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Spontaneous Breakdown of a PT-Symmetry in the Liouvillian Dynamics at a Non-Hermitian Degeneracy Point

  • Kazuki KankiEmail author
  • Kazunari Hashimoto
  • Tomio Petrosky
  • Satoshi Tanaka
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 184)

Abstract

We consider the prevalent phenomenon that a pair of eigenvalues of the Liouville-von Neumann operator (Liouvillian) changes from pure imaginary to complex values with a common imaginary part for resonance states in an extended function space outside the Hilbert space. Such a transition point is an exceptional point, where non-Hermitian degeneracy occurs and both the pairs of eigenvalues and of eigenvectors coalesce. The transition can be attributed to a spontaneous breakdown of a parity and time-reversal (PT)-symmetry. This PT-symmetry in the Liouvillian dynamics results from the microscopic dynamics based on the fundamental physical laws. The kinetic equation of the Boltzmann type for a particle weakly coupled with a bath consists of the collision term, which is similar to a Hermitian operator and has even parity, and the flow term, which is anti-Hermitian and has odd parity. As a result of the competition between the two terms, a pair of PT-symmetric eigenstates of the effective Liouvillian converts to a PT-symmetry related pair as the flow term becomes more dominant than the collision term beyond an exceptional point.

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Kazuki Kanki
    • 1
    Email author
  • Kazunari Hashimoto
    • 2
  • Tomio Petrosky
    • 3
    • 4
  • Satoshi Tanaka
    • 1
  1. 1.Department of Physical ScienceOsaka Prefecture UniversitySakaiJapan
  2. 2.Graduate School of Interdisciplinary ResearchUniversity of YamanashiKofuJapan
  3. 3.Center for Studies in Complex Quantum SystemsThe University of Texas at AustinAustinUSA
  4. 4.Institute of Industrial ScienceUniversity of TokyoTokyoJapan

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