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Linear response theory plays a prominent role in various fields of physicsand provides us with extensive information about the thermodynamics anddynamics of quantum and classical systems. Here we develop a general theory forthe linear response in non-Hermitian systems with non-unitary dynamics andderive a modified Kubo formula for the generalized susceptibility for arbitrary(Hermitian and non-Hermitian) system and perturbation. As an application, weevaluate the dynamical response of a non-Hermitian, one-dimensional Dirac modelwith imaginary and real masses, perturbed by a time-dependent electric field.The model has a rich phase diagram, and in particular, features a tachyonphase, where excitations travel faster than an effective speed of light.Surprisingly, we find that the dc conductivity of tachyons is finite, and theoptical sum rule is exactly satisfied for all masses. Our results highlight thepeculiar properties of the Kubo formula for non-Hermitian systems and areapplicable for a large variety of settings.

Kubo formula for non-Hermitian systems and tachyon optical conductivity