Solvable Critical Dense Polymers

TL;DR

This paper exactly solves a lattice model of critical dense polymers, revealing its connection to logarithmic conformal field theory, and classifies eigenvalues and representations, confirming the central charge c=-2.

Contribution

It introduces a functional inversion identity in the Temperley-Lieb algebra for the model, providing explicit solutions and fusion rules for the associated representations.

Findings

Confirmed central charge c=-2 in the scaling limit

Derived explicit finitized characters for quasi-rational representations

Classified transfer matrix eigenvalues by zero patterns in the spectral plane

Abstract

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 Delta_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.