T-9

Problem 7.2: localization-delocalization transition on the Bethe lattice

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.

1. The “localized" solution. We set $z=E+i\eta$ and $\sigma _{a}^{\text{cav}}(z)=R_{a}(z)-i\Gamma _{a}(z)$. Show that the cavity equation for the self-energies is equivalent to the following pair of coupled equations:

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

   Justify why the solution corresponding to localization, $\Gamma _{a}=0$, is always a solution when $\eta \to 0$; moreover, in the localized phase when $\eta$ is finite but small one has $\Gamma _{a}\sim O(\eta )$. How can one argue that this solution has to be discarded, i.e. that delocalisation occurs?



2. 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

   $P_{\Gamma }(\Gamma )=\int \prod _{b=1}^{k}d\epsilon _{b}\,p(\epsilon _{b})\int \prod _{c=1}^{k}d\Gamma _{b}\,P_{\Gamma }(\Gamma _{b})\delta \left(\Gamma -t^{2}\sum _{b\in \partial a}{\frac {\Gamma _{b}+\eta }{\epsilon _{b}^{2}}}\right)$

   Show that the Laplace transform of this distribution, $\Phi (s)=\int _{0}^{\infty }d\Gamma e^{-s\Gamma }P_{\Gamma }(\Gamma )$, satisfies

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



3. The stability analysis. We now wish to check the stability of our assumption to be in the localized phase, $\Gamma _{a}\sim \eta \ll 1$, which led to the identity above for $\Phi (s)$. Our assumption is that the typical value of $\Gamma _{a}$ is small, except for cases in which one of the descendants $b$ is such that $\epsilon _{b}$ is very small, in which case $\Gamma _{a}\sim 1/\epsilon _{b}^{2}$.
   • Show that if $\Gamma \sim 1/\epsilon ^{2}$ and $p(\epsilon )$ 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 fields happen to be very small.
   • Assume more generally that $P_{\Gamma }(\Gamma )\sim \Gamma ^{-\alpha }$ for large $\Gamma$ and $\alpha \in [1,3/2]$. Show that this corresponds to $\Phi (s)\sim 1-A|s|^{\beta }$ for $s$ small, with $\beta =\alpha -1\in [0,1/2]$.
   • Show that the equation for $\Phi (s)$ gives for $s$ small $1-As^{\beta }=1-Ak\int d\epsilon \,p(\epsilon ){\frac {s^{\beta }t^{2\beta }}{\epsilon ^{2\beta }}}+o(s^{\beta })$, and therefore this is consistent provided that there exists a $\beta \in [0,1/2]$ solving

     $1=k\int d\epsilon \,p(\epsilon )\left({\frac {t}{\epsilon }}\right)^{2\beta }\equiv kI(\beta ).$



4. The critical disorder. Consider now local fields $\epsilon$ taken from a uniform distribution in $[-W/2,W/2]$. Compute $I(\beta )$ and show that it is non monotonic, with a local minimum $\beta ^{*}$ in the interval $[0,1/2]$. Show that $I(\beta ^{*})$ increases as the disorder is made weaker and weaker, and thus the transition to delocalisation occurs at the critical value of disorder when $I(\beta ^{*})=k^{-1}$. Show that this gives

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

   Why the critical disorder increases with $k$?

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