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

In this Problem we 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: the real and imaginary parts of the local self energy satisfy the self-consistent equations:

$\Gamma _{a}=\sum _{b\in \partial a}t_{ab}^{2}{\frac {\Gamma _{b}+\eta }{(E-W\,V_{b}-R_{b})^{2}+(\Gamma _{b}+\eta )^{2}}},\quad \quad R_{a}=\sum _{b\in \partial a}t_{ab}^{2}{\frac {E-W\,V_{b}-R_{b}}{(E-W\,V_{b}-R_{b})^{2}+(\Gamma _{b}+\eta )^{2}}}$

These equations admit the solution $\Gamma _{a}=\Gamma _{b}=0$ when $\eta =0$, which corresponds to localization. We now determine when this solution becomes unstable.

Problem 9: an estimate of the mobility edge

1. Imaginary approximation and distributional equation. We consider the equations for $\Gamma _{a}$ and neglect the terms $R_{b}$ 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 $\Gamma _{a}\sim \eta \ll 1$. Finally, we set $t_{ab}\equiv t$ and $E=0$ for simplicity. Show that under these assumptions the probability density for the imaginary part, $P_{\Gamma }(\Gamma )$, satisfies 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 \tau=t/W}

   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 dV_b\,p(V_b)\int \prod_{b=1}^k d\Gamma_b \, P_\Gamma(\Gamma_b) \delta \left(\Gamma - \tau^2 \sum_{b \in \partial a} \frac{\Gamma_b + \eta}{ V_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

   $\Phi (s)=\left[\int dV\,p(V)e^{-{\frac {s\tau ^{2}\eta }{V^{2}}}}\Phi \left({\frac {s\tau ^{2}}{V^{2}}}\right)\right]^{k}$



2. The stability analysis. We now assume to be in the localized phase, when 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 \eta \to 0 } the 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 P_\Gamma(\Gamma) \to \delta (\Gamma)} . We wish to check the stability of our assumption. This is done by controlling the tails of the distribution $P_{\Gamma }(\Gamma )$ for finite 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 \eta } .
   • For finite 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 \eta} , we expect that typically $\Gamma _{a}\sim \eta \ll 1$, and thus 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)} should have a peak at this scale; however, we also expect some power law decay 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 $\Gamma$. These tails are contributed by the events 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 $\epsilon _{b}$ is very small, in which case 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 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 $\alpha \in (1,3/2]$. Show, using a dimensional analysis argument, that this corresponds to a non-analytic behaviour of the Laplace transform, 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) \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 $\beta =\alpha -1\in (0,1/2]$.
   • Show that the equation 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) } 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 $1-As^{\beta }=1-Ak\int d\epsilon \,p(\epsilon ){\frac {s^{\beta }V^{2\beta }}{\epsilon ^{2\beta }}}+o(s^{\beta })$, and therefore this is consistent provided that there exists a 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 (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). }




Behaviour of the integral $I(\beta )$ in the case of uniformily distributed disorder, 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 W< W_c } .
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:

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

   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 } ?

Check out: key concepts of this TD

Stability analysis, critical disorder, mobility edge.

References

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