SU(2) operators to SU(N) generators for Heisenberg XXX

by Maybe_Memorie
Tags: generators, heisenberg, operators
 P: 284 A paper I'm reading says "Our starting point is the $SU(N)$ generalization of the quantum Heisenberg model: $$H=-J\sum_{\langle i,j \rangle}H_{ij}=\frac{J}{N}\sum_{\langle i,j \rangle}\sum_{\alpha , \beta =1}^N J_{\beta}^{\alpha}(i)J_{\alpha}^{\beta}(j)$$ The $J_{\beta}^{\alpha}$ are the generators of the $SU(N)$ algebra and satisfy the usual commutation relations. ** The $SU(N)$ Heisenberg model can alternatively be written as an $SU(2)$ system with spin $S=(N-1)/2$ moments interacting via higher-order exchange processes. An exact mapping connects the conventional $SU(2)$ spin operators to the $SU(N)$ generators as follows: $$STUFF$$ The Hamiltonian can then be expressed in terms of $$STUFF$$" This is the paper http://arxiv.org/pdf/0812.3657.pdf. The stuff in question is on page 2. Sorry I didn't LaTeX the full thing but I'm using a foreign keyboard and it would've taken ages. My questions... How is ** arrived at? Presently my Lie algebra knowledge is very lacking but i'm working on it. This paper is about a square lattice. Can the result still be generalised for a 1-dim spin chain such as the Heisenberg XXX model with $SU(N)$? So essentially my real question is can I express SU(N) symmetry in terms of SU(2) symmetry with higher spin for the 1-dim spin chain, and if so how is the result arrived at? Many thanks.