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 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 local density of eigenvalues for an Hamiltonian of size is in fact given by
where 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 goes to zero when . Given the randomness, this criterion should however be formulated probabilistically. One has:
- Green functions equations. Consider an Hamiltonian split into two parts, . Show that the following general relation holds
- Cavity equations for the Green functions. We now apply this to a specific example: we consider a Bethe lattice, and choose one site 0 as the root. We then choose Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V }
to be the kinetic terms connecting the root to its Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k+1 }
neighbours ,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V= -\sum_{i=1}^{k+1} t_{0 a_i} \left( |a_i \rangle \langle 0|+ |0 \rangle \langle a_i|\right) }
For all the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_i } with we introduce the notation
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G^{\text{cav}}_{a_i} \equiv G^0_{a_i a_i}, \quad \quad \sigma^{\text{cav}}_{a_i} \equiv \sigma^0_{a_i a_i}, }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma^0 } is the self energy associated to . Show that, due to the geometry of the lattice, with this choice of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V } the Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_0 } is decoupled and is the local Green function that one would have obtained removing the root 0 from the lattice, i.e., creating a “cavity” (hence the suffix). Moreover, using the relation above show that
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G_{00}(z)= \frac{1}{z-\epsilon_0 - \sum_{i=1}^{k+1} t^2_{0 a_i}G^{\text{cav}}_{a_i}(z)} }
Iterating this argument, show that if denotes the collection of descendants of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_i} , i.e. sites that are nearest neighbours of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_i } except the root, then
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G^{\text{cav}}_{a_i}(z)= \frac{1}{z-\epsilon_{a_i} - \sum_{b \in \partial a_i}t^2_{a_i b}G^{\text{cav}}_{b}(z) } }
Problem 7.1:
The Bethe lattice is a lattice with a regular tree structure: each node has a fixed number of neighbours , where is the branching number, and there are no loops (see sketch). In these problems we consider the Anderson model on such lattice.
- Show that
Moreover, all these green functions are independent from each others.
- Finally, show that the relation above becomes the relation:; with a similar argument, show that the following recursive equation holds for the green functions. - show equation for real and imaginary part of cavity self energy - the distribution equations
Problem 7.2:
- Laplace transform - the tails - the solution for beta - the estimate for the transition
Next TD: the directed polymer treatment: KPP (es 1)
es 2: The connection to directed polymer: linearisation and stability. Glassiness vs localization
Check out: key concepts of this TD
References
- Anderson. Weak ergodicity breaking and aging in disordered systems [1]