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Interplay of localization and coherence effects
Volume 145, number 5
PHYSICS LETTERS A
16 April 1990
INTERPLAY OF LOCALIZATION AND COHERENCE EFFECTS Avinash S I N G H a n d Bala S U N D A R A M J Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA
Received 23 October 1989; revised manuscript received 29 January 1990; accepted for publication 8 February 1990 Communicated by A.R. Bishop
The interplay of disorder and coherence effects in a 1-D tight-binding system with a long-range hopping (off-diagonal) term of the form 1/r y is shown to exhibit a rich variety of localization behaviour. When 7< 2 long-range hopping strongly reduces the effective disorder strength (by reducing the density of available states to scatter into) for highly coherent states which are consequently very weakly localized. We show the existence of a crossover at 7= 2 beyond which all states are strongly (exponentially) localized. These results may be of relevance in interpreting the localization behaviour in the quantal dynamics of nonlinear systems which can be mapped to tight-binding models with long-range hopping terms.
All q u a n t u m states in a disordered, o n e - d i m e n sional system represented, for example, by the Anderson model with nearest-neighbor hopping are well known to be localized, however weak the d i s o r d e r [ 1 ]. The concept o f weak scattering, in fact, breaks down because a diffusive p a n i c l e p e r f o r m i n g a rand o m walk will, in a sufficiently long time, pass arbitrarily close to the initial site. The effective disorder is therefore always strong, a n d the localization length is o f the same o r d e r as the m e a n free path. An appealing physical picture is p r o v i d e d by the interference concept wherein the e n h a n c e d backscattering p r o b a b i l i t y due to the constructive interference o f scattering a m p l i t u d e s leads to a suppression o f diffusion [ 2 ]. Recently localization theory has been extensively used to discuss the quantal d y n a m i c s o f nonlinear systems such as the kicked rotor [ 3 ] a n d the driven surface state electron ( D S S E ) [ 4 ]. In particular, the suppression o f the classical predicted, diffusive energy growth in these two systems is seen to be a direct consequence o f q u a n t u m mechanical interference effects. In fact, using a basis o f Floquet or quasienergy states, the kicked rotor has been analytically m a p p e d onto a tight-binding, Anderson-like Present address: Theoretical Division, MSJ569, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. 232
H a m i l t o n i a n with off-diagonal or " h o p p i n g " matrix elements not necessarily restricted to nearest neighbors [ 3 ]. Also the diagonal "on-site" terms have been shown to satisfy a p s e u d o r a n d o m distribution. F o r the DSSE p r o b l e m the quasienergy states, o b t a i n e d numerically from the one-cycle time-evolution operator, have recently been used to construct an effective tight-binding H a m i l t o n i a n [5 ]. Such mappings to a tight-binding form have p r o v i d e d a d d e d insight into the quantal b e h a v i o r o f nonlinear m a p s [3,4]. Formally, our m o t i v a t i o n to introduce long-range h o p p i n g comes partly from the observation that in the m a p p i n g o f the kicked rotor p r o b l e m to the Anderson model [3], the hopping terms relate to the F o u r i e r transform o f the tangent o f ½V(O), where V(0) is the external potential, and therefore nearestneighbor or short-range coupling puts a stringent restriction on the form o f V(O). Moreover, the tightb i n d i n g equivalent o f the DSSE p r o b l e m [ 5 ] clearly reveals the long-range nature o f the off-diagonal ( h o p p i n g ) terms. F r o m a pedagogical point o f view, understanding the interplay between localization and coherence effects, arising due to long-range hopping, m a y help extend the analogy with d i s o r d e r e d electronic systems to quantized versions o f other nonlinear maps, such as the A r n o l d cat m a p [ 3 ]. In this Letter we study localization characteristics
0375-9601/90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
Volume 145, number 5
PHYSICSLETTERSA
with a long-range hopping term of the form V/r7.We report an interesting crossover behaviour, as 2) goes below 2, from strong, exponential to weak localization. We show that this crossover is because states with small k ( << 1 ) acquire a measure of rigidity from long-range hopping which makes the wavefunctions less susceptible to disorder and hence weakly localized. For these states the phase coherence is not destroyed by disorder (for Y< 2 ) as a consequence of long-range hopping. In terms of the properties of the disorderless system, this crossover originates from a drastic change in the nature of the small-k (k << l ) dispersion relation as Y goes below 2. It was suggested [3] that for Y4 1 extended states are possible in a one-dimensional system. Thus with increasing range of hopping (decreasing y) the transition from all strongly localized states to some extended states goes through a crossover at y = 2 with the emergence of some weakly localized states. We consider the following Hamiltonian,
H= ~i eia*,ai- X
V (at,aj+a]ai),
(1)
li-jl =r
where ¢i are the random on-site energies chosen independently from a uniform distribution on - ~ W to ½W. The nearest-neighbour ( N N ) case is obtained in the limit y-,oo. The energy dispersion relation in the zero disorder limit ( W= O) is obtained by Fourier transformation to momentum space,
Er(k)=-2V ~ cos(kr) r=
I
(2)
rr
For integer y a particularly convenient way to perform the sum is to differentiate Er(k) y times with respect to k to eliminate the denominator. Summing cos(kr) and sin(kr) (depending on whether y is even or odd), it is possible to reconstruct E(k). For even y this method gives results that can be expressed in terms of Bernoulli polynomials [ 6 ]. Explicitly, the energies for y of 2 and 4 are given by E 2(k) / V= - •2/3 + x k - k 2 / 2 ,
E4(k)/V=-n4/45+n2k2/3!-~k3/3!+k4/4!. (3) Odd values of y lead to a more complicated k-dependence, as illustrated by
El (k)/V=log[2( 1 - c o s k) ] ,
16 April 1990
E3(k)/V= - 2 ( ( 3 ) + 3k2-k2 logk (for small k) ,
(4)
where ( ( n ) is the Riemann zeta function. It is clear that the qualitative nature of the spectrum is modified by the long-range hopping. As expected the energy band is no longer symmetric about E = 0 as in the NN case. This is because states with small k (more coherent) are increasingly pulled down in energy with decreasing y. This reflects the gain in coherence energy from long-range hopping. The spectrum becomes unbounded from below for an infinite lattice size when y~< 1. However, for a finite size system with n sites the lower-edge energy diverges with n. The scaling form varies from a weak logarithmic divergence for y= 1 to a strong linear divergence for y=0. It should be noted that, for the generalized KR problem, the form of the potential V(O) necessary to produce the corresponding 1/r y coupling term is expressed in terms of E r by the relation V ( 0 ) = 2 tan - I [Er(0) ]. Another consequence of long-range hopping shows up in the dispersion relation for coherent states (k<< 1 ). For y~> 3, E(k) varies quadratically with k, as for N N hopping. Generally, for y> 2, the slope of E(k) vanishes as k ~ 0 . However, at y = 2 there is crossover and the energy behaves linearly with k. For y< 2, E(k) actually has a cusp at k = 0 . The small-k behaviour of E r ( k ) - E r ( 0 ) has been illustrated in fig. 1 for these three cases. Therefore, 7 = 2 marks the point where the density of states (DOS), N(E), which is proportional to the inverse of the slope, dEr(k)/dk, stops being divergent at the lower edge of the band. In fact, for general values ofy in the range l~
Volume 145, number 5
PHYSICS LETTERS A
),>2
-r=2
/
Fig. I. An illustration of the k-dependence (for small k) of the energy dispersion relation in the three regimes showing the vanishing slope at k=0 for y> 2, linear behaviour for 7=2 and cusp for 7< 2. goes elastic scattering on the average in time z. Since the scattering probability depends on availability of other m o m e n t u m states, the effective disorder strength, as measured by z-1, also depends on the density of states and is given by z - ~ ( E ) ~ N ( E ) W : . In view of the vanishing density o f states as k ~ 0 when y < 2 , it is clear that effectively the impurity scattering o f highly coherent states is diminished and quantum states with energies near the lower band edge feel the localization effect o f disorder to a much lesser degree. Using E to denote the band energy as measured from the lower band edge, for 1 ~<7~<3 and small k, we have E-E(k)-E(O)~k
~-1 .
16 April 1990
states near the bottom of the band (k small), effective impurity scattering is weakened due to their high degree o f coherence. With increasing 7 there is a crossover at 7 = 2 and when 7 > 2 the effective disorder is strong and the disorder strength actually begins to decrease with energy. This approximate analysis is borne out in our exact study of the effects of disorder in terms of the inverse participation ratio ( I P R ) [ 7 ] and the frequency-dependent conductivity, a(to). It should be noted that long-range hopping (with a falloff slower than 1/r 3) is an unphysical representation o f the kinetic energy term and that our study o f the conductivity properties is solely to extract the localization behaviour. These quantities are evaluated using the exact eigenstates obtained by diagonalization o f the Hamiltonian (eq. ( 1 ) ) with a periodic boundary condition. The configuration averaged IPR, defined as Iz=~i(~i4/(Y~i Oil) :, is plotted in fig. 2 as a function o f energy measured from the lower band edge for IV~ V= 5.0, a lattice o f length L = 150 and for several values of 7. The IPR is a measure o f the extent of spread, over the sites, of each exact eigenstate and thus contains information about the localization .6
I
'
'
'
I
'
'
'
I
'
'
'
I
'
'
'
I
O
o o°
L=,5o
.
'
'
^~
~'/
W=5.O
0
'
I
'
a
o~
-
~
oil
o
.~e~°<~
,g2.
e
o
.4 0
N • ..
0
.:.'"
t~ .2
"~-,~
h
~o
_= °Oo~
~
(5) o
Therefore, within the Boltzmann approximation, the elastic scattering rate is given by r - ' (E) ~ E (2-r)/(r- ~
0
,
,
,
I
2
,
*
,
I
4
,
,
,
I
6
Energy
,
,
,
I
8
,
,
,
I
,
10
(6)
The effective disorder strength, as measured by the elastic scattering rate, therefore vanishes as E--,0 when y < 2. This explicitly shows that for q u a n t u m 234
I
Fig. 2. Variation of inverse participation ratio with energy (measured from the lower band edge) across the band for IV~ V= 5.0 and for several values of y: (open)squares: 1.2; triangles: 1.5; circles: 2.0 and (filled) squares: 3.0.
Volume 145, n u m b e r 5
PHYSICS LETTERS A
length and hence about the effective microscopic disorder parameters - the mean free path or the elastic scattering rate. We draw attention to the behaviour of the IPR near the lower band edges. It is clear that for y= 1.2 and 1.5, states near the lower band edge are almost extended with roughly all the sites in the lattice coupled. Furthermore the IPR for y= 1.2 rises slower with energy, as expected from eq. (6). Notice also the absence of any fluctuations due to disorder near the lower band edge. The crossover at y=2 is also clear. As y goes above 2, behaviour of IPR with energy reverses and it begins to decrease with energy. For values ofy above 2, the density of states diverges at both band edges and there is a minimum which shifts towards the center of the band with increasing y. This is reflected in the IPR as shown in fig. 2 for y=3. This analysis shows that when y < 2 highly coherent states (near the lower band edge ) are very weakly localized whereas those near the upper band edge are strongly localized. Study of o(to), evaluated using the Kubo-Greenwood formula, in different sections of the band further confirms this picture. The averaging over several configurations of the random potential leads to the following expression, o(to) =nhlMIm 12N2(E)/L d ,
(7)
where M~,~ is the matrix element of the current operator between eigenstates. In terms of the exact eigenstates IMIm 12 is given by e2V2 _ 2 IMlm [2= h2 ~ (0~0Jm--q~0~) . (8) IMtml 2 is averaged for fixed l - m (for eigenvatues ordered with El increasing with l) and the appropriate frequency is simply the average of E l - E m [8]. The DOS, N ( E ) , can then be evaluated from the mean level spacing for each section. For clarity, we isolate the frequency dependence by normalizing a(m) by a ( l - m = l ) . To demonstrate the changing localization character, we segment the entire band into three sections. Fig. 3 shows the normalized a(m) evaluated for the three sections, where 1 through 3 traverse the band from the lower edge. The conductivity for section 3 falls sharply with decreasing frequency which is a clear indication that states are strongly localized in
16 April 1990
10
8
.6
b v
3 b
4
0
,
0.0
,
,
I
,
,
,
0.2
I 0.4
,
,
,
I
,
0.6
hco/v Fig. 3. Normalized low-frequency conductivity for three sections of the band for W/V= 5.0, y = 1.5 and L = 150: starting from the lower band edge, triangles: 1; squares: 2; and circles: 3. The sections are centered at energies (in units o f V) o f - 4.2, 0.2 and 2.4 respectively. The curves are drawn as guides to the eye.
this section as expected for a typical one-dimensional system [9]. This is because localized states close in energy are statistically unlikely to overlap strongly. The large fluctuation in the conductivity is typical of one-dimensional systems and indicates that the conductivity (or the square of the current matrix element) does not have a normal distribution but rather a log-normal distribution. To take this into account we have also performed the configuration average of IoglMtm 12. The conductivity evaluated in this manner gives similar frequency behaviour but with the fluctuations greatly reduced. Proceeding towards the lower band edge, where the coherence effect due to long-range hopping becomes important, one sees that in section 1 the frequency dependence of tr(to) is almost Drude-like which is a clear indication of very weakly localized states. To ascertain that states near the lower band edge are still localized (though with very large localization lengths) and not really extended, we have performed a renormalization group study using the scaling behaviour of the conductance. The dimensionless conductance is defined by g(to)=hLd-2tr(to)/e 2. The renormalization group method used is based on preserving macroscopic physical properties as the 235
Volume 145, number 5
PHYSICS LETTERS A
lattice size is varied [8]. The low-frequency conductance is evaluated for various lattice sizes, several g a m m a values ( 1 ~<7~< 2) and for d i s o r d e r strengths down to W/V= 3 (below which finite size effects begin to a p p e a r ) . In all cases the conductance scales down with increasing lattice size. This indicates that the flow is towards the insulating regime and that all states are indeed localized. In conclusion, the interplay o f effects o f d i s o r d e r and coherence in a simple 1-D system with long-range hopping exhibits a rich variety o f localization behaviour. The effective scattering strength is greatly reduced for highly coherent states by long-range hopping and arises from a reduction in the density o f available states to scatter into. W h e n 7 < 2 , this effective reduction in d i s o r d e r strength for coherent states results in very weakly (possibly power-law) localized states near the lower b a n d edge [5]. We have shown the existence o f a crossover at 7 = 2 bey o n d which all states are strongly ( e x p o n e n t i a l l y ) localized. This crossover has been shown to originate from the drastic change in the nature o f the small-k dispersion relation as 7 crosses 2. These results m a y be o f relevance in interpreting the localization be-
236
16 April 1990
haviour in the quantal d y n a m i c s o f nonlinear systems which can be m a p p e d to tight-binding models with long-range hopping terms such as the DSSE p r o b l e m [ 5 ] and the A r n o l d cat m a p [ 3 ]. This work was s u p p o r t e d in part by the N a t i o n a l Science F o u n d a t i o n grant No. PHY-8518368.
References [ 1] K. lshii, Prog. Theor. Phys. Suppl. 53 (1973) 77. [2] G. Bergmann, Phys. Rep. 107 (1984) 1; D.E. Khmelnitskii, Physica B 126 (1984) 235. [3] S. Fishman, D.R. Grempel and R.E. Prange, Phys. Rev. Lett. 49 (1982) 509; D.R. Grempel, R.E. Prange and S. Fishman, Phys. Rev. A 29 (1984) 1639. [4] G. Casati, B.V. Chirikov, D.L. Shepelyanskyand 1. Guarneri, Phys. Rep. 154 (1987) 77. [5] B. Sundaram and A. Singh, Phys. Lett. A 140 (1989) 400. [ 6 ] I.S. Gradshteyn and I.M. Ryzhik, Tables of integrals, series and products (Academic Press, New York, 1980). [7] D. Weaire and V. Srivastava, J. Phys. C 10 (1977) 4309. [8] A. Singh and W.L. McMillan, J. Phys. C 17 (1985) 2097. [9] P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57 (1985 ) 287.
PHYSICS LETTERS A
16 April 1990
INTERPLAY OF LOCALIZATION AND COHERENCE EFFECTS Avinash S I N G H a n d Bala S U N D A R A M J Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA
Received 23 October 1989; revised manuscript received 29 January 1990; accepted for publication 8 February 1990 Communicated by A.R. Bishop
The interplay of disorder and coherence effects in a 1-D tight-binding system with a long-range hopping (off-diagonal) term of the form 1/r y is shown to exhibit a rich variety of localization behaviour. When 7< 2 long-range hopping strongly reduces the effective disorder strength (by reducing the density of available states to scatter into) for highly coherent states which are consequently very weakly localized. We show the existence of a crossover at 7= 2 beyond which all states are strongly (exponentially) localized. These results may be of relevance in interpreting the localization behaviour in the quantal dynamics of nonlinear systems which can be mapped to tight-binding models with long-range hopping terms.
All q u a n t u m states in a disordered, o n e - d i m e n sional system represented, for example, by the Anderson model with nearest-neighbor hopping are well known to be localized, however weak the d i s o r d e r [ 1 ]. The concept o f weak scattering, in fact, breaks down because a diffusive p a n i c l e p e r f o r m i n g a rand o m walk will, in a sufficiently long time, pass arbitrarily close to the initial site. The effective disorder is therefore always strong, a n d the localization length is o f the same o r d e r as the m e a n free path. An appealing physical picture is p r o v i d e d by the interference concept wherein the e n h a n c e d backscattering p r o b a b i l i t y due to the constructive interference o f scattering a m p l i t u d e s leads to a suppression o f diffusion [ 2 ]. Recently localization theory has been extensively used to discuss the quantal d y n a m i c s o f nonlinear systems such as the kicked rotor [ 3 ] a n d the driven surface state electron ( D S S E ) [ 4 ]. In particular, the suppression o f the classical predicted, diffusive energy growth in these two systems is seen to be a direct consequence o f q u a n t u m mechanical interference effects. In fact, using a basis o f Floquet or quasienergy states, the kicked rotor has been analytically m a p p e d onto a tight-binding, Anderson-like Present address: Theoretical Division, MSJ569, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. 232
H a m i l t o n i a n with off-diagonal or " h o p p i n g " matrix elements not necessarily restricted to nearest neighbors [ 3 ]. Also the diagonal "on-site" terms have been shown to satisfy a p s e u d o r a n d o m distribution. F o r the DSSE p r o b l e m the quasienergy states, o b t a i n e d numerically from the one-cycle time-evolution operator, have recently been used to construct an effective tight-binding H a m i l t o n i a n [5 ]. Such mappings to a tight-binding form have p r o v i d e d a d d e d insight into the quantal b e h a v i o r o f nonlinear m a p s [3,4]. Formally, our m o t i v a t i o n to introduce long-range h o p p i n g comes partly from the observation that in the m a p p i n g o f the kicked rotor p r o b l e m to the Anderson model [3], the hopping terms relate to the F o u r i e r transform o f the tangent o f ½V(O), where V(0) is the external potential, and therefore nearestneighbor or short-range coupling puts a stringent restriction on the form o f V(O). Moreover, the tightb i n d i n g equivalent o f the DSSE p r o b l e m [ 5 ] clearly reveals the long-range nature o f the off-diagonal ( h o p p i n g ) terms. F r o m a pedagogical point o f view, understanding the interplay between localization and coherence effects, arising due to long-range hopping, m a y help extend the analogy with d i s o r d e r e d electronic systems to quantized versions o f other nonlinear maps, such as the A r n o l d cat m a p [ 3 ]. In this Letter we study localization characteristics
0375-9601/90/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland)
Volume 145, number 5
PHYSICSLETTERSA
with a long-range hopping term of the form V/r7.We report an interesting crossover behaviour, as 2) goes below 2, from strong, exponential to weak localization. We show that this crossover is because states with small k ( << 1 ) acquire a measure of rigidity from long-range hopping which makes the wavefunctions less susceptible to disorder and hence weakly localized. For these states the phase coherence is not destroyed by disorder (for Y< 2 ) as a consequence of long-range hopping. In terms of the properties of the disorderless system, this crossover originates from a drastic change in the nature of the small-k (k << l ) dispersion relation as Y goes below 2. It was suggested [3] that for Y4 1 extended states are possible in a one-dimensional system. Thus with increasing range of hopping (decreasing y) the transition from all strongly localized states to some extended states goes through a crossover at y = 2 with the emergence of some weakly localized states. We consider the following Hamiltonian,
H= ~i eia*,ai- X
V (at,aj+a]ai),
(1)
li-jl =r
where ¢i are the random on-site energies chosen independently from a uniform distribution on - ~ W to ½W. The nearest-neighbour ( N N ) case is obtained in the limit y-,oo. The energy dispersion relation in the zero disorder limit ( W= O) is obtained by Fourier transformation to momentum space,
Er(k)=-2V ~ cos(kr) r=
I
(2)
rr
For integer y a particularly convenient way to perform the sum is to differentiate Er(k) y times with respect to k to eliminate the denominator. Summing cos(kr) and sin(kr) (depending on whether y is even or odd), it is possible to reconstruct E(k). For even y this method gives results that can be expressed in terms of Bernoulli polynomials [ 6 ]. Explicitly, the energies for y of 2 and 4 are given by E 2(k) / V= - •2/3 + x k - k 2 / 2 ,
E4(k)/V=-n4/45+n2k2/3!-~k3/3!+k4/4!. (3) Odd values of y lead to a more complicated k-dependence, as illustrated by
El (k)/V=log[2( 1 - c o s k) ] ,
16 April 1990
E3(k)/V= - 2 ( ( 3 ) + 3k2-k2 logk (for small k) ,
(4)
where ( ( n ) is the Riemann zeta function. It is clear that the qualitative nature of the spectrum is modified by the long-range hopping. As expected the energy band is no longer symmetric about E = 0 as in the NN case. This is because states with small k (more coherent) are increasingly pulled down in energy with decreasing y. This reflects the gain in coherence energy from long-range hopping. The spectrum becomes unbounded from below for an infinite lattice size when y~< 1. However, for a finite size system with n sites the lower-edge energy diverges with n. The scaling form varies from a weak logarithmic divergence for y= 1 to a strong linear divergence for y=0. It should be noted that, for the generalized KR problem, the form of the potential V(O) necessary to produce the corresponding 1/r y coupling term is expressed in terms of E r by the relation V ( 0 ) = 2 tan - I [Er(0) ]. Another consequence of long-range hopping shows up in the dispersion relation for coherent states (k<< 1 ). For y~> 3, E(k) varies quadratically with k, as for N N hopping. Generally, for y> 2, the slope of E(k) vanishes as k ~ 0 . However, at y = 2 there is crossover and the energy behaves linearly with k. For y< 2, E(k) actually has a cusp at k = 0 . The small-k behaviour of E r ( k ) - E r ( 0 ) has been illustrated in fig. 1 for these three cases. Therefore, 7 = 2 marks the point where the density of states (DOS), N(E), which is proportional to the inverse of the slope, dEr(k)/dk, stops being divergent at the lower edge of the band. In fact, for general values ofy in the range l~
Volume 145, number 5
PHYSICS LETTERS A
),>2
-r=2
/
Fig. I. An illustration of the k-dependence (for small k) of the energy dispersion relation in the three regimes showing the vanishing slope at k=0 for y> 2, linear behaviour for 7=2 and cusp for 7< 2. goes elastic scattering on the average in time z. Since the scattering probability depends on availability of other m o m e n t u m states, the effective disorder strength, as measured by z-1, also depends on the density of states and is given by z - ~ ( E ) ~ N ( E ) W : . In view of the vanishing density o f states as k ~ 0 when y < 2 , it is clear that effectively the impurity scattering o f highly coherent states is diminished and quantum states with energies near the lower band edge feel the localization effect o f disorder to a much lesser degree. Using E to denote the band energy as measured from the lower band edge, for 1 ~<7~<3 and small k, we have E-E(k)-E(O)~k
~-1 .
16 April 1990
states near the bottom of the band (k small), effective impurity scattering is weakened due to their high degree o f coherence. With increasing 7 there is a crossover at 7 = 2 and when 7 > 2 the effective disorder is strong and the disorder strength actually begins to decrease with energy. This approximate analysis is borne out in our exact study of the effects of disorder in terms of the inverse participation ratio ( I P R ) [ 7 ] and the frequency-dependent conductivity, a(to). It should be noted that long-range hopping (with a falloff slower than 1/r 3) is an unphysical representation o f the kinetic energy term and that our study o f the conductivity properties is solely to extract the localization behaviour. These quantities are evaluated using the exact eigenstates obtained by diagonalization o f the Hamiltonian (eq. ( 1 ) ) with a periodic boundary condition. The configuration averaged IPR, defined as Iz=~i(~i4/(Y~i Oil) :, is plotted in fig. 2 as a function o f energy measured from the lower band edge for IV~ V= 5.0, a lattice o f length L = 150 and for several values of 7. The IPR is a measure o f the extent of spread, over the sites, of each exact eigenstate and thus contains information about the localization .6
I
'
'
'
I
'
'
'
I
'
'
'
I
'
'
'
I
O
o o°
L=,5o
.
'
'
^~
~'/
W=5.O
0
'
I
'
a
o~
-
~
oil
o
.~e~°<~
,g2.
e
o
.4 0
N • ..
0
.:.'"
t~ .2
"~-,~
h
~o
_= °Oo~
~
(5) o
Therefore, within the Boltzmann approximation, the elastic scattering rate is given by r - ' (E) ~ E (2-r)/(r- ~
0
,
,
,
I
2
,
*
,
I
4
,
,
,
I
6
Energy
,
,
,
I
8
,
,
,
I
,
10
(6)
The effective disorder strength, as measured by the elastic scattering rate, therefore vanishes as E--,0 when y < 2. This explicitly shows that for q u a n t u m 234
I
Fig. 2. Variation of inverse participation ratio with energy (measured from the lower band edge) across the band for IV~ V= 5.0 and for several values of y: (open)squares: 1.2; triangles: 1.5; circles: 2.0 and (filled) squares: 3.0.
Volume 145, n u m b e r 5
PHYSICS LETTERS A
length and hence about the effective microscopic disorder parameters - the mean free path or the elastic scattering rate. We draw attention to the behaviour of the IPR near the lower band edges. It is clear that for y= 1.2 and 1.5, states near the lower band edge are almost extended with roughly all the sites in the lattice coupled. Furthermore the IPR for y= 1.2 rises slower with energy, as expected from eq. (6). Notice also the absence of any fluctuations due to disorder near the lower band edge. The crossover at y=2 is also clear. As y goes above 2, behaviour of IPR with energy reverses and it begins to decrease with energy. For values ofy above 2, the density of states diverges at both band edges and there is a minimum which shifts towards the center of the band with increasing y. This is reflected in the IPR as shown in fig. 2 for y=3. This analysis shows that when y < 2 highly coherent states (near the lower band edge ) are very weakly localized whereas those near the upper band edge are strongly localized. Study of o(to), evaluated using the Kubo-Greenwood formula, in different sections of the band further confirms this picture. The averaging over several configurations of the random potential leads to the following expression, o(to) =nhlMIm 12N2(E)/L d ,
(7)
where M~,~ is the matrix element of the current operator between eigenstates. In terms of the exact eigenstates IMIm 12 is given by e2V2 _ 2 IMlm [2= h2 ~ (0~0Jm--q~0~) . (8) IMtml 2 is averaged for fixed l - m (for eigenvatues ordered with El increasing with l) and the appropriate frequency is simply the average of E l - E m [8]. The DOS, N ( E ) , can then be evaluated from the mean level spacing for each section. For clarity, we isolate the frequency dependence by normalizing a(m) by a ( l - m = l ) . To demonstrate the changing localization character, we segment the entire band into three sections. Fig. 3 shows the normalized a(m) evaluated for the three sections, where 1 through 3 traverse the band from the lower edge. The conductivity for section 3 falls sharply with decreasing frequency which is a clear indication that states are strongly localized in
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10
8
.6
b v
3 b
4
0
,
0.0
,
,
I
,
,
,
0.2
I 0.4
,
,
,
I
,
0.6
hco/v Fig. 3. Normalized low-frequency conductivity for three sections of the band for W/V= 5.0, y = 1.5 and L = 150: starting from the lower band edge, triangles: 1; squares: 2; and circles: 3. The sections are centered at energies (in units o f V) o f - 4.2, 0.2 and 2.4 respectively. The curves are drawn as guides to the eye.
this section as expected for a typical one-dimensional system [9]. This is because localized states close in energy are statistically unlikely to overlap strongly. The large fluctuation in the conductivity is typical of one-dimensional systems and indicates that the conductivity (or the square of the current matrix element) does not have a normal distribution but rather a log-normal distribution. To take this into account we have also performed the configuration average of IoglMtm 12. The conductivity evaluated in this manner gives similar frequency behaviour but with the fluctuations greatly reduced. Proceeding towards the lower band edge, where the coherence effect due to long-range hopping becomes important, one sees that in section 1 the frequency dependence of tr(to) is almost Drude-like which is a clear indication of very weakly localized states. To ascertain that states near the lower band edge are still localized (though with very large localization lengths) and not really extended, we have performed a renormalization group study using the scaling behaviour of the conductance. The dimensionless conductance is defined by g(to)=hLd-2tr(to)/e 2. The renormalization group method used is based on preserving macroscopic physical properties as the 235
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PHYSICS LETTERS A
lattice size is varied [8]. The low-frequency conductance is evaluated for various lattice sizes, several g a m m a values ( 1 ~<7~< 2) and for d i s o r d e r strengths down to W/V= 3 (below which finite size effects begin to a p p e a r ) . In all cases the conductance scales down with increasing lattice size. This indicates that the flow is towards the insulating regime and that all states are indeed localized. In conclusion, the interplay o f effects o f d i s o r d e r and coherence in a simple 1-D system with long-range hopping exhibits a rich variety o f localization behaviour. The effective scattering strength is greatly reduced for highly coherent states by long-range hopping and arises from a reduction in the density o f available states to scatter into. W h e n 7 < 2 , this effective reduction in d i s o r d e r strength for coherent states results in very weakly (possibly power-law) localized states near the lower b a n d edge [5]. We have shown the existence o f a crossover at 7 = 2 bey o n d which all states are strongly ( e x p o n e n t i a l l y ) localized. This crossover has been shown to originate from the drastic change in the nature o f the small-k dispersion relation as 7 crosses 2. These results m a y be o f relevance in interpreting the localization be-
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haviour in the quantal d y n a m i c s o f nonlinear systems which can be m a p p e d to tight-binding models with long-range hopping terms such as the DSSE p r o b l e m [ 5 ] and the A r n o l d cat m a p [ 3 ]. This work was s u p p o r t e d in part by the N a t i o n a l Science F o u n d a t i o n grant No. PHY-8518368.
References [ 1] K. lshii, Prog. Theor. Phys. Suppl. 53 (1973) 77. [2] G. Bergmann, Phys. Rep. 107 (1984) 1; D.E. Khmelnitskii, Physica B 126 (1984) 235. [3] S. Fishman, D.R. Grempel and R.E. Prange, Phys. Rev. Lett. 49 (1982) 509; D.R. Grempel, R.E. Prange and S. Fishman, Phys. Rev. A 29 (1984) 1639. [4] G. Casati, B.V. Chirikov, D.L. Shepelyanskyand 1. Guarneri, Phys. Rep. 154 (1987) 77. [5] B. Sundaram and A. Singh, Phys. Lett. A 140 (1989) 400. [ 6 ] I.S. Gradshteyn and I.M. Ryzhik, Tables of integrals, series and products (Academic Press, New York, 1980). [7] D. Weaire and V. Srivastava, J. Phys. C 10 (1977) 4309. [8] A. Singh and W.L. McMillan, J. Phys. C 17 (1985) 2097. [9] P.A. Lee and T.V. Ramakrishnan, Rev. Mod. Phys. 57 (1985 ) 287.