Beyond Single Tetrahedron Physics of Breathing Pyrochlore Compound Ba 3Yb 2Zn5O11 Rabindranath Bag1Sachith E. Dissanayake1Han Yan2Zhenzhong Shi1David Graf3Eun Sang Choi3Casey Marjerrison1Franz Lang4Tom Lancaster5Yiming Qiu6

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Beyond Single Tetrahedron Physics of Breathing Pyrochlore Compound Ba3Yb2Zn5O11

Rabindranath Bag,1, ∗Sachith E. Dissanayake,1, ∗Han Yan,2Zhenzhong Shi,1David

Graf,3Eun Sang Choi,3Casey Marjerrison,1Franz Lang,4Tom Lancaster,5Yiming Qiu,6

Wangchun Chen,6Stephen J. Blundell,4Andriy H. Nevidomskyy,7and Sara Haravifard1, 8, †

1Department of Physics, Duke University, Durham, NC 27708, USA

2Rice Academy of Fellows, Rice University, Houston, TX 77005, USA

3National High Magnetic Field Laboratory and Department of Physics,

Florida State University, Tallahassee, Florida 32310, USA.

4Clarendon Laboratory & Physics Department, University of Oxford,

Parks Road, Oxford OX1 3PU, United Kingdom

5Department of Physics, Centre for Materials Physics,

Durham University, Durham DH1 3LE, United Kingdom

6NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA

7Department of Physics and Astronomy, Rice University, Houston, TX 77005, USA

8Department of Materials Sciences and Mechanical Engineering, Duke University, Durham, NC 27708, USA

(Dated: October 14, 2022)

Recently a new class of quantum magnets, the so-called breathing pyrochlore spin systems, have

attracted much attention due to their potential to host exotic emergent phenomena. Here, we present

magnetometry, heat capacity, thermal conductivity, Muon-spin relaxation, and polarized inelastic

neutron scattering measurements performed on high-quality single-crystal samples of breathing py-

rochlore compound Ba3Yb2Zn5O11. We interpret these results using a simpliﬁed toy model and

provide a new insight into the low-energy physics of this system beyond the single-tetrahedron

physics proposed previously.

Frustrated quantum magnets provide a fruitful arena

to search for novel quantum phenomena [1,2]. Py-

rochlore lattice magnets, in which magnetic ions form

corner-sharing regular tetrahedra, are one of the most

studied frustrated systems in three-dimension [3–6]. In

the pyrochlore system the conventional magnetic order-

ing is suppressed by the geometrically frustrated lat-

tice, consequently resulting in emergence of exotic phases

[7–19]. Recently a new class of systems, the so-called

breathing pyrochlore magnets, have attracted much at-

tention due to their potential to host exotic phenom-

ena and topological phases [20–23]. In breathing py-

rochlore compounds the lattice inversion symmetry at

each site is broken due to the diﬀerent sizes of up-

pointing and down-pointing tetrahedra, thus resulting

in large Dzyaloshinskii-Moriya (DM) interactions on the

two tetrahedra [24] (see Fig. 1(a,b) for the structure

of breathing pyrochlores). On the theory front, recent

works have shown that breathing pyrochlore spin systems

can host novel physics including classical rank-2 U(1)

spin-liquid states [25], quantum fractons [26], competing

quantum spin liquids [27], and hedgehog lattices of mag-

netic monopoles and antimonopoles [23]. Thus, it is of

great interest to synthesize and understand breathing py-

rochlore materials. The majority of the work performed

on the breathing pyrochlore-based compounds have fo-

cused on Cr-based spinels with S= 3/2 [28–36], while

the studies performed on quantum systems with S= 1/2

remain limited to Ba3Yb2Zn5O11 in powder form [37–42].

Our recent work reported the ﬁrst comprehensive neutron

scattering studies performed on single-crystal sample of

Ba3Yb2Zn5O11 [43].

We successfully grew single crystal samples of breath-

ing pyrochlore Ba3Yb2Zn5O11 using the modiﬁed optical

ﬂoating zone technique [43]. Inelastic neutron scatter-

ing studies using our single crystal sample revealed that

the single-tetrahedron model with isolated tetrahedra

can explain the high-temperature and high-energy regime

of the collected data [43]. However, the diﬀuse neu-

tron scattering performed at low-temperature and low-

energy reveals features which cannot be understood with

this model [43]. Pair distribution function (PDF) anal-

yses performed on high quality powder neutron diﬀrac-

tion data provided evidence for the absence of chemical

disorder within experimental resolution. Single crystal

X-ray diﬀraction studies also found no evidence of site

disorder [43]. Diﬀuse neutron scattering on single crys-

tal samples [43], and the low temperature heat capacity

data collected on powder samples [39,42] suggest that

physics beyond the single tetrahedron, with small but

ﬁnite inter-tetrahedron interactions, is essential to cap-

ture the behavior of this quantum breathing pyrochlore

system. This calls for additional experimental probes

at low temperature with higher resolution, so that we

can study the subtle changes in magnetic properties of

Ba3Yb2Zn5O11 with higher accuracy.

In this letter, we report low-temperature heat capac-

ity measurements in applied ﬁeld, ultra-sensitive mag-

netic susceptibility, thermal conductivity, muon spin re-

laxation (µ+SR), and polarized inelastic neutron scatter-

ing measurements of the ytterbium based breathing py-

rochlore compound Ba3Yb2Zn5O11 in single-crystalline

form, to investigate the intrinsic low temperature mag-

arXiv:2210.06534v1 [cond-mat.str-el] 12 Oct 2022

A(t)

t(µs)

λ(µs−1)

T(K)

A(t)

t(µs)

(a)

(b)

1.8 K

250 K

0.08 K

2 mT

FIG. 1. (a) Structure of the breathing pyrochlore lattice.

Considering each A-tetrahedron (blue) as a single site and

retaining the bonds on the B-tetrahedron (red), we obtain

a face-centered cubic (FCC) lattice shown in (b). The cube

in solid black line marks the repeating unit in the left panel

in the FCC lattice. (c) Zero-ﬁeld muon asymmetry for two

temperatures. Solid lines represent ﬁts to A(t) = A(0)e−(λt)β

with A(0) ﬁxed across all temperatures. (d) The temper-

ature dependence of the ﬁtted relaxation rate λ. The inset

shows ultra-low temperature data, demonstrating the absence

of long range order at 0.08 K and the eﬀect of applying a small

2 mT longitudinal ﬁeld.

netic properties and provide a ﬁrst look into the physics

governing the low-energy regime of this system. We pro-

pose a simpliﬁed model that captures the ﬁeld depen-

dence of the heat capacity for lower ﬁeld region well

and provides a scenario beyond the previously reported

single-tetrahedron physics, with ﬁnite inter-tetrahedron

coupling necessary to interpret the experimental results.

In order to search for any trace of magnetic order,

we performed µ+SR measurements using a large powder

sample of Ba3Yb2Zn5O11 on the GPS spectrometer at

the Swiss Muon Source at PSI, and also using co-aligned

single crystal samples of Ba3Yb2Zn5O11 mounted in a

dilution refrigerator at the MuSR spectrometer of the

ISIS Muon Source. In order to understand the origin

of the contributions to the µ+SR signal, we carried out

density functional theory (DFT) calculations to locate

the most probable muon stopping sites, and assess the

degree of perturbation the muon-probe causes in the ma-

terial [44], which we ﬁnd to be small in this system (for

details see [45]). The measured muon asymmetry A(t)

in zero ﬁeld (ZF) exhibits a non-oscillating but relaxing

time dependence [see Fig. 1(c)], the relaxation rate of

which decreases on warming above around 50 K. This

relaxation can be ﬁtted to a stretched exponential form

A(t) = A(0)e−(λt)βwhere the exponent β≈0.33, indi-

cating that the spin ﬂuctuations that give rise to the re-

laxation have a range of timescales (an exponent of β= 1

would indicate a single ﬂuctuation time, and exponents

less than one are often found in systems with complex

spin dynamics, see e.g. [46]). The temperature depen-

dence of λis plotted in Fig. 1(d), illustrating the decrease

in λwith increasing temperature at high temperatures,

corresponding to the thermally-induced increase in spin-

ﬂuctuation rate (λ∝ν−1in the fast-ﬂuctuation limit,

where νis a characteristic ﬂuctuation rate [47]). There

is a maximum in λaround 35 K, the low-temperature

approach to which we attribute to the switching on of

excitations associated with the intra-tetrahedral interac-

tions (note that the energy gap of 0.38 meV [40] to the

next crystal ﬁeld level is larger than any thermal energy

in this experiment). Our experiments at milliKelvin tem-

peratures were hampered by the very small size of the sin-

gle crystals (leading to a small relaxation amplitude) but

nevertheless demonstrate that the relaxing signal persists

down to 0.08 K [see inset to Fig. 1(d)], with no appear-

ance of an oscillatory signal, ruling out the development

of long range magnetic order. It is noteworthy to add

that the ultra-low temperature persistent slow dynamics

are consistent with what is seen in many other frustrated

systems, and in this case one can argue that they might

be connected with the very weak inter-tetrahedra inter-

actions. We will explore such possibility in more details

later as we discuss the ultra-low temperature thermody-

namics results.

We show in Fig. 2(a-i) the magnetic heat capacity

for a Ba3Yb2Zn5O11 single crystal sample from 54 mK

to 1 K under diﬀerent applied magnetic ﬁelds, with

the phonon contribution subtracted using the results

of measurements made on iso-structural, non-magnetic

Ba3Lu2Zn5O11.The magnetic entropy at zero ﬁeld is

shown in Fig. 2(j). The heat capacity data were col-

lected on two diﬀerent Ba3Yb2Zn5O11 single-crystal sam-

ples (grown using diﬀerent techniques) and are compared

with the reported powder Ba3Yb2Zn5O11 sample [45].

Previous reports discussed the possibility of having de-

fects, such as structural disorder, as an underlying cause

for the peak observed at low temperatures, whereas our

heat capacity data collected on multiple single crystal

samples excludes the existence of measurable defect ef-

fects such as structural disorder. To further elaborate on

this, we show in Fig. ?? of the Supplementary Materials

[45] the results obtained for two single-crystal samples

grown with diﬀerent techniques (sample 1 and sample

2). The peak position at 110 mK remains the same for

both single-crystal samples and agrees with the reported

powder study by Haku et al. [39]. This is while the

ﬁt to the data is signiﬁcantly improved using the model

we employed to analyze the data. We explain the de-

tails of this model in the following. Additionally, here we

show the ﬁeld-dependence of the low-temperature fea-

ture which agrees reasonably with our proposed model,

in particular for the low and high ﬁeld region.

As shown in Fig. 2(j), there is 1

4Rln 2 entropy release

per Yb ion, corresponding to an eﬀective pseudo-spin-1/2

degree of freedom on each tetrahedron. As discussed by

Rau et al. [42] the experimental speciﬁc heat results col-

lected at T < 0.4 K disagree with the single tetrahedron

model, leading us, to propose that this release of entropy

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 0

0 . 4

0 . 8

1 . 2

1 . 6

Sm( T ) ( J / m o l - Y b - K )

T ( K )

R / 4 l n ( 2 )

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 0

0 . 2

0 . 4

Cm ( J / m o l - Y b - K )

T ( K )

1 2 T

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 0

0 . 4

0 . 8

1 . 2

1 . 6

Cm ( J / m o l - Y b - K )

T ( K )

1 0 T

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 0

0 . 4

0 . 8

1 . 2

1 . 6

2 . 0

Cm ( J / m o l - Y b - K )

T ( K )

8 T

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 0

0 . 4

0 . 8

1 . 2

1 . 6

2 . 0

Cm ( J / m o l - Y b - K )

T ( K )

5 T

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

1 . 4

Cm ( J / m o l - Y b - K )

T ( K )

4 T

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

Cm ( J / m o l - Y b - K )

T ( K )

3 T

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

1 . 0

1 . 2

Cm ( J / m o l - Y b - K )

T ( K )

2 T

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 0

0 . 2

0 . 4

0 . 6

0 . 8

Cm ( J / m o l - Y b - K )

T ( K )

1 T

( j )( i )( f ) ( g ) ( h )

( e )( d )( c )( b )

0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0

0 . 0

0 . 2

0 . 4

0 . 6

Cm ( J / m o l - Y b - K )

T ( K )

0 T

( a )

FIG. 2. (a)-(i) Low temperature magnetic speciﬁc heat (Cm) of Ba3Yb2Zn5O11 single crystal samples for diﬀerent ﬁelds from

0 T to 12 T. The phonon contribution was subtracted using the iso-structural, non-magnetic compound Ba3Lu2Zn5O11. Red

lines are ﬁts based on the simpliﬁed model outlined in the text [45]. (j) Magnetic entropy of Ba3Yb2Zn5O11 at zero magnetic

ﬁeld.

is related to the inter-tetrahedron interactions. This is

because in the single tetrahedron theory, the two lowest

states are robustly degenerate, and the third state lies

much higher in energy (at ∼0.5 meV). Although for a

ﬁnite external magnetic ﬁeld, the degeneracy of the two

lowest states is expected to be lifted, the energy split-

ting is much smaller than ∼0.01 meV, which cannot

explain the broad peak in heat capacity measurement in

Figs. 2(a,b).

The above observations suggest that the low-energy

properties of Ba3Yb2Zn5O11 cannot be explained by the

single tetrahedron theory, even if tuning the exchange

parameters is allowed. Instead the speciﬁc heat data

can be understood quantitatively by introducing inter-

tetrahedron interactions. To this end, we have con-

structed an eﬀective low-energy model, regarding the two

nearly degenerate lowest energy states (out of 16) on the

Atetrahedra as a pseudo-spin 1

2. This is justiﬁed by

the fact that the other states lie at much higher energies

(E3>0.3 meV) [42] relative to the range T < 1 K in our

speciﬁc heat data.

From exact diagonalization on a single tetrahedron, we

can determine the wave-function of the two lowest states

exactly, which form the two-dimensional (E) irreducible

representation of the Tdpoint group. In the limit of

vanishing intra-tetrahedron DM interaction, these states

span the two-dimensional Hilbert space of two-dimer cov-

erings of the four sites. Note that there are three such

possible dimer coverings classically, but one of them is lin-

early dependent of the other two. The small DM interac-

tion in A-tetrahedra tunes the wave-function away from

the perfect dimer-covering states [38,48], but does not lift

their degeneracy, justifying our treating them as pseudo-

spin 1

2degrees of freedom. We then consider the interac-

tion between these pseudo-spins via the weak bonds on

the B-tetrahedra [shown in red in Fig. 1(a,b)]. Shrinking

every A-tetrahedron to a point connected by these weak

bonds, the eﬀective low-energy model becomes an face-

centered cubic (FCC) lattice of pseudo-spins with nearest

neighbor interactions [c.f. Fig. 1(b)]. The eﬀective inter-

actions are the original interactions between the physical

spins projected onto the pseudo-spin Hilbert space.

We consider the simplest model of the eﬀective inter-

actions between neighboring A-tetrahedra pseudo-spins:

H=Jxy(sx

isx

j+sy

isy

j) + Jzsz

isz

j(1)

We ﬁnd that choosing ferromagnetic Jz=−0.005 meV

and antiferromagnetic Jxy = 0.0125 meV in this XXZ

model can reproduce the zero-ﬁeld speciﬁc heat very well

[Fig 2(a)] (for details, see Supplementary Material [45]).

The origin of the 110 mK peak in the speciﬁc heat is

due to the ferromagnetic ordering of the pseudo-spins,

which spontaneously lifts the two-fold degeneracy of the

single-tetrahedron model. Linear spin-wave theory then

produces pseudo-magnons of the bandwidth ∼Jxy which

propagate on the FCC lattice, as depicted in Fig. ??. It is

important to note that pseudo-magnons are not conven-

tional spin waves, but rather collective excitations of the

dimer-covering states spanned by pseudo-spin degrees of

freedom on A-tetrahedra, and hence may be challenging

to detect by inelastic neutron scattering.

The eﬀective low-energy model in Eq. (1) is expected

( c )( b )

0 . 1 1

1 0 - 3

1 0 - 2

1 0 - 1

0 . 1 1

1 0 - 2

1 0 - 1 B a 3Y b 2Z n 5O1 1

kx x (W m - 1 K- 1 )

T( K )

α= 2 . 6 3

k ~ T

µ0

∆T | | [ 1 1 1 ]

H = 0 T

kx x / T (W m - 1 K- 2 )

T( K )

∆T | | [ 1 1 1 ]

H = 0 T

1 2 3

0 . 0 2 4 K

1 T

21.6 deg

32.4 deg

43.2 deg

54.0 deg

64.8 deg

0 d e g H/ / [ 1 1 1 ]

75.6 deg

90 deg

0H( T )

109 deg

0 . 3 2 T

0 . 5 1 . 0 1 . 5

- 1 0

- 8

- 6

0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8

- 2 0

- 1 5

- 1 0

- 5

a c T D O

c' ( a r b . u n i t s )

0H( T )

1 T

0 . 3 2 T

a c

c' ( a r b . u n i t s )

0H( T )

T D O

0 . 0 2 4 K

0 . 6 6 0 K

0 . 2 3 3 K

0 . 0 7 4 K

0 . 8 8 0 K

0 . 4 5 2 K

0 . 3 1 7 K

0 . 1 9 0 K

0 . 1 4 3 K

0 . 1 0 6 K

0 . 0 4 1 K

0 . 0 2 4 K

( a )

µ0

FIG. 3. Low-Hanomalies beyond the single-tetrahedra model: (a) magnetic ac-susceptibility χ0(left axis) and TDO frequency

(f) (right axis) as a function of external magnetic ﬁeld (Hk[111]) at diﬀerent temperatures. Two anomalies are marked

by arrows. χ0is measured with an oscillating frequency of 1616 Hz. Measurements (χ0) with diﬀerent frequency (87.1 Hz)

reproduce the two anomalies at the same ﬁelds. Inset shows zoomed-in plot of χ0(left axis) and TDO f(right axis) at T=

0.024 K. (b) TDO frequency (f) data is shown for diﬀerent rotation angles between the ﬁeld and the crystal [111] axis at 0.024

K. Traces are shifted vertically for clarity. (c) Thermal conductivity data (κxx vs T) for Ba3Yb2Zn5O11 at µ0H= 0 T; δT ||

[111]. Solid red lines are power-law ﬁt to κxx data. A clear saturation of κxx at low Tis seen for Ba3Yb2Zn5O11. Inset shows

κxx/T vs Tplot.

to work in a moderately large applied magnetic ﬁeld,

provided its strength does not exceed the energy gap

(E3∼0.38 meV) to the ﬁrst excited state beyond the E-

doublet in the single-tetrahedron model [42] . The mag-

netic ﬁeld splits pseudo-spin degrees of freedom in Eq. (1)

at an energy scale much smaller than the parameters in

the eﬀective model (Fig. S3 in [45]). The main eﬀect of

the ﬁeld, from the exact diagonalization of a single tetra-

hedron, is to shift the higher-energy states downwards,

which we treat as ﬂat bands. This approximation breaks

down at a critical value of the ﬁeld Bc∼4 T when the

lowest excited state E3crosses the ground state doublet,

resulting in a phase transition. Our model (1) does not

apply in this regime or higher ﬁelds. In in the limit of

high ﬁelds B > 10 T, we are able to obtain a good match

with the experimental speciﬁc heat [see Fig. 2(i)] by using

a single tetrahedron theory, which predicts a unique non-

degenerate ground state separated by a large gap from

the higher-lying states. For intermediate ﬁeld strengths,

one cannot ignore the eﬀect of the excited states, which

result in the ground state level crossing as already noted.

The minimal model then becomes rather complicated,

with a vast range of unknown parameters, whose deter-

mination lies beyond the scope of the present work.

Further insight into the eﬀect of weak magnetic ﬁelds

can be gleaned from the magnetic ac-susceptibility, which

we measured on two separate Ba3Yb2Zn5O11 crystals.

The results are shown in Fig. 3(a), which show two

anomalies at µ0Hc1= 0.32 T and µ0Hc2= 1.0 T upon

cooling at low temperatures T'0.3 K. The two anoma-

lies are also seen in the tunnel diode oscillator (TDO)

measurements on the same crystals and do not shift ap-

preciably with ﬁeld when the crystal is rotated away from

[111] orientation [see Fig. 3(b)]. These anomalies are in-

dependent of two diﬀerent oscillation frequencies (1616

Hz and 87.1 Hz) of the ac-susceptibility measurement,

suggesting the signal is unrelated to spin freezing and

consistent with the µ+SR study showing absence thereof.

The most likely explanation for the anomalies is the level

crossing at the corresponding ﬁelds µ0Hc1and µ0Hc2.

Given the high frustration of the FCC lattice, it is possi-

ble for the system to go through several diﬀerent phases.

The exact phases and phase transitions cannot be de-

termined by current experimental data, and await fu-

ture eﬀort. Importantly, the explanation must involve

inter-tetrahedron couplings, because the single tetrahe-

dron model would predict a nonmagnetic Seﬀ = 0 ground

state at low temperatures (≤0.5 K) and thus a feature-

less magnetic susceptibility, contrary to what we see in

our measurements.

In order to further understand the nature of the low-

lying states observed in the heat capacity and magnetom-

etry data, low temperature thermal conductivity mea-

surements were carried out on single crystal sample of

Ba3Yb2Zn5O11 [45]. A power law ﬁt (κxx ∼Tα) is per-

formed on the collected data at the high-Tregion and

the value of the exponent (α) is found to be 2.63. For

a nonmagnetic insulator, κxx at very low temperature is

only due to the contribution from phonons [49,50]. How-

ever, the exponent value obtained for Ba3Yb2Zn5O11 re-

veals additional contributions coming from various quasi-

particles such as phonons, spinons and magnons, as well

as diﬀerent scattering channels for the heat current [50–

53]. Interestingly, at low T, a clear saturation of κxx

in Ba3Yb2Zn5O11 is seen. Such saturation is not ex-

pected in conventional magnets where both phonons

and magnons freeze out, but suggest itinerant fermionic

(spinon) excitations expected in gapless spin liquids can-

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摘要：

BeyondSingleTetrahedronPhysicsofBreathingPyrochloreCompoundBa3Yb2Zn5O11RabindranathBag,1,SachithE.Dissanayake,1,HanYan,2ZhenzhongShi,1DavidGraf,3EunSangChoi,3CaseyMarjerrison,1FranzLang,4TomLancaster,5YimingQiu,6WangchunChen,6StephenJ.Blundell,4AndriyH.Nevidomskyy,7andSaraHaravifard1,8,y1Departmen...

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Beyond Single Tetrahedron Physics of Breathing Pyrochlore Compound Ba 3Yb 2Zn5O11 Rabindranath Bag1Sachith E. Dissanayake1Han Yan2Zhenzhong Shi1David Graf3Eun Sang Choi3Casey Marjerrison1Franz Lang4Tom Lancaster5Yiming Qiu6.pdf

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Beyond Single Tetrahedron Physics of Breathing Pyrochlore Compound Ba 3Yb 2Zn5O11 Rabindranath Bag1Sachith E. Dissanayake1Han Yan2Zhenzhong Shi1David Graf3Eun Sang Choi3Casey Marjerrison1Franz Lang4Tom Lancaster5Yiming Qiu6

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