Extension of semi-topological quantum geometry beyond two bands

Determine whether the semi-topological nature of the quantum metric, whereby the quantum metric and Berry curvature depend only on the normalized vector of Hamiltonian coefficients, continues to hold for systems with more than two bands.

Background

The paper constructs the quantum geometric tensor for a holographic insulating system whose effective topological Hamiltonian is a two-by-two matrix. In this setting, both the quantum metric and Berry curvature depend only on the normalized coefficient vector in the Pauli-matrix decomposition, which makes the quantum geometry semi-topological and explains the agreement with the corresponding two-dimensional Dirac model.

The authors explicitly leave unresolved whether this dependence, and hence the semi-topological character of the quantum metric, persists when the effective Hamiltonian contains more than two bands. Resolving this question would clarify the scope of the proposed quantum-geometric framework for multiband holographic systems.