We compute lattice correlation functions for the model of critical dense polymers on a semi-infinite cylinder of perimeter n n . In the lattice loop model, contractible loops have a vanishing fugacity whereas non-contractible loops have a fugacity \alpha \in (0,\infty) α ∈ ( 0 , ∞ ) . These correlators are defined as ratios Z(x)/Z_0 Z ( x ) / Z 0 of partition functions, where Z_0 Z 0 is a reference partition function wherein only simple half-arcs are attached to the boundary of the cylinder. For Z(x) Z ( x ) , the boundary of the cylinder is also decorated with simple half-arcs, but it also has two special positions 1 1 and x x where the boundary condition is different. We investigate two such kinds of boundary conditions: (i) there is a single node at each of these points where a long arc is attached, and (ii) there are pairs of adjacent nodes at these points where two long arcs are attached. We find explicit expressions for these correlators for finite n n using the representation of the enlarged periodic Temperley-Lieb algebra in the XX spin chain. The resulting asymptotics as n\to \infty n → ∞ are expressed as simple integrals that depend on the scaling parameter \tau = \frac {x-1} n \in (0,1) τ = x − 1 n ∈ ( 0 , 1 ) . For small \tau τ , the leading behaviours are proportional to \tau^{1/4} τ 1 / 4 , \tau^{1/4}\log \tau τ 1 / 4 log τ ,
We compute lattice correlation functions for the model of critical dense polymers on a semi-infinite cylinder of perimeter n n . In the lattice loop model, contractible loops have a vanishing fugacity whereas non-contractible loops have a fugacity \alpha \in (0,\infty) α ∈ ( 0 , ∞ ) . These correlators are defined as ratios Z(x)/Z_0 Z ( x ) / Z 0 of partition functions, where Z_0 Z 0 is a reference partition function wherein only simple half-arcs are attached to the boundary of the cylinder. For Z(x) Z ( x ) , the boundary of the cylinder is also decorated with simple half-arcs, but it also has two special positions 1 1 and x x where the boundary condition is different. We investigate two such kinds of boundary conditions: (i) there is a single node at each of these points where a long arc is attached, and (ii) there are pairs of adjacent nodes at these points where two long arcs are attached. We find explicit expressions for these correlators for finite n n using the representation of the enlarged periodic Temperley-Lieb algebra in the XX spin chain. The resulting asymptotics as n\to \infty n → ∞ are expressed as simple integrals that depend on the scaling parameter \tau = \frac {x-1} n \in (0,1) τ = x − 1 n ∈ ( 0 , 1 ) . For small \tau τ , the leading behaviours are proportional to \tau^{1/4} τ 1 / 4 , \tau^{1/4}\log \tau τ 1 / 4 log τ ,
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.
Working in the dense loop representation, we use the planar Temperley–Lieb algebra to build integrable lattice models called logarithmic minimal models . Specifically, we construct Yang–Baxter integrable Temperley–Lieb models on the strip acting on link states and consider their associated Hamiltonian limits. These models and their associated representations of the Temperley–Lieb algebra are inherently non-local and not (time-reversal) symmetric. We argue that, in the continuum scaling limit, they yield logarithmic conformal field theories with central charges c = 1−(6( p − p ') 2 / pp '), where p , p ' = 1, 2, ... are coprime. The first few members of the principal series are critical dense polymers ( m = 1, c = −2), critical percolation ( m = 2, c = 0) and the logarithmic Ising model ( m = 3, c = 1/2). For the principal series, we find an infinite family of integrable and conformal boundary conditions organized in an extended Kac table with conformal weights Δ r , s = ((( m +1) r − ms ) 2 −1)/4 m ( m +1), r , s = 1, 2, .... The associated conformal partition functions are given in terms of Virasoro characters of highest-weight representations. Individually, these characters decompose into a finite number of characters of irreducible representations. We show with examples how indecomposable representations arise from fusion.
Abstract We investigate the bipartite fidelity for a lattice model described by a logarithmic CFT: the model of critical dense polymers. We define this observable in terms of a partition function on the pants geometry, where d defects enter at the top of the pants lattice and exit in one of the legs. Using the correspondence with the XX spin chain, we obtain an exact closed-form expression for and compute the leading terms in its 1/ N asymptotic expansion as a function of x = N A / N , where N is the lattice width at the top of the pants and N A is the width of the leg where the defects exit. We find an agreement with the results of Stéphan and Dubail for rational CFTs, with the central charge and conformal weights specialised to c = −2 and . We compute a second instance of the bipartite fidelity for d = 2 by imposing a different rule for the connection of the defects. In the conformal setting, this choice corresponds to inserting two boundary condition changing fields of weight that are logarithmic instead of primary. We compute the asymptotic expansion in this case as well and find a simple additive correction compared to , of the form . We confirm this lattice result with a CFT derivation and find that this correction term is identical for all logarithmic theories, independently of c and .