Analytic results of the excited electronic states at $\upsilon=1/3$ and the Laughlin-Jain microscopic wave function approaches

M.A. Ammar

arxiv: 1906.11562 · v1 · pith:WLNCGBRFnew · submitted 2019-06-27 · ❄️ cond-mat.str-el

Analytic results of the excited electronic states at upsilon=1/3 and the Laughlin-Jain microscopic wave function approaches

M.A. Ammar This is my paper

Pith reviewed 2026-05-25 14:37 UTC · model grok-4.3

classification ❄️ cond-mat.str-el

keywords quantum Hall effectcomposite fermionsdisk geometryanalytic Coulomb integralsfilling factor 1filling factor 1/3Laughlin-Jain states

0 comments

The pith

Exact analytic expressions for Coulomb energies at filling factor 1 are derived for disk systems of up to 10 electrons and extended to composite-fermion excited states at 1/3.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives exact analytic results for the ground-state energies of two-dimensional electrons in the integer quantum Hall regime at filling factor υ=1, for particle numbers up to Ne=10 in disk geometry. This is accomplished by transforming the Coulomb interaction integrals into closed-form expressions via the complex polar coordinates method. The resulting energies agree with earlier numerical representations of Bessel-function integrals. The same framework is then applied to the energies of composite fermions in excited states at υ=1/3, establishing a direct correspondence between the fractional and integer quantum Hall effects.

Core claim

Using the complex polar coordinates method, exact analytic expressions for the Coulomb matrix elements are obtained in disk geometry for Ne up to 10. These expressions produce ground-state energies at υ=1 that match those reported by Ciftja. The method is further used to compute composite-fermion energies for excited states at υ=1/3 and to demonstrate the correspondence between the fractional quantum Hall effect at 1/3 and the integer quantum Hall effect.

What carries the argument

The complex polar coordinates method, which converts Coulomb integrals in disk geometry into exact analytic expressions for finite electron numbers.

If this is right

Ground-state energies at υ=1 become available in closed analytic form for all Ne from 2 to 10 without numerical quadrature.
Composite-fermion energies at υ=1/3 can be evaluated analytically by mapping to the integer quantum Hall problem at υ=1.
The Laughlin-Jain wave-function description is shown to be consistent with the analytic energies for small systems at fractional filling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The closed-form expressions could serve as benchmarks for testing approximate methods in regimes where exact diagonalization becomes intractable.
The established IQHE–FQHE correspondence suggests that certain excitation spectra at 1/3 may be predicted directly from integer-filling calculations.
Extension of the same coordinate transformation to other observables, such as density profiles or pair-correlation functions, appears feasible for the same range of Ne.

Load-bearing premise

The complex polar coordinates method yields exact analytic expressions for the Coulomb matrix elements in the disk geometry for arbitrary Ne up to 10.

What would settle it

A direct comparison of the analytic ground-state energies obtained for Ne=10 at υ=1 against exact diagonalization results would confirm or refute the expressions if systematic deviations appear beyond numerical precision.

Figures

Figures reproduced from arXiv: 1906.11562 by M.A. Ammar.

**Figure 1.** Figure 1: (Colour online) The Ne − 1 quasielectron energies for Ne = 7 in disk geometry at υ = 1/3. The green thickness represents Laughlin’s states ΨL (Ne−1)qp. The dashed circle represents Jain’s states ΨCF [1,1,...,1] . The blue rectangle represents IQHE states at υ = 1. In figure 1, we can see that the results derived by the present exact analytical calculation at the filling υ = 1 and υ = 1/3 (the excited state… view at source ↗

read the original abstract

In this work we studied the properties of a two-dimensional electronic gas subjected to a strong magnetic field and cooled at a low temperature. We reported exact analytical results of energies at the ground state. The results are for systems up to $N_{e}=10$ electrons calculated in the integer quantum Hall effect (IQHE) regime at the filling factor $\upsilon=1$. To accomplish the calculation we used the complex polar coordinates method. Note that the system of electrons in the quantum Hall regime relied heavily on the disk geometry for finite systems of electrons with arbitrary values of $N_{e}=2$ to $10$ particles. The results that we obtained by analytical calculations are in good agreement with those reported by Ciftja [Ciftja O., J. Math. Phys., 2011, 52, 122105], where the representation for certain integrals of products of Bessel functions is obtained. In the end, we have studied the composite fermions energies for the excited states for several systems at $\upsilon =1/3$ and the correspondence between the fractional quantum Hall effect (FQHE) and the IQHE.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper re-derives ground-state energies at ν=1 for N≤10 that match Ciftja 2011 and adds composite-fermion calculations at 1/3, but shows no new closed forms or derivations.

read the letter

The main takeaway is that the authors use complex polar coordinates to compute ground-state energies at filling factor 1 for systems up to 10 electrons in disk geometry, and they state these match the integral representations already given by Ciftja in 2011. They also compute composite-fermion energies for some excited states at 1/3 and note the correspondence with the integer case. That extension to the fractional side is the only part that goes beyond the 2011 reference. The calculations are limited to tiny systems and serve mainly as consistency checks rather than new predictions. The agreement with prior work is a basic sanity check that the method reproduces known numbers, which is fine as far as it goes. The soft spot is the claim of “exact analytical results.” The abstract ties the energies to Ciftja’s Bessel-product integrals without exhibiting the final expressions or showing that the integrals reduce to elementary functions. If those integrals still require numerical quadrature, the results are semi-analytic at best, and the “exact” label does not hold up on the evidence given. No derivation steps, explicit matrix elements, or error estimates appear in the abstract, and the scope stays at N=10 with nothing on the thermodynamic limit or experimental links. This work would mainly interest people who run exact-diagonalization codes on small quantum-Hall droplets and want another set of benchmark numbers. A reader looking for a new framework, larger-system insight, or falsifiable predictions will not find it. I would not send it for peer review; the reproduction aspect plus the unshown steps make it a desk-reject candidate unless the full text contains previously unseen closed-form expressions that actually advance the 1/3 calculations.

Referee Report

1 major / 1 minor

Summary. The manuscript studies the 2D electron gas in a strong perpendicular magnetic field at low temperature. It claims to obtain exact analytic ground-state energies for the integer quantum Hall regime at filling factor υ=1 for systems with Ne=2 to 10 electrons in disk geometry, achieved via a complex-polar-coordinate transformation of the Coulomb matrix elements. These results are stated to agree with the Bessel-product integral representations of Ciftja (2011). The work additionally examines composite-fermion energies of excited states at υ=1/3 and discusses the correspondence between FQHE and IQHE.

Significance. If the central claim of fully closed-form, non-numerical Coulomb matrix elements holds and the resulting energies are exact, the results would supply useful analytic benchmarks for small-N systems in the IQHE regime and a controlled starting point for composite-fermion constructions at fractional filling. Such benchmarks are valuable for validating numerical diagonalizations and for testing the accuracy of approximate wave functions.

major comments (1)

[Abstract and calculation method paragraph] Abstract and paragraph on the calculation method: the assertion that the complex-polar-coordinate method 'yields exact analytic expressions for the Coulomb matrix elements' for arbitrary Ne≤10 is load-bearing for the title and abstract claim of 'exact analytical results,' yet the manuscript supplies neither the explicit closed-form expressions for ⟨m1 m2|V|m3 m4⟩ nor a demonstration that the remaining integrals after the coordinate transformation evaluate to elementary or special functions without numerical quadrature. Only agreement with Ciftja’s integral representations is stated; the matching expressions and convergence or error analysis are not exhibited.

minor comments (1)

The manuscript should clarify whether the reported energies are obtained after exact analytic summation over all matrix elements or after any truncation or numerical step.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment below.

read point-by-point responses

Referee: [Abstract and calculation method paragraph] Abstract and paragraph on the calculation method: the assertion that the complex-polar-coordinate method 'yields exact analytic expressions for the Coulomb matrix elements' for arbitrary Ne≤10 is load-bearing for the title and abstract claim of 'exact analytical results,' yet the manuscript supplies neither the explicit closed-form expressions for ⟨m1 m2|V|m3 m4⟩ nor a demonstration that the remaining integrals after the coordinate transformation evaluate to elementary or special functions without numerical quadrature. Only agreement with Ciftja’s integral representations is stated; the matching expressions and convergence or error analysis are not exhibited.

Authors: We agree that the manuscript would benefit from greater explicitness on this point. The complex-polar-coordinate transformation recasts the Coulomb matrix elements into integral forms whose analytic character is established by their exact reduction to the Bessel-product representations derived by Ciftja (2011). In the revised manuscript we will (i) display the explicit transformed expressions for ⟨m1 m2|V|m3 m4⟩ for representative values of Ne≤10, (ii) show the direct algebraic matching to Ciftja’s integrals, and (iii) include a short numerical verification confirming that the resulting energies agree with independent diagonalization results to machine precision, thereby demonstrating that no additional quadrature is required beyond evaluation of the known special functions. These additions will be placed in a new subsection of the methods and will also be referenced in the abstract and title discussion. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation relies on external validation

full rationale

The abstract describes use of the complex polar coordinates method to obtain analytic energies for Ne≤10 at υ=1, with explicit agreement stated against the independent Ciftja 2011 external reference for the Bessel-product integrals. No equations are shown that define a quantity in terms of itself, no fitted parameters are relabeled as predictions, and the sole citation is to a non-overlapping author. The derivation chain therefore remains self-contained against an external benchmark rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, new axioms, or invented entities are identifiable. The calculation relies on standard complex analysis and the disk-geometry Coulomb interaction assumed in prior quantum-Hall literature.

pith-pipeline@v0.9.0 · 5739 in / 1165 out tokens · 23012 ms · 2026-05-25T14:37:23.892915+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The CF wave function for the quasiparticle at υ=1/3... Ψ[1,...,1]CF = PLLL ...

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages · 1 internal anchor

[1]

)nTRoduCTIon A discovery of the integer quantum Hall eﬀect (IQHE) [1, 2] was the beginning of a big revolution in the ﬁeld of condensed matter. It is interesting to study theoretically and numerically the phases of the integer and fractional quantum Hall eﬀect (FQHE) in a two-dimensional geometry in order to consider various aspects of this problem. In th...

work page internal anchor Pith review Pith/arXiv arXiv 1983
[2]

Thus, any state in the lowest Landau level (LLL) is given by a polynomial equation dependent only onz

(1.5) The wave functions in the lowest Landau level are simple monomials inz. Thus, any state in the lowest Landau level (LLL) is given by a polynomial equation dependent only onz. The probability of presence of the state|0, m⟩ is maximal over a circle of radiusrm = √ 2ml0, such that the radial extension ofthewavefunctionisoftheorderof l0.When mincreases,...

work page
[3]

We considerNe = 10 electrons of charge (−e0) embedded in a uniform neutralizing background disk of an area and a positive charge Ne e0

-odEL oF InTERACTIon The many-electron system is described by the Hamiltonian ˆH = ˆK + ˆV, (2.1) where ˆK is the kinetic energy operator,ℏ is reduced Planck’s constant,ω = e0B/m is the cyclotron frequency, and the Coulomb interactionˆV projected in the LLL is obtained starting from the electron- electroninteraction,electron-backgroundandthebackground-bac...

work page
[4]

AMMAR Similarly, for theˆVeb interaction, we have ⟨ˆVeb ⟩ =−ρNe ¾ d2r1

(2.11) 23701-3 -.A. AMMAR Similarly, for theˆVeb interaction, we have ⟨ˆVeb ⟩ =−ρNe ¾ d2r1... d2rNe Ψr1,..., rNe 2 ¾ SN d2r e2 0 |r1− r|, (2.12) with ¾ SN d2r e2 0 |r1− r|= 2πRN ∞¾ 0 dq q J1(q)J0 ( q RN r1 ) , (2.13) where is theJn(x) n-th order Bessel functions. A detailed description of the ground state obtained by an analytical method is given in our e...

work page
[5]

AnALyTICAL RESuLTS FoR )1(E ATυ = 1 Inthissection,weobtaintheanalyticalexpressionsforthetotalenergyperparticle(inunits e2 0/l0)and related quantities corresponding to IQHE system of electrons in a disk geometry atυ = 1. The ground state interaction energy per particle can be written as follows: ε =εee +εeb +εbb, (3.1) whereε = ⟨ˆV ⟩ /Ne, εee = ⟨ˆVee ⟩ /Ne...

work page
[6]

quasielectrons

4HE QuASIELECTRon (qe ) EnERgIES Nowadays,therearetwouniversallyacceptedtheoriesintheﬁeldofFQHE,thetheoryofLaughlin[6] and the theory of Jain [8]. An early trial wave function proposed by Laughlin for the ground state at the ﬁlling factorυ = 1/m, m odd, turned out to work well [13]. The CF theory applies to a broader range of phenomena, while also providi...

work page
[7]

The code of the electron-background interaction energy computationεeb is shown in appendix B

2ESuLTS And dISCuSSIon Withinthiswork,weconsideredtwosystems.Theﬁrstoneforthegroundenergywithupto Ne = 10 electrons at the ﬁllingυ = 1 and the second one for the exited energy with up toNe = 7 electrons at the 23701-6 ,AUgHLIN-JAIN MICROsCOPIC WAVE FUNCTION APPROACHEs ﬁllingυ = 1/3.Severalresearchersusedthediskgeometryintheirworks[3,5,6,11].Themathematica...

work page
[8]

DIRECTORy NAME

ConCLuSIon We conclude that the analytical expression for the Coulomb interaction energy in disk geometry at ﬁllingfactor1/3permitsustoobtainacorrespondencebetweenthefractionalquantumHalleﬀectandthe integer quantum Hall eﬀect. The coinciding energies of the IQHE and the(Ne− 1) quasielectron of the FQHEstatesareexpected,wheretheFQHEandtheIQHEcanbeuniﬁed.Th...

work page
[9]

Klitzing K.V., Dorda G., Pepper M., Phys. Rev. Lett., 1980,45, 494, doi:10.1103/PhysRevLett.45.494

work page doi:10.1103/physrevlett.45.494 1980
[10]

Tsui D.C., Stormer H.L., Gossard A.C., Phys. Rev. Lett., 1982,48, 1559, doi:10.1103/PhysRevLett.48.1559

work page doi:10.1103/physrevlett.48.1559 1982
[11]

Morf R., Halperin B.I., Phys. Rev. B, 1986,33, 2221, doi:10.1103/PhysRevB.33.2221

work page doi:10.1103/physrevb.33.2221 1986
[12]

Fano G., Ortolani F., Phys. Rev. B, 1987,37, 8179, doi:10.1103/PhysRevB.37.8179

work page doi:10.1103/physrevb.37.8179 1987
[13]

Ciftja O., Wexler C., Phys. Rev. B, 2003,67, 075304, doi:10.1103/PhysRevB.67.075304

work page doi:10.1103/physrevb.67.075304 2003
[14]

Laughlin R.B., Phys. Rev. Lett., 1983,50, 1395, doi:10.1103/PhysRevLett.50.1395

work page doi:10.1103/physrevlett.50.1395 1983
[15]

Jain J.K., Composite Fermions, Cambridge University Press, Cambridge, 2007

work page 2007
[16]

Jain J.K., Phys. Rev. Lett., 1989,63, 199, doi:10.1103/PhysRevLett.63.199

work page doi:10.1103/physrevlett.63.199 1989
[17]

MorfR.H.,d’AmbrumenilN.,DasSarmaS.,Phys.Rev.B,2002, 66,075408,doi:10.1103/PhysRevB.66.075408

work page doi:10.1103/physrevb.66.075408 2002
[18]

Ciftja O., Physica B, 2009,404, 227, doi:10.1016/j.physb.2008.10.036

work page doi:10.1016/j.physb.2008.10.036 2009
[19]

Matter Phys., 2016,19, 33702: 1–9, doi:10.5488/CMP.19.33702

Ammar M.A., Bentalha Z., Bekhechi S., Condens. Matter Phys., 2016,19, 33702: 1–9, doi:10.5488/CMP.19.33702. 23701-9 -.A. AMMAR

work page doi:10.5488/cmp.19.33702 2016
[20]

Ahmed Ammar M., Méthodes Analytique et Numérique dans l’Eﬀet Hall Quantique Fractionnaire et Rôle du Spin des Électrons, PhD thesis, Abou-Bekr Belka¯ıd University, Tlemcen, Algeria, 2018

work page 2018
[21]

Jeon G.S., Jain J.K., Phys. Rev. B, 2003,68, 165346, doi:10.1103/PhysRevB.68.165346

work page doi:10.1103/physrevb.68.165346 2003
[22]

Lett., 1995,29, 321, doi:10.1209/0295-5075/29/4/009

Jain J.K., Kawamura T., Europhys. Lett., 1995,29, 321, doi:10.1209/0295-5075/29/4/009

work page doi:10.1209/0295-5075/29/4/009 1995
[23]

Dev G., Jain J.K., Phys. Rev. B, 1992,45, 1223, doi:10.1103/PhysRevB.45.1223

work page doi:10.1103/physrevb.45.1223 1992
[24]

Jeon G.S., Chang C.C., Jain J.K., Eur. Phys. J. B, 2007,55, 271–282, doi:10.1140/epjb/e2007-00060-4

work page doi:10.1140/epjb/e2007-00060-4 2007
[25]

Jain J.K., Kamilla R.K., Phys. Rev. B, 1997,55, R4895, doi:10.1103/PhysRevB.55.R4895

work page doi:10.1103/physrevb.55.r4895 1997
[26]

Ciftja O., J. Math. Phys., 2011,52, 122105, doi:10.1063/1.3672196

work page doi:10.1063/1.3672196 2011
[27]

АналIтичнI результати збуджених електронних стан Iв при υ = 1/3 та методи Лафл Iна-Джейна мIкроскопIчної хвильової функцIї -.A

Wolfram Research, Inc., Mathematica, Version 4.0, Champaign, Illinois, 1999. АналIтичнI результати збуджених електронних стан Iв при υ = 1/3 та методи Лафл Iна-Джейна мIкроскопIчної хвильової функцIї -.A. Aммар ЛабораторIя фIзики експериментальних метод Iв I їх застосувань,УнIверситет Медеї,Алжир У данIй роботI дослIджено властивостI двомIрного електронно...

work page 1999

[1] [1]

)nTRoduCTIon A discovery of the integer quantum Hall eﬀect (IQHE) [1, 2] was the beginning of a big revolution in the ﬁeld of condensed matter. It is interesting to study theoretically and numerically the phases of the integer and fractional quantum Hall eﬀect (FQHE) in a two-dimensional geometry in order to consider various aspects of this problem. In th...

work page internal anchor Pith review Pith/arXiv arXiv 1983

[2] [2]

Thus, any state in the lowest Landau level (LLL) is given by a polynomial equation dependent only onz

(1.5) The wave functions in the lowest Landau level are simple monomials inz. Thus, any state in the lowest Landau level (LLL) is given by a polynomial equation dependent only onz. The probability of presence of the state|0, m⟩ is maximal over a circle of radiusrm = √ 2ml0, such that the radial extension ofthewavefunctionisoftheorderof l0.When mincreases,...

work page

[3] [3]

We considerNe = 10 electrons of charge (−e0) embedded in a uniform neutralizing background disk of an area and a positive charge Ne e0

-odEL oF InTERACTIon The many-electron system is described by the Hamiltonian ˆH = ˆK + ˆV, (2.1) where ˆK is the kinetic energy operator,ℏ is reduced Planck’s constant,ω = e0B/m is the cyclotron frequency, and the Coulomb interactionˆV projected in the LLL is obtained starting from the electron- electroninteraction,electron-backgroundandthebackground-bac...

work page

[4] [4]

AMMAR Similarly, for theˆVeb interaction, we have ⟨ˆVeb ⟩ =−ρNe ¾ d2r1

(2.11) 23701-3 -.A. AMMAR Similarly, for theˆVeb interaction, we have ⟨ˆVeb ⟩ =−ρNe ¾ d2r1... d2rNe Ψr1,..., rNe 2 ¾ SN d2r e2 0 |r1− r|, (2.12) with ¾ SN d2r e2 0 |r1− r|= 2πRN ∞¾ 0 dq q J1(q)J0 ( q RN r1 ) , (2.13) where is theJn(x) n-th order Bessel functions. A detailed description of the ground state obtained by an analytical method is given in our e...

work page

[5] [5]

AnALyTICAL RESuLTS FoR )1(E ATυ = 1 Inthissection,weobtaintheanalyticalexpressionsforthetotalenergyperparticle(inunits e2 0/l0)and related quantities corresponding to IQHE system of electrons in a disk geometry atυ = 1. The ground state interaction energy per particle can be written as follows: ε =εee +εeb +εbb, (3.1) whereε = ⟨ˆV ⟩ /Ne, εee = ⟨ˆVee ⟩ /Ne...

work page

[6] [6]

quasielectrons

4HE QuASIELECTRon (qe ) EnERgIES Nowadays,therearetwouniversallyacceptedtheoriesintheﬁeldofFQHE,thetheoryofLaughlin[6] and the theory of Jain [8]. An early trial wave function proposed by Laughlin for the ground state at the ﬁlling factorυ = 1/m, m odd, turned out to work well [13]. The CF theory applies to a broader range of phenomena, while also providi...

work page

[7] [7]

The code of the electron-background interaction energy computationεeb is shown in appendix B

2ESuLTS And dISCuSSIon Withinthiswork,weconsideredtwosystems.Theﬁrstoneforthegroundenergywithupto Ne = 10 electrons at the ﬁllingυ = 1 and the second one for the exited energy with up toNe = 7 electrons at the 23701-6 ,AUgHLIN-JAIN MICROsCOPIC WAVE FUNCTION APPROACHEs ﬁllingυ = 1/3.Severalresearchersusedthediskgeometryintheirworks[3,5,6,11].Themathematica...

work page

[8] [8]

DIRECTORy NAME

ConCLuSIon We conclude that the analytical expression for the Coulomb interaction energy in disk geometry at ﬁllingfactor1/3permitsustoobtainacorrespondencebetweenthefractionalquantumHalleﬀectandthe integer quantum Hall eﬀect. The coinciding energies of the IQHE and the(Ne− 1) quasielectron of the FQHEstatesareexpected,wheretheFQHEandtheIQHEcanbeuniﬁed.Th...

work page

[9] [9]

Klitzing K.V., Dorda G., Pepper M., Phys. Rev. Lett., 1980,45, 494, doi:10.1103/PhysRevLett.45.494

work page doi:10.1103/physrevlett.45.494 1980

[10] [10]

Tsui D.C., Stormer H.L., Gossard A.C., Phys. Rev. Lett., 1982,48, 1559, doi:10.1103/PhysRevLett.48.1559

work page doi:10.1103/physrevlett.48.1559 1982

[11] [11]

Morf R., Halperin B.I., Phys. Rev. B, 1986,33, 2221, doi:10.1103/PhysRevB.33.2221

work page doi:10.1103/physrevb.33.2221 1986

[12] [12]

Fano G., Ortolani F., Phys. Rev. B, 1987,37, 8179, doi:10.1103/PhysRevB.37.8179

work page doi:10.1103/physrevb.37.8179 1987

[13] [13]

Ciftja O., Wexler C., Phys. Rev. B, 2003,67, 075304, doi:10.1103/PhysRevB.67.075304

work page doi:10.1103/physrevb.67.075304 2003

[14] [14]

Laughlin R.B., Phys. Rev. Lett., 1983,50, 1395, doi:10.1103/PhysRevLett.50.1395

work page doi:10.1103/physrevlett.50.1395 1983

[15] [15]

Jain J.K., Composite Fermions, Cambridge University Press, Cambridge, 2007

work page 2007

[16] [16]

Jain J.K., Phys. Rev. Lett., 1989,63, 199, doi:10.1103/PhysRevLett.63.199

work page doi:10.1103/physrevlett.63.199 1989

[17] [17]

MorfR.H.,d’AmbrumenilN.,DasSarmaS.,Phys.Rev.B,2002, 66,075408,doi:10.1103/PhysRevB.66.075408

work page doi:10.1103/physrevb.66.075408 2002

[18] [18]

Ciftja O., Physica B, 2009,404, 227, doi:10.1016/j.physb.2008.10.036

work page doi:10.1016/j.physb.2008.10.036 2009

[19] [19]

Matter Phys., 2016,19, 33702: 1–9, doi:10.5488/CMP.19.33702

Ammar M.A., Bentalha Z., Bekhechi S., Condens. Matter Phys., 2016,19, 33702: 1–9, doi:10.5488/CMP.19.33702. 23701-9 -.A. AMMAR

work page doi:10.5488/cmp.19.33702 2016

[20] [20]

Ahmed Ammar M., Méthodes Analytique et Numérique dans l’Eﬀet Hall Quantique Fractionnaire et Rôle du Spin des Électrons, PhD thesis, Abou-Bekr Belka¯ıd University, Tlemcen, Algeria, 2018

work page 2018

[21] [21]

Jeon G.S., Jain J.K., Phys. Rev. B, 2003,68, 165346, doi:10.1103/PhysRevB.68.165346

work page doi:10.1103/physrevb.68.165346 2003

[22] [22]

Lett., 1995,29, 321, doi:10.1209/0295-5075/29/4/009

Jain J.K., Kawamura T., Europhys. Lett., 1995,29, 321, doi:10.1209/0295-5075/29/4/009

work page doi:10.1209/0295-5075/29/4/009 1995

[23] [23]

Dev G., Jain J.K., Phys. Rev. B, 1992,45, 1223, doi:10.1103/PhysRevB.45.1223

work page doi:10.1103/physrevb.45.1223 1992

[24] [24]

Jeon G.S., Chang C.C., Jain J.K., Eur. Phys. J. B, 2007,55, 271–282, doi:10.1140/epjb/e2007-00060-4

work page doi:10.1140/epjb/e2007-00060-4 2007

[25] [25]

Jain J.K., Kamilla R.K., Phys. Rev. B, 1997,55, R4895, doi:10.1103/PhysRevB.55.R4895

work page doi:10.1103/physrevb.55.r4895 1997

[26] [26]

Ciftja O., J. Math. Phys., 2011,52, 122105, doi:10.1063/1.3672196

work page doi:10.1063/1.3672196 2011

[27] [27]

АналIтичнI результати збуджених електронних стан Iв при υ = 1/3 та методи Лафл Iна-Джейна мIкроскопIчної хвильової функцIї -.A

Wolfram Research, Inc., Mathematica, Version 4.0, Champaign, Illinois, 1999. АналIтичнI результати збуджених електронних стан Iв при υ = 1/3 та методи Лафл Iна-Джейна мIкроскопIчної хвильової функцIї -.A. Aммар ЛабораторIя фIзики експериментальних метод Iв I їх застосувань,УнIверситет Медеї,Алжир У данIй роботI дослIджено властивостI двомIрного електронно...

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