Imaginary Gauge Transformation in Momentum Space and Dirac Exceptional Point

An imaginary gauge transformation is at the core of the non-Hermitian skin effect. Here, we show that such a transformation can be performed in momentum space as well, which reveals that certain gain- and loss-modulated systems in their parity-time ($\mathcal{P}\mathcal{T}$) symmetric phases are equivalent to Hermitian systems with real potentials. Our analysis in momentum space also distinguishes two types of exceptional points (EPs) in the same system. Besides the conventional type that leads to a $\mathcal{P}\mathcal{T}$ transition upon the continuous increase of gain and loss, we find real-valued energy bands connected at a Dirac EP in hybrid dimensions, consisting of a spatial dimension and a synthetic dimension for the gain and loss strength.