N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, F. M. Ellis, T. Kottos

A mechanism for asymmetric transport based on the interplay between the fundamental symmetries of parity (P) and time (T) with nonlinearity is presented. We experimentally demonstrate and theoretically analyze the phenomenon using a pair of coupled van der Pol oscillators, as a reference system, one with anharmonic gain and the other with complementary anharmonic loss; connected to two transmission lines. An increase of the gain/loss strength or the number of PT-symmetric nonlinear dimers in a chain, can increase both the asymmetry and transmittance intensities.

http://arxiv.org//abs/1301.7337

Optics (physics.optics); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)

Carl M. Bender, V. Branchina, Emanuele Messina

It was shown recently that a PT-symmetric \(i\phi^3\) quantum field theory in \(6-\epsilon\) dimensions possesses a nontrivial fixed point. The critical behavior of this theory around the fixed point is examined and it is shown that the corresponding phase transition is related to the existence of a nontrivial solution of the gap equation. The theory is studied first in the mean-field approximation and the critical exponents are calculated. Then, it is examined beyond the mean-field approximation by using renormalization-group techniques, and the critical exponents for \(6-\epsilon\) dimensions are calculated to order \(\epsilon\). It is shown that because of its stability the PT-symmetric \(i\phi^3\) theory has a higher predictive power than the conventional \(\phi^3\) theory. A comparison of the \(i\phi^3\) model with the Lee-Yang model is given.

http://arxiv.org/abs/1301.6207

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

J. H. Noble, U. D. Jentschura

The PT-symmetric cubic oscillator with Hamiltonian \(H_3 = – (1/2) d_x^2 + (1/2) x^2 + i G x^3\) is a paradigmatic example of a pseudo-Hermitian, or PT-symmetric Hamiltonian with a purely real spectrum when endowed with \(L^2(R)\) boundary conditions. Eigenfunctions of the stationary Schr\”{o}dinger equation \(H_3 \psi_n(x) = E_n \psi_n(x)\) are manifestly complex, while the energy eigenvalues \(E_n\) are real and positive. Although \(H_3\) does not commute with the parity operator, we find that for a natural choice of the global complex phase of the eigenstate wave function, the real and imaginary parts of the eigenfunctions [i.e., Re \(\psi_n(x)\) and Im \(\psi_n(x)\)] are eigenstates of parity. Both the real and imaginary parts of the eigenfunctions are found to have an infinite number of zeros, even for the ground state and even for infinitesimal G, but the real and imaginary parts of psi_n(x) never vanish simultaneously when \(G>0\). Furthermore, we find that the eigenfunctions are “concentrated” in an “allowed” region where the energy \(E_n\) is larger than the complex modulus of the complex potential. PT-symmetric Hamiltonians constitute natural generalizations of Hermitian Hamiltonians as time-evolution operators. Our results suggest that PT-symmetric (pseudo-Hermitian) time evolution can naturally be interpreted as time evolution in a situation where manifestly complex “gain” and “loss” terms mutually compensate and lead to the real energy eigenvalues.

http://arxiv.org/abs/1301.5758

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Atomic Physics (physics.atom-ph)

M. I. Molina

We examine the conditions for the existence of bounded dynamical phases for finite PT-symmetric arrays of split-ring resonators. The dimer (N=2), trimer (N=3) and pentamer (N=5) cases are solved in closed form, while for \(N>5\) results were computed numerically for several gain/loss spatial distributions. It is found that the parameter stability window decreases monotonically with the size of the array.

http://arxiv.org/abs/1301.5291

Pattern Formation and Solitons (nlin.PS); Materials Science (cond-mat.mtrl-sci)

Willi-Hans Steeb, Yorick Hardy

Exceptional points are studied for non-hermitian Hamilton operators given by a hierarchy of spin operators.

http://arxiv.org/abs/1301.2900

Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Bhabani Prasad Mandal, Brijesh Kumar Mourya, Rajesh Kumar Yadav

We consider a 2d anisotropic SHO with \(\bf ixy\) interaction and a 3d SHO in an imaginary magnetic field with \(\vec\mu_l\). \(\vec B\) interaction to study the PT phase transition analytically in higher dimension.Unbroken PT symmetry in the first case is complementary to the rotational symmetry of the original Hermitian system. PT phase transition ceases to occur the moment the 2d oscillator becomes isotropic.Transverse magnetic field in the other system introduces the anisotropy in the system and the system undergoes PT phase transition depending on the strength of the magnetic field and frequency of the oscillator.

http://arxiv.org/abs/1301.2387

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

A. D. Alhaidari

We present a large class of non-Hermitian non-PT-symmetric two-component Dirac Hamiltoninas with real energy spectra. These Hamiltonians are invariant under the combined action of “charge” conjugation (two-component transpose) and space-parity. Examples are given from the two subclasses of these systems having localized and/or continuum states with real energies.

http://arxiv.org/abs/1301.2056

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

D. Dutta, O. Panella, P. Roy

We study generalized Dirac oscillators with complex interactions in \((1+1)\) dimensions. It is shown that for the choice of interactions considered here, the Dirac Hamiltonians are \(\eta\) pseudo Hermitian with respect to certain metric operators \(\eta\). Exact solutions of the generalized Dirac Oscillator for some choices of the interactions have also been obtained. It is also shown that generalized Dirac oscillators can be identified with Anti Jaynes Cummings type model and by spin flip it can also be identified with Jaynes Cummings type model.

http://arxiv.org/abs/1301.2035

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

Francisco M. Fernandez, Javier Garcia

We consider a class of one-dimensional nonhermitian oscillators and discuss the relationship between the real eigenvalues of PT-symmetric oscillators and the resonances obtained by different authors. We also show the relationship between the strong-coupling expansions for the eigenvalues of those oscillators. Comparison of the results of the complex rotation and the Riccati-Pad\(\’{e}\) methods reveals that the optimal rotation angle converts the oscillator into either a PT-symmetric or an Hermitian one. In addition to the real positive eigenvalues the PT-symmetric oscillators exhibit real positive resonances under different boundary conditions. They can be calculated by means of the straightforward diagonalization method. The Riccati-Pad\(\’e\) method yields not only the resonances of the nonhermitian oscillators but also the eigenvalues of the PT-symmetric ones.

http://arxiv.org/abs/1301.1676

Mathematical Physics (math-ph); Quantum Physics (quant-ph)

Daniel Leykam, Vladimir V. Konotop, Anton S. Desyatnikov

We study the effect of lifting the degeneracy of vortex modes with a PT symmetric defect, using discrete vortices in a circular array of nonlinear waveguides as an example. When the defect is introduced, the degenerate linear vortex modes spontaneously break PT symmetry and acquire complex eigenvalues, but nonlinear propagating modes with real propagation constants can still exist. The stability of nonlinear modes depends on both the magnitude and the sign of the vortex charge, thus PT symmetric systems offer new mechanisms to control discrete vortices.

http://arxiv.org/abs/1301.1052

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