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 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 \rangle } the wave function completely localised in site 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 } . The Anderson model has Hamiltonian

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

  where the 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_a } are random variables. We introduce the Green functions 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_{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]

  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_{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 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 } 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 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 } . They are random functions, because the Hamiltonian contains randomness. They encode properties on the spectrum of the Hamiltonian; the local density of eigenvalues 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 \rho_{a, N}(E)} for an Hamiltonian of size $N$ is in fact given by

  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 \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 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 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 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 \eta \to 0} . Given the randomness, this criterion should however be formulated probabilistically. One has:

  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 \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:

The Bethe lattice is a lattice with a regular tree structure: each node has a fixed number of 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 k+1} , 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 k } is the branching number, and there are no loops (see sketch). In these problems we consider the Anderson model on such lattice.




1. Green functions equations. Consider an Hamiltonian split into two parts, $H=H_{0}+V$. Show that the following general relation holds

   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=G^0+ G^0 V G, \quad \quad G^0 =\frac{1}{z-H_0}, \quad \quad G =\frac{1}{z-H}. }



2. 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 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 $k+1$ 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 a_i } ,

   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 $a_{i}$ 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 i=1, \cdots, k+1 } 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 $\sigma ^{0}$ is the self energy associated 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 G^0 } . 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 $G_{a_{i}}^{\text{cav}}$ 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}} }





- 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= }

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]