We present an ultra-high-precision numerical study of the spectrum of
multifractal exponents $\Delta_q$ characterizing anomalous scaling of wave
function moments $$ at the quantum Hall transition. The result
reads $\Delta_q = 2q(1-q)[b_0 + b_1(q-1/2)^2 + ...]$, with $b_0 = 0.1291\pm
0.0002$ and $b_1 = 0.0029\pm 0.0003$. The central finding is that the spectrum
is not exactly parabolic, $b_1\ne 0$. This rules out a class of theories of
Wess-Zumino-Witten type proposed recently as possible conformal field theories
of the quantum Hall critical point.Comment: 4 pages, 4 figure

A. D. Mirlin

A. Mildenberger

F. Evers

H. Obuse

R. B. Lehoucq

English

arXiv

We present an ultrahigh-precision numerical study of the spectrum of multifractal exponents Δq characterizing anomalous scaling of wave function moments ⟨|ψ|2q⟩ at the quantum Hall transition. The result reads Δq=2q(1−q)[b0+b1(q−1/2)2+⋯], with b0=0.1291±0.0002 and b1=0.0029±0.0003. The central finding is that the spectrum is not exactly parabolic: b1≠0. This rules out a class of theories of the Wess-Zumino-Witten type proposed recently as possible conformal field theories of the quantum Hall critical point

Evers, Ferdinand

Mildenberger, A.

Mirlin, A. D.

University of Regensburg Publication Server

Multifractality at the Quantum Hall Transition: Beyond the Parabolic ParadigmF. Evers,1,2 A. Mildenberger,3 and A.D. Mirlin1,2,*1Institut für Nanotechnologie, Forschungszentrum Karlsruhe, D-76021 Karlsruhe, Germany2Institut für Theorie der Kondensierten Materie, Universität Karlsruhe, D-76128 Karlsruhe, Germany3Fakultät für Physik, Universität Karlsruhe, D-76128 Karlsruhe, Germany(Received 28 April 2008; published 8 September 2008)We present an ultrahigh-precision numerical study of the spectrum of multifractal exponents qcharacterizing anomalous scaling of wave function moments hj j2qi at the quantum Hall transition. Theresult reads q ¼ 2qð1 qÞ½b0 þ b1ðq 1=2Þ2 þ   , with b0 ¼ 0:1291 0:0002 and b1 ¼ 0:00290:0003. The central finding is that the spectrum is not exactly parabolic: b1  0. This rules out a class oftheories of the Wess-Zumino-Witten type proposed recently as possible conformal field theories of thequantum Hall critical point.DOI: 10.1103/PhysRevLett.101.116803 PACS numbers: 73.43.f, 05.45.Df, 71.30.+h, 72.15.RnIntroduction.—The quantum Hall effect is a famousmacroscopic quantum phenomenon [1,2] whose discoverygave rise to one of the most active research areas incondensed matter physics of past three decades. The pla-teaus with quantized values of the Hall conductivity areseparated by quantum Hall transitions, which represent acelebrated example of a quantum critical point in a disor-dered electronic system (for a recent review, see Ref. [3]).Identification of the critical field theory of the integerquantum Hall transition remains a major unsolved problemof condensed matter physics.One of the key characteristics of the quantum Halltransition point is the multifractality spectrum governingfluctuations of amplitudes of critical wave functions.Specifically, the moments of wave functions scale withsystem size L, with a set of anomalous exponents q,hj ðrÞj2qi=hj ðrÞj2iq  Lq : (1)(The angular brackets denote the ensemble averaging.)Equivalently, one often characterizes multifractality by aclosely related set of exponents q  dðq 1Þ þ q or byits Legendre transform fðÞ (‘‘singularity spectrum’’) de-fined via q ¼ 0q, fðqÞ ¼ qq  q. Here d is the sys-tem dimensionality (while d ¼ 2 for the present case of thequantum Hall transition, we find it useful to keep it as d informulas below), and the prime denotes the q derivative.Zirnbauer [4] and Tsvelik [5,6] conjectured that theconformal theory of the quantum Hall critical point is ofthe Wess-Zumino-Witten type. These proposals imply [7]that the spectrum q is parabolic, i.e., that q definedaccording toq ¼ qqð1 qÞ (2)is in fact q-independent [8]: q ¼ . An accurate numeri-cal analysis of the multifractal spectrum plays, therefore, acrucial role for identification of the critical theory.A high-accuracy evaluation of the multifractality spec-trum was carried out in our earlier work [9]. For thispurpose, we modeled systems of a much larger size thanin preceding works and performed averaging over a largeensemble of wave functions, as well as a thorough analysisof finite-size effects. It was found that the spectrum isclose-to-parabolic [q ’ qð1 qÞ, with  ¼ 0:2620:003], thus showing that, if deviations from parabolicityare present, they are rather small (of the order of 1%).While the data of Ref. [9] were showing some indicationsfor such deviations, they were smaller than the numericaluncertainties. The latter originate from statistical noise(limited size of the data set) and from finite-size effectsaffecting the scaling relation (1) which is used to extractq.The goal of the present Letter is to determine the qspectrum with an ultrahigh precision and to give an ulti-mate answer on the question, ‘‘Is the multifractality spec-trum of the quantum Hall transition strictly parabolic?’’For this purpose, we improve upon the earlier numericalanalysis in two different ways. First, we utilize a statisticalensemble that contains approximately 10 times moresamples than the one used before [9]. Second, we employa recently discovered [10] ‘‘reciprocity relation’’q ¼ 1q (3)for a better control of finite-size corrections.Relation (3) implies a symmetry of the q spectrumaround the point q ¼ 1=2. Consequently, an expansion ofq about this point has a formq=d ¼ b0 þ b1ðq 1=2Þ2 þ b2ðq 1=2Þ4 . . . : (4)To verify or to exclude parabolicity of q, the prefactor b1of the quadratic term (and possibly those of higher-orderterms) in Eq. (4) should be determined numerically. This isthe purpose of the present work. Wewill provide numericalevidence that the corrections to parabolicity do not vanish.Specifically, we obtain b0 ¼ 0:1291 0:0002 and b1 ¼0:0029 0:0003, the nonzero b1 implying that the para-PRL 101, 116803 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending12 SEPTEMBER 20080031-9007=08=101(11)=116803(4) 116803-1  2008 The American Physical Societybolicity is not exact. The corresponding value of 0 [posi-tion of the apex of the singularity spectrum fðÞ] is 0 ¼dþ 0 ¼ 2:2596 0:0004.Method.—In order to find the critical eigenstates, weemploy the same numerical strategy that has been devel-oped before [9]. We determine the lattice time evolutionoperator U for the Chalker-Coddington network model[11,12] with periodic boundary conditions and N ¼ 2Ldnodes. For each realization of disorder, eight eigenstates ðrÞ with eigenvalues closest to unity are found with astandard sparse matrix package [13–15] from exact diag-onalization of U. (In the Chalker-Coddington model, thespatial coordinate r labels the links of the network.) Westudy systems with L ¼ 16; 32; 64; . . . ; 1024 with106 samples for the smallest sizes and 104 for thelargest ones. The statistical analysis proceeds via calculat-ing the average inverse participation ratiosPq ¼ Nhj 2jqi: (5)The symbol h. . .i in Eq. (5) indicates the combined averag-ing of the amplitude moments j ðrÞj2q over (i) all spatialpoints (links) r in the sample, (ii) the energy windowconsidered (here always eight eigenstates   per sample),and (iii) the statistical ensemble of samples with differentmicroscopic realizations of disorder. The average inverseparticipation ratios Pq thus obtained obey the scaling lawPq ¼ cqðNÞNðq1Þq=d: (6)The coefficients cq become independent of N in the limitN ! 1.As an alternative approach, we consider the ratioLq ¼ hj 2jq lnj j2ihj 2jqi  ðlnPqÞ0 ¼ qdlnN þ ðlncqÞ0;(7)whose scaling yields the exponent q ¼ 0q þ d. In thisway, the exponent q is studied directly, i.e., withoutinvoking a numerical differentiation which can signifi-cantly increase the error bars. In analogy with Eq. (4),we can expand 0q around q ¼ 1=2:0q ¼ ð1 2qÞ~q;~qd¼ a0 þ a1q 122 þ    :(8)The coefficients of both expansions are related via a0 ¼b0  b1=4, a1 ¼ 2b1  b2=2, . . ..The averages entering Eqs. (5) and (7) are readily ob-tained numerically. It is beneficial to perform the scalinganalysis of the ratio (7) in addition to that of Eq. (5) forseveral reasons. First, the curvature of q is more clearlyseen in the q derivative 0q. Second, the finite-size correc-tions are different in the cases of Eqs. (5) and (7), so that anagreement between the obtained exponents provides anadditional confirmation of the validity of the N ! 1 ex-trapolation procedure. Also, the relation a1 ¼ 2b1  b2=2allows one to extract the coefficient b2 of the quartic termin Eq. (4) out of parabolic fits for q and ~q.Numerical results.—We begin the analysis of our nu-merical results by verifying the reciprocity relation (3). Tothis end, we consider the ratiorq ¼ N2q1PqP1q¼ Nð1qqÞ=d cqðNÞc1qðNÞ : (9)The reciprocity relation (3) implies that the leading powersshould cancel, so that rq exhibits only subleading correc-tions in 1=N. The log-linear plot (Fig. 1, upper row) showsthat rq saturates in the large N limit with a very welldefined asymptotic value. Thus, we confirm reciprocityfor the exponent spectrum of the integer quantum Halleffect, as expected [10]. Since the exponent relation musthold only in the asymptotic regime, we can draw anotherconclusion which is important for the subsequent analysis:The observed saturation of rq provides evidence that ournumerically accessible sample sizes are indeed largeenough in order to be able to study the true asymptotics.Similar to rq, we consider the logarithmic derivativesq  ðlnrqÞ0 ¼2d q  1qdlnN þ ðlncqc1qÞ0;(10)which also saturates well inside the numerical window; seeFig. 1, lower row. Thus, the true asymptotics of q may bestudied by means of Eq. (7) with available system sizes,too.Having gone through important prerequisites, we nowturn to the analysis of the main data. In order to determine a1.00811.00821.00831.0084r q(Nsites)q=0.6 1.10.7160.7180.720.7220.7241.5103104105106Nsites0.0370.03750.0380.0385s q(Nsites)103104105106Nsites103104105106Nsites-0.72-0.715-0.71-0.705-0.7-0.6950.9756-0.1580.9764-0.1520.9758-0.160FIG. 1 (color online). Upper row: Ratio rq of inverse partici-pation numbers Pq and P1q [Eq. (9)] at q ¼ 0:6, 1, and 1.5(from left to right). The flat asymptotics indicates the validity ofthe reciprocity relation q ¼ 1q. Small deviations from theconstant behavior at the largest system sizes are due to residualstatistical noise. Lower row: Analogous plots for the logarithmicderivative sq ¼ ðlnrqÞ0 defined in Eq. (10).PRL 101, 116803 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending12 SEPTEMBER 2008116803-2set of relatively small exponents q to an accuracy con-siderably better than 1%, we have developed the followingprocedure. In each panel of Fig. 2, we plot for fixed q afamily of curves labeled by a parameter :FqðNÞ ¼ PqNq1þð1qÞq: (11)For the particular family member for which FqðNÞ be-comes independent of N in the limit of large N, we canconclude that  ¼ q=d. From such a procedure, we ex-tract the function q without having to resort to any (multi-parameter) fitting procedure. Similarly, by studying yetanother family of curves~F qðNÞ ¼ Lq þ lnN½1þ ð1 2qÞ~; (12)we have direct access also to the function ~q [see Eq. (8)]without the need for numerical differentiation.The functions q and ~q representing the main result ofthis Letter are displayed in Fig. 3. Also shown is q=qð1qÞ as derived from the earlier evaluation of the exponents[9] (the size of corresponding error bars is indicated bydotted lines). Figure 3 clearly shows that the curvature inq, which apparently has already left its trace in the earlierdata, now fully reveals itself thanks to the reduced errorbars. Even more pronounced is the resulting structure in thederivative ~q. It is reassuring to notice that the new dataconfirm our previous finding for b0 but provide a muchbetter accuracy: b0 ¼ 0:1291 0:0002. Our new result forthe curvature of q is b1 ¼ 0:0029 0:0003, which isclearly nonvanishing. These results are in full agreementwith those obtained by a fit to ~q (see the caption to Fig. 3).We thus conclude that, although the curvature of q isnumerically rather small (b1 is approximately 50 timessmaller than b0), it is nonzero: The multifractality spec-trum q of the quantum Hall transition is not parabolic.Surface exponents.—So far, a network model with torusgeometry (i.e., without boundary) has been considered.Recently, it has been shown [16] that wave function fluc-tuations near surfaces exhibit their own multifractal spec-trum with exponents sq, defined in full analogy to Eq. (1)viahj j2qis=hj j2iqs  Lsq ; (13)with the exception that here the average h. . .is is performedover the vicinity of the boundary only. In general, thesurface exponents sq are not related to their bulk counter-parts in any simple manner.To study the boundary exponents, we consider theChalker-Coddington network of cylinder geometry, i.e.,periodic in one direction and with hard-wall (full reflec-tion) boundary conditions in the other direction. The aver-aging h. . .is is performed over those network links that arelocated directly at the boundary (and includes the ensembleand the energy window averaging, as before). We parame-trize the surface spectrum in analogy with the bulk case[Eqs. (2), (4), and (8)] labeling the corresponding parame-ters by a superscript ‘‘s.’’ The results for sq and ~sq are1.0041.0081.012Fq(Nsites)q=0.6, 0.4 (top,bottom) q=1.1, -0.10.870.880.89q=1.5, -0.5103104105106NSites0.9961Fq(Nsites)103104105106NSites103104105106NSites1.211.221.230.9890.9861.0131.010δ=0.130δ=0.1280δ=0.1292 δ=0.1312δ=0.1312δ=0.1332FIG. 2 (color online). Family of curves FqðNÞ ¼PqNq1ðq1Þq for q ¼ 0:6, 1.1, and 1.5 (top: left, center, andright) and q ¼ 0:4, 0:1, and 0:5 (bottom: left, center, andright). Each data set is labeled by a parameter , which increasesfrom a minimum to a maximum value (given in the upper plots)in steps of 0.0004. The value of  for which FqðNÞ is flatdetermines the anomalous exponent q ¼ dqð1 qÞ. Suchsaturating data sets are marked with solid symbols (left: 4;center:; right:h); typical error in the corresponding value of does not exceed 0.001. The change in symbols (i.e., in ) forsaturating data sets illustrates a q dependence of q and thusgives direct, unprocessed evidence of nonvanishing quartic termsin q.-0.5 0 0.5 1 1.5q0.12750.130.13250.1350.13750.14γ q/d-0.5 0 0.5 1 1.5 2q0.12750.130.13250.1350.13750.14γ q/d~FIG. 3 (color online). The exponents q ¼ q=qð1 qÞ ()and ~q ¼ 0q=ð1 2qÞ () obtained from Fig. 2 and analogousanalysis for other values of q. The curvature in q and ~q impliesthat the multifractal spectrum q is not parabolic. Also shownare results of the earlier work [9] (solid line). Dotted horizontallines indicate earlier error bars in the regime 0 	 q 	 1. Dashedlines represent parabolic fits with b0 ¼ 0:1291 0:0002, b1 ¼0:0029 0:0003 (left) and a0 ¼ 0:1282 0:0001, a1 ¼0:0063 0:0005 (right). Combination of these data yields arough estimate of the quartic term b2 ¼ 0:001 0:001.PRL 101, 116803 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending12 SEPTEMBER 2008116803-3shown in Fig. 4. It is seen that the nonparabolicity of themultifractality spectrum (difference of sq and ~sq from aconstant) is present at the boundary as well and is, in fact,considerably more pronounced than in the bulk. The ratioRq ¼ sq=q has a clear q dependence, with a minimum atthe symmetry point q ¼ 1=2, where it takes the valueR1=2 ¼ 1:434 0:005. It is also worth noticing that Rq isappreciably smaller than 2, a value naturally expected forcritical theories expressed in terms of a free bosonic field.Conclusions.—In summary, we have studied numeri-cally the wave function statistics at the quantum Hallcritical point. We have verified that the reciprocity rela-tion (3) holds and have used it to control systematic errorsrelated to the finite-size effects. In combination with a verylarge size of the statistical ensemble, this has allowed us toreach unprecedented accuracy in determination of themultifractality spectrum q, with the error bars reducedby almost an order of magnitude compared to the earlierwork. The result shown in Fig. 3 reads q ¼ 2qð1 qÞ½b0 þ b1ðq 1=2Þ2 þ b2ðq 1=2Þ4 þ   , with b0 ¼0:1291 0:0002, b1 ¼ 0:0029 0:0003, and b2 ¼0:001 0:001. The obtained spectrum shows clear non-parabolicity [b1  0], thus excluding a broad class oftheories of the Wess-Zumino-Witten type as candidatesin the conformal field theory of the quantum Hall transi-tion. These results are corroborated by the analysis of thesurface multifractality.We thank A. Furusaki, I. A. Gruzberg, A.W.W. Ludwig,H. Obuse, and A. R. Subramaniam for useful discussionsand for sharing their data prior to publication. We are alsograteful to A. Tsvelik and M.R. Zirnbauer for instructivediscussions. This work was supported by the Center forFunctional Nanostructures of the DFG.Note added.—Recently, we learned about an indepen-dent study by Obuse et al. [17], who came to the sameconclusions.*Also at Petersburg Nuclear Physics Institute, 188300St. Petersburg, Russia.[1] K. von Klitzing et al., Phys. Rev. Lett. 45, 494 (1980).[2] The Quantum Hall Effect, edited by R. E. Prange and S.M.Girvin (Springer, New York, 1992).[3] F. Evers and A.D. Mirlin, arXiv:0707.4378 [Rev. Mod.Phys. (to be published)].[4] M. Zirnbauer, arXiv:hep-th/9905054v2.[5] M. J. Bhaseen et al., Nucl. Phys. B580, 688 (2000).[6] A.M. Tsvelik, Phys. Rev. B 75, 184201 (2007).[7] While rigorously proven for a number of other Wess-Zumino-Witten models, the statement of parabolicity ofq remains, strictly speaking, a plausible conjecture forthe model of Ref. [4]; see also M. Bershadsky, S. Zhukov,and A. Vaintrob, Nucl. Phys. B559, 205 (1999).[8] More accurately, the parabolicity may hold only for q <qc; its termination at q ¼ qc  ð2þ Þ=2 is related towave function normalization; see the review [3] for de-tails.[9] F. Evers, A. Mildenberger, and A.D. Mirlin, Phys. Rev. B64, 241303(R) (2001).[10] A. D. Mirlin, Y.V. Fyodorov, A. Mildenberger, and F.Evers, Phys. Rev. Lett. 97, 046803 (2006).[11] J. T. Chalker and P.D. Coddington, J. Phys. C 21, 2665(1988).[12] R. Klesse and M. Metzler, Int. J. Mod. Phys. C 10, 577(1999).[13] P. R. Amestoy et al., Comput. Methods Appl. Mech. Eng.184, 501 (2000); P. R. Amestoy et al., SIAM J. MatrixAnal. Appl. 23, 15 (2001).[14] J.W. Demmel et al., SIAM J. Matrix Anal. Appl. 20, 720(1999).[15] R. B. Lehoucq, D. Sorensen, and C. Yang, ARPACK UsersGuide (SIAM, Philadelphia, 1998).[16] A. R. Subramaniam, I. A. Gruzberg, A.W.W. Ludwig, F.Evers, A. Mildenberger, and A.D. Mirlin, Phys. Rev. Lett.96, 126802 (2006).[17] H. Obuse, A. R. Subramaniam, F. Furusaki, I. A. Gruzberg,and A.W.W. Ludwig, preceding Letter, Phys. Rev. Lett.101, 116802 (2008).-0.5 0 0.5 1 1.5q0.180.190.20.210.220.23γsq/d-0.5 0 0.5 1 1.5q0.180.190.20.210.220.23γ sq/d~FIG. 4 (color online). The surface exponents sq ¼ sq=qð1qÞ. Data are presented in the form analogous to the bulk plot(Fig. 3); the curvature is even more pronounced for the surfaceexponents. Dashed lines indicate parabolic fits with bs0 ¼0:1855 0:0005, bs1 ¼ 0:022 0:002 (left) and as0 ¼ 0:18050:001, as1 ¼ 0:048 0:003 (right). The resulting estimate for bs2is bs2 ¼ 0:008 0:01.PRL 101, 116803 (2008) P HY S I CA L R EV I EW LE T T E R Sweek ending12 SEPTEMBER 2008116803-4

Multifractality at the Quantum Hall Transition: Beyond the Parabolic Paradigm

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Multifractality at the quantum Hall transition: Beyond the parabolic
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Multifractality at the quantum Hall transition: Beyond the parabolic paradigm

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