Abstract
A new spin-chain representation of the Temperley–Lieb algebra TLn(β=0) is introduced and related to the dimer model. Unlike the usual XXZ spin-chain representations of dimension 2n, this dimer representation is of dimension 2n−1. A detailed analysis of its structure is presented and found to yield indecomposable zigzag modules.
Highlights
The classical dimer model describes perfect domino tilings or coverings of a lattice by 1 × 2 and 2 × 1 rectangles
The transfer matrix approach by Lieb [5], in particular, uses tools of statistical mechanics to describe the combinatorial problem on the square lattice and was recently revisited [6] in a study of the conformal properties arising in the continuum scaling limit of the model
Lieb’s approach is based on a map from dimer configurations to spin configurations and opens the door to study the dimer model using the machinery of spin-chains
Summary
The classical dimer model describes perfect domino tilings or coverings of a lattice by 1 × 2 and 2 × 1 rectangles. The open Heisenberg and XXZ spin-chains, in particular, are known [7] to yield representations of the Temperley–Lieb algebra TLn (β) [8,9] where n is the number of sites and β the loop fugacity. These spin-chain representations are constructed in terms of Pauli matrices acting on (C2 )⊗n and are of dimension 2n. We offer a new spin-chain representation of TLn (β = 0) It is constructed in terms of Pauli matrices, but unlike the familiar spin-chain representations, it acts on one fewer spin- 21 site and is only of dimension 2n−1.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
