Topological interactions of Nambu-Goldstone bosons in quantum many-body systems

We classify effective actions for Nambu-Goldstone (NG) bosons assuming absence of anomalies. Special attention is paid to Lagrangians invariant only up to a surface term, shown to be in a one-to-one correspondence with Chern-Simons (CS) theories for unbroken symmetry. Without making specific assumptions on spacetime symmetry, we give explicit expressions for these Lagrangians, generalizing the Berry and Hopf terms in ferromagnets. Globally well-defined matrix expressions are derived for symmetric coset spaces of broken symmetry. The CS Lagrangians exhibit special properties, on both the perturbative and the global topological level. The order-one CS term is responsible for non-invariance of canonical momentum density under internal symmetry, known as the linear momentum problem. The order-three CS term gives rise to a novel type of interaction among NG bosons. All the CS terms are robust against local variations of microscopic physics.