Momentum space metric, nonlocal operator, and topological insulators

Shunji Matsuura, Shinsei Ryu

Research output: Contribution to journalArticlepeer-review


Momentum space of a gapped quantum system is a metric space: it admits a notion of distance reflecting properties of its quantum ground state. By using this quantum metric, we investigate geometric properties of momentum space. In particular, we introduce a nonlocal operator which represents distance square in real space and show that this corresponds to the Laplacian in curved momentum space, and also derive its path-integral representation in momentum space. The quantum metric itself measures the second cumulant of the position operator in real space, much like the Berry gauge potential measures the first cumulant or the electric polarization in real space. By using the nonlocal operator and the metric, we study some aspects of topological phases such as topological invariants, the cumulants and topological phase transitions. The effect of interactions to the momentum space geometry is also discussed.

Original languageEnglish (US)
Article number245113
JournalPhysical Review B - Condensed Matter and Materials Physics
Issue number24
StatePublished - 2010

ASJC Scopus subject areas

  • Electronic, Optical and Magnetic Materials
  • Condensed Matter Physics


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