Solvable critical dense polymers

Abstract

A lattice model of critical dense polymers is solved exactly for finite strips. The model isthe first member of the principal series of the recently introduced logarithmic minimalmodels. The key to the solution is a functional equation in the form of an inversion identitysatisfied by the commuting double-row transfer matrices. This is established directly in theplanar Temperley–Lieb algebra and holds independently of the space of link states onwhich the transfer matrices act. Different sectors are obtained by acting on link states withs−1 defectswhere s = 1,2,3,... is an extended Kac label. The bulk and boundary free energies and finite-size correctionsare obtained from the Euler–Maclaurin formula. The eigenvalues of the transfer matrix areclassified by the physical combinatorics of the patterns of zeros in the complexspectral-parameter plane. This yields a selection rule for the physically relevant solutions tothe inversion identity and explicit finitized characters for the associated quasi-rationalrepresentations. In particular, in the scaling limit, we confirm the central chargec = −2 and conformalweights Δs = ((2−s)2−1)/8 for s = 1,2,3,.... We also discuss a diagrammatic implementation of fusion and show with exampleshow indecomposable representations arise. We examine the structure of theserepresentations and present a conjecture for the general fusion rules within our framework.