Spontaneous Breakdown of Translational Symmetry in Quantum Hall Systems: Crystalline Order in High Landau Levels

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by  F. D. M. Haldane

Spontaneous Breakdown of Translational Symmetry in Quantum Hall Systems:
Crystalline Order in High Landau Levels
F. D. M. Haldanea, E. H. Rezayib, and Kun Yangc
aDepartment of Physics, Princeton University, Princeton, New Jersey 08544
bDepartment of Physics, California State University, Los Angeles, California 90032
1
cNational High Magnetic Field Laboratory and Department of Physics, Florida State University, Tallahassee, Florida 32306
0
0 (January 25, 2000, revised August, 2000)
2
Wereportonresultsofsystematicnumericalstudiesoftwo-dimensionalelectrongassystemssubject

n toaperpendicularmagneticfield,withahighLandaulevelpartially filledbyelectrons. Ourresults
a are strongly suggestive of a breakdown of translational symmetry and the presence of crystalline
J
order in the ground state. This is in sharp contrast with the physicsof the lowest and first excited

6 Landau levels, and in good qualitative agreement with earlier Hartree-Fock studies. Experimental
1 implications of our results are discussed.

] 73.20.Dx, 73.40.Kp, 73.50.Jt
l
l
a
Recentlytherehasbeenconsiderableinterestinthebe- ofthere-entrantphaseandwouldfurtherexplaintheob-
h
- havior of a two-dimensional(2D) electron gas subject to served threshold in conduction [3]. However this is not
s aperpendicularmagneticfield,whenahighLandaulevel entirelyconclusiveandothermechanismsfortheconduc-
e
m (LL) (with LL index N ≥ 2) is partially filled by elec- tion threshold are also possible [3].
trons. Thisislargelyinspiredbytherecentexperimental
.
t discovery [1–3] that the transport properties of the sys-
a
m tem are highly anisotropic and non-linear for LL filling D E
0.04 0.04
fraction ν = 9/2,11/2,13/2,···. Previously, Hartree-
-
d Fock [4–6] (HF) and variational studies [7] suggested 0.003
n that, unlike the N =0 and N =1 LL’s (in which either 0.002
o 0.03 0.03
incompressible fractional quantum Hall (FQH) or com- 0.001
c
[ pressible Fermi-liquid like states are realized), in N ≥ 2 0.000
LL’s the electrons formcharge density waves(CDW). In 0.6 0.7 0.8 0.9 1.0 1.1
3 0.02 0.02
particular, at half-integral filling CDW’s break transla-
v
tional symmetry only in one direction and form stripes.
4
9 Anisotropic transport would indeed result from such a
0.01 0.01
3 striped (or related) structure [8–11].
1 We neglect LL mixing, and consider the case where
0
the LL with index N has partial filling ν˜, while LL’s
0 0.00 0.00
with lower index are completely filled (ν = 2N +ν˜). By
0
/ particle-hole symmetry of the partially-filled LL, this is 0.50 0.75 1.00 Aspect
t
a equivalent to ν = 2N + 2 − ν˜. We also assume that
m the partially-filledLL is maximally spin-polarizedat the
FIG.1. Energy levels versus aspect ratio for quarter-filled
- ν˜ we consider. Previously [12], we studied such N ≥
d N =2 Landau level with eight electrons and rectangular ge-
2 LL’s with ν˜ = 1/2 by numerically diagonalizing the
n ometry. The inset is a blow-up of the low-energy spectra for
Hamiltoniansoffinite-sizesystems;thoseresultsstrongly
o aspect ratio between 0.6 and 1.0. The points at 1.1 (open
c supported the existence of stripe order. circle plusx) correspond to hexagonal unit cell.
: An outstanding issue is the nature of the groundstate
v
i athighLL’sforfillingssufficientlyfarfromthehalf-filled In this paper, we report on new numerical results
X
level. Koulakov, Fogler and Shklovskii [4,5] (see also on systems away from half-filling using the unscreened
r Moessner and Chalker [6]) predicted a novel crystalline Coulomb interactions. Remarkably, our results suggest
a
phasecalledthe“bubble”phasewithmorethanoneelec- that CDW’s are formed at all filling factors we have
tronper unit celloutside ofthe rangeν˜=0.4−0.6. The studied, including those that would support prominent
bubble crystal has lower energy than the Laughlin state FQH states or composite fermion Fermi-liquid states in
forν =4+1/3[7]. Experimentally,are-entrantquantum the lowest or first excited Landau levels. These CDW’s,
Hall state is found near ν = 4+1/4 which is quantized however, have 2D structures and are no longer stripes
as a ν = 4 LL plateau [1–3]. Evidently the electrons in whenthefillingfactorsaresufficientlyfarawayfrom1/2.
thetop-mostLLarefrozenoutofthetransport. Pinning TheyarenotWignercrystals[13]either,unlessν˜issmall
of a crystalline structure provides a natural explanation (below 0.2). In the intermediate filling factor range, we
1
find each unit cell of the CDW contains more than one There are similarities as well as important differences
electron. Our results are in good agreement with the between these spectra and those of half-filled high LL’s
predicted bubble phase and are the first exact finite-size [12]withstripeorder. Asinthestripecase[12],thelarge
calculations which exhibit a crystalline state in a system quasi-degeneracy of the ground state manifold is an in-
with continuous translational symmetry. dication of broken translational symmetry [15]. The dif-
ferencehereisthat(i)thedegeneracyismuchlargerand
(ii) the momenta of the low-lyingstates form 2Dinstead
K of1Darrays. Thesenewfeaturesindicatethatthetrans-
2 K
y y lational symmetry is broken in both directions and the
2 1 ground state is a 2D CDW. In the stripe state, on the
0 otherhand,thetranslationalsymmetryisonlybrokenin
1 -1 the directionperpendicular to the stripes. Therefore the
K degeneracy is smaller and the momenta of the low-lying
-2 x
states form a 1D array.
-2 -1 0 1 2
0
-1 3
4 q 0
K
-2 y
x -3
2
-3 0 3
-2 -1 0 1 2
0
FIG. 2. The allowed (circles) and the ground state mani-
fold momenta (×’s). The data is for a rectangular geometry
with asp = 0.77, 8 electrons in the N = 2 LL and ν˜ = 1/4. -2
The solid line is the boundary of the BZ. The superlattice
q
reciprocal basis vectors are shown by solid arrows. The inset
-4
gives thecorresponding results for an hexagonal unit cell. x
We restrict the states of the electrons to a given LL, -6 -4 -2 0 2 4 6
andworkwithperiodicboundaryconditions(PBC,torus
geometry) as in our previous paper [12]. We also set
FIG.3. A2Dplotofthepeaksoftheprojected(orguiding
the magnetic lengthtounity. To detectintrinsicallypre-
center)chargesusceptibilityχ(q)atreciprocallatticevectors,
ferredconfigurationsweconsiderarectangularPBCunit
for a system with Ne = 8 at ν˜ = 1/4, rectangular geometry
cell and vary its aspect ratio. The PBC plays a crucial witha=0.77. Thesizeofthecirclesgiveanindicationofthe
roleinremovingcontinuousrotationalsymmetry,andse- height of the peak at that point. Only responses above 100
lecting a discrete set of possible crystal orientations. havebeenplottedassolidcircles. Thezoneboundariesarenot
In Fig.1 we plot the energy levels of systems with within therange of thefigure. Thelargest circle corresponds
N =8electronsintheN =2LLatfillingfactorν˜=1/4 to16491. Theinsetgivestheresultsforahexagonalunitcell.
e
asafunctionofthe aspectratio. We alsoshowthe levels The ×’s are theallowed wavevectors.
of a system with a hexagonal PBC unit cell at the right
The momenta of the states in the ground state mani-
side ofFig.1. Agenericfeature ofthe spectrais the exis-
fold are the reciprocal lattice vectors of the bubble crys-
tenceofalargenumberoflow-lyingstateswhoseenergies
tal. Transformingto the directlattice vectors,we obtain
are almost degenerate, which we call the ground state
manifold. The momentaof these quasi-degeneratestates a1 =π/aeˆx and a2 =π/2aeˆx+π/beˆy. For the optimum
for rectangular geometry with aspect ratio asp = 0.77 system, with asp=0.77, we obtain a1 = 8.08, a2 = 7.42,
◦
andφ=57 . Thisisveryclosetoatriangularlattice. In
andhexagonalgeometry areshownin Fig.2; they forma
the case of hexagonal PBC unit cell, both the reciprocal
2D superlattice structure, which for the rectangular ge-
ometry have the super cell vectors b1 =2aeˆx−beˆy, and superlattice and its direct lattice are triangular.
b2 = 2beˆy. Where, a = 2π/L1, and b = 2π/L2. L1, and The number ND of distinct quasi-degenerate ground
L2 are the dimensions of the unit cell (L1×L2 =2πNΦ, statesallowsthenumberNbofbubblesinthesystem,and
hence the number M = N /N of electrons per bubble,
NΦ is the total flux quanta in the system). The areaper e b
wavevector in the Brillouin zone (BZ) is ab = (2π)2/A, to be immediately obtained through the relation NbND
=N¯2, whereN¯ is the highestcommondivisorofN and
where A is the (real space) area of the system. e
NΦ. Inourcase,N¯ =Ne =8,andND =16,whichgives
2
N =4andM =2[16]. TheWignercrystalwouldcorre- alsoseeninthe“guidingcenter(GC)staticstructurefac-
b
spond to Nb =Ne and M =1. In general [17], there are tor” S0(q) [12], which we present as a 3D plot in Fig.4.
N¯2distinctvaluesofthetotalmomentumquantumnum- Here we see sharp peaks with an approximate six-fold
ber,whichdefineaBZofarea(2πN¯)2/A. Iftranslational symmetry at the primaryreciprocallattice vectors,indi-
symmetry is broken, the area of the BZ of the superlat- catingthe presenceofstrongdensity correlationatthese
tice is then (2πN¯)2/AN , which must be (2π)2/(A/N ), wave vectors in the ground state.
D b
where A/N is the area per bubble; hence N N = N¯2.
b b D
6
4
8 6 -8 2
4 4 -6 0
0 0 2 -4 -2 0 -2
-4 -2 2 4 -4
-4 6 -6
-8 -6 8
FIG. 5. Real-space “projected density” (guiding center)
FIG. 4. A 3D plot of the guiding center structure factor correlation function, derived from Fig.4. A 2D contour plot
S0(q)(samesystemasFig.3). Thesignatureofthehexagonal is also included below the3D plot.
lattice is seen in the near six-fold symmetry of thepeaks.
We next turn to the density response functions. In
Fig.3 we show the projected ground state charge sus-
ceptibility [12] χ(q) of one of the optimum rectangular
andhexagonalsystemsdescribedabove. The calculation
takes into accountthe contributions from the two lowest
energystatesineachsymmetrysubspace;thisisanexcel-
lent approximation in view of the fact that the response
functionisdominatedbylowenergystatesbecauseofthe
energydenominator. Wenotethatχ(q)exhibitsastrong
response at the reciprocal lattice vectors (Bragg condi-
tion); the background at other wave vectors (shown by
×’s in Fig.3) are negligible comparedto these responses.
6
The originofthe strongresponseliesinthe approximate 4
2
degeneracy among the states forming the ground state
-8 0
-6
manifold. The system responds very strongly to a po- -4 -2 -2
tential modulation with a wave vector that connects the 0 2 -4
4
ground state to one of the low lying states (which must 6 -6
8
be a reciprocal lattice vector) because of the small en-
FIG. 6. N = 2 Landau level full electron density correla-
ergydenominator. Thisis alsoanotherreasonwhy there
tions (relative to a guiding center), derived from Fig.4.
must be one low-lying state for each reciprocal lattice
vector. Asecondnotablefeature is the almosthexagonal
InFig.5andFig.6weplotgroundstate“projectedden-
symmetryofthe response,despitethe factthatthe PBC
sity” correlation functions in real space. These describe
geometry used in this case was rectangular. This indi-
correlations relative to the GC (not the coordinate) of
cates that the bubbles tend to form a triangular lattice,
a particle. The first is the Fourier transform (FT) of
in agreement with the predictions of HF theory. S0(q)exp(−q2/2), which is the electron density of an
The tendency toward forming a triangular lattice is
equivalent lowest-LL system, and gives information on
3
the spatialdistributionofGC’s. Thesecondisthe FTof alwayspinnedbyadisorderpotential,andinanonlinear
S0(q)[LN(q2/2)]2exp(−q2/2) with N = 2 (LN is a La- transport measurement, there should be a threshold de-
guerrepolynomial): this (plusthe uniformdensityofthe pinningfieldatwhichthereisasharpfeatureintheI−V
filled LL’s) represents the actual electron density. curve. Aweakercrystalwouldresultinmorediffusecon-
In Fig.5 the presence of four bubbles and the relative duction threshold as various portions of the crystal get
orientation of the bubbles can be clearly seen and there depinnedatdifferentcurrentvalues,whileastrongerone,
is strong crystalline order of the GC distribution. The onthe otherhand,willhavesharpconductionthreshold.
central peak contains two electrons, one of which is the ThisisconsistentwiththeobservationofCooperetal[3]
particle with the GC at the origin. For N > 0, as in that there is a sharp threshold region at about ν˜ = 1/4,
Fig.6,onlyweakorderisdisplayedbytheactualelectron but more diffuse thresholds at both higher and lower ν˜.
density, because of the averaging effect of the cyclotron In contrast to ν˜ = 1/4 and ν˜ = 1/3, the spectra for
motion around the GC’s. It is the guiding centers of the ν˜=2/5wasfoundtobeverysimilartoν˜=1/2[12]. The
electrons that form bubbles as anticipated in reference 7 momenta of the low-lying states belong to a 1D array,
(Fig.1). The electrons themselves manage to stay apart indicating formation of a 1D CDW or stripe phase; the
tolowertheCoulombrepulsion,inspiteoftheclustering weight of the HF state 11110000001111000000(N =8),
e
of their GC’s. inarectangulargeometrywithanaspectratioof0.80,is
about 65%. We conclude that the transition from stripe
to bubble phases occurs between ν˜ = 1/3 and ν˜ = 2/5,
D in qualitative agreement with HF predictions [5,6]. We
E 2
have also studied higher LL’s. The results are similar
0.03 0 and will be reported elsewhere.
We have benefited from stimulating discussions with
-2
J. P. Eisenstein, K. B. Cooper and M. P. Lilly. We
0.02 -2 0 2 thank B. I.ShklovskiiandM. M. Foglerfor helpful com-
ments. This work was supported by NSF DMR-9420560
andDMR-0086191(E.H.R.), DMR-9809483(F.D.M.H.),
0.01
DMR-9971541,andtheSloanFoundation(K.Y.). E.H.R.
acknowledgesthe hospitalityofITPSantaBarbara,sup-
0.00 ported by NSF-PHY94-07194, where part of the work
was performed.
0.5 0.6 0.7 0.8 0.9 1.0
Aspect
FIG.7. Spectraof systemswith eight electronsat ν˜=1/3
in the N = 2 LL, with rectangular geometry and various
[1] M.P.Lilly,K.B.Cooper,J.P.Eisenstein,L.N.Pfeiffer,
aspect ratios. The inset plots the momenta of the low-lying and K.W. West, Phys. Rev.Lett. 82, 394 (1999).
states for asp = 0.75, in thesame way as Fig. 2.
[2] R. R. Du, D. C. Tsui, H. L. Stormer, L. N. Pfeiffer, K.
W.Baldwin,andK.W.West,SolidStateCommun.109,
WehavealsoexploredotherfillingfactorsintheN =2 389 (1999).
LLsuchasν˜=2/5andν˜=1/3,wherethesystemwould [3] K.B.Cooper,M.P.Lilly,J.P.Eisenstein,L.N.Pfeiffer,
condenseinto prominentFQHstatesif itwasinthe low- and K.W. West, Phys. Rev.B 60, R11285 (1999).
est LL; here, however, our studies suggest formation of [4] A.A.Koulakov,M.M.Fogler,andB.I.Shklovskii,Phys.
CDW’s instead. For ν˜ = 1/3 we obtain similar behav- Rev. Lett.76, 499 (1996).
[5] M.M.Fogler,A.A.Koulakov,andB.I.Shklovskii,Phys.
ior to ν˜ = 1/4: the energy spectra as a function of the
Rev. B 54, 1853 (1996).
aspect ratio shown in Fig.7 is very similar to Fig.1 and
[6] R. Moessner and J. T. Chalker, Phys. Rev. B 54, 5006
indicates formationofa 2D CDW. Using the degeneracy
(1996).
ofthegroundstatemanifold,wefindthenumberofelec-
[7] M.M.FoglerandA.A.Koulakov,Phys.Rev.B55,9326
trons per bubble is also two. The energies of the states
(1997).
in the ground state manifold, however, are not as close [8] E. Fradkin and S. A. Kivelson, Phys. Rev. B 59, 8065
as the ν˜ = 1/4. This results in weaker peaks in χ(q) at (1999); E. Fradkin, S. A. Kivelson, E. Manousakis and
the reciprocal lattice vectors. K. Nho, Phys.Rev.Lett. 84, 1982 (2000).
We interpret this to be an indication that, in this LL, [9] H. A.Fertig, Phys.Rev. Lett. 82, 3693 (1999).
ν˜=1/4 is more favorable than ν˜=1/3 for formation of [10] A.H.MacDonaldandM.P.A.Fisher,Phys.Rev.B61,
a two-electronbubble phase. In real systems a crystal is 5724 (2000).
[11] T. Stanescu, I. Martin, and P. Phillips, Phys Rev. Lett.
4
84, 1288 (2000).
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Phys.Rev.B 19, 5211 (1979).
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editedbyR.E.PrangeandS.M.Girvin (Springer,New
York,1990).
[15] True broken translational symmetry with exact ground
state degeneracy only occurs in the infinite-sizelimit; at
finitesystem size, thisbecomes a quasi-degeneracy.
[16] This result is in overall agreement with the estimate of
reference 5: M =3ν˜N.
[17] F. D.M. Haldane, Phys.Rev.Lett. 55, 1095 (1985).
5