Fisher zeroes and dynamical quantum phase transitions for two- and three-dimensional models

Dynamical quantum phase transitions are nonanalyticities in a dynamical free energy (or return rate), which occur at critical times. Although extensively studied in one dimension, the exact nature of the nonanalyticity in two and three dimensions has not yet been fully investigated. In two dimensions, results so far are known only for relatively simple two-band models. Here, we study the general two- and three-dimensional cases. We establish the relation between the nonanalyticities in different dimensions, and the functional form of the densities of Fisher zeros. We show, in particular, that entering a critical region where the density of Fisher zeros is nonzero at the boundary always leads to a cusp in the derivative of the return rate while the return rate itself is smooth. We illustrate our results by obtaining analytical results for exemplary two- and three-dimensional models.