Even-Denominator Fractional Quantum Hall State at Filling Factor 34 Chengyu Wang1A. Gupta1S. K. Singh1Y. J. Chung1L. N. Pfeier1 K. W. West1K. W. Baldwin1R. Winkler2and M. Shayegan1

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Even-Denominator Fractional Quantum Hall State at Filling Factor ν= 3/4

Chengyu Wang,1A. Gupta,1S. K. Singh,1Y. J. Chung,1L. N. Pfeiﬀer,1

K. W. West,1K. W. Baldwin,1R. Winkler,2and M. Shayegan1

1Department of Electrical and Computer Engineering,

Princeton University, Princeton, New Jersey 08544, USA

2Department of Physics, Northern Illinois University, DeKalb, Illinois 60115, USA

(Dated: October 10, 2022)

Fractional quantum Hall states (FQHSs) exemplify exotic phases of low-disorder two-dimensional

(2D) electron systems when electron-electron interaction dominates over the thermal and kinetic

energies. Particularly intriguing among the FQHSs are those observed at even-denominator Landau

level ﬁlling factors, as their quasi-particles are generally believed to obey non-Abelian statistics and

be of potential use in topological quantum computing. Such states, however, are very rare and fragile,

and are typically observed in the excited Landau level of 2D electron systems with the lowest amount

of disorder. Here we report the observation of a new and unexpected even-denominator FQHS at

ﬁlling factor ν= 3/4 in a GaAs 2D hole system with an exceptionally high quality (mobility). Our

magneto-transport measurements reveal a strong minimum in the longitudinal resistance at ν= 3/4,

accompanied by a developing Hall plateau centered at (h/e2)/(3/4). This even-denominator FQHS

is very unusual as it is observed in the lowest Landau level and in a 2D hole system. While its

origin is unclear, it is likely a non-Abelian state, emerging from the residual interaction between

composite fermions.

Since its discovery in 1982 [1], the fractional quan-

tum Hall eﬀect has been one of the most active topics

in condensed matter physics [2]. It is observed in low-

disorder two-dimensional electron systems (2DESs) at

low temperatures and large, quantizing, perpendicular

magnetic ﬁelds, when electrons’ thermal and kinetic en-

ergies are quenched and the Coulomb interaction between

the electrons dominates. The vast majority of fractional

quantum Hall states (FQHSs) are observed in the lowest

Landau level (LL) at odd-denominator LL ﬁlling factors,

and can be mostly understood in a standard composite

fermion (CF) model [2–5]. By attaching 2mﬂux quanta

to each electron, the FQHSs at ν=p/(2mp ±1), the so-

called Jain-sequence states, can be mapped to the integer

quantum Hall states at the Lambda level (ΛL) ﬁlling fac-

tor pof the weakly interacting 2m-ﬂux CFs (2mCFs).

Thanks to intense experimental eﬀorts over the last few

decades and improvements in sample quality (mobility),

new FQHSs which cannot be explained in the standard

Jain sequence have been reported [2, 6–25]. Among these

are FQHSs observed at certain even-denominator ﬁllings,

e.g. at ν= 5/2 [6]. Although its origin is not yet entirely

clear, theory [26–28] strongly suggests that the ν= 5/2

FQHS is a spin-polarized, p-wave paired (Pfaﬃan) state

with non-Abelian statistics, rendering it a prime candi-

date for fault-tolerant, topological quantum computing

[29]. Even-denominator FQHSs have also been reported

at other ﬁlling factors, e.g., at ν= 1/2 and 1/4 in wide

GaAs quantum wells [8, 10, 14, 16, 17, 19–21, 24, 25].

The origin of these states is also unclear: some experi-

mental and theoretical results are consistent with these

being two-component, Halperin-Laughlin, Abelian states

[10, 16, 19, 30], although the latest data and calculations

suggest a one-component, Pfaﬃan, non-Abelian origin

[17, 25, 31–33].

Here we present the experimental observation of an

even-denominator FQHS in the lowest LL, at ﬁlling fac-

tor ν= 3/4, in an ultrahigh-quality GaAs 2D hole sys-

tem (2DHS). As highlighted in Fig. 1(a), our magneto-

transport measurements show a strong minimum in

the longitudinal resistance (Rxx), concomitant with a

developing Hall resistance (Rxy ) plateau centered at

(h/e2)/(3/4) to within 0.2%. Our ﬁnding is unexpected

as there is no analogue of such a FQHS in GaAs 2DESs

(Fig. 1(b)) [34] where the ground state at ν= 3/4 is a

4CF Fermi sea, ﬂanked by odd-denominator FQHSs at

nearby ﬁllings (4/5, . . . , 5/7) which ﬁt into the Jain se-

quence: ν= 1 −p/(4p±1). In our experiments, we ﬁnd

that the ν= 3/4 FQHS is fairly robust when a strong

in-plane magnetic ﬁeld is applied, but it eventually gets

weaker, with FQHSs at ν= 4/5 and 5/7 emerging on

its ﬂanks. We discuss the possible origins of this novel

FQHS based on our experimental data and available the-

ories, and suggest that it is likely a non-Abelian state.

We also observe a qualitatively similar, but somewhat

weaker, FQHS at the even-denominator ﬁlling ν= 3/8,

with likely same origin as 3/4.

The high-quality 2DHS studied here is conﬁned to a 20-

nm-wide GaAs quantum well grown on a GaAs (001) sub-

strate [35–40]. The 2DHS has a hole density of 1.3×1011

cm−2and a low temperature (0.3 K) record-high mobil-

ity of 5.8×106cm2/Vs [36]. We performed our experi-

ments on a 4 mm ×4 mm van der Pauw geometry sample.

Ohmic contacts were made by placing In/Zn at the sam-

ple’s four corners and side midpoints, and annealing at

450 °C for 4 min. The sample was then cooled down

in two diﬀerent dilution refrigerators with base temper-

atures of '20 mK. We measured Rxx and Rxy using

arXiv:2210.03226v1 [cond-mat.mes-hall] 6 Oct 2022

FIG. 1. (a) Longitudinal resistance (Rxx , in black and blue) and Hall resistance (Rxy , in red) vs perpendicular magnetic

ﬁeld B⊥traces for our ultrahigh-mobility 2D hole sample. The height of the blue trace is divided by a factor of 10. The B⊥

positions of several LL ﬁllings are marked. A strong minimum in Rxx accompanied by a developing Hall plateau is observed

at ν= 3/4. An enlarged version of the Rxy vs B⊥near ν= 3/4 at 20 mK is shown in the top-left inset. The self-consistently

calculated hole charge distribution (red) and potential (black) of the 2DHS are also shown in a top inset. (b) Rxx vs B⊥at

T'30 mK near ν= 3/4 for an ultrahigh-mobility 2D electron sample with density 1.0×1011 cm−2from Ref. [34].

the conventional lock-in ampliﬁer technique, with a low-

frequency ('13 Hz) excitation current of '10 nA.

Figure 1(a) shows the full-ﬁeld traces of Rxx and Rxy

vs perpendicular magnetic ﬁeld B⊥[41]. The B⊥po-

sitions of several LL ﬁllings are marked. A deep mini-

mum in Rxx accompanied by a developing Rxy plateau

is observed at ν= 3/4. Weaker Rxx minima are also ob-

served at other even-denominator ﬁllings ν= 3/8, as well

as 5/8 and 5/12. Numerous high-order odd-denominator

FQHSs are also seen near ν= 1/2, up to ν= 12/25 and

13/25, and near ν= 3/2, up to ν= 16/11 and 17/11 (not

marked in the ﬁgure; see Fig. 4 of [36]). These attest to

the exceptionally high quality of the 2DHS. Note that the

ν= 3/4 FQHS is the only FQHS observed between ν= 1

and 2/3. This is in sharp contrast to what is observed in

high-quality 2DESs, namely a smooth and shallow Rxx

minimum with no quantized Rxy , and ﬂanked by the

standard (Jain-sequence) odd-denominator FQHSs such

as ν= 4/5,7/9, . . . and 5/7,8/11, . . . (see Fig. 1(b)) [34].

It is also in contrast to previous GaAs 2DHSs where, be-

tween ν= 1 and 2/3, only weakly-developed FQHSs were

observed at ν= 4/5 and 5/7 [42, 43]. We also studied an-

other 2D hole sample with higher density, showing weak

Rxx minima at both ν= 3/4 and 5/7; see SM [35] for

details.

In Fig. 2 we show the temperature (T) dependence of

Rxx and Rxy between ν= 1 and ν= 2/3, measured with

a diﬀerent contact conﬁguration and in a diﬀerent cool

down. When Tis reduced, the Rxx minimum becomes

smaller but its ﬂanks rise steeply, as seen in Fig. 2(a). In

Fig. 2(b), we show an Arrhenius plot of Rxx at ν= 3/4.

The activated behavior of Rxx strongly suggests a FQHS

at ν= 3/4. An energy gap of '22 mK is deduced from

the linear ﬁt to the data points at intermediate tempera-

tures. We note that the temperature range where Rxx at

ν= 3/4 vs 1/T follows a linear ﬁt is very narrow. On the

low-temperature (large 1/T ) side, the data points start to

deviate from the linear ﬁt below '30 mK. Several factors

could be causing this deviation: (i) At very low T, the

2DHS temperature might be slightly higher than Tread

by the thermometer; (ii )Rxx at ν= 3/4 could be inﬂu-

enced by the rising background on its ﬂanks at very low

T(Fig. 2(a)); (iii ) it is also possible that the deviation is

caused by the emergence of a diﬀerent scattering mech-

anism (e.g., hopping) at very low temperatures [44]. On

the high-temperature (small 1/T ) side, the temperature

dependence of Rxx reverses its trend, and Rxx decreases

with increasing temperature above 130 mK. Similar phe-

nomenon was observed for other FQHSs [23]. While the

very narrow Trange in which we observe an activated

behavior in Fig. 2(b) limits the accuracy of the '22 mK

energy gap that we determine for the ν= 3/4 FQHS, it

is likely that this value is an underestimate and is inﬂu-

enced by the strong temperature dependence of Rxx on

the ﬂanks of ν= 3/4. The fact that the ν= 3/4Rxx

minimum survives at high temperatures (up to 188 mK)

supports this conjecture.

To further support the presence of a FQHS at ν= 3/4,

in Fig. 2(c) we show Hall traces taken at diﬀerent T.

Rxy is well quantized at the expected values at ν= 1

and 2/3 over the whole temperature range, and shows

a nearly quantized plateau at ν= 3/4 at the lowest

temperature ('20 mK). The top-left inset of Fig. 2(c)

shows an enlarged view of Rxy vs B⊥near ν= 3/4 at

FIG. 2. Temperature dependence of Rxx and Rxy . (a) Rxx vs B⊥traces near ν= 3/4, taken at diﬀerent temperatures, showing

fully developed (nearly zero) Rxx minima at ν= 1 and 2/3, as well as a strong minimum at ν= 3/4. (b) Arrhenius plot of

Rxx at ν= 3/4. An energy gap of '22 mK is obtained from the linear ﬁt to the data points at intermediate temperatures.

Inset: enlarged version of Fig. 2(a) near ν= 3/4. (c) Hall (Rxy ) traces taken at diﬀerent temperatures. Rxy is well quantized

at its expected value at ν= 1 and 2/3 in the whole temperature range, and shows a developing plateau at ν= 3/4 at the

lowest temperatures. Top-left inset: enlarged Rxy vs B⊥traces near ν= 3/4 at T'20 and 188 mK. Bottom-right inset: Hall

resistance slope dRxy /dB⊥vs Tat ν= 3/4, showing its approach to the expected (classical) value at high T(the dash-dotted

line), and to zero as Tapproaches zero, conﬁrming the Rxy quantization.

T'20 and 188 mK. At '20 mK, a plateau occurs

at exactly the expected ﬁeld position of ν= 3/4 and

is centered at Rxy = (h/e2)/(3/4) to within 0.2%. The

bottom-right inset of Fig. 2(c) shows the Hall resistance

slope dRxy /dB⊥vs Tat ν= 3/4. The dash-dotted line

represents the expected, classical, high-temperature Hall

slope based on the 2DHS density. At low temperatures,

dRxy /dB⊥exhibits a trend towards zero, consistent with

a developing Rxy plateau.

Next we study the role of an in-plane magnetic ﬁeld

(Bk) on the ν= 3/4 FQHS. Figure 3 shows the tilt angle

(θ) dependence of Rxx vs B⊥between ν= 1 and ν= 2/3

at '20 mK where θdenotes the angle between total

ﬁeld (B) and its perpendicular component (B⊥); see the

top-left inset in Fig. 3. The Rxx minimum at ν= 3/4 re-

mains fairly strong up to θ'60◦. With further increase

in θ, the 3/4 FQHS becomes signiﬁcantly weaker, while a

FQHS at ν= 4/5 starts to appear and get stronger; see

also Fig. 3 top-right inset. A hint of a FQHS at ν= 5/7

is also observed at large θ, but does not change much

with increasing θ. The simultaneous weakening of the

FQHS at 3/4 and appearance of the 4/5 and 5/7 FQHSs

imply a competition between these states. The origin of

this competition is unclear.

Our observation of a ν= 3/4 FQHS is unexpected

as no such state has been previously seen in experi-

ments or predicted by theory. While a plethora of odd-

denominator FQHSs are observed when the Fermi level

lies in the lowest (N= 0) LL, even-denominator FQHSs

in single-layer 2DESs conﬁned to diﬀerent materials such

as GaAs [6], ZnO [45], and AlAs [46] have been pre-

dominantly reported only in the excited (N= 1) LL.

In the N= 0 LL, in the CF picture and assuming

that particle-hole symmetry holds, the ground states at

ν= 3/4 and 1/4 are both expected to be Fermi seas

of 4CFs. In experiments on GaAs 2DESs, a 4CF Fermi

sea has indeed been directly observed by geometric reso-

nance near ν= 1/4 [47], and a series of standard (Jain-

sequence), odd-denominator FQHSs is seen on the ﬂanks

of ν= 1/4. A similar sequence of FQHSs is also observed

near ν= 3/4 in GaAs 2DESs (see Fig. 1(b)) [12, 34],

supporting a 4CF Fermi sea ground state. However, in

our 2DHS, a CF Fermi sea is not favored at ν= 3/4 as

evinced by the presence of a FQHS at this ﬁlling. The

obvious question is: Why is the 3/4 FQHS observed in

our GaAs 2DHS and in the lowest (N= 0) LL? While we

do not have a deﬁnitive answer, we discuss below possible

explanations.

An unlikely possibility is that the ν= 3/4 FQHS in

our experiments has an origin similar to the FQHSs re-

ported in the N= 0 LL of 2DESs in wide GaAs quantum

wells at ν= 1/2 [8, 10, 19, 25] and ν= 1/4 [14, 16, 17].

The origin of these states is in fact still unclear and the

possibilities of both a two-component, Ψ331 , Halperin-

Laughlin, Abelian state [10, 16, 19, 30], and a single-

component, Pfaﬃan, non-Abelian state [17, 25, 31–33]

have been discussed. Regardless of their origin, these

states have only been observed in 2DESs with bilayer-

like charge distributions. They are also very sensitive to

diﬀerent parameters such as carrier density, quantum well

width and symmetry [10, 19], and magnetic ﬁeld compo-

nents [48]. In GaAs 2DHSs, ν= 1/2 FQHSs are also seen

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Even-DenominatorFractionalQuantumHallStateatFillingFactor=3=4ChengyuWang,1A.Gupta,1S.K.Singh,1Y.J.Chung,1L.N.Pfeier,1K.W.West,1K.W.Baldwin,1R.Winkler,2andM.Shayegan11DepartmentofElectricalandComputerEngineering,PrincetonUniversity,Princeton,NewJersey08544,USA2DepartmentofPhysics,NorthernIllinoisUn...

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Even-Denominator Fractional Quantum Hall State at Filling Factor 34 Chengyu Wang1A. Gupta1S. K. Singh1Y. J. Chung1L. N. Pfeier1 K. W. West1K. W. Baldwin1R. Winkler2and M. Shayegan1.pdf

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Even-Denominator Fractional Quantum Hall State at Filling Factor 34 Chengyu Wang1A. Gupta1S. K. Singh1Y. J. Chung1L. N. Pfeier1 K. W. West1K. W. Baldwin1R. Winkler2and M. Shayegan1

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