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

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

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

 

[Petar Mali]

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

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

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

 

[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
$$\hat{S}^-_n - \sqrt{2S}\hat{B}_n^+$$
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?