  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.
