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 $t=1$.

We assume that the disorder variables are independent and identically distributed, uniformly in the interval

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

In one dimension, the single-particle Hamiltonian is represented by the 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

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

In the one-particle sector, this becomes a discrete Schrödinger equation.

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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.

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.

Remark

The transfer matrix formulation rewrites the problem as a recursion, starting from initial data and propagating the solution along the chain.

This corresponds to a Cauchy problem rather than a boundary value problem. Nevertheless, the growth properties of the solutions contain direct information about the nature of the eigenstates: exponential growth is associated with localization, while bounded solutions correspond to extended states.

Furstenberg theorem (physical formulation)

The exponential growth of the product of random matrices $${\mathrm{\Pi }}_n=T_n{T}_{n-1}\cdots T_1$$ can be understood as a matrix analogue of the law of large numbers. A sufficient condition for this behavior is provided by a theorem of Harry Furstenberg.

To quantify this growth, we introduce the norm of a matrix $A$ as $$\Vert A{\Vert }^2=\frac{a_{11}^2+a_{12}^2+a_{21}^2+a_{22}^2}{2}.$$

In physical terms, the theorem relies on two key ingredients:

(i) Stretching

Each matrix must be able to expand at least one direction. This means that for each $n$ there exists a vector $v$ such that $$\Vert T_{n}v\Vert ={\sigma }_{\max }(T_n)\Vert v\Vert ,{\sigma }_{\max }(T_n)>1.$$

Equivalently, ${\sigma }_{\max }^2(T_n)$ is the largest eigenvalue of the symmetric matrix $$T_n^T{T}_n.$$

(ii) Mixing of directions

At each step, the transfer matrix changes the direction of the vector $({\psi }_n,{\psi }_{n-1})$.

In the clean case, the same matrix is applied at every step, so the angular dynamics is deterministic and can be written as $${\theta }_{n+1}=F({\theta }_n).$$ Once the initial direction is fixed, the whole sequence is fixed. The direction may change, but it keeps a perfect memory of its initial value.

In the disordered case, the transfer matrix varies from step to step, and the angular dynamics becomes $${\theta }_{n+1}=F({\theta }_n,T_n).$$ The direction is then continuously reshuffled and progressively loses memory of its initial orientation.

More precisely, there is no finite set of directions that is invariant under all transfer matrices. The dynamics therefore cannot be reduced to a finite set of preferred directions.

This is what is meant by mixing of directions, and it is the key condition required by Furstenberg's theorem.

Consequence

Under these conditions, the norm of the product grows exponentially with probability one: $$\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}\log \Vert {\mathrm{\Pi }}_n\Vert =\gamma >0,$$ where $\gamma$ is the Lyapunov exponent.

This result is the matrix analogue of the fact that the logarithm of a product of independent random variables becomes self-averaging (see Exercise 15).

Verification of Furstenberg's hypotheses for the Anderson model

We now check explicitly that the transfer matrices of the one-dimensional Anderson model satisfy the two conditions stated above.

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

(i) Stretching

We first verify that each matrix expands at least one direction.

For a generic $2\times 2$ matrix, the maximal stretching factor is controlled by the largest eigenvalue of $T_n^T{T}_n$. Here $$T_n^T{T}_n=\left(\begin{array}{cc}(V_n-ϵ{)}^2+1& -(V_n-ϵ)\\ -(V_n-ϵ)& 1\end{array}\right).$$

The trace is $$\mathrm{T}\mathrm{r}(T_n^T{T}_n)=(V_n-ϵ{)}^2+2,$$ and the determinant is $$\det (T_n^T{T}_n)=\det (T_n{)}^2=1.$$

Therefore the eigenvalues satisfy $${\lambda }_{+}{\lambda }_{-}=1,$$ and the largest one obeys $${\lambda }_{+}\ge 1.$$

More precisely, as soon as $V_n-ϵ\ne 0$, one has $${\lambda }_{+}>1.$$

Thus each matrix expands at least one direction, except for the fine-tuned case $V_n=ϵ$, which has zero probability for a continuous disorder distribution. The stretching condition is therefore satisfied almost surely.

(ii) Mixing of directions

We now verify that the angular dynamics is not confined to a finite set of directions.

Define the angle $${\theta }_n=\arctan \left(\frac{{\psi }_{n-1}}{{\psi }_n}\right),$$ which represents the direction of the vector $({\psi }_n,{\psi }_{n-1})$.

The transfer matrix induces the map $${\theta }_{n+1}=\arctan \left(\frac{1}{V_n-ϵ-\tan {\theta }_n}\right).$$

For fixed ${\theta }_n$, this expression depends continuously on the random variable $V_n$. If the disorder has a continuous distribution (for instance Gaussian), the image of a given angle is not restricted to a finite set.

As a consequence, the sequence ${\theta }_n$ is not confined to a finite set of directions: under iteration of the product $$T_n{T}_{n-1}\cdots T_1,$$ the angular dynamics can access a continuum of directions.

This exploration is not required to be uniform; it is sufficient that all directions are accessible. This is precisely the mixing condition required by Furstenberg's theorem.

Conclusion

The transfer matrices of the one-dimensional Anderson model satisfy both conditions:

1. they expand at least one direction;
2. they do not confine the angular dynamics to a finite set.

As a consequence, $$\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}\log \Vert {\mathrm{\Pi }}_n\Vert =\gamma >0,$$ so the Lyapunov exponent is positive for generic disorder.

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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.