Quantum phase transition in the dioptase magnetic lattice

Abstract

The study of quantum phase transitions, which are zero-temperature phase transitions between distinct states of matter, is of current interest in research since it allows for a description of low-temperature properties based on universal relations. Here we show that the crystal green dioptase Cu_6Si_6O_18 . 6H_2O, known to the ancient Roman as the gem of Venus, has a magnetic crystal structure, formed by the Cu(II) ions, which allows for a quantum phase transition between an antiferromagnetically ordered state and a quantum spin liquid. Comment: 6 pages, 5 figures, EPL, in press

Full text

arXiv:cond-mat/0208106v1 [cond-mat.str-el] 6 Aug 2002
Europhysics Letters PREPRINT
Quantum phase transition in the dioptase magnetic lattice
Claudius Gros1, Peter Lemmens 2, K.-Y. Choi 3, G. G¨
untherodt3, M. Baenitz4
and H.H. Otto 5
1Fachbereich Physik, Postfach 151150, Universit¨at des Saarlandes, 66041 Saarbr¨ucken,
Germany
2IMNF, TU Braunschweig, D-38106 Braunschweig, Germany
3II. Physikalisches Institut, RWTH-Aachen, Templergraben 55, 52056 Aachen, Ger-
many
4Max-Planck-Institut f¨ur Chemische Physik fester Stoffe, MPI-CPfS, D-01187 Dres-
den, Germany
5Institut fr Mineralogie und Mineralische Rohstoffe, TU Clausthal, D-38678 Clausthal-
Zellerfeld, Germany
PACS. 71.20.Be – Transition metals and alloys.
PACS. 73.43.Nq – Quantum phase transitions.
PACS. 75.10.Jm – Quantized spin models.
Abstract. – The study of quantum phase transitions [1], which are zero-temperature phase
transitions between distinct states of matter, is of current interest in research since it allows for
a description of low-temperature properties based on universal relations. Here we show that
the crystal green dioptase Cu6Si6O18 ·6H2O, known to the ancient Roman as the gem of Venus,
has a magnetic crystal structure, formed by the Cu(II) ions, which allows for a quantum phase
transition between an antiferromagnetically ordered state and a quantum spin liquid.
The gem-stone dioptase Cu6Si6O18·6H2O is a transparent green mineral build up from
Si6O18 single rings on a lattice, which sandwiches six-membered water rings down the (crys-
tallographic) c-direction [2–4]. The magnetic Cu(II) ions are located between the Si6O18 rings
and form chiral chains along c, placed on an ab-honeycomb net and are there edge-sharing
connected forming Cu(II) dimers.
We illustrate in fig. 1 the sublattice of the magnetic Cu(II)-ions. This three-dimensional
magnetic lattice is characterized by only two coupling constants in between the spin-1/2
Cu(II)-moments. The magnetic sublattice is characterized by an antiferromagnetic intra-
chain J2, which couples the Cu(II)-chains and an antiferromagnetic inter-chain coupling J1,
leading for small J1/J2to a AB-type Ne´el ordered state with doubling of the unit-cell along c.
Alternatively one might consider the dioptase magnetic lattice as made up by in-plane dimers
of Cu(II)-ions, with an intra-dimer coupling strength of J1and an inter-dimer coupling along
cof J2. For small J2/J1a singlet-dimer state with a spin-gap and no long-range magnetic
order is then realized.
c
EDP Sciences

2EUROPHYSICS LETTERS
a
b
c
4
Fig. 1 – An illustration of Cu-sublattice of the dioptase crystal structure. The rhombohedral unit-
cell contains 18 equivalent Cu atoms arranged in six chains with three atoms down the cperiod.
The inter/intra-chain magnetic coupling with strength J1and J2are indicated by white/black sticks.
Left: An ab-plane. Not shown are the Si6O18 rings, located inside the 12-membered Cu-rings. The
rhombus denotes the in-plane hexagonal unit-cell. Right: two chiral chains along c.
In fig. 3 we present the phase-diagram of the dioptase magnetic lattice, which we obtained
from Quantum-Monte-Carlo (QMC) simulations, using the stochastic series expansion with
worm-updates [5, 6]. We used the parameterization J1,2=J(1 ±δ).
In order to determine the phase-diagram, an accurate estimate of the transition temper-
ature to the ordered state is necessary. For this purpose we evaluated by QMC one of the
Binder-cumulants [7], namely hm2
AF i/h|mAF |i2, where mAF is the antiferromagnetic-order pa-
rameter (the staggered magnetization). The temperature at which the cumulants for different
finite cluster intersect provide reliable estimates for the Ne´el temperature [7], see fig. 2. For
the numerical simulations we used (n, n, m) clusters with periodic boundary conditions, were
n2and mare the number of unit-cells in the ab-plane and along the c-axis respectively. We
performed simulations for (2,2,20), (3,3,30) and (4,4,40) clusters containing 1440, 4860 and
11520 Cu(II) sites respectively.
The linear raise of TNin fig. 3 occurring for small inter-chain couplings J1is a consequence
of the quantum-critical nature of the spin-1/2 Heisenberg chain realized for J1= 0. The
magnetic correlation length ξ(T) diverges as ξ(T)∼T−1for a Heisenberg-chain at low-
temperature. For small interchain couplings J1a chain-mean-field approach is valid [8] and
the transition occurs when J2≈J1ξ(TN)∼J1/TN. Consequently TN∼T J1/J2for small
J1/J2.
The critical temperature for the transition, which is in the 3D-Heisenberg universality

Claudius Gros, Peter Lemmens, K.-Y. Choi, G. G¨
untherodt, M. Baenitz and H.H. Otto :Quantum phase transition in the
0.15 0.2 0.25 0.3 0.35
1.3
1.4
1.5
1.6
Binder cummulant
11520 sites
4860 sites
1440 sites
δ=0.2
δ=0.0
T/J1
Fig. 2 – QMC-results for the dimensionless Binder-cumulant hm2
AF i/h|mAF |i2for (nnm)-clusters with
periodic boundary conditions. The lines are guides to the eye, the MC-estimates for the statistical
errors are given. Shown are the results for n= 2,m= 20 (1440 sites), n= 3,m= 30 (4860 sites) and
n= 4,m= 40 (11520 sites) and two value of δ(J1,2=J(1 ±δ)).
class, is maximal for δ≈ −0.1 and vanishes at a quantum critical point δc≈0.3. Long-range
magnetism is absent beyond this point and the ground-state is a quantum spin-liquid. For
J2= 0 the dioptase magnetic lattice decomposes into isolated dimers.
The magnitude of the singlet-triplet gap ∆ in the spin-liquid state can be estimated by a
fit of the low-temperature QMC-susceptibility to χ(T)≈(kBT /∆)d/2−1e−∆/(kBT), where d
is the dimensionality of the triplet-dispersion above the gap. For an isolated dimer d= 0, for a
spin-ladder d= 1 [9]. This analysis would predict d= 3 for the dioptase magnetic sublattice,
but fits of the QMC-results for χ(T), presented in fig. 3, favor d= 0.
In fig. 4 we present the susceptibility of green dioptase (using a crystal from Altyn Tyube,
Kazakhstan) down the He-temperatures measured with a commercial SQUID magnetometer
(Quantum Design). The data for magnetic field aligned parallel and perpendicular to the
c-axis presented in the Inset of fig. 4 show clearly a transition to Ne´el-ordered stated at
T(exp)
N= 15.5 K. The moments are aligned along cfor T < T (exp)
N.
The QMC results for the susceptibility are to be compared, due to spin-rotational in-
variance, with the directional averaged of the experimental susceptibility, presented in the
main panel of fig. 4. We have determined the Hamiltonian parameters J1=J(1 + δ) and
J2=J(1 −δ) appropriate for dioptase in the following way. For every δ < δcthe overall
coupling constant Jwas determined by fixing the transition temperature to the experimental
T(exp)
N= 15.5 K. The spin-susceptibility in experimental units is then
χ(exp)= 0.375 ∗Z∗(g2/J)∗Λmm ,(1)
where Z= 3 is number of Cu2+ -ions in the primitive unit-cell. The dimensionless
magnetization-fluctuation is Λmm = (Jβ)< m2>−< m >2, where mis the magnetiza-

4EUROPHYSICS LETTERS
−1 −0.5 0 0.5 1
δ
0
0.1
0.2
0.3
TC/J
phase diagram of Dioptase−lattice
2.0
3.0
1.0
∆/J
J1=J(1+δ)
J2=J(1−δ)
gap
Fig. 3 – Phase diagram of the dioptase magnetic sublattice as obtained by Quantum Monte Carlo
simulations. The lines are guides to the eye. The magnetic coupling constants are J1,2=J(1 ±δ)
for inter/intra-chain couplings J1/J2. At δc≈0.3 a quantum phase transition occurs. The Ne´el
temperature of the antiferromagnetically ordered phase for δ < δcis give by the left y-axis. The
antiferromagnetic order is of A-B type, with a doubling of the unit-cell along c. For δ > δca gap,
given by the right y-axis, opens in the magnetic excitation spectrum and the state is a quantum
spin-liquid.
tion. The g-factor was then determined, for every δ < δc, by adapting the right-hand-side of
eq. 1 to the experimental susceptibility at high temperatures. The results are shown in fig. 4
together with the optimal values for Jand g. We see that the optimal value g≈2.1 for the
g-factor is relatively independent of δ.
We find two possible values for the ratio of the two-coupling (antiferromagnetic) constants
J1and J2namely δ= 0.1 and δ=−0.1 which fit the experimental data equally well. Note
that δ=−0.2 does not agree well for T < T (exp)
C. We attribute the residual discrepancies in
between the theory and the experimental data to residual interactions, in addition to J1and
J2
It has been suggested previously [10] that the in-chain coupling J2might actually be
ferromagnetic. We have studied therefore also the case for negative J2and found a quantum-
phase-transition to a state with alternating ferromagnetic chains for J2≈ −0.7J1. We have
performed the corresponding analysis to the one shown in fig. 4 for the the case of ferromag-
netic J2. We found very large deviations in between experiment and theory in this case, due
to the fact that the susceptibility of ferromagnetic chains diverges for T→0.
To settle the ambiguity concerning the δparameter we investigated the magnetic Raman

Claudius Gros, Peter Lemmens, K.-Y. Choi, G. G¨
untherodt, M. Baenitz and H.H. Otto :Quantum phase transition in the
0 50 100 150
T [K]
0.005
0.01
0.015
0.02
0.025
0.03
susceptibility [emu/mol]
experiment (averaged)
δ= 0.2, J=66.0, g=2.12
δ= 0.1, J=56.6, g=2.09
δ= 0.0, J=53.4, g=2.1
δ=−0.1, J=53.3, g=2.1
δ=−0.2, J=56.1, g=2.14
TN
(exp)
0 100 T [K]
χ
Fig. 4 – QMC-results for the susceptibility (in emu/mol) for various δin comparison to the directional-
averaged experimental susceptibility (solid line). Inset: The susceptibility χfor magnetic fields
parallel/orthogonal to the c-axis (lower/upper) curve. The vertical dashed lines in the main panel
and in the inset indicates the location of the Ne´el temperature.
spectrum of dioptase as a function of temperature, as shown in fig. 5. The Raman scattering
experiments were performed in quasi-backscattering geometry with a triple grating optical
spectrometer (DILOR XY) with the λ= 514 nm laser line. Two modes at 48 and 85 cm−1
(≡69 and 122 K) are magnetic as they exhibit a temperature dependence related to the
transition. They show no anisotropy concerning the scattering selection rules. The excitation
energies 69 K and 122 K correspond, for δ= +0.1, to one and two inter-chain dimer excitation
energy J1=J(1+δ), as expected for one- and two-magnon scattering processes. The lineshape
of the magnetic two-magnon 122 K mode is very unusual, it is symmetric and not substantially
broaded by either magnon-magnon scattering or density-of-states effects, in contrast to usual
two-magnon scattering in normal 3D antiferromagnets [11]. This behavior indicates a very
small dispersion of the underlying magnon branch. We consequently conclude that dioptase
is relatively close to a quantum-critical point.
In conclusion we have presented a novel magnetic lattice structure, the dioptase magnetic
lattice, which allows for a quantum-phase transition. This lattice is realized in green dioptase
Cu6Si6O18 ·6H2O and in the recently synthesized isostructural germanate Cu6Ge6O18 ·6H2O
[12,13], a promising candidate to study further aspect of the phase diagram presented in detail
fig. 3.
We acknowledge fruitful discussions with Matthias Troyer on the stochastic series expan-
sion and Felicien Capraro for data analysis.

6EUROPHYSICS LETTERS
Fig. 5 – Low energy Raman spectrum of dioptase in xx-polarization. The modes at 48 and 85 cm−1
(≡69 and 122 K) show a strong increase of intensity for T <TN= 15.5 K and correspond to one-
and two-magnon processes. The temperature independent modes at 70 and 100 cm−1are phonons.
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