Physics Letters A 378 (2014) 1566–1570

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Physics Letters A

www.elsevier.com/locate/pla

Coherent nonlinear quantum model for composite fermions

Gilbert Reinisch a, Vidar Gudmundsson a,∗, Andrei Manolescu b

a Science Institute, University of Iceland, Dunhaga 3, IS-107 Reykjavik, Icelandb School of Science and Engineering, Reykjavik University, Menntavegur 1, IS-101 Reykjavik, Iceland

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 March 2014Accepted 28 March 2014Available online 3 April 2014Communicated by V.M. Agranovich

Keywords:Two-dimensional electron systemCoulomb interactionQuasi-particles

Originally proposed by Read [1] and Jain [2], the so-called “composite-fermion” is a phenomenologicalquasi-particle resulting from the attachment of two local flux quanta, seen as nonlocal vortices, toelectrons situated on a two-dimensional (2D) surface embedded in a strong orthogonal magnetic field.In this Letter this phenomenon is described as a highly-nonlinear and coherent mean-field quantumprocess of the soliton type by use of a 2D stationary Schrödinger–Poisson differential model with onlytwo Coulomb-interacting electrons. At filling factor ν = 1

3 of the lowest Landau level the solution agreeswith both the exact two-electron antisymmetric Schrödinger wavefunction and with Laughlin’s Jastrow-type guess for the fractional quantum Hall effect, hence providing this latter with a tentative physicaljustification deduced from the experimental results and based on first principles.

© 2014 Elsevier B.V. All rights reserved.

Perhaps the most spectacular physical concept introduced inthe description of the fractional quantum Hall effect (FQHE) isthe composite fermion (CF). It consists in an intricate mixtureof Ne electrons and vortices in a two-dimensional (2D) electrongas orthogonal to a (strong) magnetic field such that the low-est Landau level (LLL) is only partially occupied. Actually, the CFconcept provides an intuitive phenomenological way of lookingat electron–electron correlations as a part of sophisticated many-particle quantum effects where charged electrons do avoid eachother by correlating their relative motion in the energetically mostadvantageous fashion conditioned by the magnetic field. Thereforeit is picturesquely assumed that each electron lies at the center ofa vortex whose trough represents the outward displacement of allfellow electrons and, hence, accounts for actual decrease of theirmutual repulsion [1,3]. Or, equivalently, in the simplest case ofNe = 2 electrons considered in the present Letter, that two fluxquanta Φ0 = hc/e are “attached” to each electron, turning the pairinto a LLL of two CFs with a 6Φ0 resulting flux [2]. The correspond-ing Aharonov–Bohm quantum phase shift equals 2π . In addition tothe π phase shift of core electrons, this is in agreement with therequirements of the Laughlin correlations expressed by the Jastrowpolynomial of degree 3 and corresponding to the LLL filling factorν = 1

3 [4,5]. Laughlin’s guessed wavefunction for odd polynomialdegree was soon regarded as a Bose condensate [6–8] whereasfor even degree, it was considered as a mathematical artefact de-scribing a Hall metal that consists of a well-defined Fermi surface

* Corresponding author.E-mail address: vidar@hi.is (V. Gudmundsson).

http://dx.doi.org/10.1016/j.physleta.2014.03.0500375-9601/© 2014 Elsevier B.V. All rights reserved.

at a vanishing magnetic field generated by a Chern–Simons gaugetransformation of the state at exactly ν = 1

2 [9,10].Although they provide a simple appealing single-particle il-

lustration of Laughlin correlations, the physical origin of the CF,auxiliary field, and magnetic fluxes remains unclear. In particular,the way they are fixed to particles is not explained. Hence tenta-tive theories avoiding the CF concept like e.g. a recent topologicalformulation of FQHE [11]. In the present Letter, we show how astrongly nonlinear mean-field quantum model, directly deducedfrom the experimental results, provides an alternative Hamiltonianphysical description, based on first principles, of the debated CFquasiparticle.

Consider the 2D electron pair confined in the x–y plane un-der the action of the orthogonal magnetic field B. The electronsare situated at z1,2 = x1,2 + iy1,2. We adopt the usual center-of-mass coordinate z = (z1 + z2)/2 and the internal separationz = (z1 − z2)/

√2 and we select odd-m angular momenta mh in

order to comply with the antisymmetry of the two-electron orbitalwavefunction under electron interchange. The spins are alignedwith the magnetic field, and thus in a symmetric state. The eigen-state corresponding to the internal motion is Ψm(z) = um(r)eimφ

where z = reiφ . We adopt the units of length and energy givenby the cyclotron radius λc = √

h/(Mωc) and by the Larmor energyhωL = 1

2 hωc = heB/(2Mc) where M denotes the effective mass ofthe electron which may incorporate many-body effects. The eigen-state um is given by [5]:

[∇2

X + Em + m − m2

X2− X2

4− K

X

]um = 0. (1)

G. Reinisch et al. / Physics Letters A 378 (2014) 1566–1570 1567

Fig. 1. (Upper panel) The “trajectory” C|1〉(0) vs [du|1〉/dX]0 for increasing K valueswithin the interval [0,15] as indicated by the arrows, corresponding to the initialconditions of the LSP problem (5)–(7). (Lower panel) The corresponding “trajectory”defined by the initial conditions of the NSP differential system (5) and (8). Thecircle indicates the ν = 1

3 FQHE solution defined by (4). In both plots, the K = 0free-electron case is defined by the upper left point du/dX |0 = 0 and C(0) = 2.

The radial part of the 2D Laplacian operator is ∇2X = d2/dX2 +

X−1(d/dX), the energy eigenvalue is Em and X = r/λc . The dimen-sionless parameter

K = √2

e2/(ελc)

hωc, (2)

where ε is the dielectric constant of the semiconductor host, com-pares the Coulomb interaction between the two particles withthe cyclotron energy. Obviously, K = 0 corresponds to the free-particle case. Actually, the internal motion could be approximatedby the 2D free-particle harmonic oscillator eigenstate |m〉 as longas K �

√2, i.e. B � 6 T in GaAs [5]. However, in the experimental

conditions specific to the FQHE the magnetic field is much higher.In Ref. [12] the energy gaps of FQHE states related to samples Aand B, at filling factors p/(2p ± 1), between ν = 1

4 and ν = 12 , are

shown to increase linearly with the deviation of B from the respec-tive characteristic values B A

12

= 9.25 T, B A14

= 18.50 T and B B12

= 19 T

(where the superscripts refer to the samples). The correspondingslopes respectively yield the direct measures M A = 0.63, M A =0.93 and MB = 0.92 of the effective electron mass in units of theelectron mass me . Indeed, since these masses scale like λ−1

c andhence like

√B , for they are determined by electron–electron in-

teraction, we have 0.63/√

9.25 = 0.207 ≈ 0.93/√

18.50 = 0.216 ≈0.92/

√19 = 0.211. Therefore, introducing the parameter κ that ac-

counts for the above experimental results, we have [12]:

M

me≈ κ

√B; 0.207 � κ � 0.216, (3)

where B is given in Tesla. Combining now Eqs. (2)–(3) we obtainthe FQHE experimental range of the interaction parameter K :

11.07 � K = 4π3/2Mc2

εΦ3/20

√B

= κ4π3/2mec2

εΦ3/20

� 11.56. (4)

Eq. (1) is linear and hence dispersive in its free-particle angular-momentum eigenspace. Its stationary solutions are expected tospread out over more and more Slater determinants made up ofsingle particle solutions when the perturbation defined by K �= 0grows. Our approach is however different. We interpret the massM as being the effective mass of electrons, without using the CF

concept. Our model uses particles of mass M and charge e whichinteract. In this sense the mass M incorporates the interaction onlypartially.

Our first step is to consider the m = 1, lowest-energy tripletstate, and to rewrite Eq. (1) under the form of the following equiv-alent differential system:[∇2

X + C|1〉 − 1

X2− X2

4

]u|1〉 = 0, (5)

∇2X C|1〉 = Kδ(X), (6)

where δ(X) is the Dirac function and the Laplacian ∇2X is 2D in (5),

but 3D in (6). The solution of Eq. (6) is

C|1〉(X) = μ|1〉 − K G(X), (7)

where the eigenvalue μ|1〉 stands for E |1〉 + 1 corresponding to theLarmor rotation at m = 1 and G(X) = 1/X is the 3D Green func-tion.

The second step is to replace the 3D with the 2D Laplacian inthe Poisson equation (6) such that it becomes electrostatically con-sistent with the Laplacian used in the Schrödinger equation (1).This means using G(X) = − log(X) in Eqs. (5) and (7). We willcall these equations the linear Schrödinger–Poisson problem (LSP).The LSP is a differential system of second order whose solution{u(X), C(X)} is defined by four parameters, conventionally called“initial conditions”, by analogy with a dynamical system. We seek astationary solution defined by u(0), du/dX |0, C(0), and dC/dX |0.(Here, for simplicity, we dropped the subscripts.) Two of these ini-tial conditions, namely, u(0) and dC/dX |0 vanish for any valueof K . The first one due to a complete depletion in the vortextrough, and the second one due to the absence of a cusp of theinteraction potential at the origin [13]. The phase space associ-ated to the initial conditions is thus reduced to the C(0)–du/dX |0plane. A numerical solution for the LSP (with the 2D Laplacian)is constructed with a shooting method. In Fig. 1(a) we illustratethe “trajectory” in the phase space of the initial conditions for thelowest-energy and most stable free-electron vortex state |1〉 withincreasing K . In practice, to avoid divergences, we consider theinitial condition of X = 4 × 10−4 and not at X = 0. The discontinu-ities seen in Fig. 1(a) occur due to the jumps of the ground statesolution to higher values of the orbital momentum. The groundstate corresponds to m = 1 for K = 0, but this may change inthe presence of the perturbation, the solution eventually spread-ing over more and more m values when the parameter K in-creases.

Now, in the third step, we are aiming at reformulating the Pois-son equation (6), with a 2D Laplacian, such that the source termbecomes the quantum-mechanical charge distribution of one elec-tron as seen by the other electron, represented by the mean-fieldsource term u2

|1) ,

∇2X C|1) = u2|1). (8)

We distinguish the new eigenstates, which are solutions of a non-linear nature imposed by Eq. (8), by using labels with parenthe-ses instead of kets. We will call Eqs. (5) and (8) the nonlinearSchrödinger–Poisson problem (NSP). The corresponding trajectoryof the initial conditions is shown in Fig. 1(b). The spectral co-herence of the new solution – i.e. the invariance of its angularmomentum with respect to the increase of K – is obvious: in-stead of discontinuously spreading out in the phase space like inFig. 1(a) the NSP solution starts spiraling down while keeping itsm = 1 initial value [13]. No phase transition towards higher angu-lar momenta occurs for 0 � K � 15. E.g. in Fig. 1(b) the trajectoryis continuous at [du|1)/dX]0 ≈ 1.75 and C|1)(0) ≈ 1.3. The exper-imentally relevant values of K , Eq. (4), are included in the smallcircle.

1568 G. Reinisch et al. / Physics Letters A 378 (2014) 1566–1570

Fig. 2. The solution of the NSP model is illustrated by its four basic properties. The experimental range (4) of the parameter K is displayed by the two vertical marks. (a) The“nonlinear” radius X defined by (12) (continuous line) compared with Y defined by (13) (broken line). (b) The “nonlinear” eigenvalue μ|1) defined by (5) and (8) (continuousline) compared with μ|1〉 defined by (5) and (7) (broken line). (c) The wavefunction profiles |1) (left: bold green) and |1〉 (right: bold blue) at K = N|1) = 11.23 (broken-linelines in Fig. 3), compared to the modulus of Laughlin’s ν = 1

3 normalized wavefunction (9) (thin red line). (d) The “nonlinear” magnetic flux Φ|1) given by (12) and (14) incontinuous green line, compared with Φ|1〉 given by (13) and (14) in broken red line. The horizontal broken blue line at Φ/Φ0 = 6 refers to (9) whose flux does not dependon K and yields 6Φ0 [14]. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)

This phenomenon resembles the well-known soliton coherencein hydrodynamics due to the cancellation of the dispersive effectsby nonlinearity. The mathematical tool that explains the stability ofthe resulting solitary wave is the so-called nonlinear spectral trans-form. It provides a theoretical link between linear spectral – andnonlinear dynamical and/or structural properties of the wave [15].This is what our nonlinear transformation from Eq. (6) to Eq. (8)is doing. It introduces an explicit ab-initio nonlinearity that cancelsthe angular-momentum dispersion displayed by Fig. 1. Incidentally,this transformation is usually done the opposite way in classicalsoliton physics: one starts with the “real” nonlinear wave equationand ends up with its formal “spectral transformed” linear counter-part [15].

In Fig. 2(a) it is shown that the NSP ground state |1) yields anaverage spatial extension equal to the value given by ground stateLSP solution for K ≈ 13, which is quite close to the experimen-tal range which is shown by the two vertical marks in Figs. 2(a),(b), (d). Most important for the aim of the present Letter, afterthe nonlinear transformation from (6) to (8) the solution of theNSP problem for electrons yields similar properties as expected forCF’s, about gap stability and flux quantization, as shown in Fig. 2(b)and (d), respectively. The spatial extension of the nonlinear eigen-state |1) also agrees with Laughlin’s ν = 1

3 two-electron normal-ized wavefunction whose modulus |Ψ3| is derived from Ref. [16]:

Ψ3 ∝ (z1 − z2)3 e− (|z1 |2+|z2 |2)

4λc ∝ X3e3iφe− X24 , (9)

up to the unimportant factor exp[ 12 (z/λc)

2] related to the exter-nal degree of freedom. In Fig. 2(c) we compare the FQHE solutionsdefined by (4), namely |1) (nonlinear: bold green) and |1〉 (linear:bold blue), with the normalized |Ψ3| (thin red). The NSP wavefunc-tion is scaled as Ψ|1) = u|1)

√MωL/(π hN|1)) where [13]

N|1) =∫

u2|1) X dX, (10)

in order to achieve∫ ∞

0 ‖Ψ|1)(x, y)‖2 dx dy = 1. The normalizationconstant (10) is the nonlinear order parameter of our NSP model[13]. As it grows, the amplitude and width of u|1) increases whileits corresponding normalized profile u|1)/

√N|1) spreads out: see

Fig. 3. Comparing with the right-hand side of (6) we obtain

N|1) = K . (11)

Eq. (11) operates the link between the original two-particleproblem based on Eq. (1), the LSP, and mean-field NSP prob-lems. In Ref. [12] the filling factor ν = 1

3 lies at the intersectionof two slopes concerning sample A. Taking their average, we ob-tain K = 11.23 which will be our reference value in the intervalgiven by Eq. (4). Note that this value largely exceeds K = √

2and hence it corresponds to a strong Coulomb interaction [5].Similarly, according to Eq. (11), N|1) = 11.23 yields a strong non-linearity: e.g. to be compared with 2.53 in quantum-dot helium[13]. This is quite spectacularly illustrated by Fig. 3 where theLSP solution |1〉 is compared with the NSP solution |1), for in-creasing K values. The profiles defined by ν = 1

3 FQHE valueK =N|1) = 11.23 are displayed in broken lines: obviously they arestrongly modified with respect to the free-particle ones (in dottedlines).

Let us now be more specific about some of the technical pointsused in order to obtain the above results.

1. In FQHE, the zeros of the Jastrow-type many-electron wave-function proposed by Laughlin for odd polynomial degree looklike 2D charges which repel each other by logarithmic interac-tion, yielding a negative value for the energy E per electron [16].Consequently we solved the Poisson equation (8) in 2D, whichindeed ensures that the corresponding Green function becomesG(X) = − log(X). Therefore we obtain the “fully 2D” self-consistentNSP differential system (5) and (8) whose eigensolution u|1)(X) isspecified by (4), (10) and (11).

2. The radius of the NSP state |1), displayed by the continuousline in Fig. 2(a),

G. Reinisch et al. / Physics Letters A 378 (2014) 1566–1570 1569

Fig. 3. Left: The broadening and flattening of the normalized linear profile−u|1〉(X)/

√N|1〉 corresponding to the LSP eigenstate |1〉 (in bold blue and multi-

plied by −1 for the sake of clarity), compared with the normalized NSP wavefunc-tion u|1)(X)/

√N|1) (in bold red) when N|1) = K increases by integer steps from 0

(free-electron solution: dotted profiles) to 15. The NSP solution u|1)(X) is displayedin green and increases with K . The CF profiles defined by N|1) = K = 11.23 are dis-played in broken lines. (For interpretation of the references to color in this figure,the reader is referred to the web version of this article.)

X =√⟨

z2⟩|1)

= 1√2

[⟨X2⟩

|1)+ 〈X〉2

|1)

] 12 , (12)

is obtained from z = 12 (z1 + z2) by quantum-averaging X and X2 in

the state |1) (hence the subscripts) since the two electrons locatedat z1 and z2 are both in the same state |1) [17]. On the otherhand, |z1 − z2| is the diameter of the two-electron orbit defined bythe linear internal degree of freedom of the linear state |1〉 whenassuming that the external degree of freedom z is frozen in itsground state. Therefore the radius of |1〉 is calculated as

Y = 1

2

⟨|z1 − z2|⟩|1〉 = 1√

2〈X〉|1〉 (13)

and shown by the broken line in Fig. 2(a), to be compared withEq. (12). Like already emphasized, these two radii nearly coincidefor the experimental values of the interaction parameter K , whichis also a measure of the nonlinearity of the NSP solution.

3. The nonlinear energy eigenvalue μ|1) , shown by the con-tinuous line in Fig. 2(b), is obtained from Eq. (8) using the 2DGreen function G(X) = − log(X). This can be done in two ways:either from the initial condition C|1)(0) = μ|1) − ∫

G(X)u2|1)(X)dX ,

or from the asymptotic limit C|1)(X → ∞) = μ|1) − N|1)G(X).The equivalence of these two approaches constitutes a test forour numerical code, the agreement being within a relative errorof 10−7.

4. The energy eigenvalue μ|1〉 corresponding to the LSP solu-tion u|1〉 can also be obtained from (7) at the two above lim-its by simply using the explicit 2D definition G(X) = − log(X).The results are displayed in Fig. 2(b) by the broken line. Thenegative energy difference Δ = μ|1) − μ|1〉 yields the stabil-ity of |1) when compared with |1〉. Though small, the en-ergy shift is clearly visible. For the experimental FQHE valueK = 11.23 we obtain: μ|1〉 = −11.0807hωL = −0.6977e2/ελcand μ|1) = −11.7164hωL = −0.7377e2/ελc (cf. (2)). ThereforeΔ = −0.04e2/ελc , to be compared with some experimental value

Δ ≈ −0.1e2/ελc [12] or Δ ≈ −0.045e2/ελc [14]. Moreover, thenonlinear eigenenergy per particle in our Ne = 2 system is E2 =12 μ|1) = −0.37e2/ελc . In the 20 � Ne � 144 interacting elec-tron case in the disk geometry with the filling factor ν = 1

3 ,E2 ≈ −0.39e2/ελc by extrapolation to Ne = 2 of the MonteCarlo evaluation of the ground-state energy per particle [18]. Onthe other hand, the energy per particle of −0.41e2/ελc is al-most insensitive to the system size for 4 � Ne � 6 [19]. There-fore our E2 = −0.37e2/ελc seems quite acceptable in this con-text.

5. Fig. 2(d) displays the fundamental property of the presentmodel, namely, its 6Φ0 flux quantization for the LLL filling factorν = 1

3 in the experimental range defined by Eq. (4). Indeed wehave, respectively, from (12) and (13):

Φ|1)

Φ0= π Bλ2

c

Φ0X2 = 1

2X2 = 6, (14)

Φ|1〉Φ0

= π Bλ2c

Φ0Y 2 = 1

4〈X〉2|1〉 = 6, (15)

between the two vertical marks. The (blue) horizontal broken lineat Φ/Φ0 = 6 refers to Laughlin’s ν = 1

3 normalized wavefunctionansatz (9) which does not depend on K . Its flux is indeed 6Φ0[14].

In conclusion, we have described a mean field Schrödinger–Poisson nonlinear model whose solution is in a good agreementwith the wavefunction postulated by Laughlin. Our results are de-rived from first principles, i.e. combining the Schrödinger andPoisson equations. In addition we have also used a key experi-mental result, Eq. (3), which indicates that in the FQHE the ef-fective electron mass is proportional to

√B . We emphasize the

self-consistency of our FQHE nonlinear model. The spectral coher-ence of the mean-field NSP mode |1) slightly lowers the energyper electron with respect to that obtained from the two-electronLSP mode |1〉. This fact makes the NSP mode |1) energetically morefavorable for the fulfillment of the flux quantization condition (14)than the LSP mode |1〉 for (15). This property might be consideredas the attachment of two flux quanta per electron and the subse-quent transformation of this later into a nonlinear soliton-like CFwhose 3-rd remaining flux quantum makes it behave as a merequasiparticle in LLL Integral Quantum Hall Effect.

Acknowledgements

G.R. gratefully acknowledges Lagrange lab’s hospitality at theObservatoire de la Cote d’Azur (Nice, France). The authors acknowl-edge financial support from the Icelandic Research and Instru-ments Funds, the Research Fund of the University of Iceland.

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