Zero Curvature Condition for Quantum Criticality

Quantum criticality often lies beyond the scope of the conventional Landau paradigm, and a unifying framework has yet to emerge, due in part to the wide variety of quantum orders. We propose a geometric approach to quantum phase transitions that shifts focus from microscopic order to the competition between noncommuting operators. This competition is encoded in the boundary geometry of their expectation values, defining a quantum observable space. We show that quantum phase transitions occur at zero-curvature points on the quantum observable space boundary, signaling maximal commutativity and suggesting an underlying integrable structure at criticality.