T-7: Difference between revisions
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<li> <strong> Green functions and self-energies. </strong> Given a lattice <math> \Lambda </math> with sites <math>a </math>, we call <math> |a \rangle </math> the wave function completely localised in site <math> a </math>. The Anderson model has Hamiltonian | <li> <strong> Green functions and self-energies. </strong> Given a lattice <math> \Lambda </math> with sites <math>a </math>, we call <math> |a \rangle </math> the wave function completely localised in site <math> a </math>. The Anderson model has Hamiltonian | ||
<center> <math> | <center> <math> | ||
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= 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. | ||
</math> | </math> | ||
</center> | </center> | ||
We introduce the local green functions <math> G_{ab}(z) </math>: these are functions of a complex variable belonging to the upper half of the complex plane, | |||
Revision as of 17:11, 13 January 2024
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 sites , we call the wave function completely localised in site . The Anderson model has Hamiltonian
We introduce the local green functions : these are functions of a complex variable belonging to the upper half of the complex plane,
Y .
- Bouchaud. Weak ergodicity breaking and aging in disordered systems [1]
- model on the be the lattice - self energy -criterion for localization - links to ergo breaking
Problem 7.1:
the cavity equation and the linearisation