Conductance distribution in two-dimensional localized systems
EEPJ manuscript No. (will be inserted by the editor)
Conductance distribution in two-dimensional localized systemswith and without magnetic fields
J. Prior , , A. M. Somoza and M. Ortu˜no Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom Departamento de F´ısica Aplicada, Universidad Polit´ecnica de Cartagena, Cartagena 30.202, Spain Departamento de F´ısica-CIOyN, Universidad de Murcia, Murcia 30.071, Spainthe date of receipt and acceptance should be inserted later
Abstract.
We have obtained the universal conductance distribution of two-dimensional disordered systemsin the strongly localized limit. This distribution is directly related to the Tracy-Widom distribution, whichhas recently appeared in many different problems. We first map a forward scattering paths model into aproblem of directed random polymers previously solved. We show numerically that the same distributionalso applies to other forward scattering paths models and to the Anderson model. We show that most ofthe electric current follows a preferential percolation-type path. The particular form of the distributiondepends on the type of leads used to measure the conductance. The application of a moderate magneticfield changes the average conductance and the size of fluctuations, but not the distribution when properlyscaled. Although the presence of magnetic field changes the universality class, we show that the conductancedistribution in the strongly localized limit is the same for both classes.
PACS.
The hypothesis of single-parameter-scaling (SPS) [1,2] con-stitutes the main foundation of our understanding of local-ization in disordered systems. The original formulation ofthe SPS hypothesis focused on the average conductance,but it was soon realized that the full distribution functionof the conductance should be considered. Then, accordingto the SPS hypothesis, the conductance distribution func-tion should depend on a single parameter [3], for example,the mean conductance (cid:104) g (cid:105) or the mean of the logarithm, (cid:104) ln g (cid:105) .The validity of the SPS hypothesis has been thor-oughly checked in one–dimensional (1D) systems, whereit has been shown that all the cumulants of ln g scale lin-early with system size [4]. Thus, the distribution functionof ln g approaches a Gaussian form for asymptotically longsystems, which in 1D are always strongly localized. It isfully characterized by two parameters, the mean (cid:104) ln g (cid:105) andthe variance σ of ln g . In the scaling regime, both param-eters are related to each other through a universal law, σ ξ L = 1 (1)where L is the system size and ξ the localization length,defined as ξ = − lim L →∞ L (cid:104) ln g (cid:105) . (2) Eq. (1) was first derived in Ref. [5] within the so–calledrandom phase hypothesis, which assumes that there existsa microscopic length scale over which phases of complextransmission and reflection coefficients become completelyrandomized. This relation reduces the two parameters ofthe distribution to only one and provides, therefore, a jus-tification and interpretation for SPS in 1D systems.The situation in higher dimensional systems is not asclear as in 1D systems. In those dimensions is far moredifficult to do analytical calculations and numerical sim-ulations have been limited until recently to small samplesizes. In the diffusive regime, although the size of universalconductance fluctuations depend on the dimension of thesystem, the distribution function of the conductance tendsto a gaussian for all dimensions. Some people thought thatin the localized regime it could happen something similar.In the strong localization regime, ln g was claimed to benormally distributed in dimensions higher than one [6,7,8]. However, it was pointed out that the distribution isnot log-normal [9,10,11,12]. We found numerically thatthe variance behaves as [13,14] σ = A (cid:104)− ln g (cid:105) α + B (3)with the exponent α equal to 2 / α = 2 / / a r X i v : . [ c ond - m a t . m e s - h a ll ] J un J. Prior, A. M. Somoza and M. Ortu˜no: Conductance distribution in two-dimensional localized systems constants A and B are model or geometry dependent. Theprecise knowledge of the dependence of σ with (cid:104)− ln g (cid:105) made much easier the numerical verification of the SPShypothesis, which we checked for the Anderson model [13].In this paper, we concentrate in the strongly localizedregime in 2D systems. Although experimental measure-ments of coherent transport at low temperatures are diffi-cult in this regime, knowledge of the conductance distribu-tion is of interest to better understand variable range hop-ping conductance, the metal-insulator transition in threedimensions or the crossover between the diffusive and thelocalized regime in 2D systems.In the strongly localized regime, the contribution ofeach Feynman path to the tunneling amplitude betweentwo sites decays exponentially with its distance, and wecan expect that some properties of the conductance (inparticular the size dependence) should be dominated bythe shortest or forward-scattering paths (FSP). This ap-proach was introduced by Nguyen, Spivak and Shklovskii(NSS) [16] in a model to account for quantum interferenceeffects in the localized regime. Medina and Kardar [17]studied in detail the model. They computed numericallythe probability distribution for tunneling and found thatit is approximately log-normal, with its variance increas-ing with distance as r / for 2D systems. This is in con-trast with the 1D case, where the variance grows linearlywith distance, and with the implicit assumptions of someworks on 2D systems. In our opinion, this approximationdid not receive as much attention as it deserves, probablybecause in the SPS regime the localization length must bemuch larger than the lattice constant, which means thatcontributions from other paths cannot be negligible.A major step forward in the field of random systemswas done by Tracy and Widom (TW) [18] who obtainedthe distribution function of the largest eigenvalue of ran-dom matrices belonging to the gaussian ensembles, or-thogonal, unitary and symplectic. It soon became clearthat the TW distribution for the unitary ensemble alsoappears in the calculation of the length of the longestcommon subsequence in a random permutation [19] andin many other seemingly unrelated problems. For an in-troduction see Ref. [20]. Particularly important for ourproblem is the study by Johansson [21] of a specific typeof directed polymer model. He obtained the distributionfunction of the lowest energy state exactly in terms of theTracy-Widom (TW) distribution. Also relevant for us isthe one-dimensional polynuclear growth model [22], whichstudies the height fluctuations of a growing interface andis closely related to the Kardar-Parisi-Zhang equation [23].We showed that for the 2D Anderson model in thestrongly localized limit the conductance distribution is re-lated to the TW distribution [24]. Here we present resultsfor two different models in the FSP approximation, whichis important to link exact results in directed polymers withthe more realistic Anderson model. We show numericallythat for a model with disorder with both positive andnegative values (which cannot be exactly mapped to thesolvable model) the skewness of the conductance distri-bution tends without free parameters the TW value. The role of percolation versus interference is also analyzed. Wefinally study the conductance distribution function in thepresence of a magnetic field. We show that systems inthe orthogonal gaussian ensemble (Anderson model with-out magnetic field) and in the unitary gaussian ensemble.(Anderson model with magnetic field) tend to the samebehavior in the strongly localized limit.In the next section, we describe the two models thatwe have used in our calculations. In section III, we presenta mapping of the conductance in localized systems to thefree energy of directed polymers and use Johansson’s re-sults to get the distribution function. In sections IV andV, we show that the distribution function of the logarithmof the conductance in the FSP approximation and in theAnderson model, respectively, are related to the TW dis-tribution. In section VI, we analyze the effects of a mag-netic field. We finalize with a discussion and conclusionsection. We have studied numerically the Anderson model andits FSP approximation for 2D samples. For the Andersonmodel, we consider square samples of size L × L describedby the standard Anderson Hamiltonian H = (cid:88) i (cid:15) i a † i a i + t (cid:88) i,j a † j a i + h . c . , (4)where the operator a † i ( a i ) creates (destroys) an electronat site i of a cubic lattice and (cid:15) i is the energy of thissite chosen randomly between ( − W/ , W/
2) with uniformprobability. The double sum runs over nearest neighbors.The hopping matrix element t is taken equal to −
1, whichset the energy scale, and the lattice constant equal to 1,setting the length scale. All calculations with the Ander-son model are done at an energy equal to 0.01, to avoidthe center of the band.We have calculated the zero temperature conductance g from the Green functions. The conductance g is pro-portional to the transmission coefficient T between twosemi–infinite leads attached at opposite sides of the sam-ple g = 2 e h T (5)where the factor of 2 comes from the spin. ¿From now on,we will measure the conductance in units of 2 e /h . Thetransmission coefficient can be obtained from the Greenfunction, which can be calculated propagating layer bylayer with the recursive Green function method [25,26].This drastically reduces the computational effort. Insteadof inverting an L × L matrix, we just have to invert L times L × L matrices. With the iterative method we caneasily solve square samples with lateral dimension up to L = 400. We have considered ranges of disorder W equalto 13, 15 and 25, which correspond to localization lengthsof 1.12, 2.4 and 3.2, respectively, and lateral dimensions upto L = 200 for the calculation of the distribution function, . Prior, A. M. Somoza and M. Ortu˜no: Conductance distribution in two-dimensional localized systems 3 Fig. 1.
Schematic picture of the sample and the leads con-sidered: (a) narrow leads and (b) wide leads. The open cir-cles represent sites in the system and the solid dots sites inthe leads. The lines represent the hopping between sites. Thedashed lines are just a guide to the eye. which requires a huge number of independent runs to getgood statistics in the tails. For this purpose, we averageover a number of realizations larger than 6 × for eachdisorder and size. The leads serve to obtain the conductiv-ity from the transmission formula in a way well controlledtheoretically and close to the experimental situation.To consider different possible geometries, we have usedtwo types of leads: wide leads with the same width asthe lateral dimension of the samples and narrow (one-dimensional) leads. These are attached to the sample atthe centers of opposite edges, as shown in Fig 1(a). Thescheme of the wide leads is shown in Fig 1(b). In bothcases the leads are represented by the same hamiltonianas the system, Eq. (4), but without diagonal disorder. Thenarrow leads can be viewed as a simplified model of a pointcontact, while the wide leads should roughly correspondto electrodes in contact with the whole edge of the sample.We use cyclic periodic boundary conditions in the direc-tion perpendicular to the leads.The introduction of a uniform magnetic field B per-pendicular to the sample leads to complex hopping ma-trix elements. In the Landau gauge the vector potentialis A = (0 , − Bx, X direction are unchanged by the presence of thefield, while the elements in the Y direction have to be mul-tiplied by the factor exp( ± ixB ), where the sign depends on whether we are connecting a site with the upper orlower site in the same column [27].We have also studied the FSP approximation, first con-sidered by Nguyen, Spivak and Shklovskii [16] and widelystudied by Medina and Kardar [17] in 2D square samples.One can write the matrix elements of the Green functionbetween two sites a and b in terms of the locator expansion (cid:104) a | G | b (cid:105) = (cid:88) Γ (cid:89) i ∈ Γ E − (cid:15) i , (6)where the sum runs over all possible paths connecting thetwo sites a and b . In general the convergence of this seriesis very problematic, but in the strongly localized regimefor distances much larger than the localization length oneexpect that the previous sum is dominated by the FSP.Considering only directed paths is well justified in thestrongly localized regime, where the contribution of eachtrajectory is exponentially small in its length. We expectthat back-scattering paths renormalize the site energies,but they should be irrelevant in the renormalization-groupsense in the strongly localized regime. Based on this idea,the FSP approximation only considers directed paths. Inthis approximation, we will consider E = 0 and two typesof diagonal disorder: i ) (cid:15) i can only take two values W and − W , chosen at random with the same probability,which was the model originally considered in Ref. [16]; ii ) (cid:15) i = max[ | x | , x ), where x is chosen randomly in theinterval ( − W/ , W/
2) with uniform probability. The rea-son to substitute the small disorder energies, | (cid:15) i | < ± l , rather than the systemsize L . For our geometry, l = 2 L . The transmission at zeroenergy is equal to [17] T = (cid:18) tW (cid:19) l J ( l ) , (7)where the transmission amplitude J ( l ) is given by the sumover all the directed paths J ( l ) = directed (cid:88) Γ J Γ , (8)The contribution of each path, J Γ , is the product of thesigns of the disorder along the path Γ . J does not dependon W for the first type of disorder and only weakly for thesecond type, for which we take W = 10.The variance of ln g is entirely determined by J ( l ) andso it depends on l . It is convenient to quantify the mag-nitude of the fluctuations in terms of the path length l in J. Prior, A. M. Somoza and M. Ortu˜no: Conductance distribution in two-dimensional localized systems the FSP approximation. (cid:104) ln g (cid:105) is, of course, proportionalto l , but the constant of proportionality depends on thedisorder W . The assumption of directed paths facilitatesthe computational problem and makes feasible to handlesystem sizes much larger than with the Anderson hamil-tonian. The sum over the directed paths can be obtainedpropagating layer by layer the weight of all the trajectoriesin a very efficient way [17]. We have proven that for a specific FSP model the dis-tribution function of the logarithm of the conductance in2D systems is a Tracy-Widom distribution. To do so, wehave to map the problem of the conductance in stronglylocalized systems to the problem of finding the energy ofa polymer in a random environment. Each FPS path con-tributing to the conductance corresponds to a directedpolymer. The calculation of the quantum amplitude be-tween two points, given by Eq. (6), in the FSP approxi-mation is then formally similar to the calculation of thepartition function of directed polymers in a random po-tential at a very small temperature T = 1 /kβZ = (cid:88) Γ exp (cid:40) − β (cid:88) i ∈ Γ h i (cid:41) = , (cid:88) Γ (cid:89) i ∈ Γ exp {− βh i } (9) h i are random site energies and Γ runs over all possi-ble configurations of the directed polymer. We can easilymake Eqns. (6) and (9) equivalent by associating βh i withln( E − (cid:15) i ). In this case, we can map the distribution of ln g in our system to the distribution of the free energy in di-rected polymers. As in polymer physics we consider realdisorder energies h i , we then choose all the values E − (cid:15) i to be positive.The distribution function of the ground state energy H of a polymer in a disordered environment was obtained ex-actly by Johansson [21]. In his model the random site en-ergies take integer values with probabilities Pr( h i = k ) =(1 − p ) p k . The ground state energy for polymers runningbetween the origin and the point ( x, y ) is given by [21] H ( x, y ) → √ pxy + p ( x + y )1 − p (10)+ ( pxy ) / − p (cid:20) (1 + p ) + (cid:114) pxy ( x + y ) (cid:21) / χ where χ is a random variable with the TW distribution,corresponding to the distribution of the largest eigenvalueof a complex hermitian random matrix [18]. χ verifiesPr( χ < x ) = (cid:90) x ∞ f ( x (cid:48) ) dx (cid:48) = e − g ( x ) (11)where g ( x ) is the solution of the equation g (cid:48)(cid:48) ( x ) = u ( x ) , (12) which tends to zero, g ( x ) →
0, as x → ∞ . u ( x ) is theglobal positive solution of the Painlev´e II equation u (cid:48)(cid:48) = 2 u + xu (13)which tends to the Airy function, u ( x ) → Ai( x ), when x → ∞ .We can map Johansson’s result for the polymer prob-lem to obtain the distribution function of the conductancefor the localization problem by considering the FSP ap-proximation and that the disorder energies are of the form (cid:15) i = E + e βk (14)with probability Pr( k ) = (1 − p ) p k for k = 0 , , , · · · ,in the limit β → ∞ . This is a very specific model, butwe expect that their results apply in a much more gen-eral context, i.e., different disorder probabilities, as al-ready suggested by Johansson [21]. The same distribu-tion should also apply for non zero temperature, as in arenormalization-group sense temperature is an irrelevantparameter and the behavior is dominated by the zero-temperature fixed point.It is not trivial to extend the previous mapping to re-alistic problems with negative site energies, like the NSSand the Anderson model. It is then natural to study ifthe distribution function given by Eq. (11) also applies tothese models. First of all, we have studied the conductance distributionof the NSS model in order to check if it is also given by theTW function. For this model it is natural to consider ln J ,which contains all the relevant information about fluctua-tions. In Fig. 2 we plot, for several system sizes, histogramsof ln J as a function of z = (ln J − (cid:104) ln J (cid:105) ) /σ , where σ is the variance of ln J . The data are for narrow leads andeach line corresponds to a different system size, specifiedin the figure. The thick solid line is the standardized TWdistribution. We see how the results for the NSS modeltend uniformly to the TW distribution as size increases,although the convergence is relatively slow.Although the results in Fig. 2 present noticeable finitesize effects, it is clear that the TW distribution is fully con-sistent with the limiting distribution. This fact togetherwith the previous mapping permit us to suggest thatln J = αL + βL / χ (15)in the limit L → ∞ . To better guarantee that the TWdistribution is the proper limiting function, it is conve-nient to focus on the adimensional parameters of the dis-tribution, like the skewness (Sk = k /k / ) or the kurtosis(Kur = k /k ). If the previous equation is correct, thesetwo parameters must tend to the TW values as L → ∞ independently of the values of α and β . From Eq. (15) weexpect that the leading correction for the skewness, with . Prior, A. M. Somoza and M. Ortu˜no: Conductance distribution in two-dimensional localized systems 5 Fig. 2.
Histograms of ln J as a function of the standardizedvariable z for several sizes of the NSS model. The thick solidline corresponds to the TW distribution. As size increases, thedata approach the TW distribution. respect to the limiting value at L → ∞ , is proportionalto L − / . In Fig. 3 we plot the skewness as a functionof L − / for the NSS model (circles) and for our secondFSP model, with continuous disorder (triangles). The hor-izontal line corresponds to the TW value. We see how theskewness approaches the correct value and how it does soin the expected way for the two FSP models considered.Finite size effects for the NSS model are larger than forthe other FSP model considered. A similar behavior is alsoobserved for the kurtosis. It is clear from these trends thatboth models tend to the same universal distribution. Fig. 3.
Skewness vs L − / for the NSS model (black circles)and the FSP approximation with continuous disorder (greentriangles). The horizontal line corresponds to the TW value. We note that Johansson’s results imply that the con-ductivity, in our case, is likely to be dominated by themost important FSP. On the contrary, the NSS modelwas designed to maximize the interference effects and all trajectories have the same amplitude. So, considering onlythe most conducting path has no sense in this case. It isnecessary then to understand how the NSS may end upreproducing the TW distribution. In Fig. 4 we plot, for agiven sample, the value of J from the left corner to anyother point in the sample. All points in a vertical sliceare at the same hopping distance from the left corner. Foreach vertical slice the maximum value of J is plotted inblack and the minimum in white. Intermediate values areplotted on a gre y scale proportional to J . This methodgets rid of the exponential decay of J with distance. How-ever, it does not take into account that the current mustflow through the end point. Despite the fact that all singletrajectories have exactly the same weight, interference ef-fects produce that most of the current is carried throughvery few well defined paths. The interference effects play arole only at short scales, producing a kind of renormaliza-tion process, which ends up in a dominant “renormalized”path. Probably, this renormalization also explains why fi-nite size effect are larger in the NSS model than in theother FSP model considered. Fig. 4.
Plot of J from the left corner to any other pointin a given sample on a grey scale. On each vertical slice, black(white) corresponds to the maximum (minimum) conductance. Once we have shown that the conductance in the FSPapproximation follows the TW distribution, we study theAnderson model, which cannot be directly mapped to thepolymer problem. We have calculated numerically the con-ductance for the 2D Anderson model and we have obtainedits distribution function. By similarity with the FSP mod-els we first consider the narrow leads geometry. In Fig. 5we plot histograms of ln g for this model as a function of χ = (ln g − A ) /B , where A and B are chosen in orderto have the same mean and variance as the theoretical J. Prior, A. M. Somoza and M. Ortu˜no: Conductance distribution in two-dimensional localized systems distribution, the TW distribution f ( χ ). The data arefor several sizes and ranges of the disorder, W = 25 and L = 100 (squares), W = 13 and L = 200 (dots), W = 15and L = 150 (triangles) and W = 9 and L = 70 (con-nected circles). The solid line corresponds to f ( χ ). Allthe cases in the strongly localized regime, represented byunconnected solid symbols, are fitted fairly well by f ( χ ).The agreement extends over more than four orders of mag-nitude. The connected empty circles correspond to a sys-tem near the crossover regime, whose distribution is closeto a log-normal, represented by a dashed line in Fig. 5. Fornarrow leads, f ( χ ) fits the data better than the gaussianfunction for L/ξ (cid:38)
Fig. 5.
Histograms of ln g versus the scaled variable χ forseveral sizes and disorders of the Anderson model with narrowleads. The continuous line corresponds to the TW distributionand the dashed line is a gaussian with the same mean andvariance as the TW distribution. Considering the size dependence of the mean and thevariance of ln g and the excellent agreement between ourdata and the TW distribution, we conclude that in thestrongly localized regime with narrow leadsln g = − Lξ + β (cid:18) Lξ (cid:19) / χ (16)where β is a new constant and χ a random variable withthe TW distribution. In the SPS regime β is a constant,independent of the disorder, the system size or the Fermienergy. We found, from present results and previous calcu-lations on the behavior of the variance, that it is approx-imately equal 3 . β . The data set in Fig. 5 for W = 25 isoutside the SPS regime, since it corresponds to a localiza-tion length of the order of the lattice spacing ( ξ = 0 . β is3.5 in this case. Table 1.
Mean, variance, skewness, and kurtosis for the dis-tributions f and f . f f Mean -1.77109 0Variance 0.81320 1.15039Skewness 0.2241 0.35941Kurtosis 0.09345 0.28916
In table I we present the characteristic parameters ofthe distribution of f ( χ ). The fact that it has a meandifferent from zero, (cid:104) χ (cid:105) = − . (cid:104) ln g (cid:105) proportional to L / , already observed byus [12] in the Anderson model with narrow leads. This isof practical interest for the calculation of the localizationlength. Neglecting this term can cause errors of the orderof 20 % in the estimate of the localization length.We did not find any L / contribution for wide leads,which constitutes a strong indication that the conduc-tance distribution may depend on the leads, even in thestrongly localized regime. The type of leads may changethe universality class. We can expect a situation similarto polynuclear growth models [22], where the height dis-tribution was found to depend on the initial conditions.Our problem with narrow leads is directly related to thedroplet model in Ref. [22], which starts from an initialpreferential point. With wide leads we have translationalinvariance and all the initial (and final) points are equiva-lent, a problem similar to stationary growth. In this case,the height fluctuations are described by the function f whose accumulated distribution is [22] (cid:90) x ∞ f ( x (cid:48) ) dx (cid:48) = [1 − ( x + 2 f (cid:48)(cid:48) + 2 g (cid:48)(cid:48) ) g (cid:48) ] e − ( g +2 f ) (17)where f ( x ) is the solution of the equation f (cid:48) ( x ) = − u ( x ) , (18)which tends to zero when x → ∞ . g ( x ) and u ( x ) aregiven by Eqs. (12) and (13), respectively. The character-istic parameters of the distribution f are given in tableI. As required by our previous finding that there were no L / contributions to (cid:104) ln g (cid:105) , f has zero mean. In Fig 6 weshow the histograms of ln g for the Anderson model withwide leads for several disorders and sizes, W = 13 and L = 100 (squares), W = 13 and L = 200 (dots), W = 15and L = 100 (triangles) and W = 10 and L = 50 (con-nected circles). The solid symbols correspond to systemsin the strongly localized regime, while connected emptysymbols to a case near the crossover. We plot in terms of χ = (ln g − A ) /B , where A and B are chosen in order tohave the same mean and variance as the distribution f .The full line corresponds to f and the dashed line to agaussian with the same mean and variance. The agreementbetween the numerical data and the theoretical distribu-tion is again excellent, showing that ln g satisfies in thiscase ln g = − Lξ + β (cid:18) Lξ (cid:19) / χ (19) . Prior, A. M. Somoza and M. Ortu˜no: Conductance distribution in two-dimensional localized systems 7 where β is a new constant and χ a new random variablewith distribution f ( x ). In the SPS regime, β is a univer-sal constant approximately equal to 2.2. We note that, inthe data for W = 10 and L = 50, the right tail of thedistribution is cut at χ ≈
4. This is due to the existenceof a kink in the distribution for a transmission equal toone. For wide leads the distribution f ( χ ) fits the datawell when L/ξ (cid:38) Fig. 6.
Histograms of ln g versus the scaled variable χ forseveral sizes and disorders of the Anderson model with wideleads. The continuous line corresponds to f and the dashedline is a gaussian with the same mean and variance as f . Our results support the idea that the Anderson modeland its FSP approximation in 2D systems in the stronglylocalized regime will verify an equation of the form (16)or (19) for any range of parameters, type of disorder, ge-ometry or boundary conditions. The distribution of therandom variable will depend on boundary conditions, aswe have shown for two particular cases, narrow and wideleads. It becomes clear that the definition of a “univer-sality class” requires to take into account the boundaryconditions. A detailed study of these was done in the con-text of a PNG model [22]. By analogy with this model, wethink that the concept of universality class remains validas it seems that there exist only a finite set of universaldistributions. It is difficult to know the complete set ofuniversal distributions and to which of them will tend ageneral complex type of lead.Despite the existence of several universality classes,there are some results which are very robust, like the lo-calization length and the exponent of 1 / − ln f j ( x ) = d j x / x → ∞ (20)with d = 2 and d = 4. Eq. (20) is valid for the highconductance tail only. The other tail might be more sensi- tive to boundary conditions, although both distributions f and f behave as [22] − ln f j ( x ) = | x |
12 for x → −∞ . (21)As we have mentioned, we consider L/ξ (cid:38)
L/ξ ≈ g ≈ g exp( −
12) where g is close to one in units of 2 e /h . Our results can alsobe verified experimentally through the behaviour of thecumulants of the distribution. Eqs. (7) and (9) predictuniversal values for the skewness, kurtosis, etc, of the dis-tribution, given in table I. This limiting values can be ob-tained from a measurement of the cumulants in any rangeof parameter in the localized region, since Eq.(6) it is fairlywell verified even near the crossover. From the tendency ofthe second and third cumulants, for example, it is possibleto derive the asymptotic value for the skewness, which forwide leads should be A /A / = 0 .
359 (see table I).
We have studied the distribution function of ln g in thelocalized regime of the Anderson model at zero tempera-ture when we apply a magnetic field perpendicular to thesample. The field changes the mean, producing negativemagnetoresistance, and the variance of the distributionbut not the distribution itself when it is rescale in termsof the variable χ = (ln g − A ) /C , where A and C are againchosen in order to have the same mean and variance as thetheoretical distributions f and f . We found that the dis-tribution functions for both narrow and wide leads are thesame as without a magnetic field, Eqs. (16) and (19).In Fig. 7 we plot the distribution function of ln g fortwo values of the disorder W = 15 and W = 25, twosystem sizes and several values of the magnetic field B (see the legend in the figure). We have considered thetwo types of leads used before: solid symbols correspondto wide leads and empty symbols to narrow leads. Thesolid lines are our theoretical distributions f ( χ ) (nar-row leads, left curve) and f ( χ ) (wide leads, right curve).We can see that all the points are fitted pretty well bythe distributions f and f , respectively, as in the absenceof a magnetic field. We note that we have used periodicboundary conditions through out the paper, including thissection.The results indicate that the magnetic field does notchange the percolative nature of the conduction in thestrongly localized regime.The value of β in Eqs. (16) and (19) changes very littlewith the applied field, less than our uncertainty in themeasurements.We have also checked that the application of a mag-netic field in the NSS model with narrow leads produces J. Prior, A. M. Somoza and M. Ortu˜no: Conductance distribution in two-dimensional localized systems
Fig. 7.
Histogram of ln g for the Anderson model with anapplied magnetic field for narrow (empty symbols) and wide(solid symbols) leads for the values of the disorder, system sizeand magnetic field given in the figure. The continuous linescorrespond to f ( χ ) and f ( χ ). negative magnetoresistance, but does not change the dis-tribution function. To be quantitative, we have plottedin Fig. 8 the skewness as a function of L − / for severalvalues of the magnetic field, detailed in the legend of thefigure. The horizontal line corresponds to the skewness ofthe distribution f and the lower curve to the situationin the absence of a magnetic field. The skewness clearlytends to the TW value, as the system size tends to infin-ity, for all cases considered. We note that in the presenceof a field the convergence is faster than in its absence. Itis also interesting to note the dependence of the slopes inFig. 8 with magnetic field. They increase with the field,while the value in the absence of a field is larger than inthe presence of it. Fig. 8.
Skewness as a function of L − / for several values ofthe magnetic field in the NSS model with narrow leads. Thehorizontal line corresponds to the TW value. Present results confirm our previous belief that, in thestrongly localized regime, directed path models are in thesame universality class as the Anderson model [14,16,17].While the NSS model pretended to maximize interferenceeffects, Johansson’s model only considers the most impor-tant path. The agreement between both models indicatesthat it is percolation and not interference the dominanteffect in this regime. We expect that the main effect of in-terference between different paths is a renormalization ofthe disorder energies. This information may be relevant todeal with interacting systems, since for many propertiesFSP models are a good approximation to the Andersonmodel and they can be extended to many-particle systems,whose direct simulation may be feasible for relatively largesystems.The distribution functions of the conductance in theFPS approximation and the Anderson model appear inmany other problems. Our results are fully consistent withthe strong version of the SPS [29] if we take into accountthat boundary conditions (leads) are not irrelevant vari-ables in the renormalization group sense, and may changethe universality class. We have checked that the skewnessand the kurtosis obtained numerically for narrow leadstend to the theoretical predictions. We finally showed thatthe conductance distribution does not change when we ap-ply a magnetic field.The situation in 3D systems is more complex, sinceanalytical approaches valid for 2D systems do not apply.There are no hints of distribution functions for any similartype of problem. Nevertheless, the quality of the fit in 2Dsystems suggests that numerical simulations will provideuseful information.
Acknowledgements
The authors would like to acknowledge financial supportfrom the Spanish DGI, project FIS2006-11126, and Fun-dacin Seneca, projects 03105/PI/05 and 05570/PD/07 (JP).
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