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Homework Help: XXh spin chain, colleration function

  1. Oct 23, 2017 #1
    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. jcsd
  3. Oct 28, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
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