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Green's function and density of states

  1. May 29, 2015 #1
    Dear all,

    In his book chapter " Green’s Function Methods for Phonon Transport Through Nano-Contacts", Mingo arrives at the Green's function for the end atom of a one dimensional lattice chain (each atom modelled as a mass connected to neighbouring atoms through springs). He gives the green function as

    ## G = \frac{2}{\omega^2 + \sqrt{\omega^4 + 4 k \omega^2}} ##

    From this the intention is to find the spectral density of states. He directly gives it as

    ## \rho = \frac{1}{2 \pi} \frac{\sqrt{\omega^4 + 4 k \omega^2}}{2 k \omega^2} ##.

    I have not been able to show this.

    Earlier in the chapter he uses the standard representation for the spectral density of states as
    ## G - G^* = 2 \pi \rho ##.

    Using this I attempted the following way. Let ## \omega^2 = z ## (he had used such a representation earlier, hence I tried this), which gives:

    ## G(z) = \frac{2}{z + \sqrt{z^2 + 4 k z}} ##

    which can also be written as:

    ## G(z) = \frac{ z - \sqrt{z^2 + 4 k z}}{- 2 k z} ##

    Because of the ## \sqrt{z}## factor there is a branch cut along negative real axis. That's probably the only reason why ## G - G^*## would have a non-negative value - since there would be a non-zero difference in the values across the branch cut.

    So what we need is :
    ## \lim_{\delta \rightarrow 0} G(z+ i \delta) - G(z - i \delta) ##

    This gives:

    ## \rho = \lim_{\delta \rightarrow 0} \frac{1}{2 \pi} \left( \frac{ \sqrt{1 + \frac{4 k}{z+ i \delta}} - 1}{ 2 k } - \frac{ \sqrt{1 + \frac{4 k }{z - i \delta}} - 1}{ 2 k } \right) = \frac{1}{2 \pi} \left( \frac{ \sqrt{1 + \frac{4 k }{z+ i \delta}} - \sqrt{1 + \frac{4 k }{z - i \delta}}}{ 2 k } \right) ##

    If you take the limit, this just goes to zero.

    What am I doing wrong?

    Thanks!
     
  2. jcsd
  3. May 29, 2015 #2

    DrDu

    User Avatar
    Science Advisor

    I would have to take a closer look, but I think you have to write the square root as ##f(z)^{1/2}## and you have to continue it around the branch cut the square root whence it will pick up a factor -1. Summa summarum you will end up with the sum of the two square roots and not it's difference.
     
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