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  • Open Access

Inequivalent Berry Phases for the Bulk Polarization

Haruki Watanabe1,* and Masaki Oshikawa2,†

  • 1Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
  • 2Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan

  • *haruki.watanabe@ap.t.u-tokyo.ac.jp
  • †oshikawa@issp.u-tokyo.ac.jp
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Phys. Rev. X 8, 021065 – Published 15 June, 2018

DOI: https://doi.org/10.1103/PhysRevX.8.021065

Abstract

We discuss the characterization of the polarization for insulators under the periodic boundary condition in terms of the Berry phase, clarifying confusing subtleties. For band insulators, the Berry phase can be formulated in terms of the Bloch function in momentum space. More generally, in the presence of interactions or disorders, one can instead use the many-body ground state as a function of the flux piercing the ring. However, the definition of the Bloch function and the way of describing the flux are not unique. As a result, the value of the Berry phase and its behavior depend on how precisely it is defined. In particular, identifying the Berry phase as a polarization, its change represents a polarization current, which also depends on the definition. We demonstrate this by elucidating mutual relations among different definitions of the Berry phase and showing that they correspond to currents measured differently in real space. Despite the nonuniqueness of the polarization current, the total charge transported during a Thouless pumping process is independent of the definition, reflecting its topological nature.

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Popular Summary

One of the basic properties for characterizing a material is its polarization, which measures how positive and negative electric charges are distributed within the material. In recent years, polarizations have been extensively used in the study of topological insulators—exotic materials that conduct electricity on their surface but behave as insulators in their interior. Here, we explore the relationship between the bulk polarization and a property known as the Berry phase.

The Berry phase is a geometric quantity that captures part of how a material’s quantum-mechanical wave function changes when the environment undergoes a cyclic change (like a weak magnetic field oscillating in time). Previous studies have suggested that the Berry phase can characterize the polarization.

Using paper-and-pencil calculations, we show that there are actually a variety of Berry phases related to the polarization, and each one has its own physical meaning. These phases depend on how the environmental parameters are changed, and they can be used to describe transports in various systems in terms of how the polarization changes over time. We clarify the mutual relations among the Berry phases as well as their relations to the polarization.

Our discovery of multiple Berry phases should lead to a deeper understanding of material polarization, which, in turn, may accelerate the theoretical advance of topological insulators.

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Phys. Rev. X 8, 021065– Published 15 June, 2018
  • Received 27 February 2018
  • Revised 7 May 2018
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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