[Paper Review] Generalization of parity-time and partial parity-time symmetry

Francisco M. Fernández|arXiv (Cornell University)|Jul 31, 2015

Quantum Mechanics and Non-Hermitian Physics1 references3 citations

TL;DR

This paper generalizes parity-time (PT) and partial PT symmetry as specific instances of antiunitary symmetry, using group theory and perturbation theory to analyze non-Hermitian coupled harmonic oscillators. It demonstrates that antiunitary symmetry—defined via unitary operators combined with time reversal—is the most general framework, and shows that higher symmetry in the unperturbed Hamiltonian increases the likelihood of complex eigenvalues, with explicit analysis of a chain of N coupled oscillators revealing richer antiunitary structures beyond partial PT symmetry.

ABSTRACT

We show that parity-time and partial parity-time symmetries are particular cases of antiunitary symmetry. This point is illustrated by means of a recently discussed system of non-Hermitian coupled harmonic oscillators that also exhibits other types of antiunitary symmetries. We also show that a combination of group and perturbation theory is a useful tool for predicting broken antiunitary symmetry.

Motivation & Objective

• To unify PT and partial PT symmetry under the broader framework of antiunitary symmetry.
• To develop a systematic group-theoretic approach for analyzing non-Hermitian Hamiltonians with complex spectra.
• To predict the likelihood of broken antiunitary symmetry using perturbation theory.
• To demonstrate that higher symmetry in the unperturbed Hamiltonian increases the chance of complex eigenvalues.

Proposed method

• Formalize antiunitary symmetry as $ A_U = UT $, where $ U $ is a unitary operator and $ T $ is time reversal.
• Apply point-group theory to identify symmetry groups $ G_4 $ and $ G_8 $ for $ N $-dimensional coupled oscillators.
• Use perturbation theory to assess whether eigenvalues remain real under small perturbations.
• Identify unitary operators $ U_i $ that preserve $ H_0 $ and $ H' $, and $ W_j $ that reverse $ H' $, to construct antiunitary symmetries $ A_j = W_j T $.
• Analyze the $ N $-oscillator system with Hamiltonian $ H = H_0 + ig ho $, where $ H_0 $ is harmonic and $ H' $ is a quadratic coupling.
• Show that the symmetry group $ G_8 $ is isomorphic to $ C_{4v} $ for even $ N $ and $ D_{2h} $ for odd $ N $, revealing extended antiunitary symmetry beyond partial PT.

Experimental results

Research questions

• RQ1 How can PT and partial PT symmetries be unified under a more general symmetry concept?
• RQ2 What role does the symmetry of the unperturbed Hamiltonian $ H_0 $ play in determining the reality of eigenvalues in non-Hermitian systems?
• RQ3 Can group theory and perturbation theory reliably predict whether antiunitary symmetry is unbroken or broken in a given non-Hermitian Hamiltonian?
• RQ4 What are the full antiunitary symmetry structures in the $ N $-coupled oscillator model beyond partial PT symmetry?
• RQ5 Why do systems with degenerate spectra exhibit a higher likelihood of complex eigenvalues?

Key findings

• PT and partial PT symmetries are special cases of antiunitary symmetry, with the latter being the most general framework.
• The $ N $-oscillator system with $ ho = rac{1}{2} ho_{ ext{sym}} $ exhibits a full antiunitary symmetry group $ G_8 $, isomorphic to $ C_{4v} $ (even $ N $) or $ D_{2h} $ (odd $ N $), extending beyond partial PT symmetry.
• For $ N=2 $, eigenvalues are real only when $ n_1 = n_2 $, and complex otherwise, due to degeneracy and symmetry structure.
• The lowest energy state $ E_0^{(0)} $ is nondegenerate and remains real for all $ g $, consistent with perturbation theory.
• Higher degeneracy in $ H_0 $ increases the probability of complex eigenvalues, indicating a phase transition surface in parameter space.
• The use of group theory and perturbation theory provides a reliable predictive tool for the spectral behavior of non-Hermitian systems.