$U_q(\mathfrak{sl}_n)$ web models and $\mathbb{Z}_n$ spin interfaces

Abstract

This is the first in a series of papers devoted to generalisations of statistical loop models. We define a lattice model of U q (sl n ) webs on the honeycomb lattice, for n ≥ 2. It is a statistical model of closed, cubic graphs with certain non-local Boltzmann weights that can be computed from spider relations. For n = 2, the model has no branchings and reduces to the well-known O(N ) loop model introduced by Nienhuis [5]. In the general case, we show that the web model possesses a particular point, at q = e iπ/(n+1) , where the partition function is proportional to that of a Z n -symmetric chiral spin model on the dual lattice. Moreover, under this equivalence, the graphs given by the configurations of the web model are in bijection with the domain walls of the spin model. For n = 2, this equivalence reduces to the well-known relation between the Ising and O(1) models. We define as well an open U q (sl n ) web model on a simply connected domain with a boundary, and discuss in particular the role of defects on the boundary.

$U_{\mathfrak{q}}(\mathfrak{sl}_3)$ web models: Locality, phase diagram and geometrical defects

“…For a coloured web embedded in H, this agrees with (13). The total weight of a coloured web defined by the above local weights is invariant under isotopy.…”

“…We also consider applications in a few models. It follows from Section 3 and [13] that for q = e iπ/4 the critical point in the dilute phase of the n = 3 webs can be identified with spin interfaces of the critical three-state Potts model defined on the triangular lattice T, dual to H. This equivalence is analogous to the well-known identification of domain walls of the critical Ising model within the n = 2 loop model. In Section 5 we provide another mapping between the n = 3 webs and a Z 3 spin model defined on H itself, by means of a high-temperature expansion.…”

“…We first recall the definition of the Kuperberg web model, as given in our first paper [13]. The model is defined on an hexagonal lattice H made of 2M rows and 2L columns embedded in a vertical strip or a vertical cylinder.…”

“…In a recent paper [13] we have defined a series of statistical models on the hexagonal lattice H that extend this symmetry to any n ≥ 2. For the cases n > 2, these models define geometrical configurations of cubic (and bipartite, for n = 3) graphs, called webs, on H. The configurations reduce to the usual loops when n = 2, in which case bifurcations are suppressed.…”

“…The most interesting feature of loop and web models is that the partition sum carries over configurations of a set of extended, geometrical objects, whose statistical weight contains a non-local part. For the loop model (n = 2) this non-locality simply amounts to replacing each loop by a real number, while for the web model the weight results from a quite non-trivial reduction of each connected component to a set of loops which are then replaced by their corresponding weights [13,15].…”