Functional methods for correlation functions of integrable face and anyon models

Abstract

The computation of correlation functions in models of statistical mechanics is the key to comparing theoretical results with actual measurements. In the context of integrable lattice models there exists a rich literature on correlators in vertex models and related quantum-spin chains. Less is known for integrable face models and their related anyon chains.

Therefore, we define generalised transfer matrices allowing for a solution of the `inverse problem', i.e.\ we express local operators by means of objects from the Yang-Baxter algebra. This motivates the study of reduced density matrices which contain the information of all correlation functions. Instead of directly calculating them, we show that they fulfil a set of functional equations. We use these equations to study density matrices in (R)SOS models and their related anyon chains. In particular, we find integral representations for the two and three-point functions of the $r=4$ RSOS model and calculate those quantities for the $r=5$ model in the thermodynamic limit. In addition we observe a factorisation of the three-point functions into two-point functions and propose an efficient algorithm to factorise reduced density matrices for generic models.

In the last section we study density matrices for the $SO(5)_2$ face models. We examine the structure of the two-site reduced density matrices and simplify them for certain topological sectors. Since there exist different inequivalent sets of Boltzmann weights, the latter is done for each choice leading to sets of discrete functional equations.