Can Spin Comutation Relations be Simplified by Using Average Spin Values?

[Petar Mali]

[\hat{S}^+_{n},\hat{S}^-_{m}]=2S_n^z\delta_{n,m}

where n,m are position vectors of spins. Why if we have

\langle\hat{S}^z\rangle\approx S we can use

[\hat{S}^+_{n},\hat{S}^-_{m}]\approx 2S\delta_{n,m}?

where S is value of the spin, for example \frac{1}{2}.

 

[DrDu]

Its an approximation. And we know that

<br /> \langle [\hat{S}^+_{n},\hat{S}^-_{m}]\rangle \approx 2S\delta_{n,m}<br />
So it does not appear implausible.

 

[Petar Mali]

Yes ok. I don't have problem with

\langle [\hat{S}^+_{n},\hat{S}^-_{m}]\rangle \approx 2S\delta_{n,m}

but where \langle... \rangle disappear?

Why I can use this approximation?

<br /> [\hat{S}^+_{n},\hat{S}^-_{m}]\approx 2S\delta_{n,m}<br />

 

[Petar Mali]

If relation <br /> <br /> [\hat{S}^+_{n},\hat{S}^-_{m}]\approx 2S\delta_{n,m}<br /> <br />

is correct than

we can define

\hat{S}^-_n \rightarrow \sqrt{2S}\hat{B}_n^+

\hat{S}^+_n\rightarrow \sqrt{2S}\hat{B}_n

which is Bloch approximation but I don't see some reasons for

<br /> <br /> [\hat{S}^+_{n},\hat{S}^-_{m}]\approx 2S\delta_{n,m}<br /> <br />

 

[DrDu]

Why I can use this approximation?

You can always use whatever approximation you like. The question is whether it is good or bad. This depends on the system you use it for. A point about which you didn't tell us anything.

 

[Petar Mali]

Well I have some magnetic ordered system. For example ferromagnet or antiferromagnet and have some spin hamiltonian. I want to replace operators \hat{S}^{\pm},\hat{S}^z with some functions of Bose operators. Temperatures are low.

 

[DrDu]

Well, what you could try is to express the hamiltonian in terms of \hat{B}_n^+ and
<br /> \hat{S}^-_n - \sqrt{2S}\hat{B}_n^+<br />
considering terms containing the difference as a perturbation. Maybe you can show that they only lead to small corrections in the limit you are considering?