XXh spin chain, colleration function

[Illuminatio fit]

Homework Statement

So given XXh chain:
$$\hat{H} = -J \sum ( S^x_j S^x_{j+1} +S^y_j S^y_{j+1}) + h \sum S^z_j $$
Requred to find $$\langle g| S^z_{j} S^z_{j+n} | g \rangle$$, where g is ground state.
2. The attempt at a solution
Using Jordan-Wigner transform firstly I abtain:
$$\hat{H} = -\frac{J}{2} \sum ( c^+_{j+1} c^-_{j} +c^+_j c^-_{j+1}) + h \sum c^+_j c^-_{j}$$.
Then using Fourier transform, into the impulse representation:
$$\hat{c}^{\pm}_j = \frac{1}{\sqrt{N}} \sum e^{\pm pj} \hat{a}^\pm_p$$
After some algebra we get nice H:
$$\hat{H} = \sum_{p} (-J \cos p + h) \hat{a}^+_p \hat{a}^-_p - \frac{h}{2}$$.
It's easy to see that for h>J |0> is ground state, and the answer is 1/4 this case die to Wick's theorem.
To find the ground state we should "turn" every p-th state in which -Jcosp+h<0.
Lets now pick h and J so that there's only p=0 that holds the inequality (its possible for any N).
Now I'm confused firstly because I'm not sure how c-operator acts on 1 flipped momentum state. And secondly because I'm not sure if I can use Wicks theorem now.
I tried to represent all the c-operators in Fourier series but that case I'm not sure if
$$a^+ a^+ |0> = 0$$

 

[mark726]

Thank you for your interesting post. I am a fellow scientist and I would like to help you with your problem.
Firstly, your attempt at the solution seems to be on the right track. However, it is important to note that in order to find the expectation value $\langle g| S^z_{j} S^z_{j+n} | g \rangle$, we need to find the ground state $|g\rangle$ first.

To find the ground state, we can use the Jordan-Wigner transformation as you have done. However, we need to be careful when applying the Fourier transform to the creation and annihilation operators. In particular, we need to keep in mind that the operators $\hat{a}_p^+$ and $\hat{a}_p^-$ are creation and annihilation operators for fermions, which obey the anti-commutation relations:
$$\{\hat{a}_p^+,\hat{a}_q^-\} = \delta_{p,q}$$
$$\{\hat{a}_p^+,\hat{a}_q^+\} = \{\hat{a}_p^-,\hat{a}_q^-\} = 0$$
Therefore, when we apply the Fourier transform to the creation and annihilation operators, we need to take into account the anti-commutation relations. This will ensure that the ground state we obtain is a fermionic state, which is necessary for the application of Wick's theorem.

As for your second question, the action of the creation and annihilation operators on a state with flipped momentum is given by:
$$\hat{a}_p^+ |1_p\rangle = |0_p\rangle$$
$$\hat{a}_p^- |0_p\rangle = |1_p\rangle$$
where $|1_p\rangle$ represents a state with one fermion occupying the momentum state $p$. It is important to note that these operators do not change the number of particles in a given momentum state.

In conclusion, you are on the right track in your attempt to find the ground state and the expectation value. Just be careful with the application of the Fourier transform and the anti-commutation relations, and you should be able to find the correct result. I hope this helps. Good luck with your research!