How Does Tyablikov Approximation Impact Calculations in Quantum Mechanics?

[Petar Mali]

Can you tell me something more about Tyablikov approximation?

[tex]\langle\langle \hat{S}_i^z\hat{S}_j^{\pm}|\hat{B}_l\rangle\rangle
\approx \langle\hat{S}_i^z\rangle \langle\langle
\hat{S}_j^{\pm}|\hat{B}_l\rangle\rangle \qquad i\neq j[/tex]


I'm confused here? Is that approximation work in real and in inverse space or only in inverse space?


I think that if I use this approximation and than do Fourrier transformation I will get different result from the result which I will get if I first use Fourrier transf. and Tyablikov approximation?

Right?

Is

[tex]\langle\hat{S}_i^z\rangle[/tex] different than [tex]\langle\hat{S}_k^z\rangle[/tex]?

 

[BigJDubs]

Yes, the Tyablikov approximation works in real space. The approximation is used to simplify a sum over many-body terms by assuming that the terms are uncorrelated. This means that the expectation values of terms like $\langle\hat{S}_i^z \hat{S}_j^{\pm}\rangle$ can be written as the product of two expectation values $\langle\hat{S}_i^z\rangle \langle\hat{S}_j^{\pm}\rangle$. Therefore, if you perform a Fourier transformation before applying the Tyablikov approximation, then you will get a different result than if you apply the approximation first and then perform the Fourier transformation. Yes, $\langle\hat{S}_i^z\rangle$ is generally different from $\langle\hat{S}_k^z\rangle$, since these expectation values depend on the specific system you are studying.