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 $\mathrm{\Lambda }$ with 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 $G_{ab}(z)$: these are functions of a complex variable belonging to the upper half of the complex plane, and are defined by [NOTA SU STILTJIES]

  $G_{ab}(z)=⟨a|\frac{1}{z-H}|b⟩$

  The local self-energies ${\sigma }_a(z)$ are functions defined by the equality

  $G_{aa}(z)=⟨a|\frac{1}{z-H}|a⟩=\frac{1}{z-{ϵ}_a-{\sigma }_a(z)}.$ 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 density of eigenvalues ${\displaystyle }$


• A criterion for 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]