1112.3407 (A. M. Gainutdinov et al.)

Bimodule structure in the periodic gl(1|1) spin chain    [PDF]

A. M. Gainutdinov, N. Read, H. Saleur

This paper is second in a series devoted to the study of periodic super-spin
chains. In our first paper at 2011, we have studied the symmetry algebra of the
periodic gl(1|1) spin chain. In technical terms, this spin chain is built out
of the alternating product of the gl(1|1) fundamental representation and its
dual. The local energy densities - the nearest neighbor Heisenberg-like
couplings - provide a representation of the Jones Temperley Lieb (JTL) algebra.
The symmetry algebra is then the centralizer of JTL, and turns out to be
smaller than for the open chain, since it is now only a subalgebra of U_q sl(2)
at q=i, dubbed U_q^{odd} sl(2). A crucial step in our associative algebraic
approach to bulk logarithmic conformal field theory (LCFT) is then the analysis
of the spin chain as a bimodule over U_q^{odd} sl(2) and JTL. While our
ultimate goal is to use this bimodule to deduce properties of the LCFT in the
continuum limit, its derivation is sufficiently involved to be the sole subject
of this paper. We describe representation theory of the centralizer and then
use it to find a decomposition of the periodic gl(1|1) spin chain over JTL for
any even number N of tensorands and ultimately a corresponding bimodule
structure. Applications of our results to the analysis of the bulk LCFT will
then be discussed in the third part of this series.

View original: http://arxiv.org/abs/1112.3407