PHYSICAL REVIEW B
VOLUME 59, NUMBER 12
15 MARCH 1999-II
Exact results for a composite-fermion wave function
Orion Ciftja
Ames Laboratory, Iowa State University, Ames, Iowa 50011
~Received 19 October 1998!
It is noted that the radial distribution function and the interaction energy per particle can be exactly computed for a class of composite fermion wave functions obtained at the mean-field level of the Chern-Simons
theory for the fractional quantum Hall effect. These results can be quite instructive and useful to test different
numerical methods and approximations used to study these phenomena. @S0163-1829~99!50308-1#
The fractional quantum Hall effect1 ~FQHE! results from
a strongly correlated incompressible liquid state2,3 formed at
special uniform densities r~n! of a two-dimensional ~2D!
electronic system that is subjected to a very strong transverse
magnetic field BW . The dominant sequence of fractional Hall
states occurs when filling of the lowest Landau level is n
5p/(2mp11), where p51,2, . . . and m50,1, . . . are integers. Much of the theoretical work on FQHE is based on
the study of the properties of a 2D fully spin-polarized ~spinless! system of N interacting electrons embedded in a uniform positive background. The electrons with charge 2e(e
.0) and mass m e are considered confined in the x2y plane
of area V and subjected to a perpendicular magnetic field
W 5(0,0,B), which is generated from the symmetric gauge
B
vector potential AW (rW )5 @ 2(B/2)y,(B/2)x,0# . We will consider the thermodynamic limit of an infinite system defined
as the limit of N electrons in a sample of area V, where N
and V go to infinity with the density kept constant.
The many-electron system is described by the Hamiltonian Ĥ5K̂1V̂, where K̂ is the kinetic energy operator
magnetic length. The fundamental property of the CF’s is
that they experience a reduced effective field B * 5B(1
22m n ) so that the liquid of strongly correlated electrons at
B is equivalent to a liquid of weakly interacting CF’s at B * .
Since the degeneracy of each Landau level is proportional to
the magnetic field, the degeneracy N s* of each CF Landau
level will be smaller than the corresponding N s for the electrons and will be given by N s* 5N s (122m n ). As a result,
the effective filling factor of CF’s will be an integer number
n * 5 p and will correspond to stable electronic filling factors
n 5 n * /(2m n * 11), where n * 5 p51,2, . . . is the number
of filled CF Landau levels.
There are two calculational schemes based on the intuitive
physics above. One constructs explicit wave functions4 while
the second scheme employs a Chern-Simons ~CS! field
theory6 approach to investigate the CF state. Although the
two schemes are based on the same physics, a precise quantitative relationship between them is not clear. The wave
function C CS
n given by
N
C CS
n 5
N
1
W j 1eAW ~ rW j !# 2 ,
K̂5
@ 2i\¹
2m e j51
(
~1!
and
N
V̂5
(
j,k
1
N
v~ u rW j 2rW k u ! 2 r ~ n !
r~ n !2
2
E E
d 2r 1
(
j51
E
d 2 r v~ u rW j 2rW u !
d 2 r 2 v~ u rW 1 2rW 2 u !
~2!
is the total electron-electron, electron-background, and
background-background interaction potential, where v ( u rW j
2rW k u )5(1/4p e 0 )(e 2 / e u z j 2z k u ) is the interaction potential,
z j 5x j 1iy j is the location of the jth electron in complex
coordinates, and e is the dielectric constant of the background.
It has become clear in recent years that many essential
features of FQHE can be understood straightforwardly in
terms of a new kind of particle, called a composite fermion
~CF!, which is a bound state of an electron and an even
number of vortices of the many-body quantum wave
function4,5 formed at the electronic densities r ( n )
5 n / @ 2 p l 0 (B) 2 # where l 0 (B)5 A\/(eB) is the electronic
0163-1829/99/59~12!/8132~5!/$15.00
PRB 59
)
j,k
~ z j 2z k ! 2m
F ~ B*!,
u z j 2z k u 2m p
~3!
where F p (B * ) is the Slater determinant wave function of p
filled Landau levels, evaluated at the magnetic field shown in
the argument, is obtained at the mean-field level of the CS
field theory and describes electronic fillings n 5p/(2mp
11).
So few many-body problems are solvable that a certain
interest may be attached to any model that is exactly solvable. Exact solutions, when available, enable one to gauge
the efficiency of different numerical methods and to carefully test many approximations.
It seems not to have been previously observed that an
exact solution for the radial distribution function and interaction energy per particle can be obtained for the above CS
wave function. As we show in this article, the mathematics is
simple, but the results are, however, quite instructive and the
comparison with other numerical estimates is interesting. A
possible way to find the interaction energy per particle,
u n5
8132
CS
1 ^ C CS
n u V̂ u C n &
N
CS
^ C CS
n uCn &
5
r~ n !
2
E
d 2 r 12@ g n ~ r 12! 21 #v~ r 12! ,
~4!
©1999 The American Physical Society
PRB 59
EXACT RESULTS FOR A COMPOSITE-FERMION . . .
depends on the ability to compute exactly the radial distribution function g n (r 12) that in terms of C CS
n is defined as
g n ~ r 12! 5
2
N ~ N21 ! * d 2 r 3 ¯d 2 r N u C CS
n u
r~ n !2
2
* d 2 r 1 ¯d 2 r N u C CS
n u
and for the system under consideration will depend only on
the interparticle spatial distance r 125 u rW 1 2rW 2 u . For a general
many-body wave function, the radial distribution function
can be computed only approximately, but in the present case
2
2
one observes that u C CS
n u 5 u F p (B * ) u , and this corresponds
to a squared Slater determinant of p51,2, . . . filled CF Landau levels formed at the reduced magnetic field B * . At this
stage the calculation of g n (r 12) is simpler and can be performed exactly.
For a magnetic field B * applied in the z direction which,
as usual, is generated by a symmetric gauge vector potential
AW (rW ), the eigenstates of the ideal Hamiltonian
Ĥ 0 5
and its conjugate. Using a standard algebra, one can easily
prove that the contribution to r̂ p (z 1 ,z 2 ) coming from the nth
CF Landau level is
~5!
,
1
W 1eAW ~ rW !# 2
@ 2i\¹
2m e
8133
N s* 21
(
w*
n,l ~ z 1 ! w n,l ~ z 2 !
l50
5L n
S
u z 1 2z 2 u 2
2l 0 ~ B * ! 2
D(
N s* 21
w*
0,l ~ z 1 ! w 0,l ~ z 2 ! ,
l50
~12!
where L n (x)5(e x /n!)(d n /dx n )(x n e 2x ) are the Laguerre
polynomials of order n50,1, . . . . By using the formulas
given in Eqs. ~8! and ~9! one can express the second term of
Eq. ~12! as
N s* 21
(
w*
0,l ~ z 1 ! w 0,l ~ z 2 !
l50
F
u z 1u 21 u z 2u 2
1
5
exp
2
2 p l 0~ B * ! 2
4l 0 ~ B * ! 2
~6!
G( S
N s* 21
D
l
z 1* z 2
1
.
l! 2l 0 ~ B * ! 2
l50
~13!
for the various CF Landau levels n50,1,2, . . . are given by
F
GF
uzu2
]
w n,l ~ z ! 5 n exp
2 2l 0 ~ B * !
4l 0 ~ B * !
]z
A2 n!
1
H
F
3 w 0,l ~ z ! exp 2
uzu2
4l 0 ~ B * ! 2
GJ
G
n
~7!
,
where l 0 (B * ) is the CF’s magnetic length and
F
G
N s* 21
l
z
w 0,l ~ z ! 5 l
w ~ z !,
A2 l! l 0 ~ B * ! 0,0
1
F
~8!
G
uzu2
w 0,0~ z ! 5
exp
2
,
4l 0 ~ B * ! 2
A2 p l 0 ~ B * ! 2
1
~9!
and l50,1, . . . ,(N s* 21) is the angular momentum quantum
number for the CF’s. The radial distribution function g n (r 12)
will be given by
g n ~ r 12! 512
1
u l ~ r !u 2,
g s p 12
p21 N s* 21
( l50
(
n50
* ~ z 1 ! w n,l ~ z 2 ! ,
w n,l
(
l50
w*
0,l ~ z 1 ! w 0,l ~ z 2 !
5
~11!
where we are using the complex coordinates z i instead of the
two-dimensional vectors rW i . In the above equation, the sum
is extended over all occupied CF states, where w n,l (z) and
w*
n,l (z) are, respectively, the single particle wave function
G F
F
G
z*
u z 1u 21 u z 2u 2
1
1 z2
. ~14!
2 exp 2
2 exp
2 p l 0~ B * !
4l 0 ~ B * !
2l 0 ~ B * ! 2
By substituting Eq. ~14! in Eq. ~12!, one notes that, in the
thermodynamic limit, the one-body density matrix r̂ p (z 1 ,z 2 )
given from Eq. ~11! depends solely on p and not N s* . After
some algebra, the statistical exchange term is found to be
complex
~10!
where the ‘‘statistical exchange’’ factor is computed from
l p (r 12)5 r̂ p (rW 1 ,rW 2 )/ r ( n ) and the spin degeneracy g s of each
Landau state is g s 51, because we are considering fully spinpolarized ~spinless! electrons ~CF’s!. The electronic density
r~n! is the same as the CF’s density and the ~reduced! onebody density matrix r̂ p (rW 1 ,rW 2 ) that corresponds to the dynamically uncorrelated state F p (B * ) is given by
r̂ p ~ z 1 ,z 2 ! 5g s
In the thermodynamic limit both the density and the filling
factor n 5N/N s are kept constant as the number of electrons
N and the LL degeneracy N s go to infinity. Since the degeneracy N s* of each CF Landau level is directly proportional to
N s then also (N s* 21) goes to infinity in the thermodynamic
limit. At this point the summation over l in Eq. ~13! is extended from 0 to ` and one obtains
l p ~ z 1 ,z 2 ! 5
H
p21
F
1
u z 1 2z 2 u 2
Ln
p n50
2l 0 ~ B * ! 2
(
F
3exp 2
GJ
G F
UG
U
u z 1 2z 2 u 2
i
exp
~ rW 3rW ! ,
4l 0 ~ B * ! 2
2l 0 ~ B * ! 2 1 2 z
~15!
but since the radial distribution function is found by applying
Eq. ~10!, the phase factor of l p (z 1 ,z 2 ) disappears and we
obtain
F
g n ~ r 12! 512exp 2
r 212
2l 0 ~ B * ! 2
GH
p21
S
r 212
1
Ln
p n50
2l 0 ~ B * ! 2
(
DJ
2
,
~16!
where r 125 u z 1 2z 2 u is the interparticle distance. From this
result one may erroneously think that the radial distribution
ORION CIFTJA
8134
PRB 59
TABLE I. Exact results for the interaction energy per particle u n corresponding to the Chern-Simons
2
wave function, C CS
n at fillings n 5p/(2p11). The interaction energy is in units of (1/4p e 0 ) @ e / e l 0 (B) # ,
where l 0 (B) is the magnetic length of the electrons. In the fifth column we report the available variational
Monte Carlo results of Kamilla and Jain ~Ref. 7! obtained in the spherical geometry.
n5
p
2mp11
m
p
1
1
1/3
22
1
2
2/5
1
3
3/7
1
4
4/9
1
5
5/11
2 32768
1
6
6/13
2 786432
924747
p
520.408 742
26
1
7
7/15
2 2097152
2662079
p
520.410 776
30
1
8
8/17
2 67108864
91033507
p
520.412 341
34
1
9
9/19
2 19327352832
27800473833
p
520.413 583
38
1
10
10/21
130215126675
85899345920
p
520.414 592
42
1
`
1/2
Exact
p
520.361 800
6
20.3619(90)
2 16
11
p
520.385 343
10
20.3848(16)
2 128
107
p
520.395 990
14
20.3947(15)
2 2048
1971
p
520.402 064
18
20.4007(16)
35205
p
520.405 992
22
1
2
function at fillings n 5p/(2m p11) will depend only on p
and not in m, but a deeper check tells that the dependence on
m is ‘‘hidden’’ in the reduced field B * 5B/(2m p11) to
which the CF’s are subjected. We can further simplify this
FIG. 1. Exact values for the interaction energy per particle u n
corresponding to C CS
n for fillings n 5p/(2p11) ~square! and n
5 p/(4p11) ~circle!. In the p→` limit we obtain the respective
results for fillings n 51/2 ~dotted line! and n 51/4 ~dashed line!.
The energies are expressed in the standard units of
(1/4p e 0 ) @ e 2 / e l 0 (B) # .
A
A
A
A
A
A
A
A
A
A
Ref. 7
2
4
520.424413
3p
p21
L n (x)5L 1p21 (x), where
expression by noting that ( n50
L kn (x)5(21) k (d k /dx k ) @ L n1k (x) # are the associated Laguerre polynomials of order n50,1, . . . and degree k
50,1, . . . .
Since both electrons and CF’s have the same density, we
can relate the CF magnetic length l 0 (B * ) to the true electronic magnetic length l 0 (B) by the simple relation
FIG. 2. Radial distribution function g n (r) for the first two states
of the series n 5p/(2p11), p51,2, . . . and for the filling n 51/2
obtained as the p→` limit of such a series.
EXACT RESULTS FOR A COMPOSITE-FERMION . . .
PRB 59
8135
TABLE II. Exact results for the interaction energy per particle
u n corresponding to the Chern-Simons wave function C CS
n at fillings n 5p/(4p11). The interaction energy is in units of
(1/4p e 0 ) @ e 2 / e l 0 (B) # , where l 0 (B) is the magnetic length of the
electrons.
FIG. 3. Radial distribution function g n (r) for the first two states
of the series n 5p/(4p11), p51,2, . . . and for the filling n 51/4
obtained as the p→` limit of such a series.
l 0 (B * ) 2 5l 0 (B) 2 (2m p11), so finally the radial distribution
function corresponding to C CS
n for n 5p/(2m p11) is written as
F
S DG
S D GJ
r 12
1
g n ~ r 12! 512exp 2
2 ~ 2m p11 ! l 0 ~ B !
3
H
F
2
1 1
1
r 12
L p21
p
2 ~ 2m p11 ! l 0 ~ B !
2
u n 52
3
2p A2m p11
F S DG
L 1p21
x2
2
2
E
`
0
,
~17!
S D
~18!
In the p→` limit and for a nonzero value of m, the CS wave
function C CS
will correspond to the filling n 51/(2m),
n
where m51,2, . . . and, in this limit the radial distribution
function is found to be
g n 51/2m ~ r 12! 5124
F G
S
2
1
1/5
22
2
2
2/9
2 16
11
p
520.287217
18
2
3
3/13
2 128
107
p
520.290577
26
2
4
4/17
2 2048
1971
p
520.292544
34
2
5
5/21
2 32768
35205
p
520.293835
42
2
6
6/25
2 786432
924747
p
520.294748
50
2
7
7/29
2 2097152
2662079
p
520.295428
58
2
8
8/33
2 67108864
91033507
p
520.295954
66
2
9
9/37
2 19327352832
27800473833
p
520.296373
74
2
10
10/41
130215126675
85899345920
p
520.296715
82
2
`
1/4
Exact
1
2
2
x2
dx exp 2
2
1
e2
.
4 p e 0 e l 0~ B !
J1
p
2mp11
p
2
in terms of the natural dimensionless distance r 12 /l 0 (B).
The calculation of the interaction energy per particle corresponding to C CS
n for n 5p/(2m p11) follows from Eq.
~4!, where after writing r ( n )5p/ @ 2 p l 0 (B * ) 2 # we introduce
the dimensionless variable x5r 12 /l 0 (B * ) and obtain the energy in a simple one-dimensional integral form,
1
n5
m
1
r 12
Am l 0 ~ B !
1
r 12
Am l 0 ~ B !
D
2
,
~19!
where J 1 (x) is the Bessel function of the first order. It is
interesting to note that this radial distribution function corresponds to a 2D system of fully-spin polarized (g s 51) free
fermions whose Fermi radius is given by k F „n 51/(2m)…
51/@ Aml 0 (B) # .
The exact interaction energy per particle corresponding to
the n 51/(2m) state, where m51,2, . . . is computed by applying Eq. ~4! and we find that
u n 51/2m 52
A
A
A
A
A
A
A
A
A
A
p
520.280249
10
4 1
520.300105
3p &
e2
1
.
3 p Am 4 p e 0 e l 0 ~ B !
4
~20!
We calculated exactly the integrals appearing in Eq. ~18! for
the first 10 values, p51,2, . . . ,10 of the series of fillings
where m51 and 2. The interaction energy per particle u n is
shown in Fig. 1 for the series of states n 5p/(2p11)
~square! and n 5 p/(4p11) ~circle!. In the p→` limit the
results converge to the values given by Eq. ~20!, respectively, for the n 51/2 ~dotted line! and n 51/4 state ~dashed
line!. In Figs. 2 and 3 we plot the radial distribution function
corresponding to the first two values of p for the series of
fillings given, respectively, by n 5 p/(2p11) and n
5 p/(4p11) and we compare them to the radial distribution
function of the states n 51/2 and 1/4 obtained, respectively,
as the p→` limit of each of such series and given by Eq.
~19!.
A better and more instructive idea of these results is given
in Tables I and II, where we display the exact analytical
values of energy for ten states of the form n 5p/(2p11)
and n 5 p/(4p11), respectively. Since different numerical
methods and approaches have been used to study these phenomena, we believe that these exact results can be quite use-
8136
ORION CIFTJA
ful to gauge their accuracy and their dependence on the number of particles. Extrapolation of finite number calculations
in the thermodynamic limit are not always unambiguous and
the results we provide can be a good test of accuracy for
different approximation schemes. For a long time the quantitative investigations of FQHE have been limited only to
systems of a few number of particles, and only recently have
we been aware of a method developed by Kamilla and Jain,7
which enables a treatment of much bigger systems of 50
particles. By adopting this method to the spherical geometry,
they used variational Monte Carlo techniques to compute the
expectation values of the interaction and kinetic energy per
particle for different wave functions, among them the C CS
n
wave function. In the spherical geometry, they found that the
energies scale approximately linearly as 1/N where N is the
number of electrons. In Table I we compare directly our
exact results with their N→` extrapolations of the interac-
1
For a review, see The Quantum Hall Effect, edited by R. E.
Prange and S. M. Girvin ~Springer-Verlag, New York, 1990!;
The Fractional Quantum Hall Effect, edited by T. Chakraborty
and P. Pietiläinen ~Springer-Verlag, New York, 1988!.
2
D. C. Tsui, H. L. Stormer, and A. C. Gossard, Phys. Rev. Lett. 48,
1559 ~1982!.
PRB 59
tion energy per particle and as can clearly be seen, their
results are very accurate for all the relevant fillings they have
considered, but eventually the method becomes more difficult to apply and less accurate for states in the n →1/2 limit.
In summary, we have analytically computed the interaction energy per particle for all states of FQHE that are described by a CS wave function. We provide tables of exact
analytical values of the interaction energy per particle for all
these states. The data can be used to further test the reliability of different numerical methods employed on the study of
FQHE.
This work was carried out at the Ames Laboratory, which
is operated for the U.S. Department of Energy by Iowa State
University, and was supported by the Director for Energy
Research, Office of Basic Energy Sciences of the U.S. Department of Energy.
R. B. Laughlin, Phys. Rev. Lett. 50, 1395 ~1983!.
J. K. Jain, Phys. Rev. Lett. 63, 199 ~1989!.
5
J. K. Jain, Phys. Rev. B 41, 7653 ~1990!.
6
B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, 7312
~1993!.
7
R. K. Kamilla and J. K. Jain, Phys. Rev. B 55, 9824 ~1997!.
3
4