TY - JOUR

T1 - Momentum space metric, nonlocal operator, and topological insulators

AU - Matsuura, Shunji

AU - Ryu, Shinsei

N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.

PY - 2010

Y1 - 2010

N2 - 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.

AB - 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.

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U2 - 10.1103/PhysRevB.82.245113

DO - 10.1103/PhysRevB.82.245113

M3 - Article

AN - SCOPUS:78651291171

VL - 82

JO - Physical Review B-Condensed Matter

JF - Physical Review B-Condensed Matter

SN - 0163-1829

IS - 24

M1 - 245113

ER -