Particle-Hole Symmetry and the Fractional Quantum Hall States at 5/2 Filling Factor

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by  Jian Yang

Particle-Hole Symmetry and the Fractional Quantum Hall States at 5/2 Filling Factor
Jian Yang,∗
Weproposeaderivativeoperatorformedasafunctionofderivativesofelectroncoodinatesdefined
by Dm = Pf((∂∂zi−1∂∂zj)m)QNi<j(∂∂zi − ∂∂zj)m, with zj being the complex coordinate of the jth
electron, N the total number of electrons, m a positive integer, and Pf[A] is the Pfaffian of an
antisymmetric matrix A. When applied to the Laughlin wave function ΦmL =QNi<j(zi−zj)mL, a
new wave function Ψm,mL = DmΦmL in the lowest Landau level at filling factor ν = 1/(mL−m)
is generated. In spherical geometry, the relationship between the total magnetic flux number N
7 φ
1 and N is Nφ =(mL−m)N +(2m−mL). Two wave functions Ψ3,5 and Ψ1,3 with special sets of
0 values(m,mL)=(3,5)and(m,mL)=(1,3),areofparticularinterestastheybothcorrespondtoa
2 half-filledLandaulevelandarerelevanttothe5/2quantumHalleffect. ThefirstwavefunctionΨ3,5
hastheNφ andN relationship Nφ =2N+1,andthesecond wavefunction Ψ1,3 hasNφ =2N−1.
n
For systems of 4, 6, and 8 electrons in spherical geometry, it is shown that the first wave function
a
J Ψ3,5 has nearly unity overlap with the particle-hole conjugate of the Moore-Read Pfaffian wave
function, therefore together with the Moore-Read Pfaffian state forms a particle-hole conjugate
3
pair. The second wave function Ψ1,3 has essentially perfect particle-hole symmetry itself, with a
1
positiveparity whenthenumberofelectron pairs N/2is an evenintegerand anda negativeparity

] when N/2 is an odd integer. An equivalent form suggests the first wave function Ψ3,5 forms a
l f-wave pairing of composite fermions, and the second wave function Ψ1,3 forms a p-wave pairing.
e
The corresponding Non-Abelian statistics quasiparticle wave functions are also proposed.
-
r
t PACSnumbers: 73.43.Cd,71.10.Pm
s
.
t
a
m It has been thirty years since the observation of the to the 5/2 quantum Hall effect does have PH symmetry,
fractional quantum Hall effect at ν = 5/2[1], which and is identified as a symmetrized superposition of the
-
d deviates from the odd denominator filling factor rule. Pfaffian state and the anti-Pfaffian state (see also [9]).
n The Moore-Read Pfaffian state (termed the Pfaffian A possibility of spontanouse PH symmetry breaking is
o
state in the literature)[2] is considered to be the lead- also studied and ruled out for the Coulomb interaction
c
[ ing candidate for the ground state, and has been stud- even when the finite-thickness effects are included[10].

ied extensively in particular through finite size numer- Recently, a Dirac composite fermion effective field the-
1
ical calculations[4][5][6]. It is shown by adjusting the oryhasbeenproposedtodescribethelowenergyphysics
v
2 pseudopotentials [4] or fine tuning the finite thickness of where the PH symmetry is explicitly realized[11].
6 the layer that confined the two dimensional electrons[6], Althoughmuchprogresshasbeenmade, therearetwo
5 the groundstateoftheresultingColumbHamiltonianat important questions remain unanswered. In the case of
3
thesecondLandaulevelcanachievenearlyunity overlap thePHsymmetrybreaking,ifwebelievethePfaffinwave
0
with the wave function of the Pfaffian state. function provides a good description for one of the two
.
1 degenate states that are PHconjugated with eachother,
It is known that the Pfaffian state is the exact ground
0
what would the wave function look like for the other
7 state of a three-body interaction Hamiltonian[3], and
state? In the case of the PH symmetry is preservered,
1 therefore does not have particle-hole (PH) symmetry.
v: Ontheotherhand,thetwo-bodyinteractionHamitonian can we construct a wave function that is PH symmetric
in the first place and has a large overlap with the exact
i projectedintoasingleLandaulevelisinvariantbyanan-
X ground state as well as the correct parity. We wish to
tiunitary PH transformation, which requires its ground
r make contributions to the answers to both questions in
state to be PH symmetric. This apparent paradox can
a
this paper.
be resolved in two ways. One is to invoke a PH sym-
We begin by defining a derivative operator D in the
metry breaking mechanism such as Landau level mix- m
planar geometry as follows:
ing, which lifts the degeneracybetween the groundstate
described by the Pfaffian state and the other degener-
N
1 ∂ ∂
aotfethgreouPnfadffistaantestdaetsec,ritbeerdmbedy tahnetid-PisftainfficatnPsHtactoenijnugtahtee Dm =Pf((∂∂zi − ∂∂zj)m)Yi<j(∂zi − ∂zj)m (1)
literature, such that one of them becomes the ground
state[7][8]. The other resolution is the PH symmetry is where z = x + iy is the complex coordinate of the
j j j
preserved, and neither the Pfaffian state nor the anti- j electron, N is the total number of electrons, m is a
th
Pfaffian state provides a good description of the ground positiveinteger,andPf[A]isthePfaffianofanantisym-
state. RezayiandHaldane[5]intorusgeometryprovided metric matrix A. When applied to the Laughlin wave
evidencethatthenumericalgroundstatethatcorrsponds function the following new wave function Ψ is gen-
m,mL
2
erated: sameastheN andN relationshipforthePfaffianstate,
φ
which is N = 2N − 3. In other words, for the same
φ
Ψm,mL =DmΦmL (2) flux number Nφ, the number of holes of the ΨQ3,H5 state
is the same as the number of electrons of the Pfaffian
with the Laughlin wave function Φ defined by[12]
mL state. By the sametoken, the number ofelectronsof the
N |z |2 Ψ3,5 state is the same as the number of holes of the ant-
ΦmL =Y(zi−zj)mLexp(−X 4lj2 ) (3) Pfaffianstate. Encourgedbythis fact,wehaveexplicitly
i<j j B expanded the wave function Ψ3,5 and represented it in
terms of the many-body basis functions formed of the
where m >m being anoddinteger for fermions andan
L SlaterdeterminantofN singleparticlewavefunctions in
evenintegerforbosons,andl isthemagneticlength. It
B the lowestLandau level for the small number of electron
isnotedthattheproductofthePfaffianandtheJastrow
systems of N = 4 and N = 6. This representation, as
functionofthederivativesinEq.(1)shouldbecarriedout
will be illustrated in detail later, makes it very easy to
first before applying to the Laughlin wave function, this
apply the PH transformation to obtain the explicit form
way the derivatives appeared in the denominator in the of ΨQH, and to calculate the overlap between ΨQH and
Pfaffian will be cancelled out. It should also be noted 3,5 3,5
the Pfaffian state. The overlapwith the Pfaffian state is
that the D does not apply to the exponent factor of
m 0.9997 for N =6, and 0.9962 for N =8.
the Laughlin wave function exp(− |zj|2). It is clear
Pj 4l2B Thisnearlyunityoverlapprovidesanevidencethatthe
the wavefunction Ψ has the same symmetry as the
m,mL ΨQH state and the Pfaffian state, or the Ψ state and
Laughlin wave function, which is antisymmetric with re- 3,5 3,5
the anti-Pfaffian state, form PH conjugated pair states
specttothecoordinateswhenm isanoddinterger,and
L witheachother. Betweenthe Pfaffianstate andtheΨ
symmetric when m is an even interger. 3,5
L state, if one provides a good description for the ground
In Haldane’s spherical geometry[13], the D operator
m state of the two body interaction, so does the PH conju-
can be written as:
gate state ofthe other. In Fig. 1, we show the overlapof
N the exact ground state of the Coulomb interaction with
1 ∂ ∂ ∂ ∂
Dm =Pf((∂∂ui∂∂vj − ∂∂uj ∂∂vi)m)Yi<j(∂ui∂vj−∂uj ∂vi)m ttihoesPoffaffiV1a/nVs1ctartaenagnindgthfreoΨmP3,1H5 tsota1te.2r,eswpheecrteiveVl1yc fiosrtrhae-
(4) Coulomb value of V in the second Landau level. Over-
1
and the Laughlin wave function can be written as all, both states provide rather good description of the
exact ground state. When V has the value for Coulomb
N 1
ΦmL =Y(uivj −ujvi)mL (5) ifinatnerastcattioen. VA1cs,VΨP3,iH5ncrheaasselsa,rgthereoovveerrllaapptfhoarnboththe Pwfaavfe-
i<j 1
functions increases until the overlap with ΨPH reaches
3,5
where (u,v) are the spinor variables describing electron its maximum value around V /Vc =1.1 and the overlap
1 1
coordinates. Since the totalflux number Nφ correspond- withthePfaffianstatereachesitsmaximumvaluearound
ing to the Laughlin wave function is Nφ = mL(N −1), V1/V1c =1.12.
and the derivative operator D decreases the flux num-
m Now we turn our attention to the wave function Ψ .
ber bym(N−2),the relationshipbetweenthe flux num- 1,3
According to Eq.(6), the wave function Ψ has the N
ber N and the number of electrons N is given by the 1,3 φ
φ and N relationship N = 2N −1. Since the number of
following equation: φ
electrons N is related to the numeber of holes N of the
h
Nφ =(mL−m)N +(2m−mL) (6) PH conjugated state by N +Nh = Nφ+1, the number
of electrons of Ψ is the same as the number of holes
1,3
for the wave function Ψm,mL. In the thermodynamic of the PH conjugate state of Ψ1,3, represented by ΨQ1,H3
limit, this corresponds to filling factor ν =1/(mL−m). hereafter. Now the question is if the wave function Ψ1,3
In the following, we will focus on the two wave func- has the PH symmetry? To answer the question, as be-
tions Ψ3,5 andΨ1,3 describedby Eq.(2) with specialsets foreweexplicitlyexpandthewavefunctionΨ1,3 interms
of values (m,mL) = (3,5) and (m,mL) = (1,3). Both of the many-body basis functions formed of the Slater
wave functions correspond to filling factor ν =1/2. determinant of N single particle wave functions in the
According to Eq.(6), the wave function Ψ3,5 has the lowestLandaulevelforsmallnumberofelectronsystems
Nφ and N relationship Nφ =2N +1. Since the number N = 4, N = 6, and N = 8. We then construct the PH
of electrons N is related to the numeber of holes Nh of conjugatewavefunction ΨP1,H3 byapplyingthe PHtrans-
the PH conjugate state by N +Nh = Nφ +1, the re- formation, and calculate the overlap between Ψ1,3 and
lationship between the flux number and the number of ΨPH. To illustrate how this is done, we use the system
1,3
holes of the PH conjugate state of Ψ3,5, represented by N =4 as an example, as this is the simplest system and
ΨQH hereafter, is N = 2N − 3. This is exactly the the wave function Ψ is shown to have the exact PH
3,5 φ h 1,3
3
If we change the index varible M −i+1 → i, we will
1 have ΨPH = C |i>. Since the coefficients as
1,3 Pi=1 M−i+1
shown in the first coulom in Table I satisfy the equation
0.99 C = C , we have ΨQH = C |i >= Ψ . In
M−i+1 i 1,3 Pi i 1,3
other words, Ψ has perfect PH symmetry with parity
0.98 1,3
equal to +1 for a system of 4 electrons.
ap0.97 The Pfaffian State
Overl0.96 Ψ3P,H5 C-1i |1>=||0i;>1;6;7> |1>PH|=i>|2P;H3;4;5>
1 |2>=|0;2;5;7> |2>PH=|1;3;4;6>
0.95 -1 |3>=|0;3;4;7> |3>PH=|1;2;5;6>
0 |4>=|0;3;5;6> |4>PH=|1;2;4;7>
0.94 0 |5>=|1;2;4;7> |5>PH=|0;3;5;6>
-1 |6>=|1;2;5;6> |6>PH=|0;3;4;7>
0.93
1 1.05 1.1 1.15 1.2 1 |7>=|1;3;4;6> |7>PH=|0;2;5;7>
V1/V1c -1 |8>=|2;3;4;5> |8>PH=|0;1;6;7>
FIG. 1: For N = 8 and Nφ = 13. Overlap of the exact TABLEI: Coefficients and thebasis functions (and theirPH
groundstatewiththePfaffian state(solidline) andtheΨP3,H5 conjugates) as defined in Eq.(7) for N =4 system.
state (dashed line) as the function of the pseudopotential V
1
normalized by its Coulomb value Vc in the second Landau
1
level.
0.4
Ψ
symmetry. For N = 4 system, the total flux number is 1,3
0.3 ΨPH
2N −1 = 7. In the spherical geometry, the single parti- 1,3
cle wavefunction in the lowest Landau Level is specified 0.2
bythequantumnumbermoftheangularmomentumL
ttnhaokatetasttiavokanelusceoNsnφvo+efn10ie,vn1a,cl2eu,,e.sw.oe.,fwN−iφlNl2φtuo,s−espNNe22φφci+f+y1m,t.h.ei.n,ssNit2neφga.ldeFwoprahrtihtcihez- malized coefficients 0.01
clewavefunctioninthefollowingdiscussion. TheHilbert Nor−0.1
spaceofdimensionM isspannedbyM orthogonalmany- −0.2
body basisfunctions formedofthe Slater determinantof
−0.3
N (inthisexampleN =4)singleparticlewavefunctions
|i >= |N2φ +m1;N2φ +m2;N2φ +m3;N2φ +m4 >, where −0.40 10 20 30 40 50 60
i = 1,2,...,M is used to index the M basis functions. Basis function index
Usingthebasisfunctionnotation,thewavefunctionΨ
1,3
can be written in the form FIG.2: Thecoefficients(cicles)ofthewavefunctionΨ1,3and
thecoefficients(squares)ofΨPH forN =6andN =11. The
M 1,3 φ
solid line and thedashed line are used to guide theeye.
Ψ1,3 =XCi|i>. (7)
i=1
For N = 6, there are total number of 58 basis func-
where C is the coefficient for the basis function |i >. tions. Again as described in Eq.(7) we expand the wave
i
Since the wavefunctionΨ is rotationallyinvariant,its function Ψ in terms of the basis functions |i > where
1,3 1,3
totalangularmomentumiszero. Thisrequires m =0 i = 1,2,...,58, and calculated the coefficients C . The
Pi i i
or equivalently Pi(N2φ + mi) = N(2N − 1)/2 = 14. normalized coefficients Ci are plotted as cicles with the
As the result, the total number of the basis functions solid line used to guide the eye in Fig. 2. The coeffi-
M = 8 for the 4 electron system. In Table I, we list cients of the ΨQ1,H3 are also plotted as squares with the
all the 8 basis functions and the corresponding coeffi- dashed line used to guide the eye in Fig. 2. It can be
cients (unnormalized). We also listed the PH conjugate seen from the Figure, at each basis function index, Ψ1,3
of each of the basis function |i >PH. As can be seen, and ΨQH have the coefficients that are essentially the
1,3
we number the many-body basis functions in an order same in magitude but in opposite sign, indicating that
suchthat|i>PH isrelatedto |i>bythe simplerelation Ψ has the PH symmtry and the parity is −1. In fact,
1,3
|i >PH= |M −i+1 >. In other words, |i >PH can be the overlap between Ψ and ΨQH is 0.9991 for N = 6.
1,3 1,3
obtainedfrom|i>byreversingtheorder. Therefore,the The same calculation is also carried out for N = 8, the
wavefunction ΨQH = C |i>QH= C |M−i+1>. overlapbetween Ψ and ΨQH is 0.9798,and the parity
1,3 Pi i Pi i 1,3 1,3
4
is+1. Theseresultsprovidethestrongevidencethatthe This equivalent form of the wave function suggests Ψ
1,3
following equation canbeinterpretedasap-wavepairingstateofcomposite
femions.
Ψ1,3 =(−1)N2ΨP1,H3 . (8) Thirdly, as for the Moore-ReadPfaffianstate, one can
modify the the Pfaffian in Eq.(1) to construct the quasi-
is essentially true.
hole wave function:
It should be pointed out that in spherical geometry
at N = 2N −1 the nature of the ground state in the
φ n 2n
second Landau level seems to be very much N depen- (z −ξ ) (z −ξ )+(i↔j)
Q i a Q j b
dent. While we are able to identify a reasonalbe range a=1 b=n+1
Pf( ) (12)
of values of V /Vc such that Ψ has nearly unity over- ( ∂ − ∂ )m
1 1 1,3 ∂zi ∂zj
lap with the exact incompressible ground state with the
correct parity for N = 4 and N = 6 systems, we have
for the total number of 2n quasiholes located in ξ and
not been able to find such a range of V /Vc such that a
1 1 ξ , where a = 1,2,...,n and b = n+1,n+2,...,2n.
b
the exact ground state is incompressible with the same
The quasiholes carry − 1 of the electron charge,
parity of Ψ1,3 for N = 8. It is therefore interesting to andobeynon-Abelianfra2(cmtiLo−nmal)statistics 1 . The
find out if Ψ1,3 can provide a good description (correct 2(mL−m)
parity andlargeoverlap)ofthe exactgroundstate when non-Abelianquasiparticlescanbeformedinasimilarway
the finite-thickness effects are taken into account to al- by replacing zi in Eq.(12) with ∂∂zi.
ter all the pseudopotential components instead of just Finally,since Ψ is formedby applyingcoordinate
m,mL
V [6]. ItisalsointerestingtofindoutifΨ canprovide derivatives to the Laughlin wave function, one may re-
1 1,3
a good description (correct parity and large overlap) of garditasformedinthequasielectronspaceoftheLaugh-
the exact ground state using torus geometry[5], or if the lin state. It has been shown numerically by Su and the
symmetrized/antisymmetrizedsuperpositionofthe Pfaf- author[14] that the quasielectron space of the Laughlin
fian state and Ψ state can provide such a description. state provides a rather exact description for the low en-
3,5
Before closing, we would like to make a few re- ergyphysicsfrom1/3filllingfactorallthewaytothe2/5
marks. First, one may contruct the wave function filling factor. Now we have shown that its validity range
Ψ in a different way than in Eq.(2) as Ψ = may go beyond 2/5 filling factor to 1/2 filling factor.
m,mL m,mL
Φ D Φ with the understanding that D
mL−m−1 m m+1 m
only applies to the functions to its right. For example,
Ψ can be contructed as Φ D Φ . In fact, we have
1,3 1 1 2
found the the wave function Φ D Φ has an improved
1 1 2 ∗ Electronic address: [email protected]; Permanent
overlapoverD Φ withits PHconjugatedstate to unity
1 3
address: 5431ChesapeakePlace,SugarLand,TX77479,
from0.9991,whiletheparityremainsnegativeforN =6.
USA
Secondly,itisstraightfowardto showthatΨ3,5 canbe [1] R. Willett et al., Phys.Rev.Lett. 59,1776 (1987).
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N. Read and D.Green, Phys. Rev.B61, 10267 (2000).
1 N [3] M.Greiter,X.-G.Wen,andF.Wilczek,Phys.Rev.Lett.
Ψ3,5 =PLLLPf((u∗v∗−u∗v∗)3)Y(u∗ivj∗−u∗jvi∗)3Φ5 66,3205 (1991); Nucl.Phys. B374, 567 (1992).
i j j i i<j [4] R. H.Morf, Phys.Rev.Lett. 80, 1505 (1998).
(9) [5] E.H.RezayiandF.D.M.Haldane,Phys.Rev.Lett.84,
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LLL
Rev. Lett. 101,016807 (2008); Phys. Rev. B78, 155308
operator. This can further be rewritten as:
(2008).
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Ψ3,5 =PLLLPf((u∗v∗−1u∗v∗)3)Y|uivj −ujvi|6Φ2 [8] LS.e-tSt.L9e9e,, 2S3.6R8y06u,(C20.0N7)a.yak, and M. P. A. Fisher, Phys.
i j j i i<j Rev. Lett.99, 236807 (2007).
(10)
[9] Hao Wang, D. N. Sheng, and F. D. M. Haldane, Phys.
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canbeinterpretedasaf-wavepairingstateofcomposite [10] M.R.Peterson,K.Park,S.DasSarma,Phys.Rev.Lett.
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1,3
in the following equivalent form: [12] R. B. Laughlin, Phys.Rev.Lett. 50, 1395 (1983).
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N [14] J. Yang, W. P. Su, Phys. Rev. Lett. 70, 1163 (1993); J.
1
Ψ1,3 =PLLLPf(u∗v∗−u∗v∗)Y|uivj −ujvi|2Φ2 (11) Yang, Phys. Rev.B49, 16765 (1994).
i j j i i<j