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

We classify effective actions for Nambu-Goldstone (NG) bosons assuming the absence of anomalies. Special attention is paid to Lagrangians that are invariant only up to a surface term, which are shown to be in a one-to-one correspondence with Chern-Simons (CS) theories for unbroken symmetry. Without making specific assumptions on the 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 levels. The order-one CS term is responsible for the noninvariance of the 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.