This article is meant as a gentle introduction to the topological terms that often play a decisive role in effective theories describing topological quantum effects in condensed matter systems. We first take up several prominent examples, mainly from
the area of quantum magnetism and superfluids/superconductors. We then briefly discuss how these ideas are now finding incarnations in the studies of symmetry-protected topological phases, which are in a sense the generalization of the concept of topological insulators to a wider range of materials, including magnets and cold atoms.
A symmetry-protected topologically ordered phase is a short-range entangled state, for which some imposed symmetry prohibits the adiabatic deformation into a trivial state which lacks entanglement. In this paper we argue that magnetization plateau st
ates of one-dimensional antiferromagnets which satisfy the conditions $S-min$ odd integer, where $S$ is the spin quantum number and $m$ the magnetization per site, can be identified as symmetry-protected topological states if an inversion symmetry about the link center is present. This assertion is reached by mapping the antiferromagnet into a nonlinear sigma model type effective field theory containing a novel Berry phase term (a total derivative term) with a coefficient proportional to the quantity $S-m$, and then analyzing the topological structure of the ground state wave functional which is inherited from the latter term. A boson-vortex duality transformation is employed to examine the topological stability of the ground state in the absence/presence of a perturbation violating link-center inversion symmetry. Our prediction based on field theories is verified by means of a numerical study of the entanglement spectra of actual spin chains, which we find to exhibit twofold degeneracies when the aforementioned condition is met. We complete this study with a rigorous analysis using matrix product states.
