  The presence of nonanalyticity in observables is a manifestation of phase
transitions. Through the study of two paradigmatic topological models in one
and two dimensions, in this work we show that the circuit complexity based on
our specific quantification can reveal the occurrence of topological phase
transitions, both in and out of equilibrium, by the presence of nonanalyticity.
By quenching the system out of equilibrium, we find that the circuit complexity
grows linearly or quadratically in the short-time regime if the quench is
finished instantaneously or in a finite time, respectively. Notably, we find
that for both the sudden quench and the finite-time quench, a topological phase
transition in the pre-quench Hamiltonian will be manifested by the presence of
nonanalyticity in the first-order or second-order derivative of circuit
complexity with respect to time in the short-time regime, and a topological
phase transition in the post-quench Hamiltonian will be manifested by the
presence of nonanalyticity in the steady value of circuit complexity in the
long-time regime. We also show that the increase of dimension does not remove,
but only weakens the nonanalyticity of circuit complexity. Our findings can be
tested in quantum simulators and cold-atom systems.
