Kubo Formula for Non-Hermitian Systems and Tachyon Optical Conductivity

Linear response theory plays a prominent role in various fields of physics and provides us with extensive information about the thermodynamics and dynamics of quantum and classical systems. Here we develop a general theory for the linear response in non-Hermitian systems with nonunitary dynamics and derive a modified Kubo formula for the generalized susceptibility for an arbitrary (Hermitian and non-Hermitian) system and perturbation. We use this to evaluate the dynamical response of a non-Hermitian, one-dimensional Dirac model with imaginary and real masses, perturbed by a time-dependent electric field. The model has a rich phase diagram, and in particular, features a tachyon phase, where excitations travel faster than an effective speed of light. Surprisingly, we find that the dc conductivity of tachyons is finite, and the optical sum rule is exactly satisfied for all masses. Our results highlight the peculiar properties of the Kubo formula for non-Hermitian systems and are applicable for a large variety of settings.

Figure 1

(a) Dispersion relation for the non-Hermitian Hamiltonian (6) with (green) parabolic spectrum of gapped Dirac Hamiltonians at $|\mathrm{\Delta }|>m{c}^2$, (red) hyperbolic tachyon dispersion for $|\mathrm{\Delta }|<m{c}^2$, and (black) linear Dirac spectrum at $|\mathrm{\Delta }|=m{c}^2$. The dispersion is imaginary (dashed-dotted line) only in the tachyon phase for $p^2{c}^2<m^2{c}^4-{\mathrm{\Delta }}^2$. The wiggly lines denote resonant optical transitions under the applied electric field. (b) The real part of the conductivity ${\sigma }^{\prime }(\omega )$ as a function of frequency for several values of the ratio $\mathrm{\Delta }/m{c}^2$ in the case of (green) parabolic, (black) linear, or (red) hyperbolic (tachyon) dispersion. (Inset) Tachyon dc conductivity as a function of $\mathrm{\Delta }/m{c}^2$.