Cosine Edge Mode in a Periodically Driven Quantum System

TL;DR

This paper introduces an analytically solvable two-band Floquet topological insulator model showing a unique cosine-shaped edge mode spectrum that remains robust across various phases and parameter changes.

Contribution

It presents a minimal, analytically tractable model for Floquet topological phases with a novel cosine edge mode spectrum, advancing understanding of driven topological systems.

Findings

Edge spectrum always follows a cosine function, independent of parameters.

Cosine edge mode persists even in semi-metallic phases with Dirac points.

Localization length of the edge mode is quantized, containing an integer.

Abstract

Time-periodic (Floquet) topological phases of matter exhibit bulk-edge relationships that are more complex than static topological insulators and superconductors. Finding the edge modes unique to driven systems usually requires numerics. Here we present a minimal two-band model of Floquet topological insulators and semimetals in two dimensions where all the bulk and edge properties can be obtained analytically. It is based on the extended Harper model of quantum Hall effect at flux one half. We show that periodical driving gives rise to a series of phases characterized by a pair of integers. The model has a most striking feature: the spectrum of the edge modes is always given by a single cosine function, $\omega(k_y)\propto \cos k_y$ where $k_y$ is the wave number along the edge, as if it is freely dispersing and completely decoupled from the bulk. The cosine mode is robust against the change in driving parameters and persists even to semi-metallic phases with Dirac points. The localization length of the cosine mode is found to contain an integer and in this sense quantized.