Topological Entanglement and Clustering of Jain Hierarchy States

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by  N. Regnault

Topological Entanglement and Clustering of Jain Hierarchy States
N. Regnault1, B.A. Bernevig2,3, F.D.M. Haldane2
1 Laboratoire Pierre Aigrain, Departement de Physique,
ENS, CNRS, 24 rue Lhomond, 75005 Paris, France
2 Department of Physics, Princeton University, Princeton, NJ 08544 and
3 Princeton Center for Theoretical Science, Princeton, NJ 08544
(Dated: January 14, 2009)
WeobtaintheclusteringpropertiesandpartofthestructureofzeroesoftheJainstatesatfilling
9
k : theyareadirectproductofaVandermondedeterminant(whichhastoexistforanyfermionic
0 2k+1
0 state)andabosonicpolynomial atfilling k whichvanisheswhenk+1particles clustertogether.
k+1
2 We show that all Jain states satisfy a “squeezing rule” (they are “squeezed polynomials”) which
severely reduces the dimension of the Hilbert space necessary to generate them. The squeezing
n
rule also proves the clustering conditions that these states satisfy. We compute the topological
a
J entanglement spectrum of the Jain ν = 25 state and compare it to both the Coulomb ground-state
and the non-unitary Gaffnian state. All three states have very similar “low energy” structure.
4
However, the Jain state entanglement “edge” state counting matches both the Coulomb counting
1
as well as two decoupled U(1) free bosons, whereas the Gaffnian edge counting misses some of the

“edge”statesoftheCoulombspectrum. Thespectraldecompositionaswellastheedgestructureis
]
l evidencethattheJainstateisuniversallyequivalenttothegroundstateoftheCoulombHamiltonian
l
a at ν = 2. The evidence is much stronger than usual overlap studies which cannot meaningfully
5
h differentiate between theJain and Gaffnian states. Wecomputethe entanglement gap and present
- evidence that it remains constant in the thermodynamic limit. We also analyze the dependence
s
e of the entanglement gap and overlap as we drive the composite fermion system through a phase
m transition.
. PACSnumbers: 73.43.f,11.25.Hf
t
a
m
- The experimentally observed fractional quantum Hall certain pseudo-potential Hamiltonians. However, (un-
d (FQH) states in the lowest Landau level (LLL) are liketheLaughlinstates),theyarenot unique maximum-
n
thought to be described by Laughlin [1] and hierarchy density zero modes; while the zero-mode property is in-
o
states modeled by Jain’s composite fermion wavefunc- sufficientto completely determine the structure ofJain’s
c
[ tions [2]. Jain’s original model states have dramatically model wavefunctions, it provides a powerful constraint

large overlap with the true Coulomb ground states but thatenablestheir numericalconstructionatsignificantly
1
the process of flux attachment and projection to the larger N.
v
1 LLL renders them hard to analyze (Monte-Carlo meth- Thekeytechnicaladvancereportedhereistheidentifi-
2 ods havebeendevised[3]fortreating variantsof the Jain cationofthestructureofJainstatesas“squeezedpolyno-
1 states where the projection to the LLL is modified, or mials”, which means they contain only many-body free-
2
simply omitted). particle configurations obtainable from a “root” config-
.
1 The decomposition of Jain’s model states into Slater uration by a two body operation called “squeezing”, de-
0
determinants has not been obtained for N > 10 par- fined below. This drastically reduces the Hilbert space
9
ticles [4], and (unlike the Laughlin states) they have dimension and allows the generation of Jain states with
0
: not been characterized as unique ground states of some roughly twice the number of particles previously ob-
v
modelHamiltonian. Moreover,theirobservedlargeover- tained.
i
X lap with the ground-states of LLL systems with realis- Armed with this technique, we then investigate the
r tic Coulomb interactions is only empirically understood; topological entanglement spectrum [9] of the first state
a
thishasbecomemostevidentrecently,whenotherstates, inthehierarchy,theν =2/5Jainstate forupto N =16
with identical filling (and “shift”) [5, 6], as the Jain particles,andcompareitwithboththeCoulombground-
states, but exhibiting different topological order, have state and the so-called “Gaffnian” state, related to a
been found to have competitive overlaps with the true non-unitaryconformalfieldtheory(CFT)[10], whichhas
Coulomb ground-states [7]. Although these new states a Jack polynomial description[6]. We find a virtually
are conjectured to represent gapless critical points [8], identical “low energy” structure in the Schmidt spectral
their large overlap with the Jain states (thought to be decomposition of these three states, consistent with the
gapped in their interior) underscores the need to better large overlap[5, 6] of the Jain and Gaffnian states with
understandFQHstatesfromatheoreticalstandpoint. In the Coulomb ground-state. Although the Gaffnian state
this Letter we describe a previously-unrecognized “clus- is very close in both overlapand spectral decomposition
tering property” of the Jain model states which allows to Coulomb and Jain, we directly identify the “edge”
them to be (partially) characterized as zero-modes of mode structure of the Coulomb entanglement spectrum
2
andshowthatitmatchestheJainstateedgestructureas
wellasthatoftwoU(1)freebosons. Wecomputetheen-
tanglementgap,andshowevidencethatitremainsfinite
in the the thermodynamic limit. The entanglement gap
can be destroyed by tuning the ν = 2/5 FQH Coulomb
state througha phase transition. We canhowever,make
no definitive statement on the issue of whether the non-
unitary Gaffnian state is gapped or gapless [8].
Any fermionic state in the LLL can be written as a
product of a Vandermonde determinant and a symmet-
ric polynomial; for conceptual simplicity, we will focus
on the bosonic variants ofmodel FQH states whichomit
the Vandermonde factor (it is straightforward to later
go back to the fermionic states, as multiplication by
FIG. 1: Root partition in angular momentum basis for ν =
the Vandermonde factor converts symmetric “squeezed”
2,3,4... k states can bewritten as thesum of theVander-
polynomials to antisymmetric ones). We represent an 3 4 5 k+1
mondedeterminantpartition(singleSlaterdeterminant)plus
angular momentum partition λ with length ℓλ ≤ N the maximum root partition of the Determinant operator of
as a (bosonic) occupation-number configuration n(λ) = k Landau levels projected to the LLL. The root occupation
{n (λ),m = 0,1,2,...} of each of the LLL orbitals configurationcontainsk particlesink+1orbitals[1k0]when
m
φ (z) = (2πm!2m)−1/2zmexp(−|z|2/4) with angular deep in the bulk. Close to the North and South pole there
m
momentum L = m~, where, for m > 0, n (λ) is the are deviations from thisrule, as shown.
z m
multiplicity of m in λ. It is useful to identify the “dom-
inance rule” [11] (a partial ordering of partitions λ>µ) n(λ ) = N00...00N (which has the max-
with the “squeezing rule”[12] that connects configura- GenericState (cid:2)2 2(cid:3)
imum possible variance), and hence in this case the
tions n(λ) → n(µ): “squeezing” is a two-particle oper-
squeezing property is neither meaningful nor useful.
′
ation that moves a particle from orbital m to m and
1 1 However,for most “model” FQH states, the existence of
′ ′ ′
another from m to m , where m < m ≤ m < m ,
2 2 1 1 2 2 arootconfigurationdrasticallyreducestheHilbertspace
′ ′
and m +m = m +m ; λ > µ if n(µ) can be derived
1 2 1 2 necessaryforgeneratingthestateandimpliesmanyother
fromn(λ) by a sequence of “squeezings”. An interacting
special properties of the state.
LLL polynomial P indexed by a root partition λ is de-
λ WenowfindtherootconfigurationforallbosonicJain
fined as a “squeezed polynomial” if it can be expanded
states atfilling k/(k+1),defined asthe usualcomposite
in occupation-number non-interacting states (monomi-
fermion states at filling k/(2k + 1) divided by a Van-
als) of orbital occupations n(µ) obtained by squeezing
dermonde determinant. We start with the simplest of
on the root occupation n(λ):
these states, the ν = 2/3 state, defined by placing N/2
quasiparticles in the Laughlin 1/2 state. The operator
Pλ =mλ+Xvλµmµ. (1) that implements this is Jain’s operator on the plane for
µ<λ t number of quasiparticles [2]:
The vλµ are rational number coefficients. Partitions λ ∂1 ··· ∂N
can be classified by λ1, their largest part. When any Pλ  z1∂1 ··· zN∂N 
is expanded in monomials m , no orbital with m > λ . .
µ 1  . . 
is occupied. P can be interpreted as states on a sphere  . ··· . 
slaurrgreounnudminbgeraoλfmFoQnHopsotlaetewsitahrecshqauregeezeNdΦpo=lynλo1m[1i3a]l.s[6A]. ψtJqp =Det z1t−11∂1 ······ zNt−11∂N YN (zi−zj) (2)
 i<j
Thegroundstatewavefunctionsofthe ReadRezayi(RR)  z1 ··· zN 
 
Zk sequence[14] areJack polynomials (Jacks)of rootoc-  ... ··· ... 
cupationn(λ0(k,2))=[k0k0k...k0k]andJackparameter zN−t−1 ··· zN−t−1 
1 N
α = −(k+1)/(r−1) [6]. All the Jacks are known to
k,r
be squeezed polynomials [11]. For the Jacks, the coeffi- For any t, the above state is not an L~ = 0 state, and as
cients vλµ are explicitly known by recursion [11]. such,itcannotbetheground-stateatν =2/3(t=N/2),
The root configuration of a squeezed polynomial n(λ) contraryto claims inthe literature. The propercompos-
has the largest variance: ∆λ = N (λ − λ )2 of itefermionconstructioninvolveswritingdownthe above
Pi,j=1 i j
all the partitions µ ≤ λ. A generic state, for exam- operatoronthesphereandthenconstructingthestateby
ple the ground-state of the Coulomb Hamiltonian in stereographicprojection. However,Eq.(2)is sufficientto
the LLL at some arbitrary filling, has non-zero weight allowthedeterminationoftherootconfiguration,which,
on all many-body non-interacting states squeezed from per our definition, is the maximum variance configura-
3
tion of orbital occupation number in Eq.(2). The Van-
ndaernmtoonfdfeerfmaciotonricQrNio<ojt(zcio−nfizgju)riastaiosninng(lλeS)la=te[r1d1e1t..e.r1m11i-] 1 01.9 1
0 0.8 0.8
0.5
or λ0 = (N −1,N −2,N −3,N −4,...,6,5,4,3,2,1,0); 00..67 ap 0.6
one immediately recognizes in λ0 the powers (angular y 0 0.5 erl
0.4 v 0.4
dmeotmeremnitnuamn)t oopfetrhaetozriininEtqh.(e2)S,lahtoewredveetre,rhmaisnadnetr.ivaTtihvee -0.5 00..23 o 0.2 Jain CF
0.1 gaffnian
terms, whichwe denote by ∂/∂z =−1;its rootpartition -1 0 0
-1 -0.5 0 0.5 1 -0.1 -0.05 0 0.05
in angular momentum basis is λDet = (N −t−1,N − x δ V1
t−1,...,4,4,3,3,2,2,1,1,0,0,−1). There are two states
ateachangularmomentuminλ becausebothzm and
Det
zm+1∂/∂z operators contained in the determinant have FIG. 2: Left panel : Top view of the hemisphere where each
point (x,y,z) is associated to one state of the squeezed sub-
the same angular momentum m. Since the determinant
spaceatN =12,N =26withcomponents(x,y,z)(squeezed
φ
operator now acts on the Vandermonde determinant λ ,
0 subspace dimension being 3). The color code is the overlap
we could immediately add the two angular momentum between the Coulomb ground state and the squeezed state
partitions,butdoingthisblindlywouldcauseaproblem: associated to the point on the hemisphere. North pole is the
the resulting partition λ, as it describes a polynomial Gaffnian, the red cross is the Jain state and the green cross
wavefunctionψ musthaveallitscomponentspositive is the point which maximizes the overlap with the Coulomb
tqp
ground state. Right panel : Overlaps of both Gaffnian and
(the final polynomial must be analytic in z’s). As such,
Jain states with the Coulomb groundstate as a function of
thelastcomponentofλ cannot addtothelastcompo-
Det added hard-core interaction δV for N = 16 particles. A
1
nent of λ ; adding these two together would correspond
0 phase transition occurs where the overlap collapses close to
to takingthe partialderivative−1→∂/∂z ofaconstant δV ≃−0.08
1
0 → z0, and the result would vanish. As such, the next
maximumvarianceangularmomentumpartitiononecan
build is λ=(N −1,N −2,N −3,N −4,...4,3,2,1,0)+
(N − t − 1,N − t − 1,...,4,4,3,3,2,2,1,1,0,−1,0) =
(...14,13,11,10,8,7,5,4,2,0,0), where we have written again write the Vandermonde determinant λ =
onlytheangularmomentumclosetothenorthpoleinthe (...7,6,5,4,3,2,1,0)where we have written only the an-
finalpartition. When written inoccupationnumber, the gular momentum of orbitals close to the north pole.
root configuration is: n(λ) = [201011011011011011...]. The determinant operator now has 3 operators of iden-
Creating an L~ = 0 state requires that the north pole tical angular momentum, because zm,zm+1∂z,zm+2∂z2
be identical to the south pole, and hence the root all have identical angular momentum m. Moreover,
configuration number for the ν = 2/3 state reads: the determinant operator also contains 2 operators
n(λν=2) = [201011011011...0110110110102]. The bulk which lower the angular momentum by 1 unit: ∂z
occupa3tion configuration contains 2 particles in 3 or- and z∂z2, as well as 1 operator which lowers the
bitals (110), as expected for a ν = 2/3 state. For the angular momentum by 2 units: ∂z2. The maxi-
fermionic state at ν = 2/5: ψν=2/5 = ψν=23 ·QNi<j(zi − m(..u.5m,5,v5a,r4ia,n4c,e4,n3o,n3,-v3a,n2,is2h,i2n,g1,p1a,r1t,i0ti,o0n,−is1,t−he2n,−λ1D,0et).=
z ), the root occupation number reads n(λ ) =
j ν=2/5
Adding the Vandermonde occupation numbers gives an
[11001001010010100101...10100101001010010011].
orbital occupation root configuration of n(λ ) =
WecanobtaintherootoccupationnumberforallJain ν=3/4
[3010110111011101110111....011101110110103]. This
states at filling k/(k+1). We explicitly show the case
procedure and the root configuration for the ν = 4/5
k =3, with the generalization being trivial. At ν =3/4,
state, as well as the general k result are given in Fig.[1].
the Jain state is created by attaching one flux to each
electronin3occupiedLLandthenprojectingtotheLLL:
The root configuration presented in Fig.[1] for general
∂2 ... ∂2 k allowsustodetermineatleastpartoftheHamiltonian
z ∂12 ... z N∂2  forwhichtheJainstatesareexactzeromodes. Clusterk
1 1 N N
 : : :  particles at one point, which, by translationalinvariance
 ∂1 ... ∂N  (which all FQH ground-states must satisfy), we pick to
ψνJ=43 =Det z11:∂1 ....:.. zN1:∂N Yi<j(zi−zj) (3) Jpbaaeirntthiceslteoasrtiaegtiantrh.eeBsoeqrcuiageuienszereedaslulfrlttohsmeinmnmo(cid:0)onλnoνom=mkia/ia(lksl+sin1s)qc(cid:1)luu,edepezldaedciinnfrgtohmke
 z1 ... zn  [0010120130...1k−101k01k...1k01k01k−1...013012010k].
 : ... :  These monomials are proportional to QNi=k+1zi2 and
thereforethefullpolynomialwavefunctionvanisheswhen
To obtain the maximum variance partition, we ak+1’thparticleisbroughtattheorigin;sincetheorigin
4
14 14
12 12
10 10
8 8
ξ ξ
6 6
4 4
2 2
0 0
60 65 70 75 80 85 60 65 70 75 80
(cid:2) (cid:2)
(cid:0) (cid:0)
(cid:1) (cid:1)
FIG. 3: Topological entanglement of the pure Coulomb FIG. 4: Topological entanglement spectrum of the Gaffnian
groundstate (no hard-core potential added) at filling ν = 2. state. ThecountingofthelevelssatisfiesthegeneralizedPauli
5
Although the entanglement gap is smaller than the ones re- principleofnotmorethan2particlesin5consecutiveorbitals
ported for theLaughlin and Moore-Read states, one can dis- and not more than 1 particle in 1 orbital.
tinguish the topological entanglement of both the Gaffnian
(Jack - blue) and the Jain state in the “low energy” (green)
structure of the Coulomb ground state. The levels below
up to N = 16 particles on the sphere geometry (dimen-
the light blue line are almost identical to the Gaffnian levels
sionofthesqueezedHilbertspaceis99608768,compared
whereasthelevelsbelowthegreenlinearealmostidenticalto
those of the Jain state, to within 0.003−3%. to the original full size 155484150); the previous largest
size was N = 10 particles [4]. Since each component
on the E[(N +2)/4] states has its own MC error, sev-
is not special by translational invariance, we have: eral tests have been performed to test the accuracy of
this procedure. The overlap between the Jain state we
N
ψJ (z =Z,...,z =Z,z ,...,z )∼ (Z−z )2 generateusingthistechniqueandthecorrespondingana-
ν=k+k1 1 k k+1 N Y i lytical Jain state is higher than 0.9999. One can slightly
i=k+1
modify each component within its MC error bar and see
ψJ are zero modes of the pseudopotential V0 how this affects various computed quantities. Thus for
ν=k/(k+1) k+1
which eliminates the zero angular momentum state of a N =14,thetypicalrelativeerrorontheCoulombenergy
k+1 body cluster; they can be built out of Read-Rezayi islowerthan10−5whiletheoneontheentanglementgap
Z states upon the addition of quasiholes. is lower than 10−2. Notice that within the L~ = 0 sub-
k
Unfortunately, the above Hamiltonian and root par- space of squeezed polynomials, the Jain state is not the
tition do not uniquely define the Jain states. Uniform best approximation to the Coulomb ground state, but is
(ground)statesonthe spheresatisfythe conditionsL+ψ veryclosetoitasdepictedinFig.[2]. Wealsotriedtouse
= 0 (highest weight, HW) and L−ψ = 0 (lowest weight, this method to obtainhigher-orderJainstates. Unfortu-
LW) where L+ = E , and L− = N Z −E , where Z nately, the dimension of the polynomial space squeezed
0 Φ 2
≡ z , and E = zn∂/∂z . Imposing the highest fromn λ ,whilestillmuchsmallerwithrespect
Pi i n Pi i i (cid:0) ν=k/(k+1)(cid:1)
weight condition on the squeezed polynomial with the to the total number of Slater determinant coefficients, is
Jain root partition n λ results in several lin- nonetheless too large for an accurate decomposition of
ν=k/(k+1)
(cid:0) (cid:1)
early independent L~ = 0 polynomials. We pick the sim- the states.
pleststateatbosonicν =2/3orfermionicν =2/5toan- We also constructed the non-unitary Gaffnian state
alyze further. From now on, we return to the fermionic [10] for N = 16 particles (the squeezed Hilbert space
state. We conjecture that the dimension of the L~ = 0 dimension is 91736995), uniquely defined as the highest
subspace of squeezed polynomials with root occupation weightsqueezedpolynomialwithbosonicrootoccupation
n(λν=2/5)withN electronsisE[(N+2)/4]whereE[x]is n(λJack ν=2/3) = [2002002002...2002002] [6] multiplied
theintegerpartofx. Wehavehencereducedtheproblem by a Vandermonde determinant. In Fig[4] we present
ofdeterminingtheSlaterdecompositionofaJainstateto the overlapof both the Gaffnian and the Jain state with
theproblemofdeterminingE[(N+2)/4]constantsrather the ground-state of Coulomb plus delta function δV1 in-
than the order N! number of constants of each separate teractionobtainedbyexactdiagonalization. Theoverlap
Slater determinant. While the usual Monte-Carlo (MC) is above95%for both states for δV1 >−0.06. There is a
integration procedures would fail to accurately compute phase transition at around δV1 =−0.08 and the overlap
the full decomposition, they may be used to determine withboththeJainandtheGaffnianwavefunctionsdrops
the components of the Jain state on this reduced basis. dramatically.
Withthismethod,weareabletoobtaintheJainstatefor In order to better understand the remarkably large
5
the Gaffnian state, but also exhibits extra higher energy
14 levels not present in the Gaffnian. We should remark
12 that the Jain state, not being a pure CFT state (i.e.,
not obtained as a correlator of CFT primary fields, but
10
ratheroftheirderivatives[17]),hasanentanglementgap
8
ξ of its own. This means that some of the spectral lev-
6 els present in the Jain entanglement spectrum are non-
4 genericandshouldbecomeclearlygappedinthethermo-
2 dynamic limit. For example the entanglement spectrum
at the maximum L(A) = 80 is formed by one low-lying
0 z
60 65 70 75 80 eigenvalue and other high-energy ones with very little
(cid:2)
(cid:0)
weight in the Jain state. The difference between these
(cid:1)
values seems to define an entanglement gap for the Jain
state itself, although larger sizes are necessary to verify
FIG. 5: Topological entanglement spectrum of the Jain ν =
2/5stateforN =16particles. The“lowenergy”structureof this.
the Jain state (in blue) is almost identical both qualitatively The presence ofanentanglementgapin the Jainstate
and quantitatively to that of the Gaffnian (below blue line). differentiates it from “pure” CFT states, and makes the
BecausetheJainstateisnotapureCFT,ithasanentangle-
counting of the edge-state spectrum difficult. To pro-
ment gap of its own with respect to theGaffnian state.
ceed,wecountonlytheeigenvaluesoftheJainstatethat
match the eigenvalues of the Coulomb spectrum (below
the green line in Fig[3]). This should provide us with
overlap,aswellastoidentifythetopologicalorderinthe
the“universal”countingofedgestatesforourfinitesize-
JainandGaffniangroundstate,wecomputethetopolog-
system. As seen in Fig[3], this counting is 1 : 2 : 5 for
icalentanglementspectrumofthesestates. Weplaceour
∆ =0,1,2. Thismatches the countingoftwoU(1)free
statesonthesphere(comparisonispossiblebecausethey L
bosons which is also the prediction of the hierarchy or
haveidenticalfillingandshift),andcutthestateintotwo
Composite Fermion construction.
hemisphere blocks A and B. Following [9], we introduce
the entanglement spectrum ξ as λ = exp(−ξ ), where The Coulomb state follows most of the low-energy
i i
λ are the eigenvalues of the reduced density matrix ρ eigenvalues of the Gaffnian (up to ξ = 8) and Jain state
i A
of one hemisphere. The eigenvalues can be classified by (uptoξ =10). Whiletwostatesthathavealmostidenti-
the number of fermions N in the A block, and also by calspectraldecompositionnecessarilyhavelargeoverlap,
A
the totalangularmomentumL(A) ofthe Ablock. Itwas the converse is not true, as large overlap can be acci-
z
dental. Our finding of almost identical low-energy spec-
argued [9] that the low-lying spectrum ξ of the reduced
i
densitymatrixforfixedN ,plottedasafunctionofL(A), traldecompositionsindicatesthatthelargeoverlapthese
A z threestatesshowwitheachotherisnotaccidental,which
should display a structure reflecting the CFT describing
is verypuzzling consideringthe GaffnianandJainstates
theedgephysics. InFig.[3],thisCFTspectrumisdefined
represent different states of matter. In order to appre-
as every ξ below the light blue line. For interactions at
ciate the similar structure of these states, we plot the
which the FQH state provides a good description of the
overlap of several reduced density matrix eigenstates of
physics,theCFTspectrumshouldbeseparatedbyagap
the Gaffnian with Coulomb for different L(A), and find
from a higher “non-CFT” part of the spectrum. This z
Fig.[3], nearly perfect overlap up to the point where the
was shown to be the case for the ν = 5/2 state [9] as
system is driven into a phase transition by the addition
well as for the Laughlin ν = 1/3 state [15]. In our case,
of negative delta-function potential.
ν = 2/5, the entanglement gap is not extremely appar-
ent. The three states,Gaffnian, JainandCoulomb,have Asin[9],wedenotethegapbetweenthelowesttwoξi,
the same “low energy”entanglementstructure as canbe at the L(zA) value where the highest-L(zA) member of the
seen in Fig.[3], Fig.[4] and Fig.[5]. The counting of en- CFT spectrum occurs, as δ . In Fig.[6], this is the gap
0
tanglement eigenvalues for the Gaffnian at a certain an- between the lowest two states at L(A) = 80. We define
z
gularmomentumL(A) is easilyseentocorrespondtothe the quantities δ [9], as the gaps at L(A) = 79,78 val-
z 1,2 z
countingofoccupationnumber configurationsofangular ues between the values of the ξ ’s for the CFT state and
i
momentum L(A) satisfying generalizedPauliprinciple of the next Coulomb value. As noted previously, the Jain
z
[6]appliedto fermionic states: notmore than2 particles state has its own entanglement gap, equal at L(A) with
z
in5consecutiveorbitalsand,byvirtueofbeingfermions, the difference between the lowest ξ ≈ 6 and the next
i
not more than 1 particle in each orbital. The counting one at ξ ≈ 13.5 We study what happens to the spec-
i
of “edge modes” reads 1,1,3,5,10..., same as that ob- trum, in particularto the entanglementgapsδ as we
0,1,2
tained by different methods in [16]. The Jain state has tune the interaction away from the FQH state across a
a very similar “low energy” entanglementstructure with quantum phase transition. In figure 6 (lower panel), we
6
8 1 in the gaffnian sector. We notice that for N = 12, this
7 seemstobecorrelatedtothefirstexcitedstatehavingits
0.8
6 angularmomentumchangingfromL=6 to L=2 (close
5 0.6 ap to δV1 =−0.06).
δ0 4 erl In conclusion, we analyzed the topological structure
3 0.4 ov
of Jain states focusing on the ν = 2/5 state and com-
2 (cid:5) (cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12) 0.2 pared it to the Coulomb ground-state and to a recent
1 (cid:3)(cid:4) overlap
0 0 non-unitarystateatthesamefillingfactorandshift. We
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04
showedthattheJainstatesatfillingk/(2k+1)exhibita
δ V
1
squeezing property that severely reduces the size of the
2.5 1 Hilbert space needed to construct them. We found the
structure of zeroes of these states and showed that they
2 3 1 0.8
arezero modes (but not highestdensity) ofa k+1-body
2
1.5 1 0.5 0.6 ap pseudopotential. We showed that fermionic Jain states
δ1 1 0 0 0.4 verl are a direct product of a Vandermonde determinant and
-0.1 -0.05 0 0.05 o
a symmetric polynomial vanishing as the second power
0.5 (cid:13)(cid:5) (cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12) 0.2 of the difference in coordinates of a k+1-particle clus-
(cid:3) overlap
0 0 ter. We analyzed the entanglement spectrum of Jain,
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04
δ V Coulombandnon-unitary Gaffnianstate atν =2/5and
1
found a similar low-energy structure which proves that
2 1 their large overlap is not accidental; the entanglement
1.8
gap remains finite in the thermodynamic limit, but the
1.6 2 1 0.8
1.4 Jain state contains some “low-energy” levels which are
1.2 1 0.5 0.6 ap not included in the Gaffnian. Nevertheless, the similar-
δ2 0 .18 0 0 0.4 verl ity of their entanglement spectra is puzzling as the Jain
-0.1 -0.05 0 0.05 o
0.6 and Gaffnian (Jack) state correspond, in the thermody-
00..24 (cid:3)(cid:14)(cid:5) (cid:6)o(cid:7)v(cid:8)e(cid:9)(cid:10)rla(cid:11)p(cid:12) 0.2 namic limit, to different states of matter, one gapped
0 0 and the other conjectured to be gapless [8]. The overlap
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04
δ V of both states with the Coulomb groundstate vanishes
1
upon the addition of a hard-coreattraction which drives
the Coulomb system through a phase transition.
FIG. 6: Red plots : Entanglement gap of the coulomb state
B.A.B. also wishes to thank E. Rezayi, S. Simon,
for 3 different values of L(zA) = 80 (Fig (a) top), L(zA) = 79 P. Bonderson for discussions. N.R. is grateful to E.
(Fig (b) middle), L(zA) = 78 (Fig (c) bottom) as a function Bergholtz for useful comments.
of added hard-core potential. The entanglement gap is dis-
continuous (a and b) or almost vanishing (c) at roughly the
same values of δV for which theoverlap of theGaffnian and
1
Jain with the Coulomb ground-state also collapses (see in-
set Fig.[2]). The red dotted line is the entanglement gap for
the Jain state. Green plots: Overlap of the reduced density [1] R.B. Laughlin, Phys. Rev.Lett. 50, 1395 (1983).
matrix eigenstates for each of the L(zA) = 80,79,78 between [[23]] JR..KK..JKaainm,iPllhayas.ndReJv..KL.eJtat.in6,3I,n1t9.9J.(1M98o9d).. Phys. B 11,
Coulomb and Gaffnian. The green dotted line is thecalcula-
tionresultbetweenJainandGaffnian. Insets: Similarresults 2621 (1997).
[4] G. Dev and J.K. Jain, Phys. Rev.B 45, 1223 (1992).
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[5] S.H. Simon, E.H. Rezayi, N.R. Cooper, and
I. Berdnikov,PRB 75, 075317 (2007).
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100, 246802 (2008).
plot δ as a function of the pseudopotential δV for the
0 1 [7] N. Regnault, M.O. Goerbig, and Th. Jolicoeur, Phys.
ν = 2/5 case. This clearly shows a dramatic decrease of Rev. Lett.101, 066803 (2008).
the “entanglement gap” around the region of the phase [8] N. Read, arXiv:0805.2507.
transition. For values of δV1 < 0.08 the CFT-like struc- [9] H. LiandF.D.M. Haldane,Phys.Rev.Lett.101,010504
ture of the entanglement spectrum is lost. We also in- (2008).
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075318 (2007).
system size and conjecture that it remains finite in the
[11] R.P. Stanley, Adv.Math.77, 76 (1989).
thermodynamic limit. Entanglement gaps can also be
[12] B. Sutherland, Phys.Rev.A 4, 2019 (1971).
computed in the two U(1) free boson sector (see insets [13] F.D.M. Haldane, Phys. Rev.Lett. 51, 605 (1983).
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bleforvaluesofδV1 whicharelargerthanthoseinvolved [15] O. Zozulya, M. Haque, and N. Regnault,
7
arXiv:0809.1589. [17] H. Hansson, C-C. Chang, J. Jain, and S.Viefers, Phys.
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