[Paper Review] What breaks parity-time-symmetry? -- pseudo-Hermiticity and resonance between positive- and negative-action modes

Ruili Zhang, Hong Qin+2 more|arXiv (Cornell University)|Jan 5, 2018

Quantum Mechanics and Non-Hermitian Physics3 citations

TL;DR

This paper redefines PT-symmetry breaking as a resonance between positive- and negative-action modes, not eigenvalue coalescence at exceptional points (EPs). It proves that PT-symmetry breaking occurs if and only if such resonance happens, and shows that finite-dimensional PT-symmetric Hamiltonians are always pseudo-Hermitian, invalidating the conventional EP-based picture of PT breaking.

ABSTRACT

It is generally believed that Parity-Time (PT)-symmetry breaking occurs when eigenvalues or both eigenvalues and eigenvectors coincide. However, we show that this well-accepted picture of PT-symmetry breaking is incorrect. Instead, we demonstrate that the physical mechanism of PT-symmetry breaking is the resonance between positive- and negative-action modes. It is proved that PT-symmetry breaking occurs when and only when this resonance condition is satisfied, and this mechanism applies to all known PT-symmetry breakings observed in different branches of physics. The result is achieved by proving a remarkable fact that in finite dimensions, a PT-symmetric Hamiltonian is necessarily pseudo-Hermitian, regardless whether it is diagonalizable or not.

Motivation & Objective

• To challenge the widely accepted view that PT-symmetry breaking occurs at exceptional points (EPs) where eigenvalues and eigenvectors coalesce.
• To identify the true physical mechanism underlying PT-symmetry breaking across diverse physical systems.
• To establish that pseudo-Hermiticity is a necessary property of finite-dimensional PT-symmetric Hamiltonians, regardless of diagonalizability.
• To demonstrate that resonance between positive- and negative-action modes is the necessary and sufficient condition for PT-symmetry breaking.
• To resolve inconsistencies in existing literature by showing that not all EPs lead to PT-symmetry breaking, and that diagonalizability at EPs does not preclude breaking.

Proposed method

• Analyzes a 4×4 PT-symmetric Hamiltonian with real parameters to show that PT-symmetry breaking occurs only when positive- and negative-action modes resonate, not universally at EPs.
• Proves that in finite dimensions, any PT-symmetric Hamiltonian is necessarily pseudo-Hermitian, even if non-diagonalizable.
• Introduces the concept of action (or Krein signature) to classify modes as positive or negative, and defines resonance as collision of modes with opposite actions.
• Uses numerical and analytical methods to track eigenvalue trajectories in parameter space, identifying when PT-symmetry breaks due to action-antipodal mode resonance.
• Applies the theory to a coupled-oscillator system with real non-canonical Hamiltonian dynamics, confirming resonance between opposite-action modes triggers PT breaking.
• Validates the mechanism across multiple systems, including quantum and classical models, showing consistency in eigenvalue splitting and symmetry breaking.

Experimental results

Research questions

• RQ1 What is the true physical mechanism underlying PT-symmetry breaking, beyond the conventional EP coalescence picture?
• RQ2 Why do some exceptional points lead to PT-symmetry breaking while others do not?
• RQ3 Is the coalescence of both eigenvalues and eigenvectors (non-diagonalizability) a necessary condition for PT-symmetry breaking?
• RQ4 Can PT-symmetry breaking occur in the absence of eigenvector coalescence, particularly when the Hamiltonian is diagonalizable at the EP?
• RQ5 What role does the action (Krein signature) of modes play in determining whether PT-symmetry breaking occurs at an EP?

Key findings

• PT-symmetry breaking is not caused by eigenvalue or eigenvector coalescence at EPs, but by resonance between positive- and negative-action modes.
• The Hamiltonian in the example with parameters $ a = b = 0 $, $ c = 0 $ is diagonalizable at an EP but still undergoes PT-symmetry breaking, proving non-diagonalizability is not necessary.
• In the coupled-oscillator model, PT-symmetry breaking occurs at $ ho = 2\sqrt{3} $ when a positive-action mode resonates with a negative-action mode, confirmed by eigenvalue splitting across the imaginary axis.
• The system exhibits complex conjugate eigenvalue pairs after resonance, indicating broken PT symmetry, with the transition occurring precisely at the resonance point.
• All known PT-symmetry breaking phenomena across quantum and classical systems are unified under the resonance mechanism between opposite-action modes.
• Finite-dimensional PT-symmetric Hamiltonians are always pseudo-Hermitian, a fundamental property independent of diagonalizability or EP structure.