The Eight-Vertex Model and Lattice Supersymmetry

Abstract

We show that the XYZ spin chain along the special line of couplings JxJy + JxJz + JyJz = 0 possesses a hidden N = (2, 2) supersymmetry. This lattice supersymmetry is non-local and changes the number of sites. It extends to the full transfer matrix of the corresponding eight-vertex model. In particular, it is shown how to derive the supercharges from Baxter's Bethe ansatz. This analysis leads to new conjectures concerning the ground state for chains of odd length. We also discuss a correspondence between the spectrum of this XYZ chain and that of a manifestly supersymmetric staggered fermion chain.

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

“…If such a charge exists for an integrable spin chains, then we expect that its effects should manifest themselves at the level of the Bethe equations in the form of degeneracies in the energy spectrum for different lengths. Such degeneracies have indeed been observed in the case of some models for which the dynamic supersymmetry was already known [11,18]. Here we proceed in reverse: we identify models with a hidden dynamic supersymmetry by systematically looking for the respective degeneracies in the Bethe equations, for which the supercharges can then hopefully be constructed.…”

“…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.…”

“…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.…”

Order and supersymmetry at high filling zero-energy states on the triangular lattice

“…Recently, Fendley and Schoutens 2,3 proposed a many-body lattice model which exhibits what is known as supersymmetric quantum mechanics 4,5 . It consists of hard-core lattice fermions with nearest neighbor exclusion and a precisely fine-tuned interaction coupling.…”