Abstract Yang–Baxter integrable dense A 1 ( 1 ) and dilute A 2 ( 2 ) loop models are considered on the torus in their simplest physical regimes. A combination of boundary conditions ( h , v ) is applied in the horizontal and vertical directions with h , v = 0 and 1 for periodic and antiperiodic boundary conditions respectively. The fugacities of non-contractible and contractible loops are denoted by α and β respectively where β is simply related to the crossing parameter λ . At roots of unity, when λ / π ∈ ℚ , these models are the dense L M ( p , p ′ ) and dilute D L M ( p , p ′ ) logarithmic minimal models with p , p ′ coprime integers. We conjecture the scaling limits of the transfer matrix traces in the standard modules with d defects and deduce the conformal partition functions Z dense ( h , v ) ( α ) and Z dilute ( h , v ) ( α )
Abstract Yang–Baxter integrable dense A 1 ( 1 ) and dilute A 2 ( 2 ) loop models are considered on the torus in their simplest physical regimes. A combination of boundary conditions ( h , v ) is applied in the horizontal and vertical directions with h , v = 0 and 1 for periodic and antiperiodic boundary conditions respectively. The fugacities of non-contractible and contractible loops are denoted by α and β respectively where β is simply related to the crossing parameter λ . At roots of unity, when λ / π ∈ ℚ , these models are the dense L M ( p , p ′ ) and dilute D L M ( p , p ′ ) logarithmic minimal models with p , p ′ coprime integers. We conjecture the scaling limits of the transfer matrix traces in the standard modules with d defects and deduce the conformal partition functions Z dense ( h , v ) ( α ) and Z dilute ( h , v ) ( α )
A lattice model of critical dense polymers is solved exactly for arbitrary system size on the torus. More generally, an infinite family of lattice loop models is studied on the torus and related to the corresponding Fortuin–Kasteleyn random cluster models. Starting with a cylinder, the commuting periodic single-row transfer matrices are built from the periodic Temperley–Lieb algebra extended by the shift operators Ω±1. In this enlarged algebra, the non-contractible loop fugacity is α and the contractible loop fugacity is β. The torus is formed by gluing the top and bottom of the cylinder. This gives rise to a variety of non-contractible loops winding around the torus. Because of their nonlocal nature, the standard matrix trace does not produce the proper geometric torus. Instead, we introduce a modified matrix trace for this purpose. This is achieved by using a representation of the enlarged periodic Temperley–Lieb algebra with a parameter v that keeps track of the winding of defects on the cylinder. The transfer matrix representatives and their eigenvalues thus depend on v. The modified trace is constructed as a linear functional on planar connectivity diagrams in terms of matrix traces Trd (with a fixed number of defects d) and Chebyshev polynomials of the first kind. For critical dense polymers, where β=0, the transfer matrix eigenvalues are obtained by solving a functional equation in the form of an inversion identity. The solution depends on d and is subject to selection rules which we prove. Simplifications occur if all non-contractible loop fugacities are set to α=2 in which case the traces are evaluated at v=1. In the continuum scaling limit, the corresponding conformal torus partition function obtained from finite-size corrections agrees with the known modular invariant partition function of symplectic fermions.
Abstract Critical site percolation on the triangular lattice is described by the Yang–Baxter solvable dilute A 2 ( 2 ) loop model with crossing parameter specialized to λ = π 3 , corresponding to the contractible loop fugacity β = − 2 cos 4 λ = 1 . We study the functional relations satisfied by the commuting transfer matrices of this model and the associated Bethe ansatz equations. The single and double row transfer matrices are respectively endowed with strip and periodic boundary conditions, and are elements of the ordinary and periodic dilute Temperley–Lieb algebras. The standard modules for these algebras are labelled by the number of defects d and, in the latter case, also by the twist e i γ . Nonlinear integral equation techniques are used to analytically solve the Bethe ansatz functional equations in the scaling limit for the central charge c = 0 and conformal weights Δ , Δ ˉ . For the ground states, we find Δ = Δ 1 , d + 1 for strip boundary conditions and ( Δ , Δ ˉ ) = ( Δ γ / π , d / 2 , Δ γ / π , − d / 2 ) for periodic boundary conditions, where Δ r , s = 1 24 ( ( 3 r − 2 s ) 2 − 1 ) . We give explicit conjectures for the scaling limit of the trace of the transfer matrix in each standard module. For d ⩽ 8 , these conjectures are supported by numerical solutions of the logarithmic form of the Bethe ansatz equations for the leading 20 or more c
A lattice model of critical dense polymers is solved exactly on a cylinder with finite circumference. The model is the first member of the Yang–Baxter integrable series of logarithmic minimal models. The cylinder topology allows for non-contractible loops with fugacity α that wind around the cylinder or for an arbitrary number of defects that propagate along the full length of the cylinder. Using an enlarged periodic Temperley–Lieb algebra, we set up commuting transfer matrices acting on states whose links are considered distinct with respect to connectivity around the front or back of the cylinder. These transfer matrices satisfy a functional equation in the form of an inversion identity. For even N , this involves a non-diagonalizable braid operator J and an involution R = − ( J 3 − 12 J )/16 = (−1) F with eigenvalues . This is reminiscent of supersymmetry with a pair of defects interpreted as a fermion. The number of defects thus separates the theory into Ramond ( even), Neveu–Schwarz ( odd) and ( odd) sectors. For the case of loop fugacity α = 2, the inversion identity is solved exactly sector by sector for the eigenvalues in finite geometry. The eigenvalues are classified according to the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity. The finite-size corrections are obtained from Euler–Maclaurin formula. In the scaling limit, we obtain the conformal partition functions as sesquilinear forms and confirm the central charge c = − 2 and conformal weights . Here and in the even sectors with Kac labels r = 1, 2, 3,...; s = 1, 2 while in the odd sectors. Strikingly, the odd sectors exhibit a -extended symmetry but the even sectors do not. Moreover, the naive trace summing over all even sectors does not yield a modular invariant.
A lattice model of critical dense polymers is solved exactly on a cylinder with finite circumference. The model is the first member of the Yang–Baxter integrable series of logarithmic minimal models. The cylinder topology allows for non-contractible loops with fugacity α that wind around the cylinder or for an arbitrary number of defects that propagate along the full length of the cylinder. Using an enlarged periodic Temperley–Lieb algebra, we set up commuting transfer matrices acting on states whose links are considered distinct with respect to connectivity around the front or back of the cylinder. These transfer matrices satisfy a functional equation in the form of an inversion identity. For even N, this involves a non-diagonalizable braid operator J and an involution R = − (J3 − 12J)/16 = (−1)F with eigenvalues . This is reminiscent of supersymmetry with a pair of defects interpreted as a fermion. The number of defects thus separates the theory into Ramond ( even), Neveu–Schwarz ( odd) and ( odd) sectors. For the case of loop fugacity α = 2, the inversion identity is solved exactly sector by sector for the eigenvalues in finite geometry. The eigenvalues are classified according to the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields selection rules for the physically relevant solutions to the inversion identity. The finite-size corrections are obtained from Euler–Maclaurin formula. In the scaling limit, we obtain the conformal partition functions as sesquilinear forms and confirm the central charge c = − 2 and conformal weights . Here and in the even sectors with Kac labels r = 1, 2, 3,...;s = 1, 2 while in the odd sectors. Strikingly, the odd sectors exhibit a -extended symmetry but the even sectors do not. Moreover, the naive trace summing over all even sectors does not yield a modular invariant.
For general Temperley–Lieb loop models, including the logarithmic minimal models LM ( p , p ′ ) with p , p ′ coprime integers, we construct an infinite family of Robin boundary conditions on the strip as linear combinations of Neumann and Dirichlet boundary conditions. These boundary conditions are Yang–Baxter integrable and allow loop segments to terminate on the boundary. Algebraically, the Robin boundary conditions are described by the one-boundary Temperley–Lieb algebra. Solvable critical dense polymers is the first member LM ( 1 , 2 ) of the family of logarithmic minimal models and has loop fugacity β = 0 and central charge c = − 2 . Specialising to LM ( 1 , 2 ) with our Robin boundary conditions, we solve the model exactly on strips of arbitrary finite size N and extract the finite-size conformal corrections using an Euler–Maclaurin formula. The key to the solution is an inversion identity satisfied by the commuting double row transfer matrices. This inversion identity is established directly in the Temperley–Lieb algebra. We classify the eigenvalues of the double row transfer matrices using the physical combinatorics of the patterns of zeros in the complex spectral parameter plane and obtain finitised characters related to spaces of coinvariants of Z 4 fermions. In the continuum scaling limit, the Robin boundary conditions are associated with irreducible Virasoro Verma modules with conformal weights Δ r , s − 1 2 = 1 32 ( L 2 − 4 ) where L = 2 s − 1 − 4 r , r ∈ Z , s ∈ N . These conformal weights populate a Kac table with half-integer Kac labels. Fusion of the corresponding modules with the generators of the Kac fusion algebra is examined and general fusion rules are proposed.
Abstract We study the dilute <?CDATA A2(2)?> A2(2) loop models on the geometry of a strip of widthN. Two families of boundary conditions are known to satisfy the boundary Yang–Baxter equation. Fixing the boundary condition on the two ends of the strip leads to four models. We construct the fusion hierarchy of commuting transfer matrices for the model as well as itsT- andY-systems, for these four boundary conditions and with a generic crossing parameterλ. For <?CDATA λ/π?> λ/π rational and thus <?CDATA q=−e4iλ?> q=−e4iλ a root of unity, we prove a linear relation satisfied by the fused transfer matrices that closes the fusion hierarchy into a finite system. The fusion relations allow us to compute the two leading terms in the large-Nexpansion of the free energy, namely the bulk and boundary free energies. These are found to be in agreement with numerical data obtained for smallN. The present work complements a previous study (Morin-Duchesne and Pearce 2019J. Stat. Mech.1909) that investigated the dilute <?CDATA A2(2)?> A2(2) loop models with periodic boundary conditions.