Katherine Jones-Smith, Connor Wallace
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.
http://arxiv.org/abs/1407.0093
Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall)
Katherine Jones-Smith
In 1956 Dyson analyzed the low-energy excitations of a ferromagnet using a Hamiltonian that was non-Hermitian with respect to the standard inner product. This allowed for a facile rendering of these excitations (known as spin waves) as weakly interacting bosonic quasi-particles. More than 50 years later, we have the full denouement of non-Hermitian quantum mechanics formalism at our disposal when considering Dyson’s work, both technically and contextually. Here we recast Dyson’s work on ferromagnets explicitly in terms of two inner products, with respect to which the Hamiltonian is always self-adjoint, if not manifestly “Hermitian”. Then we extend his scheme to doped antiferromagnets described by the t-J model, in hopes of shedding light on the physics of high-temperature superconductivity.
http://arxiv.org/abs/1304.5689
Quantum Physics (quant-ph)
Katherine Jones-Smith, Rudolph Kalveks
We present two examples of non-Hermitian Hamiltonians which consist of an unperturbed part plus a perturbation that behaves like a vector, in the framework of PT quantum mechanics. The first example is a generalization of the recent work by Bender and Kalveks, wherein the E2 algebra was examined; here we consider the E3 algebra representing a particle on a sphere, and identify the critical value of coupling constant which marks the transition from real to imaginary eigenvalues. Next we analyze a model with SO(3) symmetry, and in the process extend the application of the Wigner-Eckart theorem to a non-Hermitian setting.
http://arxiv.org/abs/1304.5692
Quantum Physics (quant-ph)