Momentum-space Gravity from the Quantum Geometry and Entropy of Bloch Electrons

Abstract

Quantum geometry is a key quantity that distinguishes electrons in a crystal from those in the vacuum. Its study continues to provide insights into quantum materials, uncovering new design principles for their discovery. Here, we show that the quantum metric of Bloch electrons leads to a momentum-space gravity. In particular, by extending the semiclassical formulation of electron dynamics to second order, we find that the resulting velocity is modified by a geodesic term and becomes the momentum-space dual of the Lorentz force in curved space. Extending this analogy with gravity further, we find that the momentum-space dual of the Einstein field equations remains sourceless for pure states while for mixed states it acquires a source term that depends on the von Neumann entropy, for small entropies. We compare this stress-energy equation with the weakfield limit of general relativity and conclude that the von Neumann entropy is the momentumspace dual of the gravitational potential. Consequently, the momentum-space geodesic equation for mixed states is modified by a term resembling an entropic force. Our results highlight connections between quantum geometry, momentum-space gravity and quantum information, prompting further exploration of this dual gravity in quantum materials.

Superfluid weight bounds from symmetry and quantum geometry in flat bands

“…where P (k) is the N orb × N orb gauge-invariant projection matrix into the occupied bands, ∂ i is a momentum-space derivative, and the trace is over the matrix indices. The abelian Berry curvature f ij = −f ji is well studied while the positive semi-definite quantum metric g ij = g ji is an object of more recent interest [54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69][70][71][72][73] . With spatial rotation FIG.…”