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 $\Lambda$ with sites $a$, we call $|a\rangle$ the wave function completely localised in site $a$. The Anderson model has Hamiltonian $H=W\sum _{a}\epsilon _{a}|a\rangle \langle a|+\sum _{<a,b>}V_{ab}\left(|a\rangle \langle b|+|b\rangle \langle a|\right)=H_{0}+V.$

  We introduce the local 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

  $G_{ab}=\langle a|{\frac {1}{z-H}}|b\rangle$

  
  

    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]