Internal geometry of delocalised and localised states in a one-dimensional, continuous quasi-periodic potential

Physica A 189 1992 39°-402 North-Holland

Internal geometry of delocalised and localised states in a one-dimensional, continuous quasi-periodic potential Susil K u m a r M a n n a 1, Chaitali Basu and Abhijit M o o k e r j e e S.N. Bose National Centre for Basic Sciences, DB 17, Sector 1, Salt Lake City, Calcutta 700064, India

Received 5 May 1992

We study the transmittance of a continuous quasi-periodic chain through the invariant embedding approach to study the localisation-delocalisation transition. The phase development of the electron wave as it moves through the chain is studied through the Argand map of the complex transmission coefficient. The internal geometric structure of the transmittance along the chain is studied first by multiaffine and then by multifractal analysis.

I. Introduction

In recent times, one-dimensional quasi-periodic systems have evoked much interest because of their rich electronic properties. In a one-dimensional periodic system, the spectrum of the Hamiltonian is absolutely continuous and all the eigenstates are extended. In a disordered system, on the other hand, the spectrum is point-like and all states are localised for any degree of disorder [1,2]. In one dimension, no metal-insulator transition is observed in disordered systems. However, if the random potential is made inhomogeneous by introducing a spatial factor ( - x - a ) , a localisation-delocalisation transition is observed [3,4]. A periodic model can be made quasi-periodic by introducing inhomogeneity in the period of the potential. The most extensively studied quasi-periodic model is the one-dimensional tight-binding Anderson model with Aubry potential [5,6]: V. = V cos(2~rnQ) , On leave of absence from: P.T. Mahavidyalaya, Midnapore, India under F.I.P. of University Grants Commission, India. 0378-4371/92/$05.00 (~) 1992- Elsevier Science Publishers B.V. All rights reserved

S.K. Manna et al. / (De-)localised states in a quasi-periodic potential

391

where Q is incommensurate with the lattice. The other model which has aroused great interest is the Harper model [7-9]. Here the potential form is V. = V cos(27rnaQ) with 6 < 1. The most exciting feature realised from the studies of discrete quasi-periodic models is the metal-insulator transition [6-8,10-16]. The purpose of this paper is to study the behavior of an one-dimensional continuous quasi-periodic system on the basis of the invariant embedding method.

2. Formalism

2.1. The invariant embedding method for the calculation of transmittance The Schr6dinger equation for an electron in a potential V(x) is [ - d 2 / d x 2 + V(x)]W(x) = EW(x).

(1)

The potential in our study is the generalised Harper form:

V(x) = V cos(27rx6Q) .

(2)

In this model, there is no underlying discrete periodic lattice. So incommensurateness cannot be introduced through irrational values of Q, for example. The inhomogeneity in the period of the potential is introduced by the factor 6-1 x This affects the nature of the spectrum. To study the transmission properties of the system, we make use of Heinrich's approach to the invariant embedding method [17]. We consider a sample stretching from x = 0 to x = L. An incoming wave of energy E and wavevector K ( = ~ E - / ) is incident at x = 0. It is partially reflected and partially transmitted. The complex reflection and transmission coefficients are r(x) and t(x), respectively. The reflection coefficient obeys the stochastic Riccati equation [4,17] iK dr(x) dx = V(x)[r2(x) + l ] - 2[K 2 - V(x)]r(x).

(3)

The complex r(x) may be written as r(x) = r'(x) e i°~x), where r'(x) and O(x) are real functions. The transmittance T(x) is given by

392

S . K . Manna et al. / (De-)localised states in a quasi-periodic potential

T(x) =

1 - r'(x) 2 ,

(4)

differential equations dT(x)

dx

x/Svc ~ _

.-_,x,

K

V1_

(5)

T(x) T(x)

and dO(x)

d~ -

V(x) [2- T(x)] O(x)+2(K K V 1 - T(x) cos

V(x))

.

(6)

We have studied the transmittance of the continuous H a r p e r model with 6 = 0.9 as a function of energy across the spectrum, as will be reported in details in section 3. The spectrum showed localisation-delocalisation transitions at several energies. Clearly two types of states were indicated: states with finite transmittance (delocalised states) and those with zero transmittance (localised states). T h e exact nature of the states has been studied by observing transmittance as a function of the system size for different energy values in the spectrum. T o characterise these states further we have looked at the Argand m a p of the complex transmittance. The Argand m a p contains information about the phase change of the wavefunctions as they travel through the system [8]. We have also carried out multiaffine scaling and multifractal analyses of the transmittances.

2.2. Multiscaling of height correlation function As the transmittances of the two types of states obtained as a function of system sizes were clearly different in the appearance of the detailed geometric structures in them, we carried out an analysis of the height correlation function to show a non-trivial multiscaling behaviour of the transmittance [18,19]. The height correlation function is defined as

C(x) =

I r(x~)

~

-

T(x,

+ x)l,

(7)

i=1

where x scales with N, the n u m b e r of points over which the average is taken, in a very general manner, as N~x

-0 .

H e r e 4' can have any positive value with the only restriction that N----~ as

S . K . M a n n a et al. / (De-)localised states in a quasi-periodic potential

393

x---->0. This generalisation assumes that the different box sizes need not be the same (~b = 1). The partition function is the qth order height correlation function and is defined as

1L IT(xi)

Cq(x) = -~

--

T(x i -~- X )

(8)

lq .

i=l

In the limit x--->0 and N---> oo,

(9)

C q ( x ) ~ m x qH(q) .

For this limit to hold, 05 > 0 is the only choice. We restricted ourselves to 4' = 0.5 and 1.0. The height between any two points separated by x scales with x [18] as IT(x,)

T(x, + x)l

(lO)

x

The exponent % is a non-integer number and gives the local singularity strength of the measure (here it is the height difference). The number of intervals of size x having the same singularity strength is Nr(x), which scales as N

(x)

x

.

The scaling exponents y and h(y, 4') are given [18] as d

y(q) = ~ [qH(q)], uq

(11a)

h(y, 4,) = 4' + qy(q) - qH(q).

(11b)

Eq. ( l l b ) shows that H(q) and h(y, 4') are related formation. T o avoid numerical differentiation, we carry out the arrive at some expressions for y and h(y, 4') which can easily. From eq. (9) we get log Cq(x) = log A + qH(q)

by a Legendre transfollowing procedure to be computed relatively log x. So

qH(q) = - l o g A + log Cq(x) log x Differentiating both sides with respect to q and using eq. ( l l a ) we get

(12a)

S.K. Manna et al. / (De-)localised states in a quasi-periodic potential

394

C;(x) Y(q)=

OCq(x)

Cq(X) l o g x '

C;(x) -

Oq

(12b)

Using eqs. ( l l b ) , (12a) and (12b) we get h(y ,

q~) =

q5 + ~ 1

(qC'q(X) Cq(X)

log Co(x)).

(12c)

2.3. Multifractal analysis of the measure T o distinguish between the different types of states obtained in this model and also to distinguish them from Bloch states, we carried out the multifractal analysis [9,20-22]. The measure is the normalised transmittance

ei-

IT(xi)l

(13a)

E Ir(x,)l i

The partition function is the qth moment of this measure and is given by N

Z(q) = ~ Pq.

(13b)

i=l

In the limit x---~0 and N---*% Z ( q ) ~ xr(q) ~ x (q-1)D(q)

1

X : -~

(14)

Several singularities appear in this kind of measures which arise from transition in quasi-periodic systems [9]. This property is described by an exponent a by the relation Pi ~ xOti

°

(15a)

a i corresponds to the strength of the local singularities of the measure. There are many boxes with the same singularity strength a and their number scales with x as N~ (x) -- x-i(~) . T h e three exponents are related to each other by [9,19-22]

(15b)

S.K. Manna et al. / (De-)localised states in a quasi-periodic potential

395

(15c)

r( q) = a q - f ( a ) = ( q - 1 ) D ( q ) .

For numerical facility we use the a and f ( a ) values after Chhabra and Jensen

[20] as

Z'(q)

a - Z ( q ) log x '

1

f ( a ) = log x

(qZ'(q)

logZ(q))

Z(q)

(16)

3. Results Fig. 1 shows the transmittance versus energy curve for ~ = 0.9, V = 0.5 and N = 10 000. As seen, the spectrum shows two types of states - localised states which have T - - 0.0 and the other states for which T is relatively large. The two types of states exist in specific sub-bands. This transition is seen to occur at several points in the energy spectrum. Figs. 2a, b and c show transmittance versus length for E = 2.42, 4.96 and 8.4 respectively at c5 = 0.9 and V = 0.5. Fig. 2d shows the same for a periodic system with ~ = 1.0 and E = 4.5. Fig. 2a shows that transmittance falls off with system size. This is the characteristic of a localised state. Fig. 2b shows that the transmittance remains at a finite value but has an oscillatory nature with 1.000

II

0.900 0.800 0.700 0.600

LU

0.500 0.400 0.300 0.200 0.100 0.000 .......... 0.0~ 1.800

3.600

5.400

7.200

9.000

10.800 12.600 14.400 16.200

Energy Fig. 1. T r a n s m i t t a n c e versus energy for V = 0.5 and ~ = 0.9.

18.000

S.K. Manna et al. / (De-)localised states in a quasi-periodic potential

396 0

1000

2000

3000

4000

6000

12000

18000

24000

1.000

(a)

~)

0.800

i

0.400 ~.

O

I

0.200

E

=2- 0000 E

1.ooo (cO

r-

I--"

0.800

0.600

0.400

0.200 ;-

0 . 0 0 0 ",.

•.

'

. . . .

6000

'

.

.

12000

.

.

.

'

-

18000

• '

24000

'

0

.

600

.

.

.

'

• " "

1200

1SO0

2400

Length Fig. 2. T r a n s m i t t a n c e versus length for V = 0.5 and (a) E = 2.42, (5 = 0.9; (b) E = 4.96, 6 = 0.9; (c) E = 8 . 4 , 6 = 0 . 9 ; (d) E = 4 . 5 , ~ = 1 .

increasing system size. Fig. 2c also shows an asymptotic oscillatory character but with a much smaller amplitude. Fig. 2d is a typical Bloch state. Figs. 3a, b, c and d show the Argand maps of the complex transmission coefficient [8] (its imaginary part versus its real part) for the states shown in figs. 2a, b, c and d respectively. Fig. 3a shows that the map spirals in towards zero. This is the typical signature of a localised state [8]. Fig. 3b shows a complicated scattering pattern. H e r e also the map spirals inwards but its inward motion gets arrested on a limit cycle. It does not approach zero but tends to move on this cycle for larger and larger system sizes. Figs. 3c also shows the map arrested on a limit cycle. Here also, there is no collapse into the origin. Fig. 3d shows the Argand map for a Bloch state and as expected it is a regular ellipse with very small scatter about the base curve. Argand maps of critical states are always asymptotically bounded by a limiting annulus. For a localised state, the Argand map spirals into the origin whereas for a Bloch state the Argand map scatters around an ellipse Figs. 4a and b show the h(y, ~ ) versus 3' curves for E = 4.96 at ~b = 0.5 and

S.K. Manna et al. / ( De-)localised states in a quasi-periodic potential 1.0(

i

. . . . .

v - ,

i

I '''J

.....

397

i'--

-..:. ..-....,--,.=.~.,.~,-

0.6(

• . . ~.',~..~.'.k ' ":.. ~. "i ,,~'.'~ " . t Z.C . . . ~":._ :~ _ . .'_ :.; . , ',;~ ;- ~,. ,- . .,...;,. " ~.~;~x"~';"",; .',"£":'~,'~

0.20

[b)

~'¢~?~,.:~..:~",.: ",:.~,. ".'.,t'k,'r*:.. • . . . !. ".."..'." . . . . ~.,..~., ', z.. .:7..;~..:.'~' • ':"~':'" " " ' ~ A " ~ ' ~ "'" "

:'.:"" ~" 2 '~ ";.:-.':.~;";/! . •~..:~'~-.'." ~'....~,~_:,.': ".:~; .,.'-;.~;'.;'z~ L,~,;".~. "'.'.:~-;.; • :'~:',~'r.'", '"~'"'~,_~1%T_..,....S,k.....;;...:~,,:..,

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~-..,'t: ' ~'"'?"'"'":'-':'" :.,~'~ ' ~ " ~ ' .....'" .~;.. :r " . " ~.-:]b--,',: i . " - ~ . . . " ".

. ~ : .',~

".~~ ~.~"~ ~

-0.60 • .:..:T~:..~,.~:;:: • .. '...

..: #.~.-~.

,,.,,,,::k~.*.:" -. ....

.:.

~ ! ~_~ , "

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- 1.00

1.00

:;..,-

. -",:-'::'::-~-.,: 0.60

~.-.'.-..:~.-, " ~ ~

~--.%

.~.',.-,;~ ,.~ . ~ , , -

~:."-..-:,.

.

0.20

(d)

(c)

~

| ,.'.:..;

0.00 -0.20

":""~

'

•~, .~ ..;.

~

"~"-:+.

.~..

~

'.\.:-..

•?.:~"

-0.60

- 1.00

.. ...........

-1.0

.'....:~:¢"_ ~'-:.j,:,':...:...."

-0.6

Fig. 3. A r g a n d 8=0.9;

(d)

-0.2

0.0

0.2

map for V=0.5

E=4.5,

.........

0.6

1.0-1.0

and (a) E = 2 . 4 2 ,

-0.6

<5=0.9;

-0.2

0.0

(b) E = 4 . 9 6 ,

0.2 8=0.9;

0.6

1.0

(c) E = 8 . 4 ,

6=1.

1.0 respectively with increasing system size. Figs. 4c and d show the s a m e for E = 2.42 and ~b = 0.5 and 1.0, respectively• F o r E = 4.96, I T ( x i ) - T ( x i + x)l = m (a constant) for all length scales. So f r o m eq. (10) of section 2.2 we get -~ Yi -

log m log N

for all i

and h ( y , ~b) = ~b(1

qlogm]

i -fi

:

A s N---->oo, 7i---, 0 and h(7, ok)---->~b. T h e three curves are for N = 1000 (dot), 3000 (dash) and 5000 (solid). As e x p e c t e d , the p e a k of the curve shifts t o w a r d s 7 = 0.0 a n d shrinks t o w a r d s h ( y , ~b) = ~b for larger and larger system sizes. Fig. 4b shows the s a m e for ~b = 1 at E = 4.96. In fig. 4c the state is localised at

398

S.K. Manna et al. / (De-)localised states in a quasi-periodic potential

0.30 0.48 0.66 0.84 1.02 0"40 ~

0.55 0.80 1.05 1.30 1.55 1.80 I~

X"...,

i : -,,<....

o.o

0.20 O.lO ~"

~"

~

(~) 0.80

i:::

\ \ ""."".

" \ k'.... ~" ,::. o.o

\ "..

0.0 0.55!

"..

I0.00

(c)

0.250"350" ~45'0.1"5~ """""" " "

\\\~\\ \ \~

.

.

" "'"

[c~) !I.00

.]

0.050.30 0.64 0.98 1.32 ........... 1.66~".......0.50 1.10 1.70 2.30 2.90 3.50 . . . . . . . .

'

. . . . . . . . .

'

. . . . . . . . .

'

3' Fig. 4. h(3', 05) versus 3, for V= 0.5, 8 = 0.9 and E = 4.96; (a) 05 = 0.5, (b) 05 = 1. h(y, 05) versus 3' for V=0.5, ~5=0.9 and E = 2.42; (c) 05 =0.5, (d) 05 = 1.

E = 2.42. F o r an ideally localised state (localised at one point), there will be o n e box for which Cq(x)= 1 for all q and for all o t h e r boxes Cq(x)= 0. For Cq = 1, 7i = 0 and the n u m b e r o f boxes contributing to this strength being 1, h ( % ~b) = 0. F o r Cq = 0, 3' ---~oo and as N - 1 boxes contribute to this singularity h(3", ~b) = ~b. So ideally, the localised state should have two points (0, 0) and (0% th) in its h ( % ~b) spectrum. D u e to finite system size, m o r e than two points a p p e a r on the curve. With increasing length 300 (dot), 500 (dash) and 600 (solid), the curves m o v e o u t w a r d s towards (0% ~b) and the left end drops m o r e and m o r e t o w a r d s (0, 0). Fig. 4d shows similar b e h a v i o u r for th -- 1. Figs. 5a and b show h(3', 40 versus 3' and 3' versus q curves for E = 4.96 and N = 5000 at 4~ = 0.5 (solid) and th = 1.0 (dash). As expected, the nature of the curves is similar but shifted. Fig. 5a shows that the h(% th) versus 3' curve shifts to the right with an increase in the t~ value. Figs. 6a, b, c and d show f(a) versus a for transmittance at E = 2.42, 4.96, 8.4 and for a B l o c h state at E - - 4 . 5 , respectively. A t E - - 2 . 4 2 , the state being

S.K. Manna et al. / (De-)localised states in a quasi-periodic potential 0.30

0.45

0.60

0.75

0.90

1.05

1.20

1.35

1.50

399

1.65

1.80

1.0(

0.8(

0.60 \ \

0.4C

\ \

\ \

0.20

\ 0.0 1.70

I

\ 1.42

\

?

\ \

1.14

\

1.00 0.86

._~

0.58

0.30 ........ , ................... -5.00 -4.00 -3.00

, .........

-2.00

, .........

-1.00

~ .........

0.00

, .........

1.00

-' .........

2.00

,

3.00

~ .....

4.00

~...~

5.00

Fig. 5. (a) h(7, &) v e r s u s y for V = 0.5, 6 = 0.9 a n d E = 4.96 at & = 0.5 (solid) a n d 1 ( d a s h e d ) . (b) y v e r s u s q for V = 0.5, 6 = 0.9 a n d E = 4.96 at 4, = 0.5 (solid) a n d 1.0 ( d a s h e d ) .

localised, the f(a) versus a curves tend to two points (0, 0) and (0% 1) with increasing system size from 300 to 600. The curves move outwards as reported in ref. [22] for a localised state. Fig. 6b shows that f(a) versus a curves oscillate with increasing system size. Fig. 6c shows that f(a) versus a curves move inwards with increasing system size but f(Cgmax) and f(amin) both decrease continuously with increasing system size. Fig. 6d shows f(a) versus a curves for a Bloch state. As expected from the works of ref. [22], the curves tend towards the point ( i , 1) with increasing system size.

4. Discussions From the transmittance versus energy spectrum we could distinguish bet w e e n two types of states. On plotting the transmittance versus length, we found that the localised states have an exponential envelope decay after an

400

S.K. Manna et al. / (De-)localised states in a quasi-periodic potential 0.800

1.360

1.920

2.480

0.930

3.040

0.958

. . . . . . . . .

0.8321"000! ~~" '-*-""~~ i

~

'

0.986

, . . . . . . . . .

, .

.

1.014 .

.

'-,

- - ~

1.042 . . . . .

1.070

, . . . . . . . . .

.1.000

{ ......... a J ' .........

0.664

li

o.496

0.328

0.160

1.ooo

'"'

0.900

'- ~:'"'

,"'"'"

(c)

///•x

"'"

z

o°oo

,i

I'

-~0.976

x ~0.952

\\

I

I ~ ~ 0.904

...... ......... , ......... , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 0.800

0.5017

0.910

I

il.000 Idl i

0.942

0.974

1.006

1.038

0.993

0.995

0.997

0.999

1.001

1.003

Fig. 6. f(c~) versus c~ for V= 0.5 and 6 = 0.9; (a) E = 2.42, N = 300 (dot), 500 (dash), 600 (solid), (b) E = 4.96, N = 3000 (dot), 5000 (dash), 7000 (solid), (c) E = 8.4, N = 1000 (dot), 3000 (dash), 5000 (solid), (d) for a Bloch state at 6 = 1.0, E = 4.5 and N = 500 (dot) and 1000 (solid).

i n i t i a l o s c i l l a t o r y n a t u r e o v e r a significant l e n g t h scale. T h e a m p l i t u d e o f t h e s e o s c i l l a t i o n s i n c r e a s e s c o n t i n u o u s l y b e f o r e t h e t r a n s m i t t a n c e finally falls to z e r o . B u t t h e s t a t e s w i t h finite t r a n s m i t t a n c e d o n o t all h a v e t h e s a m e a p p e a r a n c e . T h e s t a t e o f fig. 2b m a y b e s i m i l a r to t h e " p a c k e t e d s t a t e " o f ref. [16]. T h e m u l t i f r a c t a l analysis o f t h e s e s t a t e s h o w e v e r i n d i c a t e s a n i n t e r n a l g e o m e t r i c s i m i l a r i t y to t h e critical s t a t e s o f ref. [9]. T h e f ( o 0 versus ~ curves h a v e an o s c i l l a t o r y n a t u r e a n d O~max O/min r e m a i n s c o n s t a n t . T h e f ( a ) v e r s u s a c u r v e s f o r t h e s t a t e at E = 8.4 m o v e i n w a r d s with i n c r e a s i n g s y s t e m size as in e x t e n d e d B l o c h s t a t e s b u t in c o n t r a s t to t h e e x t e n d e d s t a t e s , h e r e b o t h f(amax) a n d f ( a m i , ) d e c r e a s e w i t h i n c r e a s i n g s y s t e m size i n s t e a d o f c o n v e r g i n g t o w a r d s (1, 1). T h i s i n d i c a t e s t h a t a l t h o u g h t h e states a r e e x t e n d e d , t h e y still r e m a i n m u l t i f r a c t a l . This s t a t e m a y b e c a l l e d an " i n t e r m e d i a t e " state. A s t h e t r a n s m i t t a n c e has a n o n - u n i f o r m s t r u c t u r e locally, w e c a r r i e d o u t t h e scaling analysis o f t h e h e i g h t c o r r e l a t i o n f u n c t i o n s . T h i s analysis c o u l d n o t give a m e c h a n i s m to -

-

S.K. Manna et al. / (De-)localised states in a quasi-periodic potential

401

d i s t i n g u i s h b e t w e e n t h e " i n t e r m e d i a t e " a n d " p a c k e t e d " o r critical states. B u t t h e y b e l o n g to a c l e a r l y d i f f e r e n t scaling r e g i o n f r o m t h a t o f t h e l o c a l i s e d s t a t e s . H o w e v e r , t h e d i f f e r e n t scaling r e g i o n s a r e n o t c l e a r l y d i s t i n g u i s h a b l e f r o m t h e h e i g h t c o r r e l a t i o n analysis. T h e s e distinct scaling r e g i o n s h o w e v e r b e c o m e t r a n s p a r e n t if w e m a k e a m u l t i f r a c t a l analysis o f t h e t r a n s m i t t a n c e itself. A s d i s c u s s e d in s e c t i o n 2, t h e f ( a ) versus o~ curves for i n t e r m e d i a t e , critical o r " p a c k e t e d " s t a t e a n d B l o c h states b e h a v e d i f f e r e n t l y with i n c r e a s i n g s y s t e m size.

Acknowledgements O n e o f t h e a u t h o r s C B w o u l d like to t h a n k C S I R , I n d i a , for financial a s s i s t a n c e d u r i n g t h e p e r i o d this w o r k was u n d e r t a k e n . S K M w o u l d like to t h a n k D r . P.K. T h a k u r for s e v e r a l useful discussions on c a r r y i n g o u t t h e m u l t i f r a c t a l analysis. H e also t h a n k s U . G . C . , I n d i a , for financial s u p p o r t u n d e r t h e F . I . P . s c h e m e d u r i n g t h e p e r i o d o f w o r k . A M w o u l d like to a c k n o w l e d g e D S T , I n d i a for t h e p r o j e c t S P / S 2 / M - 5 6 / 8 9 u n d e r which this w o r k was d o n e .

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