The Loschmidt Index

We study the nodes of wavefunction overlap between ground states a parameter-dependent Hamiltonian. These are topological, and we can use them to analyze in unifying way both equilibrium dynamical quantum phase transitions multi-band systems. define Loschmidt index as number this discuss relationship wrapping closed auxiliary hypersurface. This allows us compute systematically, using an integral representation number. comment on other well-established topological numbers. As example, classify XY model by counting overlaps.