Statistics of anomalously localized states at the center of band in the one-dimensional Anderson localization model
We consider the distribution function of the eigenfunction amplitude at the center-of-band () anomaly in the one-dimensional tight-binding chain with weak uncorrelated on-site disorder (the one-dimensional Anderson model). The special emphasis is on the probability of the anomalously localized states (ALS) with much larger than the inverse typical localization length . Using the solution to the generating function found recently in our works [18, 17] we find the ALS probability distribution at . As an auxiliary preliminary step we found the asymptotic form of the generating function at which can be used to compute other statistical properties at the center-of-band anomaly. We show that at moderately large values of , the probability of ALS at is smaller than at energies away from the anomaly. However, at very large values of , the tendency is inverted: it is exponentially easier to create a very strongly localized state at than at energies away from the anomaly. We also found the leading term in the behavior of at small and show that it is consistent with the exponential localization corresponding to the Lyapunov exponent found earlier by Kappus and Wegner  and Derrida and Gardner .
pacs:72.15.Rn, 72.70.+m, 72.20.Ht, 73.23.-b
[August 23, 2012]
There is a long-lasting interest in localization effects [1, 2] in 1d systems -. The simplest and most widely studied model is a linear chain with a nearest-neighbor hopping and random site energies with no inter-site correlation: . The wave function at a site of this one-dimensional Anderson localization model  obeys the equation:
In the absence of disorder () the eigenstates would be plane waves, with eigenenergies determined by the wave vector : , . In the presence of the disorder, the eigenstates are random and require statistical description. Moreover, the states are localized at an arbitrary small disorder strength . For weak disorder the localization length is large as compared to the lattice constant: . This means that the “typical” magnitude of the normalized wave function near its localization center can be estimated as . However, for some realizations of the disorder, more strongly localized states, ”anomalously localized states” (ALS), are possible, with the value of the wave function maximum in the range of (the right equality would correspond to a state localized at a single lattice site). Our aim in the present paper is to study the probability distribution of such strongly localized states in a long weakly disordered chain.
We will be especially interested in the statistics of ALS in the vicinity of the so-called Kappus-Wegner center-of-band (, ) anomaly . This anomaly is a feature of a discrete chain (it is absent in the continuum model) and originates from the commensurability of the de Broglie wavelength and the lattice constant. The anomaly manifests itself [8, 9] in a sharp, finite in the limit , enhancement of the density of states (DoS) and the localization length inside a very narrow energy window (of the width ) around the band center as compared to their values
Here we have introduced the superscript “” to emphasize that the corresponding localization length is defined by the Lyapunov exponent and therefore characterizes the exponentially decaying tails of localized wave functions; for this reason it will be referred to as the extrinsic localization length. Similar anomalies have been found later [15, 16] for other physical quantities (like transmission and conductance), also related with the Lyapunov exponent.
In contrast to this set of problems, the eigenfunction statistics may provide information about an “intrinsic” spatial structure of localized wave functions including the vicinity of the center of localization. In particular, it allows to calculate the ”intrinsic” localization length , where is the inverse participation ratio.
However, studying the statistical properties of normalized eigenfunctions is a considerably more difficult theoretical problem than studying the Lyapunov exponent (the latter is related to propagation of an external wave in a semi-infinite chain and is not directly related with eigenfunctions).
The formalism for studying the eigenfunction statistics in a disordered chain (see review ), adapted recently  to the case of the center-of-band anomaly, expresses moments of the eigenfunction distribution in terms of a “generating function” of the two auxiliary variables. These variables can be loosely interpreted [8, 18] as the squared amplitude and the “phase” defined by a representation of eigenfunctions in the form: with slowly varying and . The generating function allows one to calculate all local statistics of eigenfunctions. In particular, it determines the inverse participation ratio (IPR) and higher moments , as well as the full distribution function . Also, the generating function determines (through a nonlinear integral relation Eq.(3)) the joint probability distribution of the amplitude and the phase. However, the relationship between the generating function and the normalized distribution function of the phase turns out to be remarkably simple , it is given by the limit of the generating function :
There is also a simple relationship between and the probability distribution of the reflection phase for a wave incident on a semi-infinite disordered chain. It is given by the second equality in Eq.(4). At weak disorder the phase distribution is uniform in the continuum model and outside the center-of-band anomaly but it becomes a non-trivial function of at .
A relative simplicity of calculation of such quantities as the Lyaupunov exponent (and the extrinsic localization length ) and the DoS, , is due to the fact that they can be expressed entirely in terms of the the probability distribution , i.e. involve the generating function at . For instance, the DoS, is given by [8, 18]:
On the contrary, the complexity of the problem of local eigenfunction statistics arises because it requires the full generating function of the two variables and . In particular, the statistics of relatively rare anomalously localized eigenstates of large peak amplitude which we will study in the present paper is determined by at large values of the variable .
The generating function for a long chain at the center-of-band anomaly has been found recently [17, 18] by solving exactly the corresponding second order partial differential equation Eq.(2) in and variables. The exact solution Eq.(14) to this equation reflects a hidden symmetry of the problem which has not been yet explicitly exploited. However, the solution is given in quadratures as an integral of a product of Whittaker functions over the variable which enters both the argument and the index of these functions. In this paper we perform a careful analysis of the integral and derive the asymptotic form of at large values of (from now on we omit the energy argument for brevity). It has a form:
It allows us to compute the tail of the distribution function at in a long chain of the length :
The anomalous distribution of eigenfunction amplitudes Eq.(7) should be compared with the “normal” one  valid in the continuum model and outside the center-of-band anomaly in the discrete chain :
A comparison of Eqs.(7) and (9) reveals an unexpected feature (see Fig.1). While the probability of moderately strongly localized states (with the peak intensity ) is smaller at than that away from the anomaly, very strongly localized states (with ) are more probable at the band center. Formally this re-entrant behavior is caused by the value of (see Eq.(8)) at the ”critical angle” ; for the ”normal” case one obtains and thus and .
The behavior of moderately strongly localized states is consistent with the result Eq.(3) for the Lyapunov exponent which gives an enhanced typical extrinsic localization length at . The opposite trend for very strongly localized states is perhaps due to the Bragg-mirror effect of the harmonics of the random potential which double the period of the lattice [21, 22].
A point of special interest is the distribution of small amplitudes at , as it gives an idea on the shape of the tail of the localized wave function. We found the leading term in at small and shown that it is universal for all systems with exponentially localized eigenstates.
The rest of the paper is devoted to the derivation of the announced results and is organized in the following way. In section 2 we obtain the asymptotic of the generating function at . In the subsequent section 3 we derive the asymptotic of the probability distribution function at . The behavior of at small is analyzed in Sec.4. In the last section 5 we summarize and discuss the obtained results.
2 Generating function and its asymptotic at
Sufficiently far from the ends of a long chain, the generating function becomes site independent. At the center-of-band anomaly () this stationary generating function, , obeys the partial differential equation (PDE) [17, 18]
Its solution should also meet the requirements of being a smooth periodic function of , regular, positive and non-zero at (we recall that is the phase distribution function, see Eq.(4)) and decaying at .
These requirements are rather restrictive. For instance, the solution
is not appropriate for it is not a smooth function of .
For comparison, we write down also the equation for the “normal” generating function (i.e. when the energy lies outside the anomaly region, or for the continuous model):
This equation looks like a “course-grained” PDE (2) where all the coefficients are “averaged” over the angle interval (so called “phase randomization”) which is equivalent to course-graining over the space region . The variables and in Eq.(12) are separated and one immediately finds that the only solution decaying at and remaining regular and non-zero at is given by
where is the modified Bessel function. This solution has been earlier obtained  in the continuous model. It also arises in the theory of a multi-channel disordered wire [24, 20]. The corresponding phase distribution is uniform: .
Unlike Eq.(12), the PDE (2) is not separable in the variables and . However, due to a hidden (and not well established yet) symmetry of the problem, it was possible to find new variables which allowed us to split the PDE (2) into two ordinary differential equation and thus to construct an exact general solution [17, 18]. The solution, which obeys the above requirements, is given by :
where , and is the Whittaker function (For the second index the Whittaker function can be expressed also in terms of the parabolic cylinder function, see, e.g. ). In the limit the expression (14) reproduces the phase distribution function :
which was derived earlier  in a different way. It shows that the phase distribution becomes non-uniform at the center-of-band anomaly.
Our current task is to derive an asymptotic expression for in the limit of large . The integrand in Eq.(14) is too complicated for a brute force attack. This is because both the arguments and the first indices of the Whittaker functions are large (as is shown below, the leading contribution to the integral comes from ) and the standard  asymptotic expansions of these functions are not applicable. Our approach will include three steps: first we will represent Eq.(14) in the form which allows us to find an asymptotic expression of the integrand; then we obtain the asymptotic form Eq.(7) of the generating function at large (this asymptotic expression will be obtained in the next subsection), and finally the ALS distribution function will be found by a saddle-point integration over .
The generating function Eq.(14) is periodic in (with the period ) and symmetric with respect to the change . Therefore, it is sufficient to calculate in the angular interval . We exploit the following integral representation of the Whittaker function (cf. 9.222.1 ):
valid for and . Since the integrand in Eq.(14) is an analytical function within the sector , we rotate the integration contour and introduce a new integration variable :
After these transformations, Eq.(14) takes the form:
is real, while
2.1 Asymptotic of the integrand in Eq.(18)
At the integrals Eq.(19) can be computed in the saddle-point approximation. The minimum of the action in the integrand of is achieved at the point
The integration contour goes through this point, so the corresponding saddle-point contribution is given by
For the integral the situation is more complicated as there are two saddle-points:
both lie outside the integration semi-axis . On the complex plane with two cuts, and we define an integral over a contour by
where we choose the branch of the integrand so that on the upper edge of the cut . Taking the contour as depicted in Fig.2a, one checks straightforwardly that
Thus, with the exponential accuracy we have expressed the quantity of our interest (see Eq.(18)) in terms of the contour integral . Evidently, the latter is not changed if the integration is extended to parts and comprising the closed contour (see Fig.2a). In this way we arrive at the important relation:
where the contours and are shown in Fig.2b,c; the last equality in Eq.(28) holds because the contour may be safely shifted down to the infinitely remote part of the half-plane , where vanishes (see Eqs.(26) and (21)).
Further transformations depend on the location of the saddle-points, i.e. on the value of , see Eqs.(25) and (25). For , the contour can be lifted to the upper half-plane of (see Fig.2b) to go through the saddle-point (25) which provides the minimum of the action. The corresponding saddle-point contribution to is given by ()
When , the two saddle points (25) lie on the real axis. We bent the contour so that it goes through the both points within the proper Stokes sectors (Fig.2c). The resulting saddle-point contribution to is given by
Eqs. (33) and (34) are defined on the complex plane with a cut along the ray ; branches of and are chosen to be positive on the upper edge of the cut, the (standard) branch of is defined by the requirement at and the cut along on the -plane. Accounting for Eqs. (24) and (33), we arrive at the expression for the generating function Eq.(18) in the form ():
Here the integration contour on the complex plane with the cuts along the rays and , is shown in Fig.5; the chosen branch of is positive at . The integrand in Eq.(35) is depicted (for different values of ) in Fig.3 together with the result of the direct numerical evaluation of the integrand (after switching to the -variable) in Eq.(14).
Our next step is the calculation of the integral in Eq.(35).
2.2 Saddle-point calculation of the integral in Eq.(35) for
Saddle-points of the integrand in Eq.(35) are determined by solutions to the equations
It turns out that the solutions are real and ; the solution exists for , while the solution exists for (we recall that we consider the angle interval ), see Fig.5. The critical angle is determined by the condition , i.e. the solution reaches the origin of the right cut. With this condition, the equation Eq.(2.2) results in (for the both signs):
This transcendental equation can be represented in a nice form using the parametrization
where is the solution of the equation
The value of the critical angle
arises as an important constant also in the calculation of the probability function , section 3.
In the vicinity of this critical angle we have:
Using Eq.(2.2), one can represent the saddle-point actions in a following compact form:
In a similar way, the second derivatives of the actions in the saddle point can be represented as:
It follows from Eq.(46) that the second derivative is positive at () and negative at (). Therefore, for , the contour (see Fig.5a) goes through the saddle point within the proper Stokes sectors, and the term with makes the contribution to the integral in Eq.(35). The term with in Eq.(35) does not have a saddle point at and its contribution is negligible at large .
On the contrary, for the contour is not appropriate as it goes within improper Stokes sectors of the saddle point . To overcome this obstacle, let us modify the contour by adding an additional contour which corresponds to the lower edge of the cut, see Fig.5b. It is seen easily that this operation does not change the integral Eq.(35) because its integrand is purely imaginary on . Now, the part of the modified contour can be deformed to the vertical contour which goes through the saddle point within the proper Stokes sectors, Fig.5b. The corresponding saddle-point contribution to the generating function Eq.(35) at is determined by the term in the integrand. Summarizing these results we arrive at the following asymptotic expression for the generating function:
Here the function outside of a narrow vicinity of the critical angle (see below) is given by
while the function is given by Eq.(45):
The plots of the functions and computed from Eqs.(48) and (49) are given in Fig.6 and Fig.7. Remarkably, the plots which were calculated from different expressions at and do not show any singularity at . The two pieces of the curves match perfectly at the critical angle .
In the next section we present a different calculation of the function which does not possess any critical angle by construction and coincides identically with the above saddle-point expressions. As both and are expressed through the same solutions of the saddle-point equation, smoothness of at implies also the smoothness of .
2.3 Ordinary differential equation for the exponent
Let us look for an asymptotic () solution to the original PDE (2) in the form: where and are to be determined by keeping in the PDE terms of the leading order in . We find immediately that (which is in accordance with (47)) while obeys the ordinary differential equation (ODE):
One can reduce the equation to the form convenient for numerical integration by introducing the function:
Then Eq.(54) takes the form:
It corresponds to the choice of sign ”+” in Eq.(56). This solution is a growing function of with the maximum at . Therefore it does not correspond to the saddle-point solution which has a minimum at (see Fig.7 and Eq.(50)). In fact, the solution Eq.(58) corresponds to the particular solution Eq.(11) which we have already discarded on physical grounds. Thus the relevant solution for our problem is the one which corresponds to the sign ”minus” in Eq.(56). This ODE can be transformed into the Abel’s ODE  but it does not belong to the classes with known solutions.
We solved Eq.(56) numerically applying the initial condition Eq.(57) at a point with . We checked that the solution corresponding to the sign ”plus” matches the function obtained from Eqs.(55),(58) with the same accuracy. Much less trivial is that the solution for corresponding to the sign ”minus” in Eq.(56) coincides (with the same accuracy) with the saddle-point solution given by Eqs.(45),(49). Remarkably, the solution to the particular Abel’s ODE appeared to be represented in terms of the solution to the transcendental saddle-point equation Eq.(2.2)! In this connection we would like to remind about another ”miracle” of the problem. Namely, the saddle-point solution for which we obtained from two pieces