Relationship between the transverse-field Ising model and the XY model via the rotating-wave approximation

TL;DR

This paper explores how the transverse-field Ising model can be approximately related to the XY model using the rotating-wave approximation, analyzing the conditions and limitations of this mapping through wavefunction overlap and observable comparisons.

Contribution

It provides a detailed analysis of the conditions under which the transverse-field Ising model maps onto the XY model, including the robustness and failure modes of this approximation.

Findings

The mapping is robust at short times.

The mapping fails at longer simulation times.

Trade-offs exist between simulation duration and measurement precision.

Abstract

In a large transverse field, there is an energy cost associated with flipping spins along the axis of the field. This penalty can be employed to relate the transverse-field Ising model in a large field to the XY model in no field (when measurements are performed at the right stroboscopic times). We describe the details for how this relationship works and, in particular, we also show under what circumstances it fails. We examine wavefunction overlap between the two models and observables, such as spin-spin Green's functions. In general, the mapping is quite robust at short times, but will ultimately fail if the run time becomes too long. There is also a trade-off between the length of time one can run a simulation out to and the time jitter of the stroboscopic measurements that must be balanced when planning to employ this mapping.