At small layer separations, the ground state of a nu=1 bilayer quantum Hall
system exhibits spontaneous interlayer phase coherence and has a
charged-excitation gap E_g. The evolution of this state with increasing layer
separation d has been a matter of controversy. In this letter we report on
small system exact diagonalization calculations which suggest that a single
phase transition, likely of first order, separates coherent incompressible (E_g
>0) states with strong interlayer correlations from incoherent compressible
states with weak interlayer correlations. We find a dependence of the phase
boundary on d and interlayer tunneling amplitude that is in very good agreement
with recent experiments.Comment: 4 pages, 4 figures included, version to appear in Phys. Rev. Let

A. H. MacDonald

A. Sawada

B. I. Halperin

I. B. Spielman

J. P. Eisenstein

John Schliemann

K. Moon

K. Yang

N. E. Bonesteel

R. Côte

S. He

S. M. Girvin

S. Q. Murphy

Y. N. Joglekar

English

arXiv

At small layer separations, the ground state of a ν = 1 bilayer quantum Hall system exhibits spontaneous interlayer phase coherence. The evolution of this state with increasing layer separation d has been a matter of controversy. We report on small system exact diagonalization calculations which suggest that a single-phase transition, likely of first order, separates incompressible states with strong interlayer correlations from compressible states with weak interlayer correlations. We find a dependence of the phase boundary on d and interlayer tunneling amplitude that is in very good agreement with recent experiments

Schliemann, John

Girvin, S. M.

MacDonald, A. H.

University of Regensburg Publication Server

VOLUME 86, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 FEBRUARY 2001Strong Correlation to Weak Correlation Phase Transition in Bilayer Quantum Hall SystemsJohn Schliemann, S. M. Girvin, and A. H. MacDonaldDepartment of Physics, Indiana University, Bloomington, Indiana 47405-7105(Received 20 June 2000)At small layer separations, the ground state of a n  1 bilayer quantum Hall system exhibits sponta-neous interlayer phase coherence. The evolution of this state with increasing layer separation d has beena matter of controversy. We report on small system exact diagonalization calculations which suggestthat a single-phase transition, likely of first order, separates incompressible states with strong interlayercorrelations from compressible states with weak interlayer correlations. We find a dependence of thephase boundary on d and interlayer tunneling amplitude that is in very good agreement with recentexperiments.DOI: 10.1103/PhysRevLett.86.1849 PACS numbers: 73.43.–f, 73.21.–bThe ground state of a two-dimensional monolayer elec-tron system at Landau level filling factor n  1 is a singleSlater determinant described exactly by Hartree-Fock the-ory and is a strong ferromagnet with a large gap Eg forcharged excitations [1,2]. This elementary property hasrich and interesting consequences for the physics of bilayerquantum Hall systems at the same total n, consequencesthat are readily appreciated when a pseudospin language[1,3] is used to describe the layer degree of freedom. Whenthe layer separation d goes to zero, interactions betweenlayers are identical to interactions within layers. The pseu-dospin bilayer Hamiltonian is then identical to the singlelayer Hamiltonian with spin and its ground state has pseu-dospin order and a finite charge gap. For infinite layerseparation, on the other hand, the bilayer system reduces totwo disordered, compressible, uncorrelated n  12 sys-tems. This Letter concerns the evolution of bilayer quan-tum Hall systems between these two extremes.For small layer separations the difference between in-terlayer and intralayer interactions breaks the pseudospininvariance of the Hamiltonian, resulting in an incom-pressible easy-plane pseudospin ferromagnet. In physicalterms the pseudospin order represents spontaneous phasecoherence between the electron layers. Several scenarioshave been proposed for the evolution of the ground stateas the layer separation increases further. In Hartree-Focktheory [4], spontaneous interlayer coherence is lost if thelayer separation exceeds a critical value, and the groundstate at large separations consists of weakly correlatedWigner crystals. While possibly instructive, this picture isknown to be incorrect at large d since half-filled Landaulevels do not have crystalline ground states. Workingin the other direction, Bonesteel et al. started [5] fromthe composite fermion theory of isolated compressiblen  12 layers, and concluded that coupling would leadto pairing between composite fermions in opposite layersand also, implicitly, to a charge gap. Since the pseu-dospin ferromagnet possesses particle-hole rather thanparticle-particle pairing, however, this picture still impliesthat at least one phase transition occurs as a function oflayer separation. In a numerical diagonalization study He0031-90070186(9)1849(4)$15.00 ©et al. [6] predicted, on the basis of the system parameterdependence of overlaps between exact ground states andtwo different variational wave functions, the existence oftwo distinct incompressible states separated by a regionof compressible states.Experiments, on the other hand, have tended to be con-sistent [7] with the proposal [3] that a single-phase transi-tion from an incompressible to a compressible state occurswith increasing layer separation at any value of the inter-layer tunneling amplitude. Very recently, in an intriguingnew experiment by Spielman et al. [8], the tunneling con-ductance across the layers was studied in a sample with ex-tremely small tunneling amplitude. When the ratio of layerseparation and magnetic length was lowered (at fixed fill-ing factor) below a critical value, the conductance showeda very pronounced peak around zero bias voltage betweenthe layers that provides direct evidence [9] for spontaneousinterlayer phase coherence. This is because in the coher-ent state the layer index is uncertain. Only in this casecan tunneling leave the system in or near its ground state,avoiding the orthogonality catastrophe and allowing tun-neling to occur near zero voltage.Since the critical layer separation found by Spielmanet al. is close to the one obtained earlier by Murphyet al. for the onset of the quantum Hall effect [7], experi-ment demonstrates that for vanishing tunneling amplitudethe phase transitions at which pseudospin order and thecharge gap are lost are either closely spaced or coincident.In this Letter we report on small system exact diago-nalization calculations which strongly suggest that bilayerquantum Hall systems have a single-phase transition, likelyof first order, as a function of d. Our critical layer sepa-ration is in very good quantitative agreement with thevalue measured in Ref. [8]. In the light of the experimen-tal results mentioned above, our calculations imply thatthe charge gap disappears and long-range phase coherencesimultaneously drops sharply to near zero at the phasetransition. This result is not entirely unexpected since asimple Landau-Ginzburg analysis indicates that the twoorder parameters could not vanish simultaneously withoutfine-tuning, if the transition were continuous. Also the2001 The American Physical Society 1849VOLUME 86, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 FEBRUARY 2001mean-field theory energy gap is proportional to the pseu-dospin order parameter, suggesting that these two ordersare mutually reinforcing and that a first order transition istherefore likely. Finally, we note that, experimentally, thecharge gap phase transition is sharp even at finite tunnelingbetween the layers. Since tunneling produces a pseudo-magnetic field which couples to the pseudospin order pa-rameter, this is an unusual magnetic transition which doesnot involve symmetry breaking, a fact which lends furtherweight to the suggestion that the transition is first order.We analyze bilayer quantum Hall systems numericallyby means of exact diagonalizations of finite systems usingthe spherical geometry. We have verified numerically thatthe ground state and low-lying excitations are fully spinpolarized and neglect the spin degree of freedom in thepresent discussion. The Hamiltonian is given byH  H1P 1HCoul , (1)where HCoul represents the usual Coulomb interactionwithin and between layers, and the single-particle Ham-iltonian H1P is given byH1P  2 12Xmc1m,mDytzm,m0 1 Dttxm,m0cm0,m . (2)We concentrate here on the tunneling amplitude (Dt) tunedphase transition, although bias voltage (Dy) dependenceis also interesting and often experimentally more conve-nient. m,m0 [ 1,2 run over the layer (or pseudospin)indices and a summation convention is implicit; t arethe pseudospin Pauli matrices. m [ 2Nf2, . . . ,Nf2is the z projection of the orbital angular momentum ofeach electron in the lowest Landau level, where Nf isthe number of flux quanta penetrating the sphere. In thefollowing we denote the pseudospin operators by T 12Pm c1m,m tm,m0cm0,m. The interlayer separation d ismeasured in units of the magnetic length lB ph¯ceB,and all energies are given in units of the Coulomb energyscale e2elB. We consider the case of zero well widthto enable comparison with most previous theoretical in-vestigations [10–13], and also systems consisting of tworectangular wells of finite width w [14] whose ratio to thecenter-to-center layer separation d is wd  0.65. Thisvalue corresponds to the sample used in Ref. [8].We consider systems with an even electron number Nwhich leads to a nondegenerate spatially homogeneousground state with total angular momentum L  0. Forsimplicity, let us first examine the case of vanishing biasvoltage, where both Ty	 and Tz	 are strictly zero.Figure 1 shows the interlayer phase coherence as mea-sured by the expectation value Tx	 along with the fluc-tuation DTx pTx2	 2 Tx	 Tx	 as a function of thetunneling gap for a system of 12 electrons, a layer separa-tion of d  1.80, and zero well width. At Dt  0, Tx	is necessarily zero in a finite system. With increasing tun-neling gap, Tx	 grows rapidly reaching an inflection point18500.0 0.1 0.2 0.3 0.4 0.5∆t0.01.02.03.04.05.06.00.0 0.1 0.2 0.3 0.4 0.5050100d<Tx>/d∆t<Tx>∆Tx N=12d=1.80w/d=0.00FIG. 1. The pseudospin expectation value Tx	 and the fluctua-tion DTx as a function of the tunneling gap Dt . The derivativedTx	dDt [measured in units of 1e2elB] is shown in theinset.with a very steep tangent. The differential pseudospinsusceptibility, x  1N Tx	dDt , is plotted in the in-set and shows a very pronounced peak. In the immediatevicinity of this peak, the pseudospin fluctuation DTx hasalso a pronounced maximum. In Fig. 2 the x is plotted fordifferent numbers of electrons.The rapid growth with increasing system size of the peakin this generically intensive quantity is strong evidencefor a ground state phase transition. Analogous findingsare obtained for the peak in the pseudospin fluctuation.Thus, the peaks in the susceptibility of the pseudospin andits fluctuation grow very rapidly with increasing systemsize and signal a quantum phase transition at the critical0.00 0.05 0.10 0.15 0.20∆t0.02.04.06.08.010.0χ N=12N=10N=8d=2.50w/d=0.65FIG. 2. The pseudospin susceptibility x for different systemsizes as a function of the tunneling gap. The rapidly growingpeak indicates a quantum phase transition.VOLUME 86, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 FEBRUARY 2001value of the tunneling gap. At large tunneling the systempseudospin magnetization is close to its maximum value,while at small (but also finite) tunneling the system isdisordered and the pseudospin magnetization is stronglyreduced by interactions.The two peaks described above occur at extremelynearby values of Dt at a given layer separation d, andwe consider the very tiny differences in their location asa finite-size effect. To estimate the phase diagram of thesystem we place the phase boundary at the maximum ofthe quantum fluctuations DTx .Figure 3 shows the resulting phase boundaries for dif-ferent system sizes and both cases of well width. At smalllayer separation the system is in the ordered phase and thefluctuation peak occurs exactly at Dt  0. At a criticallayer separation dcDt  0,N the phase boundary movesout rapidly to finite values of Dt and intersects the axisDt  0 with an almost horizontal tangent. This is in quali-tative agreement with earlier experimental [7] and theoreti-cal [3] estimates of the phase diagram.The critical values dcDt  0,N form a rapidlyconverging data sequence and are plotted in Fig. 4. Thesefinite-size data are accurately and consistently describedby an ansatz of the form dcN  a 1 bN2l with twofit parameters a  dcN  `, b, and a shift exponentl. The best fits to both sets of data are obtained forl  5.0 6 0.2 leading to a value of dcN  ` 1.30 6 0.03 for zero well width, and dcN  ` 1.81 6 0.03 for wd  0.65. The latter value is inexcellent agreement with the results of Ref. [8], wherethe onset of the tunneling conductance peak is observedat a layer separation of d  1.83. Thus, our numericalresults clearly indicate that the findings of the abovetunneling experiments are the signature of a quantumphase transition. The very large value of l seems incon-sistent with a diverging correlation length and suggeststhe transition is first order. A first order phase transition0.02 0.04 0.06 0.08∆t0.00 0.02 0.04 0.06 0.08∆t0.01.02.03.04.05.0d c(∆t)w/d=0.00 w/d=0.65N=6N=8 N=10N=12N=6N=8N=10N=12FIG. 3. Phase boundaries for different system sizes N and ra-tios of well width to layer separation.would explain the apparent coincidence of the appearanceof spontaneous phase coherence and the quantum Halleffect in experiment [7,8]. We note that our result for thecritical layer separation at vanishing tunneling gap agreesreasonably, at zero well width, with the point at whichthe uniform density phase coherent state first becomesunstable in the Hartree-Fock approximation [3]. At largerw, however, the Hartree-Fock estimates clearly deviatefrom the exact diagonalization result.In order to further investigate the order of the quantumphase transition, we introduce the ratiovN 2DTx2NdTx	dDtN, (3)where the subscript N refers to the system size. As wediscuss below, this type of ratio should prove to be a pow-erful general tool in the analysis of any quantum phasetransition. In classical physics this ratio of fluctuation tosusceptibility is equal to the thermal energy kBT and van-ishes at T  0. The classical relationship does not ap-ply here since the Hamiltonian fails to commute with itsderivative with respect to Dt . There is, however, a closelyrelated zero-temperature relationship with the typical exci-tation energy vN taking over the role of temperature. Thefluctuation can be written asDTx2 Xn.0jnjTxj0	j2, (4)where the sum is performed over all excited states, whilefor the derivative of the pseudospin magnetization onefinds from linear response theorydTx	dDt 2Xn.0jnjTxj0	j2En 2 E0. (5)6 8 10 12N0.01.02.03.04.0d c(∆t=0,N)w/d=0.65w/d=0.00dc(N)=α+βN5FIG. 4. The critical layer separation dcDt  0,N (filledsymbols) at vanishing tunneling as a function of the system sizeN for both cases of well width. The lines are finite-size fits tothe data with a shift exponent of l  5.0.1851VOLUME 86, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 26 FEBRUARY 2001From these equations we see thatvN is a harmonic averageof excitation energies En 2 E0, weighted by the factorsjnjTxj0	j2. In particular, vN has a vanishing thermody-namic limit if at least one state with a nonvanishing ma-trix element njTxj0	 has an excitation energy En 2 E0which extrapolates to zero for N ! `. Thus, Eq. (3) de-fines a characteristic energy scale of the system at the phaseboundary. The operator Tx naturally enters this expressionsince it couples to a control parameter driving the phasetransition.For a continuous phase transition one would clearly ex-pect vN to vanish at the phase boundary for an infinitesystem, while a finite limit limN!` vN is indicative of afinite energy scale, i.e., a first order transition. From ourfinite-size data for vN [evaluated at vanishing tunnelingand d  dcN] we conclude that this quantity extrapo-lates for N ! ` to a rather substantial nonzero value oforder 0.05e2elB  5 K for both values of w consideredhere. Along with the arguments and experimental findingsgiven so far, this result strongly suggests that the bilayerquantum Hall system at filling factor n  1 undergoes asingle first order phase transition as a function of the ra-tio of layer separation and magnetic length at all values ofthe tunneling amplitude. The phase boundary separates aphase with strong interlayer correlation (and a finite gapfor charged excitations) from a phase with weak interlayercorrelations and vanishing Eg.Finally we comment briefly on the influence of a biasvoltage between the layers. When applying a bias voltageto the system the vector  T 	 is tilted out of the xy plane witha finite z component. In this case we find numerically thatthe quantum phase transition is again signaled by the longi-tudinal fluctuation of the pseudospin magnetization and itssusceptibility, and all results concerning the phase bound-ary are qualitatively the same. This agrees with experi-mental results by Sawada et al., who found a remarkablestability of the n  1 quantum Hall state against a finitebias voltage, as compared to the behavior at other fillingfactors [15]. First order phase transitions from stronglycorrelated to weakly correlated states also occur with in-creasing bias potential. We predict measurable anomaliesin the double-layer system capacitance at bias tuned phasetransitions.We acknowledge helpful discussions with S. Sachdev.This work was supported by the Deutsche Forschungsge-meinschaft under Grant No. SCHL 539/1-1 and by the1852National Science Foundation under Grants No. DMR-9714055 and No. DMR-0087133.[1] For a review on experimental research see J. P. Eisenstein,in Perspectives in Quantum Hall Effects, edited by S. DasSarma and A. Pinczuk (Wiley, New York, 1997); for areview on theoretical research see S. M. Girvin and A. H.MacDonald, in the same volume.[2] B. I. Halperin, Helv. Phys. Acta 56, 75 (1983).[3] A. H. MacDonald, P. M. Platzman, and G. S. Boebinger,Phys. Rev. Lett. 65, 775 (1990).[4] R. Côte, L. Brey, and A. H. MacDonald, Phys. Rev. B 46,10 239 (1992).[5] N. E. Bonesteel, I. A. McDonald, and C. Nayak, Phys. Rev.Lett. 77, 3009 (1996).[6] S. He, S. Das Sarma, and X. C. Xie, Phys. Rev. B 47, 4394(1993).[7] S. Q. Murphy, J. P. Eisenstein, G. S. Boebinger, L. N. Pfeif-fer, and K. W. West, Phys. Rev. Lett. 72, 728 (1994).[8] I. B. Spielman, J. P. Eisenstein, L. N. Pfeiffer, and K. W.West, Phys. Rev. Lett. 84, 5808 (2000).[9] A. Stern, S. M. Girvin, A. H. MacDonald, and N. Ma,cond-mat /0006457; L. Balents and L. Radzihovsky,cond-mat /0006450; M. M. Fogler and F. Wilczek,cond-mat /0007403; E. Demler, C. Nayak, and S. DasSarma, cond-mat /0008137.[10] K. Moon, H. Mori, K. Yang, S. M. Girvin, A. H. MacDon-ald, L. Zheng, D. Yoshioka, and S.-C. Zhang, Phys. Rev.B 51, 5138 (1995).[11] K. Yang, K. Moon, L. Belkhir, H. Mori, S. M. Girvin, A. H.MacDonald, L. Zheng, and D. Yoshioka, Phys. Rev. B 54,11 644 (1996).[12] K. Moon, Phys. Rev. Lett. 78, 3741 (1997).[13] Y. N. Joglekar and A. H. MacDonald, Physica (Amsterdam)6E, 371 (2000).[14] To compute the pseudopotentials of the Coulomb inter-action in the case of finite well width we take the elec-tron wave function for the growth direction in each well tobe a normalized sine having its zeros at the edges of thewell. This means we neglect electron charge density out-side the wells, which is a very good approximation for nottoo large tunneling Dt & 0.2e2elB, where the Coulombenergy scale is of order 100 K. An illustrative picture ofthe charge density distribution can be found in Ref. [6].[15] A. Sawada, Z. F. Ezawa, H. Ohno, Y. Horikoshi, O. Sugie,S. Kishimoto, F. Matsukura, Y. Ohno, and M. Yasumoto,Solid State Commun. 103, 447 (1997).

Strong Correlation to Weak Correlation Phase Transition in Bilayer Quantum Hall Systems

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Strong Correlation to Weak Correlation Phase Transition in Bilayer
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