We introduce and consider the notion of *stable degeneracies* of translation invariant energy functions, taken at spin configurations of a finite Ising model. By this term we mean the lack of injectivity that cannot be lifted by changing the interaction.

We show that besides the symmetry-induced degeneracies, related to spin flip, translation and reflection, there exist additional stable degeneracies, due to more subtle symmetries. One such symmetry is the one of the Singer group of a finite projective plane.

Others are described by combinatorial relations akin to trace identities. Our results resemble traits of the length spectrum for closed geodesics on a Riemannian surface of constant negative curvature. There, stable degeneracy is defined w.r.t. Teichmüller space as parameter space.

}, author = {Knauf, Andreas}, doi = {10.1007/s00220-016-2579-x}, faupublication = {yes}, journal = {Communications in Mathematical Physics}, pages = {1432-0916}, peerreviewed = {Yes}, title = {{Stable} {Degeneracies} for {Ising} {Models}}, year = {2016} }