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Numerical study of quantum diffusion in two-dimensional disordered systems
7 November 1994 PHYSICS LETTERS A
ELSEVIER
Physics Letters A 194 (1994) 279-284
Numerical study of quantum diffusion in two-dimensional disordered systems Hiroaki Yamada, Masaki Goda Faculty of Engineering, Niigata University, Niigata 950-21, Japan Received 21 June 1994; revised manuscript received 16 August 1994; accepted for publication 1 September 1994 Communicated by A.R. Bishop
Abstract The time evolution of the wavepacket in some two-dimensional tightly binding disordered systems is numerically calculated by a sixth-order symplectic integrator with enough accuracy. The time evolution is monitored by observing the mean square displacement. The time dependence of the mean square displacement seems to exhibit an anomalous diffusion in the intermediate region between the mean free path and the localization length of the system. The mean square displacement of the packet scaled only by the localization length does not seem to have a universal feature in its time evolution, which is different from that in the case of one-dimensional disordered systems.
1. Introduction Since the appearance o f the o n e - p a r a m e t e r scaling theory o f localization ( S T L ) in d i s o r d e r e d systems [ 1 ], m a n y theoretical, numerical and experimental studies have been d e v o t e d to the inspection o f the theory [ 2 ]. The theory suggests that almost all o f the electronic eigenstates are exponentially localized in an infinite system in one- and t w o - d i m e n s i o n a l disordered systems ( D D S ) with no external field 1 However, there are rigorous proofs o f the exponential localization only for the infinite 1-DDS [4,5 ]. A m a t h e m a t i c a l l y rigorous theory o f localization for 21Recently it was reported that the disordered diagonal dimer model corresponding to a one-dimensional tightly binding binary alloy has extended states, the number of which is proportional to x/N for a finite system size N. However, it must be noticed that this is not the result of the case of a finite system. In an infinite system only one accumulation point in the energy axis corresponds to the extended states. Details are reported in Ref. [3].
D D S does not exist yet, in spite o f much effort being m a d e [ 6 ]. M a c k i n n o n et al. calculated numerically the conductance in 2-DDS based on the L a n d a u e r formula by a skillful finite size scaling m e t h o d using strips, a n d got a result which is consistent with o n e - p a r a m eter STL [ 7 ]. Recently it is also shown that in a twod i m e n s i o n a l q u a n t u m site a n d b o n d percolation p r o b l e m almost all states are exponentially localized for any a m o u n t o f d i s o r d e r by the same scaling m e t h o d [ 8 ]. However, there is still a dispute about the form o f the eigenstates suggested by the numerical simulation o f q u a n t u m diffusion. The dispute is about whether there exists a m o b i l i t y edge separating the power-law localized state from exponentially localized states and also about whether diffusion is completely suppressed in 2-DDS even with weak d i s o r d e r [ 9,10 ]. The existence o f the power-law localization suggests a t w o - p a r a m e t e r STL [ 9 ]. Longer t i m e sim-
0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI0375-9601 (94)00726-8
280
H. Yamada, M. Goda / PhysicsLettersA 194 (1994) 2 79-284
ulations in a larger system are needed for a reconciliation of the controversy. The localized eigenfunctions are directly related to the dynamical properties of a quantum particle as conduction and diffusion in the systems [4]. I f all of the eigenstates are exponentially localized, the diffusion o f a wavepacket is suppressed at about the localization length (LL) and as a result the conductivity vanishes. Though it is possible, in principle, to get a snapshot of the wavepacket at arbitrary time by several methods, for example, by direct diagonalization of the Hamiltonian matrix [ 11 ] or by numerical integration [ 12,13], it is difficult to get the numerical results with enough accuracy except for the case in 1DDS, because of the limitations of CPU time and the storage in the computer. It is also difficult to distinguish the delocalized wavepacket from the localized one numerically. Some pre-scaling studies had even suggested that there exists a transition to an extended state at finite disorder in 2-DDS [ 12,14 ]. When we use the calculational scheme of the time evolution with inadequate accuracy, the energy conservation of the system becomes insufficient [ 15 ]. The error of the numerical calculation sometimes works as a dissipation effect or an effect due to a contact with another degree of freedom [ 16 ]. Then the packet can exhibit an artificial diffusive behavior beyond the intrinsic LL. It is also interesting to investigate the correspondence between the quantum dynamics of a chaotic system and Anderson localization [ 17]. For a two-dimensional kicked rotor (2-DKR) localization takes place in the angular m o m e n t u m space and the localization length grows exponentially with respect to the mean free path numerically [ 18 ]. The rotor can be mapped on a two-dimensional tightly binding model exhibiting localization. However, it is worth noting that the 2-DKR is not an intrinsic 2-DDS and an anomalous diffusion takes place in a rotor with some kind of coupling [ 19 ]. Hitherto, there are only a few numerical calculations of the quantum diffusion with reliable enough accuracy in 2-DDS. Raedt has calculated the time evolution of the wavefunction by using a fourth-order symplectic integrator (SI) and a special initial state, which is diagonalized in the small subsystem. In this Letter, we report the property of pre-localiza-
tion of the packet in 2-DDS starting from a different type of the initial state. We use a sixth-order SI to get the time evolution of a wavepacket with enough accuracy [15,20]. The mean square displacement (MSD) of the packet exhibits a trend towards an anomalous diffusion in an intermediate regime between the ballistic propagation (BP) and the localization. We suggest numerically that the quantum diffusion in 2-DDS does not have a universal feature in its MSD that is scaled by only one parameter, different from the case of 1-DDS [21]. Our results are consistent with that in Ref. [ 10 ]. This study concerns only the transit phenomena and therefore is not able to distinguish power-law or exponential localization for the eigenstate of 2-DDS.
2. Models and numerical results
We consider a one-electron tight-binding Hamiltonian of localization described as follows, n= ~,V(n)ln)(nl+ n
~. K ( n , m ) I n ) ( m l ,
(1)
nmm
where the basis set { In ) } is an orthonormalized one, of which each base In) is localized near a two-dimensional site n, V(n) is the random on-site energy at site n and the transfer energy K(n, m) ( = - 1 ) vanishes unless the sites n, m are nearest neighbours. We use two types of random on-site energy. One is the Anderson model (A-model), in which each site energy varies at random uniformly in the range [ - IV, W], and the other is binary model (B-model), in which each site energy randomly can take only two values, - W o r IV. On numerically simulating the time development of the wavepacket some restrictions come from the following situations: The weaker the disorder strength W of the 2-DDS becomes, the faster the wave front of the packet reaches the boundary. On the other hand, as the disorder strength becomes large, the numerical accuracy becomes insufficient. We thus need a simulation of a large enough system with a higherorder time integration scheme and with a smaller time step to remove the above restrictions. Accordingly it is difficult to observe the diffusion process to the extent of fully exponential localization in 2-DDS before the wave front reaches the boundary. The numerical
H. Yamada, M. Goda / Physics Letters A 194 (I 994) 279-284
accuracy can be checked by doing a time-reversal experiment of the packet as in Ref. [ 15 ]. We use a sixlh-order SI with coefficients obtained by Yoshida [20] to calculate the time evolution of the packet by a unitary operator, e x p ( - i H t ) . The Planck constant and the size of the integration time step have been taken to be 1 and 0.025, respectively. The system size and the ensemble size are taken as N x = N y = 128 and 10, respectively, and we adapt periodic boundary conditions, because we use the FFT and inverse FFT to exchange the real space representation for the momentum space one and do the opposite operation [ 15,19 ]. We use the packet localized at the center of the system as the initial state, 7'( t = O, n) = d.,(ux/2, Ny/2)" Two characteristic scales exist in the time evolution of the packet. One is the mean free path (MFP), and the other is the LL. We observe the time evolution in the intermediate regime between MFP and LL. Fig. 1 shows some snapshots of the time evolution of a wavepacket. We can observe a process toward localization in the packet as a result of the interference effect between the scattering wave.
.18.16.14~ .12.i0.08.06.04 .02-
.050.045.040.035 = .030.025.020.015.010.005-
t=2.5
t=25
281
Next we consult the time dependence of the MSD of a packet monitoring the spread of the packet. The double bracket denotes the quantum mechanical average and the sample average. The MSDs for two kinds of site energy with the various disorder strengths W are shown in Fig. 2. It is difficult to get the analytical form of the LL and also to decide it exactly by numerical calculation in 2-DDS. It is, however, supposed that in 2-DDS almost all states are exponentially localized with a LL ~ that has the form ((R2))
~=aexp(b
-c/2)
(1~c~2) ,
(2)
where a and b are real constants. This is an extended form of the W-dependent LL which results from a numerical calculation based on one-parameter STL [ 7,8 ]. The < v z > denotes a second-order moment of random site energy. It is proportional to the square of the disorder strength W E and the inverse quantity is also almost proportional to the MFP of the system [ 18 ]. Accordingly, the LL is much longer than the MFP in 2-DDS [21 ], and we can observe the time evolution of the packet in the intermediate regime in
.018.016•014 .012.010.008.006•004 .00? -
.0090.0080.0070= .0060.0050.0040 .0030-
t=50
t=lO0
.0020.0010-
Fig. 1. Snapshots of a wavepacket at some times of development on the B-model of size 128 × 128 with disorder strength W = 1.7, at t=2.5, 25, 50, 100.
282
H. Yamada, ill. Goda I Physics Letters A 194 (1994) 2 79-284
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Fig. 2. Plots ofMSD ((R 2)) versus time for (a) the B-model and (b) the A-model, with various disorder strengths W.
Fig. 3. Log-log plots of the data in Fig. 2 for (a) the B-model and (b) the A-model.
between the above mentioned two characteristic scales, different from the case in 1-DDS, in which LL is of compatible order with the MFP [22]. Fig. 3 shows In-In plots of the data in Fig. 2. We can see the time dependence of MSD similar to that in anomalous diffusion (((R2))~t% 0 < a < 2 ) after a short period of the time evolution describing almost free or Bloch particles ( ((R 2)) ~ t 2). The power exponent a is about 0.75, 0.65 for the B-model with W= 1.5, 2.0, respectively, depending on the value of W, and 0.80 for the A-model with W=2.5. The relative error of the exponents is less than about 5%. Raedt has reported that an anomalous diffusion seems to occur in 2-DDS with a weak disorder strength under a condition different from ours [ 10 ]. Our results are consis-
tent with theirs. However, more long-time simulations are needed to get the exponent for the cases with a weaker disorder strength and to obtain a clear and comprehensive conclusion about the anomalous diffusion. We cannot decide the exponents in the system with large LL from these data because of a weak systematic change of the power-index with respect to time. Next we consider whether MSD can be scaled only by one parameter as a function of time. We assume the following scaling form as in the case of 1-DDS
[211, ((x(t)2)) =
(~/~o)2F{t(~/~o)-]},
(3)
where (o is the LL of a standard case (B-model,
H. Yamada, M. Goda / Physics Letters A 194 (1994) 279-284
W = 0 . 9 ) . This form is based on the request that the MSD behaves as t 2 with the same coefficient in the vicinity o f time t = 0 (ballistic regime). Fig. 4 shows the time dependence o f the MSD scaled by the LL. The coefficient b in Eq. (2) is selected so that the scaled MSDs are made to fit as well as possible for each value o f c in the interval [ 1,2] in the B-model. However, we cannot find a universal form o f the function F ( t ) even if we extended the unit interval o f c to the entire positive real axis. The same result is obtained also for the case o f the A-model. If the MSD has a different power-index a which depends on disorder strength in this regime, the MSD cannot have a universal form of the function F ( t ) in Eq. ( 3 ), even
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if we adapt any time-independent scaling factor instead o f adapting the form of the LL (2). To summarize, we investigated the time evolution of a wavepacket in some 2-DDSs by numerical calculation with high accuracy. We confirm the pre-localization behavior of the packet in the MSD so that it behaves as in the case of anomalous diffusion. These results suggest that the "microscopic personalities" (for example, type of model and disorder strength) o f 2-DDS characterizing the system appears in the time evolution o f the packet in the intermediate regime between the M F P and the LL. In relation to this fact we could not find a scaling form (3) in the MSD in the case of 2-DDS. However, this result does not directly claim to be the evidence supporting the twoparameter scaling theory o f localization. It is interesting to study the relation between this result and the statistical feature of probability amplitude in the wavefunction typically exemplified in Fig. 1. This problem is reported in detail elsewhere.
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Acknowledgement
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The authors would like to thank Dr. T. Okabe for visualization o f the data. The authors also acknowledge many discussions about the symplectic integrator with Professor K. Ikeda and Professor K. Takahashi. The authors are also thankful to the referee who pointed out m a n y insufficient points in the original manuscript. Numerical calculations of this study were carried out by the supercomputers at the Data Processing Center of Kyoto University.
,¢
-~1500
References
v 1000 50o 0 0
50
100
150
t
200
(,~/,~)-'
250
300
Fig. 4. Plots of scaled MSD versus scaled time on the B-model with various disorder strengths in the case of coefficients in Eq. (2) (a) c=2, (b) c= 1.
[ 1] E. Abrahams, P.W. Anderson, D.C. Licciardello and T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673. [2] For a recent review see T. Ando and H. Fukuyama, eds., Anderson localization (Springer, Berlin, 1988), and references therein. [3] P. Phillips and H.L. Wu, Science 252 ( 1991 ) 1805; J.C. Flores and M. Hilke, J. Phys. A 26 ( 1993) 11225, and references therein. [41K. Ishii, Prog. Theor. Phys. Suppl. 53 (1973) 77. [5] S. Kotani, Taniguchi Symp. SA. Katata (1982), ed. K. Ito (Kinokuniya, 1984) p. 225; M. Minami, Commun. Math. Phys. 103 (1986) 387.
284
H. Yamada, M. Goda / Physics Letters A 194 (1994) 279-284
[ 6 ] F. Delyon, B. Simon and B. Souillard, Commun. Math. Phys. 109 (1987) 157. [7] A. Mackinnon and B. Kramer, Phys. Rev. Lett. 47 ( 1981 ) 1546; Z. Phys. B 53 (1983) l; B. Bulka, M. Schreiber and B. Kramer, Z. Rev. 66 (1987) 21; M. Schreiber and M. Ottomeier, J. Phys. Condens. Matter 4 (1992) 1959. [ 8 ] C. Soukoulis and G. Grest, Phys. Rev. B 44 ( 1991 ) 4685. [9] M. Kaveh, Philos. Mag. 52 (1985) 521. [ 10] H. Raedt, Comput. Phys. Rep. 7 (1987) 1; H. Raedt and P. Vries, Z. Phys. B 77 (1989) 243. [11]H. Hiramoto and S. Abe, J. Phys. Soc. Japan 57 (1988) 230. [12] P. Prelovsek, Phys. Rev. B 18 (1978) 3657; Phys. Rev. Lett. 40 (1978) 1596. [ 13] M. Scher, J. Non-Crystalline Solids 69/70 (1983) 33. [ 14] S. Yoshino and M. Okazaki, J. Phys. Soc. Japan 43 (1977) 415.
[ 15 ] K. Takahashi and K. Ikeda, J. Chem. Phys. 99 ( 1993 ) 8680. [ 16 ] K. Ikeda, Ann. Phys. 227 ( 1993 ) 1. [17]G. Casati and L. Molinari, Prog. Theor. Phys. Suppl. 98 (1989) 287; E. Ott, T.M. Anderson Jr. and J.D. Hanson, Phys. Rev. Lett. 23 (1984) 2187. [ 18 ] E. Doron and S. Fishman, Phys. Rev. Lett. 60 ( 1988 ) 867. [ 19 ] S. Adachi, M. Toda and K. Ikeda, Phys. Rev. Lett. 61 ( 1988 ) 655; M. Toda, S. Adachi and K. Ikeda, Prog. Theor. Phys. Suppl. 98 (1989) 323. [20] H. Yoshida, Phys. Lett. A 150 (1990) 262; M. Suzuki, Phys. Lett. A 165 (1992) 387, and references therein. [ 21 ] H. Yamada, M. Goda, Y. Aizawa and M. Sano, J. Phys. Soc. Japan 61 (1992) 3050. [22] P. Maldague, Phys. Rev. B 23 (1980) 1719.
ELSEVIER
Physics Letters A 194 (1994) 279-284
Numerical study of quantum diffusion in two-dimensional disordered systems Hiroaki Yamada, Masaki Goda Faculty of Engineering, Niigata University, Niigata 950-21, Japan Received 21 June 1994; revised manuscript received 16 August 1994; accepted for publication 1 September 1994 Communicated by A.R. Bishop
Abstract The time evolution of the wavepacket in some two-dimensional tightly binding disordered systems is numerically calculated by a sixth-order symplectic integrator with enough accuracy. The time evolution is monitored by observing the mean square displacement. The time dependence of the mean square displacement seems to exhibit an anomalous diffusion in the intermediate region between the mean free path and the localization length of the system. The mean square displacement of the packet scaled only by the localization length does not seem to have a universal feature in its time evolution, which is different from that in the case of one-dimensional disordered systems.
1. Introduction Since the appearance o f the o n e - p a r a m e t e r scaling theory o f localization ( S T L ) in d i s o r d e r e d systems [ 1 ], m a n y theoretical, numerical and experimental studies have been d e v o t e d to the inspection o f the theory [ 2 ]. The theory suggests that almost all o f the electronic eigenstates are exponentially localized in an infinite system in one- and t w o - d i m e n s i o n a l disordered systems ( D D S ) with no external field 1 However, there are rigorous proofs o f the exponential localization only for the infinite 1-DDS [4,5 ]. A m a t h e m a t i c a l l y rigorous theory o f localization for 21Recently it was reported that the disordered diagonal dimer model corresponding to a one-dimensional tightly binding binary alloy has extended states, the number of which is proportional to x/N for a finite system size N. However, it must be noticed that this is not the result of the case of a finite system. In an infinite system only one accumulation point in the energy axis corresponds to the extended states. Details are reported in Ref. [3].
D D S does not exist yet, in spite o f much effort being m a d e [ 6 ]. M a c k i n n o n et al. calculated numerically the conductance in 2-DDS based on the L a n d a u e r formula by a skillful finite size scaling m e t h o d using strips, a n d got a result which is consistent with o n e - p a r a m eter STL [ 7 ]. Recently it is also shown that in a twod i m e n s i o n a l q u a n t u m site a n d b o n d percolation p r o b l e m almost all states are exponentially localized for any a m o u n t o f d i s o r d e r by the same scaling m e t h o d [ 8 ]. However, there is still a dispute about the form o f the eigenstates suggested by the numerical simulation o f q u a n t u m diffusion. The dispute is about whether there exists a m o b i l i t y edge separating the power-law localized state from exponentially localized states and also about whether diffusion is completely suppressed in 2-DDS even with weak d i s o r d e r [ 9,10 ]. The existence o f the power-law localization suggests a t w o - p a r a m e t e r STL [ 9 ]. Longer t i m e sim-
0375-9601/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI0375-9601 (94)00726-8
280
H. Yamada, M. Goda / PhysicsLettersA 194 (1994) 2 79-284
ulations in a larger system are needed for a reconciliation of the controversy. The localized eigenfunctions are directly related to the dynamical properties of a quantum particle as conduction and diffusion in the systems [4]. I f all of the eigenstates are exponentially localized, the diffusion o f a wavepacket is suppressed at about the localization length (LL) and as a result the conductivity vanishes. Though it is possible, in principle, to get a snapshot of the wavepacket at arbitrary time by several methods, for example, by direct diagonalization of the Hamiltonian matrix [ 11 ] or by numerical integration [ 12,13], it is difficult to get the numerical results with enough accuracy except for the case in 1DDS, because of the limitations of CPU time and the storage in the computer. It is also difficult to distinguish the delocalized wavepacket from the localized one numerically. Some pre-scaling studies had even suggested that there exists a transition to an extended state at finite disorder in 2-DDS [ 12,14 ]. When we use the calculational scheme of the time evolution with inadequate accuracy, the energy conservation of the system becomes insufficient [ 15 ]. The error of the numerical calculation sometimes works as a dissipation effect or an effect due to a contact with another degree of freedom [ 16 ]. Then the packet can exhibit an artificial diffusive behavior beyond the intrinsic LL. It is also interesting to investigate the correspondence between the quantum dynamics of a chaotic system and Anderson localization [ 17]. For a two-dimensional kicked rotor (2-DKR) localization takes place in the angular m o m e n t u m space and the localization length grows exponentially with respect to the mean free path numerically [ 18 ]. The rotor can be mapped on a two-dimensional tightly binding model exhibiting localization. However, it is worth noting that the 2-DKR is not an intrinsic 2-DDS and an anomalous diffusion takes place in a rotor with some kind of coupling [ 19 ]. Hitherto, there are only a few numerical calculations of the quantum diffusion with reliable enough accuracy in 2-DDS. Raedt has calculated the time evolution of the wavefunction by using a fourth-order symplectic integrator (SI) and a special initial state, which is diagonalized in the small subsystem. In this Letter, we report the property of pre-localiza-
tion of the packet in 2-DDS starting from a different type of the initial state. We use a sixth-order SI to get the time evolution of a wavepacket with enough accuracy [15,20]. The mean square displacement (MSD) of the packet exhibits a trend towards an anomalous diffusion in an intermediate regime between the ballistic propagation (BP) and the localization. We suggest numerically that the quantum diffusion in 2-DDS does not have a universal feature in its MSD that is scaled by only one parameter, different from the case of 1-DDS [21]. Our results are consistent with that in Ref. [ 10 ]. This study concerns only the transit phenomena and therefore is not able to distinguish power-law or exponential localization for the eigenstate of 2-DDS.
2. Models and numerical results
We consider a one-electron tight-binding Hamiltonian of localization described as follows, n= ~,V(n)ln)(nl+ n
~. K ( n , m ) I n ) ( m l ,
(1)
nmm
where the basis set { In ) } is an orthonormalized one, of which each base In) is localized near a two-dimensional site n, V(n) is the random on-site energy at site n and the transfer energy K(n, m) ( = - 1 ) vanishes unless the sites n, m are nearest neighbours. We use two types of random on-site energy. One is the Anderson model (A-model), in which each site energy varies at random uniformly in the range [ - IV, W], and the other is binary model (B-model), in which each site energy randomly can take only two values, - W o r IV. On numerically simulating the time development of the wavepacket some restrictions come from the following situations: The weaker the disorder strength W of the 2-DDS becomes, the faster the wave front of the packet reaches the boundary. On the other hand, as the disorder strength becomes large, the numerical accuracy becomes insufficient. We thus need a simulation of a large enough system with a higherorder time integration scheme and with a smaller time step to remove the above restrictions. Accordingly it is difficult to observe the diffusion process to the extent of fully exponential localization in 2-DDS before the wave front reaches the boundary. The numerical
H. Yamada, M. Goda / Physics Letters A 194 (I 994) 279-284
accuracy can be checked by doing a time-reversal experiment of the packet as in Ref. [ 15 ]. We use a sixlh-order SI with coefficients obtained by Yoshida [20] to calculate the time evolution of the packet by a unitary operator, e x p ( - i H t ) . The Planck constant and the size of the integration time step have been taken to be 1 and 0.025, respectively. The system size and the ensemble size are taken as N x = N y = 128 and 10, respectively, and we adapt periodic boundary conditions, because we use the FFT and inverse FFT to exchange the real space representation for the momentum space one and do the opposite operation [ 15,19 ]. We use the packet localized at the center of the system as the initial state, 7'( t = O, n) = d.,(ux/2, Ny/2)" Two characteristic scales exist in the time evolution of the packet. One is the mean free path (MFP), and the other is the LL. We observe the time evolution in the intermediate regime between MFP and LL. Fig. 1 shows some snapshots of the time evolution of a wavepacket. We can observe a process toward localization in the packet as a result of the interference effect between the scattering wave.
.18.16.14~ .12.i0.08.06.04 .02-
.050.045.040.035 = .030.025.020.015.010.005-
t=2.5
t=25
281
Next we consult the time dependence of the MSD of a packet monitoring the spread of the packet. The double bracket denotes the quantum mechanical average and the sample average. The MSDs for two kinds of site energy with the various disorder strengths W are shown in Fig. 2. It is difficult to get the analytical form of the LL and also to decide it exactly by numerical calculation in 2-DDS. It is, however, supposed that in 2-DDS almost all states are exponentially localized with a LL ~ that has the form ((R2))
~=aexp(b
-c/2)
(1~c~2) ,
(2)
where a and b are real constants. This is an extended form of the W-dependent LL which results from a numerical calculation based on one-parameter STL [ 7,8 ]. The < v z > denotes a second-order moment of random site energy. It is proportional to the square of the disorder strength W E and the inverse quantity is also almost proportional to the MFP of the system [ 18 ]. Accordingly, the LL is much longer than the MFP in 2-DDS [21 ], and we can observe the time evolution of the packet in the intermediate regime in
.018.016•014 .012.010.008.006•004 .00? -
.0090.0080.0070= .0060.0050.0040 .0030-
t=50
t=lO0
.0020.0010-
Fig. 1. Snapshots of a wavepacket at some times of development on the B-model of size 128 × 128 with disorder strength W = 1.7, at t=2.5, 25, 50, 100.
282
H. Yamada, ill. Goda I Physics Letters A 194 (1994) 2 79-284
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Fig. 2. Plots ofMSD ((R 2)) versus time for (a) the B-model and (b) the A-model, with various disorder strengths W.
Fig. 3. Log-log plots of the data in Fig. 2 for (a) the B-model and (b) the A-model.
between the above mentioned two characteristic scales, different from the case in 1-DDS, in which LL is of compatible order with the MFP [22]. Fig. 3 shows In-In plots of the data in Fig. 2. We can see the time dependence of MSD similar to that in anomalous diffusion (((R2))~t% 0 < a < 2 ) after a short period of the time evolution describing almost free or Bloch particles ( ((R 2)) ~ t 2). The power exponent a is about 0.75, 0.65 for the B-model with W= 1.5, 2.0, respectively, depending on the value of W, and 0.80 for the A-model with W=2.5. The relative error of the exponents is less than about 5%. Raedt has reported that an anomalous diffusion seems to occur in 2-DDS with a weak disorder strength under a condition different from ours [ 10 ]. Our results are consis-
tent with theirs. However, more long-time simulations are needed to get the exponent for the cases with a weaker disorder strength and to obtain a clear and comprehensive conclusion about the anomalous diffusion. We cannot decide the exponents in the system with large LL from these data because of a weak systematic change of the power-index with respect to time. Next we consider whether MSD can be scaled only by one parameter as a function of time. We assume the following scaling form as in the case of 1-DDS
[211, ((x(t)2)) =
(~/~o)2F{t(~/~o)-]},
(3)
where (o is the LL of a standard case (B-model,
H. Yamada, M. Goda / Physics Letters A 194 (1994) 279-284
W = 0 . 9 ) . This form is based on the request that the MSD behaves as t 2 with the same coefficient in the vicinity o f time t = 0 (ballistic regime). Fig. 4 shows the time dependence o f the MSD scaled by the LL. The coefficient b in Eq. (2) is selected so that the scaled MSDs are made to fit as well as possible for each value o f c in the interval [ 1,2] in the B-model. However, we cannot find a universal form o f the function F ( t ) even if we extended the unit interval o f c to the entire positive real axis. The same result is obtained also for the case o f the A-model. If the MSD has a different power-index a which depends on disorder strength in this regime, the MSD cannot have a universal form of the function F ( t ) in Eq. ( 3 ), even
(a) I
I I I I .................'""
I
"I
30002500-
283
if we adapt any time-independent scaling factor instead o f adapting the form of the LL (2). To summarize, we investigated the time evolution of a wavepacket in some 2-DDSs by numerical calculation with high accuracy. We confirm the pre-localization behavior of the packet in the MSD so that it behaves as in the case of anomalous diffusion. These results suggest that the "microscopic personalities" (for example, type of model and disorder strength) o f 2-DDS characterizing the system appears in the time evolution o f the packet in the intermediate regime between the M F P and the LL. In relation to this fact we could not find a scaling form (3) in the MSD in the case of 2-DDS. However, this result does not directly claim to be the evidence supporting the twoparameter scaling theory o f localization. It is interesting to study the relation between this result and the statistical feature of probability amplitude in the wavefunction typically exemplified in Fig. 1. This problem is reported in detail elsewhere.
~2000-
~1500-
f ~ - - . .
B-MODEL,W=0.7 B'MODEL,W=0.9 I ..... B-MODEL,W=1.1 ]---" B-MODEL, W=1.3 ..... B-MODEL, W=1.51
./.."/
1000-
f',.:~ f /~
5000 0
510
II B'MODEL'W=2"O[ .............
I 100
I 150
I 200
I 250
I 300
t (~J~)-~
1 2500
Acknowledgement
[--
///I
±
±
±
-
. ~
000 (b)
.."
The authors would like to thank Dr. T. Okabe for visualization o f the data. The authors also acknowledge many discussions about the symplectic integrator with Professor K. Ikeda and Professor K. Takahashi. The authors are also thankful to the referee who pointed out m a n y insufficient points in the original manuscript. Numerical calculations of this study were carried out by the supercomputers at the Data Processing Center of Kyoto University.
,¢
-~1500
References
v 1000 50o 0 0
50
100
150
t
200
(,~/,~)-'
250
300
Fig. 4. Plots of scaled MSD versus scaled time on the B-model with various disorder strengths in the case of coefficients in Eq. (2) (a) c=2, (b) c= 1.
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H. Yamada, M. Goda / Physics Letters A 194 (1994) 279-284
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