Signatures and conditions for phase band crossings in periodically driven integrable systems

TL;DR

This paper derives general conditions for phase band crossings in periodically driven integrable free fermionic systems, linking these crossings to system parameters, critical points, and observable correlation functions across various models.

Contribution

It provides the first comprehensive criteria for phase band crossings in driven integrable systems, including effects of critical points and implications for correlation functions.

Findings

Phase band crossings depend on the presence of critical points in the Hamiltonian.

Correlation functions exhibit characteristic maxima/minima related to phase band crossings.

Results apply to a broad class of models including Ising, XY, Kitaev, and Dirac fermions.

Abstract

We present generic conditions for phase band crossings for a class of periodically driven integrable systems represented by free fermionic models subjected to arbitrary periodic drive protocols characterized by a frequency $\omega_D$. These models provide a representation for the Ising and $XY$ models in $d=1$, the Kitaev model in $d=2$, several kinds of superconductors, and Dirac fermions in graphene and atop topological insulator surfaces. Our results demonstrate that the presence of a critical point/region in the system Hamiltonian (which is traversed at a finite rate during the dynamics) may change the conditions for phase band crossings that occur at the critical modes. We also show that for $d>1$, phase band crossings leave their imprint on the equal-time off-diagonal fermionic correlation functions of these models; the Fourier transforms of such correlation functions, $F_{\vec k_0}( \omega_0)$, have maxima and minima at specific frequencies which can be directly related to $\omega_D$ and the time at which the phase bands cross at $\vec k = \vec k_0$. We discuss the significance of our results in the contexts of generic Hamiltonians with $N>2$ phase bands and the underlying symmetry of the driven Hamiltonian.