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Hofstadter’s Cocoon

International Journal of Theoretical Physics volume 54, pages 219–226 (2015)Cite this article

Abstract

Hofstadter showed that the energy levels of electrons on a lattice plotted as a function of magnetic field form an beautiful structure now referred to as “Hofstadter’s butterfly”. We study a non-Hermitian continuation of Hofstadter’s model; as the non-Hermiticity parameter g increases past a sequence of critical values the eigenvalues successively go complex in a sequence of “double-pitchfork bifurcations” wherein pairs of real eigenvalues degenerate and then become complex conjugate pairs. The associated wavefunctions undergo a spontaneous symmetry breaking transition that we elucidate. Beyond the transition a plot of the real parts of the eigenvalues against magnetic field resembles the Hofstadter butterfly; a plot of the imaginary parts plotted against magnetic fields forms an intricate structure that we call the Hofstadter cocoon. The symmetries of the cocoon are described. Hatano and Nelson have studied a non-Hermitian continuation of the Anderson model of localization that has close parallels to the model studied here. The relationship of our work to that of Hatano and Nelson and to PT transitions studied in PT quantum mechanics is discussed.

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Authors and Affiliations

  1. Physics, Astronomy Department, Oberlin College, 110 N. Professor Street, Oberlin, OH, 44074, USA

    Katherine Jones-Smith

  2. Reed College, Portland, OR, 97202, USA

    Connor Wallace

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  1. Katherine Jones-Smith
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  2. Connor Wallace
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Correspondence to Katherine Jones-Smith.

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Jones-Smith, K., Wallace, C. Hofstadter’s Cocoon. Int J Theor Phys 54, 219–226 (2015). https://doi.org/10.1007/s10773-014-2216-4

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Keywords

  • PT quantum mechanics
  • Hofstadter butterfly
  • Harper’s Equation