Momentum space metric, non-local operator, and topological insulators

Abstract

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 non-local 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 non-local 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.

13 pages, 4 figures

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• Measurement of Topological Order based on Metric-Curvature Correspondence
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• Nontrivial quantum geometry of degenerate flat bands
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• Intrinsic nonlinear conductivity induced by quantum geometry in altermagnets and measurement of the in-plane Néel vector
• Dielectric and optical markers originating from quantum geometry
• Effective description of Chern insulators
• Hofstadter spectrum of Chern bands in twisted transition metal dichalcogenides
• Band geometry from position-momentum duality at topological band crossings
• Intrinsic nonreciprocal bulk plasmons in noncentrosymmetric magnetic systems
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• Non-local order parameters for states with topological electromagnetic response
• Detecting the spread of valence band Wannier functions by optical sum rules
• Quantum geometrical properties of topological materials
• Diagnosing Topological Phase Transitions in 1D Superconductors using Berry Singularity Markers
• Demystifying the Holographic Mystique
• Quantum geometry and elliptic optical dichroism in $p$-wave magnets
• Nonlinear Hall effects with an exceptional ring
• Quantum geometry and $X$-wave magnets with $X=p,d,f,g,i$
• Quantum geometric origin of the Meissner effect and superfluid weight marker
• Opacity of graphene independent of light frequency and polarization due to the topological charge of the Dirac points
• Localization anisotropy and complex geometry in two-dimensional insulators
• Higher-order bulk photovoltaic effects, quantum geometry and application to $p$-wave magnets
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• Quantum Hall and Light Responses in a 2D Topological Semimetal
• Quantum geometric tensors from sub-bundle geometry
• Absorbance marker: Detection of quantum geometry and spread of Wannier function in disordered 2D semiconductors