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 $|a\rangle$ the wave function completely localised in site $a$. The Anderson model has Hamiltonian

  $H=W\sum _{a}\epsilon _{a}|a\rangle \langle a|+\sum _{<a,b>}V_{ab}\left(|a\rangle \langle b|+|b\rangle \langle a|\right)=H_{0}+V$

  where the local fields $\epsilon _{a}$ 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]

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

  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:

- model on the be the lattice

the cavity equation and the linearisation

Problem 7.2:

Check out: key concepts of this TD

References

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