  Motivated by symmetry-protected topological phases (SPTs) with both spatial
symmetry (e.g., lattice rotation) and internal symmetry (e.g., spin rotation),
we propose a class of exotic topological terms, which generalize the well-known
Wen-Zee topological terms of quantum Hall systems [X.-G. Wen and A. Zee, Phys.
Rev. Lett. 69, 953 (1992)]. These generalized Wen-Zee terms are expressed as
wedge product of spin connection and usual gauge fields (1-form or higher) in
various dimensions. In order to probe SPT orders, we externally insert
"symmetry twists" like domain walls of discrete internal symmetry and
disclinations that are geometric defects with nontrivial Riemann curvature.
Then, generalized Wen-Zee terms simply tells us how SPTs respond to those
symmetry twists. Classifying these exotic topological terms thus leads to a
complete classification and characterization of SPTs within the present
framework. We also propose SPT low-energy field theories, from which
generalized Wen-Zee terms are deduced as topological response actions.
Following the Abstract of Wen-Zee paper, our work enriches alternative
possibilities of condensed-matter realization of unification of
electromagnetism and "gravity".
