Speaker
Description
We consider integrable three-state spin chain models that appear in a strongly twisted, double-scaled deformation of $\mathcal{N}=4$ Super Yang-Mills Theory. These three-parameter „eclectic" models are non-diagonalizable. Their spectrum of Jordan blocks shows surprisingly subtle yet regular patterns with regard to sizes and multiplicities.
In a first approach, we derive deformed Bethe equations. They are exactly solvable, scale with non-trivial critical exponents, exhibit completeness of states, and are consistent with Pólya's enumeration theorem. However, the quantum inverse scattering method does not account for the spectrum of Jordan blocks, as all Bethe vectors collapse to some trivial states.
In our second approach, we start with a simpler hypereclectic spin chain model, where all parameters are set to zero except for one. Despite its extreme simplicity, it appears to reproduce the full spectrum of the eclectic model, a phenomenon we call universality. Based on combinatorics and linear algebra considerations, we derive a generating function encoding the spectrum with the help of q-binomial coefficients. We treat the full state space, as well as its physically relevant restriction to cyclic states. The latter involves a certain q-deformation of the general Pólya enumeration theorem.
