The Lee-Yang property says that the partition function of a spin model has only purely imaginary zeros as a function of an external magnetic field. When it holds, it constrains the thermodynamic behavior powerfully — no phase transitions in the wrong places, analyticity where you need it, a clean mathematical structure that makes many analytic methods tractable.
For scalar spins (D = 1), this is classical. For isotropic D-dimensional rotors — vector spins that can point in any direction in D-dimensional space — a generalized Lee-Yang property has been established only for D = 2. The extension to higher dimensions has been open.
Classes and Rebenko prove it for all even D. The mathematical physics that makes D = 2 work (complex analytic structure, rotation invariance, specific integral representations) turns out to generalize when D is even but not obviously when D is odd. Even-dimensional spaces have a natural complex structure — you can pair coordinates into complex numbers — that odd-dimensional spaces lack. The partition function's zeros stay imaginary precisely because the underlying symmetry permits a complexification that keeps the algebra controlled.
The structural point: parity of dimension constrains physics through algebra. Even D works because the symmetry group SO(D) for even D admits representations with properties that the partition function's analytic structure requires. The proof doesn't merely extend D = 2 by induction — it uses the specific algebraic features of even-dimensional rotation groups. The dimension being even is not incidental; it's the reason the property holds.
(arXiv:2603.18675)