We unveil the stable ($d+1$)-dimensional topological structures underlying the quench dynamics for all of the Altland-Zirnbauer classes in $d=1$ dimension, and we propose to detect such dynamical topology from the time evolution of entanglement spectra. Focusing on systems in classes BDI and D, we find crossings in single-particle entanglement spectra for quantum quenches between different symmetry-protected topological phases. The entanglement-spectrum crossings are shown to be stable against symmetry-preserving disorder and faithfully reflect both $\mathbb{Z}$ (class BDI) and ${\mathbb{Z}}_2$ (class D) topological characterizations. As a by-product, we unravel the topological origin of the global degeneracies temporarily emerging in the many-body entanglement spectrum in the quench dynamics of the transverse-field Ising model. These findings can experimentally be tested in ultracold atoms and trapped ions with the help of cutting-edge tomography for quantum many-body states. Our work paves the way towards a systematic understanding of the role of topology in quench dynamics.

• Received 17 October 2017
• Revised 29 March 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.250601