Discrete holomorphicity in solvable lattice models

Abstract

The critical phases of two dimensional lattice models are widely believed to be described by conformal quantum field theories in the scaling limit. In the past few years, formal proofs of the conformal invariance in different formulations of the Ising model have emerged which make pivotal use of some complex lattice observables. Due to their distinctive property of discrete holomorphicity, they are considered to be the lattice counterpart of holomorphic currents in the field theories. In this thesis, we study a weakened form of discrete holomorphicity that is known to be obeyed by natural generalizations of these observables to three important families of solvable models. The main result of this thesis is that discrete holomorphicity can be seen as a requirement stronger than the conditions of integrability of the models. That is, these conditions, the inversion relations and the Yang-Baxter equations, can be derived from this form of lattice holomorphicity. This finding is proposed as an explanation for the remarkable observation that discrete holomorphicity holds only on the integrable critical manifold of the weights. For the self-dual models, the duality conditions can also be established similarly. A key role in this argument is played by the rhombic embeddings of Baxter lattices. It is noted that, by the requirement of holomorphicity on every rhombus, the conformal spins of the observables are restricted to a discrete spectrum that label the solutions in which the angles of the rhombuses are interpreted as re-scaled spectral parameters. This interpretation allows the relationships between the spectral parameters in the integrability conditions to be seen as the criteria for geometric consistency of the rhombic embedding. The crossing symmetry of the models is then related to the alignment of the rhombuses with respect to the rapidity lines. It has also been shown here that values of the observables on the boundary of a simply connected domain remain unchanged by local rearrangement of rhombuses due to the Z-invariance of the models. This provides an independent characterization of the observables besides discrete holomorphicity and enables us to define them on the equivalence class of Baxter lattices with any given braiding of the rapidity lines. For an equivalence class, the holomorphicity equations then provide linear relations among its partition functions with different boundary conditions. The results of this thesis thus point to the essential, albeit somewhat obscure, role of integrability in the rigorous proofs. Show more