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, Laplace transform, 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, independent and distributed according to . 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 identities. Consider an Hamiltonian split into two parts, . Show that the following general relation holds (Hint: perturbation theory!)
- Cavity equations. We now apply this to a specific example: we consider a Bethe lattice, and choose one site 0 as the root. We then choose to be the kinetic terms connecting the root to its neighbours ,
For all the with we introduce the notation
where is the self energy associated to . Show that, due to the geometry of the lattice, with this choice of the Hamiltonian 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
Iterating this argument, show that if denotes the collection of “descendants" of , i.e. sites that are nearest neighbours of except the root, then
- Equations for the distribution. Justify why the cavity functions appearing in the denominators in the last equations above are independent and identically distributed random variables, and therefore the cavity equations can be interpreted as self-consistent equations for the distribution of the cavity functions.
- The “localized" solution. We set and . Show that the cavity equation for the self-energies is equivalent to the following pair of coupled equations:
Justify why the solution corresponding to localization, , is always a solution when ; moreover, in the localized phase when is finite but small one has . How can one argue that this solution has to be discarded, i.e. that delocalisation occurs?
- Imaginary approximation and distributional equation. We consider the equations for and neglect the terms in the denominators, which couple the equations to those for the real parts of the self energies (“imaginary” approximation). Moreover, we assume to be in the localized phase, where . Finally, we set and for simplicity. Show that under these assumptions the probability density for the imaginary part, , satisfies
Show that the Laplace transform of this distribution, 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 \Phi(s)=\int_0^\infty d\Gamma e^{-s \Gamma} P_\Gamma(\Gamma) } , satisfies
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 \Phi(s)= \left[ \int d\epsilon\, p(\epsilon) e^{-\frac{s t^2 \eta}{\epsilon^2}} \Phi \left(\frac{s t^2 }{\epsilon^2} \right) \right]^k }
- The stability analysis. We now wish to check the stability of our assumption to be in the localized phase, 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 \Gamma_a \sim \eta \ll 1 }
, which led to the identity above for 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 \Phi(s) }
. Our assumption is that the typical value 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 \Gamma_a }
is small, except for cases in which one of the descendants 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 b }
is such 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 \epsilon_b }
is very small, in which case 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 \Gamma_a \sim 1/ \epsilon_b^2 }
.
- Show that if 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 \Gamma \sim 1/ \epsilon^2 } and is not gapped around zero, 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 P_\Gamma(\Gamma) \sim \Gamma^{-3/2}} , i.e. the distribution has tails contributed by these events in which the local fields happen to be very small.
- Assume more generally 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 P_\Gamma(\Gamma) \sim \Gamma^{-\alpha}} for large 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 \Gamma } and 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 \alpha \in [1, 3/2]} . Show that this corresponds to 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 \Phi(s) \sim 1- A |s|^\beta } for 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 s } small, with 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 \beta= \alpha-1 \in [0, 1/2] } .
- Show that the equation for 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 \Phi(s) }
gives for 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 s }
small 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 1- A s^\beta =1- A k \int d\epsilon \, p(\epsilon) \frac{s^\beta t^{2 \beta}}{\epsilon^{2 \beta}}+ o(s^\beta) }
, and therefore this is consistent provided that there exists a 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 \beta \in [0, 1/2] }
solving
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 1=k \int d\epsilon \, p(\epsilon) \left(\frac{t}{\epsilon}\right)^{2 \beta} \equiv k I(\beta). }
- The critical disorder. Consider now local fields 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 \epsilon }
taken from a uniform distribution in 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 [-W/2, W/2] }
. Compute 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 I(\beta) }
and show that it is non monotonic, with a local minimum 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 \beta^* }
in the interval 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 [0, 1/2]}
. 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 I(\beta^*) }
increases as the disorder is made weaker and weaker, and thus the transition to delocalisation occurs at the critical value of disorder when 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 I(\beta^*)=k^{-1} }
. Show that this gives
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 W_c = t \, 2 k e \log \left( \frac{W_c}{2 t}\right) \sim t \, 2 k e \log \left(k\right) }
Why the critical disorder increases with 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 } ?
- Anderson. [1]
- The model is solved in Abou-Chacra, Thouless, Anderson. A selfconsistent theory of localization. Journal of Physics C: Solid State Physics 6.10 (1973)
Problem 7.1: the Bethe lattice, recursion relations and cavity
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.
Problem 7.2: localization-delocalization transition on the Bethe lattice
We now focus on the self energies, since the criterion for localization is given in terms of these quantities. In this Problem we will determine for which values of parameters localization is stable, estimating the critical value of disorder where the transition to a delocalised phase occurs.