Dynamical quantum phase transitions on random networks

We investigate two types of dynamical quantum phase transitions (DQPTs) in the transverse-field Ising model on ensembles of Erdős–Rényi networks of size N. These networks consist of vertices connected randomly with probability p ( 0<p⩽1). Using analytical derivations and numerical techniques, we compare the characteristics of the transitions for p < 1 against the fully connected network (p = 1). We analytically show that the overlap between the wave function after a quench and the wave function of the fully connected network after the same quench deviates by at most O(N−1/2). For a DQPT defined by an order parameter, the critical point remains unchanged for all p. For a DQPT defined by the rate function of the Loschmidt echo, we find that the rate function deviates from the p = 1 limit near vanishing points of the overlap with the initial state, while the critical point remains independent for all p. Our analysis suggests that this divergence arises from persistent non-trivial global many-body correlations absent in the p = 1 limit.