A continuous transition between fractional quantum Hall and superfluid states

0.2 MB·English

by  Maissam Barkeshli

MIT-CTP 4341
A continuous transition between fractional quantum Hall and superfluid states
Maissam Barkeshli1 and John McGreevy2
1Department of Physics, Stanford University, Stanford, CA 94305
2Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139
Wedevelopatheoryofadirect,continuousquantumphasetransitionbetweenabosonicLaughlin
fractionalquantumHall(FQH)stateandasuperfluid,generalizingtheMottinsulatortosuperfluid
phasediagram of bosonstoallow for thebreakingoftime-reversal symmetry. Thedirect transition
can be protected by a spatial symmetry, and the critical theory is a pair of Dirac fermion fields
coupled to an emergent Chern-Simonsgauge field. Thetransition may beachieved in optical traps
2 of ultracold atoms by starting with a ν = 1/2 bosonic Laughlin state and tuning an appropriate
1 periodic potential to change the topology of thecomposite fermion band structure.
0
2
Introduction – One of the most celebrated examples Mott
n
Insulator
a of a continuous quantum phase transition is between a
J Mott insulator (MI) and a superfluid (SF) of bosons

0 [1, 2]. Over the last two decades, this transition has |m- | 3D XY
2
been successfully characterized, both theoretically and

experimentally. In addition to the Mott insulator and
]
l the superfluid, it is expected that a fractional quantum
e Superfluid
- Hall (FQH) state can be realized in strongly interact- FQH
r
ing bosonic systems, such as in optical traps of ultra-
t
s cold atomic gases [3].This raises a fundamental question
.
at of whether it is also possible to transition continuously (m+ ,m- )=(0,0) m+
m between FQH states and Mott insulators or superfluids.
While theories of continuous transitions between FQH
- FIG. 1: Proposed phase diagram and renormalization-group
d states and Mott insulators have been developed [4–7], it flows including the Mott insulator, superfluid, and ν = 1/2
n
has not been addressed whether the FQH state can di- Laughlin FQH state, for fixed average particle number. We
o rectly and continuously transition to a superfluid as the havedefined m± ≡m1±m2 (see eq. 7); m− is a symmetry-
c
[ kinetic energy of the bosons is increasedrelative to their breakingfield,sothedirecttransitionbetweentheFQHstate
and the SF can occur if the symmetry is preserved. The red
interaction energy.
1 points on the horizontal and vertical axes indicate the three
v Inthispaper,wedevelopatheoryofsuchacontinuous stable phases, while the blue points at the origin and the
3 transition,betweenaν =1/2bosonicLaughlinstateand diagonals indicate the unstablecritical fixed points.
9
a superfluid, thereby providinga more generalpicture of
3
the boson phase diagram (Fig. 1). Since the superfluid
4
. is described by an order parameter while the FQH state teger quantum Hall (IQH) state. An externally-applied
1
0 is a topological phase without a local order parameter, periodic potential can change the band structure of the
2 such a transition is conceptually quite exotic [8]. Realiz- composite fermions such that they occupy bands with a
1 ing it in the lab would be an experimental example of a totalChernnumberC. WhenC =1,thestateisstillthe
v: continuous quantum transition in a clean system (unlike ν = 1/2 FQH state. However, when C = 0, the result-
i QHplateautransitions)whichliesoutsidetheGinzburg- ing state is a Mott insulator, and, as we explain below,
X
Landau paradigm. Here, we will specialize to the case when C = 1, the resulting state is a superfluid. Thus
ar with fixed averageparticle number. We find that generi- the transiti−ons between these states can be understood
cally,intheabsenceofanyadditionalsymmetriesbesides as Chern number-changing transitions of the composite
particle number conservation, continuous transitions oc- fermions. The critical theories for such transitions con-
cur between the FQH state and Mott insulator or the sistofgaplessDiracfermionscoupledtoaChern-Simons
Mott insulator and the superfluid. However in the pres- (CS) gauge field.
enceofcertainspatialsymmetries,theremaybeadirect,
Effective field theory constructions – In order to de-
continuoustransitionbetweentheFQHstateandthesu-
velop our theory, we need to provide a field theoretic
perfluid.
description that can naturally interpolate between the
A simple way to understand the basic idea is through states of interest. To do this, we will use the par-
the composite fermion [9] framework. The ν = 1/2 ton/projective construction [10]. For the Laughlin FQH
Laughlin state can be understood in terms of composite state, the Mott insulator, and the superfluid, the parton
fermions attached to one flux quantum each, such that construction is essentially equivalent to the composite
the mean-field state of composite fermions is a ν =1 in- fermion construction, although the former is preferable
2
because it can describe a wider class of FQH states [11] When C = 0, (5) is simply = 1 ǫµνλa ∂ a , which
L 4π µ ν λ
and can be formulated even in the absence of a back- describes a gapped state with a unique ground state on
ground external magnetic field [12]. In this paper we all closed manifolds. The gapped f excitations are at-
2
will consider the situation where the bosons feel an ex- tachedtoaunitofflux,sotheyarebosonicexcitations. A
ternalmagneticfield,becauseitis moredirectlyrelevant carefulanalysisfollowing[10]revealstherearenogapless
to ultracold atom proposals, though the theory can be protected edge states. Such a gapped state with solely
generalized to cases without an external magnetic field. bosonicexcitationsanduniquegroundstatedegeneracies
We write the boson operator b(r) as is a Mott insulator. This result can also be cast within
the composite boson language[4], where the originalbo-
b(r)=f (r)f (r), (1)
1 2 son is considered to be a composite boson φ attached to
two units of flux. Performing the flux smearing approxi-
where f and f are charge 1/2fermions. This construc-
1 2
mation gives composite bosons in no net magnetic field.
tion introduces an SU(2) gauge symmetry [13]. Since
The φ = 0 and φ = 0 states correspond to the FQH
the fi carrycharge1/2,they effectively see half as much h i 6 h i
state and Mott insulator, respectively. This is just the
magnetic field; thus for bosons at ν = 1/2, the density
bosonizeddescriptionof the C =1 and C =0 composite
of f is suchthat their effective filling fraction is ν =1.
i fi fermion description of these states.
To describe the ν = 1/2 Laughlin state, we assume a
Sinceaisadynamicalgaugefield,todescribeagapped
mean-field ansatz that breaks the SU(2) gauge symme-
state, the gaugefluctuations mustbe gappedand, to de-
try to U(1) and where f form ν = 1 IQH states. Let-
i fi scribe a fractionalized state, the gauge theory must be
ting a denote the emergent U(1) gauge field and A the
at a deconfined fixed point. Since CS gauge theories are
background external gauge field, integrating out f and
1
relabelling a a+ 1A gives gapped [14] and representdeconfined quantum field the-
→ 2 ories[15, 16], the above constructioncan be usedto rep-
1 1 resent FQH states. However,when C = 1, from (5) we
=f†iD f f†D2f + ǫµνλa ∂ a +δ , −
L 2 0 2− 2m 2 2 4π µ ν λ L seethatthereisnoCStermfora. RestoringtheMaxwell
eff
(2) termsto(5),theeffectiveactionisperturbatively,tolow-
est order, given by
where the covariant derivative is D = ∂ ia iA ,
µ µ µ µ
− − 1 1 1 1
and δ includes additional interactions, external poten- = ǫµνλA ∂ a + f2+ F2+ fF, (6)
tials, Letc. This is the same theory obtained by the flux L 2π µ ν λ g12 g22 g32
attachment and flux smearing mean-field approximation where the Maxwell term is f2 f fµν, and similarly
µν
in the composite fermion theory, where f is the com- ≡
2 for the last two terms, and we have assumed Lorentz
posite fermion. At energies well below the gap of the
invariance for simplicity. Since there is no CS term
f1 state, a hole off1 canbe createdby inserting 2π flux; ǫµνλa ∂ a , we must reconsider whether the gauge fluc-
µ ν λ
thus,forenergiesbelowthegapofthef state,theboson
1 tuations are gapped. Without the CS term, in 2+1 di-
b can be represented by the operator
mensionalcompact U(1) gauge theory,instantons prolif-
b=Mˆf , (3) erate and condense at low energies, yielding a contribu-
2 tion e−S0Mˆ +H.c. to the effective action [17]. This in-
where Mˆ is an instanton operator that creates 2π flux ducesagapfora. Howeverthistermcannotbeaddedto
(6). From the mutual CS term ǫµνλA ∂ a , we see that
of a. Integrating out f , which is assumed to form a µ ν λ
2
ν = 1 IQH state, and relabelling a a 1A leads to flux ofa carrieselectriccharge. Mˆ, whichinstantly adds
thf2efollowingeffectiveaction,tolowest→orde−ri2nthegauge 2π flux, instantly causes a local depletion of the charge
fields and their derivatives: density; to satisfy charge conservation, it must create a
currentj δ(t), which costs an infinite action. Thus in-
2 1 1 ∼
= ǫµνλa ∂ a + ǫµνλA ∂ A . (4) stantons alone are suppressed at energies below the gap
µ ν λ µ ν λ
L 4π 24π of the fermion states [18, 19]. Since Mˆ creates a hole in
This gives the correct Hall conductance and reproduces the parton IQH states, the only possible instanton term
the correct topological degeneracies of the ν = 1/2 that might be added to the effective action at low ener-
Laughlin state [13]. gies,belowthefermiongap,isoftheformMˆf†f†+H.c..
1 2
Now suppose that δ is chosenin such a way that the The fermionoperatorsfill in the hole createdby the flux
L
lowestbandfor f has a generalChernnumber, C. Inte- insertion,thus keeping the chargedensity uniform. Such
2
gratingout the fermions results in the followingeffective a term is gauge-invariant if, under a gauge transforma-
theory, to lowest order: tionf eiγ/2f , A A ∂γ,Mˆ eiγMˆ. Suchaterm
i i
→ → − →
does not gap out the gauge field, and leads to sponta-
C+1 C C
=ǫµνλ a ∂ a + A ∂ A + A ∂ a . neous symmetry breaking of the fermionnumber conser-
µ ν λ µ ν λ µ ν λ
L (cid:20) 4π 4π 2π (cid:21)
vation[20]. Proliferationoftheseallowedinstantonsmay
(5)
be viewed as the mechanism within the gauge theory by
3
whichthe fermionnumber conservationis spontaneously Critical points occur when some m = 0.2 In the ab-
i
broken [20]. senceofanysymmetries,thegenerictransitionfromFQH
Fromthe action(6),we seethatmagnetic fluctuations to SF therefore is through the Mott insulator. However,
of a are charged under the external gauge field, which certain spatial symmetries may force m1 = m2 (see be-
implies thattheycorrespondtodensity fluctuations[18]. low), in which case there is a single tuning parameter
ThusaisdualtothesuperfluidGoldstonemode. Infact, that tunes between the superfluid and the FQH state.
(6) is dual to the standard superfluid action, as can be Integrating out a Dirac fermion with mass m coupled
seen by introducing ξµ 1 ǫµνλ∂ a and a Lagrange to a gauge field a yields a CS term sgn(m) 1 ǫµνλa ∂ a .
≡ 2π ν λ 2 4π µ ν λ
multiplierϕtoenforcetheconstraint∂ ξµ =0,andsub- Thus, we consider the following Lagrangian[23]:
µ
sequently integrating outξ . This yields (∂ϕ A)2.
µ
Alternatively,integratingoutain(6)yieldLs∝thesta−ndard N k Nf
tshuapterwflhueidnrfespfiollnssbeaLnd∝swAiµth(δµCν=−pµp12p,νt)hAeνr.eWsueltcinogncsltuadtee LNf,k = 4fπ ǫµνλaµ∂νaλ+Xi=1[ψ¯iγµDµψi+mψ¯iψi].
2 − (8)
isasuperfluid.1 IntheSupplementaryMaterials,wegive
a further discussion of how such a construction can de- TheMI-SFtransitionisdescribedby ,theFQH-MI
1,1/2
scribe a compressible state. transition is described by , andLthe FQH-SF tran-
1,3/2
L
We note that within this effective field theory descrip- sition is described by (see Fig. 1). This “fermion-
2,1/2
L
tion, a deformation of the composite fermion bandstruc- ization”ofthe3DXYtransitionwasalreadyconjectured
ture that causes the bands to overlap will result in a in [6]. A crucial point is that the FQH-MI transition is
compressible non-Fermi liquid state, with a composite differentfromthe MI-SFtransitionbecauseofthe coeffi-
fermion Fermi surface [21]. cientoftheCSterm,whichaffects thecriticalproperties
Criticaltheory–ThecriticaltheoriesbetweentheFQH [5, 6].
state, MI, and SF therefore occur when the composite The critical exponents can be computed through a
fermion f bands touch and their net total Chern num- large N expansion, which has already been performed
2 f
ber changes. The transition between the SF and the [6], motivated by the case N = 1. This is a relativistic
f
ν = 1/2 FQH state occurs when the total Chern num- transition, with dynamic critical exponent z = 1. The
ber of f changes from 1 to 1. This can happen either correlation length exponent ν is defined by ξ m−ν,
2
− ∼
at a quadratic band touching or at two Dirac cones; the whereξ is the correlationlengthandm is the tuning pa-
generic,stablecaseistwoDiraccones,becausequadratic rameter. ν can be determined by the dimension of the
band touchings are marginally unstable to repulsive in- mass term. In the large N limit, it was found to be
f
teractions [22]. To describe this, let ψ(r) be a two-
componentfermion that describes the two f2 bands that ν−1 =1+ 128 [128−(π/k)2] 1 +O(1/N2), (9)
are involved in the transition, so that at low energies, 3 [64+(π/k)2]2k2Nf f
f (r) cT(r)ψ(r), where c(r) is a two-componentscalar
2
∼ although for N = 1 the leading 1/N correction was
function of r; ie at low energies f (r) is a linear combi- f f
2
found [6] to be insufficient for accurately giving the 3D
nationofthe twobandsdescribedbyψ. Nearthetransi-
tion, at low energies ψ(r) ∼ 2i=1eiKirψi(r), where the XNY=va2lu,eko=f ν1/−21, ∼we1e.x5p.ecFtotrhethlearFgeQNH-SFexptraannssioitniotno,
Dirac points occur at momenPta Ki and ψi are the two- befmore reliable, and we get ν−1 =1.705.f..+O(1/N2).
component fermions obtained by linearizing about the f
Dirac points. The critical theory is: At low energies the boson operator is b Mˆψ, so
∼
the scaling dimension ∆ of b must be found by analyz-
b
1 ing the dimension of the monopole operator combined
= ǫµνλa ∂ a +ψ¯γµD ψ +m ψ¯ψ , (7)
L 4π µ ν λ i µ i i i i with the fermion. If there are Nf Dirac points in the
Brillouin zone, at momenta K , for i = 1, ,N , then
i f
fori=1,2,ψ¯i =ψi†σz,γ0 =σz,γx =σx,γy =σy,where ψ(r)∼ ieiKirψi(r). Sofar,thescalingdim··e·nsionofan
σi arethe Paulimatrices. When both mi <0, we obtain operatoPrlikeMˆψiisknownonlyintheNf limit. In
→∞
the superfluid state, when mi > 0, we obtain the FQH that limit, the scaling dimension of b is ∆b =∆M +∆ψ.
state, and if mi have opposite signs, then we have the Furthermore,inthelargeNf limit,∆M canbecomputed
Mott insulator (see Fig. 1).
2 Note that inaddition, chemical potential terms µiψi†ψi are rel-
1 While this appears surprising, we note that it is implicitin [6], evant operators that lead to a composite Fermi liquid. Never-
whereitwasarguedthatthe3DXYcriticalpointcanbefermion- theless, spatial symmetries can impose µi = µ, and if particle
ized. However, wherethereisoverlap,someofourresults differ number is held fixed, as in cold atoms settings, the composite
fromthoseof[6]. Similarly,[18]usesanequivalent construction Fermi liquid can be avoided and one can tune through these
inadifferentcontext, foranXY N´eelstate. transitionswithasinglemassparameter.
4
independentatthetransition. Finally,fromgeneralscal-
E ing considerationswecanconclude thatthe specific heat
scales like C T2.
v
n = 1 ∼
Physical realizations – The transition described here
is generic, and therefore can occur in principle in many
differentphysicalrealizationsinvolvingstronglyinteract-
C = -1 ingsystemsofbosons. Aparticularlypromisingvenueto
n = 0 C = 1
realize bosonic FQH states is in optical traps of ultra-
cold atoms [3], where strongly interacting bosons in a
backgroundeffectivemagneticfieldcanberealized. Now
FIG. 2: Evolution of composite fermion bands as a periodic
consider adding an external periodic potential V (r)
potential is turnedon and tuned in an appropriate way. Red pp
with flux 2πp/q per plaquette. This induces a term
labels filled states and blue labels empty states. The flat
bandsonthefarleftindicatetheLandaulevelsindexedbyn. δHpp =Vpp(r)b†(r)b(r)intheHamiltonianofthebosons.
Assuming that the composite fermion effective theory is
the correct low energy description,4 the boson is repre-
sented by b = Mˆf , and therefore b†b f†f , because
2 ∝ 2 2
fromthestate-operatorcorrespondence[24],withthere- Mˆ†Mˆ 1+αf2+ ,where f2 is the Maxwelltermfor
sult ∆M =Nf(0.265...), while ∆ψ =1+O(1/Nf). a, α is∝a constant,·a·n·d indicate higher order deriva-
···
The order parameter exponent β for the superfluid is tives of the gauge field. Therefore, to leading order, the
defined by b mβ. Following the arguments in [1], compositefermioneffectiveactionobtainsacontribution
β can be seheni t∼o obey a generalized hyperscaling rela- δ V (r)f†f (r). Such a periodic potential may
Lpp ∝ pp 2 2
tion: β = ν∆b. The large ∆b implies that the onset be used to induce the Chern number of the composite
of superfluidity is quite weak at the FQH-SF transition. fermions to change. For small V , the Landau levels
pp
Additionally, from general hyperscaling arguments, the split into p subbands. As V is increased, the top sub-
pp
superfluid susceptiblity scales like χ mν(2∆b−d−z). If band of the filled LL may eventually touch the bottom
∼
we naively plug in ∆b = 0.265Nf +1, then for Nf = 2, subband of the next empty LL, causing a change in the
χ is non-divergent at the critical point.3 total Chern number of the filled bands. Spatial sym-
The scaling of the compressibility and conductivity metries can force the Chern number to change by two
follow from current-current correlation functions, which units, causing a continuous FQH to superfluid transi-
do not acquire any anomalous dimensions, and thus are tion. The necessary spatial symmetry depends on the
similar to other two-dimensional transitions with z = 1 nature of the Vpp. There can be many ways this can
[25, 26]: happen, and the most optimal one depends on the given
experimental setup. One example is to turn on a hon-
k k eycomb lattice with 2π flux per plaquette. In the limit
Π (k)=(δ + µ µ)Π (k)+ǫµνλk Π (k). (10)
µν µν k2 e λ o of large Vpp, we can pass to the tight-binding limit with
nearest and next-nearest neighboring hopping, with two
Near the critical point, Πµν low-lyingbandswithChernnumber 1forthetwobands
ddxdt Jµ(x,t)Jν(0,0) ξ1−d. Therefore, Πe(k) ∼k [27]. If the Chern number of the bo±ttom band is 1, it is
h i ∼ ∼
aRnd Πo(k) O(k0) at the critical point. From this we possible in principle to achieve this regime without clos-
∼
conclude that the compressibility vanishes at the critical ing the energy gap. As the second neighbor hopping is
point at zero temperature, while the conductivities are tuned through zero, there will be two band touchings,
universalconstantsthatcanbecomputedinthelargeNf causing the Chern number to change directly from 1 to
limit [6]. Therefore the DC longitudinal resistivity ρxx 1. It is the C3v symmetry of the honeycomb lattice
−
is zero on either side of the transition, but is a universal that protects the two Dirac cones in this case when the
non-zero number of order h/e2 at the transition, while second neighbor hopping is zero [27].
the DC Hallresistivity ρ is zero onthe superfluidside,
xy Note that the same theory presented here can be used
h/2e2 on the FQH side, anda universalnumber of order
to develop a theory of a continuous transition between a
h/e2 at the critical point.
chiralspinliquidandanXY antiferromagnet. Also,note
The temperature dependence of the polarization ten- that the transitions considered here generically require
sor at the critical point can be found by replacing k,ω time-reversalsymmetry tobe broken,eitherexplicitly or
byT. Itfollowsthatthecompressibilityatthetransition spontaneously, in the superfluid state before the critical
scales like κ T, while the conductivity is temperature
∼
4 This may not be the case if the gap of the parton f1 and f2
3 WethankT.Senthilforemphasizingthispoint. bands arenearlyequal.
5
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the zero temperature phase diagram. [13] X.-G. Wen, Quantum Field Theory of Many-Body Sys-
tems (Oxford Univ.Press, Oxford, 2004).
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6
Supplementary Material: Compressibility of slave particle/composite fermion construction of superfluid state
Inthis sectionwe willstudy insomemore detailhow the partonconstructionofthe superfluidstatemanagesto be
compressible. As discussed in the main text, the parton construction of the superfluid state is as follows. We rewrite
the boson operator as
b(r)=f (r)f (r), (11)
1 2
where f are fermions. Next, we consider a mean-field ansatz where f forms a band insulator with Chern number 1,
i 1
while f forms a band insulator with Chern number 1, and suppose that these band insulators are created by the
2
−
application of an external periodic potential. As discussed in the main text, such a construction yields a superfluid
state for the bosons, because the emergent U(1) gauge field a is gapless and can be associated with the dual of the
superfluid Goldstone mode.
Since the fermions form band insulators due to an external periodic potential, by themselves they have a preferred
density, which is set by the number of fermions per unit cell. Therefore it is not clear that the resulting state will be
compressible, as changing the density would appear to cost a finite amount of energy. However, as we will explain
below, such a construction does indeed yield a compressible state.
The compressibility κ of a zero-temperature quantum system is defined as
∂µ ∂µ ∂2E E(N +δ) 2E(N)+E(N δ)
κ−1 = =V =V N − − , (12)
∂n ∂N ∂N2 ∼ δ2
where µ is the chemical potential, n is the density, V is the volume, and E(N) is the ground state energy for N
particles, and the above derivatives are taken at constant volume. Thus we estimate the compressibility as
N∆ (N,δ)
κ−1 2 , (13)
∼ δ2
where
∆ (N,δ) E(N +δ) 2E(N)+E(N δ). (14)
2
∼ − −
The system is incompressible if, when we take δ √N and N , N∆2(N,δ) at fixed number density. In
∼ → ∞ δ2 → ∞
other words, the system is compressible if
lim ∆ (N,√N)< . (15)
2
N→∞ ∞
The choice δ √N is for convenience; more generally, one must take the limit δ,N with δ/N 0.
∼ →∞ →
In our slave-particle construction above, it was argued in the main text that the gauge field fluctuations of a are
gapless. Therefore consider a system of fermions with a filled band with a non-zero Chern number, and subject it to
a magnetic field that can vary with essentially zero energy cost. We now consider the energy E (N,φ), which is the
f
ground state energy of the fermionic sector of the partontheory, with N particles, and with additional φ flux quanta
of a added to the system. Since the flux φ is a dynamical quantity, and the gauge field a is gapless, the ground state
energy E(N +δ) E (N +δ,δ), where the optimal φ δ is approximately the additionalnumber of particles added
f
≈ ∼
to the system.
Now, we would like to know the fate of
N∆φ(N,δ)
κ−1 2 , (16)
∼ δ2
where now
∆φ(N,δ) E (N +δ,δ) 2E (N,0)+E (N δ, δ). (17)
2 ∼ f − f f − −
When the fermions fill a Landau level, ∆φ(N,√N) < as N . This is because the ground state energy of a
filled lowest Landau level is eBN/2m, wh2ere we set ~ =∞c =1. F→ro∞m this, it follows that ∆φ(N,δ) δ2, where A is
2 ∼ A
the areaofthe system, sothatκ−1 N/A,whichis boundedasN atfixedaveragenumber densityN/A. Thus
∼ →∞
the Landau level problem gives a compressible state, if we allow the magnetic field to vary arbitrarily. This makes
7
sense, since the density is only tied to the magnetic field, and once the magnetic field can vary arbitrarily,so can the
density.
Now consider a Chern insulator, such as Haldane’s honeycomb model with the lowest band filled [27]. We would
like to know whether
lim ∆φ(N,√N)< . (18)
N→∞ 2 ∞
If so, we can then conclude that the parton Chern insulator construction of the superfulid will also be compressible
if the gauge field a is gapless.
To establishthat(18)is true forsucha situation, considera continuumsystemwith a constantmagnetic field, ie a
Landaulevel problem,and consider adding a smallperiodic potential. Let λ parametrize the strengthof the periodic
potential, and consider ∆φ(N,δ,λ), where the last argument just parametrizes the value of λ in the Hamiltonian.
2
Clearly for small λ eB/m, we must have
≪
lim ∆φ(N,√N,λ)< . (19)
N→∞ 2 ∞
Furthermore, as long as we do not close the energy gap, continuously changing λ must always preserve the above
inequality. This is because as long as we do not close the energy gap, the ground state energy in the thermodynamic
limit is analytic in λ, and so the above inequality must continue to be satisfied as λ is changed infinitesimally.
Now, we know that it is possible to, for instance, slowly turn on a honeycomb lattice potential with 2π flux per
plaquette, such that even in the limit that the periodic potential is much stronger than eB/m, we do not close the
energy gap. In this limit, we end up with two bands, and if the lower band has Chern number +1, then it is possible
to adiabatically evolve from the continuum Landau level to this situation. For the Chern insulator with the lower
band having C = 1 and 2π flux per plaquette, it follows that (18) is satisfied, because we never had to close the
energy gap as we increased the periodic potential. Flipping the sign of the second nearest neighbor hopping in such
a model can flip the Chern number. We expect therefore that as long as C =1 or C = 1 for the bottom band, that
−
(18) will remain true.
WeconcludethatCherninsulators,inadditiontofilledLandaulevels,willsatisfy(18)andarethereforecompressible
if the magnetic field is allowedto vary arbitrarily. Since the fluctuations of the emergent U(1) gauge field are gapless
in the parton construction of the superfluid, the magnetic field can indeed vary arbitrarily, so we see that the parton
construction of the superfluid state is indeed compressible when the gauge fluctuations are taken into account.