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Jordan-Wigner transformation

  1. Jul 23, 2008 #1
    How to go from this Hamiltonian :

    [tex]H= \sum^{N}_{j=1}[\sigma^{+}_{j}\sigma^{-}_{j+1}+\sigma^{-}_{j}\sigma^{+}_{j+1}] [/tex]

    to the following Hamiltonain:

    [tex]H = \sum^{N-1}_{j=1}[c^{*}_{j}c_{j+1}+c^{*}_{j+1}c_{j}]-(c^{*}_{1}c_{N}+c^{*}_{N}c_{1})exp[i\pi\sum^{N}_{j=1}c^{*}_{j}c_{j}][/tex]

    (where c* is the Hermitian conjugate of c)

    using the Jordan-Wigner transformation:
    [tex]\sigma^{+}_{j} = exp[i\pi\sum^{j-1}_{n=1}c^{*}_{n}c_{n}] c_{j}[/tex]

    [tex]\sigma^{-}_{j} = exp[-i\pi\sum^{j-1}_{n=1}c^{*}_{n}c_{n}] c^{*}_{j}[/tex]

    where are fermionic operators.

    The following is what I have calculated, please correct my mistakes if there are.

    [tex]\sigma^{+}_{j}\sigma^{-}_{j+1}=c_{j}exp[-i\pi c^{*}_{j}c_{j}]c^{*}_{j+1}=c_{j}(1-2c^{*}_{j}c_{j})c^{*}_{j+1}[/tex]

    [tex]\sigma^{-}_{j}\sigma^{+}_{j+1}=c^{*}_{j}exp[i\pi c^{*}_{j}c_{j}]c_{j+1}=c^{*}_{j}(1-2c^{*}_{j}c_{j})c_{j+1}[/tex]

    putting them into the first Hamiltonian yeilds:


    how to proceed further in order to arrive at the second Hamiltonian?
  2. jcsd
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