New family of models for incompressible quantum liquids in $d\geq 2$

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by  Chyh-Hong Chern

New family of models for incompressible quantum liquids in d ≥ 2
Chyh-Hong Chern1∗ and Dung-Hai Lee2
1ERATO-SSS, Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
2Department of Physics, University of California at Berkeley, Berkeley CA 94720, U.S.A.
Through Haldane’s construction, the fractional quantum Hall states on a two-sphere was shown
to be the ground states of one-dimensional SU(2) spin Hamiltonians. In this Letter we generalize
7
this construction to obtain a new class of SU(N) spin Hamiltonians. These Hamiltonians describes
0
center-of-mass-position conserving pair hoppingfermions in space dimension d≥2.
0
2

n Gappedquantummany-bodysystemsarestablestates dane’s work on the pseudopotential Hamiltonian for the
a of matter because they are robust against weak pertur- fractionalquantumHalleffect[6],weshallbeginbybriefly
J
bations. For fermions, the most common gapped system review it.
1
is the “band insulator” where an energy gap in the dis- If a magnetic monopole of strength 2S (S is a multi-
3
persionrelationseparatesfilledandemptysingle-particle ple of 1/2) is placed at the center of a two-sphere, the

] states. Here the insulating behavior is caused by the kinetic energy spectrum of a particle confined to move
l
e Pauli exclusion principle. We refer to a system as a on the sphere is given by Ek ∝ (S + k)(S + k + 1),
- “many-body insulator” if its energy gap is caused by where k is an non-negative integer. The kth “Landau
r
t many-body interaction rather than one-particle disper- level” is 2(S +k)+1-fold degenerate. The degeneracy
s
. sionrelation. Itiswidelybelievedthatsuchamany-body of the lowest Landau level, k = 0, is exactly the dimen-
t
a gapcanexistwhentheoccupationnumber(i.e.,the aver- sionofa spin-S SU(2) multiplet. Thus the Hilbert space
m aged number of particle per spin per unit cell) is a frac- of N spin polarized electrons in the lowest Landau level
- tion. TheMottinsulator,wheretheenergygapisdue to is the same as the exchange-antisymmetric sub-Hilbert
d
the inter-particle repulsion, is an example of many-body space of N such SU(2) spins. Haldane’s pseudopotential
n
insulator. Hamiltonian (which has the spherical version of Laugh-
o
c Contrary to the common belief, it is very difficult to lin’s ν =1/m wavefunctionas the groundstate) is given
[ find true many-body insulators. In mostcasesan energy by

4 gap at fractional occupation number is accompanied by
m−2
v the breaking of translation symmetry. After symmetry H = 1 κ P2S−q. (1)
7 breaking,theunitcellisenlargedsuchthatthenewoccu- 2X X q ij
0 i6=jq=1,odd
pation number is an integer. In a recent work Oshikawa
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7 argued that the existence of energy gap at fractional oc- Herei,j =1,..,N arethespinlabels,P2S−q projectsthe
ij
0 cupationrequiresthegroundstatetobedegenerate[1]. It product states of spin i and j onto the total spin 2S−q
6 happensthatunderusualcircumstancessuchdegeneracy
multiplet, andκ >0areparameters(as aresultEq.(1)
0 is caused by translation symmetry breaking. q
is positive-definite). In Eq.(1) the q-sum is restricted to
/
at The fractional quantum Hall state is a quantum liq- odd integers because the restriction of the Hilbert space
m uid (hence no breaking of translation symmetry) with to the total antisymmetric subspace. It can be shown
an energy gap. On the surface it is not clear what does that when N−1=2S/m≡p the Hamiltonian in Eq.(1)
-
d it have to do with the many-body insulator discussed has an unique ground state described by the following
n above. In Ref.[2–4] it is shown that, when placed on a spincoherent-statewavefunction(thesphericalversionof
o
torus,bothabelianandnon-abelianquantumHallliquids Laughlin’s wavefunction)
c
: can be mapped to many-body insulators on a one di-
v mensional ring oflattice sites. When both dimensionsof up up−1v . . vp m
i 1 1 1 1
X the torus are much bigger than the magnetic length, the (cid:12) up up−1v . . vp (cid:12)
(cid:12) 2 2 2 2 (cid:12)
r energy gap is generated by a long range center-of-mass- Ψ=(cid:12) . . . . . (cid:12) (2)
a (cid:12) (cid:12)
positionconservinghopping,ratherthandensity-density (cid:12) . . . . . (cid:12)
(cid:12) (cid:12)
interaction. (cid:12) up up−1v . . vp (cid:12)
(cid:12) N N N N (cid:12)
Partly motivated by Anderson’s proposal of spin (cid:12) (cid:12)
(cid:12) (cid:12)
liquid[5], the question of whether a many-body insula- Upon expanding the determinant, Eq.(2) can be written
tor can exist without symmetry breaking in spatial di- as a linear combination of N (mp)!/n !k !unjvkj
j=1 j j j j
mension d ≥ 2 has attracted a lot of interests. In this wheren +k =mp. EachtermQinthpislinearcombination
j j
Letter we givethis questionanaffirmativeanswerby ex- is a direct product of N, spin-S, states. An important
plicitly constructing a new class of solvable lattice mod- property of the wavefunction in Eq.(2) is that the
els that exhibit incompressible quantum liquid ground highest total spin for any pair is 2S−m. Consequently
states. Since our construction is a generalization of Hal- Eq.(2) is a zero energy state of Eq. (1). Moreover, if we
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view the 2S +1 different S states as the 2S +1 local by the following SU(3) coherent-state wavefunction
z
orbitals of a one-dimensional lattice, and write P2S−q =
ij
mP2l2=S>−−(q<2S−mq)1,Pl Sm−2=m−1S|,PESmq.1(=1−)SbCecS2oS,mm−1qe,,Ssl,la−mc1eCntS2eS,mr−-2oq,f,Sl-,ml−amss2|m2,l− (cid:12)(cid:12) uup1p2 uup1p2−−11vv12 .. .. ww12pp (cid:12)(cid:12)3
conserving pair hopping Hamiltonian.[2, 3] In the above Ψ3 =(cid:12)(cid:12) . . . . . (cid:12)(cid:12) (5)
C2S−q,l is the SU(2) Clebsch-Gordon coefficient. (cid:12)(cid:12) . . . . . (cid:12)(cid:12)
TSh,em1r,So,lle−mo1f center-of-mass position conservation in (cid:12)(cid:12) upN upN−1vN . . wNp (cid:12)(cid:12)
(cid:12) (cid:12)
producing true many-body insulators was discussed in (cid:12) (cid:12)
First we prove Eq.(5) is a ground state. Let us focus on
Ref.[2, 3].
the dependence of Eq. (5) onthe variablesof any chosen
In the following we generalize Haldane’s construction
pairofspiniandj. ForeachoftheSlaterdeterminantin
to SU(3) spins. The reasonfor doing so is SU(3), a rank
Eq.(5) the highesttotalSU(3) weightof these twospins
two Lie group, has multiplets isomorphic to two dimen-
is (2p−2,1) because of antisymmetry. As a result when
sional lattices. Hence it allows us the possibility of con-
wemultiply three determinanttogetherthe highesttotal
structingHamiltoniansformany-bodyinsulatorind=2.
SU(3)weightforspiniandj is(6p−6,3). Consequently
The irreducible representations of SU(3) are labelled by
Eq.(5)isanzero-energyeigenstateofthepositive-definite
two integers (p,q). In the following we shall focus on
Hamiltonian in Eq.(4). Thus we have found a ground
the the multiplets (k,0). The reason is because these
state.
multiplets are the only ones whose weight space is an
Astotheuniquenessletusstartwiththesimplestcase
array of non-degenerate, i.e., non-duplicated, points (see
of p=1 where the single spin Hilbert space is d(3)=10
Fig.(1)). InthefollowingweconsiderSU(3)spinseachin
dimensional and there are d(1) = 3 spins. The most
the (mp,0)representation. The dimensionofthe (mp,0)
general many-body wave function is given by
representationisd(mp)=(mp+1)(mp+2)/2. Forreason
that shall become clear later, we shall choose the num-
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ber of spins so that N =d(p)=(p+1)(p+2)/2. Under
χ= C(α ,α ,α ;β ,β ,β ;γ ,γ ,γ )
1 2 3 1 2 3 1 2 3
suchconditionthefillingfactor,f =d(p)/d(mp),is1/m2 X
{αj,βjγj=1}
in the thermodynamic (p → ∞) limit. As earlier, we
×φα1φα2φα3φβ1φβ2φβ3φγ1φγ2φγ3, (6)
will constrain the N-spin Hilbert space to be exchange- 1 1 1 2 2 2 3 3 3
antisymmetric to mimic the fermion statistics.
where φ1,2,3 = u ,v ,w . The requirement that this
The spin Hamiltonian we construct is a generalization j j j j
wavefunctionlies inthe directproductof(3,0)⊗(3,0)⊗
of Eq. (1), and is given by
(3,0)demandsthecoefficientC tobeinvariantwhenthe
q≤m−2 threeindices ofanychosenparticlearepermuted. Inad-
1
H = 2 κq Pi(j2mp−2q,q). (3) dition,theantisymmetryconstraintrestrictsC tochange
Xi6=jq=X1, odd sign upon the exchange of particle labels. Next, let us
pickanypairofspinsiandjandexaminethedependence
Here the operator P(2mp−2q,q) operates on the direct
ij of Eq. (6) on their variables. The possible total SU(3)
product states of two spins i and j, and projects them
weights of these two spins that are consistent with anti-
onto the (2mp−2q,q) multiplet, and κ > 0. For sim-
q symmetry are given by (3,0)⊗(3,0)=(4,1)⊕(0,3). In
plicity in the rest of the Letter we shall set m = 3 and
order for the wavefunction to be annihilated by Eq. (4),
p= odd integer. In this case Eq.(3) becomes
it must lie entirely in (0,3). Since there are only three
κ particles it is not hard to show that there is a unique C
H = 1 P(6p−2,1). (4)
2 ij so that the above condition holds for all pair (i,j):
X
i6=j
C(α ,α ,α ;β ,β ,β ;γ ,γ ,γ )∝ǫ ǫ ǫ .
1 2 3 1 2 3 1 2 3 α1β1γ1 α2β2γ2 α3β3γ3
Substitute the aboveequationinto Eq.(6)givesχ∼Ψ .
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Now, we consider p = 3, where the single-particle
Hilbertspaceis55dimensionalandandthere ared(3)=
10 particles. Analogous to Eq. (6) the most general 10-
particle wavefunction is given by
3 10 9
χ= C({α }) φαjn. (7)
jn j
FIG. 1: The weight space of (6,0). {αXjn=1} jY=1nY=1
We shallspend muchof the restofthe Letter to prove Here j is the particle label, n = 1,...,9 labels the nine
that Eq.(4) has a unique singlet ground state described fundamental SU(3) spinors (1,0) that make up (9,0).
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The symmetry properties of C are the same as be- that there exists a ground state solution whose C does
fore. In this case the total SU(3) weight of any two not satisfy Eq.(9) for pair (k,l). This means there must
spins that are consistent with antisymmetry are given beatleastonetripletexchange,letsay{α ,α ,α }↔
k1 k2 k3
(9,0)⊗(9,0) = (16,1)⊕(12,3)⊕(8,5)⊕(4,7)⊕(0,9). {α ,α ,α },forwhichC doesnottransformaccording
l1 l2 l3
The condition of being annihilated by the Hamiltonian to Eq. (9). However, since the wavefunction still has
requires the ground state wavefunction to lie entirely in to satisfy Eq. (8) for (k,l), we should be able to write
(12,3)⊕(8,5)⊕(4,7)⊕(0,9), i.e., C =C +C +C +C , whereC is the componentofC
3 5 7 9 q
that is odd with respect to exchange of exactly q pair of
(P(12,3)+P(8,5)+P(4,7)+P(0,9))χ=χ, ∀ (i,j). (8) indicesbetweenparticlek andl andevenwithrespectto
ij ij ij ij
the exchange of the rest. Now let us consider the effect
Eq. (8) implies that among the 9 indices spin i and j
of {α ,α ,α } ↔ {α ,α ,α } on C. Under such
k1 k2 k3 l1 l2 l3
eachpossesses,theremustbeatleast3pairs(apaircon-
operationC caneither changesignorstayinvariantde-
q
tains one index from each particle) such that C → −C
pendingonwhetheranoddorevennumber(outofq)an-
upon exchanging the indices within each pair. In addi-
tisymmetricindicesarecontainedinthespecifiedtriplets.
tion to be consistent with exchange antisymmetry, the
In other words upon {α ,α ,α }↔{α ,α ,α } we
k1 k2 k3 l1 l2 l3
number of such pairs also must be odd. Since C is in-
have
variant when the nine indices of any particle are per-
muted, we can perform permutations so that in each C →η C +η C +η C +η C , (12)
3 3 5 5 7 7 9 9
triplet (α ,α ,α ), (α ,α ,α ),(α ,α ,α ) of par-
i1 i2 i3 i4 i5 i6 i7 i8 i9
ticleiand(αj1,αj2,αj3),(αj4,αj5,αj6),(αj7,αj8,αj9)of whereηq =±1. SinceEq.(9)isnotsatisfied,η3,5,7,9 must
particle j there is an odd number of antisymmetric in- not simultaneously be −1. Now consider a new C
dices. Under such circumstance C changes sign upon
1
independent exchanges of the triplets, i.e., C′ ≡ C−η C −η C −η C −η C . (13)
3 3 5 5 7 7 9 9
2h i
C →−C upon (α ,α ,α )↔(α ,α ,α )
i1 i2 i3 j1 j2 j3 It is obvious that upon {α ,α ,α } ↔ {α ,α ,α }
k1 k2 k3 l1 l2 l3
(αi4,αi5,αi6)↔(αj4,αj5,αj6) C′ → −C′. Moreover by construction C′ only contains
(αi7,αi8,αi9)↔(αj7,αj8,αj9).(9) those Cq whose ηq = −1. Now use C′ as the starting C
and repeat the above operation until we reach a final C′
In Eq. (9) (...) ↔ (...) denotes the exchange of whole for which Eq.(9) holds for all triplet exchanges and for
group of indices. If Eq. (9) can be made true simultane- all (i,j). Since at each stage of obtaining C′ certain C
q
ously for all pairs i and j then Ψ3 is the unique solution areprojectedout,theremustbemissingqcomponentsin
of Eq. (8). This is proven as follows. the finalC. Howeverwehavealreadyproventhat anyC
Let us focus on the dependence of C on the first in- thatsatisfyEq.(9)forall(i,j)pairmustsatisfyEq.(11).
dex triplet of all particle. For each triplet of indices, However Eq. (11) contains all four q components for all
say (αi1,αi2,αi3), there are (3+2)!/(3!2!) = d(3) = 10 pair(i,j). Consequentlywehavereachedacontradiction.
inequivalent combinations. We can interpret each com- Therefore it must be possible to make Eq.(9) hold true
bination as a single-particle quantum state and C as the forallpairs(i,j)foranygroundstate solutionsatisfying
wavefunctionfor10particlestooccupythesestates. The Eq. (8).
first line of Eq. (9) allows us to interpret C as the wave- Although we have chosen p = 3 and m = 3 in the
function for fermions. For N = d(3) = 10, i.e., when abovediscussion,itshouldbeclearthatourproofcanbe
the fermion number is the same as the number of single- generalized to any odd p , any m. Thus we have proven
particle state, there is an unique wavefunctionsatisfying that Eq.(5) is the unique ground state of Eq.(4).
the antisymmetric requirement, namely, It is straightforward to prove that Eq.(3) is a center-
of-mass position conserving pair hopping model, i.e.,
C({α ,α ,α }...)∼ǫ{(α α α )} (10)
i1 i2 i3 i1 i2 i3
H =κ Fj,L,L3Fj,L,L3
where ǫ{...} is the rank 10 total antisymmetric tensor l,l3 k,k3
X XX
j,L,L3 l,l3 k,k3
with respect to the exchange of index-triplets. Sim-
ilar argument can be made to {(αi4,αi5,αi6)} and c†(l,l3)c†(j−l+21,L3−l3)c(j−k+12,L3−k3)c(k,k3). (14)
{(α ,α ,α )}, and lead to
i7 i8 i9
The central steps are 1) viewing the weight space of
C({α ,...,α })∼ǫ{(α α α )}ǫ{(α α α )} (mp,0) as a triangular lattice, and 2) decomposing the
i1 i9 i1 i2 i3 i4 i5 i6
× ǫ{(α α α )}. (11) two spin states in Eq. (3) as linear combination of prod-
i7 i8 i9
ucts of single spin state. Due to space limitation, the
Substitute Eq. (11) into Eq. (7) we obtain χ∼Ψ . resultwill be published elsewhere. Using the explicit ex-
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Now, we shall prove that Eq.(9) can indeed be made pressionfortheF’sinEq.(14)(lengthyhenceisomitted
true for all pairs i and j simultaneously. Let us assume here) we can estimate the hopping range. In Fig.(2) we
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plot the hopping range versus p for p = 30 → p = 600. LLL. Straightforwardcalculation gives,
hWohpipleintgheralningeearscdailmesenassiopn12,ofhtehnecelatthteicehoscpapliensgaisspl,otnhge- f(k)=21 v(|q|)(eq¯2k−ek¯2q)[s(q)e−|k2|2(e−k¯2q−e−q¯2k)
X
ranged as the SU(2) case. q
k¯q q¯k
+s(k+q)(e 2 −e 2 )], (17)
where v(|q|) is the Fourier transformation of the poten-
tial, which is required to be positive indicating the re-
pulsive interaction to ensure the excitation energy to be
positive. On the other hand, s(k) can be related to the
radial distribution function g(~r) by
s(k)=e−|k2|2+ρ d4re−i~k·~r[g(~r)−1]+ρ(2π)4δ4(~k)(18)
Z
where ρ is the average density and ~k=(Re(k ), Im(k ),
1 1
Re(k ), Im(k )). After some algebra, for small |k|
2 2
it can be shown that ∆(k) = (a|k |4 + b|k |2|k |2 +
1 1 2
a|k |4)/(c|k |4 +d|k |2|k |2 +c|k |4), which remains fi-
2 1 1 2 2
nite as k approaches to zero in any direction.
FIG.2: Alog-log plotof thehoppingrangeandpfrom 30to In summary, we have constructed a two dimensional
600. The straight line is the best fit. The vertical axis is the
center-of-mass conserving pair hopping model which ex-
log of thehopping rangeand thehorizontal oneis logp. The
1 hibit incompressible quantum liquid ground state. Al-
hoppingrange scales as p2
though throughout the Letter we have focused on SU(3)
whose weight space is two dimensional, our construction
Finally, we demonstrate the presence of an excitation
can easily be generalized to SU(N) giving rise to models
gap within the single mode approximation (SMA)[7].
for incompressible quantum liquid in higher dimensions.
Analogous to the SU(2) case it is possible to view the
Particularly,theSU(4)modelcanbeappliedtothefour-
(k,0)SU(3)multipletasthe“lowestLandaulevel”(LLL)
dimensionalquantumHalleffect proposedbyZhangand
of a particle running on CP2 under the action of a U(1)
Hu [10–12].
backgroundmagneticfield[8]. Ifweparameterizethefun-
We deeply appreciate the discussion with Darwin
damental SU(3) spinor as (1,z ,z )/ 1+|z |2+|z |2,
1 2 1 2 Chang. CHC is supported by ERATO-SSS, Japan Sci-
wherez =x +iy , andtaketheflat-sppacelimit(i.e.,re-
i i i ence and Technology Agency. DHL is supported by the
strict |z |<<1) the single-particle orbitals in the LLL
1,2 Directior, Office of Science, Office of Basic Energy Sci-
become
ences, Materials Sciences and Engineering Division, of
Φ (z ,z )= 1 zl1zl2e−(|z1|2+|z2|2)/4, the U.S.DepartmentofEnergyunder ContractNo. DE-
l1,l2 1 2 4π22l12l2l1!l2! 1 2 AC02-05CH11231.
p
where l are non-negative integers. We recognize that
1,2
theaboveresultistheproductoftwoLLLwavefunctions
intwospacedimensions. Thusintheflat-spacelimit,the
LLL in CP2 becomes the direct product of the LLLs in ∗ Electronic address: [email protected]
two quantum Hall planes. This reduction allows us to [1] M. Oshikawa, Phys. Rev.Lett. 84, 1535 (2000).
perform the SMA calculation pretty much in parallel to [2] D.-H. Lee and J. Leinaas, Phys. Rev. Lett. 92, 096401
that for the ordinary quantum Hall effect.[7, 9] in the (2004).
[3] A.Seidel,H.Fu,D.-H.Lee,J.M.Leinaas,andJ.Moore,
following we summarize the results. Within SMA the
Phys. Rev.Lett. 95, 266405 (2005).
excitation energy is given by [4] A. Seidel and D.-H. Lee, Phys. Rev. Lett. 97, 056804
(2006).
∆(k)=f(k)/s(k). (15)
[5] P. Anderson, Science 235, 1196 (1987).
Herek=(k ,k )wherek arethecomplexwavevectors [6] F. Haldane, Phys.Rev.Lett. 51, 605 (1983).
1 2 1,2
associated with the two quantum Hall planes, and f(k) [7] S. Girvin, A. MacDonald, and P. Platzman, Phys. Rev.
B 33, 2481 (1986).
and s(k) are given by
[8] D.KarabaliandV.Nair,Nucl.Phys.B641,533(2002).
f(k)=(1/N)<Ψ |[ρ†,[V,ρ ]]|Ψ > [9] S. Girvin, A. MacDonald, and P. Platzman, Phys. Rev.
m k k m Lett. 54, 581 (1985).
s(k)=(1/N)<Ψ |ρ†ρ |Ψ >. (16) [10] S.-C. Zhang and J. Hu,Science 294, 823 (2001).
m k k m
[11] C.-H. Chern, cond-mat/0606434 (2006).
In the above equation ρk and V are the the density op- [12] B.A.Bernevig,C.-H.Chern,J.-P.Hu,N.Toumbas,and
eratorandthe inter-particlepotentialprojectedontothe S.-C. Zhang, Annals of Physics 300, 185 (2002).