Behaviour of the electronic states in the Harper model in one dimension

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Behaviour of the electronic states in the Harper model in one dimension

Abhijit Gupta, Asok Sen

To cite this version:

Abhijit Gupta, Asok Sen. Behaviour of the electronic states in the Harper model in one dimen-

sion. Journal de Physique I, EDP Sciences, 1992, 2 (11), pp.2039-2045. �10.1051/jp1:1992265�. �jpa-

00246684�

Classification Physics Abstracts

71.50 71.30 73.20D

Short Communication

Behaviour of the electronic _states in the Harper model in

_one

dimension

Abhijit

Kar

Gupta

and Asok K. Sen

Low Temperature Physics Section, Saha Institute of Nuclear Physics, I

/AF Bidhannagar,

Cal- cutta 700 064, ^India

(Received

22 June 1992, accepted 20 August

1992)

Abstract. In contrast to _a recent claim that the metal-insulator transition in the Harper modelin _one dimension does not occur sharply at Ac

= 2t, _we find in _a numerical study involving

transmittances that there exists _a sharp boundary between the localized and extended spectra in the parameter _space

(of

_A, the amplitude of the incommensurate site

potential).

Further

more, the multifractal behaviour of the normalized transmittances _{as a} function oflength has been studied in support of _our work. The multifractal spectra along with the resistance _{as a} function of length _{prompts us} to conjecture that the nature of the extended states in _a finite sized system for values of A, _{very near} to Ac, _are different from that of typical Bloch _states.

There has been _a renewed interest in the electronic transport

properties

of almost

periodic

systems in _one dimension. One

specific example

is the

Harper model, mathematically rigorous

works _on which _were done first

by Aubry

and Andr4

ii].

This is _{a one} dimensional

tight-binding

model with

nearest-neighbour hopping:

1(u~+i

+

un-i)

+ _{vnun =} E _u~,

(1)

where _{vn =} I

cos(2xQn),

is the

amplitude

of the _on site

potential, Q

is _an irrational

number,

and t is the nearest

neighbour hopping

_energy.

Aubry

and Andr4

proved by using

the Thouless

formula [2] ^{for inverse} localization

length

and _a

duality

_property

ii]

that _a metal-insulator transition _occurs in such _an almost

periodic

system

exactly

at 1

= 2t. For 1 >

2t,

all states _are

exponentially

localized

(pure point spectrum)

and 1 <

2t,

all states _are extended

(continuous spectrum).

At 1 ₌ 2t the model is self-dual and _one gets ^critical states

(singularly

continuous

spectrum).

It _may be noted that the Thouless formula

implicitly

_assumes that the localized

states

decay exponentially

and thus the

original

results of

Aubry

and Andr6 do not allow

for slower than

exponential

localization for 1 _> 2t. That this may

give

rise to

problems

_was shown

rigorously by

Avron and Simon [3] who

proved

that if

Q

^is _a Liouville

number,

I-e-, _an irrational number which satisfies the

inequality,

_{(Q pn}

/qn(

< cn'~n, where

(pn )

and

(qn)

_are

2040 JOURNAL DE PHYSIQUE I N°11

sequences of

primes,

n is _an

integer

and

c is _a constant, then the

spectral density

in the 1 _> 2t

regime

is

only singularly

continuous.

Consequently,

for these kinds of

numbers,

_a semi-infinite domain

(2t,

_~xJ) of

"criticality"

exists for all the states without _any

exponentially

localized state

anywhere

at all.

Recently,

Sun and

Wang

_[4] have claimed that whether _{one uses a} Liouville number _or not for the inverse

period

in this

model,

there exists _a range of I around

lc

= 2t for which the spectrum ^is

intermediate,

I-e- the _states do neither behave like extended

or like localized

ones. More

explicitly

there is _no

sharp boundary

in the I domain between the extended and

localized spectra and thus their conclusions

markedly

differ from those of both references

ii, _3].

To support this

claim, they

have done numerical calculations for

Q/2x

₌

(v$ ₂₎

_on chains of

length

upto ⁶ x 10~

using

transfer matrix method

(see

_e-g- Ref. [5]) ^{and Landauer formalism} [6]. ^{To check their}

intriguing results,

_we have also undertaken similar kinds of numerical calcultations with much

larger

chain sizes

(as large

_as

10~)

and

analyzed

_our results

carefully

to avoid finite size effects. Our conclusions _are different from those of Sun and

Wang

and agree very much with those of reference

ii].

Even

though

these authors _are

quite

_vague about the

extent of their so-called intermediate range for

I,

_one notes from their work that it

certainly

includes the interval

(1.999t, 2.001t).

In _our work _we have included values of I much closer to 1

= 2t than those in reference [4] so as to show

clearly

that there is

actually

_no

region

of intermediate states except as an artefact of the

relatively

small size of the system considered.

The actual

transition,

_{as we} will _see

later,

_occurs at _a definite value of lc ₌ 2t. The

difficulty

with reference [4] seems to be that the authors decided to use

only

the

exactly diagonalized eigenstates

for their

work,

and that it is very hard to

numerically diagonalize

rather

large

size systems.

In this

cornrnunication,

_we have

always

chosen t ₌ I to set the _energy

scale, Q

=

(v$ ₁₎ /2

and the

eigenenergy

E ₌ 0

purposefully.

It has been shown

by

SokoloR

ii]

that

as I

approaches

2t from

below,

the widths of the subbands tend to zero, and

therefore,

it is

really

difficult to find _an allowed

eigenenergy

from the _{very narrow} bands of the system ^for

(lc I)

_« t. But since symmetry ^{and other} arguments ^{dictate that} E ₌ 0 is

definitely

_an

eigenstate

of the system, we have chosen to work with this energy. From the

figures

la lc it is clear that for

a small

enough

system size

(depending

_on

eigenenergy

and the _range of

I),

the state

(E

₌

0)

may behave _as

a critical _one for all chosen I

around,

but close to, 2t. But _{one can see} from the

figures

Id and If that for

larger length

systems ^the state E ₌ 0 for I

= 1.9999t

clearly

show the

signature

of _an extended state

("average"

resistance

remaining

constant, within _our numerical _accuracy, upto a size of

10~)

^{and for 1} = 2.0001 show the

signature

of _a localized

state

(average

resistance

increasing exponentially),

thus

invalidating

the claim that the

plot

of

7(E) (the exponential growth

factor for

resistance) against

I shows _no

sharp turning point

at 1= 2t.

In fact it turns out that the extended states in this _{case near}

lc

_are not

quite

Bloch-like

in the _sense that

they

have

finite,

and not

infinite, "average"

conductances

(irrespective

of

length)

which become

systematically

smaller and smaller _{as one} gets closer and closer to lc.

This is shown in _our

figure

2 in _a

plot

of _average conductance _{as a} function of I. As _{we see} from

figure If,

in the localized

region

the resistance increases

exponentially

fast without much scatter; hence _we have taken

simply

the inverse of the resistance to be the conductance in the

region

1 _> 2t. But in the

conducting region,

the conductance shows

relatively large

scatter

(Fig.

la, _or

Id).

Even in this

region, pictorially

it is

reasonably

clear what _{we mean}

by

^the

average conductance but for

quatifying

it in the 1 _< 2t

region,

_we have used the

following

approach. First,

_we calculate the _average of the

logarithm

of resistance for _a fixed I and for different

lengths

within _some chosen

length segment

^{and then} repeat ^this

procedure

^{for all the}

equal length segments

^{within the entire chain. Then} we

plot

these _average values _{as a} function

lx J0~ 3xJ0~ lx ll~~ 3x10~ _lx lo' _3x lo'

15 15

lo

5 5

0 (a)

(b)

(c) _t~

In R

35 (d) (e) (o

31~

25 5()1)

IX 9X i~l~ 6X il~~ IX (f 5X il~~ 9X1l~~

lcllgtll (L)

Fig. I. Logarithm of four-probe resistance

(R)

_versus length plot for

a system size of 5 _x 10~ _and

for three different amplitudes of site potentials:

(a)

A

= 1.9999t,

(b)

₌ 2t, and

(c)

A ₌ 2,o001t; in

(d), (e)

and

(f)

_we show the _{same as} in

(a), (b)

and

(c)

but for _a much larger system size ₌ lo~.

of their

midpoint lengths

to convince ourselves that

they

do not have _an overall increment _or decrement with

length (I.e.,

^{the inverse} localization

length

is

zero).

Next _we take the

antilog

^of

the

negative

^{of this}

quantity

and call it _{our average} conductance.

Figure

2

clearly

shows that

even

though

the resistance is finite for 1 _<

2t,

the _{states are} still extended and that _a transition

occurs

only

when the conductance for _an "infinite"

length sample

is _zero which _occurs

only

at

1 ₌ 2t.

Another crucial issue needs _to be discussed here. For the failure of the arguments ^{in reference}

ii] regarding

the existence of _a

sharp boundary

for localization-delocalization transition at

lc,

Sun and

Wang

^have

pointed

at the _use of the _Thouless formula and themselves have

consciously

avoided its _use. But

ironically

Ishii's formula [8],

7(E)

₌ Lt

(I/N)

ii

(lanl~

⁺

tan+il~)

_,

⁽²⁾

which

they

_{use, seems}

only

to be _a variant of the Thouless formula because this also _assumes

exponential

localization for N _{- ~xJ}

(note

the _presence of

a

logarithm

in that

formula).

Let

us note in this

regard

that the

Lyapunov

exponent

(here,

the inverse localization

length)

for 1 > 2t, _as obtained

by Aubry

and Andr6

using duality

and the Thouless formula for any state is

given by 7(E)

= In

(1/2t).

The

duality

obtained in reference

ii]

is

quite rigorous. Thus,

if _{we can} show

numerically

that the above formula is valid _very close to 2t, then it not

only

shows that there is _a

sharp boundary,

but also that the

assumption

of

exponential

localization

2042 JOURNAL DE PHYSIQUE I N°1

2.Ux lU~~

, , , ,

, , , ,

1.5x U~~ M I

, , , , , ,

A ,

, ⁱ

g

1.0x10-.

i fi

, ,

5.0x JU~~

A _,

, , , i i fi ,

, , ,

1.992 1.996 2.008

j~~

Fig. 2. Four-probe conductance _versus plot showing the transition in )-domain at A

= 2t.

is _{not at} stake

(at

least when the

Q's

_are not Liouville

numbers).

With that view in mind _we calculated the

slope

of In R _versus L

plot

for 1= 2.0001t

(see Fig. If)

and it _{turns out to} be very close to 5 _x

10~~

_As

one can see, this number matches

extremely

well with that obtained

directly

from the above formula. This reveals the fact that the localized states _are

actually

exponentially

localized in this system and the _use of Thouless formula is

totally justified

to obtain the

boundary

at lc ₌ 2t. The

important

_message is that the

eigenstates (for

values of

1# 2t),

which look like intermediate states, must either be extended _or localized if _one

really

goes to

large enough

system ^sizes.

Now _we present some new results

regarding

the characterization of the extended and the localized states in the

Harper

model. In order to characterize the states

properly

_we undertook

a

multifractal

analysis

for the _set of normalized transmittances for various

lengths.

These

results,

apart ^from

strengthening

the claim of _a

sharp boundary,

throw further

light

_on the nature of

extended states

(for

1 _<

2t)

which for _a finite size system _seems to be somewhat different from Bloch type as mentioned above. In _our two

previous

works

[10],

it _was demonstrated that the multifractal nature of the transmittances _may be used for

clearly distinguishing

_among

extended,

localized and critical

(or intermediate)

states. In _a sc-called _a

f(a) plot,

_one notices [10] ^that as the

length

of the system is

increased,

the support of the _measure, Do _"

peak

value of

f(a),

remains

unity

_as

expected

but the width of the spectrum

(amax am;n)

(I)

tends to

decay

to zero around «peak = I for _an extended state,

(it)

_ever increases

(as

do apeak and

amax)

^for a localized state, ^and

(iii)

tends to stabilize at _some fixed value in such _a way that all of _{drum, amax} and apeak oscillate around

respective

fixed values

(the

last _one, apeak ^> Do

[10, II]

for _a critical state.

Figures

3a-3d show the multifractal spectra for I

=

1.0t, 1.999t,

2t and 2.001t

respectively

for E

= 0 and for chains of

length spanning

_more than two decades

(10~

²

x10~).

The

behaviour of the multifractal spectra as a function of

length clearly

demonstrates the localized nature of the electronic states for 1

= 2.001t

(see Fig. 3d) according

to our criteria _as described above. We have also done _a finite size

scaling analysis

of

drum(N)

_{as a} function of

N,

and the

indication is that it stabilizes at drum = 0 with

f(drum)

₌ 0. For the critical _case 1 ₌ 2t

(Fig. 3c),

_one notes that the _amax oscillates around 2A and the _drum oscillates around 0.3

as the

length

is increased. Further the

peak position

_apeak is

quite stable,

not at 1.0 _{as we} found in _our

previous

^works [10] on different systems, ^{but close to 1.5}

(see

Tab.

I).

Also _drum is

clearly

not

tending

to _zero but its average seems to be stable around 0.3. These stabilities of the multifractal spectra are the hallmarks of _a critical state. One also notes that the

generic algebraic decay

exponent in the

Harper

model for the critical _state is about 1.5.

Very recently

Hiramoto and Kohmoto [12] have also

reported

_a multifractal

analysis

_on the _wave function of _a critical state

by using

the method of _a

progressively

^{better rational}

approximant

for

Q

_"

(V$

₁₎

/2.

Their value for apeak ^is close _to 1.3 for critical states both _near the

edge

^and

the centre of the spectrum. We stress in this

regard

that multifractal

analyses

_on two critical wavefunctions [13] ^{for about}

10,000

and

20,000

sites still

give

_am;n and apeak very close _to what

we get ^{here. It} _may ^be noted

again

^that it is

computationally

difficult to find wavefunctions of

larger

size systems and that is

why

_we work with normalized transmittances

(to

be able to

work with several decades in the chain

length

is _very essential for

doing

_a finite size

scaling analysis).

Table I.

System

Size apeak apeak

N

=

1.999t) (1

₌

2t)

10~ _1.5126 1.5382

10~ _1.4387 1.5379

5 _x 10~ _{1.3427 1.4587}

10~ 1.3159 1.5258

2 _x 10~ 1.2999 1.5332

5 _x 10~ 1.2783 1.4632

The most

interesting

situation

happens

when _we look at _an extended _state with I very close to

lc.

For the value of I

=

1.999t,

_one notices from

figure

3b that _{as one} increases the

length

of the

sample

the width of the spectrum ^decreases

(as expected

for _an extended

state)

but apeak also decreases from _a value

significantly larger

than I. This is not

quite expected

for Bloch-like

extended states [10]. ^{To check what}

actually happens

_we have also looked at the multifractal spectra for I ₌ t

(Fig. 3a),

which is

quite

far _away from lc. In this _case, the widths of the spectra ^for all the chosen chain

lengths

_are much

smaller,

and the

peaks

_are at _w > I but much closer _{to a}

= I for different

length

scales

(further they approach

towards _{a =} I _as chain

2044 JOURNAL DE PHYSIQUE ^I N°I1

0.95 1,1)5 1-lo 1,15 0.5 1-U 1.5 2.0 2.5

~~'l hi

(1.')5

I I

I

i

U.85 0.4

0.2

f a)

(cl

'j

(d)

l~.80 i U.8

l~.61) o-b

ll.41)

l~.20

l~.5 1-o 1.5 2.l~ 2.5 2 3 4 5 6 7

a

Fig. 3. Multifractal spectrum

(a f(a) plot)

for

(a)

= t,

(b)

= 1.999t,

(c)

A

= 2t, and

(d)

= 2.001t; shown for _{a set} of system ^sizes

(N)

described below:

(--)

N

= 10~, (-

-)

N

= 10~,

(-)

N

= 5 x 10~, (. N

=

lo~,

and

(-)

N

= 2 X10~.

length increases).

Our

analyses

indicate that for _any fixed chain

length

the _a

f(a)

spectrum broadens and the apeak moves away from _a

= I

(towards 1.5)

_as the value of I is chosen to be closer and closer to

lc (for

I _<

lc).

This _seems consistent with the fact that _more and _more

gaps ^start

appearing

between the bands

(Sokoloff

in Ref.

ii])

and _our observation

above,

that

the states become _more and

more resistive for the

same chain

length

_as I

-

lc.

But _as the

system size grows

(for

_a fixed

I),

the conductance

keeps

_on

maintaining

_an overall fixed average value

[which

is of _course

considerably

low for

(lc I)

«

t].

A

preliminary

finite size

scaling

with _a function of the form

(a

+

b/N')

and

using

the data in table I for I ₌ 1.999t indicates that apeak

asymptotically approaches

the value 1.0 with _an

optimum

_K st 0.I.

Finally

_we

must mention that the non-Bloch character of the extended states has also been observed _very

recently by

Hiramoto and Kohmoto

[12]. They

^{find that the} root _mean square

displacement

of

an electron released

initially

at a certain

site,

increases _as

t~,

where _~ increases

continuously

from _a value less than I

(I.e.,

non-ballistic

motion)

towards I

(ballistic)

_as t _- ~xJ

(see

their

Fig. 12).

Further this _crossover time _seems to be

longer

_as ^I _- lc. This is consistent with _our observation that the resistance of the

Harper

metallic chain increases

initially

^with

length (see Fig, la)

but

finally

becomes constant for _a

large enough length

scale

(see Fig. Id).

References

ii]

AUBRY S. and ANDRt G., Ann. Israel Phys. Soc. 3

(1980)

133;

See also the excellent review by SOKOLOFF J.B., Phys. Rep. 126

(1985)

189-224.

[2] THOULESS D.J., J. Phys. C 5

(1972)

77.

[3] ^{AVRON J. and SIMON} B., Bull. Am. Math. Soc. 6

(1982)

81.

[4] ^{SUN J. and WANG} C., Phys. Rev. B 44

(1991)

1047.

is]

GANGOPADHYAY S. and SEN A-K-, Phys. Rev. B 46

(1992)

to appear.

[6] ^LANDAUER R., Philos. Mag. 21

(1970)

863.

[7] ^SOKOLOFF J.B., J. Phys. C: Solid State Phys. 17

(1983)

1703.

[8] ^ISHII K., Frog. Theor. Phys. Suppl. 53

(1973)

77.

[9] ^{GODRtCHE C, and LUCK} J-M-, J. Phys. A: math. Gen. 23

(1990)

3769.

[10] ^BASU C., MOOKERJEE A., SEN A.K. and THAKUR P-K-, J. Phys. Condens. Matter 3

(1991)

9055;

THAKUR P-K-, BASU C., MOOKERJEE A. and SEN A.K., J. Phys. Condens. Matter

(1992)

in

press.

[11] SCHREIBER M. and GRUSSBACH H., Phys. Rev. Lett. 67

(1991)

607.

[12] ^{HIRAMOTO H. and KOHMOTO} M., lnt. J. Mod. Phys. B 6

(1992)

281.

[13] ^{THAKUR P-K-}

(private communication).