Momentum-space gravity from the quantum geometry and entropy of Bloch electrons

Quantum geometry is a key quantity that distinguishes electrons in crystal from those the vacuum. Its study continues to provide insights into quantum materials, uncovering new design principles for their discovery. However, unlike Berry curvature, an intuitive understanding of metric lacking. Here, we show Bloch leads momentum-space gravity. In particular, by extending semiclassical formulation electron dynamics second order, find resulting velocity modified geodesic term and becomes dual Lorentz force curved space. We calculate this response magic-angle twisted bilayer graphene moir\'e systems with flat bands are ideal candidates observe effect. Extending analogy gravity further, Einstein field equations remains sourceless pure states, while mixed states it acquires source depends on von Neumann entropy small entropies. compare stress-energy equation weak-field limit general relativity, conclude gravitational potential. Consequently, resembling entropic force. Our results highlight connections between geometry, gravity, information, prompting further exploration materials.