Spontaneous Breaking of Classical PT Symmetry

Abstract

The classical trajectories of the family of complex PT-symmetric Hamiltonians $H=p^2+x^2(ix{)}^{\epsilon }$ ($\epsilon \ge 0$) form closed orbits. All such complex orbits that have been studied in the past are PT symmetric (left-right symmetric). The periods of these orbits exhibit an unusual dependence on the parameter $\epsilon$. There are regions in $\epsilon$ of smooth behavior interspersed with regions of rapid variation. It is demonstrated that the onset of rapid variation is associated with strange new kinds of classical trajectories that have never been seen previously. These rare kinds of trajectories are not PT symmetric and occur only for special rational values of $\epsilon$.

16 pages, 17 figures, accepted for publication in the Journal of Mathematical Physics

References in corpus (9)

• Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry
• Complex Extension of Quantum Mechanics
• PT-Symmetric Quantum Mechanics
• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• Introduction to PT-Symmetric Quantum Theory
• Supersymmetry and the spontaneous breakdown of PT symmetry
• PT-Symmetric Extension of the Korteweg-de Vries Equation
• Classical Trajectories for Complex Hamiltonians
• Complex Trajectories of a Simple Pendulum

Cited by in corpus (20)

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• Complex Correspondence Principle
• Complexified Dynamical Systems
• Quantum effects in classical systems having complex energy
• Chaotic systems in complex phase space
• Quantum tunneling as a classical anomaly
• Probability Density in the Complex Plane
• Non-Hermitian dynamics and $\mathcal{P}\mathcal{T}$-symmetry breaking in interacting mesoscopic Rydberg platforms
• Conduction bands in classical periodic potentials
• Does the complex deformation of the Riemann equation exhibit shocks?
• Classical Particle in a Complex Elliptic Potential
• Periodic orbits for classical particles having complex energy
• Exact Isospectral Pairs of PT-Symmetric Hamiltonians
• PT-symmetric extensions of the supersymmetric Korteweg-de Vries equation
• Integrable models from PT-symmetric deformations
• Negative-energy PT-symmetric Hamiltonians
• Kondo effect in a non-Hermitian, $\mathcal{P}\mathcal{T}$-symmetric Anderson model with Rashba spin-orbit coupling
• Classical Trajectories of the Continuum States of the $\mathcal{P}\mathcal{T}$ symmetric Scarf II potential
• Quantum counterpart of spontaneously broken classical PT symmetry
• Effects of complex parameters on classical trajectories of Hamiltonian systems