In the framework of a recently developed model of interacting composite
fermions, we calculate the energy of different solid and Laughlin-type liquid
phases of spin-polarized composite fermions. The liquid phases have a lower
energy than the competing solids around the electronic filling factors
nu=4/11,6/17, and 4/19 and may thus be responsible for the fractional quantum
Hall effect at nu=4/11. The alternation between solid and liquid phases when
varying the magnetic field may lead to reentrance phenomena in analogy with the
observed reentrant integral quantum Hall effect.Comment: 4 pages, 3 figures; revised version accepted for publication in Phys.
  Rev. Let

Goerbig, M. O.

Lederer, P.

Smith, C. Morais

English

arXiv

M. O. Goerbig

P. Lederer

C. Morais Smith

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Possible Reentrance of the Fractional Quantum Hall Effect in the Lowest Landau Level

In the framework of a recently developed model of interacting composite fermions restricted to a single level, we calculate the activation gaps of a second generation of spin-polarized composite fermions. These composite particles each consist of a composite fermion of the first generation and a vortexlike excitation and may be responsible for the recently observed fractional quantum Hall states at unusual filling factors such as ν= 4/11, 5/13, 5/17, and 6/17. Because the gaps of composite fermions of the second generation are found to be more than one order of magnitude smaller than those of the first generation, these states are less visible than the usual states observed at filling factors ν= p/(2ps + 1). Their stability is discussed in the context of a pseudopotential expansion of the composite-fermion interaction potential

Goerbig, Mark O.

Morais Smith, Cristiane de

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http://doc.rero.chPossible Reentrance of the Fractional Quantum Hall Effect in the Lowest Landau LevelM. O. Goerbig,1,2 P. Lederer,2 and C. Morais Smith1,31De´partement de Physique, Universite´ de Fribourg, Pe´rolles, CH-1700 Fribourg, Switzerland2Laboratoire de Physique des Solides, UMR 8502, Universite´ Paris-Sud, F-91405 Orsay, France3Institute for Theoretical Physics, Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands(Received 2 March 2004; published 16 November 2004)In the framework of a recently developed model of interacting composite fermions, we calculate theenergy of different solid and Laughlin-type liquid phases of spin-polarized composite fermions. Theliquid phases have a lower energy than the competing solids around the electronic ﬁlling factors  4=11; 6=17, and 4=19 and may thus be responsible for the fractional quantum Hall effect at   4=11.The alternation between solid and liquid phases when varying the magnetic ﬁeld may lead to reentrancephenomena in analogy with the observed reentrant integral quantum Hall effect.DOI: 10.1103/PhysRevLett.93.216802 PACS numbers: 73.43.Nq, 71.10.Pm, 73.20.QtRecent experiments by Eisenstein et al. have revealed areentrance of the integer quantum Hall effect (IQHE)when the two spin branches of the lowest Landau level(LL) are completely ﬁlled and the ﬁrst excited LL (n  1)starts to be populated [1]. This phenomenon arises whenthe partial ﬁlling of the topmost LL is varied, and it is dueto an alternation of electron-solid phases, such as theWigner crystal (WC) or a bubble crystal, and quantumliquids, which display the fractional quantum Hall effect(FQHE) [2]. Both liquid and solid phases have their originin the Coulomb interaction between the electrons, whichis the relevant energy scale when the last occupied LL isnot completely ﬁlled. In this case, intra-LL excitationscoupling states with the same kinetic energy are possible,in contrast to the case of completely ﬁlled LLs, wheresuch excitations are prohibited by the Pauli principle.Electrons in the completely ﬁlled lower LLs may thenbe considered as an inert homogeneous background.The FQHE in the lowest LL occurs at the ﬁlling factors  nel=nB  p=2ps 1, where nel denotes the elec-tronic and nB  eB=h the ﬂux density. It may be inter-preted as an IQHE of a new type of particle, a so-calledcomposite fermion (CF) [3], which consists of an electronand a vortexlike object, which carries s pairs of ﬂuxquanta. Because of its reduced coupling eB  eB=2ps 1 to the magnetic ﬁeld B, p CF LLs are com-pletely ﬁlled when  p, where the CF-LL ﬁlling factor is related to the electronic one by   =2s  1.The similarity between the IQHE and the FQHE natu-rally raises the question whether a phenomenon analogousto the reentrant IQHE may occur for CFs when a higherCF LL (p  1) is partially ﬁlled. In this case, residualinteractions between CFs may lead to the formation ofCF-solid phases, such as stripes and bubbles [4], or toincompressible liquids, which may be interpreted in termsof a second generation of CFs (C2Fs) [5–7]. The existenceof an incompressible CF liquid is supported by the re-cent observation of a FQHE at   4=11, which corre-sponds to a CF ﬁlling factor   1 1=3, by Pan et al.[7]. Such a state had been conjectured before by Mani andv. Klitzing based on arguments concerning the self-similarity of the Hall-resistance curve [8].In this Letter, we investigate several spin-polarized CFsolid and liquid states in a recently developed model ofinteracting CFs in a partially ﬁlled CF LL [5], whichincorporates the self-similarity of the FQHE [8] by aniterative projection of the dynamics to a single CF LL.The form of the CF interaction has been derived in theframework of the Hamiltonian theory, proposed byMurthy and Shankar [9]. The basic assumptions are thatthe degenerate CF LLs remain stable in the presence ofthe residual CF interactions and that the low-energy de-grees of freedom are described by intralevel excitations,in agreement with complementary studies in the wave-function approach [4,10,11]. The energies of the compet-ing phases are calculated directly in the thermodynamiclimit. In contrast to prior investigations by Lee et al. [4],our energy calculations suggest the possibility of incom-pressible Laughlin-type liquid states of CFs. Such stateshave a lower energy than CF solids at certain ﬁllingfactors also in higher CF LLs (p  1), e.g., at  4=11; 6=17; 4=19, and 11=27. An alternation betweenthese liquid and CF-solid phases with varying CF ﬁllingoccurs in our energy studies and may lead to observablereentrance phenomena in the FQHE regime.The model Hamiltonian of CFs, the dynamics of whichare restricted to the pth CF LL, is given by [5]H^s; p  12AXqvCFs;pq CFq CFq; (1)with the measurePq  ARd2q=22 in terms of thetotal area A and the CF-interaction potential (lB 	 1)vCFs;pq  2e2qeq2l2B =2Lpq2l2B c22 c2eq2=2c2Lpq2l2B2c22; (2)where c2  2ps=2ps 1 is the vortex charge,Published in "Physical Review Letters 93: 216802, 2004"which should be cited to refer to this work. 1http://doc.rero.chlB  1=1 c2pis the CF magnetic length, and Lpxdenotes a Laguerre polynomial [9]. The potential takesinto account the form of CFs in the pth CF LL. Theprojected CF-density operators satisfy the Girvin-MacDonald-Platzman algebra [12],  CFq; CFk 2i sinq kzl2B =2 CFq k, if one replaces the elec-tronic by the CF magnetic length. This model of interact-ing CFs is similar to the one of interacting electrons in asingle LL, and this property allows for the use of thestandard theoretical approaches to describe the differentphases. Note that we consider an ideal two-dimensionalelectron gas. Finite width and impurity effects are nottaken into account. In contrast to Ref. [5], where theactivation gaps of C2F states have been discussed, weneglect inter-CF-LL mixing, which may in principle beincluded in a dielectric function and which screens theCF-interaction potential [5].The energies of the different CF-solid phases are cal-culated in the Hartree-Fock approximation (HFA), inanalogy with the electron-solid phases in higher LLs[2,13,14]. We concentrate on triangular bubble crystalswith M CFs per bubble and CF-stripe phases, whichbecome relevant at half-ﬁlled levels. The CF WC isformally treated as a bubble crystal with M  1. TheHamiltonian (1) may be rewritten in the HFA,H^HFs; p  12AXquHFs;pqh CFqi CFq;with the Hartree-Fock potentialuHFs;pq  vCFs;pq  1nBAXpvCFs;ppeipxqypyqxl2B ;where nB  1=2l2B is the renormalized ﬂux density.The second term takes into account quantum-mechanicalexchange effects. The average h CFqi plays the roleof an order parameter, q  h CFqi=nBA Rd2r reiqr=A, and is related to the local CF ﬁllingfactor r of the pth CF LL by Fourier transformation.In terms of the order parameter, the cohesive energyEcoh  hH^HFs; pi= nBA of the CF-solid phases, withthe partial CF ﬁlling factor   A1 R d2r r, isECFsolidcoh s; p;  nB2 XquHFs;pqjqj2: (3)The order parameter for the CF-bubble crystal is given interms of the Bessel function J1x,Bq 22MplBAqJ1q2MplBXjeiqRj ;where Rj are the lattice vectors of the triangular latticewith a lattice spacing B  4M=3p1=2lB. One thusobtains for the cohesive energy of the CF-bubble crystalECFBcoh s; p;M;  nB MXGl0uHFs;pGl J212Mp jGljlBjGlj2l2B;(4)where Gl are the vectors of the reciprocal lattice. Theorder parameter of the CF-stripe phase,Sq 2Lxqy;0sinqxS =2qxXjeiqxjS ;with the system extension Lx in the x direction, yields thecohesive energyECFScoh s;p;S; nB22 Xj0uHFs;pq2Sjsin2 jj2:(5)The stripe periodicity S is a variational parameter withrespect to which the energy is minimized. It scales withthe CF cyclotron radius RC  lB2p 1p , and one ﬁndsSs  1; p  1  1:95RC, Ss  1; p  2  1:8RC,and Ss  2; p  1  1:9RC for the optimal stripe pe-riodicity at   1=2.The cohesive energy of the CF solids has to be com-pared to the energy of Laughlin-type quantum liquids,which may occur around the ‘‘magical’’ ﬁlling factors  1=2~s 1, with integral ~s. Because of their strongcorrelations, these liquid phases cannot be treated in theHFA, and one has to use Laughlin’s wave functions [15],generalized to an arbitrary LL [16]. Their cohesive energyis given byELcohs; p; ~s X1m0c~s2m1V2m1s; p; (6)with Haldane’s pseudopotentials [17] V2m1s; p 2=APqvCFs;pqL2m1q2l2B  expq2l2B =2, and the ex-pansion coefﬁcients c~s2m1 characterize the wave function.In contrast to Ref. [4], where a few number of pseudopo-tentials which have been determined numerically wereused to construct a CF-interaction potential, here they areobtained to arbitrary order from the analytical expressionof the CF-interaction potential (2). As pointed out by theauthors of Ref. [4], the construction of an interactionpotential from pseudopotentials is not unique. The expan-sion coefﬁcients c~s2m1 are derived from sum rules[12,18], which are considered as a set of linear equations[19]. This method yields results deviating less than 1%from numerical studies and is thus sufﬁciently accuratefor the present investigations.Away from the magical ﬁlling factors, the energy of theCF-liquid phases is raised by the excited quasiparticles[for  > 1=2~s 1] or quasiholes [for  < 1=2~s 1],which are separated by a gap from the incompressibleliquid state. They may be interpreted as C2Fs or C2F holespromoted to the next higher C2F level [5,6]. Their energyis calculated analytically in the framework of the2http://doc.rero.chHamiltonian theory [9]. One ﬁndsqps;p~s; ~p  12AXqvCFs;pqh~pj q qj~pi  1AXqvCFs;pqX~p1j00jh~pj qjj0ij2for the quasiparticle energies andqhs;p~s; ~p   12AXqvCFs;pqh~p 1j q qj~p 1i  1AXqvCFs;pqX~p1j00jh~p 1j qjj0ij2for the quasihole energies, where q denotes the one-CF density operator, and the matrix elements (j  j0) aregiven byhjj qjj0i j0!j!s iqx  iqy~l ~c2pjj0eq2~l2~c2=4Ljj0j0q2~l2~c22 ~c21jj0eq2=2~c2Ljj0j0q2~l22~c2;with the C2F magnetic length ~l  lB=1 ~c2p, in terms of the C2F-vortex charge ~c2  2~p ~s =2~p ~s1 [5]. Here, we areinterested only in states from the Laughlin series (~p  1), and one ﬁndsqp~s  1 qp~s  2 qh~s  1 qh~s  2s  1; p  1 0.04172 0.030 65 0:015 67 0:013 09s  1; p  2 0.023 30 0.019 58 0:012 77 0:009 67s  2; p  1 0.017 27 0.013 91 0:008 59 0:006 72in units of e2=lB. The cohesive energy of the CF-liquidphases thus becomesECFliquidcoh s; p; ~s;   ELcohs; p; ~s   2~s 1  1 qp=qhs;p ~s; ~p  1; (7)where the residual C2F interactions have been neglected.This approximation is valid in the vicinity of the magicalﬁlling factors, i.e., at low C2F density. The completeenergy curve for the quantum-liquid phases would re-quire a full account of these interactions, which is beyondthe scope of the present studies.The results for the cohesive energies of the different CFphases, given by Eqs. (4), (5), and (7), are shown in Fig. 1for three speciﬁc CF LLs. Figure 1(a) shows the results forCFs carrying two ﬂux quanta (s  1) in the ﬁrst excitedCF LL (p  1). The discussion is limited to the CF ﬁllingfactor range 0< <1=2, which corresponds to a range ofthe electronic ﬁlling factor 1=3<<3=8. In analogywith the electronic case, the range 1=2<  < 1 is relatedto the shown part by the particle-hole symmetry. Our en-ergy calculations suggest that a spin-polarized Laughlin-type quantum liquid of CFs has a lower energy than thecompeting CF-solid phases around   1=3 and 1=5,which correspond to   4=11 and 6=17, respectively. Es-pecially the 4=11 state is found to be largely favored, asexpected from the form of the pseudopotentials; one ﬁndsthat V3s  p  1 is smaller than its neighboring pseu-dopotentials [5]. This stabilizes the ~s  1 Laughlin state,which has its largest weight at the corresponding angularmomentum and which screens V1 completely [17].Indeed, a spin-polarized FQHE at 4=11 has beenobserved by Pan et al. [7]. The CF WC becomes stableat lower densities,  & 0:15, as well as at larger values, *0:4. At   1=2, the CF-stripe phase has the lowestenergy, which may lead to the anisotropic longitudinalresistance at   13=8, recently observed by Fischer et al.[20]. Contrary to the electronic case in the ﬁrst excited LL[2], a CF-bubble crystal with two CFs (M  2) per latticesite never has a lower energy than the CF WC (M  1).Although screening of the CF-interaction potential af-fects the activation gaps [5] and may modify the transi-tion points between the different phases, the investigatedreentrance phenomenon persists in 1=3< < 3=8. Theeffect of screening, ﬁnite sample width, and impurities onthese phases will be discussed elsewhere [21].For s  1 and p  2 [Fig. 1(b)], i.e., in the range 2=5<< 5=12, the CF quantum liquid ceases to be the groundstate at   1=3, where the CF WC has the lowestenergy. However, an incompressible CF liquid is foundto be stable at   1=5. In contrast to the CF LL p  1, aCF-bubble crystal with M  2 has a lower energy thanthe CF WC around half ﬁlling, but a CF-stripe phase isfound to be the ground state at   1=2. For 1=5< <3=14, which corresponds to p  1 for CFs carrying fourﬂux quanta (s  2) [Fig. 1(c)], a   1=3 Laughlin stateof CFs is the lowest-energy state at   4=19, whereas theliquid state is extremely close in energy to the CF WC at  1=5. In the presence of impurities, which lower theenergy of the solid phases more signiﬁcantly than thequantum-liquid energies [2], a   1=5 Laughlin-typestate might vanish. Impurities may also affect the 6=17state. In all cases, an insulating CF WC, which is thenatural ground state at lower densities, has a lower energyalso at rather large ﬁlling factors,  * 0:4. This may leadto reentrance phenomena, which would be the CF ana-logue of the reentrant IQHE observed at lower magneticﬁelds [1].3http://doc.rero.chThe stability of C2F states has been a controversialissue during the last decade. Numerical investigationsin the Haldane-Halperin hierarchy [17,22] by Be´ran andMorf denied the stability of a spin-polarized 4=11 state,whereas they found a small gap in the absence of com-plete spin polarization [23]. Their results have been con-ﬁrmed by Wo´js and Quinn [24] and by numerical-diagonalization studies in the CF wave-function approachby Mandal and Jain [10] and Chang et al. [25]. However,more recent studies by Chang and Jain [11], which hint toa stable 4=11 state, contradict their previous results. Thesize of the diagonalized system remains too small toallow for a conclusive answer of its stability in the ther-modynamic limit.In conclusion, we have calculated the energies of com-peting CF solid and liquid phases in a recently developedmodel of interacting CFs [5]. Incompressible quantumliquids of spin-polarized CFs, which may be interpretedin terms of C2F [5,6], are found to be stable at  4=11; 6=17; 4=19, and 11=27. Around half ﬁlling of thetopmost CF LLs, a CF-stripe phase has the lowest energy,but may compete with a Pfafﬁan state [26], which hasbeen omitted in the present studies. The fact that insulat-ing CF-bubble crystals occur at different ﬁlling factors,which surround the CF-liquid phases, may lead to anexperimentally observable reentrance of the FQHE inhigh-quality samples, in analogy to the reentrant IQHEin the ﬁrst excited electronic LL [1].We acknowledge fruitful discussions with R. Mani,R. Moessner, R. Morf, and A. Wo´js. This work was sup-ported by the Swiss National Foundation for ScientiﬁcResearch under Grant No. 620-62868.00.[1] J. P. Eisenstein et al., Phys. Rev. Lett. 88, 076801 (2002).[2] M. O. Goerbig, P. Lederer, and C. Morais Smith, Phys.Rev. B 68, 241302 (2003); 69, 115327 (2004).[3] J. K. Jain, Phys. Rev. Lett. 63, 199 (1989); Phys. Rev. B41, 7653 (1990).[4] S.-Y. Lee et al., Phys. Rev. Lett. 87, 256803 (2001); Phys.Rev. B 66, 085336 (2002).[5] M. O. Goerbig, P. Lederer, and C. Morais Smith, Phys.Rev. B 69, 155324 (2004); Europhys. Lett. 68, 72 (2004).[6] A. Lopez and E. Fradkin, Phys. Rev. B 69, 155322(2004).[7] W. Pan et al., Phys. Rev. Lett. 90, 016801 (2003).[8] R. G. Mani and K. v. Klitzing, Z. Phys. B 100, 635 (1996).[9] G. Murthy and R. Shankar, Rev. Mod. Phys. 75, 1101(2003); R. Shankar, Phys. Rev. B 63, 085322 (2001).[10] S. S. Mandal and J. K. Jain, Phys. Rev. B 66, 155302(2002).[11] C.-C. Chang and J. K. Jain, Phys. Rev. Lett. 92, 196806(2004).[12] S. M. Girvin, A. H. MacDonald, and P. M. Platzman,Phys. Rev. B 33, 2481 (1986).[13] A. A. Koulakov et al., Phys. Rev. Lett. 76, 499 (1996);M. M. Fogler et al., Phys. Rev. B 54, 1853 (1996).[14] R. Moessner and J. T. Chalker, Phys. Rev. B 54, 5006(1996).[15] R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983).[16] A. H. MacDonald, Phys. Rev. B 30, 3550 (1984).[17] F. D. Haldane, Phys. Rev. Lett. 51, 605 (1983).[18] S. M. Girvin, Phys. Rev. B 30, 558 (1984).[19] M. O. Goerbig and C. Morais Smith, Phys. Rev. B 66,241101 (2002).[20] F. Fischer et al., Physica (Amsterdam) 22E, 108 (2004).[21] M. O. Goerbig, P. Lederer, and C. Morais Smith(unpublished).[22] B. I. Halperin, Phys. Rev. Lett. 52, 1583 (1984).[23] P. Be´ran and R. Morf, Phys. Rev. B 43, 12 654 (1991).[24] A. Wo´js and J. J. Quinn, Phys. Rev. B 61, 2846 (2000).[25] C.-C. Chang et al., Phys. Rev. B 67, 121305 (2003).[26] G. Moore and N. Read, Nucl. Phys. B360, 362 (1991);M. Greiter et al., Phys. Rev. Lett. 66, 3205 (1991); Nucl.Phys. B374, 567 (1992).5 10 15 20 25-0.02-0.015-0.01-0.0050.0050.20.1 0.3 0.4 0.51/3 3/86/17 4/11electronic filling factorM=2M=1quantum liquid (C  F) CF stripe2− 0.− 0.02cohesive energy1/5 1/3s=1p=1partial CF filling factor(a)5 10 15 20 25-0.004-0.003-0.002-0.0010.0010.1 0.2 0.3 0.4 0.511/27 7/17 5/12− .001− 002− .003− 004partial CF filling factorcohesive energy M=1M=2quantum liquid (C  F)CFstripe22/5electronic filling factors=1p=2 1/5 1/3(b)5 10 15 20 25-0.005-0.004-0.003-0.002-0.0010.0012−0.006− 0. 04−0. 20.50.40.30.20.14/196/29 3/141/5CFstripe1/5 1/3s=2p=1electronic filling factor(c)quantum liquid (C  F)partial CF filling factorFIG. 1. Cohesive energies of the different CF phases for(a) s  1, p  1, (b) s  1, p  2, and (c) s  2, p  1 inunits of e2=lB.4

Possible reentrance of the fractional quantum Hall effect in the lowest Landau level

Possible Reentrance of the Fractional Quantum Hall Effect in the Lowest
  Landau Level

Possible Reentrance of the Fractional Quantum Hall Effect in the Lowest Landau Level

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