1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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:

    [tex]\sum^{N}_{j=1}c_{j}(1-2c^{*}_{j}c_{j})c^{*}_{j+1}+c^{*}_{j}(1-2c^{*}_{j}c_{j})c_{j+1}[/tex]

    how to proceed further in order to arrive at the second Hamiltonian?
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?
Draft saved Draft deleted



Similar Discussions: Jordan-Wigner transformation
  1. Wigner Weyl Transforms (Replies: 6)

Loading...