Among the lattice loop models defined by Pearce et al (2006 J. Stat. Mech. P11017), the model corresponding to critical dense polymers (β = 0) is the only one for which an inversion relation for the transfer matrix D N ( u ) was found by Pearce and Rasmussen (2007 J. Stat. Mech. P02015). From this result, they identified the set of possible eigenvalues for D N ( u ) and gave a conjecture for the degeneracies of its relevant eigenvalues in the link representation, in the sector with d defects. In this paper, we set out to prove this conjecture, using the homomorphism of the TL N (β) algebra between the loop model link representation and that of the XXZ model for β = −( q + q −1 ).
Among the lattice loop models defined by Pearce et al (2006 J. Stat. Mech. P11017), the model corresponding to critical dense polymers (β = 0) is the only one for which an inversion relation for the transfer matrix D N ( u ) was found by Pearce and Rasmussen (2007 J. Stat. Mech. P02015). From this result, they identified the set of possible eigenvalues for D N ( u ) and gave a conjecture for the degeneracies of its relevant eigenvalues in the link representation, in the sector with d defects. In this paper, we set out to prove this conjecture, using the homomorphism of the TL N (β) algebra between the loop model link representation and that of the XXZ model for β = −( q + q −1 ).
A lattice model of critical dense polymers is solved exactly for finite strips. The model is the first member of the principal series of the recently introduced logarithmic minimal models. The key to the solution is a functional equation in the form of an inversion identity satisfied by the commuting double-row transfer matrices. This is established directly in the planar Temperley–Lieb algebra and holds independently of the space of link states on which the transfer matrices act. Different sectors are obtained by acting on link states with s −1 defects where s = 1,2,3,... is an extended Kac label. The bulk and boundary free energies and finite-size corrections are obtained from the Euler–Maclaurin formula. The eigenvalues of the transfer matrix are classified by the physical combinatorics of the patterns of zeros in the complex spectral-parameter plane. This yields a selection rule for the physically relevant solutions to the inversion identity and explicit finitized characters for the associated quasi-rational representations. In particular, in the scaling limit, we confirm the central charge c = −2 and conformal weights Δ s = ((2− s ) 2 −1)/8 for s = 1,2,3,.... We also discuss a diagrammatic implementation of fusion and show with examples how indecomposable representations arise. We examine the structure of these representations and present a conjecture for the general fusion rules within our framework.
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.
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.
A lattice model of critical dense polymers O(n) is considered for finite cylinder geometry. Due to the presence of noncontractible loops with a fixed fugacity $\ensuremath{\xi}$, the model at n=0 is a generalization of the critical dense polymers solved by Pearce, Rasmussen, and Villani. We found the free energy for any height N and circumference L of the cylinder. The density $\ensuremath{\rho}$ of noncontractible loops is obtained for $N\ensuremath{\rightarrow}\ensuremath{\infty}$ and large L. The results are compared with those found for the anisotropic quantum chain with twisted boundary conditions. Using the latter method, we derived $\ensuremath{\rho}$ for any O(n) model and an arbitrary fugacity.
Solvable critical dense polymers is a Yang-Baxter integrable model of polymers on the square lattice. It is the first member LM(1,2) of the family of logarithmic minimal models LM(p,p'). The associated logarithmic conformal field theory admits an infinite family of Kac representations labelled by the Kac labels r,s=1,2,.... In this paper, we explicitly construct the conjugate boundary conditions on the strip. The boundary operators are labelled by the Kac fusion labels (r,s)=(r,1) x (1,s) and involve a boundary field xi. Tuning the field xi appropriately, we solve exactly for the transfer matrix eigenvalues on arbitrary finite-width strips and obtain the conformal spectra using the Euler-Maclaurin formula. The key to the solution is an inversion identity satisfied by the commuting double-row transfer matrices. The transfer matrix eigenvalues are classified by 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 which takes the form of a decomposition into irreducible blocks corresponding combinatorially to finitized characters given by generalized q-Catalan polynomials. This decomposition is in accord with the decomposition of the Kac characters into irreducible characters. In the scaling limit, we confirm the central charge c=-2 and the Kac formula for the conformal weights Delta_{r,s}=((2r-s)^2-1)/8 for r,s=1,2,3,... in the infinitely extended Kac table.