Discretely Holomorphic Parafermions and Integrable Loop Models

Abstract

We define parafermionic observables in various lattice loop models, including examples where no Kramers-Wannier duality holds. For a particular rhombic embedding of the lattice in the plane and a value of the parafermionic spin these variables are discretely holomorphic (they satisfy a lattice version of the Cauchy-Riemann equations) as long as the Boltzmann weights satisfy certain linear constraints. In the cases considered, the weights then also satisfy the critical Yang-Baxter equations, with the spectral parameter being related linearly to the angle of the elementary rhombus.

13 pages, 7 figures Added 2 references, corrected minor typos

References in corpus (8)

• Conformal Field Theories of Stochastic Loewner Evolutions
• Discrete Riemann Surfaces and the Ising model
• Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model
• Solvable lattice models labelled by Dynkin diagrams
• Conformal restriction, highest-weight representations and SLE
• Holomorphic Parafermions in the Potts model and SLE
• Stochastic Loewner evolution for conformal field theories with Lie-group symmetries
• Discretely Holomorphic Parafermions in Lattice Z(N) Models

Cited by in corpus (25)

• Correlation functions in loop models
• Conformal boundary conditions in the critical O(n) model and dilute loop models
• Conformal invariance of lattice models
• Discrete Holomorphicity at Two-Dimensional Critical Points
• An integrable modification of the critical Chalker-Coddington network model
• Discrete stress-energy tensor in the loop O(n) model
• Corner contribution to percolation cluster numbers
• Discrete holomorphicity and integrability in loop models with open boundaries
• Exact spin quantum Hall current between boundaries of a lattice strip
• Delocalization of the height function of the six-vertex model
• Topics on abelian spin models and related problems
• Discrete holomorphic parafermions in the Ashkin-Teller model and SLE
• Critical interfaces of the Ashkin-Teller model at the parafermionic point
• Integrability as a consequence of discrete holomorphicity: the Z_N model
• Generalizations of Kitaev's honeycomb model from braided fusion categories
• Integrability as a consequence of discrete holomorphicity: loop models
• Critical behaviour of the randomly stirred dynamical Potts model: Novel universality class and effects of compressibility
• Finite size lattice results for the two-boundary Temperley--Lieb loop model
• Conformal Field Theory, Vertex Operator Algebra and Stochastic Loewner Evolution in Ising Model
• The parafermionic observable in SLE
• The free-fermionic $C_2^{(1)}$ loop model, double dimers and Kashaev's recurrence
• Discrete Holomorphic Parafermions in the Eight Vertex Model
• Boundary conformal field theories and loop models
• Self-avoiding walk on ${\mathbb{Z}}^2$ with Yang-Baxter weights: universality of critical fugacity and 2-point function
• On the construction of discrete fermions in the FK-Ising model