Continuum Limits of Homogeneous Binary Trees and the Thompson Group

Tree tensor network descriptions of critical quantum spin chains are empirically known to reproduce correlation functions matching conformal field theory (CFT) predictions in the continuum limit. It is natural to seek a more complete correspondence, additionally incorporating dynamics. On the CFT side, this is determined by a representation of the diffeomorphism group of the circle. In a remarkable series of papers, Jones outlined a research program where the Thompson group $\mathsf{T}$ takes the role of the latter in the discrete setting, and representations of $\mathsf{T}$ are constructed from certain elements of a subfactor planar algebra. He also showed that, for a particular example of such a construction, this approach only yields—in the continuum limit—a representation which is highly discontinuous and hence unphysical. Here we show that the same issue arises generically when considering tree tensor networks: the set of coarse-graining maps yielding discontinuous representations has full measure in the set of all isometries. This extends Jones’s no-go example to typical elements of the so-called tensor planar algebra. We also identify an easily verified necessary condition for a continuous limit to exist. This singles out a particular class of tree tensor networks. Our considerations apply to recent approaches for introducing dynamics in holographic codes.