Self consistent spin wave theory

[Petar Mali]

$$\hat{H}=\hat{H}_0+S\sum_{i,j}I_{i,j}(\hat{a}_i\hat{b}_j+\hat{a}_i^+\hat{b}_j^++\hat{b} ^+_j\hat{b}_j+\hat{a} ^+_i\hat{a}_i)-\sum_{i,j}I_{i,j}[\frac{1}{2}(\hat{a}_i\hat{b} ^+_j\hat{b}_j\hat{b}_j+\hat{a}^+_i\hat{a}^+_i\hat{a}_i\hat{b} ^+_j)+\hat{a} ^+_i\hat{a}_i\hat{b} ^+_j\hat{b}_j]$$

$$\hat{a}_i,\hat{a}_i^+,\hat{b}_j,\hat{b}_j^+$$ - bose operators

SCSW - theory

$$\hat{H}=\hat{H}_0+\hat{H}_2+\hat{H}_4^{SC}$$

$$\hat{H}_2=S\sum_{i,j}I_{i,j}(\hat{a}_i\hat{b}_j+\hat{a}_i^+\hat{b}_j^++\hat{b} ^+_j\hat{b}_j+\hat{a} ^+_i\hat{a}_i)$$

How is $$\hat{H}^{SC}_{4}$$ defined?

Term$$-\sum_{i,j}I_{i,j}[\frac{1}{2}(\hat{a}_i\hat{b} ^+_j\hat{b}_j\hat{b}_j+\hat{a}^+_i\hat{a}^+_i\hat{a}_i\hat{b} ^+_j)+\hat{a} ^+_i\hat{a}_i\hat{b} ^+_j\hat{b}_j]$$ represent magnon - magnon interractions.

 

[Petar Mali]

In textbook which I had

$$\hat{H}_4^{SC}=-\sum_{i,j}I_{i,j}[\hat{a}^+_i\hat{a}_i(\langle \hat{b}^+_j\hat{b}_j\rangle+\langle\hat{b}^+_j\hat{a}^+_i\rangle)+\hat{b}^+_j\hat{b}_j(\langle\hat{a}^+_i\hat{a}_i\rangle+ \langle\hat{b}_j\hat{a}_i\rangle)+\hat{a}_i\hat{b}_j(\langle\hat{a}_i^+\hat{b}_j^+\rangle+\langle\hat{b}_j^+\hat{b}_j\rangle) +\frac{1}{2}\hat{a}_i^+\hat{a}_i^+\langle\hat{b}_j^+\hat{a}_i\rangle+\frac{1}{2}\hat{a}_i^+\hat{a}_i^+\langle \hat{b}_j^+\hat{a}_i \rangle+\frac{1}{2}\hat{b}_j^+\hat{a}_i(\langle \hat{b}_j\hat{b}_j\rangle+\frac{1}{2}\langle\hat{a}_i^+\hat{a}_i\rangle)]$$

Can you explain me this?

 

[Petar Mali]

Any help?

For example

$$\hat{a}_i^+\hat{a}_i\hat{b}_j^+\hat{b}_j=\hat{a}_i^+\hat{a}_i\langle \hat{b}_j^+\hat{b}_j \rangle+\hat{b}_j^+\hat{b}_j\langle \hat{a}_i^+\hat{a}_i \rangle+\hat{a}_i\hat{b}^+_j\langle \hat{a}_i^+\hat{b}_j \rangle+\hat{a}_i^+\hat{b}_j \langle\hat{a}_i\hat{b}_j^+\rangle+\hat{a}_i^+\hat{b}_j^+\langle \hat{a}_i\hat{b}_j\rangle+\hat{a}_i\hat{b}_j\langle \hat{a}_i^+\hat{b}_j^+ \rangle$$

Correct? Can you explain me this? Thanks!

 

[kanato]

This looks like an application of Wick's theorem.
http://en.wikipedia.org/wiki/Wick's_theorem

 

[welshtill]

This is more like a Bogoliubov's method.

 

[Petar Mali]

I think that is neither of that!

 

[Petar Mali]

This is some kind of approximation.

Maybe approximation of identity

$$\hat{A}\hat{B}=\hat{A}\langle \hat{B} \rangle+\langle\hat{A}\rangle \hat{B}-\langle\hat{A}\hat{B}\rangle+(\hat{A}-\langle \hat{A}\rangle)(\hat{B}-\langle \hat{B}\rangle)$$

?

$$\hat{A}\hat{B}=\hat{A}\langle \hat{B} \rangle+\langle\hat{A}\rangle \hat{B}-\langle\hat{A}\hat{B}\rangle+(\hat{A}-\langle \hat{A}\rangle)(\hat{B}-\langle \hat{B}\rangle)$$

$$\hat{C}\hat{D}=\hat{C}\langle \hat{D} \rangle+\langle\hat{C}\rangle \hat{D}-\langle\hat{C}\hat{D}\rangle+(\hat{C}-\langle \hat{C}\rangle)(\hat{D}-\langle \hat{D}\rangle)$$

So

$$\hat{A}\hat{B}\hat{C}\hat{D}=?$$

Anybody knows?

 

[Petar Mali]

Any idea?

 

[shawl]

This is a method of mean field theory.

It is used for the weak coupled superconductor.

if there is only <a+a>, that is Hartree-Fock approximation.
If there contains <a+a+> or <aa>, that is mean field theory.

 

[Petar Mali]

I made a mistake in Hamiltonian

It looks like

$$\hat{H}_4^{SC}=-\sum_{i,j}I_{i,j}\{\hat{a}^+_i\hat{a}_i(\langle \hat{b}^+_j\hat{b}_j\rangle+\langle \hat{a}^+_i\hat{b}^+_j\rangle)+\hat{b}^+_j\hat{b}_j(\langle \hat{a}^+_i\hat{a}_i\rangle+\langle\hat{a}_i\hat{b}_j\rangle)+\hat{a}_i\hat{b}_j(\langle \hat{a}^+_i\hat{b}^+_j\rangle+\langle\hat{b}^+_j\hat{b}_j\rangle)+\frac{1}{2}\hat{a}^+_i\hat{a}^+_i\langle\hat{a}_i\hat{b} ^+_j\rangle+\frac{1}{2}\hat{b}_j\hat{b}_j\langle\hat{a}_i\hat{b} ^+_j\rangle+\frac{1}{2}\hat{a}_i\hat{b}^+_j(\langle\hat{a}^+_i\hat{a}^+_i\rangle+\langle\hat{b}_j\hat{b}_j\rangle)\}$$