Discretely holomorphic parafermions and integrable boundary conditions

TL;DR

This paper introduces a boundary condition framework for two-dimensional models with discretely holomorphic parafermions, leading to integrable boundary weights and new insights into phase transitions and boundary conditions.

Contribution

It develops a modified discrete Cauchy-Riemann equation on boundaries, connecting discretely holomorphic parafermions to integrable boundary weights and discovering new boundary conditions.

Findings

Recovered exact transition points in the O(n) loop model

Discovered a new rotation-invariant boundary condition for $Z_N$ models

Established a link between parafermions and integrable boundary weights

Abstract

In two-dimensional statistical models possessing a discretely holomorphic parafermion, we introduce a modified discrete Cauchy-Riemann equation on the boundary of the domain, and we show that the solution of this equation yields integrable boundary Boltzmann weights. This approach is applied to (i) the square-lattice O(n) loop model, where the exact locations of the special and ordinary transitions are recovered, and (ii) the Fateev-Zamolodchikov $Z_N$ spin model, where a new rotation-invariant, integrable boundary condition is discovered for generic $N$.