The tipping points of complex physical systems are actually written into the literal geometry of their energy shapes.
April 29, 2026
Original Paper
The geometric origin of criticality: a universal mechanism in mean-field rotor Hamiltonians
SSRN · 6660639
The Takeaway
Phase transitions, like water turning to steam, were long thought to be driven purely by thermodynamic math. This study shows that these changes are actually caused by a geometric instability in the system's energy shell. When the curvature of this abstract shape vanishes, the system is forced to snap into a new state. This provides a visual and geometric way to predict when a system is about to collapse or change. It could help scientists identify early warning signs for critical failures in everything from power grids to climate patterns.
From the 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 tr