An Alternative Understanding of the Fractional Quantum Hall Effect

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An Alternative Understanding of the Fractional Quantum Hall Effect

Z. Wang, Jian-Xin Zhu, Yun Wang

To cite this version:

Z. Wang, Jian-Xin Zhu, Yun Wang. An Alternative Understanding of the Fractional Quantum Hall Effect. Journal de Physique I, EDP Sciences, 1995, 5 (11), pp.1385-1388. �10.1051/jp1:1995205�.

�jpa-00247144�

Classification

Physics

Abstracts

73.40-c 05.30-d

Short Communication

An Alternative Understanding of trie fhactional Quantum Hall

Elfiect

Z. D.

Wang,

Jian-Xin Zhu and Yun

Wang

Department

of

Physics, University

of

Hong Kong,

Pokfulam

Road, Hong Kong (Received

29

May

1995, revised 16

July

1995,

accepted

11

September 1995)

Abstract. We show

that,

_m the presence of _a strong

magnetic field,

_a two-dimensional

interacting

electron gas can

eifectively

be described

by

_a Hamiltonian of'free' hard-Gore _anions

subject

to tl~e magnetic ^field.

According

to a new

understanding proposed

for _a set of

specific

anionic systems, _we

explain

the fractional quantum Hall eifect in _a very

simple

_way.

The fractional

quantum

^{Hall effect}

(FQHE) [ii

has been _a

subject

of

great

^{interest in} con-

densed matter

physics

from both the theoretical and

experimental point

of view

[2].

^A

complete understanding

of the fractional

quantization

of the Hall conductance observed in heterostruc- tures in the _presence of _an external uniform

magnetic

field

perpendicular

to the system demands

a

great

deal of theoretical

sophistication.

A _{very nice} and

convincing explanation

for the _ex- istence of _a

family

of fractional _{states at} the

filling

^factor of the Landau level _u

=

1/(2n +1) (n

₌

0,1, 2...)

_was

given

in the seminal _paper

by Laughlin [3],

ⁱⁿ which trial wavefunctions

were

proposed

to treat the interactions of two-dimensional electrons in _a

strong magnetic

^field

using

_a vanational

procedure.

The

Laughlin

state

gives

rise to _a set of

topologically interesting

vortex-like

quasipartide

excitations. As

analogies

^of electrons and

holes,

these excitations have

non-fluctuating

^fractional

charges and,

_more

intriguingly, obey

fractional statistics

[4, 5].

^The

Laughlin

wave function also admits of _a number of

generalizations

which descnbe other _more

comphcated

rational fractions. These _are the

hierarchy

schemes of Haldane [6] and

Halperin [7].

On the other

hand,

motivated

by analogies

between the

FQHE

and _a

superfluid _[8],

_as well

as the existence of

large correlated-ring-exchanges

_on

large length

scales

[9],

^{Girvin and Mac-} Donald

[loi

demonstrated that the

Laughlin

state has _a hidden form of

off-diagonal long-range

order.

They

further

suggested

_a non-local order

parameter

^for the

Laughlin

state.

Interestingly,

such _an observation is

closely

related to _a novel

picture

of anions _or fractional statistics

[5, iii.

As _a matter of

fact,

substantial efforts have been devoted to _a theoretical realization of frac- tional

quantum

statistics and to an effective

field-theory description

of the

FQHE [5,12-15].

Although

the Chern-Simons

field-theory approach

with bosons _or fermions _can

successfully

account for most of the

FQHE phenomena,

_a dear

justification

for

mapping

_a

system

^of in-

teracting

electrons _m two-dimensions in the _presence of _a

strong magnetic

field into a

system

©

Les Editions de

Physique

1995

1386 JOURNAL DE

PHYSIQUE

I N°11

of hard-core bosons _or fermions

coupled

to _a Chern-Simons gauge

field,

_1-e-, into _a

system

of 'free' hard-core anions

subject

to the _same

magnetic field,

_is ^still

lacking, despite

its obvious

fundamental

importance.

^{In this} _paper, we address this crucial issue _m the Hamiltonian for- malism. First _a

physically

reasonable

understanding

of the mapping is

presented

first. Then

we show the mathematical

validity

of this

mapping. Moreover, using

_{a new} result that for _a set of

specific

anion

systems

_a

single-partide quantum

state _can

effectively

be

occupied by

_an

integer

number of anions

[16, iii,

_{we are} able to

explain

the

FQHE

in _{a very}

straightforward

manner.

As is well

known,

_a hard-core anion _can be considered _{as a}

charged

fermion attached with _a certain 'fictitious'

magnetic

flux. When these

'composite

fermions' _are

subject

to _a very

strong magnetic

field in the two-dimensional

xv-plane,

all flux lines _are

nearly parallel

to the _z-axis.

In this _case,

apart

^{from the} energy of the _pure fermions in the

magnetic field,

there _are

also,

in

analogy

to the

magnetic

flux hnes _m

type-II superconductors _[18],

two-dimensional

repulsive

interactions among the flux lines. It is

quite

reasonable to

expect

^{this kind of} interaction

potential

to have

effectively

the _same form _as that for two-dimensional

charged

fermions.

Therefore,

it _seems

quite suggestive

that the Hamiltonian

describing

_a

_system

of hard-core anions

subject

to _a

strong magnetic

field _{may map, m}

pnnciple,

into that

describing

_a two-

dimensional electron _gas witl~ tl~e coulomb

repulsion,

_as

long

as tl~e 'fictitious' flux is

suitably

cl~osen.

We _now

proceed

to

justify

tl~e mappmg

matl~ematically

witl~in _some reasonable

approxima-

tion. In tl~e _presence of _a

strong magnetic

field

along

the

z-direction,

tl~e Hamiltonian of _a

two-dimensional electron _{gas can} be written _as

He

"

H0

+

H;nt

~Î / _{~~~~~~ ~Î}

~

Î~~~

~Î~ / /

~~~~'ÎÎ~~~'~

'

~Î

wl~ere

n(x)

=

£~ ô(x x~)

is tl~e

density

of

electrons,

A is tl~e vector

potential

wl~icl~ satisfies the relation: V _x A

= Bi with B the

magnetic field,

4lo _"

hcle

_is trie flux

quantum,

and other

quantities

^have their usual meaning.

Considering

_a

suiliciently strong magnetic field,

_we need

only

concentrate on tl~e _case wl~ere all electrons _occupy tl~e lowest Laudau level

ILLL)

witl~

spin aligned.

On tl~e

otl~erl~and,

in tl~e fermion

representation,

^the Hamiltonian for 'free' l~ard-core _anions

m a

magnetic

field _can be

represented

_as

[5,14]

Ha

=

~ _/

dx

nix) Î

+

) _(A

+

)j

^~

,

(2)

m i o

where _{a is} the 'fictitious' vector

potential charactenzing

tl~e anion's

property,

^{it satisfies the} relation V x a =

4lnix)1 12k

<

4l/4lo

< 2k +1 with k _an

mteger).

When the gauge condition V _{a =} 0 is

imposed,

it is

straightforward

to write

4l

Il ^nix')

a = dx

e~

,

2~r ÎX X' ,

where

e[(x x')

=1 x

ix x')/(x x'(

is the azimuthal unit vector with respect to

ix x').

In vie~v of that

nix')/ ix x'(

_is

non-negative everywhere,

_{we can}

apply

the _mean theorem of

the

integral

twice and rewrite _{a as}

a =

()) /

_dx'

^~~~)

~

ix

_xo

(ej

7r x x

~

j(

⁴

/ _~~j ^nlx')

27r

lx x'1

~~^~

= a4e4

,

13)

where

ejix xo)

is the azimuthal unit vector with

respect

to

ix xo),

_~

=

ix xo(/

ix

_{xi (,}

xo[nix)]

and _xi

[nix)]

_are

functional

of

nix).

After _some

algebra, equation 12)

can be

reformulated,

Ha

=

Ho

+

£ _/

dxnix)a~F~

,

14)

o

where

F~

_{= e~}

iv Ii

+

(2x/4lo )Ai

+ xa~

/4lo.

In the derivation of the above

equation,

_we have also used the relation V _a

= 0. The LLL constraint

implies [19,20]

that T7

Ii

+

(2x/4lo)A

m 0.

Thus,

_{we can} bave

F~

m

7ra~/4lo.

More

importantly, considering

that

nix)a~

is also _non-

negative everywhere,

and _usmg

again

the _mean theorem of the

integral,

_{we can} write

Ha

=

Ho

+

~~~(q / / xdx'lj~~~~~~~~

,

m4lo

x

x'( 15)

where

@

^{denotes the}

weighted

mean value. When the LLL is

fully filled, n(x)

is

nearly

uniquely

determined and

q depends essentially

_on the

magnetic

^{field B and the} number of electrons N.

Comparing equation 15)

^with

equation 11)

one sees that the Hamiltonian

Ha

_can

effectively

describe the two-dimensional

interacting

electron _gas,

particularly

in the

fully-filled

LLL. It is worthwhile to

emphasize

that to obtain

equation 15)

^the

approximation

T7

Ii

+

(2x/4lo)A

_m 0 has been

used,

which is reasonable when the

magnetic

field is

suiliciently strong

so that

only

the LLL is

occupied by

electrons. We also

expect

that when several low-

lying

Landau levels _are

involved,

this

approximation

_is fulfilled and _ouf results _are vahd.

So

fat,

_we have

justified

that the two-dimensional Coulomb

interacting

electron gas in the presence of _a

strong magnetic

field _can be described

effectively

in terms of _a 'free' _anion Hamil- tonian,

equation 12).

^After a gauge

transformation,

_{we can} write _a 'free' anion Hamiltonian in the anionic

representation directly _[5],

Ha

₌

~ / _{dxna(x) (Î}

+ ^~

i

^~

,

16)

m i c

where

noix)

is the

density

of anions.

Although

there have been several theoretical

interpreta-

tions of the

FQHE

in terms of anions

[7,12],

_we here would like to account for it

simply making

use of _a

key speculatme understanding recently proposed

that each usual

single-partide

state is

allowed to be

effectively occupied by

_an

integer

number

iM)

of hard-core anions

[16, iii.

Let

us

recapitulate

_some results of

quantum

^mechanics. In the _presence of _a

magnetic field,

the energy spectrum ^of a

charged partide

m two dimensions _is discretized and

highly degenerate.

The number of states m each level is _no

"

(eB/hc)

_per unit area. When the LLL is

fully filled,

which

gives

rise to the

plateau

in Hall conductance

against

B

[2],

^{the total number of electrons} and anions per unit _{area are}

respectively

_no and

Mno.

Since the total

charge

_per unit _area is noe, the effective

charge

of _an anion must be

q'

₌

e/M. Correspondingly,

the effective _mag- netic field for the

charge-q'

_orrions becomes B'

= MB because the

Hamiltonian, equation 16),

of these

non-mteracting

anions

depends

_on

iqB)

_{as a}

product

and is thus invanant under the

1388 JOURNAL DE

PHYSIQUE

I N°11

transformation:

je, B)

_-

(q',B').

As the textbooks teach us, the Hall conductance of'free'

aurons is _now calculated _as

~j~

=

~ (,

which

dearly

mdicates the remarkable fractional

quantization

of Hall conductance.

Acknowledgments

We

greatly

thank Dr. P. K. MacKeown for his critical

reading

of _our

manuscript.

This work

was

supported by

_a RGC

grant

of

Hong Kong

and _a CRCG research

grant

at the

University

of

Hong Kong.

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