REPORTS
◥TOPOLOGICAL MATTER
Distinguishing between non-abelian topological
orders in a quantum Hall system
Bivas Dutta, Wenmin Yang, Ron Melcer, Hemanta Kumar Kundu, Moty Heiblum*, Vladimir Umansky,
Yuval Oreg, Ady Stern, David Mross
Quantum Hall states can harbor exotic quantum phases. The nature of these states is reflected in
the gapless edge modes owing to“bulk-edge”correspondence. The most studied putative non-
abelian state is the spin-polarized filling factor (n) = 5/2, which permits different topological orders
that can be abelian or non-abelian. We developed a method that interfaces the studied quantum
state with another state and used it to identify the topological order ofn= 5/2 state. The interface
between two half-planes, one hosting then= 5/2 state and the other an integern= 3 state,
supports a fractionaln= 1/2 charge mode and a neutral Majorana mode. The counterpropagating
chirality of the Majorana mode, probed by measuring partition noise, is consistent with the particle-
hole Pfaffian (PH-Pf) topological order and rules out the anti-Pfaffian order.
T
he quantum Hall effect (QHE) state
harbors an insulating bulk and conduc-
tive edges and is the earliest known ex-
ample of a topological insulator ( 1 ). It is
characterized by topological invariants,
which are stable to small changes in the de-
tails of the system ( 2 ). Of these quantities, the
easiest to probe is the electrical Hall conduct-
anceoftheedgemode,GH=ne^2 /h, wherenis
the bulk filling factor (integer or fractional),e
is the electron charge, andhis Planck’s con-
stant. Quantum Hall (QH) states with frac-
tional filling factors support quasiparticles
with fractional charges and anyonic statistics.
The ubiquitous Laughlin states are abelian;
exchanging the positions of their quasiparti-
cles adds a phase to the ground-state wave
function ( 3 – 5 ).Inthemoreexoticnon-abelian
states ( 6 , 7 ), the presence of certain quasipar-
ticles results in multiple degenerate ground
states, and interchanging these quasiparticles
cycles the system between the different ground
states. In general, QH states permit different
topological orders, and the usual conductance
and charge measurements are not sufficient
to distinguish between the different topologi-
cal orders.
Another topological invariant, the thermal
QH conductanceGT, may help make that dis-
tinction. The thermal conductance is sensitive
to all energy-carrying edge modes, charged or
neutral, and can be expressed asGT=KT,
whereKis the thermal conductance coeffi-
cient andTis the temperature. It was pro-
posed ( 7 , 8 ) and experimentally proven thatK
of a single chiral and ballistic mode—fermionic
( 9 ), bosonic ( 10 , 11 ), or (abelian) anyonic ( 12 )—
is quantizedK=k 0 , withk 0 ¼p^2 k^2 B= 3 h, where
kBis the Boltzmann constant. However, a
fractional value ofKis expected for non-
abelian states ( 13 ). For then= 5/2 state, we
found a thermal Hall conductance coefficient,
K= 2.5k 0 , that is consistent with the non-
abelian particle-hole Pfaffian (PH-Pf) topologi-
cal order ( 14 ).
However, there is a caveat: For a QH state
that supports multiple edge modes, some mov-
ing downstream (DS; in the chirality dictated
by the magnetic field) and some upstream
(US; in opposite chirality), the theoretically
predicted thermal conductance assumes a full
thermal equilibration among all modes ( 15 – 17 ).
For example, for modes of integer thermal
conductance, the predicted value isGT=(nd–
nu)k 0 T, wherendandnuare the number of DS
and US modes, respectively. In the other ex-
treme, with no thermal equilibration,GT=(nd+
nu)k 0 T( 12 ). Therefore, the thermal equilibra-
tion length, which is in general longer than
the charge equilibration length, is of crucial
importance in interpreting thermal conduct-
ance measurements.
Moore and Read predicted that the topo-
logical order of then= 5/2 state is a Pfaffian
(Pf) state, withK= 3.5k 0 ( 18 ). However, the Pf
mode was ruled out when a US neutral mode
was observed experimentally ( 19 ) because the
Pf order does not support a topologically pro-
tected US mode. The“particle-hole conjugate”
of the Pf order, the anti-Pfaffian (A-Pf) with
K= 1.5k 0 (^20 ,^21 ), does exhibit a US mode.
Numerical studies found both Pf and A-Pf to
be highly competitive ground states in a ho-
mogenous system (neglecting disorder). The
two are degenerate within a single Landau
level, and Landau level mixing may tip the
balance between the two in either direction( 22 , 23 ). Several theoretical proposals offer pos-
sible explanations for the discrepancy between
numerical calculations and the experimen-
tally found PH-Pf order: (i) Inhomogeneity
in the density may lead to islands of local Pf
and A-Pf orders, from which a global PH-Pf
order emerges ( 24 – 28 ). (ii) A considerably
longer thermal equilibration length than the
size of the device may lead to deviation from
the expected theoretical value. In particular,
an unequilibrated Majorana mode in the A-Pf
( 15 , 29 ) order will add its contribution toK
instead of subtracting it, leading toK= 2.5k 0
( 30 – 32 ).
One may suggest to measure the thermal
conductance at a short propagation length,
where thermal equilibration is practically neg-
ligible, expectingK=(nd+nu)k 0 ; namely,
KA-Pf= 4.5k 0 orKPH-Pf= 3.5k 0. However,
“spontaneous edge reconstruction”may add
short-lived pairs of counterpropagating (non-
topological) neutral modes ( 33 ), thus increasing
the apparent thermal conductance of the states
at short distances.
We developed a method in which a studied
state is interfaced with another state, with the
interface harboring an isolated edge channel.
We used this method to probe the Majorana
modes’chirality at interfaces between then=
5/2 state andn= 2 andn= 3 integer states.
Measurements fulfill two important require-
ments: (i) thermally unequilibrated transport,
in which measurements are performed in the
intermediate length regime—longer than the
charge equilibration length (leading to the quan-
tized interface conductance) but shorter than
the thermal equilibration length (allowing un-
equilibrated modes transport) ( 15 , 29 , 34 )—
and (ii) compensating integer modes, in which
the counterpropagating integer modes at the
interface betweenn= 5/2 andn=2or3mu-
tually localize each other. This eliminates spu-
rious hotspots and allows the identification of
the topological order.
We used gated high-quality GaAs-AlGaAs
molecular beam epitaxy (MBE)–grown heter-
ostructures ( 35 ). Structures were designed to
resolve the contradictory requirements of the
doped layers, which should ensure a full quan-
tization of the fragilen= 5/2 state and, at the
same time, allow highly stable and“hysteresis-
free”operation of the gated structures ( 14 ).
Two separate gates divided the surface into
two, an upper half and lower half (Fig. 1A). The
gates were isolated from the sample and from
each other by 15- to 25-nm layers of HfO 2 [more
details of the structure are provided in ( 35 )]. A
gate-voltage in the range of–1.5 V <Vg<+0.3V
allows varying the electron density from pinch
off to 3 × 10^11 cm−^2 (fig. S1). Tuning each gate
separately controls the two interfaced fill-
ing factors, leading to the desired interface
modes’conductance. The ohmic contacts at
the physical edge of the mesa probe the fillingSCIENCEscience.org 14 JANUARY 2022•VOL 375 ISSUE 6577 193
Braun Center for Sub-Micron Research, Department of
Condensed Matter Physics, Weizmann Institute of Science,
Rehovot 76100, Israel.
*Corresponding author. Email: [email protected]
RESEARCH