Rotor/spin-wave theory for quantum spin models with U(1) symmetry

Elementary excitations of ordered quantum magnets are harmonic spin waves. Yet, when ordering breaks a continuous symmetry, excitations at zero momentum cannot really be treated as harmonic, because they correspond to the Goldstone mode that restores the symmetry in finite-size systems. Here, the authors show that zero-momentum excitations can be treated in a fully nonlinear way in terms of a macroscopic spin variable (the rotor): its spectrum reconstructs the so-called Anderson tower of states, a hallmark of continuous-symmetry breaking systems. The rotor variable can be considered as nearly separate from finite-momentum spin waves, leading to a quantitative reconstruction of the low-energy excitation spectrum of finite-size systems.