# Intrinsic new properties of a quantum spin liquid

Y. X. Yang  State Key Laboratory of Surface Physics, Department of Physics,
Fudan University, Shanghai 200433, China  Xin Li  State Key Laboratory of
Surface Physics, Department of Physics, Fudan University, Shanghai 200433,
China  Key Laboratory of Neutron Physics and Institute of Nuclear Physics and
Chemistry, China Academy of Engineering Physics (CAEP), Mianyang 621999, China
C. Tan  Z. H. Zhu  J. Zhang  Z. F. Ding  Q. Wu  C. S. Chen  State Key
Laboratory of Surface Physics, Department of Physics, Fudan University,
Shanghai 200433, China  T. Shiroka  Laboratory for Muon-Spin Spectroscopy,
Paul Scherrer Institut, 5232 Villigen, Switzerland  Y. H. Xia  Key Laboratory
of Neutron Physics and Institute of Nuclear Physics and Chemistry, China
Academy of Engineering Physics (CAEP), Mianyang 621999, China  D. E.
MacLaughlin  Department of Physics and Astronomy, University of California,
Riverside, California 92521, USA  C. M. Varma<EMAIL_ADDRESS>Department of Physics, University of California, Berkeley, California 94704,
USA  L. Shu<EMAIL_ADDRESS>State Key Laboratory of Surface Physics,
Department of Physics, Fudan University, Shanghai 200433, China  Collaborative
Innovation Center of Advanced Microstructures, Nanjing 210093, China  Shanghai
Research Center for Quantum Sciences, Shanghai 201315, China

###### Abstract

Quantum fluctuations are expected to lead to highly entangled spin-liquid
states in certain two-dimensional spin-1/2 compounds. We have synthesized and
measured thermodynamic properties and muon spin relaxation rates in the
copper-based two-dimensional triangular-lattice spin liquids Lu3Cu2Sb3O14 and
Lu3CuZnSb3O14. The former is the least disordered of this kind discovered to
date. Magnetic entropy generation at high temperatures has been ruled out
after carefully correcting for the lattice specific heat. Surprisingly,
roughly half of the magnetic entropy is missing down to temperatures of
O(10-3) the exchange energy, independent of magnetic field up to
$g\mu_{B}H\gtrsim k_{B}\Theta_{W}$, where $\Theta_{W}$ is the Weiss
temperature. The magnetic specific heat divided by temperature $C_{M}(T)/T$
and muon spin relaxation rate $\lambda(T)$ are both temperature-independent at
low temperatures, followed by logarithmic decreases with increasing
temperature. This behavior can be simply characterized by scale-invariant
time-dependent fluctuations with a single parameter. Since no cooperative
effects due to impurities are observed, the measured properties are intrinsic.
They are evidence that in Lu3Cu2Sb3O14 massive quantum fluctuations lead to
either a gigantic specific heat peak from singlet excitations at very low
temperatures or, perhaps less likely, an extensively degenerate possibly
topological singlet ground state.

††preprint: ver.7.2

##  I INTRODUCTION

The study of quantum fluctuations in interacting matter is of primary interest
in many fields of physics, encompassing fields as diverse as the
thermodynamics of black holes [1, 2], particle physics beyond the standard
model [3], the theory of quantum computation [4], and a considerable number of
phenomena in condensed matter physics. The latter allow access and control to
a wide variety of experiments, and the concepts often cut across different
fields. These range from quantum Hall effects [5] to the quantum criticality
that governs high temperature superconductivity [6, 7] to spin-liquid states
[8, 9], all of which have been intensively studied in the last three decades.

Spin liquids, in particular, have been hard to characterize beyond the fact
that quantum fluctuations prevent any conventional order in them. Despite
extensive experiments, few precise conclusions about the nature of the ground
state and low-lying excitations are available, because the results are almost
always dominated by cooperative effects, however interesting, of the
impurities [9, 10, 8].

We have synthesized the $S{=}1/2$ trigonal-lattice compounds Lu3Cu2Sb3O14
(LCSO) and Lu3CuZnSb3O14 (LCZSO). They are variations on the $R_{3}$Zn2Sb3O14
($R$ = rare earth) series of tripod-kagomé-lattice compounds [11, 12, 14, 15,
16], with $R$ = Lu and Zn completely (LCSO) or half (LCZSO) substituted by Cu.
In LCSO the Cu ions form 2D triangular sublattices in two separate layers. It
is remarkably defect-free: we estimate the magnetic impurity concentration to
be $\lesssim$0.1% and other impurity or defect concentrations $\sim$1%. In
LCZSO Cu and Zn ions predominantly occupy alternate triangular layers, but
$\sim$5% Cu/Zn site interchanges affect its properties compared to LCSO.

Thermodynamic properties and muon spin relaxation ($\mu$SR) rates of these
compounds have been measured, the latter down to 16 mK. There are no
signatures of static magnetism, ordered or disordered, or any other
cooperative effects of impurities in either compound, from 300 K down to the
lowest temperatures. A very surprising result is that in LCSO the _measured_
magnetic entropy $S_{M}(T)$ above 0.1 K, obtained from the specific heat and
magnetization (the latter from Maxwell relations), saturates at high
temperatures at roughly 40% of the total magnetic entropy $k_{B}\ln{2}$ per
spin-1/2; $\sim$60% of the magnetic entropy is _missing_. $\mu$SR measurements
effectively extend these results down to 16 mK. The missing entropy is
independent of magnetic field up to 9 T.

Furthermore, in both LCSO and LCZSO the magnetic specific heat coefficient
$C_{M}/T$ and $\lambda(T)$ track each other; both are constant at low
temperatures, followed by logarithmic decreases with increasing temperature.
These results are shown to be consistent with scale-invariant magnetic
fluctuations.

The central result of our experiments is that the missing entropy resides at
very low temperatures, or perhaps in the ground state. This requires accurate
determination of the temperature dependence of $S_{M}(T)$ over a very wide
range of temperatures. Up to about 20 K, which is of the order of the
paramagnetic Weiss temperature $\Theta_{W}$ from the susceptibility, the
measured total specific heat of magnetic LCSO is clearly larger than that of
nonmagnetic LZSO. Then $S_{M}(T)$ is obtained by subtraction, the accuracy of
which is tested by measurement of the change in entropy in a magnetic field.
This procedure yields $S_{M}(H{=}0,T)$ from 0.1 K to 20 K of only
$0.36\,k_{B}\ln{2}$ per spin.

At higher temperatures determination of $S_{M}(T)$ by subtraction becomes
difficult, due to the small difference between the lattice specific heats
$C_{\mathrm{latt}}$ in the two compounds: their masses and force constants are
not exactly the same. This difference is accounted for by scaling the
temperature dependence of $C_{\mathrm{latt}}$ in LZSO before subtraction. The
temperature scaling index $\eta$ is a free parameter at this stage, subject to
an entropy difference $S_{\mathrm{diff}}(T)$ that must exhibit zero or
positive slope and remain $\leq k_{B}\ln{2}$ per spin at high temperatures if
it is indeed $S_{M}(T)$.

Complementary information on the magnetic entropy at higher temperatures has
been obtained using reliable high-temperature expansions for the entropy and
susceptibility of a Heisenberg model on the triangular lattice with nearest-
neighbor and 2nd-nearest-neighbor exchange interactions $J_{1}$ and $J_{2}$,
respectively [13]. The expansions are expected to yield reliable results from
asymptotically high temperatures down to roughly $2J_{1}$ if $J_{2}\ll J_{1}$.
An accurate fit of the susceptibility expansion to the data determines $J_{1}$
and $J_{2}$, which are then used to deduce the entropy from its expansion. The
best estimate of $\eta$ is obtained from the requirement that the scaled
entropy at high temperatures agrees with the series expansion to within a
constant.

For $H=0$ this procedure yields a constant magnetic entropy
${\sim}0.4k_{B}\ln{2}$ per spin above $\sim$25 K. Additional entropy
generation at high temperatures is ruled out, and the missing
${\sim}0.6k_{B}\ln{2}$ per spin must therefore arise from states below
$\sim$0.1 K that do not contribute to the specific heat above that
temperature. This“ground-state” entropy (more likely from very low-lying
excited states) is a constant contribution at all experimental temperatures.

We believe it is the high purity of LCSO that allows us to characterize its
extraordinary intrinsic properties, and that these properties are
representative of a new class of spin liquids to which it belongs.

The paper is organized as follows. Structure determination and results of
specific heat, entropy, magnetization, and $\mu$SR experiments are reported in
Sec. II, with details given in the Supplemental Information (SI). In Sec. III
we discuss high-temperature series expansions for thermal quantities, Cu-ion
coordination and symmetry, evidence for two separate sublattices that are at
most weakly coupled magnetically, a scale-invariant phenomenology, theories of
spin liquids, and comparison of our results with other spin liquids. Section
IV summarizes the evidence for colossal quantum fluctuations in LCSO, which
result in completely new ground and low-lying excited states of a spin liquid.

##  II RESULTS

###  II.1 Structure

LCSO, LCZSO, and the isostructural nonmagnetic compound Lu3Zn2Sb3O14 (LZSO)
belong to the tripod-kagomé-lattice family [11, 12, 14, 15, 16], in which
kagomé lattices are formed by alternating layers of filled-shell ($S{=}0$)
Sb5+ and rare-earth/Lu3+ ions. The compounds were synthesized by solid-state
reaction. Stoichiometric amounts of Lu2O3, CuO (ZnO for LCZSO and LZSO), and
Sb2O3 were thoroughly mixed using an agate mortar, and heated to 1030∘C for 60
hours with intermediate regrinding and reheating. So far only powder samples
could be prepared.

Figure 1:  Crystal structure of LCSO. (a) Unit cell, showing alternating Lu-
Cu1 and Sb-Cu2 layers and oxygen coordinations of Cu1 3a and Cu2 3b sites.
Inequivalent triangular Cu1 and Cu2 lattices are centered in the kagomé
hexagons of (b) Lu layers and (c) Sb layers, respectively. The unit cells
(thick black lines) are the same in both layers, showing that Cu1 sites are
centered in Cu2 triangles and vice versa.

The crystal structure was determined from diffraction data taken at room
temperature using a Bruker D8 advance x-ray diffraction (XRD) spectrometer
($\lambda=1.5418$ Å), and the high-resolution neutron powder diffractometer
(NPD) ($\lambda=1.8846$ Å) at the China Mianyang research reactor [17]. XPD
and NPD spectra for LCSO and LCZSO, shown in SI Sec. S I, exhibit narrow Bragg
peaks. Parameters from Rietveld fits to the spectra are given in SI Table S1,
and the crystal unit cell is shown in Fig. 1(a).

##### LCSO.

Spin-1/2 Cu2+ ions are located at the centers of the kagomé hexagons Cu1-Lu
and Cu2-Sb, Wyckoff positions 3a and 3b respectively, and form inequivalent
triangular sublattices. The 3a and 3b sites have different oxygen
coordinations: 6-fold distorted octahedra and 8-fold distorted cubes,
respectively.

Our results indicate that LCSO is extraordinarily defect-free, with
concentrations of “orphan” spins $\lesssim 10^{-3}$ and negligible Schottky
magnetic defects. Static magnetism, ordered or disordered, would be expected
from orphan spins, perhaps at low temperatures, but ac susceptibility and
$\mu$SR experiments rule this out, the latter down to $\sim$16 mK. A weak
Schottky specific-heat anomaly has an unobservably small effect on the entropy
obtained from the specific heat.

It is important to note that when viewed along the $c$ axis, each Cu site is
centered in equilateral triangles of Cu sites in adjacent layers [Figs. 1(b)
and (c)]. Section III.2 discusses symmetry restrictions on exchange
interactions imposed by this structure.

A few percent displacement of Zn2 ions from the 3b position is observed in the
related compounds RE3Zn2Sb3O14, RE = La, Pr, Nd, Sm, Eu, and Gd [12], and of
Co1 ions around the 3a position in RE3Co2Sb3O14, RE = La, Pr, Nd, Sm–Ho [11].
The offset decreases markedly with rare-earth ionic radius [12, 18], as shown
in Fig. 2, although it does not extrapolate unambiguously to zero for Lu..

Figure 2:  Dependence of Zn2 displacement from 3b position on rare-earth ionic
radius [18] in RE3Zn2Sb3O14. Zn2 displacements from Ref. 12, ionic radii from
Ref. 18.

However, no offset is found for Cu2 in LCSO. From NPD Rietveld refinements,
the thermal parameter $B$ for the Cu2 3b site $(0,0,0.5)$ in LCSO is 1.53 with
no offset but significantly larger (2.26) assuming a 3% offset ($\sim$0.2 Å).
The assumption artificially increases $B$.

RE/Mg site disorder has been observed in the tripod kagomé systems
Dy3Mg2Sb3O14 [14] and Ho3Mg2Sb3O14 [16]. RE3Zn2Sb3O14 [15], RE = Ho, Er, and
Yb, were reported to exhibit RE/Zn disorder, but with some doubt as to the
correct model. We find no evidence for such disorder in LCSO; NPD Rietveld
fits with fixed disorder result in unphysically negative thermal factors.
Furthermore, disordered Cu ions on Lu 9e sites would be “orphan” spins, with
separate contributions to the susceptibility and specific heat; these are not
observed.

##### LCZSO.

To determine whether the observed properties are specific to the 2D layers, we
also synthesized Lu3CuZnSb3O14 (LCZSO). We find that nonmagnetic Zn ions
predominantly replace Cu2 3b ions and thus alternate with Cu1 3a layers, but
the question arises whether there is any Cu/Zn site mixing.

Cu and Zn ions have very similar coherent x-ray scattering lengths, and are
hard to distinguish in XRD. Although their neutron scattering lengths are
different (7.72 fm for Cu, 5.68 fm for Zn), we cannot obtain the Cu/Zn
occupancy in LCZSO at 3a (0, 0, 0) and 3b (0, 0, ½) sites directly from NPD.
This is because NPD intensities are different for different Cu/Zn occupancy
only with odd-$l\ G$ vectors; otherwise the geometrical factor is 1 and the
contribution of 3a and 3b sites to the structure factor is the sum of Cu and
Zn scattering lengths. Unfortunately odd-$l$ Bragg peaks in LCZSO are too weak
to determine Cu/Zn occupancy.

Instead, we use the isotropic thermal parameter $B_{\mathrm{iso}}$ obtained
from NPD Rietveld refinements, which is sensitive to Cu/Zn occupancy. Figure 3
gives the dependence of $B_{\mathrm{iso}}$ for the two sites on the assumed
Cu/Zn mixing ratio between the 3a and 3b sites.

Figure 3:  Dependence of NPD isotropic thermal parameter $B_{\mathrm{iso}}$ on
assumed Cu-3b/Zn-3a mixing ratio in LCZSO.

This ratio is 0 for 3a and 3b sites fully occupied by Cu and Zn, respectively,
1 for the opposite full occupation, and 0.5 for 0.5 Cu/0.5 Zn occupation of
each site. We obtain positive $B_{\mathrm{iso}}$ assuming Cu 3a sites and Zn
3b sites, but negative (and therefore unphysical) $B_{\mathrm{iso}}$ assuming
the opposite: cf. Fig. 3.

The fits yield physically reasonable $B_{\mathrm{iso}}$ values (0.5–1) only
for mixing ratio $\lesssim 0.1$. Refinement parameters in SI Table S1 were
obtained with this value fixed at 0.05. The 8-fold coordinated oxygen cavity
centered at the 3b site is larger than the 6-fold one at the 3a site, so that
$B_{\mathrm{iso}}$ at the 3b site should not be significantly smaller than at
the 3a site. Thus the only reasonable value is for a small mixing ratio.

We show below that a few percent impurity level is consistent with analyses of
the impurity Schottky specific heat (Sec. II.2) and the low-field dc magnetic
susceptibility (Sec. II.3). We have been unable to synthesize LCZSO with less
than this few percent of Cu/Zn site-interchange disorder, despite efforts with
different growth protocols.

###  II.2 Specific Heat and Entropy,
Muon Spin Relaxation

Specific heats were measured by the adiabatic relaxation method, using a
Quantum Design Physical Property Measurement System (PPMS) equipped with a
dilution refrigerator. Data were taken at temperatures between 50 mK and 300 K
for LCSO and LCZSO, and 0.2 K–300 K for LZSO. The availability of the
nonmagnetic isomorph LZSO is crucial to determining the LCSO magnetic specific
heat $C_{M}(T)$.

We took special care to ensure that thermal equilibrium was achieved for the
low-temperature measurements. As an example, at base temperature ($\sim$50
mK), the measurement took 70 minutes. The PPMS thermal coupling factor [[][,
Sec.~4.3.3.3.]PPMS04] between the sample and sample platform was 95% at 100 mK
and 99% for temperatures above 0.6 K. Measurements were made during cooling
down to base temperature as well as warming up. For measurements in a magnetic
field, the sample was field cooled and then measured on warming, and also
zero-field cooled, field applied at base temperature, and then measured during
warming. The results were always consistent.

A central result of this work is the observation of significantly less total
magnetic entropy than the $R\ln{2}$ expected from $S=1/2$ local moments. The
conclusion that the missing entropy resides at very low temperatures requires
compelling evidence that it is not recovered at high temperatures. The
specific heat at high temperature is, however, dominated by the lattice
contribution in both LCSO and isostructural nonmagnetic LZSO. Below $\sim$20 K
the magnetic contribution for LCSO is significant (Fig. 4), and can be
determined accurately by simple subtraction of the LZSO specific heat. Above
20 K the difference becomes very small and difficult to measure, and it cannot
be assumed that $C_{\mathrm{latt}}$ is the same for both compounds.

We therefore report our specific heat and entropy results from low- and high-
temperature regimes in two separate sections. In the following we give results
for $T\lesssim 20$ K, and discuss the regime 20–300 K in Sec. II.2.2.

####  II.2.1 $T\lesssim 20$ K

##### Specific heat and entropy.

Figure 4(a) shows the measured zero-field specific heat $C(T)$ in LCSO and
LZSO below about 20 K.

Figure 4:  Specific heat of LCSO and LZSO. (a) Measured specific heats in zero
field. (b) Intrinsic magnetic contribution $C_{M}(T,H)/T$ to the specific heat
divided by temperature for LCSO at various magnetic fields, after subtraction
of lattice, nuclear-Schottky, and impurity-Schottky contributions (see text
and SI Sec. S II)

.

The latter allows an accurate subtraction of the lattice contribution
$C_{\mathrm{latt}}(T)$. A weak “bump” in $C(T)$ at about 1 K and an increase
below 0.2 K are observed, the magnitudes and magnetic field dependencies of
which are discussed in SI Sec. S II. We attribute them to Schottky anomalies
$C_{\mathrm{nuc}}(T)$ and $C_{\mathrm{imp}}(T)$, due respectively to nuclear
spins and nonmagnetic impurities with excited magnetic states [20].
Susceptibility data (Sec. II.3) put an upper limit of ${\sim}10^{-3}$ on the
concentration of impurities with magnetic ground states in LCSO.

The intrinsic magnetic contribution $C_{M}(T)$ to the specific heat is then
obtained by subtracting $C_{\mathrm{nuc}}(T)$, $C_{\mathrm{imp}}(T)$, and
$C_{\mathrm{latt}}(T)$ from the total specific heat. The various contributions
to $C(T)/T$ at $H=0$ are separately shown in SI Fig. S2(b).

$C_{M}(T,H)$ at various fields for temperatures up to 20 K is shown for LCSO
in Fig. 4(b), and for LCZSO in SI Fig. S5(b). In LCSO $C_{M}(T)/T$ at $H=0$ is
constant below about 0.4 K, followed by an approximately logarithmic decrease
with increasing temperature at higher temperatures. These and other features
are examined in detail in Sec. III.3, where the behavior is shown to be
characterized by parameters close to the respective Weiss temperatures.

We turn next to the measurable magnetic entropy. Figure 5 shows the normalized
change in magnetic entropy $[S_{M}(T,H){-}S_{M}(0.1$K$,H)]/R\ln 2$ below 20 K,
calculated by integrating $C_{M}/T$ from 0.1 K to $T$.

Figure 5:  Change $S_{M}(T,H)-S_{M}(0.1\text{K},H)$ in magnetic entropy,
normalized to $R\ln 2$ per mol Cu, $0.1\mathrm{K}\leq T\leq 23$ K,
$0\leq\mu_{0}H\leq 9$ T.

From the proportionality of the $\mu$SR relaxation rate to $C_{M}(T)/T$ from
16 mK to 0.4 K (Sec. II.2.1 below, Fig. 6), we infer that the low-temperature
constant $C_{M}(T)/T$ also continues down to at least 16 mK. At low
temperatures the expected decrease of the magnitude of $S_{M}(T,H)-S_{M}(T,0)$
with increasing field $H$ is observed (Fig. 5). Although there is uncertainty
in determination of $C_{\mathrm{imp}}(T)$ (SI Sec. S II a), the entropy is
affected by less than 0.5% whether or not $C_{\mathrm{imp}}(T)$ is subtracted.

In LCSO for $H=0$, $S(20\text{K}){-}S(0.1\text{K})\approx 0.36\,k_{B}\ln 2$
per Cu ion, with an uncertainty of less than 2%. As discussed in Sec. III.1 in
connection with Fig. 14, for LCSO the $H=0$ entropy at $T=20$ K is found to be
about $0.95R\ln 2$ from the two-exchange high-temperature series expansion for
a triangular lattice [13]. We therefore conclude that in LCSO either $\sim$60%
of the magnetic entropy ($0.95-0.36$) resides in the ground state or, perhaps
more likely, the average $C_{M}/T$ below $\sim$16 mK is about 103 times the
measured constant value above $\sim$100 mK.

##### Specific heat and muon spin relaxation.

Muon spin relaxation measurements used the time-differential $\mu$SR technique
[21], in which the evolution of the ensemble muon-spin polarization after
implantation into the sample is monitored via measurements of the decay
positron count-rate asymmetry $A(t)$. $\mu$SR experiments were performed down
to 16 mK using the DR spectrometer on the M15 beam line at TRIUMF, Vancouver,
Canada, and the DOLLY spectrometer at the Paul Scherrer Institute, Villigen,
Switzerland. Samples were mounted on a silver cold-finger sample holder in the
DR spectrometer, to ensure good thermal contact with the mixing chamber.
$\mu$SR measurements were made from 16 mK to about 20 K in both LCSO and LCZSO
(SI Sec. S IV). Neither long-range order nor disordered spin freezing were
detected in $\mu$SR asymmetry spectra down to the lowest temperatures.
Appropriate functional forms of $A(t)$ were fit to the asymmetry data using
the musrfit analysis program [22].

Dynamic muon spin relaxation is a direct probe of low-frequency spin dynamics
[21]. The zero-field dynamic muon spin relaxation rate
$\lambda_{\mathrm{ZF}}(T)$ for LCSO is plotted as a function of temperature in
Fig. 6.

Figure 6:  Muon spin relaxation rate and specific heat in LCSO. Temperature
dependencies of zero-field muon spin relaxation rate $\lambda(T)$ (red dots:
data taken at PSI; red squares: data taken at TRIUMF) and $C_{M}(T)/T$ (blue
dots) at zero field. It is remarkable that the relaxation rate tends to a
constant value at low temperatures, and that it follows the temperature
dependence of $C_{M}/T$ over the entire temperature range. Inset: separation
of $C_{M}/T$ into contributions from the two layers (see Sec. III.3 below).
The ratios of the low-temperature constant values and the inverses of the
respective Weiss temperatures (Sec. III.3) are approximately equal. The
characteristic temperatures of the two logarithmic terms [Eqs. (3) and (4)]
are also similar to the respective $\Theta_{W}$ values. The knee between the
logarithms requires a semiclassical form (see Sec. III.1). With this fit, the
measured magnetic entropy is consistent with being the same for both layers.

Below $\sim$0.5 K $\lambda_{\mathrm{ZF}}(T)$ is essentially constant at
${\sim}0.14~{}\mu\text{s}^{-1}$, indicating persistent spin dynamics and a
high density of magnetic fluctuations at low temperatures [23, 24]. The
temperature dependence of $\lambda$ closely follows that of $C_{M}/T$ (Fig.
6).

In the motionally-narrowed limit

$\lambda(T)=\gamma_{\mu}^{2}\langle B_{\mathrm{loc}}^{2}\rangle\tau_{c}(T)\,,$
(1)

where $\gamma_{\mu}=8.5156\times 10^{4}\text{s}^{-1}\text{G}^{-1}$ is the muon
gyromagnetic ratio, $\langle B_{\mathrm{loc}}^{2}\rangle$ is the mean-square
fluctuation of the local magnetic fields at muon sites, and $\tau_{c}(T)$ is
the characteristic correlation time of the local-field fluctuations. Equation
(1) provides a rough value of $\tau_{c}$ if $B_{\mathrm{loc}}$ is estimated as
the dipolar field from a $1.85\mu_{B}$ Cu2+ moment (Sec. III.3) at distances
to the polyhedral O2- ions (1.6–1.8 Å); muon sites in transition-metal oxides
are invariably near the oxygen ions. This yields $\tau_{c}\text{ in the range
}9\text{--}25\times 10^{-13}$ s for LCSO and $\sim$40% less for LCZSO. The
relation of this estimate to the scale invariance evidenced by other
properties is discussed below in Sec. III.4.

The inset to Fig. 6 shows a fit to the specific-heat data for LCSO of the two-
component expression

$C_{M}/T=C_{1}/T+C_{2}/T\,,$ (2)

where in units J K-2 (mol-Cu)-1

$C_{1}/T\approx\left\\{\begin{array}[]{cc}0.46\,,&T<0.4\mathrm{K}\,,\\\
0.2\ln(4.2/T)\,,&0.5\mathrm{K}<T<3\mathrm{K}\,,\\\
0.2/T^{3},&T>3\mathrm{K}\,,\end{array}\right.$ (3)

and

$C_{2}/T\approx\left\\{\begin{array}[]{cc}0.09\,,&T<7\mathrm{K}\,,\\\
0.096\ln(20/T)\,,&T>8\mathrm{K}~{}\,.\end{array}\right.$ (4)

Each component is constant at low temperatures followed by a logarithmic
decrease at higher temperatures. The third term in Eq. (3) is the
semiclassical contribution from the high-temperature series expansion [ Eq.
(11)] discussed in Sect. III.1 [13]. The entropies of the two components up to
20 K are roughly equal. The good fit is evidence for the two-component scale-
invariant ansatz discussed below in Sec. III.4.

####  II.2.2 $T\gtrsim 20$ K

##### Specific Heat.

It is important to measure the magnetic specific heat to high temperatures, to
determine whether or not the missing magnetic entropy is generated there.
Above 20 K $C_{\mathrm{nuc}}(T)$ and $C_{\mathrm{imp}}(T)$ are negligible.
$C_{\mathrm{latt}}(T)_{\mathrm{LCSO}}$ is dominant [Fig. 4(a)], and
furthermore differs slightly from $C_{\mathrm{latt}}(T)_{\mathrm{LZSO}}$. This
is due to the mass difference of $\sim$0.3% between LCSO and LCZSO and
differences between Cu and Zn bonding to oxygen, which lead to a difference in
$C_{\mathrm{latt}}(T)$ that scales with a characteristic temperature $T^{*}$.
The difference has a negligible effect on
$C_{\mathrm{diff}}(T)=C(T)_{\mathrm{LCSO}}-C(T)_{\mathrm{LZSO}}$ below 20 K,
but must be taken into account at higher temperatures.

We do so by taking $C_{\mathrm{latt}}(T/T^{*})$ in the two compounds to have
the same form but different $T^{*}~{}$111We thank P. A. Lee for suggesting
this approach.. We scale the LZSO temperature $T_{Z}$ before subtraction by a
factor $1+\eta$, so that $C_{\mathrm{diff}}(T)$ is given by

$C_{\mathrm{diff}}(T)=C_{C}(T)-C^{\,\prime}_{Z}(T)\,.$ (5)

Here $C_{C}(T)$ is the specific heat of LCSO, and
$C^{\,\prime}_{Z}(T)=C_{Z}(T_{Z})$ is the specific heat of LZSO with
$T=(1+\eta)T_{Z}$. We consider $\eta$ an adjustable parameter, the calculation
of which from first principles is beyond the scope of this work. Scaling the
LZSO specific heat values rather than the temperature yields qualitatively
similar results.

Figure 7 shows the temperature dependence of $C_{\mathrm{diff}}(T)/T$ for
several values of the scaling index $\eta$.

Figure 7:  Temperature dependence of specific heat difference
$C_{\mathrm{diff}}(T)/T$ [Eq. (5)] for scaling indices $\eta$ between 0 and
0.040.

For $\eta=0$ $C_{\mathrm{diff}}(T)$ is negative above $\sim$20 K, i.e.,
$C(T)_{\mathrm{LCSO}}<C(T)_{\mathrm{LZSO}}$. This is true even if there is a
magnetic contribution for LCSO. A negative magnetic specific heat would of
course be unphysical. Positive $\eta\gtrsim 0.01$ brings $C_{\mathrm{diff}}$
to values $\gtrsim 0$.

We note that from Sec. II.2 the missing entropy is roughly $0.6R{\ln 2}$,
which corresponds to $C_{M}(T)/T\approx$ 0.02–0.03 J/mol K2 if spread out over
100–150 K. This is of the order of the error bars in Fig. 7. It is also
approximately the value of $C_{\mathrm{diff}}/T$ for $\eta=0.04$. Additional
information is necessary to determine the correct value of $\eta$.

##### Entropy from Specific Heat.

Figure 8 shows the entropy difference $\Delta
S_{\mathrm{diff}}(T)=S_{\mathrm{diff}}(T)-S_{\mathrm{diff}}(2\text{~{}K})$,
calculated by integrating the data of Fig. 7 between 2 K and $T$ 222The error
bars give the uncertainties for individual points. The scatter in the entropy
is considerably less due to the integration. and adding the measured entropy
change of $0.13R\ln{2}$ between 0.1 K and 2 K (Fig. 5).

Figure 8:  Temperature dependence of entropy difference $\Delta
S_{\mathrm{diff}}(T)$ for temperature scaling indices $\eta$ between 0 and
0.040.

The negative slope for $\eta=0$ reflects the negative values of
$C_{\mathrm{diff}}(T)$ discussed above (Fig. 7). $\Delta S_{\mathrm{diff}}$ is
roughly constant above 20 K for $\eta=0.01$ and above $\sim$150 K for
$\eta=0.015$. For $\eta=0.04$ the data increase nearly linearly with
increasing temperature above $\sim$25 K and pass through $R\ln{2}$ with no
change.

From Fig. 8 alone we cannot rule out a magnetic origin for $\Delta
S_{\mathrm{diff}}(T)$ for $\eta\approx 0.025$. This would, however, require
exchange interactions much larger than the observed Weiss temperatures from
susceptibility data (Sec. II.3 below). Also, the smooth increase of $\Delta
S_{\mathrm{diff}}(T,\eta{=}0.025)$ with increasing temperature would require a
broad inhomogeneous distribution of such interactions if it were of magnetic
origin. This is unlikely given the absence of evidence for defects in LCSO.

The general increase of slope with increasing $\eta$ in Fig. 8 is an artifact
of the temperature scaling. This is demonstrated by scaling LZSO against
itself, i.e., using LZSO specific heat data for both terms in Eq. (5). Results
for $C_{\mathrm{diff}}(T)$ and $S_{\mathrm{diff}}(T)$ are shown in Fig. 9.

Figure 9:  Temperature scaling of LZSO against itself. Dependence on scaling
index $\eta$ of (a) $C_{\mathrm{diff}}(T)/T$ and (b) $S_{\mathrm{diff}}(T)$.

Here the increase of $S_{\mathrm{diff}}$ with $\eta$ is entirely due to excess
subtraction, and is numerically the same as for LCSO (Fig. 8).

Furthermore, the measured exchange interactions (Sec. III.1.1 below) are too
small by an order of magnitude to be consistent with a magnetic origin of
$\Delta S_{\mathrm{diff}}(T,\eta{=}0.025)$ in Fig. 8. The best estimate of
$\eta$ is $\sim$0.01, so that there is a missing magnetic entropy of roughly
$0.6R\ln{2}$.

The ratio of atomic masses
$m_{\mathrm{at}}(\text{LZSO})/m_{\mathrm{at}}(\text{LCSO})$ plays a role in
determining the characteristic temperature $T^{\ast}\propto
m_{\mathrm{at}}^{-1/2}$ [[See, forexample, ][, Chaps.~22and23.]AsMe76]. This
ratio is ${\sim}1.003$, which by itself would lead to $\eta\approx 0.0015$, an
order of magnitude smaller than observed. This suggests that the lattice
binding is significantly different in the two compounds.

###  II.3 Magnetization measurements

####  II.3.1 Susceptibility

A Quantum Design Magnetic Property Measurement System (MPMS) was used to make
dc magnetic susceptibility measurements above 2 K. The ac magnetic
susceptibility was measured for frequencies 631 Hz–10 MHz and temperatures 0.1
K–4 K in a PPMS equipped with ac susceptibility and dilution refrigerator
options.

Figure 10 shows the temperature dependencies of the inverse susceptibilities
of LCSO and LCZSO.

Figure 10:  Temperature dependencies of inverse magnetic susceptibility in
LCSO and LCZSO. Solid curves: fits to sum of two Curie-Weiss terms [Eqs. (7)
and (8)]. Dashed curves: regions of inapplicability of Eqs. (7) and (8).
Arrows: Weiss temperatures $\Theta_{W}$.

Small constant Van Vleck and diamagnetic contributions have been subtracted
from the data.

Qualitatively the data exhibit Curie-Weiss behavior; quantitative
determination of exchange constants is discussed below in Sec. III.1.1. For
both LCSO and LCZSO good fits to the data are obtained using sums of two
Curie-Weiss terms:

$\displaystyle\chi(T)$ $\displaystyle=$ $\displaystyle\chi_{1}(T)+\chi_{2}(T)$
(6) $\displaystyle=$
$\displaystyle{\cal{C}}\left(\frac{1}{T+\Theta_{W1}}+\frac{1}{T+\Theta_{W2}}\right)\
\text{(LCSO)}\,,$ (7) $\displaystyle=$
$\displaystyle{\cal{C}}\\!\left(\frac{1-n_{2}}{T+\Theta_{W1}}\,+\,\frac{n_{2}}{T+\Theta_{W2}}\right)\,\text{(LCZSO)}\,.$
(8)

Here $\cal{C}$ is the Curie constant, $\Theta_{W}$’s are Weiss temperatures,
and $n_{2}$ is the concentration of defect Cu2 ions in LCZSO. The fits were
limited to data for temperatures $\gtrsim\Theta_{W}$ where the Curie-Weiss
approximation is applicable, although the agreement remains good at lower
temperatures.

Table 1 lists the results. The effective moments and Weiss temperatures are
nearly the same in the two compounds, although for LCZSO $\Theta_{W2}$ is
poorly determined because of the low Cu2 concentration.

Table 1:  Parameters from fits of Eqs. (7) and (8) to susceptibility data (Fig. 10). | LCSO | LCZSO
---|---|---
${\cal{C}}$ [emu K G-1 (mol f.u.)-1] | 0.426(1) | 0.4190(6)
Effective moment from ${\cal{C}}$ ($\mu_{B}$) | 1.846(2) | 1.830(1)
$\Theta_{W1}$ (K) | 6.7(6) | 5.4(3)
$\Theta_{W2}$ (K) | 18.9(1.2) | 22(13)
LCZSO Cu2 concentration $n_{2}$ | – | 0.07(6)

The value of $n_{2}$ for LCZSO is in agreement with that for site-interchange
Cu2 ions on the 3d Zn layers from the structure determination (Sec. II.1).
These results identify terms 1 and 2 in Eqs. (6)–(8) with Cu1 3a and Cu2 3b
sites, respectively, since Cu ions predominantly occupy the 3a site in LCZSO.

Any additional magnetic impurity concentration $n_{\mathrm{imp}}$ was
estimated by adding a Curie law ($\Theta_{W}\approx 0$, appropriate to nearly-
free impurity spins) to the fit function. This yields upper limits
$n_{\mathrm{imp}}\sim 5\times 10^{-3}$ for both compounds. Stronger estimates
of impurity concentrations are obtained from the real part
$\chi_{\mathrm{ac}}^{\prime}$ of the ac susceptibility measured over the
temperature range 0.1 K–3 K, shown in Figs. 11(a) (LCSO) and 11(b) (LCZSO).

Figure 11:  Temperature dependencies of the real part
$\chi^{\prime}_{\mathrm{ac}}$ of the ac susceptibility in (a) LCSO and (b)
LCZSO, 0.1 K $\leq T\leq 4$ K. Scales are adjusted to match
$\chi_{\mathrm{dc}}$ above 2 K. Similar temperature dependencies were found
for frequencies up to 10 kHz.

By ascribing all the temperature dependence of $\chi_{\mathrm{ac}}^{\prime}$
in this temperature range to a Curie law, we find upper limits to the
concentrations of free-spin magnetic impurities of about $10^{-3}$ in LCSO and
about twice that value in LCZSO.

A broad maximum is observed at $\sim$0.3 K for LCSO [Fig. 11(a)]. No frequency
dependence was found up to 10 kHz (data not shown) and there is no signature
of spin freezing in the $\mu$SR data (Sec. II.2.1), so that a spin-glass
transition is unlikely. The maximum is not understood, but it is interesting
that the crossover of $C(T)/T$ and the muon spin relaxation ($\mu$SR) rate
$\lambda_{\mathrm{ZF}}(T)$ from constant to logarithmic temperature dependence
(Sec. II.2.1, Fig. 6) occurs in this region. The effect is small; the area
under the maximum is only a few percent of the total area under the
susceptibility up to 3 K.

Figure 12:  Entropy of LCSO. Black symbols: from specific heat measurements.
Red symbols: from measurements of magnetization and the Maxwell relation
$(\partial S/\partial H)_{T}=(\partial M/\partial T)_{H}$. The lost magnetic
entropy is fully recovered at high temperatures, indicating that the missing
entropy is independent of the applied field.

####  II.3.2 Entropy from magnetization.

We determine characteristics of the entropy in a magnetic field up to 200 K by
an alternate method, which also gives an estimate of the accuracy of the
subtraction procedure below 20 K. In this method, the magnetization $M(H,T)$
is measured from 4 K to 300 K at various fields [SI Figs. S3(a) and (c) for
LCSO and LCZSO, respectively]. $(\partial M/\partial T)_{H}$ are shown in SI
Figs. 3(b) and 3(d), respectively. We then use the Maxwell relation

$\left(\frac{\partial S}{\partial H}\right)_{\\!T}=\left(\frac{\partial
M}{\partial T}\right)_{\\!H},$ (9)

and integrate $(\partial S/\partial H)_{T}$ to give the change $\Delta S(T,H)$
in entropy due to the magnetic field as a function of temperature. The results
are displayed as red points in Fig. 12. At high temperatures $\Delta S(T,H)\to
0$, indicating approach of the entropy to a field-independent value.

The results from direct determination by subtraction below 20 K are shown in
Fig. 12 as black points. The agreement provides a quantitative measure of the
consistency between the results of these quite different techniques, see Fig.
12 and SI Fig. S5(d). To get a measure of the consistency of results obtained
by these quite different methods, we note that the standard deviation of the
red and black points in LCSO is $\sim$0.02.

The results in Fig. 12 show that the available entropy loss due to magnetic
fields at low temperatures is monotonically reduced to zero at higher
temperatures. However, the missing entropy is field independent up to 9 T over
the whole temperature range. From this behavior at $g\mu_{B}H\leq k_{B}T$ it
follows that the missing entropy is due to purely singlet excitations. Since
it is unaffected even for $g\mu_{B}H\gtrsim k_{B}\Theta_{W}$ (Sec. III.3),
local mutually non-interacting singlet states are also ruled out because they
would be replaced by multiplet states favored by magnetic polarization. There
is no history dependence or hysteresis (Sec. II.2), so that the phenomena do
not appear to be due to metastable singlet states. The singlet states must be
collective and therefore non-local, with collective barriers against
thermally-excited magnetic states.

##  III DISCUSSION

###  III.1 High-temperature series expansions for a triangular lattice

Magnetic entropy at high temperatures could arise from stronger exchange
interactions than implied by Weiss temperatures $\Theta_{W}$ from the Curie-
Weiss fits of Sec. II.3 (Table 1), since nearest-neighbor antiferromagnetic
(AFM) and 2nd-nearest-neighbor ferromagnetic (FM) exchange tend to cancel in
$\Theta_{W}$. To examine this situation we use high-temperature
“semiclassical” truncated series expansions for the magnetic susceptibility
$\chi(T)$ and entropy $S_{M}(T)$ of the Heisenberg model on a triangular
lattice. These are available for nearest-neighbor interactions [28], and also
for nearest- and 2nd-nearest-neighbor interactions [13] with exchange
interactions $J_{1}$ and $J_{2}$, respectively.

The first three terms of the $J_{1}$-$J_{2}$ expansions are

$\displaystyle\chi(T)T/{\cal C}$ $\displaystyle=$ $\displaystyle
1-\frac{3}{2}(J_{1}+J_{2})/T$ (10)
$\displaystyle+\frac{3}{2}(J_{1}^{2}+J_{2}^{2}+3J_{1}J_{2})/T^{2}+\cdots\,,\quad$

for the susceptibility, where $\cal C$ is the Curie constant, and

$\displaystyle S_{M}(T)/R$ $\displaystyle=$
$\displaystyle\ln{2}-\frac{9}{32}\left(J_{1}^{2}+J_{2}^{2}\right)/T^{2}$ (11)
$\displaystyle+\frac{9}{96}\left(J_{1}^{3}+J_{2}^{3}+6J_{1}^{2}J_{2}\right)/T^{3}+\cdots\,,\quad$

for the entropy, where $R$ is the gas constant. The accuracy of the truncated
series is limited to temperatures above roughly twice the larger exchange
[13].

####  III.1.1 Susceptibility

We first fit high-temperature series expansions to the susceptibility data
(Sec. II.3). For LCSO the fit function consists of two terms of the form of
Eq. (10), one for each of the two layers. A fit with all four exchange
parameters free is not possible, because of the inherent ambiguity in a sum of
two terms of the same form 333In a function of the form $y=Ax+Bx$ the
parameters $A$ and $B$ cannot be fit for separately, only the sum $A+B$.. We
therefore use results from fits of Eq. (10) to the LCZSO susceptibility, which
is dominated by the contribution of the Cu1 layer (Sec. III.3). For the LCSO
fit the Cu1 layer exchanges were fixed at the LCZSO values, on the grounds
that the properties of this layer are not very different in the two compounds
(cf. Sec. II.3).

Figure 13 shows the fit to data for $T>25$ K, roughly twice the fit value of
$J_{1}$.

Figure 13:  Comparison of LCSO magnetic susceptibility with truncated high-
temperature series expansions [Eq. (10)]. Black curve: fit of Eq. (10) to data
for $T>25$ K. Other curves: series expansions for larger exchange values.
Dashed curves: regions of inapplicability of Eq. (10).

$J_{2}$ is negative (FM) and small. Values of the fit parameters are given in
Table 2.

Table 2:  Curie constant $\cal{C}$ and exchange parameters in LCSO from fits of two-exchange high-temperature series expansions to susceptibility data (Fig. 13). | LCSO444From sum of two terms of form of Eq. (10); see text. | LCZSO555From Eq. (10).
---|---|---
${\cal{C}}$ [emu K G-1 (mol f.u.)-1] | 0.4203(9) | 0.4170(2)
$J_{1}$ (K) | 12.03(12) | 3.54(6)
$J_{2}$ (K) |  $-$0.88(9) | 0.39(8)

For comparison a number of curves are shown for various larger $J_{1}$ and
$J_{2}$, such that $J_{1}+J_{2}$ in the leading term in Eq. (10) is roughly
constant. There is no agreement. This is strong evidence that exchange in LCSO
is weak and does not produce interaction phenomena at higher temperatures.

Figure 14:  (a) Specific heat difference $C_{\mathrm{diff}}(T)/T$ and (b)
entropy difference $\Delta S_{\mathrm{diff}}(T)$ from Figs. 7 and 8, compared
to truncated high-temperature expansions [Eq. (11) and its derivative]. Dash-
dotted curve: entropy expansion for $J_{1}=14$ K, $J_{2}=-2$ K with
$0.57R\ln{2}$ subtracted. Dashed curves: regions of inapplicability of Eq.
(11).

####  III.1.2 Specific heat and entropy

The high-temperature specific heat and entropy differences reported in Sect.
II.2.2 are replotted in Fig. 14, together with high-temperature expansions
with exchange values from Table 2 and Fig. 13. Our purpose is to determine the
correct value of the scaling parameter $\eta$. From Fig. 14(b) a candidate
might be $\eta=0.04$, which between 50 K and 150 K agrees roughly with the
expansion for $J_{1}=48$ K, $J_{2}=-36$ K. However, results of fits to
susceptibility data (Table 2) rule out such large exchange values. In any case
the data pass smoothly through $R\ln{2}$, which is not possible for magnetic
entropy.

For exchange values compatible with susceptibility fits, the entropy series
expansions are constant down to low temperatures. The data for $\eta=0.01$
(green circles) agree with this property, as shown in Fig. 14(b). This is
strong evidence that for those data $\Delta S_{\mathrm{diff}}(T)=\Delta
S_{M}(T)$, the magnetic entropy. Then the series expansion agrees very well
with the data after subtracting a missing entropy of $0.57R\ln{2}$ [dash-
dotted curve in Fig. 14(b)].

###  III.2 Coordination of Cu2+ ions,
symmetry of the orbitals

We consider the symmetry of the Cu orbitals where the spins reside and the
exchange path. We show by a symmetry argument that the magnetic interactions
are two-dimensional. Accordingly, all data discussed below in Sec. IV can be
separated into distinct components associated with the two lattice planes,
each characterized by a separate parameter (the Weiss temperature).
Experimental evidence for this conclusion is discussed in Secs. II.3 and
III.3.

The O2- coordination of the Cu1 3a site is a trigonally-distorted octahedron,
and that of the Cu2 3b site is a trigonally-distorted cube (SI Sec. I). The
3a-site distorted octahedron is oriented with the $c$ axis passing through the
centers of two parallel triangular faces. The [111] corners of the 3b-site
distorted cube are compressed along the $c$ axis. Thus there is 3-fold
rotation symmetry around the $c$ axis at both sites, as dictated by the
crystal symmetry.

A general symmetry argument can be made that the inter-plane coupling vanishes
despite the short interlayer distance [Fig. 1(a)]. It relies on two facts: (1)
there are common three-fold axes for the two planes, each of which passes
through a Cu ion in one layer and the center of an equilateral triangle of
three Cu ions in the adjacent layer [Figs. 1(b) and (c)]; and (2) the relevant
Cu orbitals in the two layers are different and orthogonal.

We recall standard results from crystal-field theory [[See, e.g., ]Burns93]:
the highest-energy $3d$ states, which the hole occupies in Cu2+ ions, are the
two $e_{g}$ states in octahedral coordination and the three $t_{2d}$ states in
cubic coordination. Consider a centered Cu2 ion and the equilateral triangle
of Cu1 ions in the next layer. Let $F_{1}(\theta,\phi)$ and
$F_{2}(\theta,\phi)$ be the Wannier orbitals constructed from the three
$e_{g}$-derived Cu1 wave functions and their associated oxygen polyhedra on
the triangle, and $F_{i}(\theta,\phi),~{}i=3,4,5$, the $t_{2g}$-derived Cu2
wave functions in the distorted cube. The angles $\theta$ and $\phi$ are
defined by the common three-fold axis of symmetry. All five $F_{i}$ are
second-order functions of angle.

Then the most general wave functions, which in general are nondegenerate,
projected to the base of the triangle are

$\psi_{1}=(1/N1)[F_{1}(\theta,\phi)\cos 3\phi+F_{2}(\theta,\phi)\sin 3\phi]$
(12)

for the Cu1 triangle, and

$\displaystyle\psi_{2}$ $\displaystyle=$
$\displaystyle(1/N2)[F_{3}(\theta,\phi)$ (13)
$\displaystyle+F_{4}(\theta,\phi)\cos 3\phi+F_{5}(\theta,\phi)\sin 3\phi]$

for the Cu2 ion.

Because of the orthogonality of the $F_{i}$ and the fact that they do not
contain any odd polynomials like $\sin\phi$, $\sin 3\phi$, etc., $\psi_{1}$
and $\psi_{2}$ are orthogonal. Since all potentials have three-fold symmetry,
any matrix elements of $\psi_{1}$ and $\psi 2$ with any such potential is also
zero.

This suggests that in LCSO, even though the interlayer distance between Cu
sites is shorter than the intralayer distance, superexchange interaction
between layers is likely to be absent. Then LCSO (and LCZSO) are two-
dimensional as far as the magnetic interactions between the spin-1/2 Cu2+ ions
are concerned. As a consequence LCSO has similar properties per Cu ion as
LCZSO, where Cu ions only occupy the Lu layer. However, more evidence, e.g,
from measurements of crystal-field levels in single crystals, would be
desirable to substantiate this conclusion.

As may be seen in Fig. 1(a), the superexchange path between nearest-neighbor
Cu ions in a layer is via their oxygen polyhedra and the intervening Lu or Sb
ion. This long path makes the nearest-neighbor exchange energy small. It also
means that an estimate of it (let alone 2nd-nearest-neighbor exchange or
triple exchange among three Cu’s) will be very hard to obtain from a
calculation. This is fortunately not necessary for the purposes of the present
paper.

###  III.3 Two sublattices: specific heat and susceptibility

Inequivalent Cu1 3a and Cu2 3b sites in LCSO form two separate triangular
sublattices coordinated by Lu and Sb ions, respectively. Here we present
evidence for two additive components in the magnetic specific heat and the
magnetic susceptibility that are due to the two sublattices. This is strong
evidence that their properties are two-dimensional, with little interlayer
interaction, in agreement with the preceding discussion (Sec. III.2).

##### Specific heat.

In Sec. II.2.1, we show in the inset to Fig. 6 that the measured $C_{M}(T)/T$
in LCSO can be separated into two components. The ratio of their low-
temperature constant values is approximately inversely proportional to their
respective values of $\Theta_{W}$, and they both decrease logarithmically
approximately as the corresponding $\ln(\Theta_{W}/T)$. The integrated
entropies are about the same for the two layers. These forms only pertain for
the “quantum region” below the respective $\Theta_{W}$’s. The knee region
between the two logarithms requires fit to the semiclassical region (Sec.
III.1).

A similar decomposition for LCZSO is discussed in SI Sec. S IV. There it is
shown that, even with only a few percent site interchanges of Cu and Zn (see
SI Sec. S I and below), the ratio of the entropies of the two layers is no
smaller than 30%. This emphasizes the importance of defect-free samples for
the study of intrinsic properties of spin liquids.

##### Susceptibility.

In Sec. II.3 we show results of dc and ac susceptibility measurements down to
0.1 K (Figs. 10 and 11), estimate the magnetic impurity concentrations, and
discuss two-component Curie-Weiss fits for both LCSO and LCZSO. We find
magnitudes of the components (equal in LCSO, approximately in the ratio 93:7
in LCZSO) and values of $\Theta_{W}$ (Table 1) that are strong evidence for
two weakly-interacting sublattices associated with the inequivalent layers.

It is important to note that the characteristic temperatures in the logarithms
in the specific heat [Eqs. (3) and (4)] are close to the Weiss temperatures or
characteristic exchange energies (Table 1), and that the ratio of the constant
specific heats at low temperature is approximately the inverse ratio of the
$\Theta_{W}$’s 666The break points in Eqs. (3) and (4), which were chosen for
best fit, do not scale with the $\Theta_{W}$’s.. This is evidence that in LCSO
the two Cu layers are contributing equally to both quantities.

In LCZSO there is considerable uncertainty in the Cu2 concentration from the
Curie-Weiss fit (Sec. II.3, Table 1). Nevertheless, the results are consistent
with the evidence from the structure analysis for a few percent Cu occupation
of the 3b (Zn) sublattice. The near equality of $\Theta_{W1}$ values for the
two compounds (Table 1) is evidence that $\chi_{1}$ and $\chi_{2}$ in Eq. (6)
are the contributions of the Cu1 and Cu2 layers, respectively. Our
observations support the ansatz of additive effects of the two inequivalent Cu
sublattices.

###  III.4 Scale invariance

The properties reported here can be used to specify some features of the
frequency-dependent correlation functions that a fundamental theory might
provide. We consider only the pure limit and the experimental results for
$H=0$. We show that the observed specific heat and muon spin relaxation rate
follow if there are magnetic fluctuations with a _local_ density of states
$\mathcal{A}_{M}(\omega,T)$ of a specific scale-invariant form. The ground-
state entropy requires a more singular form $\mathcal{A}_{0}(\omega,T)$.

We write

$\displaystyle\mathcal{A}_{\mathrm{loc}}(\omega,T)$
$\displaystyle\equiv\sum_{\mathbf{q}}\mathcal{A}(\mathbf{q},\omega,T)\delta(\omega-\omega_{\mathbf{q}})$
$\displaystyle=\mathcal{A}_{0}(\omega,T)+\mathcal{A}_{M}(\omega,T)\,,$ (14)

where

$\mathcal{A}_{0}(\omega,T)=S_{0}\frac{\omega}{\omega^{2}+T^{2}}e^{-\omega/T}\,,\\\
$ (15)

and

$\displaystyle\mathcal{A}_{M}(\omega,T)$
$\displaystyle=\gamma_{M}\left\\{\begin{array}[]{ll}\displaystyle\frac{\omega}{T}\left|\ln\frac{T+T_{x}^{\prime}}{T_{x}}\right|\,,&\omega\ll
T\,,\\\
\displaystyle\left|\ln\frac{\omega}{T+T_{x}}\right|\,,&T\lesssim\omega\lesssim
T_{x}\,.\end{array}\right.$ (18)

Here $T_{x}\approx\Theta_{W}$, and the experiments yield
$T_{x}^{\prime}\approx T_{x}/4$. The quantum numbers $\mathbf{q}$ of the
fluctuations are different for the two contributions.

As shown below, $\mathcal{A}_{M}(\omega,T)$ provides the measured specific
heat $C_{M}/T\approx(\gamma_{M}k_{B})$ for $T\ll T_{x}$. The local density of
states of singlet excitations $\mathcal{A}_{0}(\omega,T)$ can easily be
modified if later experiments reveal a gigantic peak in $C_{M}/T$ at very low
temperatures. $S_{0}$ and $\gamma_{M}T$ with a cutoff $T_{x}$ of order the
Weiss temperatures, are related through the sum rule that the total entropy is
$k_{B}\ln 2$ per spin at high temperatures.

The fluctuations are assumed to obey Bose-Einstein statistics with zero
chemical potential. The reason this works is that even with zero chemical
potential, as can be easily calculated, the number of excitations remains
independent of temperature with the choice of the singular density of states
of excitations. The divergent damping of the excitations implied by the
density of states also obviates a Bose-Einstein condensation. Were it to turn
out that they are hard core bosons or neutral fermions, obvious modifications
in the above hypotheses would be required. The possibility that the spectrum
represents unfamiliar particles with unfamiliar statistics should also be
entertained. The colossal degenerate fluctuating singlet state is likely to be
unstable to other states by perturbations, for example superconductivity or
itinerant charge states due to doping.

The functional form of $\mathcal{A}_{M}(\omega,T)$ is derived at criticality
in all the impurity models mentioned below in Sec. III.5, the 2D dissipative
quantum $XY$ model [32, 33, 34] and the SYK impurity model [35, 36]. In every
case, the low-energy excitations in these toy models are topological, as are
the $T\to 0$ states.

The free energy $F$ is given by

$F=-kT\ln\mathrm{Tr}\,Z\,,\quad\mathrm{Tr}\,Z=\sum_{\lambda}e^{-\beta
E_{\lambda}}\,,\ $ (19)

where ${\lambda}$ contains the quantum numbers $\mathbf{q}$ as well as their
occupation number. Summing over the occupation numbers gives, as usual,

$\ln\mathrm{Tr}\,Z=\sum_{\mathbf{q}}\ln\left(\frac{1}{2\sinh(\beta\omega_{\mathbf{q}}/2)}\right)\,.$
(20)

Let us first calculate the entropy due to $\mathcal{A}_{0}(\omega,T)$. One
finds the entropy from the free energy [$S=-(\partial F/\partial T)_{V}$]:

$\displaystyle S$ $\displaystyle=k_{B}\frac{d}{dT}T\int
d\omega\mathcal{A}_{0}(\omega,T)\ln\left(\frac{1}{2\sinh(\beta\omega/2)}\right),$
(21)
$\displaystyle=k_{B}S_{0}\int_{0}^{\infty}dx~{}x\frac{e^{-x}}{1+x^{2}}f(x)\,,$
(22) $\displaystyle\quad
f(x)=\ln\left[\textstyle{\frac{1}{2}}\csc(x/2)\right]\,.$ (23)

The value of the integral is approximately 0.275.

The local density of states function $\mathcal{A}_{M}(\omega,T)$ gives a free
energy proportional to $T^{2}$ with logarithmic corrections at high
temperatures, an entropy of the deduced form, and a measurable specific heat

$C_{M}(T)/T\approx\gamma_{M}k_{B}$ (24)

at low temperatures, with a logarithmic cutoff for temperatures above $T_{x}$.

The two layers in LCSO complicate the analysis slightly. From the fits of Sec.
III.3, $C_{M}(T)/T$ values for the two layers differ by a factor of about 5.
This is roughly the ratio of $\Theta_{W}$ values, which associates the lower
$\Theta_{W}$ and hence the larger $C_{M}/T$ with the Cu1 layer. From the fit,
for this layer $C_{M}(T)/T=$ 0.92 J K-2 per mole Cu, which yields
$\gamma_{M}\approx 8.4\times 10^{-13}$ s.

The general form of the muon spin relaxation rate is

$\lambda(T)\propto\gamma_{\mu}^{2}\lim_{\omega\to
0}\frac{T}{\omega}\sum_{\mathbf{q}}|B_{\mathrm{loc}}(\mathbf{q})|^{2}\mathrm{Im}\,\chi(\mathbf{q},\omega)\,,$
(25)

where $\sum_{\mathbf{q}}(T/\omega)\mathrm{Im}\,\chi(\mathbf{q},\omega)$ is the
spectrum of magnetic fluctuations $\mathcal{A}_{M}(\omega,T)$ at the muon
Zeeman frequency and is the correlation time $\tau_{c}$ in Eq. (1) [21]; the
same fluctuations are involved in both quantities. For
$\mathcal{A}_{M}(\omega,T)$ given by Eq. (18), the temperature dependence of
$\lambda(T)$ is seen to be the same as that of $C_{M}/T$, as in the
experimental results shown in Fig. 6. Since $\mathcal{A}(\mathbf{q},\omega)$
is the absorptive part of the magnetic fluctuation spectra, it follows that at
low temperatures $\tau_{c}\approx\gamma_{M}$.

From the low-temperature muon spin relaxation rate, $\tau_{c}$ is estimated at
${\sim}1\text{--}2\times 10^{-12}$ s (Sec. II.2.1). This is in fair agreement
with the specific-heat $\gamma_{M}$ discussed above, but must be considered
semi-quantitative at best for two reasons: $\gamma_{M}$ from $C_{M}(T)/T$ and
$\tau_{c}$ from $\lambda(T)$ are only approximately equal, and the muon sites
in the crystal structure are not known accurately.

###  III.5 Survey of theories on spin-liquid models

Detailed numerical calculations for the ground state of the $S{=}1/2$ nearest-
neighbor AFM Heisenberg model (AHM) on a triangular lattice [37] give an
ordered three-sublattice ground state, with reduction of the order parameter
by zero-point fluctuations of about 36% for the nearest-neighbor interaction
model. High-order high-temperature series expansions for the AHM [28, 38] give
a peak in the specific heat and the beginning of a decrease in the
susceptibility at about 0.5 the exchange energy. As already mentioned, these
bear no resemblance to our results.

Numerical calculations on models with substantial 2nd-nearest-neighbor AHM
interactions on a triangular lattice [39, 40] have given a quantum-disordered
state with gapless excitations conjectured to be spinons, but no ground-state
entropy or indications of a gigantic peak in exponentially low-energy singlet
excitations. Spinons have a Fermi surface and therefore a linear-in-$T$
specific heat, but the magnetic fluctuations associated with the Fermi surface
lead to a relaxation rate proportional to the density of thermal excitations.
It is therefore proportional to $T$ at low temperatures, similar to the
Korringa rate in metals and unlike the constant rate found here. Models with
ring-exchange [41] are purported to give a chiral spin-liquid state but no
low-energy singlet states. However, more work is needed on both these
variations on the AHM.

The $S{=}1/2$ Heisenberg model on a kagomé lattice is considered the most
likely spin liquid. Very careful calculations [42] and analysis of a high-
temperature series expansion up to 17th order for this model have found a
missing entropy of about 1/2 the total value down to temperatures of
$O(J/10)$, the lowest to which the calculations are reliable. These
calculations have been further substantiated [[][andreferencestherein.]YUE20].
The results are similar to what we have found, but it is not known why the
triangular and kagomé-lattice Heisenberg models should be in the same
universality class.

It is worth noting that the Kitaev model in a certain range of parameters is
similar to the toric model, which with an extreme choice of parameters
exhibits a massive density of low-energy excitations followed by a continuum
[44, 45]. The specific heat is proportional to $T^{2}$ below the peak in low-
energy excitations from two dimensional spinons, but approximately linear in
$T$ above that scale. Some gauge theory models in the large-$N$ limit, the so-
called $\mathbb{Z}_{2}$ spin liquids [46], have similar properties.

Calculations on models for ice [47] or spin ice [48] (which agree with
experiments [49, 50]), and glass or spin-glass models, possess ground-state
entropy but are obviously inapplicable here. Holographic field-theory models
[51, 52, 53] do have ground-state entropy as well as observable specific heat
with various power laws including linear. (0+1)-dimensional disordered
effective-impurity models such as the Sachdev-Ye-Kitaev (SYK) model [35, 36]
also have extensive ground-state entropy as well as gapless fermion
excitations giving a linear-in-$T$ specific heat.

The mapping of the SYK model to AdS theory of black holes has been discussed
[36]. Black holes are conjectured to be quantum-mechanical, and their physics
is fashioned parallel to the thermodynamic laws [54]. They are believed to
have an observable linear-in-$T$ entropy [1, 2].

###  III.6 Comparison with other spin liquids

Notable experimental discoveries of compounds that do not order (or do not
order down to very low temperatures compared to their $\Theta_{W}$’s) have
been discussed as spin liquids for the last 25 years. We argue that our
results (missing entropy, scale invariance) are unique to LCSO, most likely
because it is purer than any other compound in its class of spin liquids so
that its intrinsic properties are revealed. I.e., there may well be
undiscovered missing entropy in other spin liquids that is obscured by lack of
such purity.

In the following only a few representative references for each compound are
given. More complete references may be found in reviews [9, 8].

1\. Herbertsmithite is a $S{=}1/2$ kagomé lattice compound that has been
thoroughly investigated [55, 56, 57]. It has been found experimentally to have
scale-invariant magnetic excitations, but the compound suffers from
substantial disorder so that the pure limit specific heat has not been
determined. It is reported to be $\propto T^{\alpha}$, $\alpha\approx 0.6$, at
low temperatures, but this is obtained only after a contribution believed to
be exclusively due to impurities is eliminated by applying a magnetic field of
9 T. No determination of magnetic entropy is available because a nonmagnetic
counterpart, needed to subtract the lattice specific heat, has not been found.
The compound cannot be made with less than a few percent Zn/Cu site disorder,
leading to a Schottky specific heat. The low-temperature specific heat in
zero-field is dominated by impurities and a field of 9 T reduces the specific
heat to a power law $T^{\alpha}$, $\alpha\approx 0.6$. This is conjectured to
represent the “intrinsic” specific heat, but its dependence on magnetic field
cannot be determined.

The $\mu$SR rate is constant at low temperatures. It is interesting that
single crystals have been made on which neutron scattering reveals a momentum
independent continuum extending down to 0.25 meV, with $\omega/T$ scaling
proposed here and much earlier in the physics of the cuprates. NMR and $\mu$SR
results are similar to those in our compound, except at very low temperatures
where evidence for inhomogeneity is found.

2\. Some organic Cu compounds, $\kappa$-(ET)2Cu2(CN)3 [58],
EtMe3Sb[Pd(dmit)2]2 [59, 60] have a linear-in-$T$ contribution to the heat
capacity at low temperatures which, unlike our results, is field independent
and is followed at higher temperatures by a large bump. We have not located
any report in the literature of missing entropy or constant NMR rates at low
temperatures. In earlier samples a linear-in-$T$ thermal conductivity was
observed, but in more recent samples this is not found. The problems and
different results in differently prepared samples have been documented in
review articles, e.g. [9].

3\. Cs2CuCl4 [61], ZnCu3(OH)6Cl2 [62], and BaCo2(P1-xVx)2O8 [63] all show
ordering of one or the other kind at low temperatures and have Curie
contributions to the susceptibility shown by the authors to be due to several
percent orphan spins.

4\. YbMgGaO4 [64] exhibits a weakly divergent specific heat coefficient
($C/T\propto T^{-0.3}$) with less than 0.6% residual spin entropy. But a low-
temperature Curie tail is observed in the susceptibility, indicative of
impurities that are not taken into account in the specific heat analysis. Mg-
Ga site interchange is an intrinsic source of defects. The $\mu$SR relaxation
rate is constant below $\sim$0.1 K, and does not track $C/T$ [65, 66].

5\. Ba3CuSb2O9 [67] has a nominally triangular $S{=}1/2$ lattice. Entropy
saturation to only about 1/3 of $R\ln 2$ was observed using measurements on a
non-magnetic analog to subtract the lattice contribution. But the sample has
5% orphan spins. A huge peak in $C_{M}(T)/T$ occurs at about 6 K, and
$C_{M}/T$ is field independent to 9 Tesla even between 0.2 K and 1 K well
below $\mu_{B}H/k_{B}$. All this is quite different from the properties of
nearly orphan-spin-free triangular lattice LCSO. Indications are that in
Ba3CuSb2O9 there is a collective state with glassy ordering at about 6 K,
probably induced by non-local effects due to the large concentration of orphan
spins.

6\. NiGa2S4 [68] and LCSO have very different properties. Spin freezing is
observed in NMR experiments in the former below about 10 K. Relaxation-rate
temperature dependencies have various power laws unrelated to the constant
shown in LCSO.

7\. TbInO3 [69] exhibits two distinct Tb ion sites due to a ferroelectric
distortion. One forms a triangular lattice. No ordering is seen down to 0.15
K, but other properties do not resemble those in LCSO.

##  IV CONCLUSIONS

In summary, two related phenomena have been discovered in the nearly defect-
free compound LCSO.

1\. Conclusive evidence is found for missing entropy from a colossal density
of singlet excitations below an ultra-low-energy scale compared to the Weiss
temperature. If these were local singlets, they would polarize and lose
entropy in a magnetic field larger than the singlet-triplet separation. The
fact that they are not removed by a magnetic field as high as 9 T shows that
they are in a collective state where they cannot be addressed individually. At
low temperatures there must be an insurmountable barrier against modification
of this singlet characteristic. We can speculate that as in other systems,
such as crystal dislocations or superconducting vortices, the barrier is
topological in nature.

2\. Quantitatively related magnetic specific heats $C_{M}/T$ and $\mu$SR
relaxation rates $\lambda(T)$ are observed below a temperature related to the
Weiss temperature $\Theta_{W}$, followed by the same logarithmic cutoff in
both measurements at higher temperatures. The excitations necessary for these
are shown to be scale invariant. They carry finite spin quantum numbers
because their entropy for $g\mu_{B}H\lesssim k_{B}T$ is systematically reduced
in a field; this leads to temperature-independent muon spin relaxation. There
are no other measurable excitations at any temperature and up to 9 tesla. All
measured properties can be related to a single parameter in the scaling
function.

Both the specific heat and the $\mu$SR relaxation rate follow from the scale-
invariant density of states function
$\mathcal{A}_{M}(\omega,T)=\gamma_{M}f(\omega/T)$ for magnetic fluctuations
described in Sec. III.4. Not only is the temperature dependence of both
quantities given by this form, but their orders of magnitude are obtained from
the same coefficient $\gamma_{M}$. As a function of imaginary time $\tau$
periodic in inverse temperature, $\mathcal{A}_{M}(\omega,T)$ is equivalent to
an algebraic decay $\propto 1/\tau$. A ground-state entropy, which should more
accurately be called a temperature-independent entropy, requires a more
singular form $\mathcal{A}_{0}(\omega,T)$, which corresponds to a correlation
function of the singlets approximately proportional to $1/\log(\tau)$. This is
as quantum as one can get. Some conceptual questions related to this are
briefly discussed in SI Sec. IX. This form is chosen in the belief that the
missing entropy is due to a dynamical effect. The form can be modified easily
by introducing a new scale if instead there are equally unexpected colossal
ultra-low-energy excitations.

Theoretical results for spin liquids and their relation to our experimental
findings are briefly summarized in Sec. III.5. We have not found theoretical
results on any relevant model which correspond to the properties discovered
here [9, 8]. Similarly, in a close look at the literature (a summary is given
in Sec. III.6), we find that such properties have not been previously observed
in any spin-liquid candidates. We think this is because LCSO can be prepared
with fewer defects than any other spin liquid investigated so far, so that the
intrinsic behavior of its class of spin liquids is revealed. The simplicity
and the nature of the singularities in Eqs. (III.4)–(18), with which we can
parameterize all the data, invite important new theoretical developments. The
magnetic fluctuations suggested by $\mathcal{A}_{M}(\omega,T)$ should be
accessible via neutron scattering. The detection of the scalar excitations
$\mathcal{A}_{0}(\omega,T)$ poses an interesting challenge to experimental
techniques. Having no charge or magnetic moment, they are a form of dark
matter not observable by the usual spectroscopic techniques.

## Acknowledgments

We are grateful to Patrick A. Lee for extensive discussions of this work and
for suggesting temperature scaling to estimate the high-temperature magnetic
lattice specific heat, and to Rajiv R. P. Singh for providing his results for
the high-temperature series expansions. We thank B. Hitti and D. J. Arseneau
of TRIUMF and the staff of the Paul Scherrer Institute for their valuable help
during the $\mu$SR experiments. C.M.V. performed this work while a “Recalled
Professor” at UC Berkeley, and wishes to thank the members of the condensed-
matter theory group for their hospitality. This research was funded by the
National Natural Science Foundations of China, No. 12034004 and No. 11774061,
and the Shanghai Municipal Science and Technology (Major Project Grant No.
2019SHZDZX01 and No. 20ZR1405300).

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