya Ghatak, Bhabani Prasad Mandal
To develop a unitary quantum theory with probabilistic description for pseudo- Hermitian systems one needs to consider the theories in a different Hilbert space endowed with a positive definite metric operator. There are different approaches to find such metric operators. We compare the different approaches of calculating pos- itive definite metric operators in pseudo-Hermitian theories with the help of several explicit examples in non-relativistic as well as in relativistic situations. Exceptional points and spontaneous symmetry breaking are also discussed in these models.
http://arxiv.org/abs/1305.2022
Quantum Physics (quant-ph); High Energy Physics – Theory (hep-th)
Mohammad-Ali Miri, Matthias Heinrich, Demetrios N. Christodoulides
We show that the formalism of supersymmetry (SUSY), when applied to parity-time (PT) symmetric optical potentials, can give rise to novel refractive index landscapes with altogether non-trivial properties. In particular, we find that the presence of gain and loss allows for arbitrarily removing bound states from the spectrum of a structure. This is in stark contrast to the Hermitian case, where the SUSY formalism can only address the fundamental mode of a potential. Subsequently we investigate isospectral families of complex potentials that exhibit entirely real spectra, despite the fact that their shapes violate PT-symmetry. Finally, the role of SUSY transformations in the regime of spontaneously broken PT symmetry is investigated.
http://arxiv.org/abs/1305.1689
Optics (physics.optics); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
Dorje C. Brody
Wigner’s theorem asserts that an isometric (probability conserving) transformation on a quantum state space must be generated by a Hamiltonian that is Hermitian. It is shown that when the Hermiticity condition on the Hamiltonian is relaxed, we obtain the following complex generalisation of Wigner’s theorem: a holomorphically projective (complex geodesic-curves preserving) transformation on a quantum state space must be generated by a Hamiltonian that is not necessarily Hermitian.
http://arxiv.org/abs/1305.0658
Mathematical Physics (math-ph); High Energy Physics – Theory (hep-th); Differential Geometry (math.DG)
Yu.V. Bludov, R. Driben, V.V. Konotop, B.A. Malomed
We considered the modulational instability of continuous-wave backgrounds, and the related generation and evolution of deterministic rogue waves in the recently introduced parity-time (PT)-symmetric system of linearly-coupled nonlinear Schr\”odinger equations, which describes a Kerr-nonlinear optical coupler with mutually balanced gain and loss in its cores. Besides the linear coupling, the overlapping cores are coupled through cross-phase-modulation term too. While the rogue waves, built according to the pattern of the Peregrine soliton, are (quite naturally) unstable, we demonstrate that the focusing cross-phase-modulation interaction results in their partial stabilization. For PT-symmetric and antisymmetric bright solitons, the stability region is found too, in an exact analytical form, and verified by means of direct simulations.
http://arxiv.org/abs/1304.7369
Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)
Huagang Li, Zhiwei Shi, Xiujuan jiang, Xing Zhu, Tianshu Lai
We investigate light beam propagation along the interface between linear and nonlinear media with parity-time PT symmetry, and derive an equation governing the beam propagation. A novel class of two-dimensional PT surface solitons are found analytically and numerically, and checked to be stable over a wide range of PT potential structures and parameters. These surface solitons do not require a power threshold. The transverse power flow across beam is examined within the solitons and found to be caused by the nontrivial phase structure of the solitons. PT surface solitons are possibly observed experimentally in photorefractive crystals.
http://arxiv.org/abs/1304.6788
Optics (physics.optics); Pattern Formation and Solitons (nlin.PS)
Jon Links, Amir Moghaddam, Yao-Zhong Zhang
We demonstrate the occurrence of free quasi-particle excitations obeying generalised exclusion statistics in a BCS model with asymmetric pair scattering. The results are derived from an exact solution of the Hamiltonian, which was obtained via the algebraic Bethe ansatz utilising the representation theory of an underlying Yangian algebra. The free quasi-particle excitations are associated to highest-weight states of the Yangian algebra, corresponding to a class of analytic solutions of the Bethe ansatz equations.
http://arxiv.org/abs/1304.5818
Exactly Solvable and Integrable Systems (nlin.SI); Superconductivity (cond-mat.supr-con); Mathematical Physics (math-ph)
Katherine Jones-Smith
In 1956 Dyson analyzed the low-energy excitations of a ferromagnet using a Hamiltonian that was non-Hermitian with respect to the standard inner product. This allowed for a facile rendering of these excitations (known as spin waves) as weakly interacting bosonic quasi-particles. More than 50 years later, we have the full denouement of non-Hermitian quantum mechanics formalism at our disposal when considering Dyson’s work, both technically and contextually. Here we recast Dyson’s work on ferromagnets explicitly in terms of two inner products, with respect to which the Hamiltonian is always self-adjoint, if not manifestly “Hermitian”. Then we extend his scheme to doped antiferromagnets described by the t-J model, in hopes of shedding light on the physics of high-temperature superconductivity.
http://arxiv.org/abs/1304.5689
Quantum Physics (quant-ph)
Katherine Jones-Smith, Rudolph Kalveks
We present two examples of non-Hermitian Hamiltonians which consist of an unperturbed part plus a perturbation that behaves like a vector, in the framework of PT quantum mechanics. The first example is a generalization of the recent work by Bender and Kalveks, wherein the E2 algebra was examined; here we consider the E3 algebra representing a particle on a sphere, and identify the critical value of coupling constant which marks the transition from real to imaginary eigenvalues. Next we analyze a model with SO(3) symmetry, and in the process extend the application of the Wigner-Eckart theorem to a non-Hermitian setting.
http://arxiv.org/abs/1304.5692
Quantum Physics (quant-ph)
Huidan (Whitney)Yu, Xi Chen, Nan Chen, Yousheng Xu, Yogesh N. Joglekar
In recent years, open systems with equal loss and gain have been investigated via their symmetry properties under combined parity and time-reversal (\(\mathcal{PT}\)) operations. We numerically investigate \(\mathcal{PT}\)-symmetry properties of an incompressible, viscous fluid with “balanced” inflow-outflow configurations. We define configuration-dependent asymmetries in velocity, kinetic energy density, and vorticity fields, and find that all asymmetries scale quadratically with the Reynolds number. Our proposed configurations have asymmetries that are orders of magnitude smaller than the asymmetries that occur in traditional configurations at low Reynolds numbers. Our results show that \(\mathcal{PT}\)-symmetric fluid flow configurations, which are defined here for the first time, offer a hitherto unexplored avenue to tune fluid flow properties.
http://arxiv.org/abs/1304.5348
Fluid Dynamics (physics.flu-dyn); Quantum Physics (quant-ph)