Current Developments in Mathematics

Volume 2020

Classification of symmetry protected topological phases in quantum spin chains

Pages: 41 – 104



Yoshiko Ogata (Graduate School of Mathematical Sciences, University of Tokyo, Komaba, Tokyo, Japan)


We consider the classification problem of symmetry protected topological (SPT) phases on quantum spin systems. SPT phases are gapped short-range-entangled quantum phases with a symmetry $G$. We explain that in one and two-dimensional quantum spin systems, there are $H^2(G,U(1))/H^3(G,U(1))$‑valued invariant, confirming a physicists conjecture.

The author was supported in part by the Grants-in-Aid for Scientific Research, JSPS.

Published 30 September 2022