T-7: Difference between revisions
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<ul> | <ul> | ||
<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_0 + V | 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> | ||
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where the local fields <math> \epsilon_a </math> are random variables. | where the local fields <math> \epsilon_a </math> are random variables. | ||
We introduce the <ins>Green functions</ins> <math> G_{ab}(z) </math> and the <ins>local self-energies</ins> <math> \sigma_a(z)</math>: these are functions of a complex variable belonging to the upper half of the complex plane, and are defined by [NOTA SU STILTJIES] | We introduce the <ins>Green functions</ins> <math> G_{ab}(z) </math> and the <ins>local self-energies</ins> <math> \sigma_a(z)</math>: these are functions of a complex variable belonging to the upper half of the complex plane, and are defined by [NOTA SU STILTJIES] | ||
<center> <math> | <center> | ||
<math> | |||
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)}. | 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)}. | ||
</math> | </math> | ||
Revision as of 17:34, 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
where the local fields are random variables. We introduce the Green functions and the local self-energies : these are functions of a complex variable belonging to the upper half of the complex plane, and are defined by [NOTA SU STILTJIES]
It is clear that when the kinetic term in the Hamiltonian vanishes, the local self-energies vanish; these quantities encode how much the energy levels are shifted by the presence of the kinetic term . They are random functions, because the Hamiltonian contains randomness. They encode properties on the spectrum of the Hamiltonian; the density of eigenvalues is in fact given by
- A criterion for localization.
- Bouchaud. Weak ergodicity breaking and aging in disordered systems [1]
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