Applied Mathematics

Volume 14, Issue 1 (January 2023)

ISSN Print: 2152-7385   ISSN Online: 2152-7393

Google-based Impact Factor: 0.58  Citations  

A Gauge-Invariant Geometric Phase for Electrons in a One-Dimensional Periodic Lattice

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DOI: 10.4236/am.2023.141005    104 Downloads   443 Views  


Here the notion of geometric phase acquired by an electron in a one-dimensional periodic lattice as it traverses the Bloch band is carefully studied. Such a geometric phase is useful in characterizing the topological properties and the electric polarization of the periodic system. An expression for this geometric phase was first provided by Zak, in a celebrated work three decades ago. Unfortunately, Zak’s expression suffers from two shortcomings: its value depends upon the choice of origin of the unit cell, and is gauge dependent. Upon careful investigation of the time evolution of the system, here we find that the system displays cyclicity in a generalized sense wherein the physical observables return in the course of evolution, rather than the density matrix. Recognition of this 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 shortcomings of Zak’s expression, and correctly classifies the Bloch bands of the lattice. A naturally filled band extension of the Pancharatnam-Zak phase is also constructed and studied.

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Vyas, V. and Roy, D. (2023) A Gauge-Invariant Geometric Phase for Electrons in a One-Dimensional Periodic Lattice. Applied Mathematics, 14, 82-106. doi: 10.4236/am.2023.141005.

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