Table Of ContentMicroscopic theory of the quantum Hall hierarchy
E.J. Bergholtz, T.H. Hansson, M. Hermanns, and A. Karlhede
Department of Physics, Stockholm University,
AlbaNova University Center,
SE-106 91 Stockholm, Sweden
8
(Dated: February 6, 2008)
0
0
We solve the quantum Hall problem exactly in a limit and show that the ground states can be
2
organized in a fractal pattern consistent with the Haldane-Halperin hierarchy, and with the global
n phase diagram. We present wave functions for a large family of states, including those of Laughlin
a and Jain and also for states recently observed by Pan et. al., and show that they coincide with
J the exact ones in the solvable limit. We submit that they establish an adiabatic continuation of
3 our exact results to the experimentally accessible regime, thus providing a unified approach to the
hierarchy states.
]
l
l PACSnumbers: 73.43.Cd,71.10.Pm
a
h
- AkeyconceptinquantumHall(QH)physicsisthatof
s 1/3
e an incompressible electron liquid. In the integer effect,
m theformationofthisliquidcanbeunderstoodintermsof 0.3
. independent electrons moving in a magnetic field in the
t
a presence of a small but crucial amount of disorder[1]. In 0.25
m theexperimentallyverysimilarfractionaleffect,theelec-
ond- tawrceotlnilounlin.qduTeidrhsetisoforfoadrcmitnioetndeartlmhlrisqoouufgidLhsa,tuhtgheheelilsneic’mstrpwolanevs-teeloefucftnwrcohtniiocinhnsta[e2rr]e-, ~disorder 1/q0.01.25 2/7 2/5 3/7
are highly correlated quantum systems with remarkable
c 4/9
[ properties. 0.1 3/11 4/11
There are two alternative microscopic approaches to 4/13 5/13
2 5/17 6/19 6/17 7/19 7/17 8/19
v the fractional liquids[26]—the hierarchy and composite 0.05
6 fermions. In the former, successive condensation of the
1 fractionally charged quasiparticles leads to a hierarchy 0
0.25 0.3 0.35 0.4 0.45 0.5
5 of ever more complex QH-states[3, 4]. In the latter, n=p/q
2
the electrons bind magnetic flux quanta to form new
0
particles, composite fermions, that see a reduced mag-
7 Figure1: Globalphasediagram. Theglobalphasediagram
netic field, and the fractional effect is an integer effect
0 in the filling factor-disorder plane for 1/4 ≤ ν ≤ 1/2 (other
t/ of these new particles[5, 6]. However, neither compos- regionsaresimilar). Crossesmarkfractionswherewavefunc-
a ite fermions nor the hierarchy has a solid theoretical tions are constructed using conformal field theory.
m foundation, and they may well be complementary views
of the same phenomena rather than mutually excluding
-
d descriptions[7, 8]. The hierarchy construction is closely
daulevel[13]. At present,there are no agreedupon wave
n related to the global phase diagram, see Fig. 1, which
functions for these states and the proper interpretation
o
provides an overallpicture of the QH-states. This phase
c of them is under debate. In this Letter, we give theo-
diagram, which can be derived using a Chern-Simons-
: reticalsupport for the globalphase diagramand present
v Ginzburg-Landau approach[9], and exhibits an intrigu-
an explicit realization of the hierarchy in a well-defined
i
X ing modularsymmetry[10],is supportedby experiments, and solvable limit. We also report explicit and testable
but deviations have also been reported[11], and its pre-
r wave functions for the states that are obtained by suc-
a cise status remains an important open problem.
cessive condensationof quasielectrons—thesewave func-
A great advantage of the composite fermion scheme is tions agree with our exact solution in the solvable limit,
that it provides very good and explicit wave functions and naturally encompass both the Laughlin/Jain wave
for the Jain states with ν =n/(2kn±1), n,k=1,2,..., functions and states such as 4/11, reported in Ref. 12.
whichcanbeinterpretedasintegerQH-statesofcompos- For other attempts to construct hierarchy wave func-
ite fermions. Until recently, all observed states were of tions, some of them with explicit reference to composite
this type; in 2003, however, experiments on ultra clean fermions, see e.g. Refs. 8, 14.
samples revealed states at other fractions, such as 4/11 There is a simple and striking consequence of the ex-
and 6/17[12]. These new states point towards a frac- perimentinRef. 12: QH-statesareobservedatallfilling
tal, self-similar, structure of states in the lowest Lan- factors ν = p/q for q ≤ q ≈ 17 in the experimental
0
2
range of ν (here q is odd and p,q are relatively prime), as well as the fractionally charged quasiparticle excita-
see Fig. 2. A QH experiment is performed on a sample tions. Furthermore, in this limit the hierarchy construc-
with a certain amount of disorder—the lower the disor- tionoftheQH-statesismanifest,andthestabilityofthe
der is, the more fractions are observed. This defines a states decreases monotonously with q. Second, for each
notionof stability for a state: a more stable state is seen fraction where the state is obtained by (successive) con-
at higher disorder than a less stable state. Our interpre- densation of quasielectrons (as opposed to quasiholes),
tation of Ref. 12 is that: 1) there is a QH-state at any see Fig. 1, we present explicit wave functions for the
ν = p/q ≤ 1, q odd, and 2) the stability of the state groundstateandwavefunctionsforthequasiparticleex-
decreases monotonously with increasing q in agreement citations can also be obtained. These wave functions—
with the hierarchy prediction[4]. which are in the lowest Landau level—are obtained by
a unique and natural construction that exploits the in-
triguing relation between the QH effect and conformal
1/3 2/3
field theory (CFT). They are in one-one correspondence
0.3 1/q withthe exactsolutions,andreduceto these inthe solv-
able limit. Furthermore, the fractional charge and frac-
0.25 tional statistics of the quasiparticles are reflected in the
algebraicpropertiesofthe CFT-operatorsby whichthey
0.2 2/5 3/5 p/q are created. Third, we argue that this construction es-
1/q tablishes the adiabatic continuation of the results in the
solvablelimittothe experimentallyaccessibleregime(at
0.15
2/7 3/7 4/7
fractions where a QH-state is observed).
4/9 5/9
0.1 The solvable limit is obtained by considering the two-
4/11 5/11 6/11 7/11
4/13 5/13 8/13 dimensionalelectrongasonacylinderwithcircumference
5/17 6/177/17
0.05 L, and choosing the one-electron states centered along
rings around the cylinder—these states are gaussians
0 with width of order one magnetic length ℓ = ~c/eB
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p
n=p/q along the cylinder. This maps the QH problem onto
a one-dimensional lattice problem with lattice constant
2πℓ2/L. A basis of many-electron states is given by
Figure 2: Observed states and fractal structure. For
each rational filling factor, ν = p/q ≤ 1, q odd, there is a |n1n2,...i, where nk = 0,1 depending on whether site
uniquehierarchystateanditsstabilityincreasesmonotonously k is empty or occupied. When L/ℓ → 0 the overlap be-
with1/q. Thefractalstructureofstatesismanifest. Statesin tween different one-electron states vanishes and the en-
the region 2/7 ≤ ν ≤ 2/3 reported in Ref. 12 are marked by ergyeigenstatesaresimplythestateswheretheelectrons
crosses; plusses mark fractions were we infer a possible weak occupy fixed positions, |n n ,...i, and the ground state
1 2
signalfromthedatainRef. 12. Thehorizontallinemarksthe is the one that minimizes the electrostatic repulsion[15].
extentinν oftheexperimentandisalineofconstantgapand
At ν =1/3,this is obviously the state where every third
the approximate boundary for the observed states. The inset
site is occupied[16]. For general filling factor, ν = p/q,
shows the structure of hierarchy states: At each ν = p/q, q
the ground state is a gapped crystal—or Tao-Thouless
odd, there is a state with gap ∼ 1/q and quasiparticles with
(TT) state—with p electrons in a unit cell of length q.
charge ±e/q. When these condense two sequences of states
For example, at ν = 1/3,2/5,3/7,... the unit cells are
approaching p/q with decreasing gap are obtained.
100,10010≡10 10,1001010≡10 (10) ....
2 2 2
The statement that there is a QH-state for each ν = The lowest energy charged excitations at ν = p/q are
p/q ≤ 1 needs to be qualified. We introduce a parame- quasielectrons and quasiholes with charge ∓e/q, where
ter L such that the experimental regime is obtained as −e is the electron charge; in the thin limit these are do-
L→∞. WhenL→0theproblemisexactlysolvableand mainwalls in the TT-state andthe chargeis determined
thegroundstateanditsquasiparticleexcitationshaveall by the Su-Schrieffer counting argument[15, 17]. For ex-
the qualitative properties of a QH-state. If there is no ample,the excitationsatν =1/3areobtainedby insert-
phase transitionas L increases,then the QH-state is ob- ing or removing 10. When the filling factor is gradually
served at this filling factor. However, phase transitions increased away from 1/3 a gas of such quasielectrons of
may occur leading to other states such as Wigner crys- increasingdensity is formed. These repel each other and
talswheretherepulsionfreezestheelectronsinaregular condenseto formnew groundstates. Eventually,one ex-
lattice, striped states where the electron density varies citationperunitcellhasbeenaddedandthenewground
periodically or a Fermi gas (for even denominators q). state has unit cell 10010—the filling factor is then 2/5.
Togiveadetailedargumentforthepropositionsinthe However,before this happens lowerdensity condensates,
introduction, we proceed in three steps. First we solve withonequasielectronper2k−1unitcells100willform,
the interacting spin-polarized many-electron system ex- giving new ground states with unit cells (100)2k−110 at
actlyforeachfillingfactorν =p/q ≤1inacertainmath- fillingfactorsν =2k/(6k−1),k =1,2,.... Similarly,de-
ematicallywell-definedlimit. Weobtainthegroundstate creasingthefillingfactorawayfrom1/3givestheground
3
states with unit cells (100)2k−11000 at ν = 2k/(6k+1). global phase diagram of Refs. 9 and 10.
This pattern is general: The TT-state at ν = p/q is the Given the assumptions, this establishes the global
parentstatefortwosequencesofdaughterstatesthatap- phase diagram as L → 0. As L → ∞, the TT-state
proach ν = p/q from above and below and are obtained at any ν =p/q, q odd, develops into a QH-state, as will
by condensation of decreasing densities of quasielectrons be argued below. If there is no phase transition when L
and quasiholes respectively, see the inset in Fig. 2. For grows, then this state is the ground state in the experi-
details and proofs we refer to Refs. 15, 18. mentalregimeandtheQHeffectisobservedatthisfilling
To summarize, at each ν = p/q ≤ 1, q odd, there is fraction. However,phasetransitionsmayoccurandother
a TT-state, which we argue below develops into a QH- statesmaybeobserved. Webelievethatthisexplainsthe
state as L→∞, and these states are formed from other observeddeviationsfromthe globalphase diagram,such
TT-states by condensation of quasiparticles. This may as the insulating phases near Laughlin fractions[11]. In
alsobeinterpretedasthequasiparticlesfillinganeffective fact, the even denominator fractions are an example of
Landau level as one goes e.g. from ν =1/3 to ν =2/5. this phase transition scenario. Even though we have ex-
The QH effect is destroyed by disorder—at a certain cluded them from the discussion above, the gapped TT-
amount of disorder only states with a stability above states are the ground states also for these fractions as
some threshold occur. A measure of the stability is L → 0. We believe that the well-understood ν = 1/2
the energy it costs to create a quasielectron-quasihole case is representative: As L increases, the gapped TT-
pair. Inthethinlimit,thisdecreasesmonotonouslywhen state, which has unit cell 10, gives way to a state with
q increases[18]—in good agreement with the hierarchy gapless neutral excitations for L ∼ 5ℓ—a Luttinger liq-
prediction[4]. Fig. 2 shows 1/q for all hierarchy states uid; this state then develops smoothly into the observed
in the lowest Landau level. There is a unique state at gapless two-dimensional bulk state[15, 19]. This shows
eachrationalfilling factor, so this is simply a plot of 1/q that the small L limit may describe also non-hierarchy
for all rational numbers ν = p/q in the interval [0,1] states.
(where q is odd and p,q are relatively prime). This is a It is crucial that our results can be extended from the
self-similar, fractal, structure—enlarging any interval of thin cylinder, were they are established, to the experi-
ν gives an identical figure. Since the solvable limit pre- mentally relevant two-dimensional bulk case. First we
dicts that the gap increases with 1/q, states above some note that the qualitative features of the TT-states and
roughlyhorizontallineshouldbeobservedinagivensam- the QH hierarchy states are the same: they have a gap,
ple. In the figure, the states observed in the interval thesamequantumnumbersand,inparticular,theyhave
2/7 ≤ ν ≤ 2/3 for an extreme high mobility sample[12] quasiparticleswiththesamefractionalcharge. Alsonote,
are indicated. The agreementwith the prediction is sur- thatwhileanyQHstateonatorusshowsaperiodicvari-
prisingly good. The deviations seen in Fig. 2 may be ation in the density[20], the approach to homogeneity is
due to corrections to exact particle-hole symmetry and veryrapidbutdoesnotcorrespondtoaphasetransition.
to the difficulty to observea weakstatethat is closeto a The two schemes–the thin limit and the CFT approach–
much more stable state—this for instance explains why giveanidenticalhierarchyof states,this clearly suggests
theJainstateatν =10/21isobserved,butnot(yet)the an adiabatic connection.
state at ν =7/19. The latter, which we predict to be on For a short-range interaction the Laughlin states are
the verge of observation, is interesting in that it would the ground states for all L[21]; this establishes that the
be the first daughter of a non-Jain state. We conclude TT-states at these fractions develop continuously into
that Fig. 2 in general determines what states should be the bulk QH-states as L → ∞ without a phase transi-
observed at given disorder. tion. Notingthattheproblemonacylinderwithcircum-
Wenowdiscussthephasediagraminthefillingfactor- ference L can equivalently be thought of as the infinite
disorderplaneinthethinlimit,makingtwoassumptions: two-dimensional case with an L-dependent hamiltonian,
1)Thestateswithagapaboveacertaincutoffareformed we conclude that the TT-state and the bulk QH-state
atgivendisorderandthis cutoffdecreasesmonotonously are adiabatically connected for the Laughlin fractions
with decreasing amount of disorder, and 2) phase tran- (and a short range interaction). We believe this holds
sitions betweenQH-statesarecausedexclusivelyby con- generically for fractions where a QH hierarchy state is
densationofthequasiparticlesdiscussedabove. 1)means observed, ie this state is adiabatically connected to the
that Fig. 2 can be thought of as a diagram in the fill- corresponding TT-state. This claim is supported by nu-
ing factor-disorder plane and 2) implies that the phase merical simulations on small systems, where the gap to
boundaryforν =p/q mustcontainpreciselythosestates thefirstexcitedstatehasbeenshowntoremainfinitefor
thatcanbeobtainedby(successive)condensationofsuch all odd (but for no even) q ≤ 11[15]. Moreover, Jain’s
quasiparticles. For example, the 1/3-dome must extend wave functions for ν = n/(2kn± 1) reduce to the ap-
overν =p/(2p+1)→1/2andoverν =p/(4p−1)→1/4 propriate TT-states as L → 0[15], thus giving a strong
but include no larger or smaller ν. This gives the phase argumentforthe adiabatic continuityalsoforthese frac-
diagraminFig. 1,whereonlythe topologyofthe phases tions. However, the status of the fractions that can not
and their relative hights are significant. The topology is be interpreted as an integer effect of composite fermions
identicaltotheoneinthelowestLandaulevelpartofthe has until now been less clear.
4
Around 1990 it was noted that Laughlin’s wave func- where {i } is a subset of M indices, and {¯i } are the
m 2 m
tionstaketheformofcorrelationfunctionsincertaincon- remaining M = (2k −1)M indices. For k = 1, Ψ
1 2 2 2
formal field theories[22, 23], and it was also conjectured are Jain’s wave functions at ν =2/(4k +1), while k =
1 1
that this is true for general QH-states[22, 24]. Recently 1,k = 2 gives our proposal for the observed state at
2
it was shown that the composite fermion wave functions ν =4/11. The thin limits[15] of the n=2 states are the
in the Jain sequence ν = n/(2kn+ 1), k,n = 1,2... TT-states (102k1)2k2−1102k1−1.
can be constructed from correlators in a CFT with n
bosonic fields[25]. A natural extension of this construc-
tion gives, for n bosonic fields, a set of wave functions
labeled by n positive integers ki, i=1,2,...n. If ki =1 Wecomparedourcandidatewavefunctionatν =4/11
for i = 2,...n, then νn = n/(2k1n+1) and the wave with exact diagonalization results in the disc geometry.
functions are Jain’s composite fermion wave functions. For eight particles, the overlap with the exact ground
For a generalset {ki}, the wave function approachesthe state is 0.72 and the energy E4/11 = 5.317 is in good
TT-ground state in the thin cylinder limit. These frac- agreement with the exact one, E = 5.297 (in units
exact
tions, in the interval [1,1], are indicated in Fig. 1. The of e2/ǫℓ ). Preliminary results for a torus, where there
4 2
ki’sdeterminethedensitiesofthencondensatesofquasi- are no edge effects, are encouraging.
electronsthatbuildupthestate—thecompositefermion
stateistheonewhereallbutthefirstofthesecondensates
havemaximaldensity. Usingthe methods ofRef. 25one
canconstructwavefunctionsalsoforthepertinentquasi- In summary, we presented theoretical support for the
holeandquasielectronexcitations,andshowthattheex- global phase diagram and obtained an explicit realiza-
pected charge and statistics properties of these particles tion of the hierarchy of fractional QH-states in a solv-
arereflectedinthealgebraicproperties(U(1)chargeand able limit. For a large class of states, including those of
commutationrelations)ofthecorrespondinganyonicop- LaughlinandJain,wepresentedexplicitmanybodywave
erators. Detailswillbepublishedlater,andhereweonly functions that represent an adiabatic continuation from
present the n = 2 wave functions for N = 2k2M2 parti- the solvable limit to the experimentally relevant region.
cles at positions z =x +iy :
i i i
We thank Jainendra Jain, Steven Kivelson, Juha
Ψ= X (−1)Pmim∂i1···∂iM2 Y(zil−z¯im)2k1 Suorsa and Susanne Viefers for helpful discussions and
i1<i2···<iM2 il,¯im Chia-Chen Chang for help with the numerical calcula-
tions. ThisworkwassupportedbytheSwedishResearch
× Y (zil−zim)2(k1+k2)−1 Y (z¯il−z¯im)2k1+1e−Pi |z4ℓi2|2 , Council and by NordForsk.
il<im ¯il<¯im
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