PancharatnamZak phase
Abstract
Three decades ago, in a celebrated work, Zak found an expression for the geometric phase acquired by an electron in a onedimensional 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 PancharatnamZak phase. PancharatnamZak 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 PancharatnamZak phase is also constructed and studied.
 Publication:

arXiv eprints
 Pub Date:
 September 2019
 DOI:
 10.48550/arXiv.1909.00818
 arXiv:
 arXiv:1909.00818
 Bibcode:
 2019arXiv190900818V
 Keywords:

 Quantum Physics;
 Condensed Matter  Mesoscale and Nanoscale Physics;
 Condensed Matter  Superconductivity
 EPrint:
 24 pages, 4 figures