Evading Anderson localization in a one-dimensional conductor with correlated disorder

To be concrete, we consider a one-dimensional lattice with non-interacting electrons and nearest neighbor hopping. The potential at each site is either zero or is a random number between $-W$ and $W.$ The corresponding Schrödinger equation is the difference equation

$$\psi_{n-1}+V_{n}\psi_{n}+\psi_{n+1}=E\psi_{n}.$$

(1)

Here $\psi_{n}$ is the wavefunction at site $n$ and $V_{n}$ is the site potential. Note that in the absence of disorder ($V_{n}=0$ for all $n$) the solutions are plane waves $\psi_{n}=\exp(ikn)$ with energy $E=2\cos k$. Sites where $V_{n}\neq 0$ are called scatterers, and we constrain the system so that two scatterers cannot be adjacent to each other. Thus the system can be understood as a sequence of scatterers, with each successive pair of scatterers separated by a lattice segment where $V=0,$ i.e. a clean segment. Our strategy is to find the $S$-matrix for each scatterer, and combine them appropriately.

Figure 1 shows a single scatterer. We make the ansatz

	$\displaystyle\psi_{n}$	$\displaystyle=$	$\displaystyle a_{L}\exp(ikn)+a^{\prime}_{L}\exp(-ikn)\hskip 5.69054pt{\rm for}\hskip 5.69054ptn\leq 0$		(2)
		$\displaystyle=$	$\displaystyle a_{R}\exp(ikn)+a^{\prime}_{R}\exp(-ikn)\hskip 5.69054pt{\rm for}\hskip 5.69054ptn\geq 0$

Making use of the Schrödinger Eq. (1) for $n=0$ we obtain

	$\displaystyle a_{L}+a^{\prime}_{L}$	$\displaystyle=$	$\displaystyle\psi$
	$\displaystyle a_{R}+a^{\prime}_{R}$	$\displaystyle=$	$\displaystyle\psi$
	$\displaystyle(2\cos k-V)\psi$	$\displaystyle=$	$\displaystyle a_{L}e^{-ik}+a^{\prime}_{L}e^{ik}+a_{R}e^{ik}+a^{\prime}_{R}e^{-ik}.$		(3)

Solving eq (3) yields

$$\begin{pmatrix}a^{\prime}_{L}\\ a_{R}\end{pmatrix}=S\begin{pmatrix}a_{L}\\ a^{\prime}_{R}\end{pmatrix}$$

(4)

Here the $2\times 2$ $S$-matrix connects the outgoing amplitudes to the incoming amplitudes. Explicitly, we find

$$S(V,k)=-\frac{1}{2i\sin k+V}\begin{pmatrix}V&-2i\sin k\\ -2i\sin k&V\end{pmatrix}.$$

(5)

The form of the $S$-matrix for a single scatterer is powerfully constrained by general principles. Probability conservation imposes unitarity, $S^{\dagger}S=1$. Parity imposes the additional requirements that $S_{11}=S_{22}$ and $S_{12}=S_{21}$. The most general $2\times 2$ matrix consistent with these requirements may be parametrized as

$$S=e^{i\gamma}\begin{pmatrix}\cos\theta&i\sin\theta\\ i\sin\theta&\cos\theta.\end{pmatrix}$$

(6)

where $0\leq\theta\leq\pi/2$ and $0\leq\gamma<2\pi$. It follows from Eqs. (4) and (6) that the transmission coefficient is $\sin^{2}\theta$ and the reflection coefficient is $\cos^{2}\theta$. Hence we refer to the parameter $\theta$ as the opacity of the $S$-matrix.

Comparison of Eqs. (5) and (6) shows that for the tight binding model analyzed above the opacity $\theta$ and the overall phase $\gamma$ for a single scatterer are given by

	$\displaystyle\exp(i\theta)$	$\displaystyle=$	$\displaystyle\pm\frac{V-2i\sin k}{\sqrt{V^{2}+4\sin^{2}k}}$
	$\displaystyle\exp(i\gamma)$	$\displaystyle=$	$\displaystyle\mp\frac{\sqrt{V^{2}+4\sin^{2}k}}{V+2i\sin k}=-\exp(i\theta)$		(7)

where the sign on the right hand side is positive (negative) when $V$ is positive (negative), i.e. $\cos\theta>0.$ The fact that $\theta$ and $\gamma$ are related for this model is not true in general. This coincidence will play no role in our subsequent analysis.

The complete lattice can be treated as a sequence of scatterers, separated by free propagation segments of variable length. Since the lattice is not left-right symmetric, the $S$-matrix of the entire system is not as constrained as Eq.(6). Nevertheless, time reversal invariance requires that $S^{*}=S^{-1}$ which, together with the unitarity of $S,$ implies that $S=S^{T}.$ Therefore we obtain the parameterization

$$S_{N}=e^{i\gamma_{N}}\begin{pmatrix}\cos\theta_{N}e^{i\delta_{N}}&i\sin\theta_{N}\\ i\sin\theta_{N}&\cos\theta_{N}e^{-i\delta_{N}}\end{pmatrix}$$

(8)

where $S_{N}$ is the $S$-matrix of a lattice with $N$ scatterers and the parameters have the domain $0\leq\theta_{N}\leq\pi/2$, $0\leq\delta_{N}<2\pi$ and $0\leq\gamma_{N}<2\pi$.

Now suppose that a lattice with $N$ scatterers has an additional impurity attached to its end with $L$ sites between the $N^{{\rm th}}$ and $(N+1)^{{\rm th}}$ scatterer. The amplitudes of the waves in the different regions are as shown in Fig 2. In the intermediate region the forward and backward waves have amplitudes $a_{M}$ and $a^{\prime}_{M}$ respectively. The phases are chosen so that at the site immediately to the right of the $N^{{\rm th}}$ scatterer the wavefunction is $a_{M}\exp(ik)+a^{\prime}_{N}\exp(ikL)$. Hence at the site immediately to the left of the $(N+1)^{{\rm th}}$ scatterer the wavefunction is $a_{M}\exp(ikL)+a^{\prime}_{M}\exp(ik)$. Defining $\exp(i\beta)=\exp[ik(L+1)]$ we have

$$\begin{pmatrix}a^{\prime}_{M}\\ a_{R}\end{pmatrix}=S\begin{pmatrix}a_{M}e^{i\beta}\\ a^{\prime}_{R}\end{pmatrix}.$$

(9)

where $S$ is the $S$-matrix of the $(N+1)^{{\rm th}}$ impurity. On the other hand, the effect of the $N$ previous scatterers can be represented as

$$\begin{pmatrix}a^{\prime}_{L}\\ a_{M}\end{pmatrix}=S_{N}\begin{pmatrix}a_{L}\\ a^{\prime}_{M}e^{i\beta}\end{pmatrix}.$$

(10)

Our objective is to calculate $S_{N+1}$, the $S$-matrix for the combined system, which is defined by

$$\left(\begin{array}[]{c}a^{\prime}_{L}\\ a_{R}\end{array}\right)=S_{N+1}\left(\begin{array}[]{c}a_{L}\\ a^{\prime}_{R}\end{array}\right).$$

(11)

By eliminating the intermediate amplitudes from Eqs. (9) and (10), after a lengthy but straightforward calculation, we obtain

	$\displaystyle\cos\theta_{N+1}e^{i(\gamma_{N+1}+\delta_{N+1})}$	$\displaystyle=$	$\displaystyle e^{i(\gamma_{N}+\delta_{N})}\frac{\cos\theta_{N}-\cos\theta e^{i\phi}}{1-\cos\theta\cos\theta_{N}e^{i\phi}}$
	$\displaystyle\cos\theta_{N+1}e^{i(\gamma_{N+1}-\delta_{N+1})}$	$\displaystyle=$	$\displaystyle e^{i\gamma}\frac{\cos\theta-\cos\theta_{N}e^{i\phi}}{1-\cos\theta\cos\theta_{N}e^{i\phi}}$
	$\displaystyle i\sin\theta_{N+1}e^{i\gamma_{N+1}}$	$\displaystyle=$	$\displaystyle-e^{i(\gamma+\gamma_{N}+\delta_{N})/2}\frac{\sin\theta\sin\theta_{N}e^{i\phi/2}}{1-\cos\theta\cos\theta_{N}e^{i\phi}}.$

Here we have defined $\phi=2\beta+\gamma+\gamma_{N}-\delta_{N}.$ Note that $\phi$ depends on the phases of the two $S$ matrices being combined and through $\beta$ also on the distance between the new $(N+1)^{{\rm th}}$ scatterer and its predecessor.

Eq (LABEL:monster) is the main result of this section. It relates the parameters of the $N+1$ scatterer $S$-marix, $(\theta_{N+1},\gamma_{N+1},\delta_{N+1})$ to $(\theta_{N},\gamma_{N},\delta_{N})$ and $(\theta,\gamma)$ the parameters of the $S$-matrices for the first $N$ scatterers and for the $(N+1)^{{\rm th}}$ scatterer respectively.

Although we have couched our discussion in terms of a tight binding model it should be obvious that our analysis is much more general. For example it also applies to a continuum model in which the scatterers are rectangular top hat potentials of variable heights and widths separated by variable distances. The only part of the analysis that would change is Eq (7) would be replaced expressions relating the parameters of the $S$ matrix to the barrier height and width.

In our representation of the $S$-matrix, the transmission coefficient of the $N$-scatterer lattice is $\sin^{2}\theta_{N}.$ By Landauer’s formula harold this is the conductance of a system with $N$-scatterers in units of $e^{2}/h$. It follows from the third equality in Eq. (LABEL:monster) that the transmission coefficient evolves according to

$$\sin^{2}\theta_{N+1}=\frac{\sin^{2}\theta\sin^{2}\theta_{N}}{1+\cos^{2}\theta\cos^{2}\theta_{N}-2\cos\theta\cos\theta_{N}\cos\phi}.$$

(13)

Similar relations can be written down that give the phases $\gamma_{N+1}$ and $\delta_{N+1}$ in terms of the parameters of the matrices $S_{N}$ and $S$ but for the sake of brevity they are omitted. We now describe how Eq. (13) conventionally leads to Anderson localization and how suitably correlated disorder may evade it.

If the scatterers are dilute and randomly distributed then $\phi$ can be treated as a uniform random variable. Taking the reciprocal of both sides of eq (13) and averaging over disorder we obtain

$$\langle\csc^{2}\theta_{N+1}\rangle=\langle\csc^{2}\theta\rangle\langle\csc^{2}\theta_{N}\rangle+\langle\cot^{2}\theta\rangle\langle\cot^{2}\theta_{N}\rangle.$$

(14)

Note that the averages factorize because the opacity $\theta$ and the phase $\gamma$ of the $(N+1)^{{\rm th}}$ scatterer are random variables independent of the scatterers that preceded it. In context of the tight-binding model the distribution of $(\theta,\gamma)$ is fixed by Eq. (7) and the specified uniform distribution of $V$ over the interval between $-W$ and $W$; however our conclusions are not limited to this specific choice of distribution. The only assumption we have to make is that the probability of extremely opaque scatterers is small; more precisely that the probability of small opacity $\theta$ goes to zero sufficiently fast that $\langle\cot^{2}\theta\rangle$ is finite which is certainly the case for our tight binding model or any other reasonable model we might consider. Making use of trigonometric identities we may rewrite Eq. (14) as

$$\langle\csc^{2}\theta_{N+1}\rangle=[1+2\langle\cot^{2}\theta\rangle]\langle\csc^{2}\theta_{N}\rangle-\langle\cot^{2}\theta\rangle.$$

(15)

By iterating this relation it is easy to see that $\langle\csc^{2}\theta_{N}\rangle$, which has the interpretation of the mean resistance of the sample, grows exponentially with the system size $N$. This is the essence of Anderson localization. Note that our analysis only shows that the mean resistance grows exponentially which is not the same as proving that the mean Landauer conductance $\langle\sin^{2}\theta_{N}\rangle$ decays exponentially but all of this is well established lore and is not the focus of our paper (for a calculation of the full distribution of the resistance for this model see for example ref pendry ).

Now let us look for a qualitatively different fixed point for the evolution Eq. (13). To this end at first we assume that all the scatterers are identical and have the same opacity $\theta$. We also assume that the phase $\phi$ can be held constant for each successive scatterer that is added to the system. With these assumptions Eq. (13) is a simple deterministic map for $\theta_{N}$ with a fixed point $\theta^{\ast}$ given by

$$1=\frac{\sin^{2}\theta}{1+\cos^{2}\theta\cos^{2}\theta_{*}-2\cos\theta\cos\theta_{*}\cos\phi}$$

(16)

As long as $\cos\phi/\cos\theta>1,$ or equivalently $-\theta<\phi<\theta$, this equation has a solution. The condition for $\theta_{*}$ to be a stable fixed point is $-1<d\sin^{2}\theta_{N+1}/d\sin^{2}\theta_{N}<1$ at $\theta_{N}=\theta_{N+1}=\theta_{*}$ and can be verified to be always satisfied.

Now let us return to the disordered problem. In this case the opacity $\theta$ is drawn from a distribution each time Eq. (13) is evolved. To try to retain the non-trivial solution for the deterministic case found above we constrain $\phi$ to be a random variable drawn from a distribution that satisfies the solvability condition $-\theta<\phi<\theta$ noted above. Note that if we imagine building the system one scatterer at a time what we are effectively saying is that the position of the $(N+1)^{{\rm th}}$ scatterer is constrained to lie within a certain interval that is determined by the $S$-matrix of the preceding $N$ scatterers. However within that range we can place the $(N+1)^{{\rm th}}$ scatterer at random. Hence the system we are considering is random but with correlated disorder. We now show that this random system evades Anderson localization.

To this end we again take the reciprocal of both sides of Eq. (13) and average over disorder to obtain

	$\displaystyle\langle\csc^{2}\theta_{N+1}\rangle$	$\displaystyle=$	$\displaystyle\langle\csc^{2}\theta_{N}\rangle\langle\csc^{2}\theta\rangle+\langle\cot^{2}\theta_{N}\rangle\langle\cot^{2}\theta\rangle$		(17)
			$\displaystyle-2\langle\cot\theta_{N}\csc\theta_{N}\rangle\langle\cot\theta\csc\theta\cos\phi\rangle.$

We have used the fact that $\theta$ is independent of $\theta_{N}$ and $\phi$ is correlated with $\theta$, not with $\theta_{N}$. For simplicity let us assume that $\phi$ is uniformly distributed over the interval $-\theta$ to $\theta$. Performing the average of $\phi$ we then obtain

	$\displaystyle\langle\csc^{2}\theta_{N+1}\rangle$	$\displaystyle=$	$\displaystyle\langle\csc^{2}\theta_{N}\rangle\langle\csc^{2}\theta\rangle+\langle\cot^{2}\theta_{N}\rangle\langle\cot^{2}\theta\rangle$		(18)
			$\displaystyle-2\langle\cot\theta_{N}\csc\theta_{N}\rangle\langle\cot\theta/\theta\rangle.$

With some rearrangement Eq. (18) can be brought to the form

	$\displaystyle\langle\csc^{2}\theta_{N+1}\rangle-\langle\csc^{2}\theta_{N}\rangle$	$\displaystyle=$	$\displaystyle-2B\langle\csc^{2}\theta_{N}\rangle$
		$\displaystyle+$	$\displaystyle\left[\langle\frac{\cot\theta}{\theta}\rangle\langle R(\theta_{N})\rangle-\langle\cot^{2}\theta\rangle\right].$

Here $B$ is given by

$$B=\langle\frac{\cot\theta}{\theta}-\cot^{2}\theta\rangle=\langle\cot^{2}\theta\left(\frac{\tan\theta}{\theta}-1\right)\rangle$$

(20)

and is evidently a finite positive constant since $\tan\theta\geq\theta$ over the interval from zero to $\pi/2$. The specific value of $B$ will depend on the distribution chosen for the opacity $\theta$. $R(\theta_{N})=2(1-\cos\theta_{N})/\sin^{2}\theta_{N}$ is a monotonic decreasing function that goes from 1 to zero as $\theta_{N}$ goes from zero to $\pi/2$. Hence the final term in eq (LABEL:eq:andersontoo) in square brackets is finite and lies in the range between $-\langle\cot^{2}\theta\rangle$ and $B$. These observations and the form of Eq. (LABEL:eq:andersontoo) preclude Anderson localization for this disordered conductor. For a localized conductor the average resistance should grow monotonically and without bound. However if $\langle\csc^{2}\theta_{N}\rangle$ gets sufficiently large then the right hand side of Eq. (LABEL:eq:andersontoo) becomes negative contradicting the assumption that $\langle\csc^{2}\theta_{N}\rangle$ is growing monotonically without bound. Rather if the mean resistance is growing monotonically Eq. (LABEL:eq:andersontoo) shows that it must saturate to a value less than unity (in units of $h/e^{2}$). Even if we relax the assumption of monotonic behavior Eq. (LABEL:eq:andersontoo) shows that the resistance is bounded which is incompatible with Anderson localization. Moreover the finiteness of $\langle\csc^{2}\theta_{N}\rangle$ shows that the distribution $P(\theta_{N})$ must vanish as $\theta_{N}\rightarrow 0$.

We should draw attention to the fact that in order to obtain delocalization we had to build up our disordered conductor by choosing the phase $\phi$ for each successive scatterer to lie in an appropriate range of positions. The appropriate range depends on the energy parameter $k$ so the question arises whether the conductor will remain delocalized if the energy is varied. While one might hope that the conditions on the phase might be met for a range of energies in fact our numerical simulations below show that this is not the case. For a finite system therefore we will get transmission over a narrow range of energies and exponential localization away from the delocalization energy; the transmission resonance will narrow as the system grows.

Figure 3 shows numerical results for the transmission coefficient $\sin^{2}\theta_{N}.$ Histograms for $N=100$ and $N=10000$ show no significant difference, indicating that the $N\rightarrow\infty$ limit has been reached. Each scatterer has a $S$-matrix of the form of Eq.(5) with $V$ chosen uniformly over the interval $=0.3<V<0.3$ and $2\sin k=1.6.$ As discussed at the end of Section II.1, the angle $\theta$ for each scatterer is chosen to be in the first or fourth quadrant, and in Eq.(13), it can be chosen to be in the first quadrant without loss of generality. Since the allowed range of $V$ is small, all the scatterers are weak, and $\theta$ lies within a small interval near $\pi/2.$ The angle $\phi$ is chosen randomly, as described earlier. Under these conditions, one can show analytically that the distribution for $\sin^{2}\theta_{N}$ has a lower cutoff which is greater than zero, as seen in the figure.

On the other hand, if we consider $\theta$ to be a uniform random variable in the interval $(0,\pi/2]$ (with $0<\phi<\theta,$), $\langle\cot^{2}\theta\rangle$ diverges and the proof that $\langle\cot^{2}\theta_{N\rightarrow\infty}\rangle$ is finite is not valid. Nevertheless, as seen in Figure 4, $\langle\sin^{2}\theta_{N\rightarrow\infty}\rangle$ is finite; the peak at the origin of the distribution is accompanied by a broad flat tail.

From the arguments above, one might think that a random lattice that is designed to have a $O(1)$ transmission coefficient at a certain energy should have an $O(1)$ transmission coefficient (i.e. delocalized states) over a band of energy, since the $S$-matrix for each scatterer as well as the phase introduced by a path length $L$ evolve continuously as a function of the wavevector $k.$ However, this is not the case. The path-dependent phase $\phi$ associated with the interval between the $N^{{\rm th}}$ and $(N+1)^{{\rm th}}$ scatterers was defined to be $\phi=2k(L+1)+\gamma+\gamma_{N}-\delta_{N}.$ Once the length $L$ of the interval has been optimized for some $k,$ to ensure that $0<\phi<\theta,$ a small change in $k$ could change $\gamma_{N}-\delta_{N}$ by a large amount if $N$ is large, so that the condition $0<\phi<\theta$ would no longer be satisfied. Numerical simulations reveal this to be the case: although the prescription above allows one to obtain $O(1)$ transmission at any chosen energy, the transmission coefficient drops off as one moveas away from this energy, and decays as a function of $N$ for any energy other than the energy for which the structure is designed.