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.




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)