Assignment 4: The Metal Insulator Transition in Clean Systems

This the last assignment. It is designed to give you a better understanding of the tricky parts of the Fourrier transformation in a simple example of the solution of a many body problem.

The problem is to discretize and solve the discrete version of the following system of DMFT equations for G(iω_{n}), ω_{n} = (2n + 1)πT, n = −∞ … ∞.

1. G_{o}(iω_{n}) = [iω_{n} − t^{2}G(iω_{n})]^{−1}

2. G_{o}(τ) = T ∑_{n=−∞}^{∞} e^{−iω_{n}τ}G_{o}(iω_{n}))

3. ∑(τ) = −U^{2}G_{o}(τ)^{2}G_{o}(−τ)

4. ∑(iω_{r}) = \frac{1}{2} ∫_{−β}^{β} e^{iω_{n}τ} ∑(τ)dτ

5. G(iω_{n}) = \frac{1}{[G_{o}^{−1}(iω_{n})−Σ(iω_{n})]}

T U and t are parameters whose physical meaning are temperature, interaction strength and hopping integral. G(iω_{n}) ∝ iω_{n} means insulating behavior G(iω_{n}) ≃ i(sigω_{n}) as ω_{n} → 0 means metallic behavior.

You should regard 1 - 5 as a toy model system having the minimal type of non linearities necessary to model a complex physical phenomena, i.e. Mott transition.

We want to explore numerically the full solution of (1) - (5).

However first solve the trivial cases U = 0, and t = 0 analytically. Discuss the limiting behavior of the self energy Σ as the frequency goes to zero.

Then choose a discretization in τ space τ_{i}, and a cuttoff in frequency space, play with N the number of slices of [−β, β]. See how the results change when U is changed.

You are given the information that G(o^{+}) = −\frac{1}{2}, G(o^{−}) = \frac{1}{2}, so that G(τ) has a discontinuity at 0. Furthermore since the frequencies that enter (2) are of the form (2n + 1)πT, G(τ + β) = −G(τ). Also G(iω_{n}) = −G(−iω_{n}).

The strategy for solving the system 1-5 is to first discretize it and then solve it by iteration. Steps (1) (3) and (5) are trivial to implement. Step 2 involves a Fourier transform of a function with a long time tail, transform [G_{o}(iω_{n}) − \frac{1}{iω_{n}}] numerically and \frac{1}{iω_{r}} analytically T ∑_{n} \frac{e^{iωnτ}}{iω_{n}} = \{_{\frac{1}{2} −β<τ<o}^{\frac{1}{2} β>τ>o} (Check!) and observe how Fourier trades a long \frac{1}{iω} tail for a discontinuity.

Step 4 involves a Fourrier transform of a function with a known discontinuity. You first interpolate (linear interpolation is enough) and then Fourier transform the interpolation. Make sure the interpolating function has the right discontinuity and the right boundary values ∑(−β + o) and ∑(β − o). (∑(τ + β) = − ∑(τ)). Now you see how Fourier converts discontinuities into long \frac{1}{iω} tails. Now explore the high temperature regime set t = \frac{1}{2}, T = .02. Start with the U=o solution as an initial guess and solve the system 1-5 for U = .5, 1, l.5, 2, 2.5, 3 then begin with the t=o solution as an initial guess and solve (1-5), for U = 6, 5, 4, 3. [When you increase U or decrease U, use the previously obtained solution as an initial guess.] Display the G(iω_{n}) vs iω_{n} for different U’s in a graph. Is the evolution as a function of U smooth? Now repeat the same steps at much lower temperatures T = .02 paying close attention to the neighborhood of the point U=3. Is the evolution smooth?