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arXiv · 2303.00380

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

Tommaso Roscilde·Tommaso Comparin·Fabio Mezzacapo
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Abstract

The static and dynamics 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 non-linearities 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 non-equilibrium dynamics -- following a quantum quench -- dominated by the zero mode, for which a linearized approach fails. Focusing on the 2$d$ XX model with power-law decaying interactions, we compare our equilibrium predictions with unbiased quantum Monte Carlo results and exact diagonalization; and our non-equilibrium 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.

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Tommaso Roscilde, Tommaso Comparin, Fabio Mezzacapo. 2023-03-01. Rotor/spin-wave theory for quantum spin models with U(1) symmetry. https://doi.org/10.1103/physrevb.108.155130

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