Topological Invariants for Quantum Quench Dynamics

Quantum quench is nonequilibrium dynamics and its interplay with band topology gives rise to intriguing dynamical topological phenomena. We introduce the concept of loop unitary $$U_l$$ and its homotopy invariant $$W_3$$ to fully characterize the quench dynamics of arbitrary two-band insulators in two dimensions, going beyond existing scheme based on Hopf invariant which is only valid for trivial initial states. The theory traces the origin of nontrivial dynamical topology to the emergence of $$\pi$$-defects in the phase band of $$U_l$$, and establishes that $$W_3=C_f-C_i$$, i.e. the Chern number change across the quench. We further show that the dynamical singularity is also encoded in the winding of the eigenvectors of $$U_l$$ along a lower dimensional curve where dynamical quantum phase transition occurs, if the pre- or post-quench Hamiltonian is trivial. The winding along this curve is related to the Hopf link, and shown to give rise to torus links and knots for quench to Hamiltonians with Dirac points. Our framework which can be generalized to multiband systems and other dimensions paves the way to study quench dynamics and its associated topology.