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The alt boundary conditions in the RSOS model
We saw in section 3.2 that the blobbed boundary conditions, described by the transfer matrix in equation (3.32), produced the generating functions of Verma modules in the continuum limit (see equations (3.40) and (3.41)). Furthermore, it was found that by going to the RSOS representation of the model, or equivalently, by constructing an irreducible representation of the blob algebra using equation (3.42) and Figure 3.25, one could derive the generating function of CFT minimal models in terms of the generating function of the Verma module. The key takeaway was that all of the properties of the representation theory of the CFT module could be observed on the finite lattice via the representation theory of the blob algebra.
In the AF model we have a new “alt » boundary condition that can also be described by the blob algebra. We can therefore perform the same analysis as the ferromagnetic model and derive an analogous expression to equation (3.43) for the AF model, this time relating the generating function obtained from the “alt version” of the RSOS model to the discrete characters in equation (3.80), the generating function obtained from the alt boundary conditions in the loop model.
The discussion in section 3.3 provides us with a hint as to what this new identity should look like, and what should be the generating function obtained from the alt boundary conditions in the RSOS model. First of all consider the form of the leading critical exponent (equation (3.50) ) with free boundary conditions in the sector with l = 2j through lines, and the expression in (3.51) showing that these exponents are a subset of the compact parafermion theory. Furthermore, the discussion from equation (3.59) to equation (3.60) showed that all of the parafermion critical exponents were “contained » inside the Euclidean Black Hole discrete character when this theory was considered as one with central charge cPF. All of this suggests that we should be able to obtain all of these parafermion critical exponents via the alt boundary conditions in the RSOS model, and that we should be able to recover the string functions cml defined in equation (3.62) from the RSOS model also. Section 3.4.3 will present the relevant results and discuss this in more detail.
Missing string functions and alt boundary conditions
Consider the expressions in equations (3.73). Since the generating functions of the loop model on the RHS are obtained by acting with a transfer matrix written in terms of blob algebra generators, we can obtain the corresponding generating function of the RSOS model for k and r integer by finding the corresponding irreducible representation of the blob algebra, exactly as in section 3.2. For clarity, the form of the irreducible representation of the blob algebra is written again here.
Combining alt boundary conditions in the loop model
We now wish to consider the case where alt boundary conditions are imposed on both sides of the strip. The general situation is characterised by more parameters than previously. The parametrisation (3.66) of the alternatingly restricted number of Potts states on the boundary has to be made independently for both boundaries. Instead of r, we thus have r1 for the left boundary and r2 for the right boundary. The algebraic framework must be extended, so as to have blob and unblob operators—denoted b1, b2 and u1 = 1b1, u2 = 1b2 respectively—for each side. The proper algebraic framework for this situation is called the two-boundary Temperley-Lieb (2BTL) algebra [76, 56, 77]. We need to be careful with 2j > 0 through-lines, since in this case we need to define four different sectors—denoted bb, ub, bu and uu—where the left (resp. right) label specifies whether the leftmost (resp. rightmost) through-line carries the blob or unblob operator,b1 or u1 (resp. b2 or u2).
Note that even though the lattice model allows continuous values of r1 and r2, the discrete character in equation (3.80) (which played the role of the generating function when “alt” was imposed on only one side of the system) is only defined for 2J 2 N. From the correspondences in (3.77) and (3.81) we have then that r 2 N also. As we shall now see, the discrete character also arises when “alt » is placed on both sides. We hence consider only the case r1 and r2 integer. Note that when j = 0 the lattice model is more subtle since loops can touch both boundaries. In what follows we instead focus only on j > 0.
Combining alt boundary conditions in the RSOS model
We have found that the continuum limit of the AF Potts RSOS model coincides with that of the Zk2 parafermion theory and we have found an “alt” boundary condition corresponding to each of the string functions in these models. This prescription, however, was restricted to the case where the alt boundary condition is on one side only with the other side having free boundary conditions. We would expect that putting the alt boundary conditions on both sides of the lattice would correspond to the fusion of fields in the Zk2 parafermion theory. As will be shown below from our numerical results, this is indeed the case.
We can however recover this result from knowledge of the generating functions produced in the continuum limit of the loop model with the alt conditions on both sides; see eqs. (3.93)–(3.96). Section 3.4.3 used the representation theory of the blob algebra to move between the AF loop model and the AF RSOS model (similarly, section 3.2 used the same procedure to move between the two ferromagnetic models). It was found that the generating function of the irreducible representation of the blob algebra created by the infinite sum in equation (3.84) produced the RSOS representation. We can use the same method for the case with the alt condition on both sides, i.e., when there are two blob operators, to calculate the generating functions in the RSOS model produced by putting the alt condition on both sides. The relevant algebra in this case is the two-boundary Temperley-Lieb (2BTL) algebra [76, 56, 77].
Section 3.7.1 will present the numerical results of the RSOS model with the alternating boundary condition on both sides. These results will be interpreted in terms of the fusion of fields in the Zk2 parafermion theory. Section 3.7.2 will recover these results by studying the representation theory of the 2BTL algebra.
Table of contents :
1 Introduction
2 Bulk and Boundary CFT
2.1 Conformal Transformations
2.2 Conformal Invariance in Field Theories
2.2.1 The classical case
2.2.2 Quantum Conformal Field Theories
2.3 The Hilbert Space
2.4 Boundary CFT
2.5 Statistical Mechanics
3 The Potts model
3.1 The lattice model
3.1.1 Loop Model Formulation
3.1.2 The Temperley-Lieb algebra
3.1.3 The vertex representation
3.1.4 The blob algebra
3.1.5 RSOS models
3.1.6 Blobbed boundary conditions in the RSOS model
3.2 The critical ferromagnetic Potts model
3.3 The critical antiferromagnetic Potts model
3.4 New boundary conditions in the antiferromagnetic Potts model
3.4.1 The alt boundary conditions in the loop model
3.4.2 The alt boundary conditions in the RSOS model
3.4.3 Missing string functions and alt boundary conditions
3.5 Normalisability issues
3.5.1 A first-order boundary phase transition
3.6 Combining alt boundary conditions in the loop model
3.7 Combining alt boundary conditions in the RSOS model
3.7.1 Numerics
3.7.2 2BTL representation theory
3.8 Special cases: the two and three state Potts model
3.8.1 The alt boundary conditions in terms of Potts spins
3.8.2 Relationship to the six-vertex model
3.9 Odd number of sites and disorder operators
4 The Bethe Ansatz
4.1 Some general properties
4.2 Integrability: bulk and boundary
4.3 The Analytical Bethe Ansatz
4.4 The Hamiltonian Limit
4.5 The CFT limit: examples
5 Integrable boundary conditions in the Potts model
5.1 The staggered six-vertex model and the D2
5.1.1 Background
5.1.2 Review of the staggered six-vertex model
5.1.3 Mapping between the two models: General strategy
5.1.4 Deriving the Boltzmann weights
5.2 The open D2
5.2.1 Hamiltonian limit
5.2.2 Additional symmetries
5.2.3 The ! 0 limit
5.2.4 Non-zero
5.3 The Bethe Ansatz solution
5.3.1 The XXZ subset
5.3.2 Other solutions of Bethe Ansatz equations
5.4 Other Temperley-Lieb representations
5.4.1 Loop representation
5.4.2 RSOS representation
6 A Non-Compact Boundary Conformal Field Theory
6.1 New Boundary Conditions
6.1.1 The Hamiltonian limit
6.1.2 Geometry change
6.1.3 The Transfer Matrix
6.2 Finding an Exact Solution
6.2.1 The complete solution
6.2.2 Correspondence with the XXX model
6.3 The Continuum Limit
6.3.1 The Loop model
6.3.2 Back to the Bethe Ansatz
6.3.3 Discrete States
6.3.4 XXZ Subset
6.3.5 RSOS model
6.4 A Boundary RG Flow
7 Discussion
A Résumé
