Speaker
Description
Higher-order tensor renormalization group (HOTRG) methods coarse-grain a tensor network by iterated blocking, implicitly assuming that the coarse network again admits a finite periodic description. We make this assumption explicit: representing a periodic tensor network as the lift of a finite-cell voltage graph over a group $\Lambda$, an expanding finite-index endomorphism of $\Lambda$ together with connected block templates (which we call a compatible equivariant block rule), represents one HOTRG step as an exact symbolically computable map between finite voltage data. We partially characterize when a compatible block rule can exist: it forces polynomial growth of the group (by Gromov's theorem these groups are virtually nilpotent). But we also note that polynomial growth is not sufficient, as the characteristically nilpotent Dixmier-Lister algebra has polynomial growth but no such endomorphism.
Our mathematical framework is able to represent several renormalizable geometries: triangular, square and hexagonal lattice. We validate exact partition-function agreement on these lattices. Moreover, lifts of wallpaper symmetry groups such as $p4,p4m,p6,p6m$ as well as the discrete Heisenberg group give rise to interesting new non-abelian geometries represented by the same framework. We show some (work-in-progress) tensor network calculations on each of these lattices to verify our HOTRG-inspired contraction procedure and investigate renormalizability of these geometries.