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Subband Engineering Even-Denominator Quantum Hall States
V.W. Scarola1, C. May2, M. R. Peterson3, and M. Troyer2
1 Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA
2Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland
3 Condensed Matter Theory Center, Department of Physics,
University of Maryland, College Park, MD 20742, USA
(Dated: April 12, 2010)
Proposed even-denominator fractional quantum Hall effect (FQHE) states suggest the possibility
of excitations with non-Abelian braid statistics. Recent experiments on wide square quantum wells
observe even-denominator FQHE even under electrostatic tilt. We theoretically analyze these struc-
tures and develop a procedure to accurately test proposed quantum Hall wavefunctions. We find
that tilted wells favor partial subband polarization to yield Abelian even-denominator states. Our
results show that tilting quantum wells effectively engineers different interaction potentials allowing
exploration of a wide variety of even-denominator states.
PACS numbers: 73.43.-f,71.10.Pm
Experimental evidence for more than seventy odd-
denominator FQHE states [1] demonstrates the ubiquity
of incompressible quantum Hall electron liquids in high
mobility semiconductor quantum wells. The Laughlin
wavefunctions [2] describe a few of these states, how-
ever, the composite fermion (CF) theory includes Laugh-
lin states into a comprehensive framework that captures
the essential physics of the observed lowest Landau level
(LLL) FQHE states [3, 4].
Observations of rare even-denominator FQHE states,
exceptions to the odd-denominator rule, occur in multi-
component systems [5] and at half filling of the second
LL[6], i.e., filling factor 5/2. Spin, layer, or subband de-
grees of freedom allow combinations of single-component
odd-denominator states to yield even-denominator frac-
tions [7]. But at half filling of the second LL theoretical
analyses [8] suggest that the Coulomb interaction hap-
pens to favor the formation of a single component paired
CF state, the Pfaffian [9], a profoundly distinct state.
Proposed FQHE states possess topologically non-
trivial excitations. The common CF states carry excita-
tions that obey Abelian anyonic braid statistics. Braid-
ing two Abelian excitations changes the overall wavefunc-
tion by a phase [4, 10]. In contrast, excitations above the
CF paired Pfaffian state obey non-Abelian braid statis-
tics; a braid operates on the overall wavefunction by a
matrix. Non-Abelian excitations have potential applica-
tion in topological quantum information processing mak-
ing observation of non-Abelian excitations a key goal [11].
Experimental parameters (width and density) in nar-
row quantum well samples allow only limited tunabil-
ity. Recent diagonalization studies [12] indicate that in a
single-layer system the Pfaffian state is likely (unlikely)
to be the ground state for ν = 5/2(1/2) within the simple
Coulomb interaction model. By chance, including finite
thickness of realistic quantum wells marginally favors the
formation of the Pfaffian state in the second LL [13]. A
route towards the exploration of putative non-Abelian
phases and their phase boundaries requires construction
of tunable high mobility quantum wells designed to favor
these fragile states by engineering the effective electron-
electron interaction.
Recent experiments with wide square quantum wells
demonstrate even-denominator LLL FQHE in a regime
with a surprisingly strong inter-layer tunneling [14, 15]
suggesting that explanations in terms of the usual
Abelian multicomponent states need to be revisited. Do
these observations suggest that the new samples effec-
tively tailor the interaction to favor a LLL Pfaffian state
[15, 16]? The competition between the (non-Abelian)
Pfaffian and (Abelian) multicomponent states remains a
subtle issue [16, 17]. Furthermore, it is currently un-
known whether these or similar samples will favor a sec-
ond LL Pfaffian or entirely new FQHE states.
In this Letter we show that recent experiments observ-
ing even-denominator FQHE [14, 15] in wide quantum
wells occur in a regime that can favor partial subband oc-
cupancy and therefore offer the ability to engineer a wide
array of even-denominator FQHE states. We develop a
protocol that combines a modified high field local density
approximation (LDA) of quantum well subbands with
FQHE wavefunctions in the plane of the quantum well.
Focusing on the experimental parameters of Ref. [15] we
argue that evidence of even-denominator FQHE in tilted
samples arises from partially subband polarized Abelian
FQHE states [18] and, in turn, demonstrate the remark-
able ability to tune among subband balanced and imbal-
anced FQHE states. Our protocol also applies to the sec-
ond LL where subband engineering can be used to tune
and explore second LL even-denominator states [17].
We begin with a model of square quantum wells of wide
width where at most two subbands are populated. We
construct a procedure to energetically minimize the fol-
lowing three dimensional model in a strong perpendicular
magnetic field, B:
H3D = K + VW + V +Hb, (1)

2whereK is the electron kinetic energy, VW is the quantum
well confinement potential, V =
∑
i6=j e2/(2|ri − rj |) is
the three dimensional Coulomb interaction,  is the di-
electric constant, and Hb accounts for the energy of a
plane of rigid positive background charges. In what fol-
lows we consider a tilted square quantum well to model a
specific set of experimental parameters [15] as an applica-
tion of our procedure: VW(z⊥) = αz⊥ +V0θ(|z⊥|−w/2).
The extra potential, αz⊥, tilts the well. We choose
V0 = 270 meV and a well width of w = 55 nm.
To determine the tilt parameter α we first consider the
B = 0 limit. To solve for α we assume a Fermi sea in
the plane and apply usual LDA methods [19]. Here we
assume that the electron spins are unpolarized and use
the Hedin-Lundqvist exchange correlation energy [20] at
a planar density of ρ = 1.72 × 1011 cm−2. We find that
the density imbalance (7.8 × 1010 cm−2) and subband
splitting measured in Ref. [15] (∆B=001 = 41 K) are re-
produced with α = 0.3 meV/nm.
We use this value of α to connect to the high field
experiments of Ref. [15] and now consider the high
field, LLL limit of H3D. The planar and perpendic-
ular coordinates separate and we assume that in the
xy plane the electrons are in the LLL with a quenched
kinetic energy and basis states given by: φm(z/l0) =
(z/l0)m exp (−|z|2/4l20)/(l0
√
2pi2mm!), where z = x − iy
and l0 =
√
~c/eB is the magnetic length. We also as-
sume that the real spins are fully polarized (the experi-
ments at total filling ν = 1/2 in Ref. [15] are performed
for B ≈ 14T ). We then use LDA with a spin polarized
exchange correlation energy [21] to model the electron
wavefunction perpendicular to the plane.
In wells of wide width the subband polarization, γ =
(N0−N1)/N , is a key unknown connecting the energetics
of the plane to the energetics arising from perpendicular
coordinates. Here N0 (N1) denotes the number of elec-
trons in the lowest (first) subband with N = N0 + N1.
At zero field γ is determined by the energetics of the xy-
plane Fermi surface [19] but this is not obviously accurate
in wide quantum wells in the LLL.
To determine the ground state we minimize the to-
tal energy per particle, Etotalγ = E⊥γ +Exyγ , where E⊥γ =
(N0E0+N1E1)/N , the weighted sum of both subband en-
ergies, arises from the quantum well confinement and Exyγ
is the correlation energy due to the LLL planar Coulomb
interaction. We calculate Etotal with the following pro-
cedure: For each value of γ we use LDA to compute
the confinement energy, E⊥γ = ∆γµ/2−γ∆γ01/2, rewritten
here in terms of the γ dependent subband energy differ-
ence, ∆γ01, and an effective chemical potential, ∆γµ. The
LDA also yields ξσ, where ξ0 and ξ1 denote the lowest
and first subband wavefunctions, respectively. The LDA
output is used to construct an effective 2D model of spin
polarized LLL fermions, Heffγ = H⊥γ +Hxyγ , one at each
γ. Finally, we compute Exy with two-component varia-
tional FQHE wavefunctions in the subband basis at fixed
0
0.3
0.6
E⊥
[m
eV
]
0
0.0182
0.0365
E⊥
[e2
/ε
l]
0 0.25 0.5 0.75 1
Subband Polarization (γ)
0
1
2
3
En
er
gy
[m
eV
]
0
0.0608
0.122
0.182
En
er
gy
[e2
/ε
l]
∆µ/2
∆01/2
∆01
B=0/2
/2∆µ
∆01 /2
µ
N
N
1
0
Figure 1: Top: Confinement energy plotted as function of
subband polarization for parameters corresponding to tilted
wide well samples of Ref. [15]. The right axis shows Coulomb
units in terms of the fixed magnetic length, l, obtained for
ν = 1/2 and ρ = 1.72×1011cm−2. Bottom: The same but for
the subband energy difference (dashed) and the chemical po-
tential offset (solid). The dotted line plots the B = 0 subband
energy difference. Inset: Schematic showing partial subband
occupancy captured by an effective chemical potential.
γ thus allowing global minimization of Etotalγ in the space
of competitive FQHE wavefunctions. Our procedure re-
sults in a set of models, Heffγ , that are very sensitive to the
quantum well parameters showing that experiments can
tune through a large set of multicomponent interactions.
To model the perpendicular degrees of freedom we in-
troduce an effective model that captures the subband
energetics:
H⊥γ =
∑
m
[∆γ01
2 (nm,1 − nm,0)−
∆γµ
2 (nm,1 + nm,0)
]
,
where nm,σ = c†mσcmσ and c†mσ creates an electron in the
state ξσφm. H⊥ parameterizes interacting electrons at
fixed γ in the perpendicular direction as non-interacting
fermions, with an effective chemical potential. The first
term is the subband splitting commonly used in models
of very wide wells but the second term is added to capture
the energetics of partial subband occupation in wide wells.
The inset of Fig. (1) shows that ∆µ > 0 favors partial
subband occupancy. We obtain the very wide well limit
(two fully-occupied subbands) for ∆µ  ∆01 and the
narrow well limit (one subband occupied) with ∆µ 
∆01.
In the top panel of Fig. (1) we see that the lowest
energy is obtained for subband polarization γ ≈ 0.59.
The large upturn near γ = 0 arises because the well
tilt strongly penalizes occupancy of the second subband.
The upturn near γ = 1 arises from the Coulomb cost
of putting all charges in the lowest subband. The bot-
tom panel of Fig. (1) plots ∆γµ and ∆γ01 versus γ to show
that they vary in comparison to the subband splitting

30 1 2 3 4
R/l0
-0.3
-0.25
-0.2
Ex
y [e
2 /ε
l 0]
(3/7,1/5|1) γ=0.5
(1/2,1/2|2) γ=0.5
(1/2,1/2|2)Pf γ=0.5
-4 -2 0 2 4
z⊥ [nm]
R/l0=2.55
Figure 2: In-plane correlation energies of several states,
Eqs. (3) and (4), at γ = 0.5 plotted versus the inter-subband
interaction parameter R. The arrow indicates the parame-
ter relevant for the tilted well samples of Ref. [15] where the
(3/7, 1/5|1) state is shown to have the lowest correlation en-
ergy at γ = 0.5. Inset: Normalized lowest (solid) and first
(dashed) subband wavefunctions plotted versus position in
the tilted quantum well for several γ between 0 and 1. In the
limit γ → 0 the subbands localize to mimic layer states.
obtained from the zero field calculation, ∆B=001 . The
non-linearity demonstrates a Stoner-like dependence on
the subband-pseudospin occupancy that arises from the
competition between the kinetic energy, well tilt, and
Coulomb interaction along the direction perpendicular
to the plane.
We now use output from our γ dependent LDA cal-
culation, ξσ, to construct an effective interaction within
the plane. The LLL interaction contains matrix elements
of the form: V σ1σ2σ3σ4m1m2m3m4 = 〈m1,m2|V{σ}(r)|m3,m4〉.
The effective planar interaction is given by: V{σ}(r) =
〈ξσ1 , ξσ2 |V |ξσ3 , ξσ4〉. The tilt breaks the symmetry in the
interaction, e.g., V1010 6= V0101. The inset of Fig. 2 plots
the subband wavefunctions. The tilt localizes the sub-
band states on either side of the well.
For the tilted wide well we find that we can ignore
V1100 and V0011 in the regime 0 ≤ γ . 0.7 and γ = 1 to
yield a simple effective model:
Heffγ ≈ H⊥γ +
1
2
∑
σ,σ′,{m}
V σσ′σσ′{m} c†m1σc
†
m2σ′cm4σ′cm3σ. (2)
The off-diagonal terms give Haldane pseudopotentials
[22] that are smaller by more than a factor of 7 than the
diagonal terms. We assume that diagonal repulsion dom-
inates the inter-subband energetics and that the omission
of these small off-diagonal terms does not change the en-
ergetic ordering of trail wavefunctions computed below
using Eq. (2).
We model the planar interaction computed with the
subband wavefunctions by a modified Zhang-Das Sarma
[23] potential:
V MZDS =
(
r2 + d2{σ}/l20
)−1/2
+ c{σ}
(
r2 +W 2{σ}/l20
)−1/2
where the parameters d, c, and W are determined by fit-
ting lowest pseudopotentials of V MZDS to those of the
interaction computed using the subband wavefunctions.
We have found that V MZDS gives good fits for a variety of
quantum wells in the subband basis. For non-tilted wide
wells the c term must be retained. We find largest uncer-
tainty (< 5%) in the m = 1 pseudopotential. We have
checked, by varying fitting parameters, that 5% varia-
tion in pseudopotentials does not qualitatively alter our
results.
For the tilted well sample of Ref. [15] we focus on
the regime 0 ≤ γ . 0.7 and γ = 1 and keep only the
Vσσ′σσ′ terms. We then require only three fitting param-
eters: d0(γ) ≡ d0000, d1(γ) ≡ d1111, and R(γ) ≡ d1010
to approximately match all pseudopotentials. For ex-
ample, we find the lowest Etotal to occur at: d0(0.5) =
1.16l0, d1(0.5) = 1.37l0, and R(0.5) = 2.55l0.
Equation (2) resembles models of bilayers and very
wide wells but there are crucial differences: (i) There
is a ∆γµ term that favors partial subband polarization,
(ii) All terms are functions of the subband polarization,
γ, (iii) The interaction within each subband differs, e.g,
V1111 6= V0000, and (iv) In the absence of tilt we must
keep off-diagonal terms of the form Vσσσ′σ′ .
We now construct competitive two-component varia-
tional wavefunctions expected to minimize the energy
of Eq. (2) and therefore H3D. The single component
LLL CF wavefunctions at filling n/(2pn± 1), ψn/(2pn±1),
are given in the literature [3, 4]. The single component
CF wavefunctions can be generalized to capture two-
component states [18]:
Ψ(ν0,ν1|m) =
∏
r,j
(zj − wr)mψν0 [{zk}]ψν1 [{ws}], (3)
where the fully antisymmetric wave function ψν is a sin-
gle component state at filling factor ν. Halperin’s wave
functions [7] are obtained as special cases for ν0 = 1/m′
and ν1 = 1/m′′. In order to ensure that the electrons
of each component occupy the same area, N0 and N1
must be related by N0ν−10 +mN1 = N1ν−11 +mN0, thus
ν = (ν−10 +mN1/N0)−1 + (ν−11 +mN0/N1)−1. Some of
the γ = 0 states of Eq. (3) have been shown to be energet-
ically competitive with favorable ground states at several
different filling factors in bilayer systems (small tunnel-
ing and equal subband population) [18]. Table I lists a
larger set. The constituent wavefunctions, ψν , with the
lowest energy gap yield an upper-bound for the energy
gap of each multicomponent state. For example, the gap

4(ν0, ν1|m) n0 n1 γ
(1/3, 1/3|1) 1 1 0
(3/7, 1/5|1) 3 1 1/2
(5/11, 1/7|1) 5 1 2/3
(1/2, 1/2|2) ∞ ∞ any
(1/4, 1/4|0) ∞ ∞ 0
Table I: Several possible states, Eq. (3), at half filling. The
last column, polarization, shows that two of the incompress-
ible states are locked at partially polarized configurations.
of the (3/7, 1/5|1) state is no larger than that of the ψ1/5
state while (1/2, 1/2|2) is a gapless CF Fermi sea [4]. We
also include the multicomponent Pfaffian states in our
comparison:
Ψ(ν0,ν1|m)Pf =
∏
r,j
(zj − wr)mψPf[{zk}]ψPf[{ws}], (4)
where ψPf is the Pfaffian wavefunction [9].
We compute the ground state energies of several can-
didate wavefunctions in the subband basis at ν = 1/2
for the parameters of Ref. [15] using variational Monte
Carlo. We use the spherical geometry [4] to compute en-
ergies in finite size systems and extrapolate our results to
the thermodynamic limit by adding the background en-
ergy: Eb = −
[
N21F (d1) +N20F (d0) + 2N0N1F (R)
]
/2,
where F (x) ≡ (e2/2l0)(−x/Rs +
√
4 + x2/R2s) and Rs
is the radius of the sphere in units of the magnetic length.
Fig. 2 shows a representative comparison of Exy for
three of the lowest energy states at γ = 0.5. Here the
parameter R is varied to test the robustness of the lowest
energy state. The Monte Carlo error bars are smaller
than the line width. The lowest energy state at γ = 0.5
is found to be the incompressible (3/7, 1/5|1) state. The
compressible Fermi sea state, (1/2, 1/2|2), is nearby in
energy. The Pfaffian state becomes lower in energy for
the unphysical regime of R < lo.
We have compared wavefunctions at several subband
polarizations to globally minimize the total energy. We
expect partially subband polarized states to be favored
by the competition between the Coulomb interaction
along the direction perpendicular to the plane and the
quantum well tilt. We find that the small gain in E⊥ fa-
vors the (3/7, 1/5|1) state over all others in a parameter
window corresponding to the parameters relevant for the
tilted well experiments of Ref. [15], α = 0.3 meV/nm, see
Table II. We also conclude that small changes in α will
favor the partially subband polarized compressible state,
(1/2, 1/2|2).
Our procedure provides the following physical picture
of the experiments in Ref. [15] at ν = 1/2. A symmetric
wide well favors the compressible (1/2, 1/2|2) state with
γ ≈ 0 (or the incompressible γ = 0 (1/3, 1/3|1) state
(ν1, ν2|m) γ Exy[e2/l0] E⊥[e2/l] Etotal[e2/l]
(1/3, 1/3|1) 0 -0.24858(2) 0.045 -0.20358(2)
(3/7,1/5|1) 1/2 -0.24542(4) 0.00036 -0.24506(4)
(1/2, 1/2|2) 1/2 -0.2440(4) 0.00036 -0.2436(4)
(1/2, 1/2|2)Pf 1/2 -0.2214(4) 0.00036 -0.2210(4)
(1/2, 0|0) 1 -0.246394(4) 0.0035 -0.242894(4)
(1/2, 0|0)Pf 1 -0.24338(5) 0.0035 -0.23988(5)
Table II: Comparison of planar correlation energies (third col-
umn) and confinement energies (fourth column) for several
ν = 1/2 states. The energies were computed for parameters
relevant for the tilted well sample of Ref. [15]. The lowest to-
tal energy is found for the incompressible (3/7, 1/5|1) state.
depending on ρ and w). As the density is made more
asymmetric via a tilt the CF Fermi sea state continu-
ously depopulates its higher subband (it is a subband-
paramagnet). Near α ≈ 0.3 meV/nm each subband be-
comes localized on either side of the well to approximate
layer-like behavior but with asymmetric parameters in
the subband basis. The incompressible asymmetric state,
(3/7, 1/5|1), is then favored in a narrow parameter win-
dow. We find that this state is responsible for the indi-
cations of FQHE observed under tilt in Ref. [15]. As the
state is made more asymmetric the higher subband state
becomes further depopulated to again favor (1/2, 1/2|2).
Our analysis shows that the samples of Ref. [15] allow
tuning among partially subband polarized multicompo-
nent states.
We have shown that quantum wells with two active
subbands can exhibit a wide variety of partially subband
polarized, even-denominator FQHE states. Our proce-
dure can be used to find parameter regimes where incom-
pressible partially subband polarized FQHE states arise
as a function of sample density, width, and tilt (e.g., a
mixed paired state [24]). Eq. (3) also provides candidates
for recently observed [14, 15] LLL ν = 1/4 features (e.g.,
(3/13, 1/7|3)), and the second LL FQHE where tilt can
be used to tune between a variety of subband polariza-
tions thereby allowing one to engineer a larger class of
interaction potentials. Furthermore, Eq. (3) offers sev-
eral candidate partially polarized real-spin FQHE states.
We thank S. Das Sarma, J.K. Jain, and M. Shayegan
for helpful discussions. Work at University of Maryland
(MRP) was supported by Microsoft Q.
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