T-8

Goal: the goal of this set of problems is to derive a criterion for localization on a peculiar lattice, the Bethe lattice.
Techniques: green functions, recursion relations, cavity method.

A criterion for localization: vanishing decay rate

• Anderson model and return probability. Consider a lattice with $N$ sites $x$. We call $|x\rangle$ the wave function completely localised in site $x$. The Anderson model has Hamiltonian:

  $H=W\sum _{x}V_{x}|x\rangle \langle x|-\sum _{<x,y>}t_{xy}\left(|x\rangle \langle y|+|y\rangle \langle x|\right)\equiv H_{0}+T$

  where the local fields $V_{x}$ are random variables, independent and distributed according to some distribution $p(V)$. Usually, the hopping amplitude $t_{xy}=t$ is chosen to be uniform.

  To see whether localization occurs, consider a quantum particle initialised on the site $x_{0}$ at $t=0$. The return probability amplitude , i.e. the probability amplitude to find the particle on the same site at later time, is

  ${\mathcal {A}}_{x_{0}}(t)=\theta (t)\langle x_{0}|e^{-itH}|x_{0}\rangle .$

  When the system is not localized, the wave function of the particle spreads in time, and the return probability decays.



• Green functions and self-energies. We introduce two functions on the complex plane: the Green functions $G_{xy}(z)$ and the local self-energies $\sigma _{x}(z)$, where $z=E+i\eta ,\quad \quad \eta >0$ belongs to the upper half of the complex plane. The functions are defined by

  $G_{xy}(z)=\langle x|{\frac {1}{z-H}}|y\rangle ,\quad \quad G_{xx}(z)=\langle x|{\frac {1}{z-H}}|x\rangle ={\frac {1}{z-V_{x}-\sigma _{x}(z)}}.$

  They are random functions, because the Hamiltonian contains randomness. The Green functions and the self-energies encode properties on the spectrum of the Hamiltonian; for example, the local density of eigenvalues $\rho _{x,N}(E)$ for an Hamiltonian on a lattice of size $N$ is indeed 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_{x,N}(E)=-\frac{1}{\pi}\lim_{\eta \to 0} \Im G_{xx}(E+ i \eta) = \sum_{\alpha=1}^N |\langle E_\alpha| x\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 full Hamiltonian $H$ 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 |E_\alpha \rangle } the corresponding eigenstates. 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 T } in the Hamiltonian vanishes, the local self-energies vanish. they encode how much the energy levels $V_{x}$ (that are the eigenvalues 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 V=0} ) 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 T } .



• Green functions and self-energies. Given a lattice with $N$ sites 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 } , 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 $a$. The Anderson model has 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= W \sum_{a} \epsilon_a |a \rangle \langle a| - \sum_{<a, b>} V_{ab} \left(|a \rangle \langle b|+ |b \rangle \langle a| \right)\equiv H_0 + V }

  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, independent and distributed according to some distribution $p(\epsilon )$. 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 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_a(z):} these are functions of a complex variable belonging to the upper half of the complex plane,

  $z=E+i\eta ,\quad \quad \eta >0$

  and are defined 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 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)}. }

  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. They are random functions, because the Hamiltonian contains randomness.

  
  



• Self-energies and return probabilities. The local self-energies encode some information on the system’s dynamics, and thus on whether localization occurs. Consider a quantum particle initialised on the site $a$ at 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 t=0 } . The return probability amplitude , i.e. the probability amplitude to find the particle on the same site at later time, is

  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 \mathcal{A}_a(t)= \theta(t) \langle a| e^{-i t H} | a \rangle =\lim_{\eta \to 0} \int \frac{ dz}{2 \pi i}e^{-i t z} G_{aa}(z)=\lim_{\eta \to 0} \int \frac{ dz}{2 \pi i}e^{-i t z} G_{aa}(z)=\lim_{\eta \to 0} \int \frac{ dz}{2 \pi i}\frac{e^{-i t z}}{z- \epsilon_a-\sigma_a(z)}. }

  If the self-energy has a non-zero imaginary part (when $N\to \infty$):

  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_a(z)= R_a(z)-i \Gamma_a(z), }

  then one can easily show (Residue theorem) that the return probability amplitude 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 \mathcal{A}_a(t)} decays exponentially

  ${\mathcal {A}}_{a}(t)\sim A(t)e^{-\gamma t}+B(t),\quad \quad \quad \gamma =\Gamma _{a}(\epsilon _{a})+O(V^{4}).$

  Therefore, the system is not localized, since the probability to find it, at 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 t \gg 1} , in the same configuration where it was at 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 t=0 } decays very fast.

  
  

• A criterion for localization. Motivated by the reasoning above, one can claim 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, as

  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} }

Showing that at strong enough disorder this condition is satisfied is the core of Anderson’s 1958 work. 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 (like depinning!). 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).


Problems

Problem 8: the Bethe lattice, recursion relations and cavity


A Bethe lattice 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=2 } .

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 identities. Consider an Hamiltonian split into two parts, 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= H_0 + V } . Show that the following general relation for the Green functions 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 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 $a_{i}$,

   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} V_{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 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}, }

   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 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 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} } 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} V^2_{0 a_i}G^{\text{cav}}_{a_i}(z)} }

   Iterating this argument, 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 \partial a_i } 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}V^2_{a_i b}G^{\text{cav}}_{b}(z)}, \quad \quad \sigma^{\text{cav}}_{a_i}(z)=\sum_{b \in \partial a_i}V^2_{a_i b}G^{\text{cav}}_{b}(z)=\sum_{b \in \partial a_i} \frac{V^2_{a_i b}}{z- \epsilon_b - \sigma^{\text{cav}}_{b}(z)} }



3. 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.


4. The “localized" solution. We set 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 z=E+ i \eta } 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 \sigma^{\text{cav}}_{a}(z)= R_a(z) -i \Gamma_a(z)} . Show that the cavity equation for the self-energies is equivalent to the following pair of coupled equations:

   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= \sum_{b \in \partial a} V_{ab}^2 \frac{\Gamma_b + \eta}{(E- \epsilon_b - R_b)^2+ (\Gamma_b +\eta)^2}, \quad \quad R_a = \sum_{b \in \partial a} V_{ab}^2 \frac{E- \epsilon_b - R_b}{(E- \epsilon_b - R_b)^2+ (\Gamma_b +\eta)^2} }

   Justify why the solution corresponding to localization, 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=0 } , is always a solution 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 } ; moreover, in the localized phase 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 } is finite but small 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 \Gamma_a \sim O(\eta) } . How can one argue that this solution has to be discarded, i.e. that delocalisation occurs?







Check out: key concepts of this TD

Green functions, self-energies, return probability amplitude, decay rates, trees and cavity method, the criterion for localization.

References

• The Anderson model was formulated by P. W. Anderson in 1958, in the paper Absence of diffusion in certain random lattices , Phys. Rev. 109, 1492.