Understanding Magnetisation for S=1/2: Exploring the Concept and Its Limitations

[Petar Mali]

If we have case

$$\sigma=\frac{1}{2}-\frac{1}{N}\sum_{\bf{k}}\langle\hat{S}^-\hat{S}^+\rangle_{\bf{k}}$$

where $$\sigma$$ is magnetisation. How we know that $$\sigma$$ must be less than $$\frac{1}{2}$$. Or why is

$$\frac{1}{N}\sum_{\bf{k}}\langle\hat{S}^-\hat{S}^+\rangle_{\bf{k}}>0$$

Thanks for your answer.

 

[daveyrocket]

Just look at the matrix elements of the spin raising and lowering operators.
Or alternatively, multiply out the spin operators to get

$$S^+ S^- = S^2 \sigma^+ \sigma^- = S^2 (\sigma_x^2 + \sigma_y^2 + 2\sigma_z)$$

The $$\sigma_i^2$$ matrix is the identity, so its expectation value is 1. $$\sigma_z$$ has matrix elements of +1 and -1, so the quantity in the parentheses has to be between 0 and 4. S^2 is 1/4, so the result is between 0 and 1.

 

[Petar Mali]

Just look at the matrix elements of the spin raising and lowering operators.
Or alternatively, multiply out the spin operators to get

$$S^+ S^- = S^2 \sigma^+ \sigma^- = S^2 (\sigma_x^2 + \sigma_y^2 + 2\sigma_z)$$

The $$\sigma_i^2$$ matrix is the identity, so its expectation value is 1. $$\sigma_z$$ has matrix elements of +1 and -1, so the quantity in the parentheses has to be between 0 and 4. S^2 is 1/4, so the result is between 0 and 1.

You use if I see well

$$\hat{S}^+=S\hat{\sigma}^+$$

$$\hat{S}^-=S\hat{\sigma}^-$$

and how you define $$\hat{\sigma}^+$$ and $$\hat{\sigma}^-$$?

 

[daveyrocket]

$$\sigma^{\pm} = \sigma_x \pm i\sigma_y$$