Systematic Effective Field Theory Analysis of the D=2+1 Quantum XY Model at Low Temperatures

The low-temperature properties of the (2+1)-dimensional quantum XY model are studied within the framework of effective Lagrangians up to three-loop order. At zero temperature, the system is characterized by a spontaneously broken spin rotation symmetry, O(2) $\to$ 1, where the corresponding Goldstone bosons are the spin waves or magnons. Even though there is no spontaneously broken order at finite $T$, the low-temperature behavior of the system is still governed by the spin waves. The partition function is evaluated and various thermodynamic quantities, including the order parameter, are derived in the presence of a weak external field. In particular, we show that the spin-wave interaction is repulsive at low temperatures, its magnitude depending on the strength of the external field. We compare our results with those for the (2+1)-dimensional antiferromagnetic Heisenberg model which, at T=0, exhibits the spontaneously broken spin rotation symmetry O(3) $\to$ O(2).