Dynamical topology of quantum quenches in two dimensions

Haiping Hu, Erhai Zhao

Published 2019-11-06Version 1

The quench dynamics following a sudden change in the Hamiltonian of a quantum system can be very complex. For band insulators, the global properties of a quantum quench may be captured by its topological invariants over the space-time continuum, which are intimately related to the static band topology of the pre- and post-quench Hamiltonian. 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 $W_3=\mathcal{C}_f-\mathcal{C}_i$, 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 gives rise to torus links and knots for quench to Hamiltonians with Dirac points. This framework can be generalized to multiband systems and other dimensions.