Scarring of Dirac fermions in chaotic billiards

Xuan Ni, Liang Huang, Ying-Cheng Lai, Celso Grebogi

Research output: Contribution to journalArticle

18 Citations (Scopus)

Abstract

Scarring in quantum systems with classical chaotic dynamics is one of the most remarkable phenomena in modern physics. Previous works were concerned mostly with nonrelativistic quantum systems described by the Schrodinger equation. The question remains outstanding of whether truly relativistic quantum particles that obey the Dirac equation can scar. A significant challenge is the lack of a general method for solving the Dirac equation in closed domains of arbitrary shape. In this paper, we develop a numerical framework for obtaining complete eigensolutions of massless fermions in general two-dimensional confining geometries. The key ingredients of our method are the proper handling of the boundary conditions and an efficient discretization scheme that casts the original equation in a matrix representation. The method is validated by (1) comparing the numerical solutions to analytic results for a geometrically simple confinement and (2) verifying that the calculated energy level-spacing statistics of integrable and chaotic geometries agree with the known results. Solutions of the Dirac equation in a number of representative chaotic geometries establish firmly the existence of scarring of Dirac fermions.

Original languageEnglish
Article number016702
Number of pages8
JournalPhysical Review. E, Statistical, Nonlinear and Soft Matter Physics
Volume86
Issue number1
DOIs
Publication statusPublished - 11 Jul 2012

Keywords

  • time-reversal symmetry
  • topological insulators
  • graphene
  • systems
  • eigenfunctions
  • colloquium
  • transport
  • lattice
  • orbits
  • scars

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