Dynamics of the Topological States of Matter

Topological phase transitions defy the conventional understanding on the typical continuous phase transitions which are driven by the spontaneous symmetry breaking and naturally described by local order parameter. Topological phase transitions involve the change in internal topology. Topological states are then classified by topological quantum numbers which are mostly discrete numbers such as topological Chern number in quantum Hall states and the number of gapless boundary states in topological insulators and superconductors. The issue in the dynamics of the topological states is that the discrete topological numbers are inadequate to describe the topological order during dynamical phase transition. In particular, the topological quantum numbers concern about only the ground state even though the time evolution should involve the excited states. In order to describe the full aspects of the time evolution of the topological order, one should define the order with respect to the full wave function, which is expected to be necessarily nonlocal. In this article, we briefly review recent studies and developments of the dynamics of the topological states, focusing on one-dimensional topological superconductors and their edge states, Majorana bound states.