An interplay of topology and quantized geometric phase for two different symmetry-class Hamiltonians

Study of symmetry, topology and geometric phase can reveal many new and interesting results on the topological states of matter. Here we present a completely new and interesting result of symmetry, topology and quantization of geometric phase along with the physical explanation for two different symmetry classes. We present a detailed study of the auxiliary space for two different symmetry classes of Hamiltonians. We show explicitly that the origin of the auxiliary space inside the curve is only a necessary condition but it is not a sufficient condition for the topological state. One of the most interesting results is that same symmetry-class Hamiltonians show different behaviour in topology and quantized geometric phase.