Cell Natural Orbital Decomposition of Fermi Surfaces

Quantum geometry, defined via overlaps between the cell-periodic parts of the Bloch wavefunctions, plays a crucial role in many emergent properties of correlated materials. It manifests as form factors in projected two-body operators. Naturally, these form factors appear in susceptibilities and are expected to modify the weak-coupling analysis of a given Fermi surface. However it is unclear if taking the form factors into account leads to a qualitatively new instability or small corrections. To clarify this role of quantum geometry in the low-energy phenomenology of metals, we introduce the Cell Natural Orbital (CNO) decomposition on the Fermi surface. We utilise the CNOs as the natural basis that decomposes the form factors and show that this decomposition quantitatively ranks the importance of quantum geometry at the Fermi surface. Our formalism can be applied to either tight-binding models or continuum models. In particular, we show that CNOs distinguish the two different van Hove singularities in Kagome metals, and allow us to obtain low-energy Hamiltonians to capture the essential geometric content of twisted WSe2.