Competing orders, nonlinear sigma models, and topological terms in quantum magnets

A number of examples have demonstrated the failure of the Landau-Ginzburg-Wilson (LGW) paradigm in describing the competing phases and phase transitions of two-dimensional quantum magnets. In this paper, we argue that such magnets possess field theoretic descriptions in terms of their slow fluctuating orders provided certain topological terms are included in the action. These topological terms may thus be viewed as what goes wrong within the conventional LGW thinking. The field theoretic descriptions we develop are possible alternates to the popular gauge theories of such non-LGW behavior. Examples that are studied include weakly coupled quasi-one-dimensional spin chains, deconfined critical points in fully two-dimensional magnets, and two-component massless QED₃. A prominent role is played by an anisotropic O (4) nonlinear sigma model in three space-time dimensions with a topological theta term. Some properties of this model are discussed. We speculate that similar sigma model descriptions might exist for fermionic algebraic spin liquid phases.