L-8

Goal. We introduce the Anderson model and study the statistical properties of its eigenstates. In one dimension disorder leads to localization of all eigenstates, which can be understood using products of random matrices.

Anderson model (tight-binding model)

We consider non-interacting particles hopping between nearest-neighbour sites of a lattice in the presence of disorder.

The Hamiltonian reads

$$H=-t\sum _{⟨i,j⟩}(c_i^{†}{c}_j+c_j^{†}{c}_i)+\sum _i{V}_i{c}_i^{†}{c}_i.$$

The random variables $V_i$ represent on-site disorder.

For simplicity we set

$t=1$.

The disorder variables are independent random variables drawn from the box distribution

$$V_i\in (-\frac{W}{2},\frac{W}{2}).$$

In one dimension the single-particle Hamiltonian corresponds to a tridiagonal matrix

$$H=\left(\begin{array}{ccccc}{V}_1& -1& 0& 0& \dots \\ -1& V_2& -1& 0& \dots \\ 0& -1& V_3& -1& \dots \\ 0& 0& -1& \ddots & -1\\ \dots & \dots & \dots & -1& V_L\end{array}\right).$$

We study the statistical properties of the eigenvalue problem

$$H\psi =ϵ\psi ,\sum _{n=1}^L|{\psi }_n{|}^2=1.$$

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Density of states

Without disorder the dispersion relation is

$$ϵ(k)=-2\cos k,k\in (-\pi ,\pi ).$$

The energy band is therefore

$$-2<ϵ<2.$$

The density of states is

$$\rho (ϵ)=\underset{-\pi }{\overset{\pi }{\int }}\frac{dk}{2\pi }\delta (ϵ-ϵ(k))=\frac{1}{\pi \sqrt{4-{ϵ}^2}}(ϵ\in (-2,2)).$$

In the presence of disorder the density of states broadens and becomes sample dependent.

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Transfer matrices

The discrete Schrödinger equation reads

$${\psi }_{n+1}+{\psi }_{n-1}+V_n{\psi }_n=ϵ{\psi }_n.$$

It can be rewritten as

$$\left(\begin{array}{c}{\psi }_{n+1}\\ {\psi }_n\end{array}\right)=T_n\left(\begin{array}{c}{\psi }_n\\ {\psi }_{n-1}\end{array}\right)$$

with

$$T_n=\left(\begin{array}{cc}{V}_n-ϵ& -1\\ 1& 0\end{array}\right).$$

Iterating gives

$$\left(\begin{array}{c}{\psi }_{n+1}\\ {\psi }_n\end{array}\right)={\mathrm{\Pi }}_n\left(\begin{array}{c}{\psi }_1\\ {\psi }_0\end{array}\right),{\mathrm{\Pi }}_n=T_n{T}_{n-1}\cdots T_1.$$

Thus the wavefunction is controlled by a product of random matrices.

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Lyapunov exponent

Define

$$\Vert {\mathrm{\Pi }}_n{\Vert }^2=\frac{{\pi }_{11}^2+{\pi }_{12}^2+{\pi }_{21}^2+{\pi }_{22}^2}{2}.$$

Furstenberg's theorem ensures

$$\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}\ln \Vert {\mathrm{\Pi }}_n\Vert =\gamma .$$

The quantity $\gamma$ is the Lyapunov exponent.

Without disorder

$$\gamma =0(ϵ\in (-2,2)).$$

For generic disorder

$$\gamma >0.$$

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Localization length

The transfer-matrix recursion corresponds to fixing the wavefunction at one boundary and propagating it through the system.

Typical solutions grow exponentially

$$|{\psi }_n|\sim e^{\gamma n}.$$

However a physical eigenstate must satisfy boundary conditions at both ends of the system. Matching two such solutions leads to exponentially localized eigenstates

$$|{\psi }_n|\sim e^{-|n-n_0|/{\xi }_{\text{loc}}}.$$

The localization length is

$${\xi }_{\text{loc}}(ϵ)=\frac{1}{\gamma (ϵ)}.$$

Thus in one dimension arbitrarily weak disorder localizes all eigenstates. This result is consistent with the scaling theory of localization discussed earlier, which predicts that for $d\le 2$ disorder inevitably drives the system toward the insulating regime.

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Fluctuations

Quantities such as

$|{\psi }_n|,\Vert {\mathrm{\Pi }}_n\Vert ,G$

show strong sample-to-sample fluctuations, while their logarithm is self-averaging.

For instance

$$\ln |{\psi }_n|\sim \gamma n+O(\sqrt{n})$$

so that the logarithm of the wavefunction performs a random walk with a positive drift.