Correlated composite approach to fractional quantum Hall effect via edge current
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Correlated composite approach to fractional quantum Hall effect via edge current
Jongbae Hong , , Soo-Jong Rey , Research Institute of Basic Sciences,Incheon National University,Yeonsu-gu, Incheon 22012 korea Asia Pacific Center for Theoretical Physics,Pohang, Gyeongbuk 37673 korea
Email: [email protected] Quantum Advanced Research Korea (Quantum ARK)
Institute,Seoul 06642 korea The Korea Academy of Science & Technology,Seongnam 13630 korea
Email: [email protected] (Dated: February 18, 2021)The fractional quantum Hall effect (FQHE) is extensively studied, but the explanation for Hallplateau widths and excitation energy gaps remains elusive. We study the effective theory of FQHEbuilt upon experimental inputs of Hall current distribution, edge dynamics, and many-body corre-lations. We argue that correlated composites of integer spin, comprising electrons and their images,localized at the edge of the incompressible strip are the basic transport entity. We show in the lowestLandau level that Zeeman interactions of these composites produce all odd denominator plateausand effective fractional charges. Utilizing field-dependent chemical potential and effective g-factor,we fully explain the observed Hall resistivity curve and excitation energy gaps of the half-fillingfamily. The plateau heights are systematically generated by multi-particle correlations, whereas theplateau widths and excitation energy gaps are determined by the correlation strengths. We explic-itly show that the Drude-like behavior at half-filling follows from equal strength of multi-particlecorrelations.
Two-dimensional electron system is a source of richphysics ( ), hallmarked by the fractional quantum Halleffect (FQHE) ( ); at low temperature and under strongmagnetic field, plateaus of fractionally quantized Hall re-sistivity are observed. The Laughlin wave function ( )provides an explanation to the effect and the compos-ite fermion model ( ) provides a picture to the observedplateau patterns, yet both fall short of quantitatively ex-plaining fundamental properties like plateau widths ofobserved Hall resistivity curve ( ) and excitation energygaps ( ).Resistivity is phenomenon of current flow. Therefore,information on where and how the Hall current flowsinside a Hall bar is essential for the understanding ofFQHE. Halperin ( ) first argued that Hall current flowsalong the boundary of Hall bar, due to Landau levelbending by confining potential within the Hall bar. How-ever, previous studies (
9, 10 ) of the edge current fell shortfor explaining the FQHE. Recently, Hall current flow dis-tribution was reported in fractional quantum Hall system(FQHS) around filling fraction ν = by measuring Hallpotential profile using the scanning probing microscope( ). The result reveals two important features that theHall current flows (i) macroscopically over about 5 µ mwidth, three orders of magnitude wider than the mag-netic length scale ℓ B = (¯ hc/eB ) , and (ii) symmetrically on the left and right sides of Hall bar.The first feature is explainable by introducing multi-ple stacks of incompressible strip (IS), where the Hallcurrent flows, distributed over the current flow region,as shown in the inset of Fig. 1A. For integral quantum Hall system (IQHS), multiples of insulating IS and con-ducting compressible strips were predicted (
12, 13 ) andobserved ( ) near the boundary of Hall bar. Figure1 illustrates alternating incompressible and compressiblestrips, distinguished by carrier density profile.To explain the second feature, one needs informationon how the Hall current actually flows within each IS.Under external perpendicular magnetic field, electronsperform a skipping motion at the IS edge and a deformedcycloidal motion away from the edge, as sketched in Fig.1B. The skipping motion is faster than the deformed cy-cloidal motion and the direction of drift is opposite atopposite side. The difference in speed and orientation ofthe guiding center implies the presence of a well-type po-tential V C ( y ) (red line in Fig.1B) across the IS becausethe y -slope of potential reflects electron’s x -group veloc-ity. The potential V C ( y ) is basically formed by free elec-trons accumulated at the edge of adjacent compressiblestrips that is in conducting state. We then see that, un-der the Hall bias V H , the tilted V C ( y ) sketched in Fig.1Cnaturally produces symmetric currents (two long arrows)on both sides of the IS loop. The skipping motion andwell-type potential within IS, which constitute essentialaspects of the symmetric Hall current ( ), provide thebasis of our approach.For more comprehensive understanding of the FQHEobserved in a Hall bar, one must explain existing ex-periments such as Hall resistivity curve ( ), excitationenergy gaps ( ), and fractional charges ( ) alongwith predictions for field-dependent chemical potentialand effective g -factor. Clarifying all of these issues is thepurpose of this study.We begin with replacing the well-type potential inFig. 1B by an image electron of the same spin orien-tation, as depicted in Fig. 1D. This ensures electronwave function nodes at IS edges via the Pauli principle,thereby eliminates the potential. The skipping electronand its image electron form a bound state, as illustratedin Fig. 1E; a dipolar composite (dashed ellipse in Fig. 1D)of spin-state | χ ↑↑ i , forming total spin of S = 1 in the low-est Landau level (LLL) in which spins are fully polarized( ).The electron-electron Coulomb interaction causesmany-particle correlations and leads to higher compos-ites. We sketch two correlated electrons and their imagesin Fig. 1F; such correlations likely form only among elec-trons drifting with equal group velocity, as correlation isstrongly affected by the distance between them. The spinstate of two correlated composites is given by | χ ↑↑↑↑ i withtotal spin S = 2 in the LLL. One easily pictures the semi-classical motion of higher multiples of correlated compos-ites and their total spins in the same manner. Therefore,LLL electrons confined in the IS are effectively described Image electron
A BD Region
L_2_ y Compressible (cid:15)(cid:71)(cid:15)(cid:71) (cid:101)
S=1 (cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32) x (cid:32) (cid:32)(cid:32)(cid:32) y yx S=2 (cid:32)(cid:32) C on c en t r a t i on E l e c t r on StripIncompressible (cid:101)
Confining Potential X B E F __ B ( T ) (cid:582)(cid:148) V (y) c C Under Hall Bias VV HC B yy x x FIG. 1: Semiclassical picture of electron motion in an FQHS.(A) Electron concentration in the left half of a Hall bar ofwidth L is presented by the dark blue line ( ). ISs (yellow),compressible regions (white) and a depletion region (blue)are shown. The inset depicts a schematic of ISs in the regionof Hall current (brown) ( ). (B) Electron motions and well-type potential within an IS from (A). (C) Electron motions inan IS loop under Hall bias V H . The longer arrows show that V H induces a left–right symmetric Hall current. Hot spots(red dots) are observed at drain and source points ( ). (D)A composite (dashed ellipse) comprising a real electron (blue)and its image (gray) with the same spin. (E) Dynamics of acomposite in an IS, depicting skipping and deformed cycloidalmotions. The composite has spin S = 1. (F) Skipping motionof two correlated composites, having total spin S = 2. by correlated composites of spin S = 1 , , · · · . The com-posites are free, as Coulomb interactions of electrons arealready taken into account in many-body correlation.The FQH states at filling ν of odd denominators im-mediately follow from the Zeeman interaction of com-posites under external magnetic field. The compositesof total spin S yield (2 S + 1) split levels and produceplateaus at odd-denominator filling factors ν = p/ (2 S +1)where p = 1 , · · · , (2 S + 1). Figure 2 illustrates how LLLis split by such Zeeman interactions, where the odd-denominators start to appear, and how the split levelsare distributed symmetrically about ν = 1 /
2. We showbelow that specifics of Zeeman splitting play core role inexplaining the observed FQHE phenomena.We now construct effective theory for the correlatedcomposites introduced above. The Hamiltonian of aFQHS is composed of electron’s kinetic energy and Zee-man energy as well as electron-background and electron-electron Coulomb interactions. In strong magnetic field,the Coulomb interaction is responsible for both the for-mation of IS (
12, 13 ) and multi-particle correlations in-side IS as shown in Fig. 1F. As n correlated electrons con-tribute the Zeeman energy − ~µ cpn · ~B , where ~µ cpn denotes (cid:47)(cid:47)(cid:47) (cid:149)(cid:100)(cid:89)(cid:149)(cid:100)(cid:90)(cid:149)(cid:100) > (cid:49)(cid:32)(cid:19) > > (cid:32) (cid:32) (cid:145) δ δ δ (cid:32) e_3 5e_ ⋯ (cid:149)(cid:100)(cid:88)
15 45 (cid:32) (cid:32)(cid:32) δ Number ofcorrelatedelectrons:
25 35 E n . . . _ Level Spiltting by Correlation: > >
47 57 n (cid:100)(cid:151)(cid:86)(cid:94) n (cid:100)(cid:151)(cid:86)(cid:90)(cid:83)(cid:88)(cid:86)(cid:89) AB Edge state chargeelectron (cid:47)(cid:47)(cid:47) (cid:149)(cid:100)(cid:88)(cid:149)(cid:100)(cid:90) > (cid:49)(cid:32)(cid:19) > > δ (cid:149)(cid:100)(cid:89)
15 45 (cid:32)(cid:32) δ Number ofcorrelatedelectrons:
15 45 (cid:32) δ (cid:32) C (cid:47)(cid:47)(cid:47) (cid:149)(cid:100)(cid:88)(cid:149)(cid:100)(cid:89) > (cid:49)(cid:32)(cid:19) δ (cid:149)(cid:100)(cid:90) Number ofcorrelatedelectrons: δ n (cid:100)(cid:151)(cid:86)(cid:92) . _ . . _ . _ . . _ (cid:32)(cid:32) σ=1 > > (cid:32) δ (cid:32) FIG. 2: Schematics of LLL splitting in the IS edge by Zeemaninteractions of correlated composites. (A) ν = p/ / ν = p/
5. (C) ν = p/
7. Level of correlation order n and corresponding effectiveelectron charge (green) in the edge state are shown. The δ n denotes split by n -th order correlation. the magnetic moment operator of n -correlated compos-ite, each electron attains an additional energy − ~µ cpn · ~B/n on average. Such description is most accurate near the ISedge, which in fact is the region where the FQH transportsubstantially takes place.Hence, the effective Hamiltonian in the single particledescription is given by H = X i " ( ~p i + e ~A i ) m ∗ c − g ∗ µ B ~s i ¯ h · ~B ! − ∞ X n =1 ~µ cpni n · ~B , (1)where i runs over all electrons in IS edges, m ∗ c is thecyclotron effective mass, ~p and ~A are the electron mo-mentum and external vector potential, respectively, g ∗ is the effective g -factor, µ B = ¯ he/ m is the Bohr mag-neton where m is the bare electron mass, and ~s i is theelectron spin operator. The effect of multi-particle cor-relations is perturbatively added in the last term.The magnetic moment of n -correlated composite isgiven as ~µ cpn = α νn ~J n / ¯ h where α νn / ¯ h and ~J n are the gyro-magnetic ratio at filling ν and total angular momentumoperator of n -correlated composite, respectively. We as-sume that the correlation effects are strong, yet the mag-nitude of last term in Eq.(1) is small compared to thefirst two terms. We also truncate out residual Coulombinteractions, which is subdominant at large B .The Hamiltonian Eq.(1) is readily diagonalized: for ~B = B ˆ z , J zn | Ψ i = j n ¯ h | Ψ i where | Ψ i is the total wavefunction and j n = − n, · · · , n because the orbital mo-menta of electron and its image are cancelled each otherand J n = S n = n ¯ h . The energy spectrum reads E νN,σ,j n = ¯ hω c N + 12 − ζ ν σ − ∞ X n =1 δ νn j n ! , (2)where ω c = eB/m ∗ c , ζ = ( | g ∗ | / m ∗ c /m ), σ = ±
1, and δ νn = ( α νn /µ ∗ B ) / n with µ ∗ B = ¯ he/ m ∗ c , respectively. Inother words, δ νn represents the amount of magnetic mo-ment of n -correlated composite per particle expressed inunits of effective Bohr magneton at filling ν , and it car-ries information on the strength of n -particle correlation.The effect of Coulomb interaction between the electronand its image is incorporated in δ νn .We truncate to the LLL by setting N = 0 , σ = 1 ofdegeneracy D L = BA/ ( h/e ), where A is the area of ISregion. The effective electron charges at edge state aredetermined by level splits in Fig. 2 sharing equal fractionof the underlying LLL states. Normal splitting proce-dure according to perturbed energy scales denoted by δ n is shown in Fig. 2A which is applicable to fillings ν = p/ /
2, not to ν = p/ p/ D LF = D L for ν = p/ / D LF = D L for ν = p/
5, and D LF = D L for ν = p/
7, where the subscript F denotes first splitting.This suggests that the electron at filling ν = p/ (2 q + 1),where q = 1 , , · · · , carries fractional charge e/ (2 q + 1)( ) through the IS edge. Higher order fractional chargescomposed of mixed correlations are shown in Fig. 2.We now obtain the Hall resistivity and the excita-tion energy gap just based on the energy spectrumEq.(2), which depends on the filing fraction ν . The en-ergy spectrum, along with chemical potential and effec-tive g factor, determines the carrier density ρ c ( B ) = A P E D L f ( E ), where f ( E ) is the Fermi distributionfunction. The Hall resistivity then follows from the car-rier density, ρ νxy = ( B ν /e ) /ρ c . Toyoda ( ) exploredsimilar idea but for ad hoc energy spectrum. The energyspectrum also determines the excitation energy gap thatis given by E Gν = ξ ∆ ν B ν where ξ = ¯ hω c /B , set by thejump ∆ ν from the level of filling ν to the lowest emptyone, as shown in Fig. 3A. Note that ∆ ν is composed of δ νn . We shall determine µ and g ∗ as well as δ n from theexperimental data for Hall resistivity and excitation en-ergy gap for each plateau state. We restrict ourselvesto filling sequences ν = q/ (2 q ± ν = .It turns out that the entire Hall resistivity curve andexcitation energy gaps follow from only a finite numberof parameters, δ n , as well as µ , g ∗ at each filling. We firstset the chemical potential µ linearly increasing over therange of magnetic field and drop at the transition in eachplateau, as shown in Fig. 3C, which has been observed inexperiments for both IQHS ( ) and FQHS ( ). Theamounf of drop is equal to the corresponding excitationenergy gap ( ). We extrapolate µ at zero magnetic fieldto the limiting value µ / for the sequence ν = q/ (2 q − µ / for the sequence ν = q/ (2 q + 1). Then, we de-termine g ∗ and δ n to obtain the observed Hall resistivity,and those δ n also determine the excitation energy gapsusing ∆ ν given in Table I. Extension to other sequencessuch as ν = q/ (4 q ±
1) or (3 q ± / (4 q ±
1) ( ) is anopen problem.We plot our theoretical ρ νxy superimposed on the exper-imental data ( ) (Fig. 3B), where we set g ∗ and µ as inFig. 3C, in which the amount of chemical potential dropat transition is equal to our theoretical E Gν presented inFig. 3D, along with the experimental data ( ). Takingthe values of δ n for each ν using the data in Table I and T = 10 mK, and setting ¯ hω c = (1 . / T) B for m ∗ c /m = 0 .
059 (which is close to that of GaAs), agree-ments with the experimental data over the entire rangeof magnetic field are remarkable. Note that the energylevels populate densely about ν = (Fig. 2), which givesrise to rapidly vanishing ∆ ν as ν approaches , and sothe excitation energy gap vanishes about the half filling.See Fig. 3D. We stress that our excitation energy gap isproportional to B itself, not ( B − B ) or any other shiftedvalues. Detailed procedure and further explanations aregiven in the Supplementary Materials.Finally, we discuss the half-filling region ( ). We firstnote that the Hall resistivity (orange line) in Fig. 3Bagrees with the linear Drude-like line (black dashed) once δ n is systematically given as δ n = 0 . / [3 · · · · (2 n +1)], as shown in Table I. From the expression δ n =( α n /µ ∗ B ) / n , where α n = g n µ nB , and Kohn’s theorem( ) asserting that the cyclotron mass is unaffected byinteractions, one may write δ n = ( g n / n )( e ∗ ( n ) /e ), where e ∗ ( n ) is the single electron effective charge of n -correlatedcomposite. Hence, e ∗ ( n ) /e = 1 / [3 · · · · (2 n + 1)] (Fig. 2A),and the numerator 0.3446 corresponds to ( g n / n ) thatcontains the correlation effect of n -correlated compos-ite. This argument implies that many-body correlationsare equally weighed to all correlation orders at half-filling, i.e., the state of correlation “white noise”, whichis equivalent to noninteracting electrons, thereby produc-ing the Drude-like Hall resistivity and constant chemicalpotential (flat orange line), as shown in Figs. 3B and3C, respectively. In fact, both are profiles of infinitelymany plateaus as the filling fraction approaches 1/2, pro-vided by the aforementioned dense energy levels shownin Fig. 2A.In conclusion, we showed that plateau heights in FQHEare automatically quantized just by presence of multi- particle correlations (Fig.2), while plateau widths andexcitation energy gaps are determined by the strength ofmulti-particle correlations δ νn , independent of Laughlinwave function or composite fermion model. Extension to ν > [1] K. von Klitzing, G. Dorda, M. Pepper, Phys. Rev. Lett. , 494-497 (1980).[2] D. C. Tsui, H. L. Stormer, A. C. Gossard, Phys. Rev.Lett. , 1559-1562 (1982).[3] J. G. Bednorz, K. A. M¨uller, Zeit. f¨ur Phys. B
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Phys. Rev. Lett. , 216804 (2003).[21] I. V. Kukushkin, K. v. Klitzing, K. Eberl, Phys. Rev.Lett. , 3665-3668 (1999).[22] T. Toyoda, Sci. Rept. , 12741 (2018).[23] Y. Y. Wei, J. Weis, K. v. Klitzing, K. Eberl, Appl. Rev.Lett. , 2514 (1997).[24] V.S. Kharapai et.al. , Phys. Rev. Lett. , 196805(2008).[25] V.S. Kharapai et.al. , Phys. Rev. Lett. , 086802 (2007).[26] W. Pan et.al., Physica E9 , 9-16 (2001).[27] B. I. Halperin, P. A. Lee, N. Read, Phys. Rev. B ,7312-7343 (1993).[28] W. Kohn, Phys. Rev. , , 1242 (1961). TABLE I: Values of | g ∗ | , δ n , and magnetic field along with energy gap parameter ∆ ν . λ =34.46. ν | g ∗ | · δ λ /3 11.73 12.06 12.30 13.40 14.30 3.25010 · δ λ /15 2.540 3.100 2.400 2.400 3.000 0.653510 · δ λ /105 0.657 0.350 0.110 0.560 0.300 0.075610 · δ λ /945 0.024 0.014 0.000 0.000 0.000 0.000 δ δ − − −− δ δ δ − δ − − − ∆ ν − δ − δ δ δ − δ − δ − − −− δ − δ − δ − δ − δ − δ − − − B (T) 28.50 23.75 22.12 19.00 16.57 15.71 14.10 13.30 12.00 9.50 BDCA (cid:46)(cid:46)(cid:46) > >> (cid:32) δ δ (cid:32) n (cid:32)(cid:32) δ E (cid:105) >> Excitation B(T)
123 100 20 30 (cid:101)(cid:88) (cid:88) n < n (3/4) B(T) E ne r g y G ap ( K ) q2q-1 n (cid:100) q2q+1 n (cid:100) (cid:582) (cid:79) (cid:148) (cid:140) (cid:125) (cid:80)
10 20 30
B(T) x (cid:543)(cid:71) δ δ δ ( (cid:543)(cid:71) =( Energy Gap E G = g * (3/4) FIG. 3: Hall resistivity curve and excitation energy gaps. (A) Illustration of the excitation energy gap E / G at filling ν = 1 / ), the dashed gray line is our result for ν = , and the red dashed line is its extrapolation. Thegreen region belongs to the 3 / and fillings. (C) Chemical potential and g ∗ determined inobtaining the Hall resistivity in (B). They are matched to the Hall resistivity intervals according to color. The gray dashed linesintercept at µ / or µ / . (D) Excitation energy gaps. Our results (red dots) E Gν = ξ ∆ ν B ν are superimposed on experimentaldata (black squares) ( ). The blue line is our prediction. The flattening near half filling originates from dense level distribution,illustrated in Fig. 2. r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Supplementary Materials for ”Correlated compositeapproach to fractional quantum Hall effect via edgecurrent”
Jongbae Hong , * , Soo-Jong Rey , Research Institute of Basic Sciences, Incheon National University,Yeonsu-gu, Incheon 22012 KOREA Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 37673 KOREAEmail: [email protected] Quantum Advanced Research Korea (Quantum ARK) Institute, Seoul 06624 KOREA The Korea Academy of Science & Technology, Seongnam 13630 KOREAEmail: [email protected]
We first set the chemical potential µ linearly increasing and drop at transition point over therange of magnetic field in each plateau. The amount of drop at transition is set to the exci-tation energy gap, as reported in Ref. ( ) via experimental study. Linearly increasing anddrop behavior of chemical potential has been observed experimentally in integral quantum Hallsystem (IQHS) ( ) and fractional quantum Hall system (FQHS) ( ). In FQHS, we predictthat the value of µ at zero magnetic field, µ , is that of half filling, µ / = 6 . meV, for thesequence ν = q/ (2 q − . However, we found that a different µ is needed for the sequence ν = q/ (2 q + 1) . We choose µ / = 4 . meV for the chemical potential at 1/4 filling, whichwe determined analogously from the data given in Ref. ( ). Since we fixed µ and the amount * Corresponding Author
1f drop or jump at the transition point, the field-dependent chemical potential µ ( B ) is drawn asa sawtooth function of the external magnetic field B of the form µ ν ( B ) = ( κ ν B + µ ν ) meV for B = ( B ν − , B ν + ) , where the superscript labels each Hall plateau of filling factor ν , κ ν is a constant coefficient, µ ν denotes chemical potential at zero magnetic field, and B ν ± are the lower and upper mag-netic fields corresponding to the transition points associated with the plateau at filling ν . Theparameter values κ ν and µ ν are given in Table S1, and µ ( B ) is explicitly shown in Fig. 3C.Table S1: Values of κ , µ , | g ∗ | , ζ , and δ n ( λ =34.46). ν · κ µ | g ∗ | · ζ · δ λ /3 11.73 12.06 12.30 13.40 14.30 3.250 · δ λ /15 2.540 3.100 2.400 2.400 3.000 0.6535 · δ λ /105 0.657 0.350 0.110 0.560 0.300 0.0756 · δ λ /945 0.024 0.014 0.000 0.000 0.000 0.000In Table S1, we also included the values for ν = 5 / , / , and in the last three columnsto obtain Hall resistivity curves. The former two fillings, ν = 5 / and / , are members of thesequences ν = (3 q − / (4 q − and ν = (3 q + 1) / (4 q + 1) , respectively. We predict that thevalue of µ for the sequence ν = (3 q − / (4 q − is the chemical potential at 1/2 filling, while µ for the sequence ν = (3 q + 1) / (4 q + 1) is that at 3/4 filling, which we expect it . meV.Filling ν = 1 is special because it is the highest energy level among the split levels shownin Fig. 2 and Fig. 3A, and energy excitation involves the next Landau level ( N = 0 , σ = − )2nlike other fillings in the lowest Landau level (LLL) as well as a member of integral quantumeffect. Therefore, ν = 1 is excluded in the sequence ν = q/ (2 q − . We choose µ = 9 . meV for ν = 1 , which gives relatively small effective g factor, | g ∗ | = 5 . .Since experimental data for excitation energy gaps at fillings / , / , and are unavailable,determining δ n values is restrictive. For this reason, the δ n values for these fillings given in TableS1 are less reliable than those for the sequences ν = q/ (2 q ± . Therefore, we omit chemicalpotential, effective g factor, and excitation energy gap in Figs. 3C and 3D in the text. We presentonly Hall resistivity curves for fillings / , / , and for continuation (Fig. 3B). We obtain the Hall resistivity curve separately for each plateau region because each plateaustate has different characteristics such as κ , | g ∗ | , and δ n representing n -body correlation. Forthe LLL, the Hall resistivity formula is given by ρ νxy = ( h/e ) . X { j n } [1 + exp { ( E ν ,σ =1 ,j n − µ ν ) /k B T } ] − , where E ν ,σ =1 ,j n = ¯ hω c (cid:0) − ζ ν − P ∞ n =1 δ νn j n (cid:1) , and P { j n } abbreviates the sum over correlatedstates X { j n } = 13 × × · · · +1 X j = − X j = − X j = − · · · , in which the prefactor is to render the upper bound of Fermi distribution function unity.We illustrate the explicit expression of ρ xy for the second-order truncation considering upto two-particle correlations, namely n = 1 and . The corresponding Hall resistivity expressionis explicitly written as ρ νxy = 3 × h/e ) P +1 j = − P +2 j = − (cid:2) e { ¯ hω c (0 . − ζ ν − j δ ν − j δ ν ) − µ ν } /k B T (cid:3) − , E ν ,σ =1 ,j n . Explicit expression of the denominator is given by (1 + αe − ∗ δ γ e − ∗ δ γ ) − + (1 + αe − ∗ δ γ e − ∗ δ γ ) − + (1 + αe − ∗ δ γ e ∗ δ γ ) − + (1 + αe − ∗ δ γ e ∗ δ γ ) − + (1 + αe − ∗ δ γ e ∗ δ γ ) − + (1 + αe ∗ δ γ e − ∗ δ γ ) − + (1 + αe ∗ δ γ e − ∗ δ γ ) − + (1 + αe ∗ δ γ e ∗ δ γ ) − + (1 + αe ∗ δ γ e ∗ δ γ ) − + (1 + αe ∗ δ γ e ∗ δ γ ) − + (1 + αe ∗ δ γ e − ∗ δ γ ) − + (1 + αe ∗ δ γ e − ∗ δ γ ) − + (1 + αe ∗ δ γ e ∗ δ γ ) − + (1 + αe ∗ δ γ e ∗ δ γ ) − + (1 + αe ∗ δ γ e ∗ δ γ ) − , where α = e { ¯ hω c (0 . − ζ ) − µ } /k B T with ζ = ( | g ∗ | / m ∗ c /m ) , and γ = ¯ hω c /k B T . We omit thesuperscript ν in ζ and δ for our convenience. If one considers up to four-particle correlation,the denominatior has terms.We calculated the Hall resistivity as follows. At each plateau region of filling fraction ν , wefirstly calculate ρ νxy by using the chemical potential, parameter values ζ , and δ n given in TableS1 (blue lines in Fig. S1) that fit the best to the corresponding region in the precision experiment(6). We then choose the transition point at which two neighboring Hall resistivity curves meet.Thus, we obtain the Hall resistivity curve ρ xy ( B ) over the entire domain of the magnetic field, B = 10 − T, as shown in the lowermost panel in Fig. S1.4
B(T)
123 100 20 305 ρ xy [ h / e ] n =1/3 ρ xy [ h / e ] n =2/5 ρ xy [ h / e ] n =1/2 Fig. S1: Hall resistivity curves for each plateau (upper three panels) and their sum (bottompanel). The thick gray line is experimental data (6), and the blue line drawn over the entiredomain of the magnetic field is calculated Hall resistivity curve. Note that at filling / ( / ),plateau heights of fillings / and / ( / , / , and /5