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

The static and dynamical properties of finite-size lattice quantum spin models which spontaneously break a continuous U(1) symmetry in the thermodynamic limit are of central importance for a wide variety of physical systems, from condensed matter to quantum simulation. Such systems are characterized by a Goldstone excitation branch, terminating in a zero mode whose theoretical treatment within a linearized approach leads to divergencies on finite-size systems, revealing that the assumption of symmetry breaking is ill defined away from the thermodynamic limit. In this work we show that, once all its nonlinearities are taken into account, the zero mode corresponds exactly to a U(1) quantum rotor, related to the Anderson tower of states expected in systems showing symmetry breaking in the thermodynamic limit. The finite-momentum modes, when weakly populated, can be instead safely linearized (namely treated within spin-wave theory) and effectively decoupled from the zero mode. This picture leads to an approximate separation of variables between rotor and spin-wave ones, which allows for a correct description of the ground-state and low-energy physics. Most importantly, it offers a quantitative treatment of the finite-size nonequilibrium dynamics-following a quantum quench-dominated by the zero mode, for which a linearized approach fails after a short time. Focusing on the 2d XX model with power-law decaying interactions, we compare our equilibrium predictions with unbiased quantum Monte Carlo results and exact diagonalization; and our nonequilibrium results with time-dependent variational Monte Carlo. The agreement is remarkable for all interaction ranges, and it improves the longer the range. Our rotor/spin-wave theory defines a successful strategy for the application of spin-wave theory and its extensions to finite-size systems at equilibrium or away from it.