Fermionic path integrals and correlation dynamics in a 1D XY system

TL;DR

This paper develops a rigorous fermionic path integral approach to compute time-dependent correlation functions in a 1D XY spin model, enabling accurate analysis of dynamics under various conditions.

Contribution

It introduces a consistent method for constructing fermionic path integrals in the XY model, addressing previous inconsistencies and extending analysis to driven magnetic fields.

Findings

Reproduces known static correlation functions

Derives formulas for driven transverse magnetic fields

Confirms results with the Kibble-Zurek mechanism

Abstract

We derive time dependent correlation functions in an one dimensional XY spin model with the use of generating functionals, the latter being defined as path integrals over fermionic coherent states. We focus on the proper construction of the aforementioned integrals in order to avoid the inconsistencies usually encountered in the literature. The static limit of our results successfully reproduces the known ones, confirming the validity of our construction and allowing for further investigation of the dynamics. In the same context, we examine the case of a general driven transverse magnetic field, for which case we derive formulas for the equal-time correlation functions and confirm the consistency of our results with the Kibble-Zurek mechanism.