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Diagonalize the hamiltonian!

  1. Apr 14, 2009 #1
    1. The problem statement, all variables and given/known data
    I am trying to solve a problem of 1D electron system.
    Given [tex]a,a^\dagger,b,b^\dagger[/tex] annihilation and creation operator which satisfy the fermion commutation relations diagonalize the following hamiltonian:

    [tex]H=v_F\sum_{k>0}k(a^\dagger_ka_k-b^\dagger_kb_k)+\Delta\sum_k(b^\dagger_{k-k_F}a_{k+k_F}+a^\dagger_{k+k_F}b_{k-k_F})[/tex]

    where [tex]v_F,\Delta[/tex] are c-numbers.

    Prove that the spectrum is given by:
    [tex]E=v_Fk_F\pm v_F(\Delta^2+k^2)^{1/2}[/tex]

    2. The attempt at a solution
    I try to define the following operators (that form an su(2) algebra):
    [tex]J_3=\frac{1}{2}(a^\dagger_{k+k_F}a_{k+k_F}-b^\dagger_{k-k_F}b_{k-k_F})[/tex]
    [tex]J_+=a^\dagger_{k+k_F}b_{k-k_F}[/tex]
    [tex]J_-=b^\dagger_{k-k_F}a_{k+k_F}[/tex]
    and to calculate the adjoint action but I don't know how to continue.
    Please help me, thank you.
     
  2. jcsd
  3. Feb 2, 2015 #2
    The spectrum cannot be given by
    [tex]E=v_Fk_F\pm v_F(\Delta^2+k^2)^{1/2}[/tex]
    because the units of [tex] \Delta [/tex] and [tex] k [/tex] are not the same. I tried rewriting the first term (linear kinetic energy) using operators shifted by [tex] k_F [/tex] and then diagonalizing it, and I got
    [tex] E=\frac{1}{2} (v_F k_F \pm (v_F^2 k^2 + 4\Delta^2)^{1/2})[/tex]
    Still may not be correct, but at least the units match.
     
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