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arXiv:1409.7848 · Physical Review E

Noncontractible loops in the dense O(n) loop model on the cylinder

Francisco Castilho Alcaraz, J.G. Brankov, V. B. Priezzhev, V. Rittenberg, Alex Rogozhnikov

Published Nov 18, 2014• 3 citations
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Abstract

A lattice model of critical dense polymers $\mathrm{O}(n)$ 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 $\mathrm{O}(n)$ model and an arbitrary fugacity.

DOI
10.1103/physreve.90.052138

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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.
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 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 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.
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 relation between the average length of loops and their free energy is obtained for a variety of O( n )-type models on two-dimensional lattices, by extending to finite temperatures a calculation due to Kast. We show that the (number) averaged loop length stays finite for all non-zero fugacities n , and in particular it does not diverge upon entering the critical regime ( n 2 + ). Fully packed loop (FPL) models with n = 2 seem to obey the simple relation = 3 L min , where L min is the smallest loop length allowed by the underlying lattice. We demonstrate this analytically for the FPL model on the honeycomb lattice and for the 4-state Potts model on the square lattice, and based on numerical estimates obtained from a transfer matrix method we conjecture that this is also true for the two-flavour FPL model on the square lattice. We present, in addition, numerical results for the average loop length on the three critical branches (compact, dense and dilute) of the O( n ) model on the honeycomb lattice, and discuss the limit n 0. Contact is made with the predictions for the distribution of loop lengths obtained by conformal invariance methods.