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SU(2) operators to SU(N) generators for Heisenberg XXX 
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#1
Aug114, 06:23 PM

P: 284

A paper I'm reading says
"Our starting point is the [itex]SU(N)[/itex] generalization of the quantum Heisenberg model: [tex]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) [/tex] The [itex]J_{\beta}^{\alpha}[/itex] are the generators of the [itex]SU(N)[/itex] algebra and satisfy the usual commutation relations. ** The [itex]SU(N)[/itex] Heisenberg model can alternatively be written as an [itex]SU(2)[/itex] system with spin [itex]S=(N1)/2[/itex] moments interacting via higherorder exchange processes. An exact mapping connects the conventional [itex]SU(2)[/itex] spin operators to the [itex]SU(N)[/itex] generators as follows: [tex]STUFF [/tex] The Hamiltonian can then be expressed in terms of [tex]STUFF [/tex]" 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 1dim spin chain such as the Heisenberg XXX model with [itex]SU(N)[/itex]? So essentially my real question is can I express SU(N) symmetry in terms of SU(2) symmetry with higher spin for the 1dim spin chain, and if so how is the result arrived at? Many thanks. 


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