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, stability analysis.

A criterion for localization

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

  $H=W\sum _a{ϵ}_a|a⟩⟨a|+\sum _{<a,b>}{V}_{ab}(|a⟩⟨b|+|b⟩⟨a|)=H_0+V$

  where the local fields ${ϵ}_a$ are random variables. 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]

  $G_{ab}(z)=⟨a|\frac{1}{z-H}|b⟩,G_{aa}(z)=⟨a|\frac{1}{z-H}|a⟩=\frac{1}{z-{ϵ}_a-{\sigma }_a(z)}.$

  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 ${ϵ}_a$ 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

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

  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:

  $\underset{\eta \to 0}{\lim }\underset{N\to \mathrm{\infty }}{\lim }\mathbb{\mathbb{P}}(-\mathrm{\Im }{\sigma }_a(E+I\eta )>0)=0⟹\text{Localization}$


Y . - model on the be the lattice - self energy -criterion for localization - links to ergo breaking

Problem 7.1:

the cavity equation and the linearisation




Problem 7.2:

Check out: key concepts of this TD

References

• Bouchaud. Weak ergodicity breaking and aging in disordered systems [1]