The observation of quantum ﬂuctuations in a kagome Heisenberg antiferromagnet Fangjun Lu1Long Yuan1Jian Zhang1Boqiang Li1Yongkang Luo1yand Yuesheng Li1z 1Wuhan National High Magnetic Field Center and School of Physics

2025-05-06 0 0 2.58MB 21 页 10玖币

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The observation of quantum ﬂuctuations in a kagome Heisenberg antiferromagnet

Fangjun Lu,1, ∗Long Yuan,1, ∗Jian Zhang,1Boqiang Li,1Yongkang Luo,1, †and Yuesheng Li1, ‡

1Wuhan National High Magnetic Field Center and School of Physics,

Huazhong University of Science and Technology, 430074 Wuhan, China

(Dated: October 25, 2022)

Abstract

The search for the experimental evidence of quantum

spin liquid (QSL) states is critical but extremely challeng-

ing, as the quenched interaction randomness introduced

by structural imperfection is usually inevitable in real ma-

terials. YCu3(OH)6.5Br2.5(YCOB) is a spin-1/2 kagome

Heisenberg antiferromagnet (KHA) with strong coupling

of hJ1i ∼ 51 K but without conventional magnetic freez-

ing down to 50 mK ∼0.001hJ1i. Here, we report a Br

nuclear magnetic resonance (NMR) study of the local spin

susceptibility and dynamics on the single crystal of YCOB.

The temperature dependence of NMR main-line shifts and

broadening can be well understood within the frame of

the KHA model with randomly distributed hexagons of

alternate exchanges, compatible with the formation of a

randomness-induced QSL state at low temperatures. The

in-plane spin ﬂuctuations as measured by the spin-lattice

relaxation rates (1/T1) exhibit a weak temperature depen-

dence down to T∼0.03hJ1i. Our results demonstrate that

the majority of spins remain highly ﬂuctuating at low tem-

peratures despite the quenched disorder in YCOB.

Introduction

Quantum spin liquid (QSL) is a state of matter that ex-

hibits exotic fractional excitations and long-range en-

tanglement without symmetry breaking [1–4]. Since Ander-

son’s proposal of the prototype, i.e., resonating-valence-bond

(RVB) state, in 1973 [5], QSL has been attracting researchers

for decades, due to its key role in understanding high-

temperature superconductivity [6] and the possible realization

of the topological quantum computation [7]. Experimentally,

many prominent two-dimensional QSL candidate compounds

have been extensively studied (the one-dimensional scenario

of QSL is qualitatively different [2]), including the kagome-

lattice ZnCu3(OH)6Cl2(herbertsmithite) [8–17], triangular-

lattice κ-(ET)2Cu2(CN)3[18, 19], EtMe3Sb[Pd(dmit)2]2[20,

21], YbMgGaO4[22, 23], etc., all of which generally exhibit

gapless QSL behaviors [9, 11, 14, 16, 18–20, 22, 24–27], but

without evident magnetic thermal conductivity [28–32].

Despite the progress, the existing experimental evidence for

QSL remains circumstantial and strongly depends on theoreti-

cal interpretation. The root cause lies in the quenched interac-

tion randomness introduced by structural imperfection that is

inevitable in all real materials [2, 17, 33]. Therefore, great

efforts are being devoted to exploring for ultrahigh-quality

candidate materials, which is extremely challenging [2]. On

the other hand, disorder-free QSL, even if successfully pre-

pared, is usually very fragile. For instance, the most frus-

trated kagome Heisenberg antiferromagnet (KHA) falls back

to conventional long-range magnetic ordering in the presence

of a weak next-nearest-neighbor coupling |J2| ≥ 0.03J1[34]

or Dzyaloshinsky-Moriya interaction |D| ≥ 0.012J1[35].

These constrictions further compress the “living space” of

disorder-free perfect QSL compounds.

Alternatively but more realistically, one could ﬁrst ﬁnd

out whether the inherent randomness is fatal or vital to the

QSL physics [4]. In fact, this same question can also be

raised for high-temperature superconductivity, as it is gener-

ally believed that Cooper pairs naturally form once the RVB

states are charged upon chemical doping [6, 36]. The pres-

ence of quenched vacancies in the KHA can lead to a va-

lence bond glass ground state (GS) [37]. Further, Kawamura

et al. found that randomness-induced QSL GSs instead of

spin glasses form in both KHA and triangular Heisenberg an-

tiferromagnet with strong bond randomness, ∆J/J1≥0.4

and 0.6 [38, 39], respectively, which may explain the gapless

behaviors observed in ZnCu3(OH)6Cl2,κ-(ET)2Cu2(CN)3,

EtMe3Sb[Pd(dmit)2]2[33], etc. Later similar scenarios have

been generally applied to the gapless QSL behaviors ob-

served in the strongly-spin-orbital-coupled triangular-lattice

YbMgGaO4[40, 41] with the mixing of Mg2+/Ga3+ [26, 42],

as well as in other relevant materials [43]. Despite the growing

interest in theory, the key issue is whether the paramagnetic

phase conspired by frustration and randomness in real materi-

als is relevant to the exotic QSL/RVB state with strong quan-

tum ﬂuctuations, or simply a trivial product state of quenched

random singlets. To address this issue, local and dynamic

measurements on QSL candidates with quantiﬁable random-

ness are particularly needed.

Recently, a S= 1/2 KHA YCu3(OH)6.5Br2.5(YCOB) has

been proposed, without any global symmetry reduction of the

kagome lattice (space group P¯

3m1, see Fig. 1d) [44–46].

Neither long-range magnetic ordering nor spin-glass freezing

was observed down to 50 mK ∼0.001hJ1i, as evidenced by

speciﬁc heat [46], thermal conductivity [47], and ac suscep-

tibility [44, 45] measurements. The observed power-law T

dependence of low-Tspeciﬁc heat suggests the appearance of

gapless spin excitations [44, 46]. Unlike other known QSL

materials (e.g. ZnCu3(OH)6Cl2[14–16, 48, 49]), the mixing

between Cu2+ and other nonmagnetic ions is prohibited due

to the signiﬁcant ionic difference, thus defect orphan spins

are essentially negligible [44]. Further, the antisite mixing of

the polar OH−and nonpolar Br−causes 70(2)% of randomly

distributed hexagons of alternate exchanges (e.g. Fig. 1b) on

the kagome lattice (Fig. 1d), which accounts for the measured

arXiv:2210.12627v1 [cond-mat.str-el] 23 Oct 2022

Br1-

Br2-

Cu2+

O22-

H2+

Y23+

a b c

-37.8 eV/FU

108.2o

∠CuOCu

~ 116.6o∠CuOCu

~ 115.5o

H1+

O12- Y13+

-35.7 eV/FU

J1a

J1c J1b

Vzz

Vxx Vyy

Vzz

Vxx Vyy

Y2 (~70%)

Y1 (~30%)

Br2 (~50%)

OH2 (~50%)

Br1

FIG. 1. Crystal structure of YCu3(OH)6.5Br2.5around the kagome layer of Cu2+.The Br1 nuclear (81 Br or 79Br ) spin detects three

equidistant Cu2+ electronic spins of each triangle on the kagome lattice (a), and the inset deﬁnes the coordinate system for the spin components.

Whereas, the Br2 nuclear spin mainly probes the nonsymmetric hexagon of spins with alternate exchanges (b), instead of the symmetric

hexagon with almost uniform exchange (c). The formation energies of band cstacking sequences are listed, and the different exchange paths

of Cu-O-Cu (J1a,J1b, and J1c, depending on the bond angles) are marked. The principal axes of the electric ﬁeld gradient calculated by density

functional theory at the Br1 (a) and Br2 (b) sites are displayed by arrows scaled by the modulus of the tensor element, Vzz ∼ −2Vxx ∼ −2Vyy

(see Supplementary Note 6). In our measurements, the external magnetic ﬁeld is always applied along the zaxis, and thus the second-order

quadrupole shifts of the main NMR lines are negligibly small. (d) The crystal structure determined by the single-crystal x-ray diffraction [44].

The thin lines mark the unit cell.

thermodynamic properties above T∼0.1hJ1i[44]. There-

fore, YCOB provides an excellent platform for a quantita-

tive study of the aforementioned randomness-induced QSL

physics.

Herein, the effect of inherent randomness on the QSL prop-

erties was investigated locally by 81Br and 79Br nuclear mag-

netic resonance (NMR) measurements on high-quality YCOB

single crystals. We successfully identify the two NMR sig-

nals originating from the intrinsic kagome spins (Br1, see

Fig. 1a) and spins of hexagons with alternate exchanges on the

kagome lattice (Br2, see Fig. 1b). Simulations of the random

exchange model show good agreement with the measured T

dependence of NMR line shifts and broadening, which sug-

gests the formation of the randomness-induced QSL phase at

low T. The measured in-plane spin ﬂuctuations of the kagome

spin system exhibit a weak power-law Tdependence down

to T∼0.03hJ1idespite the quenched exchange randomness,

thus supporting the survival of strong quantum ﬂuctuations in

YCOB.

Results

NMR spectra. Figure 2a shows the 81Br NMR spectra mea-

sured on the crystal S1(Supplementary Fig. 1) at µ0Hk∼

10.75 T (∼0.14hJ1i). Two well-separated peaks are ob-

served, originating respectively from two different Wyckoff

positions of 2d(Br1) and 1a(Br2). Above ∼15 K, the

ratio between the integrated intensities of these two peaks

IBr2/IBr1 ∼0.2 is well consistent with the stoichiometric ra-

tio of Br2 and Br1 determined by single-crystal x-ray diffrac-

tion (XRD) fBr2/fBr1 = 0.22(1) [44] (see inset of Fig. 2b).

Below 15 K, the weight of Br1 NMR line decreases drastically

and IBr2/IBr1 increases sharply, due to the reduced spin-spin

relaxation times (Supplementary Note 4 and Supplementary

Fig. 5) [50]. Neither Br1 nor Br2 peak splits down to 1.7

K, suggesting the absence of conventional magnetic ordering

within the ability of our resolution.

The NMR shift of Br1 (K1) detecting three (z1= 3) equidis-

tant spins of each triangle on the kagome lattice (Fig. 1a), fol-

lows the bulk susceptibility χkmeasured at the same magnetic

ﬁeld strength in the full temperature range (Fig. 2b), com-

patible with the absence of defect orphan spins in YCOB. In

contrast, the NMR shift of Br2 (K2) shows an obvious devi-

ation from the bulk susceptibility below ∼100 K ∼2hJ1i.

Generically, the NMR shift consists of a T-dependent term

proportional to the local susceptibility and a T-independent

term (K0). Above 100 K, the NMR line broadening is in-

- 1 . 4 - 0 . 7 0 . 0 0 . 7

0 . 0 0

0 . 0 3

0 . 0 6

- 0 . 8

- 0 . 4

0 . 0

0 . 0 0

0 . 0 3

0 . 0 6

0 100 200 300

0 . 0

0 . 3

0 . 6

0 100 200 300

0 . 0

0 . 2

0 . 4

0 . 6

122 123 124

- 0 . 4

- 0 . 2

0 . 0

0 . 0 0 . 2 0 . 4 0 . 6

0 . 0

0 . 1

0 . 2

0 . 3

300 K

200 K

230 K

200 K

165 K

100 K

130 K

8 0 K

6 0 K

5 0 K

4 0 K

3 0 K

2 0 K

1 5 K

1 0 K

6 K

S p i n e c h o i n t e n s i t y ( a r b . u n i t s )

f/f0- 1 ( % )

1 . 7 K

〈Sz〉

K1 ( % )

〈Sz〉

T ( K )

K2 ( % )

IB r 2 /IB r 1

T ( K )

X R D , fB r 2 /fB r 1

f ( M H z )

K1 ( % )

100 K

K2 ( % )

M||/H|| ( 0 . 0 1 B/ T )

300 K

FIG. 2. Nuclear magnetic resonance spectra of YCu3(OH)6.5Br2.5.(a) Frequency-sweep spectra measured on the sample S1at a ﬁeld

µ0Hk∼10.75 T (the reference frequency f0=81γnµ0Hk∼123.64 MHz). The shifts of two lines, K1and K2, are marked by solid and

hollow triangles, respectively. Temperature dependence of K1(b) and K2(c), with the bulk magnetization (hSzi) measured at µ0Hk= 10.75

T for comparison. The inset of bshows the ratio between the integrated intensities of 81Br2 and 81 Br1 lines (IBr2/IBr1 ), as well as the

stoichiometric ratio from x-ray diffraction [44]. Inset of cdisplays K1and K2shifts vs bulk susceptibility χk(i.e. Mk/Hk). The red bars

display the normalized frequency regions where the intensity is larger than half of the maximum value in band c, and error bars on IBr2/IBr1

show a standard error from the ﬁt.

signiﬁcant (see Fig. 2) and the calculated local magnetization

is nearly spatially homogeneous (see Fig. 3a), and thus one ex-

pects a scaling law K=Ahf χk+K0, where Ahf presents the

hyperﬁne coupling between Br (Br1 or Br2) nuclear and Cu2+

electronic spins. By ﬁtting the experimental data (see inset of

Fig. 2c), we obtain Ahf1 =−0.68(2) T/µB,K01 =−0.015(7)%

and Ahf2 = 0.55(3) T/µB,K02 = 0.02(1)%. The presence of

both negative and positive hyperﬁne couplings of the same

nuclear species is surprising, and the underlying mechanism

must be complex, including both the positive and negative

contributions.

The NMR shift of Br2 probes spins of hexagons on the

kagome lattice (Fig. 1b,c), but is obviously smaller than

Ahf2χk(K02 K2) below 100 K (Fig. 2c). In fact, the

formation energy of the optimized nonsymmetric Br2-OH2

stacking sequence (Fig. 1b) (−37.8 eV/FU), is ∼2.5×104

K/FU lower than that of the symmetric Br2-Br2 conﬁguration

(Fig. 1c) (−35.7 eV/FU). Therefore, Br2 ions actually prefer

the nonsymmetric local environments, and Br2 nuclear spins

mainly probe the nonsymmetric hexagons with alternate ex-

changes [44] (J1a > J1c) as illustrated in Fig. 1b. Intuitively,

the nonsymmetric hexagon tends to locally release the frustra-

tion and form three nonmagnetic singlets along the stronger

couplings J1a (see Fig. 3e), which accounts for the relatively

smaller Br2 shifts observed below 100 K (Fig. 3a). More-

over, the Br1 line detects the site susceptibility/magnetization

of all the Cu2+ spins with exchange couplings J1a,J1b, and

J1c, whereas the Br2 line mainly probes the hexagons of spins

only with J1a and J1c. The Br2 line probes less kinds of Cu2+

spins, and thus is narrower than the Br1 line.

Quantitatively, the temperature dependence of both Br1

and Br2 shifts can be reproduced by the average magnetiza-

tion of all the triangles and nonsymmetric hexagons on the

kagome lattice, viz. K1∼Ahf1gkhSz

ti/(µ0Hk)and K2∼

Ahf2gkhSz

nhi/(µ0Hk), respectively (see Fig. 3a), with J1a =

89 K, J1b = 48 K, and J1c =16K(hJ1i ∼ 51 K) experi-

mentally determined by bulk susceptibilities (Supplementary

Note 1 and Supplementary Fig. 2). Taking all the triangles

and nonsymmetric hexagons into account, we are able to sim-

ulate the broadening of both Br1 and Br2 lines by introduc-

ing the distributed density ∝dnS(hSzi)/dhSzi(see Fig. 3b,c),

where dnS(hSzi)is the number of triangles or nonsymmetric

hexagons with the local magnetization (per site) ranging from

hSzito hSzi+dhSzi.hSziis thermally averaged, so the dis-

tributed density is a function of T.

The simulations also enable us to revisit the correlation

functions hSi·Sjiin this KHA system with randomness. Com-

pared to the ideal case (Fig. 3f), a small fraction (∼3/54) of

well-deﬁned singlets with hSi·Sji∼−0.7 → −0.75 are frozen

a b

- 0 . 8

- 0 . 4

0 . 0 0 50 100 150 200 250 300

0 . 0

0 . 3

0 . 6

V S M 〈Snhz〉

〈Stz〉 〈Ss h z〉

0 . 0 0

0 . 0 3

0 . 0 6

- 1 . 0

- 0 . 5

0 . 0 0 . 0

0 . 4

0 . 8

0 . 1 1

- 0 . 5

0 . 0

012345

b 3

b 5

b 4

b 2

J1 a

J1 b

J1 c

- 0 . 5

0 . 0

0 . 5

〈Szi〉

- 0 . 7 5

- 0 . 2 5

0 . 2 5

〈Si⋅Sj〉

b 1

K1 ( % )

B r 1

T ( K )

K2 ( % )

B r 2

〈Sz〉

0 100 200 3000 . 0 0

0 . 0 3

0 . 0 6

〈Stz〉

T ( K )

0 . 0 0 . 1 0 . 2

d e n s i t y

K1 ( % )

0 100 200 3000 . 0 0

0 . 0 3

0 . 0 6

〈Snhz〉

T ( K )

0 . 0 0 . 1 0 . 2

d e n s i t y

T ( K )

K2 ( % )

b 1

b 2

b 3

b 4

b 5

〈Si⋅Sj〉

T/〈J1〉

FIG. 3. Simulations of the spatially inhomogeneous susceptibility probed by nuclear magnetic resonance (NMR) spectra. (a) Temper-

ature dependence of NMR shifts K1and K2, as well as bulk hSzimeasured in a vibrating sample magnetometer at a ﬁeld µ0Hk= 10.75 T.

The colored lines show the calculated average hSziover all the triangles (hSz

ti), nonsymmetric hexagons (hSz

nhi), and symmetric hexagons

(hSz

shi). (b,c) The ﬁnite-temperature Lanczos diagonalization results of local magnetization of triangles (hSz

ti) and nonsymmetric hexagons

(hSz

nhi), along with the measured K1and K2, respectively. (d) Calculated local correlations of the selected spin pairs (see e,f) as function of

normalized temperature T/hJ1i. (e) Calculations of a sample within the random kagome Heisenberg antiferromagnet (KHA) model at T=

0.1hJ1iand µ0Hk= 10.75 T. The solid circles and squares stand for local magnetization hSz

iiat the kagome site iand correlation function

hSi·Sjiof the nearest-neighbor spin pair hiji, respectively. (f) The same calculations as in e, but within the ideal KHA model. The dashed

lines mark the clusters with periodic boundary conditions, and J1a,J1b ,J1c, and J1present the exchange couplings. The color scale in band

cquantiﬁes the distributed density, whereas the ones in eand fquantify the local magnetization (circles) and correlation function (squares),

respectively. The bars in a-cshow the regions where the intensities are larger than half of the maximum value.

at low Tin YCOB (Fig. 3e), which is a signature of releas-

ing frustration due to the quenched randomness. These local

singlets might conﬁne the mobile spinons [27, 37, 51, 52],

which might be responsible for the absence of large mag-

netic thermal conductivity observed in nearly all of the ex-

isting gapless QSL candidates, including the well-known

ZnCu3(OH)6Cl2[28, 29], EtMe3Sb[Pd(dmit)2]2[30, 31], etc.

However, the majority of antiferromagnetic interactions re-

main not fully satisﬁed at low T(see Fig. 3d,e), and the GS

wavefunction should be represented by a superposition of var-

ious pairings of spins. It is worth to mention that the weights

of different pairings should be different due to the quenched

randomness. The survival of strong frustration in the S= 1/2

random KHA speaks against the product GS wavefunction of

randomly distributed singlets, and may still give rise to strong

quantum ﬂuctuations. To testify this, we turn to the spin dy-

namics of YCOB mainly probed by the spin-lattice relaxation

rates as follow.

Spin dynamics. The representative spin-lattice relaxation

data measured on YCOB are displayed in Figs 4a and 4b,

which can be well ﬁtted to the single-exponential function for

the central transition of I= 3/2 nuclear spins, i.e.,

M(t) = M0−2M0F(1

10e−t

T1+9

10e−6t

T1),(1)

where M0and Fare scale parameters for intensity. Alter-

natively, the relaxation data may be ﬁtted with the stretched-

exponential one,

M(t) = M0−2M0F[1

10e−(t

T1)β+9

10e−(6t

T1)β].(2)

c d

FIG. 4. Nuclear spin-lattice relaxation of YCu3(OH)6.5Br2.5.(a) A representative T= 2.8 K spin-lattice relaxation of Br1 measured by

an inversion recovery method, ﬁtted to the stretched-exponential function with ﬁxing β= 1 (red) and tuning β(blue). The pulse sequence for

T1measurements is depicted in the inset. (b) The same ﬁts to the relaxation data measured at other selected temperatures. The inset shows

the ﬁtted stretching exponent β. (c) Temperature dependence of 81Br1 nuclear spin-lattice relaxation rate 1/T1measured on the sample S1

at a ﬁeld µ0Hk= 10.75 T, as well as 81Br1 and 79 Br1 1/T1measured on S2at µ0Hk= 10.75 and 11.59 T, respectively. The colored lines

present the power-law ﬁts to the experimental data below ∼10 K ∼0.2hJ1i, the dashed black and blue lines are the 1/T1and energy (per

site), respectively, calculated by using the random kagome Heisenberg antiferromagnet model of YCu3(OH)6.5Br2.5. (d) The Curie (∼T−1,

black lines) and critical (∼(T−Tc)−α0−1, the critical temperature Tc=−0.1±0.4 K, green lines) ﬁts to 1/(T1T)at 1.7 ≤T≤300 K. The

blue line is the low-Tpower-law dependence as shown in c. The inset shows the T1Tvs Tplot, where the dashed violet line displays the

antiferromagnetic Curie-Weiss behavior (T1T∼T+hJ1i, with coupling hJ1i ∼ 50 K). The T1data presented in cand dare obtained from the

single-exponential ﬁts (i.e., β= 1, see aand b), and the stretched-exponential ﬁts are made only for comparison in aand b. Error bars on the

experimental data points show a standard error from ﬁt, and the error bars in aand bare small.

Here, the stretching exponent βslightly decreases at low tem-

peratures (see inset of Fig. 4b), but remains large down to the

lowest temperature of 1.7 K, β > 0.8. Moreover, the ﬁt to

the single T1function (i.e. β= 1) is still good even at 1.7 K

(Fig. 4b), with the adj. R2= 0.997 [53]. The inclusion of

the additional ﬁtting parameter βonly slightly improves the

ﬁt (the adj. R2increases to 0.998), and even makes the stan-

dard error on T1larger. Therefore, all the following T1data

are obtained with the single-exponential ﬁts.

Figure 4c shows the results of Br1 nuclear spin-lattice re-

laxation rates (1/T1). Br1 1/T1is highly sensitive to the elec-

tronic spin ﬂuctuations perpendicular to the applied magnetic

ﬁeld on the kagome layer via the hyperﬁne coupling [39, 54–

56],

∼T

hf0X

χ00(q, f0)∼πγ2

nA2

hf1g2

⊥

q,m,m0

e−Em

kBT

× |hm0|S⊥

q|mi|2δ(f0−Em0−Em

h),(3)

where the qdependence of Ahf1 is neglected, f0=γnµ0Hk

(kBT/h)→0 presents the NMR frequency, S⊥

qis

the Fourier transform of S⊥

iover all triangles, and Z=

Pmexp(−Em/kB/T )the partition function. In our simu-

lation of 1/T1, the delta function is replaced by a Gaussian

distribution with narrow width ∼10−4hJ1i/h [39, 56].

At high temperatures (T hJ1i), the Moriya para-

magnetic limit yields the T-independent 1/T ∞

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摘要：

TheobservationofquantumuctuationsinakagomeHeisenbergantiferromagnetFangjunLu,1,LongYuan,1,JianZhang,1BoqiangLi,1YongkangLuo,1,yandYueshengLi1,z1WuhanNationalHighMagneticFieldCenterandSchoolofPhysics,HuazhongUniversityofScienceandTechnology,430074Wuhan,China(Dated:October25,2022)AbstractThesearchf...

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The observation of quantum ﬂuctuations in a kagome Heisenberg antiferromagnet Fangjun Lu1Long Yuan1Jian Zhang1Boqiang Li1Yongkang Luo1yand Yuesheng Li1z 1Wuhan National High Magnetic Field Center and School of Physics.pdf

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The observation of quantum ﬂuctuations in a kagome Heisenberg antiferromagnet Fangjun Lu1Long Yuan1Jian Zhang1Boqiang Li1Yongkang Luo1yand Yuesheng Li1z 1Wuhan National High Magnetic Field Center and School of Physics

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