Large-S Spin-Wave Theory for the 3D Quantum XY Model

The low-temperature behavior of spin systems with a continuous symmetry is expected to be governed by Gaussian fluctuations around the ground state, known as spin waves. For classical models this picture has been rigorously established by Bricmont-Fontaine-Lebowitz-Lieb-Spencer (1982) for the O(2) model and, more recently, by Giuliani-Ott (2025) for general O(N) systems. In the quantum setting, however, the situation is more subtle: the low-energy behavior arises from a nontrivial interplay between thermal and quantum excitations, and a rigorous justification of spin-wave theory, understood as a semiclassical expansion in the spin size S, has remained open. In this thesis we investigate this problem for the three dimensional quantum XY model in the large-S regime. The main result is the construction of the large-S formal expansion of the zero-temperature spontaneous magnetization to arbitrary order. The analysis is based on the Holstein-Primakoff representation, which maps spin operators to bosonic creation and annihilation operators and naturally suggests a semiclassical expansion in powers of 1/S. After performing a formal expansion of the resulting bosonic Hamiltonian, the model is analyzed using perturbative methods combined with renormalization group techniques. Infrared divergences are controlled by a multiscale analysis and by exploiting a Ward identity associated with the underlying O(2) symmetry of the model. The results clarify the structure of the large-S spin-wave expansion and provide a rigorous construction of its formal series for the quantum XY model.