Geometric orbital susceptibility: quantum metric without Berry curvature

The orbital magnetic susceptibility of an electron gas in a periodic potential depends not only on the zero field energy spectrum but also on the geometric structure of cell-periodic Bloch states which encodes interband effects. In addition to the Berry curvature, we explicitly relate the orbital susceptibility of two-band models to a quantum metric tensor defining a distance in Hilbert space. Within a simple tight-binding model allowing for a tunable Bloch geometry, we show that interband effects are essential even in the absence of Berry curvature. We also show that for a flat band model, the quantum metric gives rise to a very strong orbital paramagnetism.