- #1
Maybe_Memorie
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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)<br />
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<br />
The Hamiltonian can then be expressed in terms of
STUFF<br />"
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.
"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)<br />
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<br />
The Hamiltonian can then be expressed in terms of
STUFF<br />"
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.
