T-9

Goal: the goal of this problem is to determine when the solution of the distributional equations corresponding to localization is unstable, providing an estimate of thee mobility edge on the Bethe lattice.
Techniques: stability analysis, Laplace transforms.

Problems

We now focus on the self energies, since the criterion for localization is given in terms of these quantities. In this Problem we will determine for which values of parameters localization is stable, estimating the critical value of disorder where the transition to a delocalised phase occurs. Recall the results of Problem 8:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma_a= \sum_{b \in \partial a} V_{ab}^2 \frac{\Gamma_b + \eta}{(E- \epsilon_b - R_b)^2+ (\Gamma_b +\eta)^2}, \quad \quad R_a = \sum_{b \in \partial a} V_{ab}^2 \frac{E- \epsilon_b - R_b}{(E- \epsilon_b - R_b)^2+ (\Gamma_b +\eta)^2} }

Problem 9: an estimate of the mobility edge

1. Imaginary approximation and distributional equation. We consider the equations for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma_a } and neglect the terms Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_b } in the denominators, which couple the equations to those for the real parts of the self energies (“imaginary” approximation). Moreover, we assume to be in the localized phase, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma_a \sim \eta \ll 1 } . Finally, we set Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_{ab} \equiv V } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E=0 } for simplicity. Show that under these assumptions the probability density for the imaginary part, $P_{\Gamma }(\Gamma )$, satisfies

   Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_\Gamma(\Gamma)= \int \prod_{b=1}^k d\epsilon_b\,p(\epsilon_b)\int \prod_{b=1}^k d\Gamma_b \, P_\Gamma(\Gamma_b) \delta \left(\Gamma - V^2 \sum_{b \in \partial a} \frac{\Gamma_b + \eta}{ \epsilon_b^2} \right) }

   Show that the Laplace transform of this distribution, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(s)=\int_0^\infty d\Gamma e^{-s \Gamma} P_\Gamma(\Gamma) } , satisfies

   $\Phi (s)=\left[\int d\epsilon \,p(\epsilon )e^{-{\frac {sV^{2}\eta }{\epsilon ^{2}}}}\Phi \left({\frac {sV^{2}}{\epsilon ^{2}}}\right)\right]^{k}$



2. The stability analysis. We now wish to check the stability of our assumption to be in the localized phase, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma_a \sim \eta \ll 1 } , which led to the identity above for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Phi(s) } . Our assumption is that the typical value of $\Gamma _{a}$ is small, except for cases in which one of the descendants Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b } is such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon_b } is very small, in which case $\Gamma _{a}\sim 1/\epsilon _{b}^{2}\gg 1$.
   • Show that if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma \sim 1/ \epsilon^2 } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(\epsilon)} is not gapped around zero, then $P_{\Gamma }(\Gamma )\sim \Gamma ^{-3/2}$, i.e. the distribution has tails contributed by these events in which the local random potential happen to be very small.
   • Assume more generally that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_\Gamma(\Gamma) \sim \Gamma^{-\alpha}} for large Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Gamma } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha \in [1, 3/2]} . Show that this corresponds to $\Phi (s)\sim 1-A|s|^{\beta }$ for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s } small, with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta= \alpha-1 \in [0, 1/2] } .
   • Show that the equation for $\Phi (s)$ gives for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s } small Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1- A s^\beta =1- A k \int d\epsilon \, p(\epsilon) \frac{s^\beta V^{2 \beta}}{\epsilon^{2 \beta}}+ o(s^\beta) } , and therefore this is consistent provided that there exists a $\beta \in [0,1/2]$ solving

     Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1=k \int d\epsilon \, p(\epsilon) \left(\frac{V}{\epsilon}\right)^{2 \beta} \equiv k I(\beta). }



3. The critical disorder. Consider now local fields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \epsilon } taken from a uniform distribution in $[-W/2,W/2]$. Compute Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I(\beta) } and show that it is non monotonic, with a local minimum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta^* } in the interval $[0,1/2]$. Show that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I(\beta^*) } increases as the disorder is made weaker and weaker, and thus the transition to delocalisation occurs at the critical value of disorder when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I(\beta^*)=k^{-1} } . Show that this gives the following estimate for the critical disorder at which the transition to delocalisation occurs:

   $W_{c}=V\,2ke\log \left({\frac {W_{c}}{2V}}\right)\sim \,2Ve\log \left(k\right)$

   Why the critical disorder increases with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k } ?

References

• Abou-Chacra, Thouless, Anderson. A selfconsistent theory of localization. Journal of Physics C: Solid State Physics 6.10 (1973)