Integer quantum Hall effect of interacting electrons dynamical scaling and critical conduct

a r X i v :c o n d -m a t /9807054v 2 [c o n d -m a t .m e s -h a l l ] 6 J u n 1999

Integer quantum Hall e?ect of interacting electrons:dynamical scaling and critical

conductivity

Bodo Huckestein and Michael Backhaus

Institut f¨u r Theoretische Physik,Universit¨a t zu K¨o ln,D-50937K¨o ln,Germany

(February 1,2008)

We report on a study of interaction e?ects on the polarization of a disordered two-dimensional electron system in a strong magnetic ?eld.Treating the Coulomb interaction within the time-dependent Hartree-Fock approximation we ?nd numerical evidence for dynamical scaling with a dynamical critical exponent z =1at the integer quantum Hall plateau transition in the lowest Landau level.Within the numerical accuracy of our data the conductivity at the transition and the anomalous di?usion exponent are given by the values for non-interacting electrons,independent of the strength of the interaction.PACS:73.40.Hm,71.30.+h,72.15.Rn

While the occurrence of quantized plateaus in the Hall conductivity of two-dimensional systems in strong mag-netic ?elds is the most striking aspect of the quantum Hall e?ect (QHE)[1],it is the transitions between these plateaus that have recently attracted a lot of attention,both experimentally and theoretically [2].Our under-standing of the QHE is not complete without an under-standing of the plateau transitions.Furthermore,the plateau transitions are among the most extensively stud-ied examples of quantum phase transitions.The origin of the plateaus is the localization of the electrons due to dis-order,with the plateau transitions corresponding to the localization-delocalization transitions in the disorder po-tential [2].Theoretical studies have focused on the e?ect of disorder on non-interacting electrons in a strong mag-netic ?eld.From numerical and analytical calculations the following picture emerged for weak disorder [2]:in ev-ery Landau level exists a critical energy E c at which the localization length ξdiverges as a power law |E ?E c |?νwith a localization length exponent ν=2.35±0.03in-dependent of Landau level index and correlation length of the disorder potential [3].At all other energies elec-trons are exponentially localized.In the region of local-ized states the longitudinal conductivity vanishes and the Hall conductivity is quantized in multiples of e 2/h .At the critical energies the Hall conductivity jumps by e 2/h and the longitudinal conductivity takes on a ?nite value σc with σc ≈0.5e 2/h in the lowest Landau level [4,5].The eigenstates at the critical energies show multifractal ?uctuations leading to correlation functions being char-acterized by an anomalous di?usion exponent η≈0.4[4].Dynamical correlation functions show scaling behavior as a function of the variable qL ω,where q is the wave-vector and L ω=1/

√

to obtain the critical quantities z ,σc ,and ηfrom the scaling limit of the polarization.Let us ?rst sketch our approach.As the starting point for our numerical cal-culations

we use

the

self-consistent

HF

approximation

[11,12].The corresponding conserving approximation for the two-particle Green’s function is the time-dependent Hartree-Fock (TDHF)approximation [13].Within this approximation we evaluated the dynamical polarization Πirr (q,ω)including vertex corrections.The disorder-averaged polarization is extrapolated to in?nite system size and vanishing q and ω.In this limit,the polarization obeys the scaling form

Πirr

(q,ω)=

σ?q 2/e 2

2π3e 2lκ?1e q

2l 2

/4

I 0(q 2l 2/4)

(4)

and hence the HF interaction

V HF (q,q ′)=δq,?q ′(V (q )+V ex (q ))

(5)

become local (I 0is the Bessel function)[17,18].

Perhaps the most characteristic feature of the HF spec-trum is the formation of a Coulomb gap in the single-particle DOS [19].In particular,the DOS at the Fermi energy ρ(E F )decreases linearly with l/γL .This is shown in Fig.1[11,12].The suppression of the DOS is not re-lated to the phase transition but occurs for all values of the Fermi energy.Only when the Fermi energy coin-cides with the critical energy does it change the critical dynamics of the system.

The bare polarization Π0(q ,q ′,ω)is given by

Π0(q ,q ′,ω)=

α,β

(α|β)q (β|α)q ′

×

f α(1?f β)

ˉ

h ω+?β??α+iδ

,(6)

where (β|α)q = β|e ?i qr |α ,f αis the Fermi function (we take T =0in our calculations)and δis an in-?nitesimal parameter of the order of the mean level spacing.After averaging over disorder (we use between

FIG.2.Feynman (top)and the screened polarization (bottom)in time-dependent Hartree-Fock approximation.

13and 100realizations of the disorder)Π0becomes translationally and rotationally invariant Π0(q ,q ′,ω)=δ(q +q ′)Π0(q,ω).The real part of the static bare po-larization coincides in the limit q →0with the

single particle DOS,lim q →

0ReΠ0(q,ω=0)=χ00=ρ(E F )[15].The TDHF approximation for the irreducible and the screened polarizations are shown in Fig.2.In the lowest Landau level the ladder sum for the vertex corrections can be transformed into a RPA-like bubble sum with the Coulomb potential replaced by V ex (q )[20,15].Similarly,the screened polarization Πscr is related to the bare po-larization through a bubble sum containing the HF po-tential V HF (q )[15],

Πirr (q ,ω)=Π0(q ,ω) 1?V ex (q )Πirr (q ,ω)

,(7a)Πscr (q ,ω)=Π0(q ,ω)(1?V HF (q )Πscr (q ,ω)).

(7b)

In principle,the vertex corrections should be applied to Π0(q ,q ′)since it becomes diagonal only after disorder averaging.We checked for small system sizes that using Eq.(7)only introduces errors comparable to the statis-tical uncertainties for the relevant parameter range.Since the approximations leading to Πirr and Πscr are conserving and due to Eq.(7),the polarizations can be parameterized in the limit of in?nite system size and van-ishing q and ωthrough a susceptibility χq and a di?usion coe?cient D (q,ω),

ΠA

(q,ω)=

χA q D A q

2

σ?x ?ie 2/ˉh

,(10)

with x =q 2/χirr q ˉ

h ω.If dynamical scaling holds then σ?x is a function of q z /ω.In order to extract the scaling be-havior from our numerical data for ?nite system sizes,

0.05

0.10.15

0.200.51 1.5

χq i r r l 2

Γ

ql/γ

FIG.3.Static irreducible susceptibility for N Φ=900and γ=0.2,0.4and 0.6.The ?nite intercept at q =0scales like l/L to zero.

12

3

4

0.02

0.040.060.08

I m Πi r r (q ,ω)/

χi r r

q

2πωχirr q

FIG.4.Extrapolation of ImΠirr (q,ω)/χirr q as described in the text.x/2π=0.5,1,1.5,2,4,6(bottom to top).

we ?rst calculate the static susceptibility χirr q from the real part of the irreducible polarization.Due to the oc-currence of the linear Coulomb gap (cf.Fig.1),χirr q ∝q for small q in the thermodynamic limit,as is shown in Fig.3.

Next,the imaginary part of Πirr is divided by the nu-merically obtained χirr q .The resulting function is then simultaneously extrapolated to in?nite system size and q and ωto zero for ?xed value of x as is shown in Fig.4[4,15].This choice of scaling variable allows for reason-able extrapolations,while other choices like q 2/ρ0ˉh ωdo

not give satisfactory results.Since χirr

is asymptotically linear in q ,x ∝q/ωin the thermodynamic limit and the dynamical critical exponent z =1.

Figure 5shows the resulting conductivity σ?as a func-tion of x .In the limit x →0,σ?coincides with the DC Kubo conductivity σc .We ?nd that σc does not depend on the strength of the interaction up to γ=0.6and is

1

10

σ?x [

]e /h 2/ 2π

FIG.5.Conductivity σ?as a function of the scaling vari-able x =q 2/χirr q ˉ

h ωfor γ=0.2,0.4and 0.6(bottom to top).The curves are shifted vertically.

given by σc =0.5±e 2/h .In the opposite limit x →∞,the conductivity re?ects multifractal density ?uctuations and scales like x 1?η[15].A numerically more accurate determination of ηis possible from the scaling of the static polarization [15].We ?nd η=0.4±0.1indepen-dent of the interactions and in agreement with published data for non-interacting electrons [4,14].Comparing our result to [12]we see that the irreducible polarization re-?ects the multifractal exponent D 2of the HF eigenstates similar to the case of non-interacting electrons.

Let us make a few remarks on our results.Although the conductivity σc is ?nite at the transition,the van-ishing susceptibility χirr q implies non-di?usive behavior of the HF-particles,D (q,ω)∝q ?1for q →0.A cen-tral issue is the validity of the HF approach that has recently been questioned [21].The change in the dynam-ical critical exponent is due to the use of the unscreened Coulomb interaction.The dielectric function κis given by κ(q,ω)=1+V (q )Πirr (q,ω).From eq.(8)it follows that screening behaves di?erently in the limits of small and large x .In the limit of small x ,that is relevant to transport,screening is ine?cient and κ→1.The use of the unscreened HF approximation thus might be justi?ed for the calculation of the frequency-dependent conductiv-ity.On the other hand,in the limit of large x we observe static screening,κ(q,ω)=1+χirr q V (q ).For slow pro-cesses,during which the ground state has time to relax,the statically screened HF approximation is more appro-priate.It leads to a ?nite susceptibility and is described by z =2[15].A ?nal judgment on the validity of the HF approach to the critical quantum Hall system might only by possible if one succeeds in a renormalization group treatment of the neglected correlations [10,22].

In summary,we have presented results on the in?u-ence of Coulomb interactions on the critical properties of the integer quantum Hall system.Within the time-

dependent Hartree-Fock approximation we ?nd the criti-cal conductivity σc and the anomalous di?usion exponent

ηare unchanged by interactions while the dynamical crit-ical exponent z changes from 2to 1.The origin of this be-havior is the non-critical reduction of the single-particle density of states.

This work was performed within the research program of the SFB 341of the DFG.