Dynamical Localization and Delocalization in Polychromatically Perturbed Anderson Map
aa r X i v : . [ c ond - m a t . d i s - nn ] M a r Dynamical Localization and Delocalization in Polychromatically Perturbed AndersonMap
Hiroaki S. Yamada and Kensuke S. Ikeda Yamada Physics Research Laboratory, Aoyama 5-7-14-205, Niigata 950-2002, Japan College of Science and Engineering, Ritsumeikan University Noji-higashi 1-1-1, Kusatsu 525-8577, Japan (Dated: March 11, 2020)In the previous paper[arXiv:1911.02189], localization and delocalization phenomena in the poly-chromatically perturbed Anderson map (AM) were elucidated mainly from the viewpoint oflocalization-delocalization transition (LDT) on the increasing of the perturbation strength ǫ . In thispaper, we mainly investigate the disorder strength W − dependence of the phenomena in the AMwith a characetristic disorder strength W ∗ . In the completely localized region the W − dependenceand ǫ − dependence of the localization length show characteristic behavior similar to those reportedin monochromatically perturbed case [PRE 97,012210(2018)]. Furthermore, the obtained resultsshow that even for the increase of the W , the critical phenomena and critical exponent are foundto be similar to those in the LDT caused by the increase of ǫ . We also investigate the diffusiveproperties of the delocalized states induced by the parameters. PACS numbers: 05.45.Mt,71.23.An,72.20.Ee
I. INTRODUCTION
In recent years, localization of wave packet in quantummap systems has been extensively studied experimentally[1, 2] and theoretically [3–6]. Experimentally, Chabe etal. observed the critical phenomena of the localization-delocalization transition (LDT) for cold atoms in an opti-cal lattice, which corresponds to perturbed standard map(SM). The results are interpreted based on the equiv-alence to Anderson transition in the three-dimensionaldisordered tight-binding system [7, 8].We have proposed Anderson map (AM) that becomesthe one-dimensional Anderson model in a certain limit,and investigated the parameter dependence of the LDTin the AM with the time-quasiperiodic perturbation com-posed of M − color modes [9, 10]. Let us define W asthe disorder strength and ǫ as the perturbation strength,respectively. The characteristic ǫ − dependence and the W − dependence for the LDT in the perturbed AM wereclarified in comparison with the SM under the same per-turbation. A schematic representation of the resultingphase diagram for the LDT of the dichromatically per-turbed AM ( M = 2) is shown in Fig.1. The critical curve ǫ c ( W ) is shown as the log ǫ on the vertical axis and thelog W on the horizontal axis.In the previous paper [9, 10], we investigated the LDTwith increasing the perturbation strength ǫ along theline L or line L in the Fig.1, and obtained the pa-rameter dependence of the critical exponent of the local-ization length and critical strength ǫ c by using finite-time scaling analysis of the LDT. Roughly speaking,the W − dependence of the critical curve ǫ c ( W ) greatlychanges around W ≃ W ∗ . Table I summarizes the differ-ence between the time-evolution of the initially localizedwave packet in W < W ∗ and W > W ∗ when ǫ changesalong L and L passing the LDT points ǫ = ǫ c .The purpose of this paper is to give some new results FIG. 1: (Color online) The illustrating of the critical curve ǫ c ( W ) in the phase plane (log ǫ, log W ) for the dicchromat-ically perturbed AM ( M = 2), where W = W ∗ is shownby dotted black line. Yellow region represents the critical re-gion of LDT, gray represents the region where the localizationlength obey W − law, and light blue represents the local-ized region deviated from the law. Some typical paths fromthe localized state to the delocalized state via the LDT dueto parameter changes are represented by L n ( n = 1 , , , P n ( n = 1 , , ,
4) represents the transition points in each case,and P indicates the intersection with W ∗ when W is in-creased along L . If the vertical axis is replaced to ǫ ( M − M >
2) [9]. and data complementary to the previous paper that havenot been clearly shown yet for the localization and/ordelocalization phenomena in the polychromatically per-turbed AM ( M ≥ ǫ is small ( ǫ < ǫ c ) and W is also small. In the pa- ǫ < ǫ c Ballistic → (Diffusive ) → Localization
W < W ∗ ǫ = ǫ c Ballistic → (Diffusive ) → Subdiffusion( L ) ǫ > ǫ c Ballistic → (Diffusive ) → Diffusion ǫ < ǫ c (Ballistic) → Diffusive → Localization
W > W ∗ ǫ = ǫ c (Ballistic) → Diffusive → Subdiffusion( L ) ǫ > ǫ c (Ballistic) → Diffusive → DiffusionTABLE I: The image of time-evolution of the spread for the initially localized wave packet in the order of the arrows. Thetransition from localization to diffusion is shown divided into two regions,
W < W ∗ and W > W ∗ , corresponding the Fig.6 andFig.9, respectively. ( ... ) represents the behavior observed in the very short time domain. per [II], we have already investigated the W − dependenceof the localization length (LL) of a monochromaticallyperturbed AM. As a result, the characteristics changesaround W ≃ W ∗ . In this paper, we investigated changesin the LL depending on W and ǫ in the fully localizedregion (corresponding to the gray region in Fig.1). Forboth regions, W < W ∗ and W > W ∗ , the LL shows anexponential increase with respect to ǫ , but has the param-eter dependence two kinds of behavior depending on theregion of the W , similar to the result already reported inRef.[11]. When W < W ∗ , the LL shows the W − − decay,but for the region of W > W ∗ , it can be seen that the LLincreases as the disorder strength W increases, regardlessof the number of the modes M ( ≥ ǫ is fixed, and the disorder strength W is var-ied along the line L or line L . Let the critical disorderstrength W c . In particular, in the case of W ∗ < W c , thereis a change in the localization process (transient regionof time toward the localization) at W = W ∗ before theLDT occurs at W = W c , even if LDT finally occurs dueto the increase of W .Finally, we show the diffusive property of the delocal-ized states for ǫ > ǫ c and/or W > W c by using the diffu-sion coefficient D . The diffusion coefficient D behaves as D ∝ W − for W < W ∗ , and it makes minimum diffusioncoefficient around W ≃ W ∗ and it gradually increasestowards a constant value for W >> W ∗ regardless of M ( ≥ W − dependency for the W >> W ∗ regioncan be explained by ballistic model without localizationeven for the unperturbed case.The organization of this paper is as follows. In the nextsection, we introduce the perturbed Anderson map withquasiperiodic modulation and the Maryland transform[12]. We show the localization property of the system inthe Sect.III. The W − dependence of LDT is shown in theSect.IV. In the Sect.V, we show the diffusive property ofthe delocalized states.Also, considering the LDT along the line L1 in Fig.1,we consider the relationship between the LL in the yel-low critical region and the LL in the grey region (stronglocalization) in appendix A. In addition, we investigatedhow polychromatic perturbation affects quantum statesin ballistic model without localization in appendix B. II. MODELS
The time evolution from m th step to ( m + 1)th stepfor the wave packet | Ψ > is described by | Ψ( m + 1) > = ˆ U m | Ψ( m ) > . (1)The one-step time-evolution operator of the following An-derson map isˆ U m = e − if ( m ) W v ( q ) / ~ e − iT ( p ) / ~ , (2)where T ( p ) = 2 cos( p/ ~ ) = e − d/dq + e + d/dq is the ki-netic energy term and v ( q ) = P n δ ( q − n ) v q | q >< q | israndom on-site potential. Here ˆ p and ˆ q are momentumand position operators, respectively. The v n is uniformlydistributed over the range [ − , W denotes thedisorder strength. It is a quantum map version of theAnderson model defined on the discretized lattice q ∈ Z [18]. The quasiperiodic modulation f ( t ) is given as, f ( t ) = ǫ √ M M X j cos( ω j t ) (3)where M and ǫ are number of the frequency componentand the strength of the perturbation, respectively. Notethat the strength of the perturbation is divided by √ M so as to make the total power of the long-time average in-dependent of M , i.e. f ( t ) = 1+ ǫ /
2, and the frequencies { ω i } ( j = 1 , ..., M ) are taken as mutually incommensuratenumber of O (1).We can regard the harmonic perturbation as the dy-namical degrees of freedom. To show this we introducethe classically canonical action-angle operators ( ˆ J j = − i ~ ∂ j ∂ j φ j , φ j ) representing the harmonic perturbation asthe linear modes, and we call them the “color modes”hereafter. We consider the Hamiltonian H aut so as toinclude the color modes, H aut (ˆ p, ˆ q, { ˆ J j } , { ˆ φ j } ) = T (ˆ p ) + W v (ˆ q ) ǫ √ M M X j cos φ j δ t + M X j =1 ω j ˆ J j , (4)where δ t = P ∞ m = −∞ δ ( t − m ). One can easily check thatby Maryland transform the eigenvalue problem of thequantum map system interacting with M -color modescan be transformed into d (= M + 1)-dimensional latticeproblem with disorder. Let us consider an eigenvalueequation e − i ˆ A e − i ˆ B e − i ˆ C | u i = e − iγ | u i , (5)where ˆ A = ( W v (ˆ q ) + P Mj ω i ˆ J j ) / ~ , ˆ B = v (ˆ q ) ǫW √ M P Mj cos φ j / ~ , ˆ C = 2 cos(ˆ p/ ~ ) / ~ (6)for the time-evolution operator. γ and | u i are the quasi-eigenvalue and quasi-eigenstate. Here, if the eigenstaterepresentation of ˆ J j is used, ˆ J j | m j i = m j ~ | m j i ( m j ∈ Z ),we can obtain the following ( M + 1) − dimensional tight-binding expression by the Maryland transform [9]: D ( n, { m j } ) u ( n, { m j } ) + X n ′ , { m ′ j } h n, { m j }| ˆ t AM | n ′ , { m ′ j }i u ( n ′ , { m ′ j } ) = 0 , (7)where { m j } = ( m , ...., m M ). Here the diagonal term is D ( n, { m j } ) = tan " W v n + ~ P Mj m j ω j ~ − γ , (8)and the ˆ t AM of the off-diagonal term isˆ t AM = i e − i ǫW √ M v (ˆ q )( P Mj cos φ j ) / ~ − e i p/ ~ ) / ~ e − i ǫW √ M v (ˆ q )( P Mj cos φ j ) / ~ + e i p/ ~ ) / ~ . (9)It follows that the ( M + 1) − dimensional tight-bindingmodels of the AM have singularity of the on-site en-ergy caused by tangent function and long-range hoppingcaused by the kick δ t .If the off-diagonal term does not change qualitatively,there exists W ∗ = 2 π ~ ≃ .
78 where the effect of thethe fluctuation width of the diagonal term is saturatedfor the change of the disorder strength W . It can beseen that for W > W ∗ the diagonal fluctuation widthis saturated, and the effect of the W is effective onlyfor hopping term in the form of ǫW . Therefore, it issuggested that for W > W ∗ the phenomenon related tothe transition phenomenon can be scaled in the form of ǫW . III. LOCALIZATION PROPERTIES IN THEPOLYCHROMATICALLY PERTURBEDANDERSON MAP
We use an initial quantum state < n | Ψ( t = 0) > = δ n,N/ and monitor the spread of the wave packet by themean square displacement (MSD), m ( t ) = < Ψ( t ) | (ˆ n − N/ | Ψ( t ) > . (10) In the unperturbed cases ( ǫ = 0), it is known that the AMshows the localization in the real lattice space and the lo-calization can be retained even when the monochromaticperturbation mode is added ( ǫ = 0, M = 1). We havereported that in the monochromatically perturbed case( M = 1) the W − -dependence of the localization length(LL) is stably maintained even for ǫ = 0 and W < W ∗ ,but the LL increases with increase of W for W > W ∗ at least in the weak perturbation limit ǫ <<
1. In thecase with M ≥
2, the LDT occurs by increasing ǫ and W . However, the localization characteristics in the poly-chromatically perturbed AM with small ǫ ( < ǫ c ) have notbeen investigated yet. The purpose of this section is toshow this. A. Localization length
We compute the LL of the dynamical localization, ξ = p m ( ∞ ) for the polychromatically perturbed cases,after numerically calculating the MSD for long-time,where m ( ∞ ) is numerically saturated MSD. Figure 2shows time-dependence of the MSD for different valuesof the perturbation strength ǫ and the disorder strength W when M = 2 ,
3. It can be seen that the LL increasesas ǫ increases when the W is fixed, and the LL for thecases when the ǫ is fixed behaves somewhat complicatedbecause of the existence of the W ∗ .The W − dependence of the LL is over-plotted inFig.3(a) for M = 2 and M = 3. We devid the W − dependence into two regions, i.e., W < W ∗ and W > W ∗ , to clarify their characteristics. It follows thatfor W ≤ W ∗ , the LL shows W − − decays like the case of M = 1, and it increases with respect to W in the egion W > W ∗ .Figure 3(b) shows the result of the ǫ − dependence inthe the perturbed AM( M = 1 , ,
3) for W = 1 . > W ∗ ).It is obvious that the LL grows exponentially as the per-turbation strength ǫ increases in the all cases, i.e., ξ ∼ e cǫ , (11)where c is a growth rate. Furthermore, as shown inFig.4(a),(c), the exponential growth of the LL ξ ( ǫ ) is alsoconfirmed by changing the disorder strength W . Figures4(b) and (d) shows the plots of the (a) and (c) on the hor-izontal axis scaled as ǫ → ǫW , respectively. We can seethat it stands very well in one straight curve. As a result,regardless of the number of color modes M the parame-ter dependence of the localization length for W > W ∗ isrepresented as ξ ∼ e c ǫW . (12)This form is also the same as in the monochromaticallyperturbed AM. It follows that the LL for W < W ∗ be-haves ξ ∼ e c ǫ with W − independent coefficient c asshown in Fig.4(c). m t ε=0.005 ε=0.01 ε=0.02 ε=0.03 (a) M=2, W=1 m t ε=0.001 ε=0.002 ε=0.003 ε=0.004 (b) M=3, W=110 m t W=0.6 W=0.7 W=1.2 W=1.5 (d) M=3, ε=0.01 m t W=0.6 W=0.7 W=1.2 W=1.5 (c) M=2, ε=0.01
FIG. 2: (Color online) The plots of m ( t ) as a function of timefor different values of the ǫ and W in the polychromaticallyperturbed AM. (a) M = 2, W = 1 .
0, (b) M = 3, W = 1 . M = 2, ǫ = 0 .
01 and (d) M = 3, ǫ = 0 .
01. Note that thehorizontal axes are in the logarithmic scale.
Therefore, if ǫ <<
1, the ǫ − dependence of the LL ξ ( ǫ )of the polychromatically perturbed cases also increasesexponentially as: ξ ( ǫ, W ) ≃ (cid:26) cW − exp { c ǫ } ( W < W ∗ ) ξ ∗ exp { c W ǫ } ( W > W ∗ ) , (13)where the c and c are coefficients that increase with M .Note that for the larger ǫ , the LL ξ ( ǫ ) seems to increasemore strongly than the exponential function in the casesof M ≥ ǫ c of the LDT. See appendixA for the more details. B. Scaling of the dynamical localization
In this section, we recheck one-parameter scaling forthe transient process to the localization. For the
W
M=1, ε=0.01
M=2, ε=0.01
M=2, ε=0.02
M=3, ε=0.01 slope -2 ξ -3 ε M=1 M=2 M=3 (b)W=1.0
FIG. 3: (Color online) (a)Localization length of the polychro-matically perturbed AM as a function of disorder strength W for various values of M and ǫ . The unperturbed case ( ǫ = 0)and monochromatically perturbed case ( ǫ = 0 .
01) are alsoplotted by the dotted line with circles and filled circles, re-spectively. Note that the axes are in the logarithmic scale.(b)Localization length of some perturbed AM ( M = 1 , , ǫ with W = 1 . We use the following scaled MSD as the scaling function:Λ( t ) ≡ m ( t ) t = F (cid:18) tξ (cid:19) . (14)In this case, the fact that the localization process is scaledby the one-parameter ξ in the entire time region meansthe following asymptotic form of the scaling function F ( x ): F ( x ) ∼ ( const x → x x → ∞ . (15)This type of the scaling analysis has been performed toinvestigate LDT phenomena at the critical point for poly-chromatically perturbed disordered systems [ ? ]. Theestablishment of one-parameter scaling means that theasymptotic shape is smoothly connected by a single curveif the localization length is used even in the case of variousparameters. Figure 5(a) shows the typical result of the ξ -3 W ε (d) M=3 W=0.7 1.0 1.2 1.5 ξ -3 ε (c) M=3 W=0.7 1.0 1.2 1.5 ξ -3 ε M=2 (a)
W=0.7 1.0 1.2 1.5 W=0.5 ξ -3 W ε (b) W=0.7 1.0 1.2 1.5
M=2
FIG. 4: (Color online) (a)Localization length ξ of the dicchro-matically perturbed AM ( M = 2) as a function of perturba-tion strength ǫ for the relatively large W . (b)Plot of ξ as afunction of ǫW for the panel (a). (c)Localization length ξ ofthe dicchromatically perturbed AM ( M = 3) as a function ofperturbation strength ǫ . (d)Plot of ξ as a function of ǫW forthe panel (c). Note that the all the vertical axes are in thelogarithmic scale. localization phenomenon in (time-continuous) Andersonmodel by scaling the time with the localization length forvarious disorder strength W determined by the numericaldata of MSD. It follows that the Λ( t ) roughly overlapsfrom the ballistic to the localized region. The horizontalaxis x = t/ξ → ∞ shows a decrease of the slope − m ∼ t to m ∼ t is scaled by one-parameter when thelocalization occurs in the system.Let examine the scaling for the perturbed AM. Figure5(b) and (c) show the result of applying the same scalingto the cases with M = 1 and M = 2 with small ǫ . Thelocalized states are obtained by changing various W s, andΛ as a function of t/ξ are displayed. Since the x → ∞ sideis localized, it is obvious that it is asymptotic to Λ s ( x ) ∼ x − . Actually, the data with varying W are also plotted,but the scaling is not bad. However, if you look closely,it follows that the scaling works well for the cases for W < W ∗ , but we can see the shift to the different curvesfor W > W ∗ in the transient region to the localization.(The inset in the Fig.5(c) is an enlarged view.) In otherwords, this is an existence effect of W ∗ , and when W isincreased along the line L in Fig.1, these features occurat the point P denoted by the white circles. In addition,as shown in the Fig.5(d), it can be seen that even for -7 -6 -5 -4 -3 -2 -1 Λ ξ W=0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.6 (a) 10 -4 -3 -2 -1 Λ ξ W=0.2 0.4 0.6 0.8 2.0
M=1 ε =0.03(b)10 -4 -3 -2 -1 Λ ξ W=0.3 0.4 0.5 0.6 0.7 1.8 2.0
M=2, ε =0.01(c)0.11 1 10 -4 -3 -2 -1 Λ ξ M=0 M=1 M=2 M=3
W=0.4 ε =0.01(d) FIG. 5: (Color online) Scaled MSD Λ( t ) as a function ofthe scaled time t/ξ ( W, ǫ ), where ξ ( W, ǫ ) are determined byMSD for several parameter sets (
M, W, ǫ ). (a)Unperturbedtime-continuous Anderson model ( ǫ = 0) with several valuesof the W . (b)Monochromatically perturbed cases ( M = 1, ǫ = 0 .
03) with several values of the W . (c)Dichromaticallyperturbed cases ( M = 2, ǫ = 0 .
03) with several values ofthe W . (d)Perturbed cases ( M = 0 , , ,
3) of the case with W = 0 . ǫ = 0 .
01. Note that all axes are in logarithmicscale. different M the similar scaling curves are obtained forthe region W < W ∗ . IV. LOCALIZATION-DELOCALIZATIONTRANSITION IN THE POLYCHROMATICALLYPERTURBED AM
In the previous paper [9], we analyzed in detail LDTcaused with increasing ǫ for a fixed W along Line L inFig.1. In this section, we confirm the LDT by changingthe disorder strength W for the fixed ǫ along the L or L in the Fig.1. A. subdiffusion of LDT
We can numerically determine the critical value ǫ c and/or W c of the LDT so that the MSD becomes subd-iffusion, m ∼ t α (0 < α < . (16)It is known from numerical calculation that the diffusionindex α is determined by the number of colors M as, α ≃ M + 1 , (17)regardless of the LDT produced by changing ǫ or bychanging W . This is consistent with the prediction dueto the one-parameter scaling (OPS).Figures 6(a) and (b) show the m ( t ) divided into tworegions, W < W ∗ and W > W ∗ , respectively, in the dic-chromatically perturbed AM with ǫ = 0 .
05. This casecorresponds to L in the Fig. 1. In the case of W < W ∗ ,the MSD increases as the W decreases, but the shape ofthe curve is the same and no transition to the delocaliza-tion is seen. On the other hand, in the case of W > W ∗ ,the LDT occurs around W c ≃ . m ∼ t / at the critical point. Itmay seem strange that the localized quantum state delo-calizes as the disorder width W grows, but this is one ofthe features of the perturbed AM with the characteristicvalue W ∗ ( < W c ). Of course, it can be seen that when the W is made larger ( W >> W c ), the wavepacket spreadscloser to the normal diffusion. We can also see how thegrowth of MSD changes with the value of W . In the re-gion W < W ∗ , the ballistic increase is remarkable at thevery short-time stage ( t < W > W ∗ , afterthe ballistic growth of the short time ( t < ǫ = 0 .
05 in which W c is largerthan W ∗ ( W c > W ∗ ), the time-dependence of scaledMSD, Λ( t, W ) = m ( t ) t / is plotted for various W s. When W is increased the curve moves from top to bottom andrises again (denoted by green → red → blue) toward theLDT. In the region W < W ∗ , even if the W is increased,the MSD decreases, but there is no transition yet, andwhen W further increases from the W ≃ W ∗ at the bot-tom, the LDT occurs at W c . We can see the trumpet-shaped change of the scaling function Λ( t ) around thecritical region W ≃ W c . It should be noted that the twotypes of the decreasing curve (denoted by green and redlines, respectively) are different in the shape (quality).Accordingly, it follows that they cannot be overlapped inthe one-parameter scaling analysis.On the other hand, Fig.7(b) shows m ( t ) along the line L in the perturbed AM with ǫ = 0 .
065 with increasing W in the case of W c < W ∗ . The LDT occurs after theMSD decreases with the increase of W , but there is al-most no trumpet-shaped lower critical region. B. Scaling analysis of the LDT
Similar to the case of the LDT with increasing ǫ in theprevious paper, here the critical exponent of the local-ization length can be determined by performing finite-time scaling analysis in the critical region of the LDTwith changing W . This corresponds to the finite-size m t W=0.7 0.8 0.9 1.0 1.1 1.2 (b) ε=0.05 slope 1slope 2/310 m t W=0.2 0.3 0.4 0.6 (a) ε=0.05 slope 2
FIG. 6: (Color online) The double-logarithmic plots of m ( t )as a function of t for M = 2 with ǫ = 0 .
05 including the shorttime region. (a)
W < W ∗ , (b) W > W ∗ . scaling analysis for Anderson transition in the higher-dimensional random system [13–16].Figure 8 displays the results of the finite-time scal-ing analysis for dicchromatically perturbed AM ( M = 2)with ǫ = 0 .
05 corresponding to Fig.6. We choose thefollowing quantity as a scaling variableΛ s ( W, t ) = log (cid:20) m ( t ) t α (cid:21) . (18)For W > W c , the Λ s ( t ) increases and the wave packetdelocalizes with time. On the contrary, for W < W c ,Λ s ( t ) decreases with time and the wave packet turns tothe localization. In the vicinity of the LDT, for Λ s ( t ),OPST is assumed with localization length ξ s ( W ) as theparameter. Then, Λ s ( W, t ) can be expressed as,Λ s ( W, t ) = F ( x ) , (19)where x = | W c − W | t α/ ν . (20) Λ ( t ) t M=2, ε =0.05, W_c=0.9 (a) Λ ( t ) t M=2, ε =0.065, W_c=0.7 (b) FIG. 7: (Color online) The double-logarithmic plots of scaledΛ(
W, t ) = m ( t ) /t / as a function of time in the dicchromat-ically perturbed AM ( M = 2). (a)The case with ǫ = 0 . W along the L in the Fig.1, where the critical case W c ≃ . ǫ = 0 . W c ≃ . W, t ) can be observed inthe panel (a) because W ∗ < W c . F ( x ) is a differentiable scaling function and the α is thediffusion index. Figure 8(b) shows the plot of Λ s ( t ) asa function of W at several times t , and it can be seenthat this intersects at the critical point W c . In addition,Fig.8(c) shows the plot of s ( t ) = Λ s ( W, t ) − Λ s ( W c , t ) | W c − W | (21) ∝ t α/ ν . (22)as a function of t , and the critical exponent ν = 1 .
48 ofthe LDT is determined by best fitting this slope whichis similar result obtained in previous paper for the LDT[9].In Fig.8(a), we plot Λ s as a function of x = t α/ /ξ s ( W )for different values of W by using the obtained the crit-ical exponent ν . It is well scaled and demonstrates the validity of the OPS. The upper and lower curves repre-sent the delocalized and localized branches of the scalingfunction, respectively. The establishment of OPS showsthe equivalence of the time change and the parameterchange in the m ( t, ǫ, W ).Around the critical point of the LDT, the localizationlength ξ s is supposed to diverge as ξ s ≃ ξ | W c − W | − ν (23)at W = W c . ( ξ s depends on the number of modes M ,but the subscript M is omitted.) Λ s ( W , t ) log x (a) 1.91.81.71.61.5 l og s ( t ) Λ s ( W , t ) log W (b) FIG. 8: (Color online) The results of the critical scalinganalysis for dicchromatically perturbed AM ( M = 2) with ǫ = 0 .
05. (a)The scaled MSD Λ s ( W, t ) with α = 0 .
65 as afunction of x = ξ | W c − W | − ν t α/ ν for some values of W .(b)The same scaled MSD Λ s ( W, t ) as a function of W forsome some pick up time. The crossing point is W c ≃ . s ( t ) = log(Λ( W, t ) / Λ( W c , t )) / ( W c − W ) as a function of t . The critical exponent ν ≃ .
48 is determined by a scalingrelation Eq.(22) by the least-square fit for data in (c).
V. DELOCALIZED STATES AND NORMALDIFFUSION
In this section, we evaluate how the delocalized statesspread due to the changes in W and ǫ ( > ǫ c ). Is thistrue that if ǫ > ǫ c the delocalized states asymptoticallyapproach the normal diffusion as t → ∞ ? Little is knownabout the models of quantum systems in which normaldiffusion occurs without any stochastic fluctuation [17]. A. Delocalized states
Figure 9 shows the long-time behavior of the MSD for ǫ > ǫ c in the polychromatically perturbed AM ( M =4 , M the MSD approachesto the normal diffusion as m ∼ t when ǫ is large. It canbe expected that the normal diffusion occurs in a longertime even if ǫ is relatively small if ǫ > ǫ c because thescaling curve for various ǫ s neatly fit on one curve. Theinset is an enlarged view in a short-time domain ( t < ( ≡ t ) ) before the perturbation starts to work. Here,it can be seen that since W = 2( > W ∗ ) the perturbationworks after passing through the diffusive time domain( t < t ). m
1 100 t t slope 1 m t (a) M=4, W=2 slope 1slope 2/5 m t M=6 W=2 (b) slope 2/7slope 1
1 100 t FIG. 9: (Color online) The double-logarithmic plots of m ( t )as a function of t for different values of ǫ in the polychromat-ically perturbed AM with W = 2 including the detail shorttime region. (a) M = 4 and (b) M = 6. The insets are theenlarged view of the short-time region t < . B. W − dependence of the diffusion coefficient As shown in the Fig.6, it can be expected that m ( t )asymptotically approaches to the normal diffusion as the W ( > W c ) increases for the fixed ǫ . Indeed, Fig.10 showsthe time-dependence of the MSD by changing W in thecase of the polychromatically perturbed AM with ǫ = 0 . >> ǫ c ). As shown in the Fig.10(a), the m ( t ) behavesthe normal diffusion, m ( t ) ≃ Dt, (24)for M = 2, where D is the diffusion coefficient. As shownin the Fig.10(b), similar results are confirmed even for thecase of M = 6. Figure 11 shows the W − dependence ofthe diffusion coefficient numerically estimated. Regard-less of the number of colors M , in the region W << W ∗ it behaves D ∝ W . (25)However, when the W increases, it does not decreasemonotonously, but it shows the minimum of D around W ∗ and gradually increases towards the constant valueand it becomes D ∼ const. (26)for W >> W ∗ . This behavior can be inferred from theMaryland transform as follows. For W ≃ W ∗ , the effectof the randomness of the diagonal term saturates, and therange of the random hopping term still increases even for W > W ∗ when the ǫ = 0 .
2. This tendency does notdepend on the number of colors M after the LDT.The above behavior of the diffusion coefficient for theweak disorder region ( W < W ∗ ) can be explained fromthe W − dependence of the localization length in the lo-calized side. First, the W − dependence of the localizationlength is assumed to be ξ ( W ) ∝ /W for W <<
1. Asseen in the Table 1, for ǫ > ǫ c , the MSD changes from theballistic motion to the diffusion as the time elapses. Ifthe duration (characteristic time) of this ballistic motionis τ , it can be estimated as τ ≃ ξ ( W )2 π . (27)This is equivalent to considering the time evolution of thewave packet in Brownian motion, assuming that memoryis lost at τ , for the weak disorder region ( W < W ∗ ). Byusing the characteristic time τ , the time-dependence of m ( t ) is expressed as m ( t ) ∼ ξ tτ , (28)if we consider ξ spreads every time τ = ξ/ π . Then ifthe ǫ is fixed, the diffusion coefficient becomes D = lim t →∞ m ( t ) t = ξ τ (29) ∼ W . (30)As a result, the W − dependence of the localization lengthpropagates to that of the diffusion coefficient. m t W=0.3 0.4 0.5 0.6 0.7 0.8 1.0 2.0 3.0 4.0 5.0 (b) M=6 m t (a) M=2 W=0.3 0.5 0.7 1.0 3.0 5.0 m t slope 1 FIG. 10: (Color online) The plots of m ( t ) as a function of t for different values of W in the polychromatically perturbedAM with ǫ = 0 .
2. (a) M = 2 and (b) M = 6. Note that theaxes are in the real scale. The inset of the panel (a) is inthe logarithmic scale. Black dotted line shows m ( t ) ∝ t forreference. C. ǫ − dependence of the diffusion coefficient Next, let’s examine the ǫ − dependence of the diffusioncoefficient of the delocalized states. The ǫ − dependenceat ǫ > ǫ c for M = 2 ,
3, fixed at W = 1, is shown inFig.12. It can be seen that the diffusion coefficient of thedelocalized state increases gradually with increasing ǫ ,and saturates beyond ǫ ≃
1. This tendency also does notdepend on the number of colors if the LDT occurs when M ≥
2. What does this value ǫ ≃ f ( t ) of the potential with g ( t ) = " ǫ √ M M X i cos( ω i t + θ i ) . (31)This case can be called a ballistic model because there isno localization in the unperturbed case ( ǫ = 0). Actually,the ǫ − dependence of the diffusion coefficient in the ballis-tic model is also plotted in the Fig.12. It shows D ∝ ǫ − for ǫ < D M=2 M=3 M=6 slope -2 ε=0.2
W=W * FIG. 11: (Color online) The diffusion coefficient D of thequantum diffusion as a function of W in the polychromaticallyperturbed AM with ǫ = 0 . M = 2 , ,
6. Note that the axesare in the logarithmic scale. D ∝ W − and W ∗ = 0 .
78 areshown by black line and black dotted lines, respectively, forreference. value when the perturbation strength ǫ increases. This isa reasonable result because the effect of the presence of”1” that gives localization at the case of f ( t ) relativelyweakens, and the time-varying term becomes more dom-inant. As a result, for ǫ >> W − dependenceof the normal diffusion in the ballistic model. VI. SUMMARY AND DISCUSSION
We investigated the localization-delocalization transi-tion (LDT) of the AM (Anderson map) which are dy-namically perturbed by polychromatically periodic oscil-lations for the initially localized quantum wave packet.The W − dependence and ǫ − dependence of the localiza-tion length (LL) in the completely localized region havecharacteristics similar to those of the monochromaticallyperturbed AM. Since the characteristic value W ∗ exists,the MSD shows a peculiar change with respect to thechange of W , but the critical behavior of the LDT forchanging W is similar to that in the LDT for changing ǫ around the critical point. In the region W ∗ < W c , theshape of the scaling function changes, but the localiza-tion remains the same at W = W ∗ when W increases,and the increasing W causes the LDT at a certain value W = W c .We also studied the delocalized states for ǫ > ǫ c . The W − dependence of the diffusion coefficient of the delocal-ized states decreases in a form D ∝ W − for the region W < W ∗ . The D gradually increases towards D ≃ const D ε M=2 ballistic model M=2 Anderson map M=3 ballistic model M=3 Anderson map ε=1 slope -2
FIG. 12: (Color online) The diffusion coefficient D of thequantum diffusion as a function of ǫ in the polychromaticallyperturbed AM with W = 1 of M = 2 ,
3. The correspondingresults for the ballistic model are also provided. Note thatthe axes are in the logarithmic scale. D ∝ ǫ − and ǫ = 1 areshown by black line and black dotted lines, respectively, forreference. for W >> W ∗ after it becomes minimum diffusion coef-ficient around W ≃ W ∗ .Roughly speaking, since the time-continuous systemcorresponds to W ∗ → ∞ in the AM, it is expected thatthere is no LDT along the line L and L in Fig.1, onlyLDT corresponding to the region W < W ∗ along L willbe observed in the time-continuous Anderson model withthe time-quasiperiodic perturbation [18–20]. Appendix A: Critical phenomena of LDT and thelocalization length
In this appendix we consider the relation of the lo-calization length ξ ( ǫ ) obtained by direct calculation for ǫ << ǫ c and the LL by finite-time scaling analysis around ǫ ≃ ǫ c in the polychromatically perturbed AM W = 0 . L but the sameargument holds qualitatively for cases with the other pa-rameters. The localization length ξ ( M ) s ( ǫ ) obtained indi-rectly from the critical scaling analysis around the criticalvalue ǫ ( M ) c is expressed, ξ ( M ) s ( ǫ ) = ξ ( M )0 | ǫ ( M ) c − ǫ | − ν ( M ) , (A1)where ξ ( M ) s , ǫ ( M ) c and ν ( M ) are the the localization length, critical strength and the critical exponent of the LDT,respectively. A ( M )0 ( W ) = ξ ( ǫ ( M ) c ) ν can be determinedby the LL of the unperturbed case ξ = ξ ( ǫ = 0). Theorder of the localization length for the different M satis-fies ξ (2) s ( ǫ ) < ξ (3) s ( ǫ ) < ξ (4) s ( ǫ ) ... (A2) In addition, the following relations of the critical strengthand critical exponent holeds, ǫ (2) c > ǫ (3) c > ǫ (4) c > ...., (A3) ν (2) > ν (3) > ν (4) > ..... (A4)It are established from the results and theory of the finite-time scaling analysis. In other words, considering the M − dependence of this transition, the critical value ǫ c becomes smaller and the critical exponent ν also becomessmaller as the number of modes M increases, and thedivergence of the LL around the critical value becomesmild as the M increaes.In Fig.13, we compare the localization length ξ ( M ) s ( ǫ )decided indirectly by OPST in the critical region ǫ ≃ ǫ c with ξ ( M ) ( ǫ ) decided directly by the saturated MSD datawhich are precisely calculated for ǫ ’s much less than thecritical region. The ǫ − dependence of these two localiza-tion lengths, ξ ( M ) s ( ǫ ), ξ ( M ) ( ǫ ), seem to connect continu-ously, which implies unexpected wideness of the criticalregion in which the OPST works [21].On the other hands, as seen in main text, even if M = 2and M = 3, the localization length increases exponen-tially as ǫ increases at least for ǫ << ǫ c , according to theEq.(13). Furthermore, it seems that when ǫ increases, thelocalization length increases with increasing of the ǫ morestrongly than the exponential growth. This is natural be-cause ǫ grows closer to ǫ c and leads to the divergence ofthe localization length at ǫ = ǫ c T. Appendix B: Diffusive characteristics of Ballisticmodel
Here we consider a model without the localization byreplacing the time-dependent part f ( t ) of Eq.(3) with g ( t ) of Eq.(31). The system describes the free parti-cle scattered by the quasiperiodically oscillating irregularpotential. This model is also interesting as it is con-nected with the problem of a ballistic electrons scatteredby dynamical impurities. We refer to this model as theballistic model in this paper. The AM with the time-dependent part f ( t ) approaches to the ballistic model ina limit ǫ → ∞ , and the system has completely parameterdependence in the form of ǫW .In the ballistic model, a simple interpretation by Mary-land transform is possible. The diagonal term is a con-stant value that does not depend on the site. If W = 0(equivalent to M = 0), the hopping terms are also con-stant, and the Hamitonian describes the tight-bindingsystem with periodic potential. Accordingly, the dynam-ics of the unperturbed case exhibits the ballistic spreadinstead of the localization such as, m ( t ) ∼ t . (B1)Such a ballistic motion is suppressed and changes intoother kind of motion by introducing the dynamically os-cillating part ( W = 0, ǫ = 0).1 ξ s , ξ ε direct M=1 direct M=2 scaling M=2 direct M=3 scaling M=3 scaling M=4 direct M=4 ε c(3) ε c(2) ε c(4) (a) ξ s , ξ ε c (M) - ε (b)W=0.5 M=2M=3 M=4
FIG. 13: (Color online) (a)The localization length ξ ( ǫ ), ξ s ( ǫ )as a function of ǫ for perturbed AM ( M = 1 , , ,
4) with W = 0 .
5. The filled symbols denote the numerical data ob-tained by ξ ( ǫ ) = p m ( t → ∞ ) in the long-time limit. Theopen symbols indicate the localization length ξ s ( ǫ ) obtainedby the OPST in the critical region. Note that the vertical axisis in the logarithmic scale. (b) ξ n ( ǫ ) and ξ s ( ǫ ) as a functionof ǫ ( M ) c − ǫ . Figure 14 shows the time-dependence of the MSDwith changing ǫ and M as the perturbation is applied.As shown in the Fig.14 (a), the MSD changes from m ( t ) ∼ t to m ( t ) ∼ t when the perturbation ternon. Similarly, as shown in the Fig.14(b), there is no sub-diffusion even in the cases of M = 1 ∼
5, and it behavesnormal diffusion, m ( t ) ≃ Dt for t >>
1, where D is thediffusion coefficient.The ( ǫW ) − dependence of the diffusion coefficient D is shown in the Fig.15. It can be seen that when theparameter ǫW becomes larger than ( ǫW ) ∗ = 0 .
78 , the D gradually approaches a certain value. It follows thateven in the monochromatically perturbed AM, there is nolocalization and normal diffusion is achieved. These phe-nomena can also be interpreted based on the Marylandtransform. For ǫW >
0, it becomes diffusive by the hop-ping term including the disorder. When the W farther m t ε =0.02 0.05 0.08 0.10 0.20 0.50 1.00 2.00 M=2 slope 2 slope 1 (a) 10 m t M=1 M=2 M=3 M=4 M=5 slope 1 (b) ε =0.80 FIG. 14: (Color online) The double-logarithmic plots of m ( t )as a function of t for different values of ǫ in the polychromati-cally perturbed ballistic model W = 1. (a) M = 2 with variousvalues of ǫ s, and (b) M = 1 ∼ ǫ = 0 .
8. Black dottedlines show m ( t ) ∝ t and m ( t ) ∝ t for reference. increases, the diffusion is suppressed due to the disorderand the diffusion coefficient is reduced. Even if W is fur-ther increased, the disorder effect seems to be saturatedbecause the off-diagonal term is also tangent-type. D ε W M=1 M=2 M=3 M=5 ( ε W) * slope -2 FIG. 15: (Color online) The diffusion coefficient D of thequantum diffusion as a function of ǫW in the polychromati-cally perturbed ballistic model ( M = 1 ∼ ǫW ) ∗ = 0 .
78 is shown byblack dotted lines for reference.
Acknowledgments
This work is partly supported by Japanese people’s taxvia JPSJ KAKENHI 15H03701, and the authors wouldlike to acknowledge them. They are also very gratefulto Dr. T.Tsuji and Koike memorial house for using thefacilities during this study.2 [1] J.Chabe, G.Lemarie, B.Gremaud, D.Delande, andP.Szriftgiser, Phys. Rev. Lett. , 255702(2008).[2] J. Wang and A. M. Garcia-Garcia, Phys. Rev. E ,036206(2009).[3] G. Lemarie, H.Lignier, D.Delande, P.Szriftgiser, and J.-C.Garreau, Phys. Rev. Lett. , 090601(2010).[4] C. Tian, A. Altland, and M. Garst, Phys. Rev. Lett. ,074101(2011).[5] M.Lopez, J.F.Clement, P.Szriftgiser, J.C.Garreau, andD.Delande, Phys. Rev. Lett. , 095701(2012).[6] M. Lopez, J.-F. Clement, G. Lemarie, D. Delande,P. Szriftgiser, and J. C. Garreau, New J. Phys. ,065013(2013).[7] K.Ishii, Prog. Theor. Phys. Suppl. , 77(1973).[8] L.M.Lifshiz, S.A.Gredeskul and L.A.Pastur, Introduc-tion to the theory of Disordered Systems , (Wiley, NewYork,1988).[9] H.S. Yamada and K.S. Ikeda, arXiv:1911.02189 [cond-mat.dist-nn] (2019).[10] H.S. Yamada, F. Matsui and K.S. Ikeda, Phys. Rev. E , 062908(2015).[11] H.S. Yamada, F. Matsui and K.S. Ikeda, Phys. Rev. E , 012210(2018).[12] S. Fishman, D.R.Grempel, R.E.Prange, Phys. Rev. Lett. , 509 (1982); D.R. Grempel, R.E. Prange and S. Fishman, Phys. Rev. A , 1639(1984); R.E. Prange,D.R. Grempel, and S. Fishman, Phys. Rev. B , 6500-6512(1984).[13] P. Markos, Acta Phys. Slovaca , 561(2006).[14] Antonio M. Garca-Garcia and Emilio Cuevas, Phys. Rev.B ,174203(2007).[15] Yoshiki Ueoka, and Keith Slevin, J. Phys. Soc. Jpn. ,084711(2014).[16] E. Tarquini, G. Biroli, and M. Tarzia, Phys. Rev. B ,094204(2017).[17] H.S.Yamada and K.S.Ikeda, Eur. Phys. J. B ,195(2012); ibid , 208(2014).[18] H.Yamada and K.S.Ikeda, Phys. Rev. E , 5214(1999); ibid , , 046211(2002).[19] H.Yamada and K.S.Ikeda, Phys. Rev. E65