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The fractional charges in quantum hall effect and the possibility of their observation
Solid State C o ~ u n i c a t i o n s , Vol.57,No.3, pp.159-]60, ]986. Printed in Great Britain.
0038-]098/86 $3.00 + .00 Pergamon Press Ltd.
THE FRACTIONAL CHARGES IN QUANTUM HALL EFFECT AND THE POSSIBILITY OF THEIR OBSERVATION
V.L.
Pokrovsky
L.D. Landau Institute The Academy of Sciences
(Received:
The minimal quantum fluid is gap on the value theory.
20
and
_
N
1
2S
m
September
1985
Is]
A.
Zawadowski)
F i r s t l e t us c o n s i d e r the case of . Then the wave function of the quasihole, proposed b y L a u g h l i n 2, h a s t h e form
v=i/m
N
=
H ( z i - ~ ) ~ L ( Z l , . . . , z N) i=l
(3)
where N
N
~L= ~ (z -z~) m exp(- Z 1~i2/4) ~<8
(i)
(4)
£=i
Here z are the complex coordinates of electrons, ~ is a c o o r d i n a t e of the quasih o l e . It h a s b e e n s h o w n i n S , 5, t h a t (3) is an e x a c t g r o u n d s t a t e w a v e f u n c t i o n when the Coulomb interaction between the electrons is r e p l a c e d by a short-range interaction.
Laughlin h a s s h o w n t h a t if t h e n u m ber of flux quanta differs by a unity from that obtained b y Eq. (i) a t t h e g i v e n N, t h e n e x t r a o r m i s s i n g flux quant u m is j o i n e d w i t h t h e l o c a l c h a n g e in the electron density. This local change of the electron d e n s i t y c a n b e t r e a t e d as a quasiparticle floating in the incompressible quantum fluid. Using the formal analogy with the one-component plasma Laughlin obtained the quasiparticle charge e , which modulus is e q u a l t o =
by
charge of quasiparticles in the incompressible calculated. A strong dependence of the energy of this charge makes it possible to verify the
w h e r e N is t h e n u m b e r o f e l e c t r o n s , 2S is t h e n u m b e r o f t h e f l u x q u a n t a p a s s i n g through the area occupied by electrons, m is an o d d i n t e g e r .
fe*l
Talapov
for Theoretical Physics of the USSR, Moscow, USSR
The fractional quantum Hall effect, discovered b y T s u i , e t al I is a s u b j e c t of intense investigation. To account for this effect Laughlin 2 has proposed a concept of quasiparticles with fractional c h a r g e s . He h a s a s s u m e d that the elect r o n s in a s t r o n g m a g n e t i c field on the f i r s t L a n d a u l e v e l at s o m e c o n c e n t r a t i o n s f o r m an i n c o m p r e s s i b l e quantum fluid. These concentrations are v
A.L.
L e t us c o n s i d e r a large closed contour F surrounding n electrons. The quasihole motion along this contour changes the phase of the wave function by value (-2~n). O n t h e o t h e r h a n d , it f o l l o w s from the gauge invariance t h a t t h e p h a s e c h a n g e is ( e * / ~ c ) ~ , w h e r e ~ is t h e m a g n e t i c flux enc l o s e d b y r. A t v = i / m m flux quanta ~o = =2w~c/e is p e r e a c h e l e c t r o n . Comparing phase changes obtained by two ways we find
(2)
m -2~n
w h e r e e is t h e e l e c t r o n charge. The same result has been obtained in 3 w i t h t h e a i d of g a u g e i n v a r i a n c e . The value of quasiparticle c h a r g e is v e r y e s s e n t i a l for the calculation o f t h e H a l l c o n d u c t i v i t y 4. Therefore we simplify arguments o f 3 to get quasiparticle charge for v=i/m. Moreo v e r , we f i n d t h e m i n i m a l c h a r g e of q u a s i particles arising near other rational fillings v=~/q, which differs from that, obtained i n 3 . We a l s o d i s c u s s t h e p o s s i b i l i t y to c h e c k o u r r e s u l t s for quasiparticle c h a r g e s in t h e e x p e r i m e n t .
Hence,
we
e = ~-~- n m ~ o get
the
=
2nnm
result
e ~-
(5)
(2).
To calculate the charge of quasipart i c l e s i t is n o t n e c e s s a r y to k n o w t h e i r exact wave functions. Suppose the ground state wave function is n o n d e g e n e r a t e at some rational f i l l i n g v = p / q . L e t us change the number of flux quanta by ±i . If t h e a d d i t i o n a l quantum is c o n n e c t e d with a local variation of electron density, then the total charge of this local formation can be found. While the electron
159
Vol. 57, No. 3
THE FRACTIONAL CHARGES IN QUANTUM HALL EFFECT
160
moves along the contour encircling the localized quantum, the w a v e f u n c t i o n p h a s e c h a n g e s b y + 2 ~ . H e n c e , w h e n the l o c a l i z e d q u a n t u m p a s s e s a r o u n d the e l e c t r o n , the p h a s e c h a n g e s by ~ 2 n . R e p e a t i n g consideration, presented a b o v e , we g e t t h a t the c h a r g e , r e l a t e d to t h e l o c a l i z e d quantum o f the f l u x , is e q u a l to + e ~ . In c a s e ~ i / m , say ~=p/ (mp±l), where m is o d d i n t e g e r , p - e v e n i n t e g e r , it is a more complex process to o b t a i n q u a s i particles with minimal charges. According to H a l d a n e 7 a n d H a l p e r i n 8 s u c h f i l l i n g s a r i s e f r o m the L a u g h l i n states with ~=i/m provided quasiholes or q u a s i e l e c t r o n s themselves f o r m the L a u g h l i n incompressible f l u i d w i t h the c o n c e n t r a t i o n i/p. N e w q u a s i p a r t i c l e s with a minimal c h a r g e a p p e a r w h e n we a d d s i m u l t a n e o u s l y one flux quantum and one quasielectron w i t h the c h a r g e e / m . The e
minimal
*
m,p
= ±
In p a r t i c u l a r , t r o n c h a r g e is v a l u e of e is derations show
possible e
m (rap+-i)
charge
The energy s p e c t r u m o f the h a s the o r d e r
g a p in t h e e x c i t a t i o n electron f l u i d at v = p / q
(e*) 2 A
(7)
.
< £H w h e r e < is the d i e l e c t r i c is t h e m a g n e t i c length
*
=
constant,
~
£~
(8)
It s h o u l d be n o t e d t h a t A ~ (e*) 5/2 For different fillings v , w h i c h are c l o s e to e a c h o t h e r , the v a l u e s of t h e energy gap A may differ markedly. For example A (~=2/5) A (~=i/3)
1 5/2 ([)
= 0.02
(9)
is (6)
near ~=2/5 the quasielece / 1 5 , a n d n e a r ~=2/7 t h e e/21. Analogous consithat e*=e/35 f o r ".)=3/7.
Thus, measuring the e n e r g y g a p o n e c a n verify whether the quasiparticle charges r e a l l y o b e y Eq. (6) at 9 = p / (mp±l) The value of A also determines maximal temperature at w h i c h a s t e p e x i s t s on a H a l l c h a r a c t e r i s t i c .
the
REFERENCES 1. 2. 3. 4.
D . C . T s u i , H.L. S t 6 r m e r , A.C. Gossard, P h y s . R e v . L e t t . 48, 1 5 5 9 (1982) R.B. Laughlin, P h y s . R e v . L e t t . 50, 1395 (1983) D. A r o v a s , J . R . S c h r i e f f e r , F.Wilczek, P h y s . R e v . L e t t . 53, 722 (1984) V.L. Pokrovsky, A.L. Talapov, Pisma v Z h E T F 42, 68 (1985)
5. 6. 7. 8.
V.L. Pokrovsky, A.L. Talapov, J. P h y s . C S.A. T r u g m a n , S. K i v e l s o n , Phys. Rev. B31, 5 2 8 0 (1985) F.D.M. Haldane, P h y s . R e v . L e t t . 51, 6 0 5 (1983) B.I. H a l p e r i n , P h y s . R e v . L e t t . 52, 1583 (1984)
0038-]098/86 $3.00 + .00 Pergamon Press Ltd.
THE FRACTIONAL CHARGES IN QUANTUM HALL EFFECT AND THE POSSIBILITY OF THEIR OBSERVATION
V.L.
Pokrovsky
L.D. Landau Institute The Academy of Sciences
(Received:
The minimal quantum fluid is gap on the value theory.
20
and
_
N
1
2S
m
September
1985
Is]
A.
Zawadowski)
F i r s t l e t us c o n s i d e r the case of . Then the wave function of the quasihole, proposed b y L a u g h l i n 2, h a s t h e form
v=i/m
N
=
H ( z i - ~ ) ~ L ( Z l , . . . , z N) i=l
(3)
where N
N
~L= ~ (z -z~) m exp(- Z 1~i2/4) ~<8
(i)
(4)
£=i
Here z are the complex coordinates of electrons, ~ is a c o o r d i n a t e of the quasih o l e . It h a s b e e n s h o w n i n S , 5, t h a t (3) is an e x a c t g r o u n d s t a t e w a v e f u n c t i o n when the Coulomb interaction between the electrons is r e p l a c e d by a short-range interaction.
Laughlin h a s s h o w n t h a t if t h e n u m ber of flux quanta differs by a unity from that obtained b y Eq. (i) a t t h e g i v e n N, t h e n e x t r a o r m i s s i n g flux quant u m is j o i n e d w i t h t h e l o c a l c h a n g e in the electron density. This local change of the electron d e n s i t y c a n b e t r e a t e d as a quasiparticle floating in the incompressible quantum fluid. Using the formal analogy with the one-component plasma Laughlin obtained the quasiparticle charge e , which modulus is e q u a l t o =
by
charge of quasiparticles in the incompressible calculated. A strong dependence of the energy of this charge makes it possible to verify the
w h e r e N is t h e n u m b e r o f e l e c t r o n s , 2S is t h e n u m b e r o f t h e f l u x q u a n t a p a s s i n g through the area occupied by electrons, m is an o d d i n t e g e r .
fe*l
Talapov
for Theoretical Physics of the USSR, Moscow, USSR
The fractional quantum Hall effect, discovered b y T s u i , e t al I is a s u b j e c t of intense investigation. To account for this effect Laughlin 2 has proposed a concept of quasiparticles with fractional c h a r g e s . He h a s a s s u m e d that the elect r o n s in a s t r o n g m a g n e t i c field on the f i r s t L a n d a u l e v e l at s o m e c o n c e n t r a t i o n s f o r m an i n c o m p r e s s i b l e quantum fluid. These concentrations are v
A.L.
L e t us c o n s i d e r a large closed contour F surrounding n electrons. The quasihole motion along this contour changes the phase of the wave function by value (-2~n). O n t h e o t h e r h a n d , it f o l l o w s from the gauge invariance t h a t t h e p h a s e c h a n g e is ( e * / ~ c ) ~ , w h e r e ~ is t h e m a g n e t i c flux enc l o s e d b y r. A t v = i / m m flux quanta ~o = =2w~c/e is p e r e a c h e l e c t r o n . Comparing phase changes obtained by two ways we find
(2)
m -2~n
w h e r e e is t h e e l e c t r o n charge. The same result has been obtained in 3 w i t h t h e a i d of g a u g e i n v a r i a n c e . The value of quasiparticle c h a r g e is v e r y e s s e n t i a l for the calculation o f t h e H a l l c o n d u c t i v i t y 4. Therefore we simplify arguments o f 3 to get quasiparticle charge for v=i/m. Moreo v e r , we f i n d t h e m i n i m a l c h a r g e of q u a s i particles arising near other rational fillings v=~/q, which differs from that, obtained i n 3 . We a l s o d i s c u s s t h e p o s s i b i l i t y to c h e c k o u r r e s u l t s for quasiparticle c h a r g e s in t h e e x p e r i m e n t .
Hence,
we
e = ~-~- n m ~ o get
the
=
2nnm
result
e ~-
(5)
(2).
To calculate the charge of quasipart i c l e s i t is n o t n e c e s s a r y to k n o w t h e i r exact wave functions. Suppose the ground state wave function is n o n d e g e n e r a t e at some rational f i l l i n g v = p / q . L e t us change the number of flux quanta by ±i . If t h e a d d i t i o n a l quantum is c o n n e c t e d with a local variation of electron density, then the total charge of this local formation can be found. While the electron
159
Vol. 57, No. 3
THE FRACTIONAL CHARGES IN QUANTUM HALL EFFECT
160
moves along the contour encircling the localized quantum, the w a v e f u n c t i o n p h a s e c h a n g e s b y + 2 ~ . H e n c e , w h e n the l o c a l i z e d q u a n t u m p a s s e s a r o u n d the e l e c t r o n , the p h a s e c h a n g e s by ~ 2 n . R e p e a t i n g consideration, presented a b o v e , we g e t t h a t the c h a r g e , r e l a t e d to t h e l o c a l i z e d quantum o f the f l u x , is e q u a l to + e ~ . In c a s e ~ i / m , say ~=p/ (mp±l), where m is o d d i n t e g e r , p - e v e n i n t e g e r , it is a more complex process to o b t a i n q u a s i particles with minimal charges. According to H a l d a n e 7 a n d H a l p e r i n 8 s u c h f i l l i n g s a r i s e f r o m the L a u g h l i n states with ~=i/m provided quasiholes or q u a s i e l e c t r o n s themselves f o r m the L a u g h l i n incompressible f l u i d w i t h the c o n c e n t r a t i o n i/p. N e w q u a s i p a r t i c l e s with a minimal c h a r g e a p p e a r w h e n we a d d s i m u l t a n e o u s l y one flux quantum and one quasielectron w i t h the c h a r g e e / m . The e
minimal
*
m,p
= ±
In p a r t i c u l a r , t r o n c h a r g e is v a l u e of e is derations show
possible e
m (rap+-i)
charge
The energy s p e c t r u m o f the h a s the o r d e r
g a p in t h e e x c i t a t i o n electron f l u i d at v = p / q
(e*) 2 A
(7)
.
< £H w h e r e < is the d i e l e c t r i c is t h e m a g n e t i c length
*
=
constant,
~
£~
(8)
It s h o u l d be n o t e d t h a t A ~ (e*) 5/2 For different fillings v , w h i c h are c l o s e to e a c h o t h e r , the v a l u e s of t h e energy gap A may differ markedly. For example A (~=2/5) A (~=i/3)
1 5/2 ([)
= 0.02
(9)
is (6)
near ~=2/5 the quasielece / 1 5 , a n d n e a r ~=2/7 t h e e/21. Analogous consithat e*=e/35 f o r ".)=3/7.
Thus, measuring the e n e r g y g a p o n e c a n verify whether the quasiparticle charges r e a l l y o b e y Eq. (6) at 9 = p / (mp±l) The value of A also determines maximal temperature at w h i c h a s t e p e x i s t s on a H a l l c h a r a c t e r i s t i c .
the
REFERENCES 1. 2. 3. 4.
D . C . T s u i , H.L. S t 6 r m e r , A.C. Gossard, P h y s . R e v . L e t t . 48, 1 5 5 9 (1982) R.B. Laughlin, P h y s . R e v . L e t t . 50, 1395 (1983) D. A r o v a s , J . R . S c h r i e f f e r , F.Wilczek, P h y s . R e v . L e t t . 53, 722 (1984) V.L. Pokrovsky, A.L. Talapov, Pisma v Z h E T F 42, 68 (1985)
5. 6. 7. 8.
V.L. Pokrovsky, A.L. Talapov, J. P h y s . C S.A. T r u g m a n , S. K i v e l s o n , Phys. Rev. B31, 5 2 8 0 (1985) F.D.M. Haldane, P h y s . R e v . L e t t . 51, 6 0 5 (1983) B.I. H a l p e r i n , P h y s . R e v . L e t t . 52, 1583 (1984)