Hofstadter's Cocoon

TL;DR

This paper explores a non-Hermitian extension of Hofstadter's model, revealing complex eigenvalue structures called the Hofstadter cocoon and analyzing symmetry-breaking transitions and their relation to PT quantum mechanics.

Contribution

It introduces a non-Hermitian Hofstadter model exhibiting a novel eigenvalue bifurcation sequence and the Hofstadter cocoon structure, expanding understanding of non-Hermitian quantum systems.

Findings

Eigenvalues undergo double-pitchfork bifurcations as non-Hermiticity increases.

The real parts of eigenvalues form a Hofstadter butterfly pattern.

Imaginary parts create a new intricate structure called the Hofstadter cocoon.

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.