$U_q(\mathfrak{sl}_n)$ web models and $\mathbb{Z}_n$ spin interfaces - Archive ouverte HAL Access content directly
Journal Articles Journal of Statistical Mechanics: Theory and Experiment Year : 2021

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


This is the first in a series of papers devoted to generalisations of statistical loop models. We define a lattice model of $U_q(\mathfrak{sl}_n)$ webs on the honeycomb lattice, for $n \ge 2$. 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. In the general case, we show that the web model possesses a particular point, at $q=e^{i \pi/(n+1)}$, where the partition function is proportional to that of a $\mathbb{Z}_n$-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 $U_q(\mathfrak{sl}_n)$ web model on a simply connected domain with a boundary, and discuss in particular the role of defects on the boundary.

Dates and versions

hal-03122299 , version 1 (26-01-2021)



Augustin Lafay, Azat M. Gainutdinov, Jesper Lykke Jacobsen. $U_q(\mathfrak{sl}_n)$ web models and $\mathbb{Z}_n$ spin interfaces. Journal of Statistical Mechanics: Theory and Experiment, 2021, ⟨10.1088/1742-5468/abfcb3⟩. ⟨hal-03122299⟩
160 View
0 Download



Gmail Facebook X LinkedIn More