The geometric origin of criticality: a universal mechanism in mean-field rotor Hamiltonians

Abstract

We introduce a universal criterion for criticality in mean-field rotor Hamiltonians based on the geometric structure of the constant-energy shell. Rather than characterizing the onset of a phase transition through the conventional thermodynamic singularities alone, we show that the relevant information is already encoded in the way the geometry of the shell reorganizes along distinguished collective directions. For a broad class of finite-dimensional trigonometric mean-field interactions, the trace of the Weingarten operator (representing the principal curvatures) admits a universal collective expansion in terms of the order-parameter amplitudes. This expansion defines an energy-dependent quadratic form whose eigenmodes identify the geometrically unstable channels of the system. Criticality is then associated with the vanishing of the corresponding curvature coefficients, yielding a direct geometric selection principle for the modes that become unstable at the transition. In this way, the phase transition in mean-field systems (usually of first- or second-order) is reformulated as a geometric instability phenomenon intrinsic to the microcanonical energy shell. The resulting framework is geometrically universal within the class considered, independent of model-specific details except for a finite set of collective couplings. Moreover, our approach recovers the known critical channels in standard mean-field rotor models while extending naturally to multimode and spectrally coupled cases. These results support a view in which critical behavior can be understood as reorganizations of energy-shell geometry triggered by a collective restructuring of the underlying energy-shell geometry.