We unveil the role of the long time average of Loschmidt echo in the characterization of nonequilibrium quantum phase transitions by studying sudden quench processes across quantum phase transitions in various quantum systems. While the dynamical quantum phase transitions are characterized by the emergence of a series of zero points at critical times during time evolution, we demonstrate that nonequilibrium quantum phase transitions can be identified by nonanalyticities in the long time average of Loschmidt echo. The nonanalytic behaviours are illustrated by a sharp change in the long time average of Loschmidt echo or the corresponding rate function or the emergence of divergence in the second derivative of rate function when the driving quench parameter crosses the phase transition points. The connection between the second derivative of rate function and fidelity susceptibility is also discussed.

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We investigate the topological properties of a ladder model of the dimerized Kitaev superconductor chains. The topological class of the system is determined by the relative phase θ between the inter-and intra-chain superconducting pairing. One topological class is the class BDI characterized by the Z index, and the other is the class D characterized by the Z 2 index. For the two different topological classes, the topological phase diagrams of the system are presented by calculating two different topological numbers, i.e., the Z index winding number W and the Z 2 index Majorana number M , respectively. In the case of θ = 0, the topological class belongs to the class BDI, multiple topological phase transitions accompanying the variation of the number of Majorana zero modes are observed. In the case of θ = π/2 it belongs to the class D. Our results show that for the given value of dimerization, the topologically nontrivial and trivial phases alternate with the variation of chemical potential.