Numerical Study of Quantum Hall Bilayers at Total Filling

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As Published http://dx.doi.org/10.1103/PhysRevLett.119.177601

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Terms of Use Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. week ending PRL 119, 177601 (2017) PHYSICAL REVIEW LETTERS 27 OCTOBER 2017

Numerical Study of Quantum Hall Bilayers at Total Filling νT =1: A New Phase at Intermediate Layer Distances

Zheng Zhu,1 Liang Fu,1 and D. N. Sheng2 1Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2Department of Physics and Astronomy, California State University, Northridge, California 91330, USA (Received 2 April 2017; revised manuscript received 1 August 2017; published 23 October 2017)

We study the phase diagram of quantum Hall bilayer systems with total filing νT ¼ 1=2 þ 1=2 of the lowest Landau level as a function of layer distances d. Based on numerical exact diagonalization calculations, we obtain three distinct phases, including an exciton superfluid phasewith spontaneous interlayer coherence at small d, a composite Fermi liquid at large d, and an intermediate phase for 1.1 < d=lB < 1.8 (lB is the magnetic length). The transition from the exciton superfluid to the intermediate phase is identified by (i) a dramatic change in the Berry curvature of theground state under twisted boundary conditions on the two layers and (ii) an energy level crossing of the first excited state. The transition from the intermediate phase to the composite Fermi liquid is identified by thevanishing of the exciton superfluid stiffness. Furthermore, from our finite-size study, the energy cost of transferring one electron between the layers shows an even-odd effect and possibly extrapolates to a finite value in the thermodynamic limit, indicating the enhanced intralayer correlation. Our identification of an intermediate phase and its distinctive features shed new light on the theoretical understanding of the quantum Hall bilayer system at total filling νT ¼ 1.

DOI: 10.1103/PhysRevLett.119.177601

Introduction.—The multilayer quantum Hall systems On the experimental side, transport measurements indicate demonstrate tremendously rich physics when tuning the a transition between an exciton condensed interlayer interlayer interaction by changing the layer distance d.One coherent incompressible quantum Hall effect state and of the prominent examples is the bilayer systems [1–4] at a compressible liquid with varying the layer distance total filling νT ¼ 1 (ν ¼ 1=2 in each layer) with negligible [43–46]. At a smaller layer distance, the total Hall conduct- tunneling. Experimentally, the bilayer systems can be real- ance is quantized to e2=h. A strong enhancement in the zero- ized in single wide quantum wells, double quantum wells, or bias interlayer tunneling conductance [47] and the vanishing – bilayer graphenes [5 9]. Theoretically, the quantum states in of the Hall counterflow resistance [46,48] provide evidence d small and large limits have been well understood. When the for interlayer coherence [4]. Above a critical distance d ≈ layer distance is small, the strong interlayer Coulomb 1.6–2 (in units of magnetic length lB), which depends on the interaction drives the electron system into a pseudospin quantum well thickness, a compressible liquid state is found (layer) ferromagnetic long range order (FMLRO) state with [4,43–50]. However, the nature of the state at the intermediate the spontaneous interlayer phase coherence and interlayer – distance is unsettled after numerous investigations [4]. superfluidity [10 14]. The FMLRO can also be described as Motivated by this unsolved issue, we perform an an exciton condensation state as an electron in an orbit of one extensive ED study of ν ¼ 1=2 þ 1=2 bilayer system on layer is always bound to a hole in another layer forming an torus [51–53] up to 20 electrons, the phase diagram is exciton pair. This excitonic superfluid state can be described summarized in Fig. 1. We identify signatures of two phase by the Haplerin “111 state” wave function [15,16].Inthe transitions between the exciton superfluid and the CFL at limit of infinite layer separation, the bilayer system reduces to critical distances dc ≈ 1.1 and dc ≈ 1.8, respectively. For two decoupled composite Fermi liquids (CFL) [17–21]. 1 2 layer distance dfermion state stiffness. The drag Hall conductance is quantized to zero [40,41] is proposed. Until now it remains controversial for with no singularity in the Berry curvature, while the the phase at intermediate distances. total Hall conductance remains exactly quantized to e2=h.

0.12 K =K y 0 αðβÞ¼1 (a) K =K Here, , 2 are indices of two layers (which are the two d y 0 2

] =0 K =K 1=2 V ðqÞ¼2πe =ðεqÞ y 0 B xy components of a pseudospin ), α;α and l |K -K |=2 /N / y 0 2 −qd 2 0.08 V12ðqÞ¼V21ðqÞ¼2πe =ðεqÞe are the Fourier trans- Ky=0 K =0 y formations of the intralayer and interlayer Coulomb inter- (d) [e 0 actions, respectively. d is the distance between two layers and 0.04

(d)-E Exciton R i n CFL α;i is the guiding center coordinate of the th electron in E Superfluid α dc1 layer . In the present work, we consider the physical systems 0.00 with two identical 2D layers (with zero width) in the absence 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 of electron interlayer tunneling while spins of electrons are 0.16 (b) N=14 N=16 fully polarized due to the strongly magnetic field. Exciton CFL We use the ED algorithm to study the energy spectrum 0.12 Superfluid and state information on the torus. In order to study the s 0.08 physics of the pseudospin sector, we generalize the peri- odical boundary condition to twisted boundary condition 0 α 0.04 s with phase 0 ≤ θλ ≤ 2π along the λ direction in the layer α. d =0 c2 s By a unitary transformation, one can getP theP periodic wave 0.00 α α function Ψ on torus with jΨi¼exp½−i α iððθx=LxÞxi þ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 α α ðθy=LyÞy ÞjΦi d/lB i . Then the Berry curvature is defined by α β α β β α Fðθx;θyÞ¼Imðh∂Ψ=∂θxj∂Ψ=∂θyi−h∂Ψ=∂θyj∂Ψ=∂θxiÞ. FIG. 1. The phase diagram of ν ¼ 1=2 þ 1=2 quantum Hall The integral over the boundary phase unit cell leads bilayers with varying layer distance d=lB. We identify three phases: to the topological Chern number matrix Cα;β ¼ exciton superfluid phase, the intermediate phase, and composite R 1=2π dθαdθβFðθα; θβÞ Fermi liquid phase. (a) The transition from exciton superfluid to x y x y , which contains topological infor- d ≈ 1 1 – intermediate phase near c1 . is identified by the drag Hall mation for the bilayer quantum Hall state [38,55 60]. d conductance σxy and the energy level crossing. Here, the ground Numerically, applying common and opposite boundary state is in the momentum sector K0 ¼ π and N ¼ 16. (b) The phases on two layers, one can obtain the Hall conductances d ≈ 1 8 transition from intermediate phase to CFL phase near c2 . is in the layer symmetric and antisymmetric channel, c 2 s 2 identified by the exciton superfluid stiffness ρs [see Eq. (2)]. denoted by C ðe =hÞ and C ðe =hÞ, respectively. The drag d c s 2 Hall conductance, defined by σxy ¼ðC − C Þðe =2hÞ¼ ðC þ C Þðe2=hÞ The quantum phase transition between the exciton con- 1;2 2;1 , can be obtained directly by calculat- densed state and intermediate phase is identified by a ing C1;2 (or C2;1), corresponding to twisting boundary dramatic change in the Berry curvature of the ground state phases along the x direction in one layer and along the y under twisted boundary conditions on the two layers, and direction in another layer. One can also obtain the exciton the level crossing with a change of the nature of the low- superfluid stiffness when applying twisted boundary d ¼ d phases [38]. lying excitations at c1 . The fact of level crossing near d Energy spectrum and pseudospin excitation gap.—In c1 is consistent with previous studies [37,39,42]. The second transition between the intermediate phase and the Fig. 2(a), we show the lowest energies in each momentum sector for different layer distances d. For smaller layer CFL is characterized by the vanishing of the exciton d ≲ 1 1 superfluid stiffness. Further discussions of the finite size separations . , indeed we find the low energy exci- effect of numerical simulation can be found in the tation has the form of the linear dispersing Goldstone mode for small momenta [61]. One can also measure the Supplemental Material [54]. Model and method.—We consider bilayer electron sys- pseudospin excitation gap directly, which represents the tems subject to a magnetic field perpendicular to the two- energy cost of moving one electron from one layer to another layer and is defined as ΔpsðdÞ ≡ E0ðN↑;N↓;dÞ− dimensional (2D) planes. We use torus geometry with the 2 E0ðN=2;N=2;dÞþdSz=Nϕ N↑ ¼ N=2 þ ΔN length vectors Lx and Lx, and an aspect angle θ between . Here, and L ¼ L ¼ L θ ¼ π=2 N↓ ¼ N=2 − ΔN denote the number of electrons in two them. Here, x y and pffiffiffiffiffiffiffiffiffiffiffiffiffiffifor most of calcu- l ≡ ℏc=eB ≡ 1 layers for Sz ¼ ΔN ¼ 1; 2; … excitation. The energy shift lations. The magnetic length B is set to be 2 dSz=Nϕ is the charge energy induced by the imbalance of the unit of the length and Nϕ represents the number of electron number in two layers with total pseudospin Sz magnetic flux quanta determined by jLxLy sin θj¼2πNϕ.In [62]. As shown in Fig. 2(b), the finite size scaling of ΔpsðdÞ the presence of strong magnetic field, the Coulomb inter- S ¼ 1 action, projected onto the lowest Landau level, is written as for z goes to zero in the thermodynamic limit for d ≲ 1.1. X X d ≳ 1 1 1 2 As for layer distance . , the low energy linear −q =2 iq·ðRα;i−Rβ;jÞ V ¼ VαβðqÞe e : ð1Þ dispersion spectrum moves up in energy [see Fig. 2(a)] with 2πNϕ i

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0.07 d/l =0 (a) B d/l =0 0.20 d=1.3 (b) B (c) 0.1 (d) d/lB=0.2 d/lB=0.1 N=10 d/lB=0.4 0.06 0.15 d/l =0.6 d/lB=0.2 0.4 B N=12 d/l =0.3 d/lB=0.8 B

] 0.05 ] 0.15 N=14

d/lB=1 B d/l =0.4 B

l B l ] d/l =1.2 N=16 B B ] / / l d/lB=0.5 B 2 2

d/l =1.4 l B 0.04 0.0 0.10 / d/lB=0.6 2 d/lB=1.6 0.00 0.05 0.10 / 2 d/l =1.8 d/lB=0.7 0.10 B 0.03 1/N d/lB=0.8 0.2 [e =1) [e =1) d/l =0.9 (K) [e z tot B z tot (d) [e

0.02 d/l =1 d/lB=1.2 E 0.05 min B (S (S 0.05 d/l =1.3 E B ps ps ps d/l =1.4 0.01 B d/lB=1.5 d/l =1.6 0.0 0.00 0.00 B 0.00 02468 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 |K -K |/(2 /N ) y 0 1/N 1/N d/lB

FIG. 2. (a) The energy dispersion curves of lowest-energy excitations at each momentum sector. Here, the ground state is in the momentum sector K0 ¼ π. (b) and (c) show finite size scaling of the single pseudospin excitation gap Δps by using a parabolic function for layer distance d=lB < 1.1 (b) and d=lB > 1.1 (c). The inset of (c) indicates the even-odd effect in the intermediate phase up to N ¼ 20. (d) The energy spectrum gap ΔE ≡ E1ðdÞ − E0ðdÞ as a function of d=lB. The cusp near d=lB ≈ 1.1 indicates the level crossing for the excited states. sectors for d ≳ 1.1 as shown in Fig. 1(a). For the layer curvature associated with gapless points in low energy distance d ≈ 1.1, the energy spectrum shows the level spectrum. Figures 3(a) and 3(b) show the Berry curvatures K ¼ π ddc — d ≈ 1 1 1 boundary conditions. The transition near c1 . can regime, where a small gap opens to separate the lowest two α β also be identified by the Berry curvature Fðθx; θyÞ and the states, indicating the existence of the pseudospin gap. energy spectrum under twisted boundary conditions. Based on the above analysis, we confirm that the pseudo- Physically, a gap state has a well-defined smooth Berry spin Berry curvature also indicates the phase transition d curvature, while a gapless state may have singular Berry taking place near c1.

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(a) N=16,d/l =0.8 ]

] -4.7 B B l B l (c) N=16,d/l =0.8 (e) N=16,d/lB=3 B / / 2 0.5 2 [e

y -4.35 [e n n -4.8

x 0.0

) y

E ( , ) E ( ,0) , -4.40 0 0 x -0.5 -4.9 E ( , ) 0 1 E1( , )

F ( F E (0, ) 0 0 E0( , ) 0 E0( , ) -1.0 1.5 E (0, ) E (0,0) 0.5 E ( , ) 1 1.0 1 -4.45 1

1.0 E Spectrum Energy E Spectrum Energy 0.5 -5.0 / / 048121620 1.5 y 048121620 x -4.35 ]

E ( , ) B -4 (b) N=16, d/l =1.2 0 l (f) N=16 d/l increase B ] (d) N=16, d/lB=1.4 B B / l E1( , ) 2 / 2

0.10 -4.40 [e 0 [e

n -5

y 0.05 x -4.45

0.00

) y d/l =0.4

-6 B , 5

x -0.0 -4.50 d/lB=0.2

F ( F -0.10 1.5 d/lB=0 0.5 1. E Spectrum Energy 0 -4.55 -7

1.0 E Spectrum Energy 0.5 / 048121620 0.00.51.01.52.0 / 5 y x 1. 10 + [ ] 10 x y x+ y [ ]

α β FIG. 3. The Berry curvature Fðθx ; θyÞ for d=lB ¼ 0.8 (a), d=lB ¼ 1.2 (b). Here, Δθx and Δθy are the interval of mesh in phase space. It has strong fluctuation in the FMLRO phase (a), while it is smooth in the intermediate phase (b). (c) to (e) are the energy spectrum of the N ¼ 16 system with twisted boundary phases for d=lB ¼ 0.8 (c), d=lB ¼ 1.4 (d), and d=lB ¼ 3 (e). By fitting the energy spectrum with twisted phases, one can get the exciton superfluid stiffness ρs [see Eq. (2)] (f), which decreases with the layer distance and finally d>d d=l d=l ¼ 0 vanishes for c2. (f) From bottom to top, B increases from B with interval 0.2.

Exciton superfluid stiffness.—To study the evolution of the exciton superfluid phase and CFL phase are exciton superfluidity with the layer distances, we obtain the separated by an intermediate phase, which exhibits exciton superfluid stiffness ρs by adding a small twisted finite exciton superfluid stiffness, flat Berry curvature, boundary phase [38], which is proportional to the super- zero drag Hall conductance, and the even-odd effect of fluid density and identifies the energy cost when one rotates pseudospins. the order parameter of the magnetically ordered system by a The theoretical interpretation of the intermediate phase small angle. In our ED calculation, the exciton superfluid may start from two well-known limits. Starting from the stiffness can be obtained according to infinite distance, it is natural to choose the composite fermion (CF) picture [31–34,63]. Recently, a fully gapped 1 2 4 interlayer pairing phase is proposed based on the random- EðθtÞ=A ¼ Eðθt ¼ 0Þ=A þ ρsθt þ Oðθt Þ; ð2Þ 2 phase approximation calculation [33], which is consistent with our numerical findings of flat Berry curvature as well Eðθ Þ where t is the ground-state energy with twisted as gapped spin-1 and charge excitations, but the explan- (opposite) boundary phases θt between two layers θt ¼ ation of finite exciton superfluid stiffness is lacking. The 1 2 1;2 θx − θx (θy ¼ 0), A ¼jLx × Lxj is the area of the torus other candidate, the interlayer coherent CFL (ICCFL) [31] surface. Figures 3(c) to 3(e) show the energy spectrum as a state, has finite pseudospin stiffness due to interlayer Uð1Þ function of twisted phases for different layer distance. At phase fluctuations and possesses quantized Hall conduct- smaller layer separation, one can find the ground state ance in the antisymmetric channel, which is consistent with energy increases with tuning the twisted phases [see our ED findings on finite pseudospin gap and flat Berry Figs. 3(c) and 3(d)]. By fitting the energy curve using curvature. However, ICCFL indicates compressible prop- the quadratic function [see Fig. 3(f)], we get the exciton erty with respect to the symmetric current, while our superfluid stiffness ρs, which decreases with the increase of numerical data indicate a finite charge gap as well as the layer distance, and finally falls down to a negligible enhanced intralayer correlation (see the Supplemental d>d value for c2 [see Fig. 1(b)]. As shown in Fig. 3(e), the Material [54]). To understand the physics in the charge energy almost does not change with the twisted phases for channel better, one may start from the small distance limit larger distances, indicating the vanishing of superfluidity in the composite boson (CB) picture [28,42,64] and assume d>d ν ¼ 1 and the decoupling of two layers for c2, correspond- the system is the integer quantum Hall state. Based ing to the CFL states. on a recent proposed wave function [64], the SUð2Þ d Discussion.—We study the phase diagram of ν ¼ symmetry for CBs emerges near c1, leading to the 1=2 þ 1=2 quantum Hall bilayers on a torus and find that level crossing of the first excited state [see Fig. 1(a)].

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The low-lying charge excitation is dominated by interlayer [9] P. Maher, L. Wang, Y. Gao, C. Forsythe, T. Taniguchi, K. dFermions (Cambridge University T. Senthil, L. J. Zou, M. Zaletel, Z. Papi , S. D. Geraedts, Press, Cambridge, England, 2007); J. K. Jain, Phys. Rev. H. Isobe, and Y. Z. You. Z. Z. and L. F. are supported by the Lett. 63, 199 (1989); Annu. Rev. Condens. Matter Phys. 6, David and Lucile Packard foundation. L. F. is also sup- 39 (2015). ported by the DOE Office of Basic Energy Sciences, [21] D. T. Son, Phys. Rev. X 5, 031027 (2015). Division of Materials Sciences and Engineering under [22] R. Cote, L. Brey, and A. H. MacDonald, Phys. Rev. B 46, Grant No. DE-SC0010526. D. N. S. is supported by the 10239 (1992). U.S. Department of Energy, Office of Basic Energy [23] N. E. Bonesteel, I. A. McDonald, and C. Nayak, Phys. Rev. Sciences under Grants No. DE-FG02-06ER46305. Lett. 77, 3009 (1996). D. N. S. was also supported in part by the Gordon and [24] Y. B. Kim, C. Nayak, E. Demler, N. Read, and S. Das Betty Moore Foundation’s EPiQS Initiative, Grant Sarma, Phys. Rev. B 63, 205315 (2001). [25] Y. N. Joglekar and A. H. MacDonald, Phys. Rev. B 64, No. GBMF4303, during her visit at MIT. Part of the 155315 (2001). simulation were performed by using the Extreme [26] A. Stern and B. I. Halperin, Phys. Rev. Lett. 88, 106801 Science and Engineering Discovery Environment (2002). (XSEDE), which is supported by National Science [27] M. Y. Veillette, L. Balents, and M. P. A. Fisher, Phys. Rev. B Foundation Grant No. ACI-1548562. 66, 155401 (2002). [28] S. H. Simon, E. H. Rezayi, and M. V. Milovanovic, Phys. Rev. Lett. 91, 046803 (2003). [29] D.-W. Wang, E. Demler, and S. Das Sarma, Phys. Rev. B 68, [1] S. M. Girvin and A. H. MacDonald, Perspectives in Quan- 165303 (2003). tum Hall Effects, edited by A. Pinczuk and S. Das Sarma [30] R. L. Doretto, A. O. Caldeira, and C. M. Smith, Phys. Rev. (Wiley, New York, 1997). Lett. 97, 186401 (2006); R. L. Doretto, C. Morais Smith, [2] J. P. Eisenstein and A. H. Macdonald, Nature (London) 432, and A. O. Caldeira, Phys. Rev. B 86, 035326 (2012). 691 (2004). [31] J. Alicea, O. I. Motrunich, G. Refael, and M. P. A. Fisher, [3] B. N. Narozhny and A. Levchenko, Rev. Mod. Phys. 88, Phys. Rev. Lett. 103, 256403 (2009). 025003 (2016), and references therein. [32] R. Cipri and N. E. Bonesteel, Phys. Rev. B 89, 085109 [4] J. P. Eisenstein, Annu. Rev. Condens. Matter Phys. 5, 159 (2014). (2014), and references therein. [33] H. Isobe and L. Fu, Phys. Rev. Lett. 118, 166401 (2017). [5] Y. W. Suen, L. W. Engel, M. B. Santos, M. Shayegan, and [34] I. Sodemann, I. Kimchi, C. Wang, and T. Senthil, Phys. Rev. D. C. Tsui, Phys. Rev. Lett. 68, 1379 (1992); B 95, 085135 (2017). [6] J. P. Eisenstein, G. S. Boebinger, L. N. Pfeiffer, K. W. West, [35] A. C. Potter, C. Wang, M. A. Metlitski, and A. Vishwanath, and S. He, Phys. Rev. Lett. 68, 1383 (1992). arXiv:1609.08618. [7] D.-K. Ki, V. I. Fal’ko, D. A. Abanin, and A. F. Morpurgo, [36] J. Schliemann, S. M. Girvin, and A. H. MacDonald, Phys. Nano Lett. 14, 2135 (2014). Rev. Lett. 86, 1849 (2001); J. Schliemann, Phys. Rev. B 67, [8] A. Kou, B. E. Feldman, A. J. Levin, B. I. Halperin, K. 035328 (2003). Watanabe, T. Taniguchi, and A. Yacoby, Science 345,55 [37] N. Shibata and D. Yoshioka, J. Phys. Soc. Jpn. 75, 043712 (2014). (2006).

177601-5 week ending PRL 119, 177601 (2017) PHYSICAL REVIEW LETTERS 27 OCTOBER 2017

[38] D. N. Sheng, L. Balents, and Z. Wang, Phys. Rev. Lett. 91, [50] A. D. K. Finck, J. P. Eisenstein, L. N. Pfeiffer, and K. W. 116802 (2003). West, Phys. Rev. Lett. 104, 016801 (2010). [39] K. Park, Phys. Rev. B 69, 045319 (2004). [51] F. D. M. Haldane, Phys. Rev. Lett. 55, 2095 (1985). [40] G. Möller, S. H. Simon, and E. H. Rezayi, Phys. Rev. Lett. [52] E. H. Rezayi and F. D. M. Haldane, Phys. Rev. Lett. 84, 101, 176803 (2008). 4685 (2000). [41] G. Möller, S. H. Simon, and E. H. Rezayi, Phys. Rev. B 79, [53] D. Yoshioka, Phys. Rev. B 29, 6833 (1984). 125106 (2009). [54] See Supplemental Material at http://link.aps.org/ [42] M. V. Milovanović, E. Dobardzic, and Z. Papić, Phys. Rev. supplemental/10.1103/PhysRevLett.119.177601 for B 92, 195311 (2015). numerical details and additional numerical evidence. [43] R. D. Wiersma, J. G. S. Lok, S. Kraus, W. Dietsche, K. [55] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den von Klitzing, D. Schuh, M. Bichler, H.-P. Tranitz, and W. Nijs, Phys. Rev. Lett. 49, 405 (1982). Wegscheider, Phys. Rev. Lett. 93, 266805 (2004). [56] Q. Niu, D. J. Thouless, and Y.-S. Wu, Phys. Rev. B 31, 3372 [44] S. Q. Murphy, J. P. Eisenstein, G. S. Boebinger, L. N. (1985). Pfeiffer, and K. W. West, Phys. Rev. Lett. 72, 728 (1994). [57] X. G. Wen and A. Zee, Phys. Rev. B 44, 274 (1991). [45] P. Giudici, K. Muraki, N. Kumada, and T. Fujisawa, Phys. [58] Kun Yang and A. H. MacDonald, Phys. Rev. B 63, 073301 Rev. Lett. 104, 056802 (2010); P. Giudici, K. Muraki, N. (2001). Kumada, Y. Hirayama, and T. Fujisawa, Phys. Rev. Lett. [59] D. N. Sheng, Z.-Y. Weng, L. Sheng, and F. D. M. Haldane, 100, 106803 (2008). Phys. Rev. Lett. 97, 036808 (2006). [46] M. Kellogg, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, [60] D. N. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Nat. Phys. Rev. Lett. 93, 036801 (2004); 90, 246801 (2003). Commun. 2, 389 (2011). [47] I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, and K. W. [61] I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, and K. W. West, Phys. Rev. Lett. 84, 5808 (2000). West, Phys. Rev. Lett. 87, 036803 (2001). [48] E. Tutuc, M. Shayegan, and D. A. Huse, Phys. Rev. Lett. 93, [62] A. H. MacDonald, P. M. Platzman, and G. S. Boebinger, 036802 (2004). Phys. Rev. Lett. 65, 775 (1990). [49] S. Luin, V. Pellegrini, A. Pinczuk, B. S. Dennis, L. N. [63] Y. You, arXiv:1704.03463. Pfeiffer, and K. W. West, Phys. Rev. Lett. 94, 146804 [64] B. Lian and S.-C. Zhang, arXiv:1707.00728. (2005). [65] M. V. Milovanović, Phys. Rev. B 95, 235304 (2017).

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