Presence and Absence of Delocalization-localization Transition in Coherently Perturbed Disordered Lattices
aa r X i v : . [ c ond - m a t . d i s - nn ] N ov Presence and Absence of Delocalization-localization Transition in CoherentlyPerturbed Disordered Lattices
Hiroaki S. Yamada
Yamada Physics Research Laboratory, Aoyama 5-7-14-205, Niigata 950-2002, Japan
Kensuke S. Ikeda
College of Science and Engineering, Ritsumeikan University Noji-higashi 1-1-1, Kusatsu 525-8577, Japan (Dated: November 17, 2020)A new type of delocalization induced by coherent harmonic perturbations in one-dimensionalAnderson-localized disordered systems is investigated. With only a few M frequencies a normaldiffusion is realized, but the transition to localized state always occurs as the perturbation strengthis weakened below a critical value. The nature of the transition qualitatively follows the Andersontransition (AT) if the number of degrees of freedom M + 1 is regarded as the spatial dimension d ,but the critical dimension is not d = M + 1 − d = 3. PACS numbers: 05.45.Mt,71.23.An,72.20.Ee
Introduction-
Since the proposal of Anderson, the lo-calization of electron in disordered lattices has been oneof the most fundamental problems associated with theessence of electron conduction process [1–3]. No matterhow high the spatial dimension may be, the Andersonlocalized state exist prior to the delocalized conductingstate, and a transition from localized state to the de-localized state, the so called Anderson transition (AT),occurs as the relative strength of disorder decreases [4–8].Theoretical predictions have been obtained by using sev-eral theoretical tools such as the one-parameter scalinghypothesis, the self-consistent theory, and so on [9, 10].On the other hand, in the study of chaotic systems theergodic transition of quantum maps is equivalent to theAT of disordered lattice [11–13]. Upon this equivalence,the dynamical AT has been first experimentally observedfor the quantum map systems implemented on the opticallattice [14, 15]. In this case the number of dynamicaldegrees of freedom corresponds to the number of spatialdimension of the disordered lattices, and so the featuresof AT in high-dimensional lattices can be explored by thequantum maps.The dynamical interaction among the degrees of free-dom thus enables the delocalization transition. Then thefollowing question immediately follows: can the Ander-son localization in the disordered lattices be destroyedas it is perturbed by dynamical degrees of freedom suchas phonon modes? The perturbation by infinitely manyphonon modes can be modeled by a stochastic pertur-bation, and it is well-known that the stochastic pertur-bation destroys the localization and induces a normaldiffusion [16–19]. However, the effect of dynamical per-turbation composed of finite number of coherent modeshas still been unanswered. In the previous papers, weinvestigated the effect of finite-number harmonic pertur-bations on one-dimensional disordered lattice (ODDL),and showed that the ODDL exhibits a normal diffusionat least on a finite time scale [20–22]. On the other hand,numerical and mathematical studies claim that the lo-calization is persistent for finite-number harmonic per- turbations Refs.[23, 24], and which of localization anddelocalization dominates has still been open to question.It is quite interesting whether or not a coherent dynami-cal perturbation composed of finite number of harmonicmodes can dynamically destroy the localization. In thisletter, we present novel results replying the question.
Model-
We consider tightly binding ODDL perturbedby coherent periodic perturbations with different incom-mensurate frequencies. It is given by i ~ ∂ Ψ n ( t ) ∂t = Ψ n − ( t ) + Ψ n +1 ( t ) + V L ( n, t )Ψ n ( t ) , (1)where V L ( n, t ) = V ( n )[1 + f ( t )]. The coherently time-dependent part f ( t ) is given as, f ( t ) = ǫ √ M M X i cos( ω i t ) , (2)where M and ǫ are the number of frequency componentsand the strength of the perturbation, respectively. Notethat the long-time average of the total power of the per-turbation is normalized to f ( t ) = ǫ /
2. The frequencies { ω i } ( i = 1 , ..., M ) are taken as mutually incommensu-rate numbers of order O (1). The static on-site disorderpotential takes random value V ( n ) uniformly distributedover the range [ − W, W ], where W denotes the disorderstrength.It is important to note that the harmonic source can beinterpreted as the quantum linear oscillator of the Hamil-tonian P Mi ωJ i interacting with the irregular lattice withthe quantum amplitude ǫ √ M P Mi cos φ i instead of theclassical force f ( t ), where ( J i , φ i ) = ( − i∂/∂φ i , φ i ) are theaction-angle operators of the i -th oscillator. Each quan-tum oscillator has the action eigenstates | n i > with theaction I i = n i ~ ( n i :integer) and the energy n i ~ ω i . Thusthe system (1) is regarded as a quantum autonomoussystem of ( M + 1)-degrees of freedom spanned by thequantum states | n > Q Mi =1 | n i > [23].We take a lattice-site eigenstate as the initial state | Ψ( t = 0) i and numerically observe the spread of thewavepacket measured by the mean square displacement(MSD), m ( t ) = P n ( n − n ) (cid:10) | Ψ( n, t ) | (cid:11) .First, we consider the limit M → ∞ . In this case f ( t ) can be identified with the delta-correlated stochasticforce < f ( t ) f ( t ‘ ) > = Γ δ ( t − t ‘ ), where Γ ∝ ǫ is a noisestrength. The localization is surely destroyed and thenormal diffusion m ( t ) = Dt with the diffusion constant D is recovered for t → ∞ [21, 22], as was first pointedout by Haken and his coworkers [16, 17]. They predictedanalytically for the white Gaussian noise D = lim t →∞ m ( t ) t ∝ ΓΓ + W / , (3)for weak enough ǫ . If W ≫ Γ, D ∝ W − but recentlyit was shown that D ∝ W − for strong disorder region W ≫ M , f ( t ) can no longer be replacedby the random noise, and it plays as a coherent dynami-cal perturbation, and the system is a quantum dynamicalsystem with ( M + 1)-degrees of freedom. The main pur-pose of this study is to investigate how does the natureof the quantum dynamics of the irregular lattice changesas the number M decreases from ∞ to 0. Delocalized states and normal diffusion-
We show typ-ical examples of time evolution of MSD for M = 7 and M = 3 in Fig.1(a) and (b), respectively. If ǫ is largeenough, it is evident that MSD follows asymptoticallythe normal diffusion m = Dt , which means that only afinite number of coherent periodic modes plays the samerole as the stochastic perturbation in the disordered lat-tice. The W − and ǫ − dependence of the diffusion con-stant D depicted in Fig.1 follow the main feature of thestochastically induced diffusion constants: as shown inthe Fig.1(c) the W − dependence changes from D ∝ W − for weak W in Eq.3 to D ∝ W − W ≫
1, followingthe theoretical prediction of stochastic perturbations [18].Moreover, as depicted in the Fig.1(d), even for M = 3the ǫ − dependence reproduces the characteristic behav-ior of the stochastically induced D , which first increasesbut finally decreases with ǫ after reaching to a maximumvalue. It is a remarkable feature of ODDL that a normaldiffusion, which mimics the one induced by a stochasticforce composed of infinite number of frequencies, is self-generated by a coherent perturbation composed of onlythree incommensurate frequencies.On the other hand, the coherently perturbed ODDL al-ways undergoes a definite phase transition from the dif-fusing state to a localized state as ǫ decreases crossingover a critical value ǫ c . The transition is quite similar tothe AT of high-dimensional disordered lattice. As shownin Fig.2(a), at ǫ = ǫ c , the MSD exhibits a subdiffusion m ∼ t α with a critical diffusion index α (0 < α < ǫ c , typical critical transient phenomena are ob- m t ε =0.06 0.10 0.13 0.15 0.20(a) M=7 m t ε =0.20 0.23 0.25 0.30 (b) M=3 D W slope -2 (c) M=3, ε=0.3 Μ=4,ε=0.3 Μ=6,ε=0.3 Μ=6,ε=0.2 slope -4 D ε M=3 M=5 M=7 M=10 Gaussian noise
W=2 slope -2 (d)
FIG. 1: (Color online) The m ( t ) as a function of time for(a) M = 7 and (b) M = 3 some values of the perturbationstrength ǫ increasing from ǫ = 0 . ǫ = 0 . M = 7 and from ǫ = 0 . ǫ = 0 . M = 3,respectively. Note that the axes are in the real scale. (c)Thediffusion coefficient D as a function of W and (d) the D as afunction of ǫ for several M , determined by the least-square-fitfor the m ( t ) for t >>
1. The system and ensemble sizes are N = 2 ∼ and 10 ∼
40, respectively, throughout thispaper. We used 2nd order symplectic integrator with timestep size ∆ t = 0 . ∼ .
05, and take ~ = 0 .
125 as the Planckconstant. served. To show them we define the function Λ( t ) as thescaled MSD Λ( t ) ≡ m ( t ) t α (4)divided by the critical MSD. In Fig.2(b) the Λ( t ) at var-ious ǫ close to ǫ c are displayed for M = 7, which form acharacteristic fan pattern spreading outward.As are demonstrated in Fig.3(a), the index of the crit-ical subdiffusion decreases with M , following the resultof one-parameter scaling hypothesis α = 2 d = 2 M + 1 (5)for the d -dimensional disordered lattice, if we regard d asthe total number of degrees of freedom of our system, ie,. d = M + 1, which seems to be quite reasonable.The localization in the side of ǫ < ǫ c is characterizedby the localization length ξ M , which diverges at ǫ c as m t ε =0.025 0.033 0.038 0.045 0.050 0.063 t (a) M=7 Λ ( t ) t(b) M=7 FIG. 2: (Color online) The double-logarithmic plots of(a) m ( t ) and (b)the scaled Λ( ǫ, t ) as a function of time forsome values of the perturbation strength ǫ increasing from ǫ = 0 . ǫ = 0 . α is determined by the least-square-fit for the m ( t )with the critical case, in the perturbed 1DDS of M = 7 with W = 2 .
0. The data near the critical value ǫ c are shown bybold black lines. The blue dashed curves show the results for ǫ < ǫ c in the panels. ǫ c ≃ . α ≃ / .
25. Note thatthe axes are in the logarithmic scale. ξ M ( ǫ ) ∼ ( ǫ − ǫ c ) − ν with the critical exponent ν ( > t ) are represented by two unified curvesdepending whether ǫ > ǫ c or ǫ < ǫ c by using the scalingvariable x = ξ M ( ǫ ) t α/ ν , as demonstrated by Fig.3(b) for M = 3. The d = M + 1 − dependence of the critical index ν is shown in Fig.3(d).Such a remarkable critical sub-diffusion exists at ǫ c foran arbitrary M , but the critical value ǫ c decreases with M following the diffusion index α : ǫ c ∝ M − δ , δ ≃ , (6)which does not depend upon W as shown in Fig.3(c).Thus the ODDL is always localized if ǫ is small enough,but no matter how small ǫ may be, a normal diffusionmimicking a stochastically induced diffusion is realized if M is taken large enough.It is quite interesting that the dependencies of both α and ǫ c upon M are the same as those of the AT observedfor the quantum maps simulating the high-dimensionaldisordered lattice [25–27]. If we are allowed to extrapo-late the above results for the smaller M , ǫ c diverges at M = 1, at which the critical diffusion index becomes α = 1. This fact implies that for M = 1 the criti-cal subdiffusion is realized at ǫ = ǫ c = ∞ as a normaldiffusion; namely, that M = 1 is the critical dimensionof the delocalization-localization transition (DLT), whichhas been established for the quantum maps and high-dimensional disordered lattices. However, our numericalresults reject the above conjecture. Number of critical modes ( M = 2 )- If the above conjec-ture is correct, M = 2 ( d = 3) should exhibit the critical m ( t ) t M=3 M=4 M=5 slope 2/4slope 2/5slope 2/6 (a) l og Λ ( ε , t ) log x (b) M=3 ε c M-1 slope -1 Set (c) c r i t i c a l e x p . ν d, M+1 this study ν VW ν G (d) FIG. 3: (Color online) (a)The double-logarithmic plots of m ( t ) as a function of time near the critical pints ǫ c inthe polychromatically perturbed 1DDS ( M = 3 , , s ( ǫ, t ) as a function of x = | ǫ c − ǫ | t α/ ν . The delocalized(localized) regime is upper(lower)branch. (c)The critical perturbation strength ǫ c as a functionof ( M − { ω i } is alsoentered. Note that the axes are in the logarithmic scale. Theline with slope − M + 1) = d dependence of the critical exponent ν which characterizes the critical dynamics. The red solidline and green dashed line are the results of the analyticalprediction by ν V W and ν G , respectively. phenomenon. In Fig.4(a) the log-log plot of MSD curvesfor M = 2 are shown for various values of ǫ . Surely, somecurve follows the expected critical subdiffusion of the ex-ponent α = 2 / α ( t ) = d log m ( t ) d log t . (7)If DLT happens at a finite ǫ = ǫ c , then the α ( t ) shouldkeep a constant value α ( ǫ c ) <
1. Above ǫ c , as time passes, α ( t ) increases up to the exponent 1 indicating the normaldiffusion, while it decreases to zero indicating the local-ization below ǫ c . Indeed, the expected feature is evidentfor the α ( t ) plot of M = 3 shown in Fig.4(b) The samefeature is observed also for M ≥ α ( t )-plots of M =2 shows a quite different feature. No curves follow thecritical behavior α ( t ) =const <
1, and all the curves tendsto decrease from the initial values, which approaches to1 as ǫ increases. As α ( t ) comes close to 1, the time scalebeyond which α ( t ) begins to decrease becomes longer.Certainly it seems as if the normal diffusion α ( t ) = 1,which would be realized in the limit ǫ → ∞ , were thecritical diffusion. These facts indicates that the DLT doesnot exists for M = 2 in contradiction with the predictionof the Eqs.(5) and (6), and that M = 2( d = 3) is thecritical dimension. m t ε =0.10 0.20 0.60 0.30 0.65 0.40 0.70 0.50 0.80 0.55 1.00 1.30 (a) M=2 slope 1 α ξ Μ ε M=3 M=2 M=2(scaling) M=1 ε =0.18 ε =0.6 FIG. 4: (Color online) (a)The double-logarithmic plots of m ( t ) as a function of time for some values of the perturbationstrength ǫ increasing from bottom to top in the trichromati-cally perturbed 1DDS of M = 2. The panels (b) and (c) arethe diffusion exponent α ( t ) as a function of time in the cases M = 3 and M = 2, respectively. (e)Localization length as afunction of ǫ for M = 1 , , W = 2. Some LL of M = 2are obtained by scaling relation m ( t ) ∼ ξ M ( ǫ ) F ( t/ξ M ( ǫ ) )for ǫ ≥ .
5. Note that the horizontal axis is in logarithmicscale. The dashed lines show ξ M ∝ e . ǫ and ξ M ∝ e . ǫ ,respectively. The lines ǫ = 0 .
18 and ǫ = 0 . Comparison by localization length-
Localization, ofcourse, occurs with M = 1. Then what is the differenceof the localization between the case of M = 1 ( d = 2) andthe case of M = 2 ( d = 3). In both cases of M = d − M = d − ǫ is small enough, which coincides with the case of the d = 2 ordinary irregular lattice.However with a further increase of ǫ , ξ M begins to de-crease steeply for M = 1. Such a behavior is a directreflection that the inter-site transfer is suppressed by therandom potential enhanced with the increasing couplingstrength ǫ . Let us remember that as shown in Fig.1(d)even the recovered diffusion constant of the the systemof d = M + 1 ≫ ǫ ( > ǫ c ). This is the reason why, unlike the ordinary d − dimensional irregular lattice, d = 2 can not be thecritical dimension of our system. The growth of ξ M with ǫ takes place only by increasing the dimension from d = 2to 3.Indeed, for d = M +1 = 3 the localization still remains,but ξ increases with ǫ exponentially. The exponentialgrowth rate is further enhanced and a super-exponentialgrowth occurs as ǫ increases beyond O (1), as is depictedin Fig.4(d). And it is for M = 3 that the divergence of ξ M is first observed at a finite ǫ c . Summary and discussion-
In the present paper, we in-vestigated the delocalized and the localized motion in 1Dirregular lattice coherently perturbed by the harmonicmodes. In order to induce a delocalized motion thestochastic perturbation composed of infinite number ofharmonic mode is not necessary: the diffusive motion isalways induced only by a few number of harmonic modesif the perturbation strength is strong enough. On theother hand, the transition to the localized phase occurshow many the mode number M may be, and the criti-cal perturbation strength as well as the critical diffusionexponent follows those of the Anderson transition in thehigh-dimensional lattice and many dimensional quantummaps. However, the critical number of the degrees offreedom is not d = M + 1 = 2 but d = 3 in our system.Thus our system provides with an example demonstrat-ing that the critical dimension of the DLT may be largerthan d = 2 and depend upon the nature of recovereddiffusion, as summarized in the TABLE I.The threshold number of degrees of freedom, for whichthe delocalized motion, and more generally the ergodicmotion, takes place in quantum systems, is a quite in-teresting theoretical problem. We hope such approachesmay contribute to elucidate the origin of quantum statis-tical behavior from the side of the study of small quantumsystems. TABLE I: Dimensionality of the DLT. For 4 ≤ M < ∞ theresult is same as the case of M = 3. The lower lines is resultof the d − dimensional disordered systems. Loc: exponentiallocalization, Diff:Normal diffusion. d (= M + 1) 1 2 3 4 5 ... ∞ this study Loc Loc Loc DLT DLT ... DiffAM[27] Loc Loc DLT DLT DLT ... DiffAnderson model Loc Loc DLT DLT DLT ... DLT [1] P. W. Anderson, Phys. Rev. 109, 1492-1505 (1958).[2] L.M.Lifshiz, S.A.Gredeskul and L.A.Pastur, Introduc-tion to the theory of Disordered Systems , (Wiley, NewYork,1988).[3] E. Abrahams (Editor),
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