Scale invariant transfer matrices and Hamiltionians

Understand

Given a direct system of Hilbert spaces $s\mapsto \mathcal H_s$ (with isometric inclusion maps $\iota_s^t:\mathcal H_s\rightarrow \mathcal H_t$ for $s\leq t$) corresponding to quantum systems on scales $s$, we define notions of scale invariant and weakly scale invariant operators.

• Is some cases of quantum spin chains we find conditions for transfer matrices and nearest neighbour Hamiltonians to be scale invariant or weakly so.
• Scale invariance forces spatial inhomogeneity of the spectral parameter.
• But weakly scale invariant transfer matrices may be spatially homogeneous in which case the change of spectral parameter from one scale to another is governed by a classical dynamical system exhibiting fractal behaviour.