ANDERSON Localization on One-dimensional Lattices

An important subject in solid state physics is the investigation of electronic motion in disordered solids at low temperature. ANDERSON initiated a particular type of research in this field, focusing on one-dimensional lattices as model systems for the crystal lattices of solids [And58,And61,And78]. (A more recent review may be found in [LR85].) Here, I discuss only those aspects of the theory that are relevant for the understanding of localization phenomena on such lattices that can be taken as simple model systems for disordered solids.

The concept of transfer matrices [Pen94] is an essential ingredient of the theory. As an introductory example for the use of these matrices I consider a simplified discussion of a stationary quantum wave function within the one-dimensional potential

$$% V(x) V_{\mbox{\scriptsize A}}(x) \; = \; \sum_{m=-\infty}^\infty w_m v(x-md), % \quad b_m\in\RR,$$ (5.7)

consisting of a bi-infinite sequence of attractive potentials at the equally spaced lattice sites $md$, with the lattice constant $d$. For simplicity, the individual potentials are all assumed to be of the same shape given by $v(x)$, but the weight factors $w_m\in\mathbb{R}$ may vary. The single site potential $v(x)$ is assumed to be sufficiently localized around $x=0$, such that the overlap of potentials belonging to different sites is small and $V_{\mbox{\scriptsize A}}(x)$ may be considered zero in the inter-site regions. In other words, an electron moving within such a lattice essentially interacts at most with one lattice atom at any given time, and it propagates freely while not being scattered on-site. A lattice of this type may serve as a model system for weakly bound electrons moving within a disordered solid in the framework of a single particle approximation; the lack of order due to impurities (e.g. different types of atoms [Fis96]) or other perturbations of the crystal lattice is described by the varying $w_m$, leading to reflection and transmission properties that vary from site to site.^5.3See figure 5.5 for a schematic sketch of such an ANDERSON lattice.^5.4

In the zero potential region between the $(m-1)$-st and $m$-th site the quantum wave function can be written in the form

$$\left< x \left\vert \psi \right> \right. \; = \; A_me^{ikx}... ...kx} % , \quad \mbox{for} \quad md+\delta x<x<(m+1)d-\delta x,$$ (5.8)

with coefficients $A_m,B_m\in\mathbb{C}$, the wave number $k\in\mathbb{R}$ and some suitable $\delta x\in\mathbb{R}$. As the model does not depend on time explicitly it suffices to consider stationary states. The values of the coefficients in neighbouring flat regions can be related by the $(2,2)$ transfer matrix $\cal{T}_m$, which in this context is defined implicitly by

$${A_{m+1} \choose B_{m+1}} \; = \; \cal{T}_m \, {A_m \choose B_m}.$$ (5.9)

In this way the wave function in any of the inter-site regions of the lattice, given by $\left(A_m,B_m\right)^t$, can be constructed systematically by specifying a suitable ``initial condition'' $\left(A_0,B_0\right)^t$ and iterating the discrete ``dynamical system'' (5.17). The particular properties of $V_{\mbox{\scriptsize A}}(x)$ enter into the matrix elements of $\cal{T}_m$ via the reflection and transmission coefficients $T_m^{(\varsigma)},R_m^{(\varsigma)}\in\mathbb{C}$, $\:\varsigma=\mbox{l,r}$, at the $m$-th site:

$$\cal{T}_m \; = \; \frac{1}{T_m^{(\mbox{\scriptsize r})}} ... ...mbox{\scriptsize l})} & \displaystyle 1 \end{array} \right).$$ (5.10)

The phases of these coefficients and the transmission probability $T_m^2:=\big\vert T_m^{(\varsigma)}\big\vert^2=1-\big\vert R_m^{(\varsigma)}\big\vert^2,\;$ $\varsigma=\mbox{l,r}$, are subject of the shape of the $m$-th potential $w_mv(x-md)$ and depend on its width and depth, for example. Due to the time-reversal symmetry of the system defined by the potential (5.15), $\cal{T}_m$ can be simplified to give

$$\cal{T}_m \; = \; \frac{1}{T_m} \left( \begin{array}{cc} ... ...2)}} & \displaystyle e^{-i\phi_m^{(1)}} \end{array} \right),$$ (5.11)

where the only remaining parameters are $T_m\in\mathbb{R}$ and two phases $\phi_m^{(1)},\phi_m^{(2)}$.

For a periodic potential $V_{\mbox{\scriptsize A}}(x)$ with $w_m=w \; \forall m$ the conditions for complete transmission through a lattice site are identical for all sites and can be satisfied by adjusting the energy of the electron. This is a consequence of BLOCH's theorem which asserts the existence of delocalized motion [Mad78].

For aperiodic potentials, on the other hand, which -- as discussed above -- provide a more appropriate model for disordered solids than periodic potentials, the situation is different and more conveniently discussed in terms of another class of systems, giving rise to another type of transfer matrices. Here, the role of the time-independent SCHRÖDINGER equation is taken by the discrete SCHRÖDINGER equation [And58,And78]:

$$% \epsilon_m u_m + \sum_{m'\in\ZZ} W_{m'} u_{m-m'} \epsilo... ...y W_{mm'} \, u_{m'} \; = \; \eta \, u_m. % , \quad m\in\ZZ.$$ (5.12)

It describes a particle with energy $\eta\in\mathbb{R}$, moving on a one-dimensional lattice of equidistant sites, and is to be solved for $u_m\in\mathbb{C}$, the probability amplitude of finding the particle at the $m$-th site. At each site a random potential $\epsilon_m\in\mathbb{R}$ acts as a diagonal energy; the way in which this potential term becomes ``randomized'' is discussed extensively below. The off-diagonal matrix elements $W_{mm'}\in\mathbb{C}$ describe the coupling of the sites labelled by $m$ and $m'$, respectively. Often the $W_{mm'}$ are referred to as interaction energies (cf. [Zim79]), although in general they cannot be restricted to take on real values only.

The discrete SCHRÖDINGER equation (5.20) can be derived from its continuous counterpart, the time-independent SCHRÖDINGER equation for a state $\left\vert \psi \right>$ with energy $\eta$,

and the electron mass $m_{\mbox{\footnotesize e}}$, in the following way. The potential (5.15) again has been assumed to be composed of well localized, nonoverlapping individual potentials $v(x)$. This assumption is often called the tight binding approximation(see for example [Stö99]) -- although some authors also refer to tight binding with respect to a special case of the model that I discuss below on page .

Let the eigenstates of $v(x)$ be denoted by $\left\vert \psi^{(\alpha)} \right>$, with $\alpha$ labelling the different eigenstates. While a general state needs to be constructed by superposition of all of these $\left\vert \psi^{(\alpha)} \right>$, for the model system to be discussed here it suffices to consider a single eigenstate, $\left\vert \psi^{(0)} \right>$, taken to be normalized. It may be looked upon as that single atom state interacting strongest with the passing electron. Obviously, the restriction to a single eigenstate is a further assumption on the system, but allowed in the present context, since the resulting model system is still powerful enough to explain some of the key features of electronic states in disordered crystalline lattices. The localization property of $v(x)$ carries over to $\left\vert \psi^{(0)} \right>$, such that the following LCAO ansatz (linear combination of atomic orbitals, [Zim79]) for an eigenstate $\left\vert \psi \right>$ of the complete lattice can be made,

$$% \Bracket{x}{\psi} \left\vert \psi \right> \; = \; \sum_... ...infty u_{m'} \, \hat{T}(m'd) \, \left\vert \psi^{(0)} \right>,$$ (5.12)

characterizing $\left\vert \psi \right>$ as a linear combination of the eigenstates $\left\vert \psi^{(0)} \right>$ shifted to each lattice site by means of the translation operator $\hat{T}(x')$:

$$\left\vert x+x' \right> \; = \; \hat{T}(x') \left\vert x \r... ...hbar}x'{\hat{p}}} \left\vert x \right>, \quad x'\in\mathbb{R};$$ (5.13)

$\hat{T}(\cdot)$ is a special case of the more general translation operator $\hat{D}(\cdot,\cdot)$ of equation (4.1b): $\hat{T}(x')=\hat{D}(x',0)$.

Inserting the expansion (5.22) into the SCHRÖDINGER equation (5.21), and making use of the orthonormality relation

$$\left< \! \psi^{(0)} \! \left\vert \, \big(\hat{T}(md)\big)... ... \, \right\vert \! \psi^{(0)} \! \right> \; = \; \delta_{mm'},$$ (5.14)

which expresses the vanishing of the overlap between eigenstates belonging to different sites, I obtain the discrete SCHRÖDINGER equation (5.20) once the matrix elements

$$% W_{mm'} \; := \; \Int_{-\infty}^\infty \! \dop x \, W_{m... ...e A}} \, \hat{T}(m'd) \, \right\vert \! \psi^{(0)} \! \right>$$ (5.15)

and the diagonal energies

$$\epsilon_m \; := \; W_{mm}$$ (5.16)

have been defined.

Essentially, the expectation values $\epsilon_m$, taken with respect to $\hat{T}(md)\left\vert \psi^{(0)} \right>$, depend on the weight factors $w_m$ of $V_{\mbox{\scriptsize A}}$ only,

$$\epsilon_m \; = \; \frac{1}{2m_{\mbox{\footnotesize e}}} \... ...^{(0)} \left\vert \,\hat{v} \, \right\vert \psi^{(0)} \right>,$$ (5.17)

as the two expectation values in equation (5.27), taken with respect to $\left\vert \psi^{(0)} \right>$, do not depend on $m$ any more. This observation is useful when discussing the way in which disorder or ``randomness'' is introduced into the theory.

Up to this point, the dynamics on the lattice as given by equation (5.20) is completely deterministic, and its parameters are all fixed by specifying $V_{\mbox{\scriptsize A}}$. On the other hand, since a priori the eigenstates $\left\vert \psi^{(0)} \right>$ are unknown and the calculation of the $W_{mm'}$ and $\epsilon_m$ might be a difficult task, it is much simpler to choose these quantities in an appropriate way, making the model as simple as possible, while still retaining its essential characteristics. In the following I discuss the most important of the possible choices, which is known as diagonal or site disorder. Here, the matrix elements $W_{mm'}$ are assumed to be ``constant'', i.e. they are not treated as varying much with $m$, $m'$. Frequently, most of the $W_{mm'}$ are taken to be zero, and just a few of them take on a very limited number of nontrivial values. A typical example for such a choice is given below in equation (5.30). The disorder part of the case of site disorder is constituted by assuming the diagonal energies to vary with $m$, in such a way that the $\epsilon_m$ are statistically distributed according to a given distribution and nothing more is known about them. It is a nice feature of this particular model that, as indicated by equation (5.27), the assumptions concerning the distribution of the $\epsilon_m$ carry over to the $w_m$, and vice versa. In this way it is possible to adjust the disorder properties of the model by specifying the weight factors of the ``random potential'' $V_{\mbox{\scriptsize A}}$ in the very beginning of the discussion.^5.5

Often, the model potential $V_{\mbox{\scriptsize A}}$ is defined in such a way that the resulting discrete SCHRÖDINGER equation is at least approximately translation invariant with the lattice constant $d$, i.e. the matrix elements $W_{mm'}$ depend on the distance of the two respective sites $m$ and $m'$ only. In these cases the simplified discrete SCHRÖDINGER equation

$$\epsilon_m u_m + \sum_{ {\scriptstyle m'=-\infty \atop \sc... ...le m'\neq m} }^\infty W_{m'-m} \, u_{m'} \; = \; \eta \, u_m$$ (5.18)

can be used, where the hopping matrix elements

$$% W_{mm'} \; := \; W_{m'-m} W_{m'} \; := \; W_{m,m+m'}$$ (5.19)

specify the interaction of the particle at a given site with its $m'$-th nearest neighbour site. An especially simple example of such a model is defined by a potential $V_{\mbox{\scriptsize A}}$ for which all weight factors $w_m$ are identical; but in the present context this example expresses an oversimplification that rules out ANDERSON localization and therefore is not considered in the following.

Depending on $V_{\mbox{\scriptsize A}}$, the model may be simplified even further. In order to study ANDERSON localization it suffices to consider the special case where only the interaction with nearest neighbours is taken into account and assumed to be symmetric,

The restriction (5.30a) is also frequently referred to as the tight binding approximation [Fis93]. (Cf. the discussion of tight binding on page .) The resulting tight binding equation is

$$\epsilon_m u_m + W_1 \left(u_{m-1}+u_{m+1}\right) \; = \; \eta \, u_m.$$ (5.19)

By scaling the energies $\epsilon_m$ and $\eta$, $W_1$ can be set to unity without loss of generality. What is left is a model system that contains just a single external parameter $\eta-\epsilon_m$, fluctuating with $m$ in a random fashion but in accordance with some given distribution of values.

Again, this time as a result of the tight binding approximation (5.30a), the problem can be formulated using a transfer matrix approach by setting

$$\vec{u}_m \; := \; {u_m \choose u_{m-1}}.$$ (5.20)

Defining the transfer matrix as

$$\cal{T}_m \; = \; \left( \begin{array}{cc} \eta-\epsilon_m & -1 \\ [0.2cm] 1 & 0 \end{array} \right),$$ (5.21)

the tight binding equation (5.31) becomes

$$\vec{u}_{m+1} \; = \; \cal{T}_m \, \vec{u}_m.$$ (5.22)

Every point (here: corresponding to the index $m+1$) to the right of some initial point (with index $0$) on the lattice can be reached by repeated application of the transfer matrix:

$$\vec{u}_{m+1} \; = \; \cal{T}_m \cal{T}_{m-1} \cdots \cal{T}_1 \cal{T}_0 \, \vec{u}_0, \quad m\geq 0.$$ (5.23)

Similarly, with

$$% \vec{v}_n := {u_{n+m} \choose u_{n+m+1}}, \quad n,m\in\ZZ \vec{v}_m \; := \; {u_{m-1} \choose u_m} % , \quad m\in\ZZ$$ (5.24)

every point (with index $m-2$) to the left of another initial point (with index $-1$) is obtained by

$$\vec{v}_{m-1} = \cal{T}_m \, \vec{v}_m$$ (5.25)

$$= \cal{T}_m \cal{T}_{m+1} \cdots \cal{T}_{-1} \cal{T}_0 \, \vec{v}_0, \quad m\leq 0.$$

In such a setting FURSTENBERG's theorem [Fur63] can be applied, which deals with the more general class of unimodular random matrices. Obviously the $\cal{T}_m$ are unimodular for all $m$,

$$\det(\cal{T}_m) = 1,$$ (5.26)

and $\eta-\epsilon_m$ is a random variable for all $\eta$ if and only if $\epsilon_m$ is random. The theorem then ensures that with probability one the limit

$$\gamma \; := \; \lim_{m\to\infty} \frac{\log\left\Vert\vec{... ...\lim_{m\to\infty} \frac{\log\left\Vert\vec{v}_m\right\Vert}{m}$$ (5.27)

for the ``LIAPUNOV exponent $\gamma$ of the transfer matrix'' [Fis93] exists and that $\gamma$ is positive.^5.6(See appendix B for a more detailed exposition of FURSTENBERG's theorem.) This means that in general -- i.e. for generic values of the energy $\eta$ -- $\left\Vert\vec{u}_m\right\Vert$ and $\left\Vert\vec{v}_m\right\Vert$ essentially grow exponentially with $m$. For boundary conditions I choose the values of the components of two vectors $\vec{u}_{m_1}$, $\vec{v}_{m_2}$ describing distant points on the lattice, i.e. with $m_1\ll m_2$. While by the above theorem both sequences $\left\{\left\Vert\vec{u}_m\right\Vert\right\}$ and $\left\{\left\Vert\vec{v}_m\right\Vert\right\}$ generically grow exponentially -- thus yielding no physical (normalizable or at least improper) states -- it is still possible to adjust the energy in such a way that the wave functions iterated from both ends match somewhere in between; this was first observed in [Bor63] and rigorously proven in [DLS85a,DLS85b] to be the generic case. As a result, one obtains almost surely a discrete point spectrum of energy eigenvalues with eigenstates which are exponentially localized around a site $m_{\rm c}\in\mathbb{Z}$. They can be written as

$$u_m \; = \; A \: f_m \: e^{-\gamma \vert m-m_{\rm c}\vert}$$ (5.28)

with a normalization constant $A\in\mathbb{R}$, and with coefficients $f_m\in\mathbb{C}$ the absolute values of which do not exceed unity and typically oscillate rapidly with $m$. Note that in this way the LIAPUNOV exponent $\gamma$ is identified as the inverse of the localization length of the state described by (5.40). See figure 5.6 for a sketch of the envelope $\left\vert u_m\right\vert _{\rm max} = A e^{-\gamma \vert m-m_{\rm c}\vert}$ of an ANDERSON-localized wave function.

A necessary condition for this explanation to work is that the diagonal energies $\epsilon_m$ are randomly distributed under variation of $m$. The definition of a tight binding model thus has to be completed by specifying the distribution function $p(\epsilon_m)$ for the values of $\epsilon_m$. Actually, in order to ensure that FURSTENBERG's theorem is applicable, $p(\epsilon_m)$ must be ``sufficiently well-behaved'' in a way that is described in some more detail in appendix B. (In [FMSS85,CKM87] ANDERSON localization is even proved under assumptions which are somewhat weaker than those of FURSTENBERG's theorem, as discussed in appendix B.) Several different choices for $p(\epsilon_m)$ satisfying these requirements have been applied successfully; choosing a Lorentzian distribution function -- in which case the model is often referred to as the ANDERSON-LLOYD model --,

$$p(\epsilon_m) \; = \; \frac{\delta}{\pi(\delta^2+\epsilon_m^2)}, \quad \delta\in\mathbb{R},$$ (5.29)

has the advantage that in this case an explicit expression for the localization length $1/\gamma$ can be derived analytically:

$$% Gleichung gibts in Fish93 auch allgemeiner MIT V_1-Abhaen... ...t{(2+\eta)^2+\delta^2} + \sqrt{(2-\eta)^2+\delta^2} \right);$$ (5.30)

i.e. it depends on the energy $\eta$ and the half width $2\delta$ of the Lorentzian only. A proof of this statement is reviewed in [Haa01].

To summarize, a discrete SCHRÖDINGER equation describing electronic motion within the framework of a one-dimensional lattice model of a disordered solid has been derived, and I have demonstrated that typical solutions exhibit exponential localization.

Footnotes

... site.^5.3

The model can also be used to describe light propagation in a randomly stratified transparent medium, where the medium is found to reflect the light perfectly. In this way an optical realization of ANDERSON localization is obtained [BK97].

... lattice.^5.4

It is also possible to choose $v(x)$ as a narrow rectangular potential barrier, thus automatically guaranteeing good localization of $v(x)$ [PGF85]. But this choice requires a different physical interpretation and is not used in the present context.

... discussion.^5.5

Another way to introduce disorder into the theory is to consider bond disorder. In this case, the $W_{mm'}$ are assumed to be statistically distributed, and the $\epsilon_m$ are constant.

... positive.^5.6

Note that FURSTENBERG's theorem can also be utilized to prove chaoticity of the classical kicked rotor for sufficiently large $V_0$, by reformulating the linearized standard mapping (5.4) in the form of equation (5.35) and thus establishing positivity of the larger LIAPUNOV exponent of the system [Haa01].