The adiabatic approximation for quantum spin systems with a spectral gap

Summary

We estimate the accuracy of the adiabatic approximation in predicting the time evolution of local observables for an XY quantum magnet with a slowly variable external magnetic field. The system evolves according to the natural Hamiltonian dynamics and the spectral gap produced by the magnetic field is assumed to be large with respect to the term inducing quantum fluctutions. The proof is based on a finite order truncation of a time dependent cluster expansion in inverse powers of the time scale τ. In the analytic case, we show that the accuracy of this truncated expansion is of order$O(e^{ - \alpha e\tau ^{\tfrac{1}{\alpha }} } )$ for any α>1. If the time dependent perturbation is suddenly switched on at time zero and switched off at time τ, the accuracy of the adiabatic approximation is proven to be of orderO(τ ⁻¹.