L-8

Goal: we will introduce the Anderson model, discuss the behaviour as a function of the dimension. In 1d localization can be connected to the product of random matrices.

Anderson model (tight binding model)

We consider disordered non-interacting particles hopping between nearest neighbors sites on a lattice. The hamiltonian reads:

$H=-t\sum _{<i,j>}(c_{i}^{\dagger }c_{j}+c_{j}^{\dagger }c_{i})\sum _{i}\epsilon _{i}c_{i}^{\dagger }c_{i}$

The single particle hamiltonian in 1d reads

$H={\begin{bmatrix}V_{1}&-t&0&0&0&0\\-t&V_{2}&-t&0&0&0\\0&-t&V_{3}&-t&0&0\\0&0&-t&\ldots &-t&0\\0&0&0&-t&\ldots &-t\\0&0&0&0&-t&V_{L}\end{bmatrix}}$

For simplicity we set the hopping $t=1$. The disorder are iid random variables drawn, uniformly from the box $(-{\frac {W}{2}},{\frac {W}{2}})$.

The final goal is to study the statistical properties of eigensystem

$H\psi =\epsilon \psi ,\quad {\text{with}}\sum _{n}|\psi _{n}|^{2}=1$

Density of states (DOS)

In 1d and in absence of disorder, the dispersion relation 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 \epsilon(k) = -2 \cos k, \quad k \in (-\pi, \pi), -2< \epsilon(k)< 2 } . From the dispersion relation, we compute the density of states (DOS) :

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(\epsilon) =\int_{-\pi}^\pi \frac{d k}{2 \pi} \delta(\epsilon-\epsilon(k))=\frac{1}{\pi } \frac{1}{\sqrt{4-\epsilon^2}} \quad \text{for } \epsilon \in (-2,2)}

In presence of disorder the DOS becomes larger, and display sample to sample fluctuations. One can consider its mean value, avergaed over disorder realization.

Eigenstates

In absence of disorder the eigenstates are plane waves delocalized along all the system. In presence of disorder, three situations can occur and to distinguish them it is useful to introduce the inverse participation ratio, IPR

$IPR(q)=\sum _{n}|\psi _{n}|^{2q}\sim L^{-\tau _{q}}$

The normalization imposes 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 \tau_1 =0 } . 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 q=0} , $|\psi _{n}|^{2}=1$, hence, 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 \tau_0 =-d } .

• Delocalized eigenstates In this 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 |\psi_n|^{2} \approx L^{-d} } . Hence, we expect

$IPR(q)=L^{d(1-q)}\quad \tau _{q}=d(1-q)$

• Localized eigenstates In this 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 |\psi_n|^{2} \approx 1/\xi_{\text{loc}}^{1/d} } 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 \xi_{\text{loc}}^{d}} sites and almost zero elsewhere. Hence, we expect

$IPR(q)={\text{const}},\quad \tau _{q}=0$

• Multifractal eigenstates. At the transition( the mobility edge) an anomalous scaling is observed:

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 IPR(q)=L^{D_q(1-q)} \quad \tau_q=D_q(1-q) }

Here 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 D_q} is q-dependent multifractal dimension, smaller than $d$ and larger than zero.

Transfer matrices and Lyapunov exponents

Product of random variables and Central limit theorem

Consider a set of positive iid random variables 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 x_1,x_2,\ldots x_N} with finite mean and variance and compute their product

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 \Pi_N= \prod_{n=1}^N x_i, \quad \text{or} \; \ln \Pi_N= \sum_{n=1}^N \ln x_i }

For large N, the Central Limit Theorem predicts:

$\log \Pi _{N}=\gamma N+\gamma _{2}{\sqrt {N}}\chi$

• 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 \chi} is a Gaussian number of zero mean and unit variance
• 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, \gamma_2} are N indepent and can be written as

$\gamma ={\overline {\ln x}}<\ln {\overline {x}},\quad \gamma _{2}={\sqrt {{\overline {(\ln x)^{2}}}-({\overline {\ln x}})^{2}}}$

Log-normal distribution

The distribution 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 \Pi_N} is log-normal

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(\Pi_N) d \Pi_N = \frac{1}{ \sqrt{2 \pi \gamma_2^2 N}} \exp\left[-\frac{(\ln(\Pi_N)-\gamma N)^2}{2 \gamma_2^2 N}\right] \frac{d\Pi_N}{\Pi_N} }

Quenched and Annealed averages

To compute the moments of the log-normal distribution, it is convenient to introduce the variable

$X\equiv \ln(\Pi _{N})$

which is Gaussian distributed:

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(X) = \frac{1}{ \sqrt{2 \pi \simga^2}} \exp\left[-\frac{(X-\mu)^2}{2 sigma^2}\right] }

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 \mu =\gamma N} and $\sigma ^{2}=\gamma _{2}^{2}N$

The moments 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 \Pi_N} can be easily computed: 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 \overline{\Pi_N^n} = \int dX \, e^{nX} p(X) = \exp\left[\mu n +(\sigma^2 n^2)/2 \right] }

For the log-normal distribution the mean ${\overline {\Pi _{N}}}=\exp[(\gamma +\gamma _{2}^{2}/2)N]$ is larger than the median value 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 \Pi_N^{\text{median}} = \exp(\gamma N)} (which is larger than the mode 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 \overline{\Pi_N^{mode}} = \exp[(\gamma-\gamma_2^2) N]} ). Hence, $\Pi _{N}$ is not self averaging, while 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 \ln \Pi_N } is self averaging. This is the reason why in the following we will take quenched averages.

Product of random matrices

Let's consider again the Anderson Model in 1d. The eigensystem is well defined in a box of size L with Dirichelet boundary condition on the extremeties of the box.

Here we will solve the second order differential equation imposing instead Cauchy boundaries on one side of the box. Let's rewrite the previous eigensystem in the following form

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 \begin{bmatrix} \psi_{n+1} \\ \psi_{n} \end{bmatrix} = \begin{bmatrix} V_n -\epsilon & -1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} \psi_{n} \\ \psi_{n-1} \end{bmatrix} }

We can continue the recursion

${\begin{bmatrix}\psi _{n+1}\\\psi _{n}\end{bmatrix}}={\begin{bmatrix}V_{n}-\epsilon &-1\\1&0\end{bmatrix}}{\begin{bmatrix}V_{n-1}-\epsilon &-1\\1&0\end{bmatrix}}{\begin{bmatrix}\psi _{n-1}\\\psi _{n-2}\end{bmatrix}}$

It is useful to introduce the transfer matrix and their product

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_n =\begin{bmatrix} V_n -\epsilon & -1 \\ 1 & 0 \end{bmatrix}, \quad \Pi_n= T_n \cdot T_{n-1} \cdot\ldots T_1 }

The Schrodinger equation can be written 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 \begin{bmatrix} \psi_{n+1} \\ \psi_{n} \end{bmatrix} = \Pi_n \begin{bmatrix} \psi_{1} \\ \psi_{0} \end{bmatrix} = \begin{bmatrix} \pi_{11} & \pi_{12} \\ \pi_{21} & \pi_{22} \end{bmatrix} \begin{bmatrix} \psi_{1} \\ \psi_{0} \end{bmatrix} }

Fustenberg Theorem

We define the norm of a 2x2 matrix:

$\|\Pi _{n}\|^{2}={\frac {\pi _{11}^{2}+\pi _{21}^{2}+\pi _{12}^{2}+\pi _{22}^{2}}{2}}$

For large N, the Fustenberg theorem ensures the existence of the non-negative Lyapunov exponent, namely

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_{n\to \infty} \frac{\ln \|\Pi_n\|}{n} = \gamma \ge 0 }

In absence of disorder 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 =0 } for $\epsilon \in (-2,2)$. Generically the Lyapunov is positive, 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 >0 } , and depends on 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 } and on the distribution of $V_{i}$.

Consequences

Localization length

Together with the norm, also 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 |\psi_n|^2} grows exponentially with n. We can write

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 \ln |\psi_n| \sim \gamma n + \gamma_2 \chi \sqrt{n}}

which means 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 \ln |\psi_n| } is performing a random walk with a drift.

However, our initial goal is a properly normalized eigenstate at energy 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 } . We need to switch from Cauchy, where you set the initial condition, to Dirichelet or vonNeuman, where you set the behaviour at the two boundaries. The true eigenstate is obtained by matching two "Cauchy" solutions on the half box and imposing the normalization. Hence, we obtain a localized eigenstate and we can identify

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 \xi_{\text{loc}}(\epsilon) \equiv \gamma^{-1}(\epsilon) }

Fluctuations

We expect strong fluctuations on quantites like 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 |\psi_n|, \|\Pi_n\|, G, \ldots } , while their logarithm is self averaging.