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arXiv:cond-mat/0010071v2

[cond-mat.mes-hall]5Dec2000

Comment on Cooper instability of composite fermions

N. ReadDepartment of Physics, Yale University, P.O. Box 208120, New Haven, CT 06520-8120

(October 4, 2000)

We comment on the recent paper by Scarola, Park, and Jain [Nature 406, 863 (2000)] on a trialwavefunction calculation of pairing in a fractional quantum Hall system at = 5/2. We point outtwo errors that invalidate the claimed calculations of a binding energy for Cooper pairs and of an

energy gap for charged excitations.

The old problem of the incompressible fractional quan-tum Hall state observed at filling factor = 5/2 [1] hasreceived renewed attention recently. In particular, a pa-per by Scarola, Park, and Jain (SPJ) [2] presents a nu-merical calculation based on trial wavefunctions whichfinds that composite fermions [3,4] bind to form Cooperpairs at half-filling of the first excited (or N = 1) Lan-dau level ( = 5/2), but not at half-filling of the lowest(N= 0) Landau level (= 1/2).

First, earlier work on this problem along similar linesshould be properly pointed out. The idea that composite

(or neutral) fermions at even denominator filling factors(such as 1/2, 5/2) can form an incompressible state bypairing as in Bardeen-Cooper-Schrieffer (BCS) theory [5]was pointed out in Ref. [6]. The existing spin-singletHaldane-Rezayi state [7] was interpreted in this way, anda spin-polarized state, termed the Pfaffian state, was con-structed as another example [6]. Later, Greiter et al. ex-plored this idea further, and suggested that the Pfaffianstate might apply to = 5/2 [8]. The incompressibil-ity of the paired state may be rephrased by invoking theMeissner effect, see for example Ref. [9]. This is ofcourse a composite boson type of explanation for in-compressibility [1013].

In the early days, mainly because of Ref. [14], it was as-sumed in most theoretical work (an exception being thesuggestion in Ref. [8]) that the N = 1 Landau level isspin-unpolarized at = 5/2including the earlier work,Ref. [15]. Recently, convincing evidence that the N= 1Landau level is polarized in these states, even when theZeeman energy is turned off, has been presented in diago-nalization studies [16]. This is assumed without commentin SPJ. The recent, very extensive, numerical studies findthat the ground state at 5/2, unlike that at 1/2, is an in-compressible state with the quantum numbers of, and alarge overlap with, the Pfaffian state (or its projection toa particle-hole symmetric state), and that phase transi-tions to compressible states occur when the interactionis modified away from the Coulomb interaction [16,17].Thus, the evidence already favored a paired state of com-posite fermions at = 5/2, whereas for = 1/2, similarstudies showed Fermi-liquid-like behavior [20,17], consis-tent with Ref. [4].

Further, for = 5/2, Morf [16] obtained a gapfor charged excitations (well-separated quasiparticle andquasihole) of = 0.025 in units of e2/l0; SPJ find a

number 5 times smaller, but do not make the compari-son with Morfs result. Experimental results [18,19] aresmaller still, about a factor of 5 smaller than the SPJresult in the case of Ref. [19].

Technically, SPJ use trial wavefunctions for N elec-trons on the sphere at magnetic flux N = 2(N 1),consisting of Slater determinants of spherical harmonics(representing fermions in zero net magnetic field) times aLaughlin-Jastrow factor, projected to the lowest Landaulevel, to represent ground states at = 1/2 (the = 5/2problem can be mapped to this, provided the Coulomb

interaction appropriate to the N = 1 Landau level isused as the Hamiltonian). The first use of these func-tions for the = 1/2 problem at this sequence of systemsizes was in Ref. [20]. In a comment, Jain claimed thatthe = 1/2 state (with N= 0 Coulomb interaction) inthe thermodynamic limit could not be approached in thisway [21]. Also, the consistency of the angular momentumof the ground states for each N with Hunds rule was aresult in Ref. [20]; this was dismissed as wild fluctua-tions [21], yet all these results are now accepted in SPJ,without citing the earlier discussion. (The correspond-ing energies were used to estimate the composite fermioneffective mass at 1/2 by Morf and dAmbrumenil, who

also examined = 5/2 [22].) We do not believe that, forthe Fermi-liquid state, the thermodynamic limit must beapproached along the subsequence N = n2 (where filledshells occur [20]) as SPJ suggest. The reason is thatthe fluctuation in the total angular momentum of theground state caused by partially-filled shells of compos-ite fermions is relatively small (only order

N particles

are involved), and correlation functions of local operatorsat fixed separations will be insensitive to this fluctuationas N .

Now we turn to more substantive errors in SPJ. Thefirst concerns the order of magnitude of an interactionmatrix element. The trial wavefunctions are supposedto represent two fermions in the n + 1th shell, each ofangular momentum l = n, and the lower filled shellsform a Fermi sea [20]. The distinct states can be la-beled by their total angular momentum L, which is 1, 3,. . . , 2n 1. SPJ interpret the difference in energy EL(calculated by Monte Carlo) between the L = 2n 1 andL = 1 trial states as an interaction energy for the twoadded fermions. When E1 E2n1 is negative, the pairare trying to form a Cooper pair. We will consider this

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model problem of two fermions outside a Fermi sea (inzero magnetic field) on a sphere, where they interact onlywith each other, not with the Fermi sea (similar to theCooper problem [23]). In the model, all the states ofthe two fermions each with angular momentum n are de-generate, in the absence of the interaction, and are splitfrom other l values by an effective kinetic energy.

The problem arises because each L eigenstate of two

fermions, each with angular momentum n, is a linearcombination of only of order

N states (since N =

n2 + 2). Because the spherical harmonics of angularmomentum n are extended over the sphere, this is toofew to obtain a bound state wavefunction of fixed sizeas N . The smallest wavepacket that can be con-structed covers an area of order

N in the relative coor-

dinate (the unit length is the magnetic length of the orig-inal quantum Hall problem, and the system area 4R2

is of order N). A true bound state would have a size oforder one; for this, order N basis states are needed.

Related to this, the expectation value of the interac-tion energy of two particles in such a state built fromonly order N basis states is of order 1/N. For anyphysically-sensible interaction, including long-range in-teractions such as the Coulomb interaction, the matrix el-ements of the interaction between normalized, extended,basis states of fixed single-particle linear momentum, areof order 1/N (or 1/R2). Here we take the N limitat fixed linear momenta of the in- and out-going parti-cles; for the sphere, linear momentum k (as in the plane)corresponds to l/R. For l of order

N R (to main-

tain fixed density of particles), this is of order one, as weexpect since it is of order the Fermi wavevector. This be-havior of the matrix elements is familiar for plane waves

and periodic boundary conditions in flat space (see e.g.Ref. [24] for the three-dimensional analog), but is alsotrue for other cases including the sphere. When thesematrix elements are used to calculate the expectation ofthe interaction for two particles each of angular momen-tum l = n in the state of total angular momentum L = 1(or any other value up to the limit 2n 1), the resultis of order 1/

N, because there are order N diagonal

and off-diagonal terms, and we divide byN to normal-

ize the state. (The Clebsch-Gordan coefficients used informing the L = 1 state scale uniformly as N1/4, as onecan see from explicit expressions [25].) The same depen-dence can be obtained easily on the plane by restricting

each particle to a set of N single-particle states k justoutside the Fermi wave vector, and pairing k with k.Order N basis states would be needed to obtain an

expectation value of the interaction of order 1. Since, inthe model, such states are not degenerate in the absenceof interaction, and the ground state is not determinedsolely by symmetry as it was before, a correct calculationthen involves solving the Schrodinger equation, as firstdone by Cooper [23] (and easily adapted to the presentcase). If it is claimed that, contrary to our analysis, the

expectation of the interaction in the state in the smallerbasis set is independent of system size, then this givesa contradiction with Coopers result for weak coupling,which was that the binding energy is exponentially smallfor weak coupling, not linear in the coupling as it is inthe smaller basis set.

SPJ take their numerical data and plot it versus 1/N,using a linear extrapolation to a nonzero value. We find

that, within the interpretation used, this is inappropri-ate, since as an interaction matrix element it should ap-proach zero as 1/

N. This is the first error. In the plots

of SPJ, the data does appear linear in 1/N. This mustmean either that it will eventually turn over to 1/

N,

or that the trial states should not be interpreted as twofermions just outside the Fermi sea as in Refs. [20,2] andthe analysis here.

The second error comes when SPJ compare theirbinding energy with the gap for charged excitationsin the quantized Hall system at = 5/2. We leave asidethe obvious question of whether such a binding energywould give accurately the energy gap in the many-particleground state. More importantly, the fermions obtainedby breaking a pair are neutral [6]; this is true even inthe Fermi liquid state without pairing [9,2629]. Thecharged excitations are vortices which, because of pair-ing, effectively contain a half quantum of flux each, andso have charge 1/4 (in electronic units) at = 5/2 [6].There appears to be no expected relation between the ex-citation energy gaps for these two very different types ofexcitations. This again invalidates the comparison withexperimental results.

Another way to understand this point is in terms ofwhat change must be made in the ground-state quan-

tum numbers to create certain excitations. SPJ addtwo particles to a filled shell, staying on the sequenceN = 2(N 1), and this means their excitation, whena pair is broken, is neutral. Of course, such a state maycontain charged particles of opposite charges, but thisis not obvious. For this reason, charged excitations areusually obtained by adding or subtracting flux or parti-cles so as to leave a given sequence. This is a well-knowntechnique, used for example by Morf to obtain a charge-excitation gap for the 5/2 state [16]. Since the elemen-tary charged excitations contain a half flux quantum inthe present case, Morf divides the number obtained thatway by 2, after subtracting the interaction energy for two

1/4 charges and extrapolating to the thermodynamiclimit, to obtain his result 0.025.In spite of these errors, the calculation by SPJ is still

an interesting trial wavefunction calculation of somethinglike an interaction matrix element, which is attractive inthe N = 1, but not in the N = 0, Landau level, a factconsistent with earlier results about the different spectrain these two cases [16,17,20,22].

I thank R. Morf for helpful correspondence. My re-search is supported by the NSF, under grant no. DMR-

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