Topological lattice field theories from intertwiner dynamics

TL;DR

This work develops a framework of 2D lattice models encoding intertwiners of quantum groups, with triangulation-invariant fixed points that illuminate the continuum limit problems of spin foams and spin nets. By deriving and solving 2-2 Pachner move–based recursion relations for fixed-point amplitudes, the authors produce a broad family of topological fixed points parameterized by the quantum group level $k$ and a maximal spin $J$, many of which admit matrix-product-state representations and correspond to AKLT-like phases. The fixed points exhibit rich phase structure, including symmetry-protected topological order and ground-state degeneracy in some sectors, and they admit an interpretation in terms of anyon condensation into an effective vacuum, consistent with Frobenius algebra classifications. The construction also connects to 3D topological models via Beidenharn–Elliott identities, providing a geometric view through Regge-like asymptotics and offering a controlled setting to study the role of diffeomorphism symmetry in discretized gravity-like theories.

Abstract

We introduce a class of 2D lattice models that describe the dynamics of intertwiners, or, in a condensed matter interpretation, the fusion and splitting of anyons. We identify different families and instances of triangulation invariant, that is, topological, models inside this class. These models give examples for symmetry protected topologically ordered 1D quantum phases with quantum group symmetries. Furthermore the models provide realizations for anyon condensation into a new effective vacuum. We explain the relevance of our findings for the problem of identifying the continuum limit of spin foam and spin net models.

Figures (13)

• Figure 1: An intertwiner model defining an intertwining map: ${\cal H}_b\rightarrow {\cal H}_t$. Here all the vertices are 'fat vertices'.
• Figure 2: The 2--2 move relations (\ref{['36']}) and their geometrical interpretation as re--gluing two triangles.
• Figure 3: The two left panels show a simple triangulation of a cylinder and its dual graph. The right panel shows how to identify the bottom and top boundary of the cylinder to obtain a torus. The corresponding graph (with fat vertices) can be evaluated within diagrammatical calculus and defines the partition function of the torus.
• Figure 4: A refined triangulation of the cylinder and the associated dual graph. We apply the 2--2 move to reach the graph on the right .
• Figure 5: The diagrammatic equations on the left show how to evaluate the graph associated to the torus, obtained by closing the cylinder in figure \ref{['tor2']}. The figure on the right shows a choice of dual graph for the cylinder, giving a projector map.