Pancharatnam-Zak phase

TL;DR

This paper introduces the Pancharatnam-Zak phase, a corrected geometric phase for electrons in periodic lattices that overcomes previous flaws and accurately characterizes topological properties.

Contribution

It provides a new formulation of the geometric phase, called Pancharatnam-Zak phase, addressing flaws in Zak's original expression and establishing a more consistent topological classification.

Findings

Pancharatnam-Zak phase is gauge invariant and origin independent.

The new phase correctly classifies Bloch bands.

Extension to filled bands is developed.

Abstract

Three decades ago, in a celebrated work, Zak found an expression for the geometric phase acquired by an electron in a one-dimensional periodic lattice as it traverses the Bloch band. Such a geometric phase is useful in characterizing the topological properties and the electric polarization of the periodic system. Unfortunately Zak's expression suffers from two flaws: its value depends upon the choice of origin of the unit cell, and is gauge dependent. Here we explain that these flaws in Zak's expression arise from the assumption that the electron's adiabatic motion is cyclic in the sense of recurrence of the density matrix in course of time evolution. We find through a careful investigation that the system displays cyclicity in a generalized sense wherein the physical observables return in the course of evolution. This notion of generalized cyclicity paves the way for a correct and consistent expression for the geometric phase in this system, christened as Pancharatnam-Zak phase. Pancharatnam-Zak geometric phase does not suffer from the flaws inherent in Zak's expression, and correctly classifies the Bloch bands of the lattice. A natural filled band extension of the Pancharatnam-Zak phase is also constructed and studied.