T-7

Goal: the goal of this set of problems is to derive an estimate for the transition point for the Anderson model on the Bethe lattice.
Techniques: cavity method, Laplace transform, stability analysis.

A criterion for localization

• Green functions and self-energies. Given a lattice with $N$ sites $a$, we call $|a\rangle$ the wave function completely localised in site $a$. The Anderson model has Hamiltonian

  $H=W\sum _{a}\epsilon _{a}|a\rangle \langle a|-\sum _{<a,b>}t_{ab}\left(|a\rangle \langle b|+|b\rangle \langle a|\right)$

  where the local fields $\epsilon _{a}$ are random variables, independent and distributed according to $p(\epsilon )$. We introduce the Green functions $G_{ab}(z)$ and the local self-energies $\sigma _{a}(z)$: these are functions of a complex variable belonging to the upper half of the complex plane, and are defined by [NOTA SU STILTJIES]

  $G_{ab}(z)=\langle a|{\frac {1}{z-H}}|b\rangle ,\quad \quad G_{aa}(z)=\langle a|{\frac {1}{z-H}}|a\rangle ={\frac {1}{z-\epsilon _{a}-\sigma _{a}(z)}}.$

  It is clear that when the kinetic term $V$ in the Hamiltonian vanishes, the local self-energies vanish; these quantities encode how much the energy levels $\epsilon _{a}$ are shifted by the presence of the kinetic term $V$. They are random functions, because the Hamiltonian contains randomness. They encode properties on the spectrum of the Hamiltonian; the local density of eigenvalues $\rho _{a,N}(E)$ for an Hamiltonian of size $N$ is in fact given by

  $\rho _{a,N}(E)=-{\frac {1}{\pi }}\lim _{\eta \to 0}\Im G_{aa}(E+i\eta )=\sum _{\alpha =1}^{N}|\langle E_{\alpha }|a\rangle |^{2}\delta (E-E_{\alpha }),$

  where $E_{\alpha }$ are the eigenvalues of the Hamiltonian. [NOTA SU PLEMELJI]



• A criterion for localization. The local self-energies encode some information on whether localization occurs. More precisely, one can claim [CITE] that localization occurs whenever the imaginary part of $\sigma (E+i\eta )$ goes to zero when $\eta \to 0$. Given the randomness, this criterion should however be formulated probabilistically. One has:

  $\lim _{\eta \to 0}\lim _{N\to \infty }\mathbb {P} \left(-\Im \sigma _{a}(E+i\eta )>0\right)=0\quad \Longrightarrow \quad {\text{Localization}}$

Notice that in this criterion, the probability plays the role of an order parameter (like the magnetization in ferromagnets, or the overlap in spin glasses), and the imaginary part $\eta$ plays the role of a symmetry breaking field (like the magnetic field in the ferromagnet, or the coupling between replicas in spin glasses). However, the localization transition has nothing to do with equilibrium, i.e., it is not related to a change of structure of the Gibbs Boltzmann measure; rather, it is a dynamical transition. Pushing the analogy with equilibrium phase transitions, one can say that the localised phase corresponds to the disordered phase (the one in which symmetry is not broken, like the paramagnetic phase). The symmetry in question is time-reversal symmetry.


Problem 7.1: the Bethe lattice, recursion relations and cavity

The Bethe lattice is a lattice with a regular tree structure: each node has a fixed number of neighbours $k+1$, where $k$ is the branching number, and there are no loops (see sketch). In these problems we consider the Anderson model on such lattice.




1. Green functions identities. Consider an Hamiltonian split into two parts, $H=H_{0}+V$. Show that the following general relation holds (Hint: perturbation theory!)

   $G=G^{0}+G^{0}VG,\quad \quad G^{0}={\frac {1}{z-H_{0}}},\quad \quad G={\frac {1}{z-H}}.$



2. Cavity equations. We now apply this to a specific example: we consider a Bethe lattice, and choose one site 0 as the root. We then choose $V$ to be the kinetic terms connecting the root to its $k+1$ neighbours $a_{i}$,

   $V=-\sum _{i=1}^{k+1}t_{0a_{i}}\left(|a_{i}\rangle \langle 0|+|0\rangle \langle a_{i}|\right)$

   For all the $a_{i}$ with $i=1,\cdots ,k+1$ we introduce the notation

   $G_{a_{i}}^{\text{cav}}\equiv G_{a_{i}a_{i}}^{0},\quad \quad \sigma _{a_{i}}^{\text{cav}}\equiv \sigma _{a_{i}a_{i}}^{0},$

   where $\sigma ^{0}$ is the self energy associated to $G^{0}$. Show that, due to the geometry of the lattice, with this choice of $V$ the Hamiltonian $H_{0}$ is decoupled and $G_{a_{i}}^{\text{cav}}$ is the local Green function that one would have obtained removing the root 0 from the lattice, i.e., creating a “cavity” (hence the suffix). Moreover, using the relation above show that

   $G_{00}(z)={\frac {1}{z-\epsilon _{0}-\sum _{i=1}^{k+1}t_{0a_{i}}^{2}G_{a_{i}}^{\text{cav}}(z)}}$

   Iterating this argument, show that if $\partial a_{i}$ denotes the collection of “descendants" of $a_{i}$, i.e. sites that are nearest neighbours of $a_{i}$ except the root, then

   $G_{a_{i}}^{\text{cav}}(z)={\frac {1}{z-\epsilon _{a_{i}}-\sum _{b\in \partial a_{i}}t_{a_{i}b}^{2}G_{b}^{\text{cav}}(z)}},\quad \quad \sigma _{a_{i}}^{\text{cav}}(z)=\sum _{b\in \partial a_{i}}t_{a_{i}b}^{2}G_{b}^{\text{cav}}(z)=\sum _{b\in \partial a_{i}}{\frac {t_{a_{i}b}^{2}}{z-\epsilon _{b}-\sigma _{b}^{\text{cav}}(z)}}$

3. Equations for the distribution. Justify why the cavity functions appearing in the denominators in the last equations above are independent and identically distributed random variables, and therefore the cavity equations can be interpreted as self-consistent equations for the distribution of the cavity functions.




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 $k$?




Check out: key concepts of this TD

References

• Anderson. [1]
• The model is solved in Abou-Chacra, Thouless, Anderson. A selfconsistent theory of localization. Journal of Physics C: Solid State Physics 6.10 (1973)