Method of computation of energies in the fractional quantum Hall effect regime
In a previous work, we reported exact results of energies of the ground state in the fractional quantum Hall effect (FQHE) regime for systems with up to Nе = 6 electrons at the filling factor ν = 1/3 by using the method of complex polar coordinates. In this work, we display interesting computation...
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| Cite this: | Method of computation of energies in the fractional quantum Hall effect regime/ M.A. Ammar, Z. Bentalha, S. Bekhechi // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33702: 1–9. — Бібліогр.: 28 назв. — англ. |
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Ammar, M.A. Bentalha, Z. Bekhechi, S. 2019-06-18T10:08:26Z 2019-06-18T10:08:26Z 2016 Method of computation of energies in the fractional quantum Hall effect regime/ M.A. Ammar, Z. Bentalha, S. Bekhechi // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33702: 1–9. — Бібліогр.: 28 назв. — англ. 1607-324X PACS: 73.43.-f, 73.43.Cd, 71.10.Ca, 02.70.Wz DOI:10.5488/CMP.19.33702 arXiv:1609.04713 https://nasplib.isofts.kiev.ua/handle/123456789/156220 In a previous work, we reported exact results of energies of the ground state in the fractional quantum Hall effect (FQHE) regime for systems with up to Nе = 6 electrons at the filling factor ν = 1/3 by using the method of complex polar coordinates. In this work, we display interesting computational details of the previous calculation and extend the calculation to Ne = 7 electrons at ν = 1/3. Moreover, similar exact results are derived at the filling ν = 1/5 for systems with up to Ne = 6 electrons. The results that we obtained by analytical calculation are in good agreement with their analogues ones derived by the method of Monte Carlo in a precedent work. У попереднiй роботi ми отримали точнi результати для енергiй основного стану у режимi дробового квантового ефекта Холла (FQHE) для систем з Ne = 6 електронiв включно при коефiцiєнтi заповнення ν = 1/3, використавши метод комплексних полярних координат. В цiй роботi ми представляємо цiкавi обчислювальнi деталi попереднiх розрахункiв i розширюємо нашi обчислення до Ne = 7 електронiв при ν = 1/3. Крiм того, отримано подiбнi точнi результати при заповненнi ν = 1/5 для систем з Ne = 6 електронiв включно. Отриманi результати за допомогою аналiтичних обчислювань добре узгоджуються з їхнiми аналогами, отриманими методом Монте Карло в данiй роботi. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Method of computation of energies in the fractional quantum Hall effect regime Метод розрахунку енергiй у режимi дробового квантового ефекта Холла Article published earlier |
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Method of computation of energies in the fractional quantum Hall effect regime |
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Method of computation of energies in the fractional quantum Hall effect regime Ammar, M.A. Bentalha, Z. Bekhechi, S. |
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Method of computation of energies in the fractional quantum Hall effect regime |
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Method of computation of energies in the fractional quantum Hall effect regime |
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Method of computation of energies in the fractional quantum Hall effect regime |
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Method of computation of energies in the fractional quantum Hall effect regime |
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method of computation of energies in the fractional quantum hall effect regime |
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Ammar, M.A. Bentalha, Z. Bekhechi, S. |
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Ammar, M.A. Bentalha, Z. Bekhechi, S. |
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Метод розрахунку енергiй у режимi дробового квантового ефекта Холла |
| description |
In a previous work, we reported exact results of energies of the ground state in the fractional quantum Hall
effect (FQHE) regime for systems with up to Nе = 6 electrons at the filling factor ν = 1/3 by using the method of
complex polar coordinates. In this work, we display interesting computational details of the previous calculation
and extend the calculation to Ne = 7 electrons at ν = 1/3. Moreover, similar exact results are derived at the
filling ν = 1/5 for systems with up to Ne = 6 electrons. The results that we obtained by analytical calculation
are in good agreement with their analogues ones derived by the method of Monte Carlo in a precedent work.
У попереднiй роботi ми отримали точнi результати для енергiй основного стану у режимi дробового квантового ефекта Холла (FQHE) для систем з Ne = 6 електронiв включно при коефiцiєнтi заповнення ν = 1/3,
використавши метод комплексних полярних координат. В цiй роботi ми представляємо цiкавi обчислювальнi деталi попереднiх розрахункiв i розширюємо нашi обчислення до Ne = 7 електронiв при ν = 1/3.
Крiм того, отримано подiбнi точнi результати при заповненнi ν = 1/5 для систем з Ne = 6 електронiв
включно. Отриманi результати за допомогою аналiтичних обчислювань добре узгоджуються з їхнiми
аналогами, отриманими методом Монте Карло в данiй роботi.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/156220 |
| citation_txt |
Method of computation of energies in the fractional quantum Hall effect regime/ M.A. Ammar, Z. Bentalha, S. Bekhechi // Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33702: 1–9. — Бібліогр.: 28 назв. — англ. |
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| first_indexed |
2025-11-25T04:43:29Z |
| last_indexed |
2025-11-25T04:43:29Z |
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1850506834295652352 |
| fulltext |
Condensed Matter Physics, 2016, Vol. 19, No 3, 33702: 1–9
DOI: 10.5488/CMP.19.33702
http://www.icmp.lviv.ua/journal
Method of computation of energies in the fractional
quantum Hall effect regime
M.A. Ammar1, Z. Bentalha2, S. Bekhechi2
1 Environment Department, University of Medea, 26000 Media, Algeria
2 Theoretical Physics Laboratory, University of Tlemcen, B.P. 230, 13000 Tlemcen, Algeria
Received March 12, 2016, in final form April 18, 2016
In a previous work, we reported exact results of energies of the ground state in the fractional quantum Hall
effect (FQHE) regime for systems with up to Ne = 6 electrons at the filling factor ν= 1/3 by using the method of
complex polar coordinates. In this work, we display interesting computational details of the previous calculation
and extend the calculation to Ne = 7 electrons at ν = 1/3. Moreover, similar exact results are derived at the
filling ν = 1/5 for systems with up to Ne = 6 electrons. The results that we obtained by analytical calculation
are in good agreement with their analogues ones derived by the method of Monte Carlo in a precedent work.
Key words: quantum Hall effect, 2D electron gas, many-body wave function, strongly correlated system
PACS: 73.43.-f, 73.43.Cd, 71.10.Ca, 02.70.Wz
1. Introduction
The discovery of the fractional quantum Hall effect (FQHE) [1] was the beginning of a big revolution
in the field of condensed matter. Since then, new concepts of matter state have been raised such as the
incompressible quantum fluid [2], composite fermions [3–6], composite bosons [7] and anyons [8, 9], all
emanating from elegant theories with sophisticated mathematics. Nowadays, there are two world wide
accepted theories in the field of FQHE, the theory of Laughlin [2] and the theory of Jain [3–6]. The former
describes the ground state as an incompressible quantum fluid which successfully clarified the nature of
states at the filling factors ν= 1/3,1/5,1/7, . . . . The latter (theory) is built upon the concept of composite
fermions that are topological entities caricaturing the idea of electrons embracing a number (even) of
quantized vortices and gives satisfactory results regarding the ν = p/(2mp +1) states, integer m and p.
In Laughlin theory, the incompressible quantum fluid consists of strongly correlated electrons interacting
with a strongmagnetic fieldwhereas in Jain theory it consists of weakly correlated composite fermions in-
teracting with a reduced magnetic field. The aim of determining ground state energies for FQHE electron
systems has been the object of many investigations with various computational methods such as exact
diagonalization [10–14], density matrix renormalization group [15] or Monte Carlo simulations [16–18].
However, it is worth notifying that at the level of quasiparticle state, certain discrepancy is observed
between the results of [19] using spherical geometry and the results of [20] using disk geometry, while
the authors of [20] have found the reason of the discrepancy to be unclear. Also, all these methods are
numerical and one may wonder whether it is possible to perform analytical methods that would serve
as reliable comparison instruments even for small systems of electrons. The most pronounced analyt-
ical method in this line of research was given by the author of [21], where ordinary polar coordinates
(including ordinary Jacobi coordinates) are employed but technical calculational difficulties arise for sys-
tems with Ne > 4 electrons, and one can see an explicit dependence of the integrands upon the angles of
the particles (see equation (21) in [21]). This apparent difficulty can be overcome by using complex polar
coordinates, being the main contribution of [22]. By the way, it should be notified that ordinary polar
coordinates are also used in the original paper by Laughlin [2] within a Monte Carlo study. Regarding the
©M.A. Ammar, Z. Bentalha, S. Bekhechi, 2016 33702-1
http://dx.doi.org/10.5488/CMP.19.33702
http://www.icmp.lviv.ua/journal
M.A. Ammar, Z. Bentalha, S. Bekhechi
work [22], we developed an analytical method based on complex polar coordinates and explicitly calcu-
lated the energies of the ground state in the FQHE regime for systems with up to six electrons at the filling
ν= 1/3. The use of polar coordinates in a complex form has been the key tool which greatly simplifies the
calculation of some complicated expressions involving integrals over many variables in [22]. The aim of
this work is to show all the necessary computational steps and details underlying the analytical method
of [22] so as to make its application possible in other areas of condensed matter physics, especially in
2D Coulomb systems such as 2D Dyson gas wherein expressions involving integrals over many variables
are encountered such as the partition function or the mean energy. Thus, we obtain new exact analytical
results concerning the energy of the ground state for systems with Ne = 7 electrons at the filling ν= 1/3,
and Ne = 5,6 electrons at the filling ν= 1/5.
The paper is organized as follows. In section 2, the theoretical background is presented. In section 3,
the electron-electron interaction energy is calculated. In section 4, the method of computation of the
electron-background interaction energy is shown. In section 5, we give the results of our calculus. Sec-
tion 6 is devoted to the conclusion.
2. Theoretical background
We consider Ne(> 2) electrons of charge (−e0) embedded in a uniform neutralizing background disk
of positive charge Ne e0 and area SNe = πR2
Ne
, RNe is the radius of the disk. We also assume that the disk
is a part of the X Y plane subjected to a strong uniform magnetic field, in the z direction, B = B ez. The
physics of the FQH fluid is then governed by a full interaction potential
V =Vee+Veb+Vbb (2.1)
with Vee,Veb and Vbb denoting the electron-electron, electron-background and the background-back-
ground interaction potentials, respectively. Their corresponding expressions are given by
Vee =
N∑
i< j
e2
0
| ri − r j |
, (2.2)
Veb =−ρ
N∑
i=1
∫
SN
d2r
e2
0
| ri − r | , (2.3)
and
Vbb = ρ2
2
∫
SN
d2r
∫
SN
d2r ′ e2
0
| r− r′ | , (2.4)
where ri (or r j ) indicate the electron vector position while r and r′ are background coordinates. SNe (B)
is the area of the disk and ρ(B) is the density of the system (the number of electrons per unit area) that
can also be defined by
ρ = ν
2πl 2
0
, (2.5)
where l0(B) = √ħc/(e0B) is the magnetic length, c is the speed of light, B is the magnetic field strength,
and ν = 1/m is the filling factor, m = 3,5, . . . . The background-background interaction potential can be
classically calculated without using the wave function of the electron system. Its value is simply deter-
mined by calculating the elementary defined integral (2.4) and is given by [21]
Vbb =
8e2
0
3π
Ne
RNe
(2.6)
with RNe =
p
2Nem l0. It remains to calculate the energies corresponding to the electron-electron and
the electron-background potentials which depend on the nature of the wave function characterizing the
system of electron.
33702-2
Method of computation in FQHE
For a given wave functionΨ(r1,r2, . . . ,rNe ), these energies are determined using the following formu-
lae
〈Vee〉 = 〈Ψ|Vee|Ψ〉
〈Ψ|Ψ〉 , (2.7)
〈Veb〉 = 〈Ψ|Veb|Ψ〉
〈Ψ|Ψ〉 . (2.8)
In an explicit manner, we have
〈Ψ|Vee|Ψ〉 = Ne(Ne−1)
2
∫
d2r1 . . .d2rNe
e2
0
| r1 − r2 |
|Ψ(r1, . . . ,rNe ) |2, (2.9)
〈Ψ|Veb|Ψ〉 =−ρNe
∫
d2r1 . . .d2rNe |Ψ(r1, . . . ,rNe ) |2
∫
SNe
d2r
e2
0
| r1 − r | , (2.10)
〈Ψ|Ψ〉 =
∫
d2r1 . . .d2rNe |Ψ(r1, . . . ,rNe ) |2 (2.11)
with ([23, 24]) ∫
SNe
d2r
1
| r1 − r | = 2πRNe
∞∫
0
dq
q
J1(q)J0
(
q
RN
r1
)
, (2.12)
where Jn(x) are n-th order Bessel functions.
However, as shown in [22], the best way of making numerous simplifications in subsequent calcula-
tions amounts to replacing real polar coordinates by complex polar coordinates. Thus, let us apply the
change rk −→ zk = (xk + iyk = rk e iϕk )k=1,...,N to localize the electrons.
3. The 〈Vee〉 calculation
Now, for a demonstrative calculation, we will focus on the case of Ne = 4 electrons andm = 3. Let Ψ
be the wave function of Laughlin for Ne = 4 electrons andm = 3
Ψ(4) = P (4) exp
(
−∑
k
| zk |2
4l 2
0
)
, (3.1)
where P (4) is the Jastrow part of the wave function that is given by
P (4) = (z1 − z2)3(z1 − z3)3(z1 − z4)3(z2 − z3)3(z2 − z4)3(z3 − z4)3. (3.2)
In complex coordinates, the expressions of equation (2.9) and equation (2.11) transform into
〈Ψ|Vee|Ψ〉 = Ne(Ne−1)
2
∫
d2z1 . . .d2zNe
e2
0
| z1 − z2 |
|Ψ(z1, . . . , zNe ) |2, (3.3)
〈Ψ|Ψ〉 =
∫
d2z1 . . .d2zNe |Ψ(z1, . . . , zNe ) |2 . (3.4)
Now, we should perform a Jacobi transformation with complex coordinates instead of real coordinates
[22] so as to get rid of the term | z1−z2 | in the denominator of the integrand of the expression (3.3), which
is done using the following:
Z1 = z1 − z2, (3.5)
Z2 = z1
2
+ z2
2
− z3, (3.6)
Z3 = z1
3
+ z2
3
+ z3
3
− z4, (3.7)
Z4 = z1
4
+ z2
4
+ z3
4
+ z4
4
. (3.8)
33702-3
M.A. Ammar, Z. Bentalha, S. Bekhechi
Then, the inter-particle coordinates (zi − z j ) can be written in terms of Jacobi coordinates as follows:
z1 − z2 = Z1, (3.9)
z1 − z3 = Z2 + Z1
2
, (3.10)
z1 − z4 = Z3 + Z2
3
+ Z1
2
, (3.11)
z2 − z3 = Z2 − Z1
2
, (3.12)
z2 − z4 = Z3 + Z2
3
− Z1
2
, (3.13)
z3 − z4 = Z3 −2
Z2
3
. (3.14)
Now, to write P (4) in terms of the Zi Jacobi coordinates, we just recast (3.14) into (3.2), then there holds
the polynomial
P J (4) = Z 3
1
(
Z2 + Z1
2
)3 (
Z3 + Z2
3
+ Z1
2
)3 (
Z3 + Z2
3
+ Z1
2
)3 (
Z2 − Z1
2
)3 (
Z3 + Z2
3
− Z1
2
)3 (
Z3 −2
Z2
3
)3
. (3.15)
Similarly, the wave function becomes
ΨJ = P J (4)exp
(
−|Z1|2
8l 2
0
− |Z2|2
6l 2
0
− 3|Z3|2
16l 2
0
− |Z4|2
l 2
0
)
. (3.16)
It is possible to develop (3.15) in terms of Z n
1 , where n belongs to the set {3,5, . . . ,15} for the case of Ne = 4
electrons, thus we have
P J (4) =
15∑
n=3
Cn(Z2, Z3) Z n
1 , (3.17)
where Cn are functions of the only variables Z2 and Z3 that can be extracted from (3.15) by the use of
Cn(Z2, Z3) = 1
πΓ(1+n)
15∑
m=3
Cm(Z2, Z3)
∫
d2Z1 Z m
1 Z̄1
n
e−Z1 Z̄1 , (3.18)
wherein the integration is determined with the help of the key rule [25]∫
d2Z Z m Z̄ n e−Z Z̄ =πδmn Γ(1+n). (3.19)
In computing | Ψ(Z1, . . . , Z3) |2, we will encounter the expression P J (4)P̄ J (4), where P̄ J is the complex
conjugate of P J , by requiring to satisfy the rule (3.19), there only remain the terms with the same power
in Zi and Z̄i . Thus, the integrand of equation (3.3) has no dependence on the angles. This independence
upon the angles is the key tool that greatly facilitates the exact calculation of complicated expressions
involving integrals over many variables (see the work [22]). This is the most prominent advantage of the
method of complex coordinates. For instance, the integral (3.3) can be reduced to a simple form
〈Ψ|Vee|Ψ〉 = e2
0
Ne(Ne−1)
2
15∑
n=3
Fn
∫
d2Z1(Z1 Z̄1)n− 1
2 exp
(
−|Z1|2
4l 2
0
)
(3.20)
with the factorFn given by
Fn =
∫
d2Z2d2Z3d2Z4CnC̄n exp
(
−|Z2|2
3l 2
0
− 3|Z3|2
8l 2
0
− 2|Z4|2
l 2
0
)
. (3.21)
Similarly, the norm 〈Ψ|Ψ〉 is of the form
〈Ψ|Ψ〉 =
15∑
n=3
Fn
∫
d2Z1(Z1 Z̄1)n exp
(
−|Z1|2
4l 2
0
)
. (3.22)
33702-4
Method of computation in FQHE
Now, dividing (3.20) by (3.22), we get the (e-e) interaction energy for a system with Ne = 4 electrons,
Eee = 1.310596(e2
0/l0),
which coincides with an analogous result in [21].
4. The 〈Veb〉 calculation
In the case of 〈Veb〉 calculation, there is no need to use Jacobi coordinates, it suffices to work with the
polynomial P (4) of equation (3.3) directly. So, let us expand P (4) in powers of z1, that is
P (4) =
9∑
n=0
Cn(z2, z3, z4) zn
1 , (4.1)
where n = 0 (n = 9) denotes the minimum (maximum) power in z1, the wave function is, therefore,
written as follows:
Ψ= P (4)exp
(
−|z1|2
2l 2
0
− |z2|2
2l 2
0
− |z3|2
2l 2
0
− |z4|2
2l 2
0
)
. (4.2)
Furthermore, it is possible to write 〈Ψ|Veb|Ψ〉 like
〈Ψ|Veb|Ψ〉 = −2N 2
e
Re
9∑
n=0
G (n)
∫
Dp z CnC̄n exp
(
−|z2|2
2l 2
0
− |z3|2
2l 2
0
− |z4|2
2l 2
0
)
(4.3)
with Dp z = d2z2d2z3d2z4,
G (n) =
∫
dr1
∫
dq
q
J1(q)J0
(
q
RNe
r1
)
r 2n+1
1 exp
(
− r 2
1
2l 2
0
)
, (4.4)
and |zi | = ri . One can verify that [26]
G (n) = (
2l 2
0
)n+1
(
1
4
)
MeijerG
[{
{1}, {1}
}
,
{
{1/2,n +1}, {−1/2}
}
, Nem
]
. (4.5)
In the present demonstrative calculation, Ne = 4 and m = 3. MeijerG is the Meijer G function [26]. The
expression of the Meijer G function in (4.5) can also be written as
MeijerG
[{
{1}, {1}
}
,
{
{1/2,n +1}, {−1/2}
}
, Nem
]=G21
23
(
Nem
∣∣∣1,1
1
2 ,n+1,− 1
2
)
.
As concerns the norm 〈Ψ|Ψ〉, in the z-coordinates, it will take the following form
〈Ψ|Ψ〉 =
9∑
n=0
∫
Dp z CnC̄n exp
(
−|z2|2
2l 2
0
− |z3|2
2l 2
0
− |z4|2
2l 2
0
)∫
d2z1(z1 z̄1)n exp
(
−|z1|2
2l 2
0
)
. (4.6)
As in the case of Vee calculation, the electron-background interaction energy is determined using
Eeb = 〈Ψ|Veb|Ψ〉
〈Ψ|Ψ〉 , (4.7)
which gives the value Eeb =−5.638272(e2
0/l0) as in the work [21]. The dependence in angles is simplified
when dividing the quantity 〈Ψ|Veb|Ψ〉 by 〈Ψ|Ψ〉. This is because the result of the integration upon the
angle variables is the same for both quantities and is equal to (2π)4
.
33702-5
M.A. Ammar, Z. Bentalha, S. Bekhechi
5. Results and discussion
In this paragraph, we present our results concerning the ground state energy for systems with up to
Ne = 7 electrons at the filling ν= 1/3 and Ne = 6 electrons at the filling ν= 1/5. We also make a compar-
ison with other works such as [17, 21, 27]. Our analysis is summarized in the tables blow. It should be
notified that for Ne = 4 electrons, the authors in [17] and with Monte Carlo calculations, have derived for
the energy of the ground state the value −1.55536e2
0/l0 at the filling ν= 1/3 and the value −1.28636e2
0/l0
at the filling ν= 1/5, which well agree with the values we derived by the exact analytical calculation, see
table 1 and table 2. Table 3 allows us to compare our results concerning the (e-b) and (e-e) interaction
energies with those derived in [27] at the filling ν= 1/5.
In tables 1 and 2, our results regarding the energy of the ground state are given in the fifth column
whereas in the sixth column there are given those of [21] and [27], respectively. At this point, it should be
emphasized that in table 1, table 2 and table 3, the comparison is carried out between analytical methods
depending on whether ordinary or complex polar coordinates are used. Moreover, this presentation of
tables allows one to clearly show the advantage of using polar coordinates in the complex form.
Table 1. Ground-state energy E = Eee+Eeb+Ebb (in units of e2
0/l0) obtained in the Laughlin state at the
filling ν = 1/3 for systems with up to Ne = 7 electrons. The results EA are the values of the ground state
energy obtained also by an exact analytical calculation in [21] at the filling ν= 1/3.
Ne Ebb Eeb Eee E EA
2 0.98014 −2.02115 0.276946 −0.764064 −0.764064
3 1.800633 −3.679464 0.719316 −1.159515 −1.159515
4 2.772256 −5.638272 1.310596 −1.55542 −1.55542
5 3.874345 −7.856335 2.030715 −1.951275 ———
6 5.092956 −10.306452 2.864394 −2.349102 ———
7 6.417859 −12.968494 3.802267 −2.748368 ———
Table 2. Ground-state energy E = Eee+Eeb+Ebb (in units of e2
0/l0) obtained in the Laughlin state at the
filling ν = 1/5 for systems with up to Ne = 6 electrons. The results EA are the values of the ground state
energy obtained also by an exact analytical calculation in [27] at the filling ν= 1/5.
Ne Ebb Eeb Eee E EA
3 1.394763 −2.911356 0.554745 −0.961848 −0.961848
4 2.14738 −4.4429 1.009184 −1.286336 −1.286312
5 3.001055 −6.17325 1.566615 −1.60558 ———
6 3.944988 −8.082144 2.209278 −1.927878 ———
Table 3. The electron-electron (e-e) and electron-background (e-b) interaction energies are given (in units
of e2
0/l0) at the filling ν= 1/5, namely Eee and Eeb. Analogous results that we designate by EeeA and EebA,
are derived in [27] at the filling ν= 1/5.
Ne Eeb EebA Eee EeeA
3 −2.911356 −2.911356 0.554745 0.554745
4 −4.4429 −4.442876 1.009184 1.009184
5 −6.17325 ——— 1.566615 ——–
6 −8.082144 ——— 2.209278 ——–
33702-6
Method of computation in FQHE
Figure 1. (Color online) Exact analytical results for the ground state energy E using the method of com-
plex polar coordinates in disk geometry for the Laughlin state at ν = 1/5. The ground state energy E is
plotted as a function of 1/Ne for systems with Ne = 3,4,5 and 6 electrons. The stars represent our result,
the disks are the results derived in [17] using the method of Monte Carlo. Energies are in units of e2
0/l0.
In figure 1, we can see that the results derived by the present exact analytical calculation at the filling
ν= 1/5 compare well with the results of [17] obtained using the method of Monte Carlo.
6. Concluding remarks
In this work we have exposed all the necessary steps that permit to make an analytic computation
of the energies of the ground state for FQHE systems of electrons at ν = 1/m, m odd. The electron-
electron and electron-background interaction energies are calculated separately. The results we derived
are in perfect accordance with previous calculations such as the exact analytical calculation of [21, 27]
or Monte Carlo simulations of [17]. In a broader view, the method of complex polar coordinates de-
scribed in [22] may be useful and efficient in analytically calculating the ground state or excited state
energies for various quantum Hall systems of electrons with filling factors other than ν = 1/m, m odd,
such as (ν= 2/5,3/7, . . .). We expect that the method of complex polar coordinates has some relevance to
2D Coulomb systems. For instance, it can be seen, for 2D Dyson gas, that the method of complex coordi-
nates may be useful and practical in analytically evaluating, withmany simplifications, the key quantities
such as the partition function or the mean energy. The issue of finding links for the approach described
in [22] with other areas of condensed matter physics remains to be extensively investigated. The calcu-
lation can be extended to larger systems with Ne > 7 electrons depending on the performance of the
machine. This will make it possible to derive exact analytical bulk regime values for key quantities, such
as various interaction energies. A part of the code [28] of the electron-electron interaction energy com-
putation Vee is given in the Appendix.
33702-7
M.A. Ammar, Z. Bentalha, S. Bekhechi
Appendix
* Part of the code of the Vee calculation in MATHEMATICA SOFTWARE (Ne=4 electrons)*
PartialVee=Block[{PolyJaco, CoefPoly, CoefPolyMin, PolyExpand, RePoly, CoefPolyList, Clist,
Inner1, Inlist, I1, I2, I3,Vee },
PolyJaco=Z[1]^3(-Z[1]/2 + Z[2])^3(Z[1]/2 + Z[2])^3(-2 Z[2]/3 + Z[3])^3
(-Z[1]/2 + Z[2]/2 + Z[3])^3(Z[1]/2 + Z[2]/2 + Z[3])^3;
CoefPoly=Exponent[PolyJaco, Z[1]];
CoefPolyMin=Exponent[PolyJaco,Z[1],Min];
PolyExpand=Expand[PolyJaco];
RePoly=Flatten[Table[Coefficient[PolyExpand,Z[1],i],{i,CoefPolyMin,CoefPoly,2}]];
CoefPolyList=Plus@@RePoly;
Clist=CoefPolyList/.Plus->List;
Inner1=Inner[Times,Clist/.{Z[2]-> 1, Z[3]-> 1},Clist,Plus];
Inlist=Inner1/.Plus->List/.{Z[2]-> r[2]^2, Z[3]-> r[3]^2};
I1 = Integrate[Inlist*r[2]*r[3]*Exp[-(3 r[3]^2/(8l[0]^2))],
{r[3],0,Infinity}, Assumptions -> (l[0]>0)];
I2 = Integrate[I1*Exp[-(r[2]^2/(3l[0]^2))],{r[2],0,Infinity},
Assumptions -> (l[0]>0)];
I3 = Integrate[I2*r[4]*Exp[-(2 r[4]^2/(l[0]^2))],{r[4],0,Infinity},
Assumptions -> (l[0]>0)];
Vee = Reverse[Plus @@ I3 /. Plus -> List]]]
PartialVee
{1688579923968000 l[0]^36, 3170189352960 l[0]^32,
3024980640 l[0]28, 2095200 l[0]^24, 122715/128 l[0]^20,
189/1024 l[0]^16), 3/262144 l[0]^12}
II[i_] := Integrate[r[1]^(2i)*Exp[(-r[1]^2)/(4l[0]^2)], {r[1], 0,
Infinity}, Assumptions -> l[0] > 0)];
JJ[i_] := Integrate[r[1]^(2i+1)*Exp[(-r[1]^2)/(4l[0]^2)], {r[1], 0,
Infinity}, Assumptions -> l[0] > 0)];
list1 = Flatten[Table[II[i], {i, 3, 15, 2}]];
list2 = Flatten[Table[JJ[i], {i, 3, 15, 2}]];
num = Inner[Times, PartialVee, list1, Plus];
denom = Inner[Times, PartialVee, list2, Plus];
FullVee = Divide[num, denom];
Eee = N[Times[FullVee, (12/2) e[0]^2], 6];
Eee
1.310596 e[0]^2 / l[0]
33702-8
Method of computation in FQHE
References
1. Tsui D.C., Stormer H.L., Gossard A.C., Phys. Rev. Lett., 1982, 48, 1559; doi:10.1103/PhysRevLett.48.1559.
2. Laughlin R.B., Phys. Rev. Lett., 1983, 50, 1395; doi:10.1103/PhysRevLett.50.1395.
3. Jain J.K., Phys. Rev. Lett., 1989, 63, 199; doi:10.1103/PhysRevLett.63.199.
4. Jain J.K., Composite Fermions, Cambridge University Press, New York, 2007.
5. Jain J.K., Phys. Rev. B, 1990, 41, 7653; doi:10.1103/PhysRevB.41.7653.
6. Jain J.K., Science, 1994, 266, 1199; doi:10.1126/science.266.5188.1199.
7. Simon S.H., Rezayi E.H., Milovanovic M.V., Phys. Rev. Lett., 2003, 91, 046803; doi:10.1103/PhysRevLett.91.046803.
8. Wilczek F., Phys. Rev. Lett., 1982, 49, 957; doi:10.1103/PhysRevLett.49.957.
9. Arovas D.P., Schrieffer J.R., Wilczek F., Phys. Rev. Lett., 1984, 53, 722; doi:10.1103/PhysRevLett.53.722.
10. Haldane F.D.M., Phys. Rev. Lett., 1983, 51, 605; doi:10.1103/PhysRevLett.51.605.
11. Haldane F.D.M., Rezayi E.H., Phys. Rev. Lett., 1985, 54, 237; doi:10.1103/PhysRevLett.54.237.
12. Fano G., Ortolani F., Colombo E., Phys. Rev. B, 1986, 34, 2670; doi:10.1103/PhysRevB.34.2670.
13. D’Ambrumenil N., Morf R., Phys. Rev. B, 1989, 40, 6108; doi:10.1103/PysRevB.40.6108.
14. He S., Simon S.H., Halperin B.I., Phys. Rev. B, 1994, 50, 1823; doi:10.1103/PhysRevB.50.1823.
15. Feiguin A.E., Rezayi E., Nayak C., Das Sarma S., Phys. Rev. Lett., 2008, 100, 166803;
doi:10.1103/PhysRevLett.100.166803.
16. Caillol J.M., Levesque D., Weis J.J., Hansen J.P., J. Stat. Phys., 1982, 28, 325; doi:10.1007/BF01012609.
17. Ciftja O., Wexler C., Phys. Rev. B, 2003, 67, 075304; doi:10.1103/PhysRevB.67.075304.
18. Morf R., Halperin B.I., Phys. Rev. B, 1986, 33, 2221; doi:10.1103/PhysRevB.33.2221.
19. Melik-Alaverdian V., Bonesteel N.E., Phys. Rev. B, 1998, 58, 1451; doi:10.1103/PhysRevB.58.1451.
20. Jeon G.S., Jain J.K., Phys. Rev. B, 2003, 68, 165346; doi:10.1103/PhysRevB.68.165346.
21. Ciftja O., Physica B, 2009, 404, 227; doi:10.1016/j.physb.2008.10.036.
22. Bentalha Z., Moumen L., Ouahrani T., Cent. Eur. J. Phys., 2014, 12, 511; doi:10.2478/s11534-014-0476-5.
23. Ciftja O., Phys. Lett. A, 2010, 374, 981; doi:10.1016/j.physleta.2009.12.017.
24. Ciftja O., J. Comput.-Aided Mater. Des., 2007, 14, 37; doi:10.1007/s10820-006-9035-8.
25. Shakirov Sh., Phys. Lett. A, 2011, 375, 984; doi:10.1016/j.physleta.2011.01.004.
26. Gradshteyn I.S., Ryzhik I.M., Table of Integrals, Series, and Products, Academic Press, New York, 1980.
27. Ciftja O., Physica B, 2009, 404, 2244; doi:10.1016/j.physb.2009.04.018.
28. Wolfram Research, Inc., Mathematica, Version 4.0, Champaign, Illinois, 1999.
Метод розрахунку енергiй у режимi дробового квантового
ефекта Холла
M.A. Аммар1, З. Бенталха2, С. Бехечi2
1 Вiддiлення охорони довкiлля, Унiверситет м.Медеа, 26000Медеа, Алжир
2 Лабораторiя теоретичної фiзики, Унiверситет м. Тлемсен, B.P. 230, 13000 Тлемсен, Алжир
У попереднiй роботiми отримали точнi результати для енергiй основного стану у режимi дробового кван-
тового ефекта Холла (FQHE) для систем з Ne = 6 електронiв включно при коефiцiєнтi заповнення ν= 1/3,
використавши метод комплексних полярних координат. В цiй роботi ми представляємо цiкавi обчислю-
вальнi деталi попереднiх розрахункiв i розширюємо нашi обчислення до Ne = 7 електронiв при ν= 1/3.
Крiм того, отримано подiбнi точнi результати при заповненнi ν = 1/5 для систем з Ne = 6 електронiв
включно. Отриманi результати за допомогою аналiтичних обчислювань добре узгоджуються з їхнiми
аналогами, отриманими методом Монте Карло в данiй роботi.
Ключовi слова: квантовий ефект Холла, 2D електронний газ, багаточастинкова хвильова функцiя,
сильно скорельована система
33702-9
http://dx.doi.org/10.1103/PhysRevLett.48.1559
http://dx.doi.org/10.1103/PhysRevLett.50.1395
http://dx.doi.org/10.1103/PhysRevLett.63.199
http://dx.doi.org/10.1103/PhysRevB.41.7653
http://dx.doi.org/10.1126/science.266.5188.1199
http://dx.doi.org/10.1103/PhysRevLett.91.046803
http://dx.doi.org/10.1103/PhysRevLett.49.957
http://dx.doi.org/10.1103/PhysRevLett.53.722
http://dx.doi.org/10.1103/PhysRevLett.51.605
http://dx.doi.org/10.1103/PhysRevLett.54.237
http://dx.doi.org/10.1103/PhysRevB.34.2670
http://dx.doi.org/10.1103/PysRevB.40.6108
http://dx.doi.org/10.1103/PhysRevB.50.1823
http://dx.doi.org/10.1103/PhysRevLett.100.166803
http://dx.doi.org/10.1007/BF01012609
http://dx.doi.org/10.1103/PhysRevB.67.075304
http://dx.doi.org/10.1103/PhysRevB.33.2221
http://dx.doi.org/10.1103/PhysRevB.58.1451
http://dx.doi.org/10.1103/PhysRevB.68.165346
http://dx.doi.org/10.1016/j.physb.2008.10.036
http://dx.doi.org/10.2478/s11534-014-0476-5
http://dx.doi.org/10.1016/j.physleta.2009.12.017
http://dx.doi.org/10.1007/s10820-006-9035-8
http://dx.doi.org/10.1016/j.physleta.2011.01.004
http://dx.doi.org/10.1016/j.physb.2009.04.018
Introduction
Theoretical background
The "42683AD Vee"52693AE calculation
The "42683AD Veb"52693AE calculation
Results and discussion
Concluding remarks
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