Relaxation of the entanglement spectrum in quench dynamics of topological systems

We study how the entanglement spectrum relaxes to its steady state in one-dimensional quadratic systems after a quantum quench. In particular we apply the saddle point expansion to the dimerized chains and 1-D p-wave superconductors. We find that the entanglement spectrum always exhibits a power-law relaxation superimposed with oscillations at certain characteristic angular frequencies. For the dimerized chains, we find that the exponent $\nu$ of the power-law decay is always $3/2$. For 1-D p-wave superconductors, however, we find that depending on the initial and final Hamiltonian, the exponent $\nu$ can take value from a limited list of values. The smallest possible value is $\nu=1/2$, which leads to a very slow convergence to its steady state value.