Abstract
We initiate a study of the boundary version of the square-lattice Q-state Potts antiferromagnet, with Q ∈ [0, 4] real, motivated by the fact that the continuum limit of the corresponding bulk model is a non-compact CFT, closely related with the SL(2, ℝ)k/U(1) Euclidian black-hole coset model. While various types of conformal boundary conditions (discrete and continuous branes) have been formally identified for the the SL(2, ℝ)k/U(1) coset CFT, we are only able in this work to identify conformal boundary conditions (CBC) leading to a discrete boundary spectrum. The CBC we find are of two types. The first is free boundary Potts spins, for which we confirm an old conjecture for the generating functions of conformal levels, and show them to be related to characters in a non-linear deformation of the W∞ algebra. The second type of CBC — which corresponds to restricting the values of the Potts spins to a subset of size Q1, or its complement of size Q − Q1, at alternating sites along the boundary — is new, and turns out to be conformal in the antiferromagnetic case only. In the following, we refer to these new boundary conditions as “alt” boundary conditions. Using algebraic and numerical techniques, we show that the corresponding spectrum generating functions produce all the characters of discrete representations for the coset CFT. The normalizability bounds of the associated discrete states in the coset CFT are found to have a simple interpretation in terms of boundary phase transitions in the lattice model. In the two-boundary case, with two distinct alt conditions, we obtain similar results, at least in the case when the corresponding boundary condition changing operator also inserts a number of defect lines. For sqrt{Q} = 2 cos frac{pi }{k} , with k ≥ 3 integer, we show also how our boundary conditions can be reformulated in terms of a RSOS height model. The spectrum generating functions are then identified with string functions of the compact SU(2)k−2/U(1) parafermion theory (with symmetry Zk−2). The new alt conditions are needed to cover all the string functions. We provide an algebraic proof that the two-boundary alt conditions correctly produce the fusion rules of string functions. We expose in detail the special case of Q = 3 and its link with three-colourings of the square lattice and a corresponding boundary six-vertex model. Finally, we discuss the case of an odd number of sites (in the loop model) and the relation with wired boundary conditions (in the spin model). In this case the RSOS restriction produces the disorder operators of the parafermion theory.
Highlights
The critical antiferromagnetic Potts model on the square lattice enjoys remarkable properties which are not fully understood, despite years of work on the topic [1,2,3,4,5]
It is natural to wonder how the relation between models persists in the boundary case, and what this might tell us about issues ranging, for instance, from the microscopic interpretation of the non-compact degrees of freedom in the continuum limit of the antiferromagnetic (AF) Potts model to the possible conformally invariant boundary conditions in the Euclidian black hole conformal field theory (CFT) [16]
There, we identify new conformal boundary conditions specific to the AF Potts model — which we nickname “alt” because they involve the alternation of certain variables — together with the associated partition functions, which turn out to be combinations of discrete characters of the SL(2, R)/U(1)
Summary
The critical antiferromagnetic Potts model on the square lattice enjoys remarkable properties which are not fully understood, despite years of work on the topic [1,2,3,4,5]. It is natural to wonder how the relation between models persists in the boundary case, and what this might tell us about issues ranging, for instance, from the microscopic interpretation of the non-compact degrees of freedom in the continuum limit of the antiferromagnetic (AF) Potts model to the possible conformally invariant boundary conditions in the Euclidian black hole CFT [16]. ΛdJ,M — discrete character of the SL(2, R)/U(1) coset model, λr,j — generating function of levels in the loop model with “alt” boundary conditions, cml — string function, i.e., the generating function of levels in the Zk−2 parafermion CFT
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