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A research problem has led me to calculate a Green's function of a tight binding model with both onsite disorder and hopping amplitudes which vary in space. Since so much is known about tight binding models I thought I'd ask if this or other similar problems have been encountered before, since it doesn't make much sense to re-invent the wheel.

More concretely: I have in mind a finite tight binding model with open boundary conditions, with a Hamiltonian given by

$$H = \sum_{j=0}^{N}\left(\epsilon_{j}|j\rangle\langle j| - t_{j} |j+1\rangle\langle j| -t_{j} |j\rangle\langle j+1| \right)$$

To restrict to $N$ sites, we set $t_{N}=0$. In a particular problem, we can determine the potentials $\epsilon_{j}$ and hopping amplitudes $t_{j}$, but this is not really the point; all that matters is that these parameters don't have any symmetries. However, we can assume that the hopping amplitudes vary slowly for sufficiently large $j$, and in this regime the potentials also vary slowly. This should correspond to something like a gradient expansion in a suitable continuum limit.

What I would now like to calculate is the Green's function defined by

$$H|G\rangle = |0\rangle$$

which is to say, the vector $|G\rangle$ which, when acted on by the Hamiltonian, projects onto the first tight binding site. I imagine this can be converted into a continuum differential equation which can be treated by something like WKB methods, but before spending a ton of time working that through I wanted to see if this kind of problem is well-known to the stack exchange community. Suggestions for how to proceed are very welcome.

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The problem is that of inverting a Jacobi (tri-diagonal matrix). For numerical case of a specific realization of the disorder you can use Haydock recursion. For the general theory there are number of articles reprinted in the book "Mathematical Physics in One Dimension Exactly Soluble Models of Interacting Particles" By Elliott H. Lieb, Daniel C. Mattis.

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  • $\begingroup$ Thanks for the reference, I will take a look. As for your comment about Jacobi matrices - indeed, I am familiar with the recursion method, orthogonal polynomials, etc., and this is useful when looking at particular numerical problems. I'd like to move in a more analytic direction making use of the fact that the tight binding parameters behave smoothly at large $j$, where I think there should be a continuum limit. $\endgroup$
    – miggle
    Jan 3, 2023 at 14:45
  • $\begingroup$ I think the Leib and Mattis book will be useful then. I forget the authors of the exact one-d studies that are reprinted in there, but there is analytical work proving localization etc. I doubt thata there will be a simple ODE if the site-diagonal energies are truly random though. $\endgroup$
    – mike stone
    Jan 3, 2023 at 14:51

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