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

1. ### Maybe_Memorie

351
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.