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

Abstract

Supersymmetry operators that change a spin chain's length have appeared in numerous contexts, ranging from the AdS/CFT correspondence to statistical mechanics models. In this article, we present, via an analysis of the Bethe equations, all homogeneous, rational and trigonometric, integrable gl(n|m) spin chains for which length-changing supersymmetry can be present. Furthermore, we write down the supercharges explicitly for the simplest new models, namely the sl(n|1) spin chains with the (n − 1)-fold antisymmetric tensor product of the fundamental representation at each site and check their compatibility with integrability.

Lattice supersymmetry in the open XXZ model: an algebraic Bethe Ansatz analysis

“…The type of lattice SUSY displayed by (1.1) has now been found and studied, primarily by Fendley and Hagendorf and collaborators, in a growing range of open and closed quantum spin chains [3,[10][11][12][13][14][15]. In particular, the manifestation of SUSY as pairings of sets of roots of the Bethe Equations was observed and studied for the closed 8-vertex model along its combinatorial line in [11].…”

Lattice SUSY for the DiSSEP at λ2 = 1 (and λ2 = −3)

“…The supercharge Q L resulting from equation (14) can be further elaborated to give a limited class of non-identical boundary terms [5]. The approach readily generalises to higher spin models [4,8] and N M ( | ) gl Hamiltonians [9].…”

The nineteen-vertex model and alternating sign matrices

“…The supersymmetry is therefore dynamic in the sense that the length of the chain changes through the action of its supercharges. The precise definition of Q and further technical can be found in [9], and a recent, very concise and general discussion of dynamic lattice supersymmetry in (super)spin chains in [25]. For our purposes, it is sufficient to know that the subsectors of V where the supersymmetry exists are the eigenspaces of the twisted translation operator S with eigenvalue (−1) N +1 .…”

“…The result is sometimes called a scattering operator as its effect is to drag the inhomogeneity at position j through all others in a cyclic manner. We equate both expressions, use the exchange relation (25), and find after elimination of some trivial factors the following relation for all j = 1, . .…”