Berezinskii–Kosterlitz–Thouless-Type Transitions in d = 2 Quantum O(2) and O(2) × O(2) Nonlinear Sigma Models

We discuss the d = 2 quantum O(2) × O(2) nonlinear sigma model as a low-energy theory of phase reconstruction near a quantum critical point. We first examine the evolution of the Berezinskii–Kosterlitz–Thouless (BKT) transition as the quantum limit is approached in the usual O(2) nonlinear sigma model. Then we go on to review results on the ground-state phase diagram of the O(2) × O(2) nonlinear sigma model, and on the behaviour of the O(2) × O(M) nonlinear sigma model with M > 2 in the classical limit. Finally, we present a conjectured finite-temperature phase diagram for the quantum version of the latter model in the O(2) × O(2) case. The nature of the finite-temperature BKT-like transitions in the phase diagram is discussed, and avenues for further calculation are identified.