A Curious Mapping Between Supersymmetric Quantum Chains

Abstract

We present a unitary transformation relating two apparently different supersymmetric lattice models in one dimension. The first [1] describes semionic particles on a 1D ladder, with supersymmetry moving particles between the two legs. The second [2] is a fermionic model with particle-hole symmetry and with supersymmetry creating or annihilating pairs of domain walls. The mapping we display features non-trivial phase factors that generalise the sign factors occurring in the Jordan-Wigner transformation.We dedicate this work to our friend and colleague Bernard Nienhuis, on the occasion of his 65-th birthday.

Topological quantum pump of strongly interacting fermions in coupled chains

“…Different types of boundary conditions will be considered. This Hamiltonian is a modification of a model introduced by Fendley and Schoutens [22,23], which we will refer to as  ( ) H 1 because it is equal to the above Hamiltonian with all λ j =1. This  ( ) H 1 is a supersymmetric (SUSY) lattice model whose spectrum is completely solvable by nested Bethe ansatz.…”

Ground states of Nicolai and ${\mathbb{Z}_2}$ Nicolai models

“…A particle-hole symmetric version of the 1D M 1 model [10]: Surprisingly, this model turns out to be equivalent to the FS model-an explicit unitary map was presented in [11].…”

Free fermions beyond Jordan and Wigner

“…Another interesting feature of the Cooper-pair chain is that it exhibits a N = 2 lattice supersymmetry [6,7] where the Hamiltonian can be written as a square of either of two supersymmetry generators. Subsequently, the Cooper-pair model with open boundary conditions was shown to be equivalent (via a unitary transformation) [8] to another fermion chain with lattice supersymmetry, the "DFNR" model [9].…”