Quantum geometry (condensed matter)

This article is about the condensed-matter usage of quantum geometry based on Bloch-band wavefunctions and the quantum geometric tensor. For the quantum-gravity usage, see Quantum geometry.

Quantum geometry in condensed matter physics refers to gauge-invariant geometric properties of quantum states as functions of external parameters—most commonly the crystal momentum of Bloch-band eigenstates in a periodic solid. It provides a geometric language for how a band wavefunction changes across parameter space and how its phase twists under parallel transport, with consequences for semiclassical transport, topological band invariants, localization, and superconductivity in multiband and flat-band systems.

In many settings, quantum geometry is encoded in the quantum geometric tensor (QGT), a complex tensor field over parameter space. For a single isolated quantum state, the real (symmetric) part of the QGT defines a Riemannian metric called the quantum metric (equivalently, the pullback of the Fubini–Study metric on projective Hilbert space), while the imaginary (antisymmetric) part is the Berry curvature.

In crystalline solids, Berry curvature governs geometric contributions to semiclassical dynamics (including anomalous velocity terms) and underlies topological invariants such as Chern numbers that control quantized Hall responses. The quantum metric measures distances on the space of states and is tied to wavefunction localization, Wannier function spreads, interband matrix elements, and geometric contributions to response functions (for example, current noise and superfluid weight).

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