Spin Chains with Dynamical Lattice Supersymmetry

Abstract

Spin chains with exact supersymmetry on finite one-dimensional lattices are considered. The supercharges are nilpotent operators on the lattice of dynamical nature: they change the number of sites. A local criterion for the nilpotency on periodic lattices is formulated. Any of its solutions leads to a supersymmetric spin chain. It is shown that a class of special solutions at arbitrary spin gives the lattice equivalents of the N = (2, 2) superconformal minimal models. The case of spin one is investigated in detail: in particular, it is shown that the Fateev-Zamolodchikov chain and its off-critical extension possess a lattice supersymmetry for all its coupling constants. Its supersymmetry singlets are thoroughly analysed, and a relation between their components and the weighted enumeration of alternating sign matrices is conjectured.

Dynamic Lattice Supersymmetry in $$\mathfrak {gl}\left( {n}|{m}\right) $$ gl n | m Spin Chains

“…In particular the a 1 = 2, or spin 1, is the only finite dimensional gl(2) representation that exhibits dynamic supersymmetry in the rational limit and its supercharges can be found in [19]. We also find solutions in the q-deformed case starting from a 1 = 5 with q a1−2 = 1.…”

“…Mainly, we shall review here the results of [18,19], adapted to the language of superspaces. The reader is referred to the original publications for proofs.…”

“…We begin by reviewing the sl(2|1) case for which the solution was already given in [19] and which we generalize to the q-deformed case. From this special case, it is then straightforward to construct the local length-changing operators for all sl(n|1) cases and compare the resulting Hamiltonian density with the one derived from integrability.…”

“…Investigating lattice fermion models with exclusion rules that are constructed to possess supersymmetry [15][16][17], it was found that some of these models map to integrable spin chains for which the supersymmetry generators become length-changing. These results were later generalized to elliptic and higher spin models [18,19], in a fashion that made some of the underlying general structure more transparent. A closer look shows that these models are similar to the ones arising in some subsectors 1 of the gauge theories at one-loop, where the manifest spin chain symmetry algebra g commutes with the gl(1|1) algebra formed out of the Hamiltonian H, the two length-changing supercharges Q, Q † and the length-measuring operator.…”

“…The present article is structured as follows. We first introduce the reader in section 2 to the basic elements of length-changing or "dynamic" supersymmetry in spin chains and review the main elements of [18,19], adapted to graded representations.…”