Distribution of Fisher zeros in dynamical quantum phase transitions of two-dimensional topological systems

The short time evolution of a topological system after a sudden quench is considered, calculating the Loschmidt rate and studying in detail the distribution of Fisher zeros. Singular behavior in the Loschmidt rate has been predicted and found in several systems and, in particular, in one dimension (1D) one finds sharp features in the time evolution of the Loschmidt rate. These are associated with critical times that may be seen as the imaginary part of the Fisher zeros, under the condition that their real part vanishes. In the complex plane one finds lines of zeros that cross the imaginary axis at certain points. In a 2D system, in general, the Fisher zeros occupy areas instead of lines, as shown before, with the consequences that a continuous set of critical times results along the imaginary axis and that one does not find a well defined singular behavior in the Loschmidt rate. Here we show that several 2D topological systems also display very narrow regions (effectively lines) of zeros, in addition to finite extent areas of zeros. As a consequence, well defined sharp critical times are found, under appropriate conditions. An example is a $2\text{D}\phantom{\rule{4pt}{0ex}}p$-wave superconductor in the presence of a magnetic field, but other systems share the same property, such as a Haldane insulator, a system with an anomalous Hall effect, and topological insulators. Details on the momentum distribution of the real and imaginary parts of the Fisher zeros are discussed, clarifying the existence of the lines of zeros and taking as an example the $p$-wave superconductor in magnetic field. The time dependence of the Loschmidt rate phase and of the Pancharatnam geometrical phase is considered, for different times after the quench and as a function of momentum. Pairs of vortices and antivortices in the phase associated with the geometric phase and the Loschmidt rate phase, that have been proposed for some 2D systems, are also shown to appear in the regime where we find lines of zeros.