# Homework Help: XXh spin chain, colleration function

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1. Oct 23, 2017

### Illuminatio fit

1. The problem statement, all variables and given/known data
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
$$a^+ a^+ |0> = 0$$

2. Oct 28, 2017

### PF_Help_Bot

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