International Journal of Theoretical Physics manuscript No. (will be inserted by the editor) Hofstadter’s 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 re-ferred to as “Hofstadter’s butterfly”. We study a non-Hermitian continuation of Hofstadter’s model; as the non-Hermiticity parameter g increases past a se-quence of critical values the eigenvalues successively go complex in a sequence of “double-pitchfork bifurcations ” wherein pairs of real eigenvalues degener-ate and then become complex conjugate pairs. The associated wavefunctions undergo a spontaneous symmetry breaking transition that we elucidate. Be-yond 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