IRF MODELS, A^(1)_(n−1) FACE MODELS AND STOCHASTIC VERTEX MODELS

Jocelyne Z. Bisagny, Department of Physics, ETH Zürich; Prof. Giovanni Felder, Department of Mathematics, ETH Zürich

Master Thesis, Autumn 2017. (Full Version available upon request

Contact: jocelyne.bisagny@alumni.ethz.ch)

Abstract.

We introduce the dynamical stochastic higher spin six vertex models that are related to representations of Felder’s elliptic quantum group E_(τ,η)(sl_2). These models are dynamical in the sense of Borodin’s stochastic Interaction-Round-a-Face (IRF) models. We first consider the IRF model in a quadrant and derive the Boltzmann weights in the 'trigonometric limit' for the spin 1/2 six vertex model and show how to construct Boltzmann weights for the higher spin six vertex model based on the fusion procedure proposed by Date et al. We then introduce a family of solvable lattice models called A^(1)_(n-1) face models in order to generalize the Boltzmann weights to the sl_n representation and show how to recover the Boltzmann weights for the sl_2 representation. We finally establish the stochastic weights finding stochastic corrections for the previously introduced Boltzmann weights and show that the ’stochastically corrected’ weights satisfy the Yang-Baxter equation (YBE).

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Outline

The search for integrable (or exactly solvable) systems has been a central theme in mathematical physics. More precisely, the Yang-Baxter integrability has long been studied in quantum, statistical, mathematical Physics. Of fundamental importance for such systems is the The Yang-Baxter equation, which, is the local version of the commutativity of transfer matrices. An important prototype of many solvable lattice models defined on two dimensional lattices, is the six vertex model, which consists of six vertex configurations to which we can can assign a Boltzmann weight. We introduce the IRF (or SOS) model following mainly the formalism of Borodin [1], who treats the dynamical stochastic higher spin vertex model, which is a solvable model. To solve dynamical stochastic processes, the idea is to use elliptic symmetric functions, as the integrability of the model is due to the properties of elliptic symmetric functions which result from fused representations of Felder’s elliptic quantum group E_(τ,η)(sl_2). Borodin studies elliptic weight functions, which he views as partition functions for vertex models with an additional dynamical parameter. We introduce the Boltzmann weights by taking the trigonometric limit of the elliptic symmetric functions. These weights are solution to the Yang-Baxter equation. We treat the spin 1/2 representation first and explain that through a method called the fusion procedure, one can ’fuse’ these weights and obtain higher spin Boltzmann weights. We then show that the higher spin Boltzmann weights satisfy a different version of the YBE, called YBE of type (p, q, s). At the end of the section, we introduce the stochastic weights for the (dynamical) higher spin six vertex model which were introduced by Borodin in [1]. This will motivate us to find a way of fusing stochastic weights. Up to this point the cases we treat are in the sl_2 representation. However, Jimbo et al. [11] have established a correspondence principle between one point functions of solvable lattice models and irreducible decomposition of character for affine Lie algebra, introducing IRF models which correspond to the level l representation of the affine Lie algebra A^(1)_(n−1). We introduce the A^(1)_(n−1) face models. The basic idea behind this section is to introduce the generalisation where A^(1)_(n−1) reaplces A^(1)_(1) . This generalisation is the result of the generalised R-matrix, called generalised quantum R-matrix. The Boltzmann weights in this case are defined in terms of vectors, and appear in the generalised quantum R-matrix. Once introduced the Boltzmann weights in the sl_n representation, we show that with the right formalism, setting n = 2 we can recover the Boltzmann weights defined in the sl_2 representation. The last section is motivated by Borodin’s work [1], where he explains that the IRF model can be degenerated to (1 + 1)-dimensional interacting particle systems known as dynamic exclusion processes (more precisely, he treats the dynamic symmetric simple exclusion process (SSEP) and the dynamic asymmetric simple exclusion process (ASEP)). This motivates the interest in stochastic processes, where the Boltzmann weights need to be transformed so as to add up to one at each vertex. In section 7, we thus show that the weight functions which we have introduced in the previous sections can be ’corrected stochastically’ with some corrections that we call ’extra weights’. To do so, we define a transformation given by the extra weights F(a,b)F(d,a)/F(d,c)F(c,b), with F(a,b)= ̸= 0

for |a − b| = 1, in which case a,b are said to be admissible, and apply it to the previously defined Boltzmann weights. We show that under such transformation, the resulting Boltzmann weights sum to one, without influencing the integrability of the model. More precisely, we prove that resulting stochastic weights are a solution of the YBE. We find that the stochasticity relations for the sl_2 representation can be generalised to the sln case. The main results of our work are Theorem 7.1 (s. Thesis) which establishes the stochastic weights in the spin 1/2 representation, given by a stochastic correction of the type F(a,b)F(d,a)/F(d,c)F(c,b) and Theorem 7.2 (s. Thesis) which establishes the stochastic weights in the sl_n representation.

This work provides a small step towards the recent developments in the connection between IRF models and the dynamic of interacting particle systems.

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