We continue our work on lattice models of webs, which generalise the well-known loop models to allow for various kinds of bifurcations [1,2]. Here we define new web models corresponding to each of the rank-two spiders considered by Kuperberg [3]. These models are based on the <math><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub></math>, <math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math> and <math><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub></math> Lie algebras, and their local vertex configurations are intertwiners of the corresponding q-deformed quantum algebras. In all three cases we define a corresponding model on the hexagonal lattice, and in the case of <math><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub></math> also on the square lattice. For specific root-of-unity choices of q, we show the equivalence to a number of three- and four-state spin models on the dual lattice.The main result of this paper is to exhibit integrable manifolds in the parameter spaces of each web model. For q on the unit circle, these models are critical and we characterise the corresponding conformal field theories via numerical diagonalisation of the transfer matrix.In the <math><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub></math> case we find two integrable regimes. The first one contains a dense and a dilute phase, for which we have analytic control via a Coulomb gas construction, while the second one is more elusive and likely conceals non-compact physics. Three particular points correspond to a three-state spin model with plaquette interactions, of which the one in the second regime appears to present a new universality class. In the <math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math> case we identify four regimes numerically. The <math><msub><mrow><mi>B</mi></mrow><mrow><mn>2</mn></mrow></msub></math> case is too unwieldy to be studied numerically in the general case, but it found analytically to contain a simpler sub-model based on generators of the dilute Birman-Murakami-Wenzl algebra.

France

Finland

Augustin Lafay

Azat M. Gainutdinov

Jesper Lykke Jacobsen

Integrability of rank-two web models

We continue investigating the generalisations of geometrical statistical models introduced in [13], in the form of models of webs on the hexagonal lattice <math><mi>H</mi></math> having a <math><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math> quantum group symmetry. We focus here on the <math><mi>n</mi><mo>=</mo><mn>3</mn></math> case of cubic webs, based on the Kuperberg <math><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub></math> spider, and illustrate its properties by comparisons with the well-known dilute loop model (the <math><mi>n</mi><mo>=</mo><mn>2</mn></math> case) throughout. A local vertex-model reformulation is exhibited, analogous to the correspondence between the loop model and a three-state vertex model. The <math><mi>n</mi><mo>=</mo><mn>3</mn></math> representation uses seven states per link of <math><mi>H</mi></math>, displays explicitly the geometrical content of the webs and their <math><msub><mrow><mi>U</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>sl</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math> symmetry, and permits us to study the model on a cylinder via a local transfer matrix. A numerical study of the effective central charge reveals that for <math><mi>q</mi><mo>=</mo><mo>−</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>γ</mi></mrow></msup></math>, in the range <math><mi>γ</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>/</mo><mn>3</mn><mo>]</mo></math>, the web model possesses a dense and a dilute critical point, just like its loop model counterpart. In the dense <math><mi>q</mi><mo>=</mo><mo>−</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>π</mi><mo>/</mo><mn>4</mn></mrow></msup></math> case, the <math><mi>n</mi><mo>=</mo><mn>3</mn></math> webs can be identified with spin interfaces of the critical three-state Potts model defined on the triangular lattice dual to <math><mi>H</mi></math>. We also provide another mapping to a <math><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub></math> spin model on <math><mi>H</mi></math> itself, using a high-temperature expansion. We then discuss the sector structure of the transfer matrix, for generic <math><mi>q</mi></math>, and its relation to defect configurations in both the strip and the cylinder geometries. These defects define the finite-size precursors of electromagnetic operators. This discussion paves the road for a Coulomb gas description of the conformal properties of defect webs, which will form the object of a subsequent paper. Finally, we identify the fractal dimension of critical webs in the <math><mi>q</mi><mo>=</mo><mo>−</mo><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>π</mi><mo>/</mo><mn>3</mn></mrow></msup></math> case, which is the <math><mi>n</mi><mo>=</mo><mn>3</mn></math> analogue of the polymer limit in the loop model.