Statistical properties of the localization measure in a finite-dimensional model of the quantum kicked rotator
aa r X i v : . [ n li n . C D ] M a r Statistical properties of the localization measure in a finite-dimensional model of thequantum kicked rotator
Thanos Manos ∗ CAMTP - Center for Applied Mathematics and Theoretical Physics,University of Maribor, Krekova 2, SI-2000 Maribor, SloveniaSchool of Applied Sciences, University of Nova Gorica, Vipavska 11c, SI-5270 Ajdovˇsˇcina, Slovenia andInstitute of Neuroscience and Medicine Neuromodulation (INM-7),Research Center J¨ulich, D-52425 J¨ulich, Germany
Marko Robnik † CAMTP - Center for Applied Mathematics and Theoretical Physics,University of Maribor, Krekova 2, SI-2000 Maribor, Slovenia (Dated: July 2, 2018)We study the quantum kicked rotator in the classically fully chaotic regime K = 10 and for variousvalues of the quantum parameter k using Izrailev’s N -dimensional model for various N ≤ N → ∞ tends to the exact quantized kicked rotator. By numerically calculating theeigenfunctions in the basis of the angular momentum we find that the localization length L for fixedparameter values has a certain distribution, in fact its inverse is Gaussian distributed, in analogyand in connection with the distribution of finite time Lyapunov exponents of Hamilton systems.However, unlike the case of the finite time Lyapunov exponents, this distribution is found to beindependent of N , and thus survives the limit N = ∞ . This is different from the tight-bindingmodel of Anderson localization. The reason is that the finite bandwidth approximation of theunderlying Hamilton dynamical system in the Shepelyansky picture (D.L. Shepelyansky, Phys. Rev.Lett. , 677 (1986)) does not apply rigorously. This observation explains the strong fluctuationsin the scaling laws of the kicked rotator, such as e.g. the entropy localization measure as a functionof the scaling parameter Λ = L /N , where L is the theoretical value of the localization length inthe semiclassical approximation. These results call for a more refined theory of the localizationlength in the quantum kicked rotator and in similar Floquet systems, where we must predict notonly the mean value of the inverse of the localization length L but also its (Gaussian) distribution,in particular the variance. In order to complete our studies we numerically analyze the relatedbehavior of finite time Lyapunov exponents in the standard map and of the 2 × Phys. Rev. E ,062905 (2013)). PACS numbers: 05.45.Mt,05.45.Ac,05.60.Cd
I. INTRODUCTION
Time-periodic (Floquet) quantum systems, whose clas-sical analog is fully chaotic and diffusive, typically exhibitdynamical localization [1, 2], if a certain semiclassicalcondition is satisfied, as explained below. We study theperiodically kicked rotator in the classically fully chaoticregime K = 10 using Izrailev’s N -dimensional model [3–6] for various N ≤ N → ∞ tendsto the quantized kicked rotator. We restrict our analy-sis to the case K = 10 because this is empirically themost typical uniformly chaotic regime, apparently free ofany islands of stability or acceleration modes [7]. Dueto the finiteness of N the observed (dimensionless) local-ization length of the eigenfunctions in the space of theangular momentum quantum number does not possess asharply defined value, but has a certain distribution in-stead. Its reciprocal value is almost Gaussian distributed. ∗ [email protected] † [email protected]
This might be expected on the analogy with the finitetime Lyapunov exponents in the Hamiltonian dynamicalsystems. In order to corroborate the theoretical findingson this topics we perform in Secs. IV and VI the nu-merical analysis of the finite time Lyapunov exponentsin the standard map (classical kicked rotator), especiallythe decay of the variance. Indeed, in the Shepelyanskypicture [8] the localization length can be obtained as theinverse of the smallest positive Lyapunov exponent of afinite 2 k -dimensional Hamilton system associated withthe band matrix representation of the quantum kickedrotator, where k is the quantum kick parameter (to beprecisely defined below). In this picture, N plays therole of time. However, unlike the chaotic classical mapsor products of transfer matrices in the Anderson tight-binding approximation, where the mean value of the fi-nite time Lyapunov exponents is usually equal to theirasymptotical value of infinite time and the variance de-creases inversely with time, as we also carefully checked(see Secs. V and VI), here the distribution is found to beindependent of N : It has a nonzero variance even in thelimit N → ∞ . The reason is that the quantum kickedrotator at N = ∞ cannot be exactly modeled with finitebandwidth (equal to 2 k ) band matrices, but only approx-imately, such that the underlying Hamilton system ofthe Shepelyansky picture has a growing dimension with N , implying asymptotically an infinite set of Lyapunovexponents and behavior different from the finite dimen-sional Hamiltonian systems. The observation of the dis-tribution of the localization length around its mean valuewith finite variance also explains the strong fluctuationsin the scaling laws of the kicked rotator, such as e.g. theentropy localization measure as a function of the theoret-ical scaling parameter Λ, to be discussed below. On theother hand, the two different empirical localization mea-sures, namely the mean localization length as extracteddirectly from the exponentially localized eigenfunctionsand the measure based on the information entropy ofthe eigenstates, are perfectly well linearly connected andthus equivalent. Therefore these results call for a refinedtheory of the localization length in the quantum kickedrotator and similar systems, where we must predict notonly the mean value of the inverse localization length butalso its (Gaussian) distribution, in particular the vari-ance. This paper is a follow-up paper of our recent work[9] (Manos and Robnik 2013).The time-independent and time-periodic systems havemuch in common when discussing the localization prop-erties of the chaotic eigenstates and of the correspondingenergy spectra. The main result of stationary quantumchaos (or wave chaos) [1, 2, 10] is the discovery that inclassically fully chaotic, ergodic, autonomous Hamiltonsystems with the purely discrete spectrum the fluctua-tions of the energy spectrum around its mean behaviorobey the statistical laws described by the Gaussian Ran-dom Matrix Theory (RMT) [11, 12], provided that weare in the sufficiently deep semiclassical limit. The lattersemiclassical condition means that all relevant classicaltransport times are smaller than the so-called Heisen-berg time, or break time, given by t H = 2 π ~ / ∆ E , where h = 2 π ~ is the Planck constant and ∆ E is the meanenergy level spacing, such that the mean energy leveldensity is ρ ( E ) = 1 / ∆ E . This statement is known asthe Bohigas - Giannoni - Schmit (BGS) conjecture andgoes back to their pioneering paper in 1984 [13], althoughsome preliminary ideas were published in [14]. Since∆ E ∝ ~ f , where f is the number of degrees of freedom(= the dimension of the configuration space), we see thatfor sufficiently small ~ the stated condition will always besatisfied. Alternatively, fixing the ~ , we can go to highenergies such that the classical transport times becomesmaller than t H . The role of the antiunitary symme-tries that classify the statistics in terms of GOE, GUE orGSE (ensembles of RMT) has been elucidated in [15], seealso [16], and [1, 2, 10, 11]. The theoretical foundationfor the BGS conjecture has been initiated first by Berry[17], and later further developed by Richter and Sieber[18], arriving finally in the almost-final proof proposedby the group of F. Haake [19–22].Here it must be emphasized again that considering the chaotic eigenstates and their dynamical localiza-tion properties there are strong analogies between thetime-periodic systems (like the kicked rotator) and time-independent systems (like static billiards) [23], where theBrody distribution [24, 25] plays a key role, as discussedin [9, 26–29].The paper is organized as follows: In Sec. II we de-fine the model, in Sec. III we present the evidence forand the description of the distribution of the localizationmeasures, in Sec. IV we study the finite time Lyapunovexponents of the classical standard mapping as a genericexample of a chaotic area preserving mapping, in Sec. Vwe study the finite time Lyapunov exponents of the prod-uct of two-dimensional random symplectic matrices de-scribing the tight-binding model of Anderson localiza-tion, in Sec. VI we present the high precision numericalresults about the decay of the variance of the distribu-tion of the finite Lyapunov exponents of Sec. IV and V,and in Sec. VII we conclude and discuss the results inthe broader theoretical perspective. II. THE KICKED ROTATOR, THE IZRAILEVMODEL AND THE DYNAMICALLOCALIZATION
The kicked rotator was introduced by Casati, Chirikov,Ford and Izrailev in 1979 [30]. Here we follow our nota-tion [9]. The Hamiltonian function is H = p I + V δ T ( t ) cos θ. (1)Here p is the (angular) momentum, I the moment ofinertia, V is the strength of the periodic kicking, θ ∈ [0 , π ) is the (canonically conjugate, rotation) angle, and δ T ( t ) is the periodic Dirac delta function with period T . Between the kicks the rotation is free, and thus thedynamics can be reduced to the standard mapping, p n +1 = p n + V sin θ n +1 , θ n +1 = θ n + TI p n , (2)as introduced in [31–33]. The quantities ( θ n , p n ) referto their values just immediately after the n -th kick. Byusing new dimensionless momentum P n = p n T /I , we get P n +1 = P n + K sin θ n +1 , θ n +1 = θ n + P n , (3)where the system has now a single classical dimensionless control parameter K = V T /I .The quantum kicked rotator (QKR) is the quantizedversion of Eq. (1), namelyˆ H = − ~ I ∂ ∂θ + V δ T ( t ) cos θ. (4)The Floquet operator ˆ F acting on the wavefunctions(probability amplitudes) ψ ( θ ), θ ∈ [0 , π ), upon each pe-riod (of length T ) can be written as (see e.g. [1], Chapter4) ˆ F = exp (cid:18) − iV ~ cos θ (cid:19) exp (cid:18) − i ~ T I ∂ ∂θ (cid:19) , (5)where now we have two dimensionless quantum controlparameters k = V ~ , τ = ~ TI , (6)which satisfy the relationship K = kτ = V T /I , K beingthe classical dimensionless control parameter of Eq. (3).By using the angular momentum eigenfunctions h θ | n i = a n ( θ ) = 1 √ π exp( i n θ ) , (7)where n is any integer, we find the matrix elements of ˆ F ,namely F m n = h m | ˆ F | n i = exp (cid:18) − iτ n (cid:19) i n − m J n − m ( k ) , (8)where J ν ( k ) is the ν -th order Bessel function. For awavefunction ψ ( θ ) we shall denote its angular momen-tum component (Fourier component) by u n = h n | ψ i = Z π a ∗ n ( θ ) ψ ( θ ) dθ == 1 √ π Z π ψ ( θ ) exp( − inθ ) dθ. (9)The QKR has very complex dynamics and spectral prop-erties. As the phase space is infinite (cylinder), p ∈ ( −∞ , + ∞ ) , θ ∈ [0 , π ), the spectrum of the eigenphasesof ˆ F , denoted by φ n , or the associated quasienergies ~ ω n = ~ φ n /T , introduced by Zeldovich [34], can be con-tinuous, or discrete [35–38].The asymptotic localized eigenstates are exponentiallylocalized . The (dimensionless) theoretical localizationlength in the space of the angular momentum quantumnumbers is given below, and is equal (after introducingsome numerical correction factor α µ ) to the dimension-less localization time t loc [Eq. (12), given below]. Wedenote it unlike in reference [6] and [9] by L . Therefore,an exponentially localized eigenfunction centered at m inthe angular momentum space [Eq. (7)] has the followingform | u n | ≈ L exp (cid:18) − | m − n |L (cid:19) , (10)where u n is the probability amplitude [Eq. (9)] of the lo-calized wavefunction ψ ( θ ). The argument leading to t loc in Eq. (12) given below originates from the observationof the dynamical localization by Casati et al [30], and inparticular from [39], and is well explained in [1], in case of normal diffusion, whilst for general anomalous diffusionwe gave a theoretical argument in [9]. We shall denote σ = 2 / L , and will later on determine the σ ’s directlyfrom the individual numerically calculated eigenstate.The question arises, where do we see the phenom-ena (spectral statistics, namely Brody-like level spacingdistribution) analogous in the quantum chaos of time-independent bound systems with discrete spectrum? Tosee these effects the system must have effectively finitedimension, because in the infinite dimensional case wesimply observe Poissonian statistics. Truncation of theinfinite matrix F mn in Eq. (8) in tour de force is notacceptable, even in the technical case of numerical com-putations, since after truncation the Floquet operator isno longer unitary.The only way to obtain a quantum system which shallin this sense correspond to the classical dynamical sys-tem [Eqs. (1), (2) and (3)] is to introduce a finite N -dimensional matrix, which is symmetric unitary, andwhich in the limit N → ∞ becomes the infinite dimen-sional system with the Floquet operator [Eq. (5)]. Thesemiclassical limit is k → ∞ and τ →
0, such that K = kτ = constant. As it is well known [6], for thereasons discussed above, the system behaves very simi-larly for rational and irrational values of τ / (4 π ). Such a N -dimensional model [40] will be introduced below.The generalized diffusion process of the standard map(3) is defined by h (∆ P ) i = D µ ( K ) n µ , (11)where n is the number of iterations (kicks), and the ex-ponent µ is in the interval [0 , P , θ and K are dimensionless. Here D µ ( K ) is the general-ized classical diffusion constant . The averaging h . i is over an ensemble of initial conditions with fixed P ,specifically in our case P = 0. In case µ = 1 we havethe normal diffusion, and D ( K ) is then the normal dif-fusion constant, whilst in case of anomalous diffusion weobserve subdiffusion when 0 < µ <
1, or superdiffusionif 1 < µ ≤
2. In case µ = 2 we have the ballistic trans-port which is associated with the presence of acceleratormodes (see below).Following [9] we find that the dimensionless Heisen-berg time, also called break time or localization time,denoted by t loc , in units of kicking period T , is equal tothe dimensionless localization length LL ≈ t loc = (cid:18) α µ D µ ( K ) τ (cid:19) − µ . (12)where α µ is a numerical constant to be determined em-pirically, and in case of normal diffusion µ = 1 is close to1 / µ = 1, considered in thepresent paper, the theoretical value of D ( K ) is given inthe literature, e.g. in [6] or [41], D ( K ) = ( K [1 − J ( K ) (1 − J ( K ))] , if K ≥ . . K − K cr ) , if K cr < K ≤ . , (13)where K cr ≃ . J ( K ) is the Bessel func-tion. Here we neglect higher terms of order K − . Inthe present paper we shall consider exclusively the case K = 10, which has been carefully checked to be fullychaotic, without any regular islands, and well describedby the normal diffusion µ = 1, so that the above formulaapplies very well [7].The motion of the QKR [Eq. (4)] after one period T ofthe ψ wavefunction can be described also by the followingsymmetrized Floquet mapping, describing the evolutionof the kicked rotator from the middle of a free rotationover a kick to the middle of the next free rotation, asfollows ψ ( θ, t + T ) = ˆ U ψ ( θ, t ) , (14)ˆ U = exp (cid:18) i T ~ I ∂ ∂θ (cid:19) exp (cid:18) − i V ~ cos θ (cid:19) exp (cid:18) i T ~ I ∂ ∂θ (cid:19) . Thus, the ψ ( θ, t ) function is determined in the middle ofthe rotation, between two successive kicks. The evolutionoperator ˆ U of the system corresponds to one period.In the case K ≡ kτ ≫ K = 10 certainly without anyregular islands of stability, and also there are no acceler-ator modes, so that the diffusion is normal ( µ = 1). Wehave carefully checked that the case K = 10 is the clos-est to the normal diffusion µ = 1 for all K ∈ [0 , k → ∞ , τ → K = const. We shall consider theregimes on the interval 3 ≤ k ≤
20, but will concentratemostly on the semiclassical regime k ≥ K , where τ ≤ N of levels [3–6, 40], which we refer to asIzrailev model u n ( t + T ) = N X m =1 U nm u m ( t ) , n, m = 1 , , ..., N . (15)The finite symmetric unitary matrix U nm determines theevolution of an N -dimensional vector, namely the Fouriertransform u n ( t ) of ψ ( θ, t ), and is composed in the follow-ing way U nm = X n ′ m ′ G nm ′ B n ′ m ′ G n ′ m , (16)where G ll ′ = exp (cid:0) iτ l / (cid:1) δ ll ′ is a diagonal matrix corre-sponding to free rotation during a half period T /
2, and the matrix B n ′ m ′ describing the one kick has the follow-ing form B n ′ m ′ = 12 N + 1 × N +1 X l =1 (cid:26) cos (cid:20) ( n ′ − m ′ ) 2 πl N + 1 (cid:21) − cos (cid:20) ( n ′ + m ′ ) 2 πl N + 1 (cid:21)(cid:27) × exp (cid:20) − ik cos (cid:18) πl N + 1 (cid:19)(cid:21) . (17)The Izrailev model in Eqs. (15-17) with a finite numberof states is considered as the quantum analogue of theclassical standard mapping on the torus with closed mo-mentum p and phase θ , where U nm describes only theodd states of the systems, i.e. ψ ( θ ) = − ψ ( − θ ), pro-vided we have the case of the quantum resonance, namely τ = 4 πr/ (2 N + 1), where r is a positive integer. The ma-trix (17) is obtained by starting the derivation from theodd-parity basis of sin( nθ ) rather than the general angu-lar momentum basis exp( inθ ).Nevertheless, we shall use this model for any value of τ and k , as a model which in the resonant and in thegeneric case (irrational τ / (4 π )) corresponds to the clas-sical kicked rotator, and in the limit N → ∞ approachesthe infinite dimensional model [Eq. (14)], restricted tothe symmetry class of the odd eigenfunctions. It is ofcourse just one of the possible discrete approximationsto the continuous infinite dimensional model.The difference of behavior between the generic caseand the quantum resonance shows up only at very largetimes, which grow fast with (2 N + 1), as explained in[9]. It turns out that also the eigenfunctions and thespectra of the eigenphases at finite dimension N of thematrices that we consider do not show any significantdifferences in structural behavior for the rational or ir-rational τ / (4 π ), which we have carefully checked. In-deed, although the eigenfunctions and the spectrum ofthe eigenphases exhibit sensitive dependence on the pa-rameters τ and k , their statistical properties are stableagainst the small changes of τ and k . This is an advan-tage, as instead of using very large single matrices forthe statistical analysis, we can take a large ensemble ofsmaller matrices for values of τ and k around some cen-tral value of τ = τ and k = k , which greatly facilitatesthe numerical calculations and improves the statisticalsignificance of our empirical results. Therefore our ap-proach is physically meaningful. Similar approach wasundertaken by Izrailev (see [6] and references therein).In Fig. 1 of paper [9] we show the examples of stronglyexponentially localized eigenstates by plotting the natu-ral logarithm of the probabilities w n = | u n | versus themomentum quantum number n , for two different matrixdimensions N . By calculating the localization length L from the slopes σ = 2 / L of these eigenfunctions usingEq. (10) we can get the first quantitative empirical local-ization measure to be discussed and used later on. Thenew finding of this paper is that σ has a distribution,which is close to the Gaussian (but cannot be exactlythat, because σ is a positive definite quantity). It doesnot depend on N and survives the limit N → ∞ . There-fore also L has a distribution whose variance does notvanish in the limit N → ∞ .Following [9] and [6] we introduce another measure oflocalization. For each N -dimensional eigenvector of thematrix U nm the information entropy is H N ( u , ..., u N ) = − N X n =1 w n ln w n , (18)where w n = | u n | , and P n | u n | = 1. We denote H GOEN = ψ (cid:18) N + 1 (cid:19) − ψ (cid:18) (cid:19) ≃ ln (cid:18) N a (cid:19) + O (1 /N ) , (19)where a = − γ ) ≈ .
96, while ψ is the digamma func-tion and γ the Euler constant ( ≃ . ... ). We thusdefine the entropy localization length l H as l H = N exp (cid:0) H N − H GOEN (cid:1) . (20)Indeed, for entirely extended eigenstates l H = N . Thus, l H can be calculated for every eigenstate individually.However, all eigenstates, while being quite different indetail, are exponentially localized, and thus statisticallyvery similar. Therefore, in order to minimize the fluctu-ations one uses the mean localization length d ≡ h l H i ,which is computed by averaging the entropy over alleigenvectors of the same matrix (or even over an ensem-ble of similar matrices of the same N but nearby k ) d ≡ h l H i = N exp (cid:0) h H N i − H GOEN (cid:1) . (21)The localization parameter β loc is then defined as β loc = dN ≡ h l H i N . (22)The parameter that determines the transition from weakto strong quantum chaos is neither the strength param-eter k nor the localization length L , but the ratio of thelocalization length L to the size N of the system in mo-mentum p Λ = L N = 1 N (cid:18) α µ D µ ( K ) τ (cid:19) − µ , (23)where L ≈ t loc , the theoretical localization lengthEq. (12), was derived in [9]. Λ is the scaling parameterof the system. The relationship of Λ to β loc is discussedin section VII of [9]. III. THE DISTRIBUTION OF THELOCALIZATION LENGTH AND OTHERLOCALIZATION MEASURES
In this section we present the main results of the paper.First we demonstrate that the localization measures 2 /σ / < l H > < s > f(x)=0.28x+0.002 FIG. 1. We show h σ i versus 2 / h l H i for matrices of dimension N = 3000, for 7 nearby values of k , namely k = k ± jδk ,where j = 0 , , , δk = 0 . k = 3 , , , . . . , and l H are very well defined, linearly related and thusequivalent. In Fig. 1 we show this in the diagram of themean h σ i versus 2 / h l H i , where both averagings are overall eigenfunctions for matrices of dimension N = 3000,for 7 nearby values of k around k , namely k = k ± jδk , where j = 0 , , , δk = 0 . k =3 , , , . . . , L in Eq. (12) and the mean value of the empirical2 / h σ i for k = 3 , , , ...,
19. It is clearly seen in Fig. 2(a)that there are strong fluctuations which we attribute tothe fact that 2 /σ has a certain distribution with nonva-nishing variance, to be presented and described below,and that the theory of L is too simple, as it correspondsonly roughly to the value of 2 / h σ i . On the other hand,in Fig. 2 (b) we see again that the two empirical local-ization measures are exactly linearly related. We shouldmention that in the cases of larger k >
19 the slopes σ areso small, and the localization too weak, that we cannotget reliable results, thus in this work we limit ourselvesto the interval 3 ≤ k ≤ K = 10 and k ) have a distributionwith nonvanishing variance, which is out of the scope ofcurrent semiclassical theories, as they do not predict thisdistribution and the corresponding variance. This findingas the central result of the present paper is demonstratedin Fig. 3. The distributions are clearly seen to be closeto a Gaussian, but cannot be exactly that as σ is alwaysa positive definite quantity. Its inverse, the localizationlength equal to 2 /σ , has a distribution whose empiricalhistograms are much further away from a Gaussian, sothat in this sense σ is the fundamental quantity. Indeed,as we will see, it corresponds to the finite time Lyapunov L s > (a) K=10 f(x)=0.798x-2.359g(x)=0.748x-2.375 / ( < s > N ) b loc (b) K=10 f(x)=0.40x-0.0018
FIG. 2. [Color online] (a) We show L versus 2 / h σ i for matricesof dimension N = 1000 (crosses and solid fit line) and formatrices of dimension N = 3000 (stars and dashed fit line), for7 nearby values of k , namely k = k ± jδk , where j = 0 , , , δk = 0 . k = 3 , , , ...,
19. (b) We plot themean value of 2 / ( N h σ i ) versus β loc for k = 3 , , , . . . , N = 3000 with k = k ± jδk ,where j = 0 , , , δk = 0 . exponent known in the theory of dynamical systems.As l H and 2 /σ are equivalent localization measures,the former one is expected also to have a distribution,which we demonstrate in the histograms of Fig. 4.We have also analyzed how the localization measuresvary in the semiclassical limit of the increasing valueof the quantum parameter k , at fixed classical param-eter K = 10. Indeed, the theoretical estimate of L inEq. (12), at fixed K , and remembering k = K/τ , showsthat approximately the mean value of the localizationlength should increase quadratically with k , or equiva-lently, the slope σ should decrease inversely quadraticallywith k . This prediction is observed, and is demonstratedin the Table I, and also in Fig. 5. It is also in agreementwith the prediction based on the tight-binding approxi-mations in reference [42] [Eq. (6)]. We give, in Table I,the mean slope σ and the standard deviation of σ , as wellas the mean value of the related quantity 2 /l H and itsstandard deviation for various k = k = 3 , , , . . . , k , namely k = k ± jδk , where j = 0 , , , δk = 0 . N = 3000. Each histogram s (a) k=5 s (b) k=9 s (c) k=13 s (d) k=17 FIG. 3. We show the histograms of the slopes σ for four sys-tems, matrices of dimension N = 3000, for each of them withseven different values of k close to k = 5 , , ,
17, namely k = k ± jδk , where j = 0 , , , δk = 0 . k = 5, (b) k = 9, (c) k = 13 and (d) k = 17. l H (a) k=10 H (b) k=10 FIG. 4. We show the histograms of l H in (a) and 2 /l H in (b) for the system k = 10 described by the matrices of dimension N = 3000. In both cases we show the Gaussian best fit. −1.7−1.6−1.5−1.4−1.3−1.2−1.1−1−0.9−0.8−0.7−0.6 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 l og ( < s > ) ( ce n t e r) log (k) (a) g(x)=-2.077x+1.038 −2.1−2−1.9−1.8−1.7−1.6−1.5−1.4−1.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 l og ( S D s ) log (k) (b) f(x)=-1.937x+0.394 FIG. 5. We show log-log plots in (a) the mean slope h σ i as a function of k , and in (b) the standard deviation of σ as a functionof k . The fitting by a straight line is only on the semiclassical interval 10 ≤ k ≤
19. In the former case the behavior is roughlyas 1 /k , in agreement with the theoretical estimate 1 /k of Eq. (12), and in the latter case also like 1 /k , surely not as thetheoretical estimate 1 /k based on the Lyapunov exponents method in the reference [42] [Eq. (9)]. for all k was fitted with the Gaussian distribution andthen the mean values and the standard deviations wereextracted. All four quantities decrease to zero with in-creasing k , meaning that in the semiclassical limit thelocalization lengths monotonically increase to infinity, sothat in this limit we have asymptotically extended states(no localization), and their standard deviation also goesto zero as 1 /k , which is different from the tight-bindingapproximations in reference [42] [Eq. (9)].Next we want to study how does the distribution ofthe localization measure σ behave as a function of thedimension N of the Izrailev model Eqs. (15-17). Since inthe limit N → ∞ the model converges to the infinitelydimensional quantum kicked rotator, we would at firstsight expect that following the Shepelyansky picture [8] σ should converge to its asymptotic value, which is sharplydefined in the sense that the variance of the distributionof σ goes to zero inversely with N . Namely, at fixed K and k Shepelyansky reduces the problem of calculat-ing the localization length to the problem of the finitetime Lyapunov exponents of the approximate underly-ing finite dimensional Hamilton system with dimension 2 k . The localization length is then found to be equal tothe inverse value of the smallest positive Lyapunov ex-ponent. In our case, the dimension of the matrices N ofthe Izrailev model plays the role of time. As it is known,and analyzed in detail in the next sections, the finite timeLyapunov exponents have a distribution, which is almostGaussian, and its variance decays to zero inversely withtime. Thus on the basis of this we would expect that thevariance of σ decays inversely with N .However, this is not what we observe. In the Table IIwe clearly see that at constant K = 10 and k = 10 themean value of σ is constant and obviously equal to itsasymptotic value of N = ∞ , while the variance of σ does not decrease with N , as 1 /N , but is constant in-stead, independent of N . This is in disagreement withthe banded-matrix models of the tight-binding approxi-mations and thus disagrees with the Eq. (9) of reference[42], and also disagrees with the Shepelyansky picture.The reason is that the associated Shepelyansky’s Hamil-ton system is only approximate construction, becausewith increasing N the matrix elements of the Floquetpropagator (matrix) outside the diagonal band of width TABLE I.
The mean value and the standard deviation of theslopes σ and /l H as a function of k = k = 3 , , , . . . , . For each k = k we used N = 7 × slopes σ (see text). All quantities decayto zero in the semiclassical limit. K = 10 – N = 7 × N = 3000 (2 /l H ) k < σ > SD σ < /l H > SD /l H The mean value and the variance of the slope σ as afunction of the matrix dimension N for a fixedsystem with K = 10 and k = 10 . Both are obviouslyconstant. K = 10 – k = 10 N h σ i var σ
500 0.102624 0.001132241000 0.101170 0.001125582000 0.100066 0.001155753000 0.102217 0.00110438 k become important, and thus the dimension of theHamilton system cannot be considered finite, constantand equal to 2 k , but increases with N . As a consequencewe have the constant value of the variance of σ , and thusconstant variance of the localization length L = 2 /σ , andtherefore the localization length has a distribution withnonvanishing variance even in the limit N = ∞ . This isprecisely the reason why the semiclassical prediction ofthe localization length in Eq. (12) fails in detail and wefind strong fluctuations in the plot of L against the 2 /σ ofFig. 2. The proper theory of the localization length mustpredict its distribution rather than just its approximatemean value. IV. NUMERICAL STUDY OF FINITE TIMELYAPUNOV EXPONENTS FOR THECLASSICAL STANDARD MAP
Finite time Lyapunov exponents of chaotic systems isa subject of not very much intense research. Taking anensemble of uniformly distributed initial conditions of auniformly chaotic (ergodic) system (with no islands ofstability) we of course expect that for any finite timethe Lyapunov exponents will have a certain distribution.With increasing time the mean value of each of them isexpected to converge to the asymptotic Lyapunov expo-nent, and since the asymptotic Lyapunov exponent mustbe the same for all initial conditions, the distributionmust converge to the Dirac delta distribution. Some earlyresults on this topic go back to the 1980s, in the worksof Fujisaka [43], reviewed and summarized by Ott [44].Some details are not so important, as it turns out that thedistribution becomes Gaussian very fast with increasingtime, which we want to demonstrate in this section.In Fig. 6 we show the histograms of the positive finitetime Lyapunov exponent for the standard map [Eq. (3)]with K = 10, for the finite times (=number of itera-tions) t = 50 , , , ×
200 on a grid,have been taken uniformly distributed over the square2 π × π . Already at t = 50 the distribution is quite closeto a Gaussian, and this trend increases very fast. Atlonger times like t = 2000 , , , /t , as it is demonstrated and analyzed inSec. VI. V. NUMERICAL STUDY OF FINITE TIMELYAPUNOV EXPONENTS FOR THE PRODUCTOF RANDOM SYMPLECTIC 2D MATRICES
As it is well known the problem of quantum or dynam-ical localization is related to the Anderson localizationmodel, within the framework of the tight-binding approx-imation, with hopping transitions between the nearestneighbors only. This goes back to the pioneering work ofFishman, Grempel and Prange [45], as discussed in [1, 2],and also reviewed in [46]. Assuming the nonvanishingnearest neighbor interaction only and the site disorder,the governing Schr¨odinger equation is [1] a n +1 + E n a n + a n − = Ea n (24)where E is the eigenenergy of the eigenfunction, while E n is the fluctuating on-site potential, varying from siteto site, with a certain probability distribution. Thereforewe have the equation (cid:18) a n +1 a n (cid:19) = T n (cid:18) a n a n − (cid:19) (25) MLE (standard map) (a) t=50
MLE (standard map) (b) t=100
MLE (standard map) (c) t=500
MLE (standard map) (d) t=1000
FIG. 6. We show the histograms of the positive finite timeLyapunov exponents for the standard map [Eq. (3)] with K =10 and times (number of iterations) t = 50 , , , ×
200 on the square [0 , π ) × [0 , π ). In allcases we show the Gaussian best fit. where the 2 × T n is given by T n = (cid:18) E − E n −
11 0 (cid:19) (26)The determinant is equal to one, and W = E − E n isdrawn from a distribution, defined by a given model.Therefore the asymptotic properties of the eigenfunctioncoefficients a n as a function of n are determined by thebehavior of the product of the random transfer matri-ces, T = T n T n − . . . T T . Everything is determined bythe trace B = T rT . If | B | > T are real reciprocals, λ > /λ <
1. Typically λ grows exponentially with n , and M n = n − ln λ > n , has certain distribution for each finite n ,and the limit M = lim n →∞ M n exists. The latter isknown as Furstenburg theorem [47]. Thus, for genericinitial condition ( a , a ) the a n will grow exponentiallywith n and only for a special initial condition, they willdecrease exponentially with the rate M n as n → + ∞ ,but still will increase exponentially in the backward di-rection n → −∞ . There are then exactly the eigenener-gies E for which the a n decrease exponentially in bothdirections n → ±∞ . In such case then M n , the finitetime Lyapunov exponent , is precisely the inverse value ofthe localization length in the n -space. Thus, for the finitesystem n < ∞ , we shall have a certain distribution of theLyapunov exponents M n . Indeed, this is observed in ournumerical experiments shown in Fig. 7, for the box distri-bution of W , namely within the interval W ∈ [ − , +2],for four values n = 50 , , , W , and we see that the Gaussian approximationis very good, and becomes perfect for longer values of n ,such as n = 2000 , , , W by other distributions, and con-vinced ourselves that the dependence on the details ofthe distribution of W is very weak, as the distributionof the finite time Lyapunov exponents is always Gaus-sian. In Fig. 8(a) we show the histogram of the finitetime Lyapunov exponents for n = 100 with the Gaussiandistribution of W with zero mean and standard deviationequal to one.One might expect that things will be changed drasti-cally if the distribution of W is different, with divergingvariance. In Fig. 8(b) we show the result for the Cauchy-Lorentz distribution of W defined as follows P ( W ) = 1 π bW + b , (27)where b is the halfwidth at the half maximum, and wehave chosen b = 1. We have taken the values insidethe cut-off interval [ − , +2] and n = 100, and then thesame thing for the interval [ − , +100] and n = 100 inFig. 8(c). We clearly see that the distribution is alwaysGaussian.0 MLE (random matrix) (a) t=50
MLE (random matrix) (b) t=100
MLE (random matrix) (c) t=500
MLE (random matrix) (d) t=1000
FIG. 7. We show the histograms of the positive finite timeLyapunov exponents for the product of random matrices[Eq. (26)] with W = E − E n uniformly distributed in a box W ∈ [ − , +2], for n = 50 , , , MLE (random matrix − Gaussian N(0,1)) (a) t=100
MLE (random matrix − Cauchy w ˛ [−2,2] ) (b) t=100 MLE (random matrix − Cauchy w ˛ [−100,100] ) (c) t=100 FIG. 8. The histograms of the positive finite time Lyapunovexponents for the product of random matrices [Eq. (26)] at n = 100 with (a) W = E − E n Gaussian distributed withzero mean and unit variance, (b) Cauchy-Lorentz distribution[Eq. (27)] with W in the cut-off interval [ − , +2], and (c) thesame as (b) but W ∈ [ − , +100]. In all cases we show theGaussian best fit which is excellent. VI. NUMERICAL STUDY OF THE DECAY OFTHE VARIANCE OF THE DISTRIBUTION OFTHE FINITE TIME LYAPUNOV EXPONENTS
Finally, in this section we present numerical evidencefor the theoretical expectation [44] that the finite timeLyapunov exponents have approximately Gaussian dis-tribution whose variance decreases inversely with time t (the number of iterations in the case of the standard map;Sec. IV) and n in the case of the product of random ma-1 −2.6−2.4−2.2−2−1.8−1.6−1.4−1.2−1−0.8−0.6 1 1.5 2 2.5 3 3.5 4 L og ( s t anda r d d ev i a ti on ) Log (t) MLEs (standard map)f(x)=0.504*x+0.563MLEs (random matrix)g(x)=0.504*x+0.383
FIG. 9. [Color online] The standard deviation of the positivefinite time Lyapunov exponents for the standard map (stars)and for the product of random transfer matrices with thebox distribution of W (empty boxes), as a function of time inlog − log presentation, and their best fits. The slope is exactly-1/2. trices in the context of the unimodular transfer matricesof the tight-binding approximation to describe the An-derson localization, expounded in section V. Indeed, theevidence is overwhelming, as shown in Fig. 9, where weplot the standard deviation as a function of time in log-log plot, showing that it decays inversely with the squareroot of time.In the context of our Izrailev model the dimension N of the matrix plays the role of time. The width of thediagonal band is equal to 2 k . Shepelyansky reduces theproblem of the localization length to the determinationof the smallest positive Lyapunov exponent (its inverseis the localization length) of the underlying finite dimen-sional Hamilton system with dimension 2 k . Then, thefinite time Lyapunov exponent should have some almostGaussian distribution, whose mean tends to the asymp-totic Lyapunov exponent with N → ∞ and the varianceshould decrease to zero as 1 /N .If this picture were exact, then the mean localizationlength as a function of N should converge to the asymp-totic value, which we do observe in Table II of Sec. III,while the variance does not decay to zero, but ratherremains constant, independent of N . From this we con-clude that even in the limit N → ∞ the localizationlength has a certain distribution with nonvanishing vari-ance, or more precisely, its inverse (the slope σ ) has analmost Gaussian distribution with nonvanishing variance.We believe that this is the cause of the strong fluctua-tions observed for example in Fig. 2(a) of Sec. III. Thesame observation applies to the scaling laws of the Figs. 9and 10 of our previous paper [9]. VII. SUMMARY
The main conclusion of this paper is the empirical factbased on our numerical computations of the eigenfunc- tions of the N -dimensional Izrailev model, that the local-ization length has a distribution with nonvanishing vari-ance not only for finite N , but even in the limit N → ∞ .This is the reason, we believe, for the strong fluctuationsin the scaling laws which involve the empirical localiza-tion measures and the theoretical semiclassical value ofthe localization length. In the Shepelyansky picture [8]this might seem to be a contradiction, but the resolu-tion of the puzzle is that in the limit of large N thefinite dimensional Hamilton system extracted from theFloquet propagator of the quantum kicked rotator is notgood enough, and therefore the matrix elements outsidethe main diagonal band of width 2 k play a role, mak-ing the Hamilton system effectively infinite dimensional,with infinitely many Lyapunov exponents. This findingis a challenge for the improved semiclassical theory ofthe localization length, to derive and explain the discov-ered distribution function. On the other hand, the simplemodel of the Anderson localization based on the tight-binding approximation, with only the nearest neighborinteractions, described by the product of 2 × /n . Thesame conclusion applies to such a model with a finitenumber of interacting neighbors. Indeed, according tothe references [42, 48] the variance of σ should vanishas V ar ( σ ) ∝ / ( N k ), but our work shows that in thequantum kicked rotator this is not observed: the vari-ance does not depend on N , and decays with k fasterthan 1 /k , namely as 1 /k . Thus, here we found someimportant differences between the dynamical localizationin the quantum kicked rotator and the Anderson tight-binding model of localization, and the Shepelyansky pic-ture, which rest upon the banded matrix models withfinite bandwidth.To summarize: We do not have yet a theory to de-scribe this behavior, namely the theory of the distribu-tion of the localization length, including the variance,rather than just its average value, as explained in thepaper, but only the clear understanding of what is thereason for this behavior: The fact that the banded ma-trix model for the QKR is not good enough, one has totake into account also the (small but many) matrix ele-ments outside the main diagonal band, and therefore theShepelyansky picture and approximation breaks down,meaning that the finite dimensional Hamiltonian systemcannot capture the correct behaviour of the QKR. Thus,the problem is open for the future work. ACKNOWLEDGEMENTS
This work was supported by the Slovenian ResearchAgency (ARRS).2 [1] H. J. St¨ockmann,
Quantum Chaos - An Introduction (Cambridge: Cambridge University Press, 1999).[2] F. Haake,
Quantum Signatures of Chaos (Berlin:Springer, 2001).[3] F. M. Izrailev, Phys. Rev. Lett. , 541 (1986).[4] F. M. Izrailev, Phys. Lett. A , 250 (1987).[5] F. M. Izrailev, J. Phys. A: Math. Gen. , 865 (1989).[6] F. M. Izrailev, Phys. Rep. , 299 (1990).[7] T. Manos and M. Robnik, Phys. Rev. E , 022905(2014).[8] D. L. Shepelyansky, Phys. Rev. Lett. , 677 (1986).[9] T. Manos and M. Robnik, Phys. Rev. E , 062905(2013).[10] M. Robnik, Nonl. Phen. in Compl. Syst. (Minsk) , 1(1998).[11] M. L. Mehta, Random Matrices (Boston: AcademicPress, 1991).[12] T. Guhr, A. M¨uller-Groeling, and H. Weidenm¨uller,Phys. Rep. , 4 (1998).[13] O. Bohigas, M. J. Giannoni, and C. Schmit, Phys. Rev.Lett. , 1 (1984).[14] G. Casati, F. Valz-Gris, and I. Guarneri, Lett. NuovoCimento , 279 (1980).[15] M. Robnik and M. V. Berry, J. Phys. A: Math. Gen. ,669 (1986).[16] M. Robnik, Lect. Notes Phys. , 120 (1986).[17] M. V. Berry, Proc. Roy. Soc. Lond. A , 229 (1985).[18] M. Sieber and K. Richter, Phys. Scr. T90 , 128 (2001).[19] S. M¨uller, S. Heusler, P. Braun, F. Haake, and A. Alt-land, Phys. Rev. Lett. , 014103 (2004).[20] S. Heusler, S. M¨uller, P. Braun, and F. Haake, J. Phys.A:Math. Gen. , L31 (2004).[21] S. M¨uller, S. Heusler, P. Braun, F. Haake, and A. Alt-land, Phys. Rev. E , 046207 (2005).[22] S. M¨uller, S. Heusler, A. Altland, P. Braun, andF. Haake, New J. of Phys. , 103025 (2009).[23] T. Prosen, in Proceedings of the International School ofPhysics “Enrico Fermi”, Course CXLIII , edited by G.Casati and I. Guarneri and U. Smilyanski (Amsterdam:IOS Press, 2000) p. 473.[24] T. A. Brody, Lett. Nuovo Cimento , 482 (1973).[25] T. A. Brody, J. Flores, J. B. French, P. A. Mello,A. Pandey, and S. S. M. Wong, Rev. Mod. Phys. ,385 (1981). [26] B. Batisti´c and M. Robnik, J. Phys. A: Math. Gen. ,215101 (2010).[27] B. Batisti´c, T. Manos, and M. Robnik, Europhys. Lett. , 50008 (2013).[28] B. Batisti´c and M. Robnik, J. Phys. A: Math. Theor. ,315102 (2013).[29] B. Batisti´c and M. Robnik, Phys. Rev. E , 052913(2013).[30] G. Casati, B. Chirikov, J. Ford, and F. M. Izrailev, Lect.Notes Phys. , 334 (1979).[31] J. B. Taylor, Culham Laboratory Progress Report, CLM-PR-12 (1969).[32] C. Froeschl´e, Astron. Astrophys. , 15 (1970).[33] B. Chirikov, Phys. Rep. , 263 (1979).[34] Y. B. Zeldovich, Eksp. Teor. Fiz. , 1942 (1966).[35] F. M. Izrailev and D. L. Shepelyansky, Dokl. Akad. NaukSSSR , 1103 (1979).[36] F. M. Izrailev and D. L. Shepelyansky, Sov. Phys. Dokl. , 996 (1979).[37] F. M. Izrailev and D. L. Shepelyansky, Teor. Mat. Fiz. , 417 (1980).[38] F. M. Izrailev and D. L. Shepelyansky, Theor. Math.Phys. , 553 (1980).[39] B. V. Chirikov, F. M. Izrailev, and D. L. Shepelyansky,Sov. Sci. Revv. C 2 , 209 (1981).[40] F. M. Izrailev, Phys. Lett. A , 13 (1988).[41] A. J. Lichtenberg and M. A. Lieberman, Regular andChaotic Dynamics (New York: Springer Verlag, 1992).[42] T. Kottos, A. Politi, F. Izrailev, and S. Ruffo, Phys. Rev.E , R5553 (1996).[43] H. Fujisaka, Prog. Theor. Phys. , 1264 (1983).[44] E. Ott, Chaos in Dynamical Systems (Cambridge Uni-versity Press, 1993).[45] S. Fishman, D. Grempel, and R. Prange, Phys. Rev.Lett. , 509 (1982).[46] R. Prange, D. Grempel, and S. Fishman, Como Confer-ence on Quantum Chaos, G. Casati, ed. (Plenum, NewYork, 1984).[47] A. Crisanti, G. Paladin, and A. Vulpiani,
Products ofRandom Matrices in Statistical Physics (Springer-Verlag,Berlin Heidelberg, 1993).[48] T. Kottos, F. Izrailev, and A. Politi, Physica D131