# Ground state and excitation properties of the quantum kagom\'{e} system ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$ investigated by local probes

**ABSTRACT** We characterize the ground state and excitation spectrum of the $S=1/2$, nominally pure and perfect kagom\'{e} system ZnCu$_{3}$(OH)$_{6}$Cl$_{2}$ using the following measurements: magnetization, muon spin rotation frequency shift $K$, transverse relaxation time $T_{2}^{\ast}$, and zero field relaxation, and Cl nuclear spin-lattice relaxation $T_{1}$. We found no sign of singlet formation, no long range order or spin freezing, and no sign of spin-Peierls transition even at temperatures as low as 60 mK. The density of states has $E^{1/4}$ energy dependence with a negligible gap to excitation.

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**ABSTRACT:**Motivated by the unabating interest in the spin-1/2 Heisenberg antiferromagnetic model on the Kagome lattice, we investigate the energetics of projected Schwinger boson (SB) wave functions in the $J_1$--$J_2$ model with antiferromagnetic $J_2$ coupling. Our variational Monte Carlo results show that Sachdev's $Q_1=Q_2$ SB ansatz has a lower energy than the Dirac spin liquid for $J_2\gtrsim 0.08 J_1$ and the ${\bf q=0}$ Jastrow type magnetically ordered state. This work demonstrates that the projected SB wave functions can be tested on the same footing as their fermionic counterparts.Physical review. B, Condensed matter 03/2011; 84. - SourceAvailable from: arxiv.org[show abstract] [hide abstract]

**ABSTRACT:**We present transverse field muon spin rotation/relaxation measurements on single crystals of the spin-1/2 kagome antiferromagnet Herbertsmithite. We find that the spins are more easily polarized when the field is perpendicular to the kagome plane. We demonstrate that the difference in magnetization between the different directions cannot be accounted for by Dzyaloshinskii-Moriya-type interactions alone and that anisotropic axial interaction is present.Journal of Physics Condensed Matter 04/2011; 23(16):164207. · 2.36 Impact Factor - SourceAvailable from: arxiv.org[show abstract] [hide abstract]

**ABSTRACT:**We use the density matrix renormalization group to perform accurate calculations of the ground state of the nearest-neighbor quantum spin S = 1/2 Heisenberg antiferromagnet on the kagome lattice. We study this model on numerous long cylinders with circumferences up to 12 lattice spacings. Through a combination of very-low-energy and small finite-size effects, our results provide strong evidence that, for the infinite two-dimensional system, the ground state of this model is a fully gapped spin liquid.Science 06/2011; 332(6034):1173-6. · 31.20 Impact Factor

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arXiv:cond-mat/0610540v2 [cond-mat.str-el] 28 Dec 2006

Ground state and excitation properties of the quantum kagom´ e system

ZnCu3(OH)6Cl2investigated by local probes.

Oren Ofer and Amit Keren

Physics Department, Technion, Israel Institute of Technology, Haifa 32000, Israel

Emily A. Nytko, Matthew P. Shores, Bart M. Bartlett, and Daniel G. Nocera

Department of Chemistry, Massachusetts Institute of Technology, Cambridge, MA 02139 USA

Chris Baines and Alex Amato

Paul Scherrer Institute, CH 5232 Villigen PSI, Switzerland

We characterize the ground state and excitation spectrum of the S = 1/2, analytically pure and

perfect kagom´ e system ZnCu3(OH)6Cl2 using the following measurements: magnetization, muon

spin rotation frequency shift K, transverse relaxation time T∗

T1. We found no sign of singlet formation, no long range order or spin freezing, and no sign of spin-

Peierls transition even at temperatures as low as 60 mK. The density of states has an E1/4energy

dependence with a negligible gap to excitation.

2, and Cl nuclear spin-lattice relaxation

The study of spin 1/2 quantum magnetism on the

kagom´ e lattice is very intriguing and lively because dif-

ferent investigation strategies have led to fundamentally

different predictions regarding the ground state and ex-

citations of this system. Theories based on numerical

or approximate diagonalization of the Heisenberg Hamil-

tonian favors non-magnetic (singlet) ground state [1, 2]

with full symmetry of the Hamiltonian [3], negligible cor-

relation length [4], and an upper limit on excitation gap

of J/20 [5]. These strategies provide conflicting reports

on the possibility of a spin-Peierls state [3, 6]. In con-

trast, semiclassical treatments based on the thermal [7] or

quantum [8] “order from disorder” approach gives rise to

a ground state with a broken symmetry of the

type, which is stable even against tunneling [10], with

gapless magnon excitations [9], and no spin-Peierls type

distortion [11]. In these circumstances, one expects ex-

periments to provide some guidance. However, the exper-

imental situation is equally confusing since most of the

experimental studies of kagom´ e-like materials suffered

from several kinds of shortcomings, which made com-

parison with theoretical models difficult. Some material,

such as the jarosites [12] have spin S > 1/2. Others have

S = 1/2 but with a non perfect kagom´ e structure, such

as the volborthite Cu3V2O7(OH2)2H2O (CVO) [15]. A

third class of materials such as SrCr9pGa12−9pO19 [13]

and Ba2Sn2ZnGa10−7pCr7pO22 [14] were hampered by

disorder or strong third direction interaction. In light

of these difficulties, the recent synthesis [16] of herbert-

smithite, ZnCu3(OH)6Cl2, with its Cu-based quantum

S = 1/2, analytically pure and perfect kagom´ e lattice,

should put the field on a new course.

√3 ×√3

In this letter we present a comprehensive study of

ZnCu3(OH)6Cl2 using local probes.

address four questions which are at the heart of the in-

vestigation of the quantum kagom´ e system: Do S = 1/2

spins on kagom´ e lattice freeze? Is the ground state mag-

netic? What is the density of excited states, and is there

a gap in the spin energy spectra? Finally, does the lat-

In our study, we

tice distort in order to accommodate spin-Peierls state?

We address these questions in the present work using

nuclear magnetic resonance (NMR) and muon spin res-

onance (µSR) local probes. We also use magnetization

measurements to calibrate the local probes.

ZnCu3(OH)6Cl2was prepared by hydrothermal meth-

ods performed at autogenous pressure.

teflon liner was charged with 16.7 g Cu2(OH)2CO3(75.5

mmol), 12.2 g of ZnCl2(89.5 mmol), and 350 mL water,

capped and placed into a custom-built steel hydrother-

mal bomb under ambient room atmosphere. The tight-

ened bomb was heated at a rate of 1◦C/min to 210◦C

and the temperature was maintained for 48 h. The oven

was cooled to room temperature at a rate of 0.1◦C/min.

A light blue powder was isolated from the base of the

liner by filtration, washed with water, and dried in air

to afford 21.0 g product (49.0 mmol, 97.4% yield based

on starting Cu2(OH)2CO3). Magnetic and pXRD data

were consistent with those previously reported for her-

bertsmithite [16].

DC magnetization m measurements were performed

on a powdered sample using a Cryogenic SQUID mag-

netometer at temperatures T ranging from 2 to 280 K

and fields H varying from 2 kG to 60 kG. In the inset

of Fig. 1 we present mT/H versus T. The data collapse

onto a single line, especially at low T, meaning that the

susceptibility is field-independent in our range of temper-

atures and fields. Also, no peak in the susceptibility is ob-

served, indicating the absence of magnetic ordering. The

only indication of interactions between spin in these mea-

surements is the fact that mT/H decreases upon cooling

whereas in an ideal paramagnetic system this quantity

should be constant. Fits of the susceptibility data at

high temperatures and lower fields than presented here

reveal a Curie-Weiss temperature of ΘCW= −314 K [17].

The frustration parameter TF/|ΘCW| ≈ 0.22, where TF

∼ 70 K is the temperature at which χ−1is no longer a lin-

ear function of T, indicates strong geometric frustration.

For comparison, in CVO TF/|ΘCW| ≈ 1 [18].

A 800 mL

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FIG. 1: The muon shift K against susceptibility. In the inset,

normalized magnetization versus temperature.

Muon spin rotation and relaxation (µSR) measure-

ments were performed at the Paul Scherrer Institute,

Switzerland (PSI) in the GPS spectrometer with an He

cryostat, and in the LTF spectrometer with a dilution re-

frigerator. The measurements were carried out with the

muon spin tilted at 45◦relative to the direction of the

beam. Positron data were collected in both the forward-

backward (longitudinal) and the up-down and right when

available (transverse) detectors simultaneously.

were collected at temperatures ranging from 60 mK to

200 K with a constant field of 2 kG. In GPS we used a

resistive magnet with a calibrated field; in LTF we used

a superconducting magnet. We took data at overlapping

temperatures to calibrate the LTF field. The longitudi-

nal data, which measure muon spin T1, were found to be

useless since the T1in our field range is much longer than

the muon lifetime and showed no temperature variations.

In the lower inset of Fig. 2 we show real and imagi-

nary transverse field [TF] data taken at H = 2 kG and

T = 100 K. The data are presented in a rotating reference

frame (RRF) at a field of 1.9 kG. The TF asymmetry is

best described by ATF= A0exp?−t2/(2T∗2

where T∗

2is the transverse relaxation time, and ω is the

frequency of the muon. The quality of the fit is repre-

sented by the solid line.

In Fig. 1 we depict the frequency shift, K = (ω0−

ω)/ω0 where ω0 is the free muon rotation frequency in

the RRF. The difference in frequency between free and

implanted muons is a consequence of the sample magneti-

zation; therefore, the shift is expected to be proportional

to the susceptibility. Indeed, as shown in the main panel

of Fig. 1, there is a linear relation between K and the

susceptibility χ = m/H, with the temperature as an im-

plicit parameter; some representative temperatures are

shown on the upper axis. In the upper inset of Fig. 2 we

present the field dependence of the shift at T = 10 K.

Data

2)?cos(ωt+φ)

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FIG. 2: A plot of the muon shift K, transverse relaxation

time σ, versus temperature. In the upper inset, the muon

shift K versus external field H. In the lower inset, real and

imaginary transverse field asymmetry for T = 100 K.

Within the error bars, the shift in the RRF does not de-

pend on the external field. This is in agreement with

the susceptibility results. Therefore, Fig. 2 could serve

as a conversion graph from muon spin frequency shift to

susceptibility.

In the main panel of Fig. 2, we depict K as a function

of temperature. An additional axis is presented where

K has been converted to χ as discussed above. We find

that K (and hence χ) increases with decreasing temper-

atures and saturates below T ∼ 200 mK at a value of

χ = 15.7(5) × 10−3cm3/mol Cu; the error is from the

calibration. It should also be pointed out that the en-

ergy scale associated with spin 1/2 in a field of 2 kG is

200 mK, and the saturation could be a consequence of

the external field. The saturation of χ is a strong ev-

idence for the lack of impurities in our sample. More

importantly, it indicates the lack of singlet formation or

spin freezing. The last conclusion is also in agreement

with neutron scattering measurements [17] and zero field

µSR [19].

The muon transverse relaxation rate 1/T∗

sented in Fig. 2. Roughly speaking, it has the same

temperature behavior as the shift (and as the suscepti-

bility). T∗

2relaxation is a result of defects in the sample

causing a distribution of muons to electronic spin cou-

pling constants or a distribution of susceptibilities. It

has been shown that when the muon relaxation rate be-

haves similarly to the shift [20] (or susceptibility [21])

upon cooling, it indicates quenched distribution of either

the coupling constants or susceptibilities. In this case

the relaxation increases simply because the average mo-

ment size increases. Since the coupling constants and

susceptibility are functions of distances between muon

and electronic spin or between two electronic spins, our

results are consistent with a lack of lattice deformation

in ZnCu3(OH)6Cl2.

Since muons could not reveal dynamic T1information,

2is also pre-

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we performed37Cl and35Cl NMR experiments on the

same sample. Using the two isotopes, we are able to

determine the origin of T1. The first step in such a mea-

surement is to find the line shape and to identify the

isotopes and transitions. This measurement was done at

a constant applied frequency of νapp = 28.28 MHz and

a varying external field H. A standard spin-echo pulse

sequence, π/2 − τ − π, was applied, and the echo sig-

nal was integrated for each H. In Fig. 3 we show a

field sweep for both Cl isotopes obtained at T = 100 K.

A rich spectrum is found and is emphasized using five

x-axis and one y-axis breakers. This rich spectrum is

a consequence of the Cl having spin 3/2 for both iso-

topes. In the case where the nuclei reside in a site with

non cubic local environment and experience an electric

field gradient, their spin Hamiltonian could be written as

H = −hνlI·(1+K)·?H+(hνQ)/6?3I2

the anisotropy parameter, K is the shift tensor, and

νl= γH/(2π). The powder spectrum of such nuclei has

two satellite peaks corresponding to the 3/2 ←→ 1/2 and

−3/2 ←→ −1/2 transitions, and a central line from the

1/2 ←→ −1/2 transition, which is split due to the pow-

der average. The transition names are presented in the

figure. The satellite peaks at T = 100 K of35Cl are at

6.52 and 7.07 T, and for37Cl at 7.91 and 8.41 T. The

lack of singularity in the satellite spectrum indicates that

the Cl resides in a site with η > 0, namely, with no xy

symmetry. These findings are consistent with a 3m site

symmetry of the Cl ion in the space group R¯3m.

In contrast to the two satellites, the spliting of the

central lines at T = 100 K is clear, and appear for the

35Cl at 6.778 and 6.801 T and for the

and 8.161 T. Under some assumptions these values could

be used to determine the parameter of the nuclear spin

Hamiltonian [13]; assuming that the nuclear spin opera-

tors, Ix, Iyand Izare colinear with the principal axes of

the shift tensor, and that the in-plane shift is isotropic

with K⊥ = (Kx+ Ky)/2, we find for both isotopes,

K⊥ ≃ −0.0017(5), Kz ≃ 0.035(9) and η = 0.4, and

35νQ= 3.75 MHz and37νQ= 2.55 MHz. The ratio of νQ

is as expected from the ratio of the quadrupole moments.

Due to the assumptions, the value of Kzshould only be

considered as an order of magnitude. Nevertheless, it

is not very different from that of the muon shift (in the

laboratory frame) at the same temperatures. This means

that both probes experience a similar field generated by

the Cu spins, which for muons is usually a dipolar field.

Temperature dependence field sweeps of the35Cl cen-

tral line are shown in Fig. 4. The intensities are in arbi-

trary units for clarity. The ±1/2 ←→ ∓1/2 transitions

are easily observed at T = 300 and 100 K (indicated by

the arrows in the figure) but are smeared out at lower

T. In fact, the lines become so broad that the NMR

shift cannot be followed to low temperature; hence the

importance of the µSR results.

Finally, we measured the37Cl spin-lattice relaxation

rate T−1

1

to determine spin gap and excitation spec-

z− I2+ η?I2

x− I2

y

??

where νQ is the quadrupole frequency, 0 ≤ η ≤ 1 is

37Cl at 8.148

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FIG. 3: A field sweep of35Cl and37Cl.

trum. The data were taken at a field of 8.15 T which

corresponds to the low field peak of the central line.

We use a saturation recovery pulse sequence.

5 we depict T−1

1

normalized by γ2where37γ = 3.476

MHz/T on a semi-log scale. T−1

ing down to 50 K and then sharply decreases. We also

present35Cl (T1γ2)−1where35γ = 4.172 MHz/T below

50 K in order to determine the origin of the dynamic

fluctuations. These measurements were done under the

same conditions as37Cl. When considering all temper-

atures we find that T35

1

netic relaxation mechanism we expect this ratio to equal

(37γ/35γ)2= 0.69. From a quadrupole based mechanism

we anticipate (37Q/35Q)2= 0.62 where Q is the nuclear

quadrupole moment. Our finding is in favor of relaxation

mediated by a magnetic mechanism as indicated by the

In Fig.

1

increases upon cool-

1/T37

= 0.75(10). From a mag-

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FIG. 4:

peratures. The arrows indicate the central line singularities

observed at high-T but smeared out at low T.

35Cl field sweep (ν = 28.28MHz) at different tem-

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FIG. 5: A semi-log plot of the Cl inverse spin-lattice relax-

ation, (γ2T1)−1, versus temperature. Inset, a linear plot of

the low-temperature region. The black line is a fit to Eq. 1.

The dashed line is straight.

overlapping (T1γ2)−1data points in Fig. 5.

In the inset of Fig. 5 we zoom in on the low T data us-

ing a linear scale. A first glance suggests that at low tem-

perature 1/T1is a linear function of T as indicated by the

dashed line. A remanent relaxation at zero temperature

1/Tn

1could be due to magnetic fluctuations form other

nuclear moments such as the protons or copper, since

they continue to fluctuate even when the electronic mo-

ments stop. Thus the spin lattice relaxation due to elec-

tronic contribution only 1/Te

1/(Te

1T) = const. This relation is expected in the case

of free fermions and might be related to recent theories

[22, 23].

A different approach to T1interpretation is in terms of

magnon Raman scattering where

?∞

∆

1≡ 1/T1(T) − 1/Tn

1obeys

1

T1(T) =

1

Tn

1

+γ2A2

ρ2(E)·n(E)·[n(E) + 1]dE (1)

with ρ being the density of states, ∆ the gap, A is a

constant derived from the hyperfine coupling, and n(E)

the Bose-Einstein occupation factor [24]. This expres-

sion is constructed from the population of magnons be-

fore and after the scattering, with the associated density

of states and the assumption that they exchanged negli-

gible amount of energy with the nuclei since its Zeeman

splitting is much less than a typical magnon energy. How-

ever, in frustrated magnets the magnon might not be the

proper description of the excitations [13, 14]. Neverthe-

less, we use Eq. 1 since it is expected for any kind of

bosonic excitations, and since there is no other available

theory. We assume ρ(E) ∼ Eα, with α and ∆ as fit pa-

rameters. The fit of Eq. 1 to the data is presented as the

solid line in Fig. 5, and in its inset. We find α = 0.23(1)

and ∆ = 0.5(2) K. Comparing to J = 209 K [17], this is

a negligibly small gap. It indicates that most likely there

is no gap in the spin energy spectra, in agreement with

Ref. [17], and ρ(E) ∼ E1/4.

To conclude, susceptibility measurements down to

60 mK suggest that there is no freezing and only a sat-

uration of susceptibility, namely, no singlet formation.

The data also do not support the presence of lattice de-

formation.Finally, Cl NMR T1 measurements find a

negligibly small magnetic gap and the density of states

ρ ∼ E1/4. Thus, ZnCu3(OH)6Cl2 is an exotic magnet

with no broken continuous symmetry but gapless exci-

tations. It might be an example of algebraic spin liquid

[23].

We would like to thank the PSI facility for support-

ing the µSR experiments and for continuous high quality

beam, and the NATO CollaborativeLinkage Grant, refer-

ence number PST.CLG.978705. We acknowledge helpful

discussions with Young. S. Lee and Philippe. Mendels,

Peter Mueller, Joel Helton, and Kittiwit Matan.

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