2006/10/31 by Paul A. Pearce, Jørgen Rasmussen, Jorgen Rasmussen · 2 citations
Materials Science · Mathematics · Physics and Astronomy · #Complex plane #Conformal map #Conjecture #Diagrammatic reasoning #Eigenvalues and eigenvectors #Geometry #Indecomposable module #Lattice (music) #Markov Chains and Monte Carlo Methods #Material Dynamics and Properties #Mathematical analysis #Mathematics #Physics #Pure mathematics #Quantum mechanics #Random matrix #Scaling #Scaling limit #Theoretical and Computational Physics #Transfer matrix #cond-mat.stat-mech #hep-th
paper · pdf · doi:10.1088/1742-5468/2007/02/p02015
published as J.Stat.Mech.0702:P02015,2007 · 35 pages, v2: comments and references added
arxiv created 2007/02/06 · openalex publication_date 2007/02/19 · arxiv updated 2011/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.