Scaling and localization lengths of a topologically disordered system
We consider a noninteracting disordered system designed to model particle diffusion, relaxation in glasses, and impurity bands of semiconductors. Disorder originates in the random spatial distribution of sites. We find strong numerical evidence that this model displays the same universal behavior as the standard Anderson model. We use finite-size-scaling to find the localization length as a function of energy and density, including localized states away from the delocalization transition. Results at many energies all fit onto the same universal scaling curve.
The disorder-induced transition from extended to localized states in non-interacting quantum systems has been a rich source physics insight for over fifty years Anderson58 . It is relevant for a broad range of transport properties Lee85 , glass formation Amir10 , conductivity of composites Ambrosetti10 , random walks *[; p.29.]Mulken11, as well as for non-radiative recombination in intermediate-band photovoltaics Luque06 ; *Lopez11. Scanning-tunneling and Bose-Einstein condensate experiments are increasingly able to probe localization properties directly Morgenstern02 ; *Hashimoto08; *Richardella10; *Clement06; *Billy08. The delocalization transition is usually studied using the standard Anderson model, which considers a non-interacting tight-binding lattice with uniform nearest-neighbor coupling and random on-site energies Anderson58 . For systems in which the disorder originates in the random configuration of the sites rather than, e.g., random local fields, it is better to consider the so-called topologically disordered or Lifshitz model, in which sites are distributed randomly in space with no on-site energies and all pairs of sites are connected by hopping terms with amplitude exponentially decaying with the distance between them Lifshitz65 . As the density of sites increases, a localization/delocalization transition occurs, just as in the standard Anderson model. The density of states Lifshitz65 ; Matsubara61 ; Ching82 ; *Logan85; *Logan86; Shklovskii84 ; Mezard99 ; Amir08 ; Amir10 and localization properties Ching82 ; *Logan85; *Logan86; Priour10 ; Amir10 of this model have received much attention. Here we obtain the localization lengths of this model quantitatively as a function of wavefunction energy and site density. We adapt the finite-size scaling method, which has been successfully applied to the standard Anderson model MacKinnon81 ; MacKinnon83 ; Slevin99 ; *Slevin03; Rodriguez10 ; Markos06 , and give strong numerical evidence that the topologically disordered model displays the same universal behavior.
We consider particles confined to identical lattice sites distributed randomly in space, with density . Particles can hop between sites with an exponentially decaying hopping coefficient. The Hamiltonian is
where are the site wavefunctions, is the distance between sites and , is an effective Bohr radius giving the decay of the wavefunctions, is an overall energy scale, and the sum runs over all pairs of sites. This model is called topologically disordered because there is no ordered structure describing which lattice sites are strongly coupled to each other. This model can describe the Hamiltonian of impurities with hydrogenic wavefunctions in a semiconductor, where the hopping originates in overlaps of their effective atomic wavefunctions. The dimensionless density is . Experimentally, such systems are found to have a metal-insulator transition at Edwards78 . If the diagonal components of are set such that each column sums to zero, the model can describe glass dynamics Amir08 ; Amir10 and continuous time quantum walks Mulken11 , but many properties are similar.
Ii Localization Lengths from Quasi-1D Scaling
The localization length of a localized eigenstate is determined by the asymptotic decay of the wavefunction , where is some location of high wavefunction amplitude. Our goal is to find the localization length as a function of and . Finite-size scaling techniques allow us to study computationally tractable small systems and systematically extrapolate to results for the true infinite system. We assume, and find strong numerical evidence to confirm, that the topologically disordered system is controlled by the same fixed point as the standard Anderson model. The critical exponent and other critical parameters of this fixed point have been well-determined by previous studies MacKinnon94 ; Slevin99 ; Slevin03 ; Rodriguez10 , and we will use these results to aid our study of the localization lengths. While studies interested in the critical exponents have focused on systems close to the localization transition, we are interested in localization lengths across the range of energies and densities relevant to experiments, which includes not only values close to the transition but also ones in the strongly localized regime. As we move away from the critical point, corrections to scaling should become more important, and we test our results’ sensitivity to these corrections.
The quasi-1D scaling method was introduced for the standard Anderson model by MacKinnon and Kramer, who considered several wires of varying widths and long lengths MacKinnon81 . We adapt the recursive Green function technique to find the localization length for equivalent wires in our system, see inset to Fig. 1a MacKinnon81 ; MacKinnon83 . The key idea is to divide a long wire into slices, where the sites in each slice are directly coupled only to each other and to sites in the immediately adjacent slices. This is not strictly possible for the system of Eq. 1, as all sites are directly coupled. We take, however, a cutoff and set for . We start with a single slice of width , length and periodic boundary conditions in two dimensions. We recursively add slices adjacent to the wire and find the portion of the Green function that connects any site in slice 1 to any site in slice at fixed energy . We then use the standard method to determine the localization length of the long wire at energy MacKinnon81 ; MacKinnon83 ; Markos06 . The width of each wire must be large enough that there is a sufficient probability that each slice will have sites within of sites in the neighboring slices, otherwise the wire is disconnected Taylor89 . Due to the varying number of particles in each slice and the varying off-diagonal matrix elements, the transfer matrix method (reviewed in Ref. Markos06, ) cannot be used.
The statistical error in can be estimated by assuming that the estimate of after slices is the mean of independent and identically distributed samples chosen from a normal distribution, so the sampling error goes down as Markos06 . This is rigorously proved for the standard Anderson model with Oseledec’s theorem Markos06 , and we assume that similar statistics hold here. We choose a maximum error of 1%, which requires to slices, depending on the parameters.
After determining for a range of widths , MacKinnon and Kramer (and many subsequent authors) use the one-parameter scaling theory Abrahams79 to extrapolate to infinite size systems. The scaling can be expressed in terms of the relevant variable , with greater (less) than zero for extended (localized) states. Then the dimensionless quantity is a universal function of and , with critical exponent , Slevin99
The correlation length is .
The hypothesized universality of the metal-insulator transition with time reversal symmetry has been confirmed in a variety of different models Slevin99 ; Markos06 . If we assume that this topologically disordered model is controlled by the same fixed point as the standard Anderson model, then we expect Eq. 2 to hold in our system, with the same . Previous work has expanded in polynomials in order to fit the data Slevin99 ; Slevin03 ; Rodriguez10 . We have not found this to be a successful strategy, at least in part because the underlying functions are strongly non-polynomial away from the transition region (see below). In general in such fits, there is a large number of parameters Slevin99 ; Rodriguez10 , so it is helpful to constrain as many of them as possible. We assume universality and use four pieces of information from previous work: and , as shown for the standard Anderson model Slevin99 . Further, we have asymptotic limits MacKinnon83 . We choose to fit to . Then we have and . The asymptotic limits are obeyed by the function . We then fit our data to where and .
The scaling form, Eq. 2, applies only for sufficiently large. At small , corrections to scaling modify Eq. 2, and they have proven essential for accurate determination of and Slevin99 . Corrections to scaling require introduction of some number of irrelevant variables , which have no effect on the scaling in the limit . We illustrate with only one irrelevant variable , but more are easily added. We rewrite our scaling equation as , where is another critical exponent. We expand in powers of ,
We are interested in finding the localization length at many values of , unlike the usual choice of . The Green functions in the determination of must be evaluated separately for each choice of , increasing the computational requirements. We choose 14 values of density and 40 values of energy satisfying , which are focused around the relevant energies for the metal insulator transition near . We do not study because isolated sites always produce eigenvalues with , causing divergences in the Green functions. We take and . At each , we take from in increments of up to the largest system size we find computationally tractable. For the 14 densities, the maximum studied are , from lowest to highest density. Previous studies of mobility edges and critical exponents have performed separate scaling at each Bulka85 ; *Kramer90. In this work, all of the data collapse onto a single scaling curve, see Fig. 1, showing universality independent of energy. These calculations used approximately 24 CPU-years of computing time.
We approximate as a sum of three Gaussians, fixed to have , with eight free parameters. We use a direct search algorithm as implemented in MATLAB’s fminsearch function to find the least-squares optimal choice of these eight parameters. Within the search loop, for each proposed , we find (and possibly ) independently at each value of , giving several hundred additional parameters. It would be preferable to have a functional form for , but we did not find a parametrization that permitted accurate fits. To ensure that these fits are not overdetermined, we include in our dataset only with at least 4 values of ; the average number of values of at each is 13. In fits including corrections to scaling, we approximate as a second order polynomial with , which fixes the scale of .
Fig. 1a shows the scaled data, with one correction to scaling. The upper branch shows the extended states ( as ) and the lower branch shows localized states. The scaling is excellent, which justifies the use of and from the standard Anderson model studies. The consistency of with previous work indicates that the lattice spacing in the standard Anderson model is equivalent to in this model. The extended state curve does not approach its asymptote due to a lack of data in the computationally demanding high density regime. Fig. 1b shows the same fit with and the irrelevant corrections subtracted. The dashed lines show the constituent Gaussians in , which are not well-constrained in the fits. See supplementary information for other fits. Fig. 2 shows the resultant correlation length . At each , the correlation length falls smoothly as moves away from the delocalization transition, just as we expect. These results give confidence that the are not simply arbitrary parameters that happen to produce good scaling fits but rather are determined by the underlying physics. We find, however, that (and thus ) is quantitatively determined only for localized states.
To assess the quality of the scaling curves produced, we consider the statistic. Previous high accuracy numerical works on the standard Anderson model have evaluated their fits with the quality of fit (also known as the p-value) from the statistic and the number of fitting parameters Slevin99 ; Slevin03 . All fits shown here have , indicating that statistical fluctuations of the numerical procedure alone are insufficient to explain the deviations of the data from the model. This is not, however, surprising. Our study has several thousand degrees of freedom, , where is the number of data points, which gives it statistical power to detect relatively small deviations between the model and the data. In the true scaling function, is not actually a sum of three Gaussians, and we are sensitive to the deviations between the true and its model. We should be able to add more Gaussians to to better approach the universal function, but fitting becomes difficult. If we are not entirely interested in the exact shape of , then this deviation is not a concern. If we limit our dataset to contain only close to the delocalization transition, as in previous studies of the standard Anderson model, we can obtain good values of ; this occurs because the number of data points is significantly reduced and the scaling function can be accurately approximated as a polynomial.
Since we are interested in extracting correlation lengths , a better determination of the quality and confidence of the fits is to compare the resultant for different fitting procedures. We compare fits with and without corrections to scaling and with the smallest being 22, 25, or 28, in units of . Depending on initial guesses, fits can arrive at a number of different local minima. We use a multistart procedure for the fitting, starting with 100 widely varying parameters for and . We find that the “best fits,” judged solely by minimizing , have highly oscillatory , discontinuous , large deviations from the asymptotic form even for strongly localized states, or large corrections to scaling; they generally have multiple of these features. If we exclude the fits with these four characteristics, for the best fits without (with) corrections to scaling have . Including the anomalous fits, we can find . We find from 92 different fits and, independently at each , find the mean and standard deviation , shown in Fig. 2b. The values of are found to vary by less than 10% in the localized regime, except for the points closest to the delocalization transition. For , we can consider the localization lengths to be given quantitatively by the scaling method. Due to the lack of strongly-extended states, is not well-determined, and the fits show a range of different shapes. It is then no surprise that in the delocalized region varies widely. Accumulating more data in the extended regime should fix this problem.
This scaling technique quantitatively gives the localization lengths at nearly any localized we care to study. Application of this method to systems with on-site disorder, in addition, should shift the mobility edges “inwards” so that more states are localized. These localization lengths allow insight into the properties of a range of material systems, and in future work we will consider their effects on intermediate band photovoltaics.
Acknowledgements.We acknowledge fruitful conversations with Bertrand Halperin, Kristin Javaras, Mark Winkler, Justin Song, Man-Hong Yung, Ari Turner, and Mauricio Santillana and use of Odyssey, supported by the FAS Research Computing Group. We acknowledge support of the Ziff Environmental Fellowship of the Harvard University Center for the Environment, NSF DMR-0934480, and DARPA.
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APPENDIX: Scaling curves without corrections to scaling
As discussed in the main text, several fitting procedures can be used to produce plots similar to those in Figs. 1 and 2. We illustrate here that the choice of fitting procedure does not significantly change the localization lengths extracted. Fig. 3 shows fits equivalent to those in Fig. 1, without including corrections to scaling. The scaling still looks very good, though Fig. 3b shows that the data points do not fall on the scaling curve as precisely as in Fig. 1b. The overall shape of the scaling curve deviates from that in Fig. 1 mostly in the extended region, , which is not well determined from the data. Additionally, though the overall curve is very similar in shape to that in Fig. 1b, the underlying Gaussians are different, which illustrates the difficulty in converging the fits of .
The correlation lengths extracted from these fits are shown in Fig. 4a. They are clearly very similar to those in Fig. 2a, showing that the scaling fits robustly determine , regardless of whether corrections to scaling are used. Fig. 2b shows the deviations in the fitted values of for 92 different fits with and without corrections to scaling.
Fits including corrections to scaling have larger disagreements in . Fig. 4b shows the deviations for 65 fits without corrections to scaling (blue) and 27 fits with corrections to scaling (green). The fits without corrections to scaling have a smaller number of fit parameters, which seems to constrain the fits more. In any of the cases studied, the states with are quantitatively determined within 10% by the fits. The consistency of the fits without corrections to scaling argues for use of that method in future studies, despite the large values of produced by ignoring the corrections to scaling.