We develop a bosonization scheme for the collective dynamics of a spinless
two-dimensional electron gas (2DEG) in the lowest Landau level. The system is
treated as a continuous elastic medium, and quantum commutation relations are
imposed between orthogonal components of the elastic displacement field. This
theory provides a unified description of bulk and edge excitations of
compressible and incompressible phases, and explains the results of recent
tunneling experiments at the edge of the 2DEG.Comment: 4 pages, includes 1 figur

Conti, S.

Vignale, G.

English

arXiv

http://arxiv.org/abs/cond-mat/9801318v1We develop a bosonization scheme for the collective dynamics of a spinless two- dimensional electron gas (2DEG) in the lowest Landau level. The system is treated as a continuous elastic medium, and quantum commutation relations are imposed between orthogonal components of the elastic displacement field. This theory provides a unified description of bulk and edge excitations of compressible and incompressible phases, and explains the results of recent tunneling experiments at the edge of the 2DEG.This work was supported by NSF grant No. DMR-
9706788

Conti, Sergio

Vignale, Giovanni, 1957-

University of Missouri: MOspace

arXiv:cond-mat/9801318v1  [cond-mat.mes-hall]  30 Jan 1998Bosonization of the two-dimensional electron gas in the lowest Landau levelS. ContiMax-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, GermanyG. VignaleDepartment of Physics, University of Missouri, Columbia, Missouri 65211(February 1, 2008)We develop a bosonization scheme for the collective dynamics of a spinless two-dimensional electrongas (2DEG) in the lowest Landau level. The system is treated as a continuous elastic medium, andquantum commutation relations are imposed between orthogonal components of the elastic displace-ment field. This theory provides a unified description of bulk and edge excitations of compressibleand incompressible phases, and explains the results of recent tunneling experiments at the edge ofthe 2DEG.73.40.Hm, 73.20.Mf, 71.10.PmBosonization [1] is a powerful mathematical techniqueto describe the long-wavelength dynamics of correlatedelectronic systems, in which the motion of the electronsis highly collectivized. This technique has proved veryuseful in studies of the behavior of interacting electronsin one-dimensional metals. Higher-dimensional general-izations have also been proposed [2]. More recently, it hasbeen realized that the edges of a two-dimensional electrongas (2DEG) in a strong magnetic field behave as a “chi-ral Luttinger liquid” [3] and are therefore amenable tobosonization [3,4]. A collective description of the uni-form incompressible 2DEG in a strong magnetic fieldsuch that all the electrons reside within the lowest Lan-dau level (LLL) [5] was pioneered by Girvin et al. [6],and an effective bosonic theory was developed shortly af-terwards [7]. The similarity between the local structureof the 2DEG in the LLL and that of a classical Wignercrystal was exploited by Johansson and Kinaret [8] toset up an “independent boson model” for the electronicspectral function; an equivalent model was derived byHaussmann [9] from diagrammatic theory. Finally, sev-eral authors have used bosonization schemes to describethe dynamics of a smooth compressible edge in a quan-tum Hall liquid [10–12], or of the electron gas in a weakmagnetic field [13].In this Letter we formulate a bosonic theory of the lin-ear dynamics of a general uniform or nonuniform distri-bution of electrons in the LLL. Unlike previous theories[7–13], the present one treats bulk and edge excitations ofcompressible and incompressible states on an equal foot-ing. One of our main results is a simple explanation forrecent experimental observations of one-electron tunnel-ing characteristics in the edge [14] of the 2DEG.The basic idea of our approach is to treat the electronsin the LLL as a continuous elastic medium characterizedby an equilibrium density ρ0(~r), and by a local displace-ment field ~u(~r, t), such that the time-dependent densityis (to first order in ~u) ρ(~r, t) = ρ0(~r)− ~∇· [ρ0(~r)~u(~r, t)] ≡ρ0(~r) + δρ(~r, t). We ignore the spin in this paper. Underthe assumption that both the equilibrium density andthe displacement are slowly varying on the scale of themagnetic length l = (h¯c/eB)1/2 (where B is the magneticfield) the effective long-wavelength Hamiltonian isH=12∫d~r∫d~r′δρ(~r)(e2|~r − ~r′|− χ˜−1(~r, ~r′))δρ(~r′)+∫d~rµ [ρ0(~r)]∑α,β[sαβ(~r)−12δαβ ~∇ · ~u(~r)]2, (1)where χ˜−1(~r, ~r′) is the inverse of the proper static density-density response function [15], µ is the shear modulus(discussed below), sαβ ≡ [∂uα(~r)/∂rβ+∂uβ(~r)/∂rα]/2 isthe strain tensor, and α, β are cartesian indices. Eq. (1)generalizes the Hamiltonian of classical elasticity theory[16] to take into account the long range of the Coulombinteraction, and of χ˜(~r, ~r′) in the incompressible states ofthe 2DEG [6].The second term of Eq. (1) is the crucial new featureof our theory. Ordinarily, the shear modulus of a liquid istaken to be zero, but this is only true for shear deforma-tions which occur slowly compared to the thermalizationtime τ of the system. At frequencies higher than 1/τ thecorrelated electron liquid behaves as a Wigner glass, andtherefore exhibits a nonvanishing shear modulus. The es-sential assumption is that, in this system, the relaxationfrequency scale 1/τ is much smaller than the character-istic frequency of the collective dynamics.The algebra of the displacement operators ux, uy, inthe LLL, is deduced from the canonical quantization con-dition for the hydrodynamical momentum and displace-ment fields [pα(~r), uβ(~r′)] = −iδ(~r−~r′)δαβ , by projectingout the higher Landau levels, i.e. averaging over the fastcyclotron motion. This leads to[uα(~r), uβ(~r′)] = −iǫαβδ(~r − ~r′) l2/ρ0(~r) , (2)where ǫαβ is the two-dimensional Levi-Civita tensor.This is consistent with the commutation relation between1projected density fluctuations [6] in the long wavelengthlimit, namely [ρ(~r), ρ(~r′)] = −i~∇r × ~∇r′ρ0(~r)l2δ(~r − ~r′).Bosonization of (1) is accomplished byuα(~r) =lρ1/20 (~r)∑n>0[bng∗nα(~r) + b†ngnα(~r)], (3)where bn and b†n are boson operators, satisfying theusual algebra [bn, b†n′ ] = δnn′ etc.. The functions gnα(~r)(g∗nα(~r)), with n > 0, are the positive (negative) fre-quency solutions of the eigenvalue problem∫Hαβ(~r, ~r′)gnβ(~r′)d~r′ = ωngnα(~r) (4)which is equivalent to the classical equation of motion.(We adopt the convention of summing over repeatedcartesian indices.) The operator Hαβ is given byHαβ(~r, ~r′) = −il2ρ1/20 (~r)ǫαγ[∂γ∂′βχ−1(~r, ~r′)]ρ1/20 (~r′)+il2ρ0(~r)ǫαγ[∂β∂′γ − ∂γ∂′β + δγβ∂η∂′η] [δ(~r − ~r′)µ], (5)where χ−1(~r, ~r′) ≡ χ˜−1(~r, ~r′)−e2/|~r− ~r′|, and ∂α and ∂′βdenote derivatives with respect to rα and r′β respectively.It is hermitian with respect to the scalar product (f, g) ≡∫f∗α(~r)iǫαβgβ(~r)d~r, and its eigenvalues are therefore real.For each eigenfunction gn with positive eigenvalue ωn,there is a complex conjugate one g∗n ≡ g−n with negativeeigenvalue −ωn. These eigenfunctions can be chosen tosatisfy orthonormality relations (gn, gm) = sgn(n)δnm (nandm can now have either sign) [17]. The “completenessrelation” has the form∑n>0[gnα(~r)g∗nβ(~r′)− g∗nα(~r)gnβ(~r′)] = iǫαβδ(~r − ~r′). (6)These equations guarantee that the commutation re-lation (2) is satisfied. Then, substitution of Eq. (3)into Eq. (1) yields the Hamiltonian in the desired formH =∑n>0(b†nbn + 1/2)ωn.Our next step is to construct the electron tunnelingoperator ψ(~R) associated with the LLL coherent stateorbital centered at ~R. This operator must satisfy thecommutation relation[ψ(~R), ρ(~r)] = ψ(~R) exp[−|~r − ~R|2/2l2]/2πl2 , (7)i.e., destroy a gaussian density near ~R. Eq. (7) alonedoes not uniquely determine ψ(~R). In order to do thiswe must also specify its commutator with the vorticityρv(~r) ≡ −ǫαβ∂α[ρ0(~r)uβ(~r)]. For the purpose of study-ing tunneling, we choose [ψ(~R), ρv(~r)] = 0, because theincoming electron is not expected to change the vorticityat the initial time. The solution has the formψ(~R) = exp[−∑n>0(M∗n(~R)ωnbn −Mn(~R)ωnb†n)](8)where the “electron-phonon” matrix elementsMn(~R) arewritten as Mn(~R) =∫exp[−|~r − ~R|2/2l2]M˜n(~r)d~r/2πl2,andM˜n(~r)ωn=i2π∫ǫαβgnα(~r′)ρ1/20 (~r′)l∂β log |~r − ~r′|d~r′ . (9)The calculation of the local Green’s function G(~R, t) =−i〈Tψ(~R, t)ψ†(~R, 0)〉 can be carried out by standardtechniques [18,8]. Here we simply report the zero tem-perature result for the integral equation connecting theelectronic spectral function A(~R, ω) (the Fourier trans-form of G(~R, t)) to the collective excitation spectrum:ωA(~R, ω) =∫ ω0g(~R,Ω)A(~R, ω − Ω)dΩ (10)where ω > 0 andg(~R,Ω) =1Ω∑n>0|Mn(~R)|2δ(Ω− ωn) (11)is closely related to the local dynamical structure factorof the liquid. The above equations constitute a completescheme for the calculation of single particle and collectiveproperties of a general distribution of electrons in theLLL. We now discuss some specific examples.(1) The uniform 2DEG. Let ρ0 be the uniform density.The normalized eigenfunctions of Eq. (4) have the formg~qL(~r) = iql(µ2ωqρ0)1/2ei~q·~r, (12)andg~qT (~r) =1ql(ωqρ02µ)1/2ei~q·~r, (13)whereωq = µ1/2(v(q) +K(q) + µρ20)1/2(ql)2 (14)is the frequency of the mode at wavevector ~q, v(q) isthe Fourier transform of the electron-electron interaction,K(q) = −ρ20χ˜−1(q) is the q-dependent bulk modulus, andthe labels L (longitudinal) and T (transverse) refer tothe components parallel and perpendicular to ~q. Theelectron-phonon matrix element of Eq. (9) has the formMq(~r) =[v(q) +K~r(q) + µρ20]e−q2l2/2ρq(~r) , (15)where ρq ≡ −iqlρ1/20 gqL(~r) and K~r(q) ≡ −ρ20∫χ˜−1(~r −~r′) exp[i~q · (~r′ − ~r)]d~r′. (The reason for the apparentlyunnecessary subscript ~r will become clear below).We now distinguish two cases: (i) Compressible case.limq→0 K(q) = K is finite. The values of K and µ can2be approximated by those of a classical Wigner crystal,namely, µ ∼ µc = 0.09775ρ0(e2/l)ν1/2 and K ∼ Kc =6µc [19]. The long wavelength modes have frequenciesωq ∝ q3/2 for Coulomb-like interaction (v(q) = 2πe2/q)and ωq ∝ q2 for short-range interaction (v(q) = e2d).We have now all the necessary input for calculating thespectral function from Eqs. (10,11). The low-frequencybehavior (where “zero” frequency is the chemical poten-tial) is found to be A(ω) ∝ ω−5/4e−γ/ω1/2for Coulombinteraction and A(ω) ∝ ω(7+(2.7ν)1/2d/l)1/2/4ν−1 for short-range interaction. Complete numerical results are shownin Fig. 1. They are in good agreement with previous cal-culations [8,9,20] as well as with experimental data [21].(ii) Incompressible case. Within the single-pole ap-proximation K(q → 0) ≃ ∆ρ0/2αq4, where ∆ is thecollective excitation gap at q = 0, and α = (1 − ν)/8νis derived from the Laughlin wavefunction at filling fac-tors ν ≡ 2πl2ρ0 = 1/odd integer [6]. Using Eq. (14) wefind at long wavelength ωq = (∆µ/2αρ0)1/2 independentof q. This formula can be used to deduce the value ofµ if ∆ = limq→0 ωq is known. Alternatively, one cansubstitute µ = µc from the classical Wigner crystal andobtain ∆ν = 0.391ν3/2(1− ν)−1e2/l, which gives ∆1/3 =0.11e2/l, ∆1/5 = 0.044e2/l and ∆1/7 = 0.025e2/l. Theseresults are in good agreement with variational estimates[6,22], with the exception of the first one which is al-most 30% lower than the variational one, but comparesfavorably with exact diagonalization studies [23].The calculation of the spectral function is more sub-tle. Straightforward application of the linear responseformulas (10-15) is incorrect, because the addition (or re-moval) of charge at point ~R creates a compressible regionin the middle of the liquid, changing the topology of theincompressible region from simply to doubly connected.The change in topology can be taken into account in thefollowing manner. We stipulate that the bulk moduluskernel −ρ20χ˜−1(~r, ~r′) has the form characteristic of an in-compressible liquid (namely, |~r − ~r′|2 log |~r − ~r′|) whenboth ~r and ~r′ are within the incompressible region, butit is given by the local form Kcδ(~r − ~r′) when either ~ror ~r′ are within the compressible “core” of the excita-tion. Because the size of the core region is microscopic,its presence does not affect significantly the frequenciesof the long wavelength modes. On the other hand, in the“electron-phonon” matrix element, given by Eq. (15), wemust use K~r(q) ≃ Kc, because ~r is inside the core re-gion. Within this scheme, the calculation of the spectralfunction can be straightforwardly carried out, withoutadjustable parameters. In Fig. 1, we plot the results forν = 1/3. We have used q-dependent K and µ in or-der to fit accurately the collective mode dispersion andstructure factor [6] at finite q. The essential difference be-tween this and the compressible case is the appearanceof a δ-function peak at ω = 0, which now correspondsto µ+ (recall that in the incompressible liquid the chem-FIG. 1. Spectral function for a compressible liquid atν = 0.3 with short range interactions (d = 10l, dashed curve),with long-range interactions (dotted curve) and for an incom-pressible liquid at ν = 1/3 (full curve, the δ-function peak atω = 0 contains around 2% of the spectral strength).ical potentials for addition or removal of charge, µ+ andµ−, are different [24]). The strength of the δ-functionis Z = exp[−∑q |Mq|2/ω2q ], which would have vanishedin the compressible case, and does not vanish here be-cause of the gap. The peak at µ+ reflects the ability ofthe incompressible liquid to accommodate the incomingelectron in the ground state as topological defects of theinitial incompressible state, without creating collectiveexcitations. The incoherent part of the spectral func-tion, at higher frequencies, corresponds to the creationof additional collective excitations [25].(2) Edge dynamics. The case of a smooth compressibleedge has been treated in previous publications [11]. Herewe focus on the case of a sharp edge, which is directlyrelevant to the interpretation of recent lateral tunnelingexperiments by Chang et al. and Grayson et al. [14]. Letus consider a straight edge along the y axis, and let eExˆbe the gradient of the confinement potential at the edge.The density ρ0 is assumed to be uniform for x < 0, andzero for x > 0. The presence of the edge electric fieldbreaks the rotational symmetry of the Hamiltonian (1),and must be taken into account with the additional termHedge = e∫ρ(~r)~u(~r) · ~E d~r . (16)With this term included into the eigenvalue problem (4),we obtain a new set of solutions, which satisfy the condi-tions ~∇·~g = 0 and ~∇×~g = 0 for x < 0. Neglecting the log-arithmic correction arising from the long-range Coulombinteraction, they obey the equation of motion− iωgα(~r) = vǫαβ∂βgx(~r) , (17)3where v = cE/B is the classical drift velocity. The or-thonormal solutions are (for x < 0)~gq(~r) = q1/2eqx−iqy(xˆ− iyˆ) , (18)where q > 0 is a one-dimensional vector along the edge,and the eigenvalues are ωq = vq. These are analogous togravity waves on the surface of a liquid. Because there isneither density change nor shear strain in the interior ofthe system, these solutions do not depend on the valuesof the bulk elastic constants. The use of elasticity theoryis justified at small q since the displacement field ~u isslowly varying.The effective edge dynamics can be derived from thefull dynamics by projecting the latter onto the subspacespanned by the edge-wave solutions of Eq. (18). Withinthis subspace, we can define an “edge density” operatorρedge(y) ≡∫δρ¯(x, y)dx = ρ0u¯x(0, y), where the bar de-notes projection onto the edge-wave subspace, i.e., forinstance, u¯x(0, y) ≡ (l/ρ1/20 )∑q>0 q1/2[bqeiqy + b†qe−iqy].It is easy to verify that the edge density satisfies the stan-dard Kac-Moody algebra [3] [ρedgeq , ρedge−q′ ] = (νq/2π)δqq′where ν is the bulk filling factor. Thus, we have deducedthe dynamics of the chiral Luttinger liquid at the edgefrom a projection of the canonical dynamics of displace-ment fields in the bulk. The edge tunneling operator,obtained by imposing [ψ(y), ρedge(y′)] = ψ(y)δ(y− y′) inthe edge-wave subspace, is still given by Eq. (8), withthe sum running over the edge modes, and the matrix el-ement Mq(y) = v(2πq/ν)1/2e−iqy. Use of Eqs. (10) and(11) then leads to the conclusion that the tunneling cur-rent must vanish, at low bias V , as V 1/ν . This result isin good agreement with the experimental findings. 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Bosonization of the two-dimensional electron gas in the lowest Landau level

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