In these lectures, we explore some of the aspects of the six-vertex model :
- we start with proofs of the existence and the condensation of Bethe roots for the Bethe Ansatz equation associated with the six-vertex model with periodic boundary conditions and an arbitrary density of up arrows (per line) in the regime Δ<1. As an application, we provide a short fully rigorous computation of the free energy of the six-vertex model on the torus, as well as an asymptotic expansion of the six-vertex partition functions when the density of up arrows approaches 1/2.
- We then explain the link between the six-vertex model and the Fortuin-Kasteleyn percolation.
- We study the height function associated with the six-vertex model and prove that it is localized when Δ <-1, and delocalized when -1< Δ < 1/2. Finally, we prove that in the delocalized regime above, the height function is rotationally invariant in the scaling limit.
The Heisenberg-Ising chain is the basic example of a solvable model in quantum statistical mechanics. Its physical observables -- thermodynamic quantities and correlation functions -- can be represented as limits of sequences of partitions functions of inhomogeneous six-vertex models with certain cuts and toroidal boundary conditions. We shall explain this connection in the more general context of fundamental Yang-Baxter integrable models. Varying the inhomogeneities different statistical ensembles can be realized. We shall concentrate on the case of the grand canonical ensemble, describing the coupling to a heat bath at finite temperature T. As an example we consider the calculation of the free energy per lattice site of the Heisenberg-Ising chain. This requires to derive the Bethe Ansatz equations and Bethe eigenvectors and eigenvalues of the vertical transfer matrix of the inhomogeneous six-vertex model. We show how the solutions of the Bethe Ansatz equations are related to solutions of a non-linear integral equation and express the dominant eigenvalue that determines the free energy per lattice site in terms of its solution.
We will see how probability distributions of random matrix origin (Gaussian Unitary Ensemble, Tracy-Widom distribution) appear in the asymptotics of the six-vertex model. On our way we will encounter the Izergin-Korepin determinant, Schur polynomials, and determinantal point processes.