Supersymmetry on Jacobstahl lattices

TL;DR

This paper explores supersymmetric systems derived from Jacobstahl lattices, revealing their spectral properties and relation to the XXZ chain, with implications for understanding supersymmetry in lattice models.

Contribution

It demonstrates that supersymmetric systems constructed via Yang and Fendley's method correspond to Jacobstahl lattice systems, not the open XXZ chain, and analyzes their spectral continuum limits.

Findings

Ground states are unique for each Jacobstahl system.

Spectra match the $U_q(sl(2))$ invariant XXZ chain in the continuum limit.

Relation between Jacobstahl systems and open XXZ chain is clarified.

Abstract

It is shown that the construction of Yang and Fendley (2004 {\it J. Phys. A: Math.Gen. {\bf 37}} 8937) to obtainsupersymmetric systems, leads not to the open XXZ chain with anisotropy $\Delta =-{1/2}$ but to systems having dimensions given by Jacobstahl sequences.For each system the ground state is unique. The continuum limit of the spectra of the Jacobstahl systems coincide, up to degeneracies, with that of the $U_q(sl(2))$ invariant XXZ chain for $q=\exp(i\pi/3)$. The relation between the Jacobstahl systems and the open XXZ chain is explained.