Diverse methods for integrable models

This thesis is centered around three topics, sharing integrability as a common theme. This thesis explores different methods in the field of integrable models. The first two chapters are about integrable lattice models in statistical physics. The last chapter describes an integrable quantum chain. Integrable lattice models and quantum chains are closely related, both by the motivation to study them and by method to probe them. Typically, the Hamiltonian of the quantum chain is the logarithmic derivative of the transfer matrix of the corresponding lattice model.
One reason which makes these models interesting is exact solvability. They are typically strongly correlated systems which are impossible to approach with perturbation theory in interaction strength. In the context of statistical physics, integrable models can be used to compute universal quantities. They are also used as toy-models to learn about new phenomena, e.g. spin-charge separation is observed in the one-dimensional Hubbard-model. Low-dimensional systems with tunable parameters are realized in laboratories recently, giving direct experimental relevance to integrable systems.
In the first chapter, a spin-1 current in computed for the dilute O(n = 1) loop model. The quantum Knizhnik-Zamolodchikov equations are used to compute exact results for small systems. Recursion relations are used to prove the proposed expression.
The second chapter explores the continuum limit of the fused Restricted Solid-on-Solid models. The Corner Transfer Matrix method is used to compute the finitized characters relevant to the question.
In the third chapter, a one dimensional supersymmetric fermionic model is solved by the means of nested Bethe ansatz.