$\mathcal{K}_{\textsf{DPnP},\eta}$ also has the same eigenvalues as these two,
we conclude that $\mathcal{K}_{\textsf{DPnP},\eta}$ is a self-adjoint compact
operator on $L^{2}(\pi_{\eta})$, of whom $1$ is a simple eigenvalue. Moreover,
any other eigenvalue of $\mathcal{K}_{\textsf{DPnP},\eta}$ has modulus no
larger than $\lambda$.

#### Step 2: establishing the contractivity of
$\mathcal{K}_{\textsf{DPnP},\eta}$ in $L^{2}(\pi_{\eta})$.

It is easy to verify that the constant function $\mathbf{1}$, which takes
value $1$ for any $x\in\mathbb{R}^{d}$, is a eigenfunction of
$\mathcal{K}_{\textsf{DPnP},\eta}$ associated to the simple eigenvalue $1$,
thus is the only (up to scaling) eigenfunction associated to that eigenvalue.
It is also a unit-length eigenfunction, since
$\|\mathbf{1}\|_{L^{2}(\pi_{\eta})}=(\int 1\cdot\pi_{\eta}(x)dx)^{1/2}=1$.
Therefore, the operator
$\mathcal{K}_{\textsf{DPnP},\eta}-\mathbf{1}\mathbf{1}^{\top}$ is a self-
adjoint operator whose eigenvalues have modulus no larger than $\lambda$,
where $\mathbf{1}\mathbf{1}^{\top}$ is the orthogonal projection onto
$\mathbf{1}$ in $L^{2}(\pi_{\eta})$, defined by

$\mathbf{1}\mathbf{1}^{\top}f(x)\equiv\int
f(x^{\prime})\pi_{\eta}(x^{\prime})\mathrm{d}x^{\prime},\quad\forall
x\in\mathbb{R}^{d}.$

Using the fact that
$\mathcal{K}_{\textsf{DPnP},\eta}\mathbf{1}\mathbf{1}^{\top}=\mathbf{1}\mathbf{1}^{\top}\mathcal{K}_{\textsf{DPnP},\eta}=\mathbf{1}\mathbf{1}^{\top}$,
one may show
$(\mathcal{K}_{\textsf{DPnP},\eta}-\mathbf{1}\mathbf{1}^{\top})^{N}=\mathcal{K}_{\textsf{DPnP},\eta}^{(N)}-\mathbf{1}\mathbf{1}^{\top}$
by expanding the product, see e.g. Saloff-Coste, (1997). Consequently,
$\mathcal{K}_{\textsf{DPnP},\eta}^{N}-\mathbf{1}\mathbf{1}^{\top}$ is a self-
adjoint operator whose eigenvalues have modulus no larger than $\lambda^{N}$,
i.e.,

$\left\|\mathcal{K}_{\textsf{DPnP},\eta}^{N}-\mathbf{1}\mathbf{1}^{\top}\right\|_{L^{2}(\pi_{\eta})\to
L^{2}(\pi_{\eta})}\leq\lambda^{N},$ (49)

where $\|\cdot\|_{L^{2}(\pi_{\eta})\to L^{2}(\pi_{\eta})}$ denotes the
operator norm on $L^{2}(\pi_{\eta})$.

#### Step 3: bounding the inner product of
$p\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}K_{\textsf{DPnP},\eta}^{(N)}-\pi_{\eta}$
with any square-integrable function.

Note that when $\chi^{2}(p\,\|\,\pi_{\eta})=\infty$, the conclusion is
trivially true. For the rest part of the proof, we assume
$\chi^{2}(p\,\|\,\pi_{\eta})<\infty$. Now, for any $f\in L^{2}(\pi_{\eta})$,
by applying Lemma 10 iteratively, we obtain

$\displaystyle\int
p\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}K_{\textsf{DPnP},\eta}^{(N)}(x)f(x)\mathrm{d}x$
$\displaystyle=\int
p(x^{\prime})\mathcal{K}_{\textsf{DPnP},\eta}^{N}f(x^{\prime})\mathrm{d}x^{\prime}$
$\displaystyle=\int
p(x^{\prime})\cdot(\mathcal{K}_{\textsf{DPnP},\eta}^{N}-\mathbf{1}\mathbf{1}^{\top})f(x^{\prime})\mathrm{d}x^{\prime}+\int
p(x^{\prime})\mathbf{1}\mathbf{1}^{\top}f(x^{\prime})\mathrm{d}x^{\prime}$
$\displaystyle=\int
p(x^{\prime})\cdot(\mathcal{K}_{\textsf{DPnP},\eta}^{N}-\mathbf{1}\mathbf{1}^{\top})f(x^{\prime})\mathrm{d}x^{\prime}+\int
f(x^{\prime})\pi_{\eta}(x^{\prime})\mathrm{d}x^{\prime},$

where the last line follows from the definition of
$\mathbf{1}\mathbf{1}^{\top}$ and $\int p(x^{\prime})\mathrm{d}x^{\prime}=1$.
Rearrange the terms to see

$\int\big{(}p\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}K_{\textsf{DPnP},\eta}^{(N)}(x)-\pi_{\eta}(x)\big{)}f(x)\mathrm{d}x=\int
p(x^{\prime})\cdot(\mathcal{K}_{\textsf{DPnP},\eta}^{N}-\mathbf{1}\mathbf{1}^{\top})f(x^{\prime})\mathrm{d}x^{\prime}.$
(50)

In particular, taking $p=\pi_{\eta}$ yields

$0=\int\pi_{\eta}(x^{\prime})\cdot(\mathcal{K}_{\textsf{DPnP},\eta}^{N}-\mathbf{1}\mathbf{1}^{\top})f(x^{\prime})\mathrm{d}x^{\prime}.$
(51)

Substract (51) from (50), and then take absolute value, we obtain

$\displaystyle\left|\int\big{(}p\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}K_{\textsf{DPnP},\eta}^{(N)}(x)-\pi_{\eta}(x)\big{)}f(x)\mathrm{d}x\right|$
$\displaystyle=\left|\int\big{(}p(x^{\prime})-\pi_{\eta}(x^{\prime})\big{)}\cdot(\mathcal{K}_{\textsf{DPnP},\eta}^{N}-\mathbf{1}\mathbf{1}^{\top})f(x^{\prime})\mathrm{d}x^{\prime}\right|$
$\displaystyle\leq\left(\int\frac{\big{(}p(x^{\prime})-\pi_{\eta}(x^{\prime})\big{)}^{2}}{\pi_{\eta}(x)}\mathrm{d}x\right)^{1/2}\cdot\left\|(\mathcal{K}_{\textsf{DPnP},\eta}^{N}-\mathbf{1}\mathbf{1}^{\top})f(x^{\prime})\right\|_{L^{2}(\pi_{\eta})}$
$\displaystyle\leq\sqrt{\chi^{2}(p\,\|\,\pi_{\eta})}\cdot\lambda^{N}\|f\|_{L^{2}(\pi_{\eta})}.$
(52)

#### Step 4: choosing an appropriate square-integrable function.

Now, set

$f(x)=\frac{p\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}K_{\textsf{DPnP},\eta}^{(N)}(x)-\pi_{\eta}(x)}{\pi_{\eta}(x)}.$

It is easily checked that

$\displaystyle\int\big{(}p\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}K_{\textsf{DPnP},\eta}^{(N)}(x)-\pi_{\eta}(x)\big{)}f(x)\mathrm{d}x$
$\displaystyle=\chi^{2}(p\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}K_{\textsf{DPnP},\eta}^{(N)}~{}\|~{}\pi_{\eta}),$
$\displaystyle\|f\|_{L^{2}(\pi_{\eta})}$
$\displaystyle=\sqrt{\chi^{2}(p\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}K_{\textsf{DPnP},\eta}^{(N)}~{}\|~{}\pi_{\eta})}.$

Plug these equations into (52), we obtain

$\chi^{2}(p\mathbin{\mathchoice{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}{\vbox{\hbox{$\scriptscriptstyle\circ$}}}}K_{\textsf{DPnP},\eta}^{(N)}~{}\|~{}\pi_{\eta})\leq\lambda^{2N}\chi^{2}(p\,\|\,\pi_{\eta}),$

as claimed. ∎


# Spin liquid behavior of a three-dimensional magnetic system Ba3NiIr2O9 with
$S$ = 1

Siddharth Kumar  Department of Physics, Indian Institute of Science, Bengaluru
560012, India  S. K. Panda  Department of Physics, Bennett University, Greater
Noida 201310, Uttar Pradesh, India  Manju Mishra Patidar  UGC-DAE Consortium
for Scientific Research, University Campus, Khandwa Road, Indore 452017, India
Shashank Kumar Ojha  Department of Physics, Indian Institute of Science,
Bengaluru 560012, India  Prithwijit Mandal  Department of Physics, Indian
Institute of Science, Bengaluru 560012, India  Gangadhar Das  Chemistry and
Physics of Materials Unit, Jawaharlal Nehru Centre for Advanced Scientific
Research, Jakkur, Bengaluru, 560064 India  J. W. Freeland  Advanced Photon
Source, Argonne National Laboratory, Argonne, Illinois 60439, USA  V. Ganesan
UGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road,
Indore 452017, India  Peter J. Baker  ISIS Pulsed Neutron and Muon Source,
STFC Rutherford Appleton Laboratory, Harwell Campus, Didcot OX11 0QX, United
Kingdom  S. Middey<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Science, Bengaluru 560012, India

###### Abstract

The quantum spin liquid (QSL) is an exotic phase of magnetic materials where
the spins continue to fluctuate without any symmetry breaking down to zero
temperature. Among the handful reports of QSL with spin $S\geq$1, examples
with magnetic ions on a three-dimensional magnetic lattice are extremely rare
since both larger spin and higher dimension tend to suppress quantum
fluctuations. In this work, we offer a new strategy to achieve 3-D QSL with
high spin by utilizing two types of transition metal ions, both are
magnetically active but located at crystallographically inequivalent
positions. We design a 3-D magnetic system Ba3NiIr2O9 consisting of
interconnected corner shared NiO6 octahedra and face shared Ir2O9 dimer, both
having triangular arrangements in a-b plane. X-ray absorption spectroscopy
measurements confirm the presence of Ni2+ ($S$=1). Our detailed thermodynamic
and magnetic measurements reveal that this compound is a realization of
gapless QSL state down to at least 100 mK. Ab-initio calculations find a
strong magnetic exchange between Ir and Ni sublattices and in-plane
antiferromagnetic coupling between the dimers, resulting in dynamically
fluctuating magnetic moments on both the Ir and Ni sublattice.

Experimental realization and theoretical description of the highly entangled
quantum spin liquid phase remain challenging topics of quantum many-body
physics Anderson (1973). Over the last fifteen years, several spin-1/2 systems
with two-dimensional frustrated lattice have been reported as probable
candidates with QSL behavior Balents (2010); Savary and Balents (2016); Zhou,
Kanoda, and Ng (2017); Knolle and Moessner (2019); Broholm  _et al._ (2020).
Since experimental reports of QSL with either spin $(S)\geq$ 1 Cheng  _et al._
(2011); Chamorro  _et al._ (2018) or three dimensional arrangement of spins
Okamoto  _et al._ (2007a); Koteswararao  _et al._ (2014); Balz  _et al._
(2016); Gao  _et al._ (2019); Chillal  _et al._ (2020) are very few, it can be
easily anticipated that chances of having a QSL with both higher spin and 3-D
geometry is extremely low Plumb  _et al._ (2019). Many of the attempts to
obtain spin-1 QSL have been focused on stabilizing Ni within various
structural network Nakatsuji  _et al._ (2005); Cheng  _et al._ (2011); Lu  _et
al._ (2018); Chamorro  _et al._ (2018); Plumb  _et al._ (2019); Medrano  _et
al._ (2018). However, the magnetic behavior of these $S$ = 1 systems at low
temperature differs widely from each other, even in the compounds with similar
structural geometry. For example, unlike to the well known 120∘ spin structure
Starykh (2015), as observed in $A_{3}$NiNb2O9 Lu  _et al._ (2018), Ba3NiSb2O9
shows characteristic spin liquid behavior Cheng  _et al._ (2011); Fåk  _et
al._ (2017); Quilliam  _et al._ (2016), whereas NiGa2S4 hosts a spin nematic
phase Bhattacharjee, Shenoy, and Senthil (2006). The interaction of such Ni-
based $S$ = 1 triangular lattice with another magnetically active sublattice
might result in an exotic magnetic phase in three-dimensional materials.
However, only very few 3-D compounds with such feasibility exist Treiber,
Kemmler-Sack, and Ehmann (1982); Lightfoot and Battle (1990); Ferreira  _et
al._ (2018).

Six-layered hexagonal (6$H$) perovskite $A_{3}MM^{\prime}_{2}$O9 (Fig. 1(a))
with magnetic $M$ and $M^{\prime}$ ions constitutes a 3-D spin system. Both,
$M$O6 octahedral units and face-shared $M^{\prime}_{2}$O9 dimers form a
triangular lattice in $a$-$b$ plane (Fig. 1(b)-(c)) and would become
geometrically frustrated in the presence of antiferromagnetic interaction.
Moreover, the $M$-O-$M^{\prime}$ connectivity constitutes a buckled honeycomb
lattice (Fig. 1(d)), which could host a Kitaev spin liquid phase in case of
spin-1/2 ions with strong spin-orbit coupling (SOC) Kitaev (2006); Takagi  _et
al._ (2019). In search of SOC driven elusive nonmagnetic $J$ = 0 state and
excitonic magnetism in $d^{4}$ system, several Ba${}_{3}M$Ir2O9 compounds with
nonmagnetic $M^{2+}$ have been investigated recently Khaliullin (2013); Nag
_et al._ (2016, 2018a); Khan  _et al._ (2019). However, the comparable
strength of SOC, non-cubic crystal field, Hund’s coupling, and superexchange
interaction gives rise to a small but finite moment on the Ir sublattice.
Moreover, interdimer hopping results in a spin-orbital liquid phase in
Ba3ZnIr2O9 Nag  _et al._ (2016). Replacing the nonmagnetic Zn2+ by an
isovalent magnetic ion such as Ni2+ should provide a unique opportunity to
investigate the magnetic response of a triangular lattice with $S$ = 1 in
presence of the interconnected Ir sublattice with fluctuating magnetic
moments. If both Ni and Ir moments of Ba3NiIr2O9 fluctuate dynamically, then
it would offer a new route to realize 3-D QSL by involving two different
magnetic ions. Also, it would be a 3-D QSL with a new type of structural
network as compared to all existing examples with pyrochlore Gao  _et al._
(2019); Plumb  _et al._ (2019), hyperkagome Okamoto  _et al._ (2007a), and
hyper hyperkagome structure Koteswararao  _et al._ (2014); Chillal  _et al._
(2020).

Figure 1:  (a) Unit cell of 6$H$ $A_{3}MM^{\prime}$O9 without any disorder.
(b), & (c) shows the triangular lattice arrangement of $M^{\prime}_{2}$O9
dimers, $M$O6 octahedra, respectively, in a-b plane. (d) $M$-O-$M\prime$
connectivity forms buckled honeycomb lattice. $A$, $M$ and $M^{\prime}$
corresponds to Ba, Ni, and Ir, respectively, for BNIO. Different magnetic
exchange pathways ($J_{i}$) are also shown in (a)-(d). (e) Observed and
refined powder XRD pattern of BNIO. (f) XAS spectrum of Ni $L_{3,2}$-edge of
BNIO along with NiO and NdNiO3 for comparison. The XAS data for NdNiO3 has
been adapted from Ref. Freeland, Van Veenendaal, and Chakhalian, 2016.

In this paper, we report on the electronic and magnetic behavior of Ba3NiIr2O9
(BNIO). The phase purity and absence of any cationic site disorder have been
demonstrated by powder X-ray diffraction (XRD) measurement. X-ray absorption
spectroscopy (XAS) experiments have confirmed the desired +2 oxidation state
of Ni. Persisting spin fluctuations down to 100 mK have been revealed by
magnetization, specific heat and muon spin-rotation ($\mu$SR) measurements. We
have also investigated BNIO by density functional theory calculations
including Hubbard $U$ and SOC within the framework of LSDA+$U$ (local spin
density approximation + $U$) approach. We have found not only appreciable
magnetic exchange between Ni and Ir-sublattice but also antiferromagnetic
coupling in the triangular sublattice of Ir. This geometrical frustration
prohibits any long-range magnetic ordering and makes BNIO a rare example of
three-dimensional QSL involving $S$ = 1.

## Results

Polycrystalline BNIO was synthesized by solid state synthesis route. Members
of the $A_{3}MM^{\prime}_{2}$O9 series can have site disorder between face
shared, and corner shared octahedra Middey  _et al._ (2011). It is also well
known that structural disorder often jeopardizes QSL behavior, resulting in
magnetic order or spin glass freezing Zhong  _et al._ (2019). All peaks of the
powder XRD pattern of BNIO (Fig. 1(e)) can be indexed and refined with 6$H$
structure having space group $P6_{3}/mmc$. The refinement also confirms that
all corner-shared (face-shared) octahderal units are occupied by Ni (Ir)
without any Ni-Ir site disorder. The structural parameters obtained from the
refinement have been listed in SM sup . The temperature dependent XRD
measurements down to 15 K also rules out any structural transition. Having
confirmed that both Ni and Ir ions form triangular lattices in a-b plane
without any disorder, we want to verify whether Ni has indeed $S$ = 1 state.
For this purpose, XAS measurements were carried out Freeland, Van Veenendaal,
and Chakhalian (2016). The comparison of Ni $L_{3,2}$ XAS line shape and
energy position of BNIO, Ni2+O, and NdNi3+O3 (Fig. 1(e)) testifies the desired
+2 oxidation of Ni in present case. The octahedral crystal field of Ni2+
($d^{8}$: $t_{2g}^{6}$, $e_{g}^{2}$) ensures $S$ = 1 on Ni sublattice.

Electrical measurement demonstrates insulating nature of the sample (inset of
Fig.2 (a)), which can be fitted using Mott’s variable range hopping (VRH) Hill
(1976) model in three-dimensions ($\rho=\rho_{o}\exp(T_{o}/T)^{1/4}$) as shown
in Fig.2 (a). We also note that the insulating behavior of Ba3ZnIr2O9 is well
described by VRH in two-dimensions. This difference between two compounds is a
manifestation of electron hopping paths along the Ni-O-Ir bonds. Our
electronic structure calculations (see SM sup ) further demonstrate that the
insulating state can be obtained only by considering both electron correlation
and SOC, implying that BNIO is a SOC driven Mott insulator Kim  _et al._
(2008).

Figure 2:  (a) Fitting of $\rho$ by 3-D variable range hopping model for BNIO.
Inset shows $\rho$ vs. $T$. (b) $\chi$ vs T  on left axis and
$1/(\chi-\chi_{0})$ along with fitting on right axis. (c) M-H at 2 K.

The temperature dependent field cooled and zero-field cooled magnetic
susceptibility does not differ from each other and also do not show any
anomaly (Fig. 2(b)). This strongly implies absence of any long range magnetic
ordering and spin glass behavior. We have fitted the data by a modified Cuire-
Weiss law ($\chi$ = $\chi_{0}$ \+ $\frac{C_{W}}{T-\theta_{CW}}$) where
$\chi_{0}$, $C_{W}$ and $\theta_{CW}$ represents temperature independent
susceptibility contribution, Curie constant and Curie temperature,
respectively. The fitting, shown as plot of 1/($\chi$-$\chi_{0}$) vs. $T$ in
right axis of Fig. 2(b), results a $\theta_{CW}\sim$ -15 K. Negative values of
Curie-Weiss temperature signify net antiferromagnetic interaction among the
spins of BNIO. The relatively smaller value of $\theta_{CW}$ is related to the
presence of multiple exchange pathways, which will be discussed in later part
of this paper. The effective magnetic moment $\mu_{eff}$ (= $\sqrt{8C_{W}}$)
is found to be $\sim$ 3.65 $\mu_{B}$ from the fitting. Interestingly, the
effective magnetic moment of a similar compound Ba3NiSb2O9, with nonmagnetic
Sb5+ was reported to be around 3.54 $\mu_{B}$ Cheng  _et al._ (2011). This
gives an estimate of $g$-factor $\sim$ 2.5, similar to other Ni2+ based
systems carlin_paramagnetism_1986 . If we assume similar value for the present
compound, the effective Ir moment turns out to be 0.9 $\mu_{B}$ per dimer (=
$\sqrt{\mu_{eff}^{2}-\mu_{Ni}^{2}}$) i.e. 0.64 $\mu_{B}$/Ir. This is very
similar to the Ir moment (0.5 - 0.6 $\mu_{B}$) reported for Ba3MgIr2O9 Nag
_et al._ (2018a), though a nonmagnetic $J$ = 0 state is expected for Ir5+ from
a pure ionic picture. Thus, our analysis highlights both Ni2+ and Ir5+
participate in magnetism of BNIO compound. The estimated magnetic moments from
our LSDA+U+SOC calculations (shown in SM sup ) are also in good agreement with
our experimental results. Fig. 2 (c) shows $M$-$H$ done at 2 K between $\pm$ 9
T. The absence of any hysteresis again confirms absence of ferromagnetism and
spin glass freezing at 2 K. The presence of antiferromagnetic interaction
without any long range magnetic ordering or spin glass transition strongly
indicates that BNIO is a favorable candidate of QSL.

Figure 3: (a) $C_{p}$ vs T curves at various fields. (b) Magnetic specific
heat ($C_{m}$) extracted by subtracting lattice contribution. (c) Low
temperature part of $C_{m}$ plotted in log-log plot. (d) Magnetic entropy for
zero field.

In order to further investigate the magnetic nature of the sample, we have
measured specific heat ($C_{p}$) from 100 mK to 200 K. $C_{p}$ not only probes
the presence/absence of any long-range magnetic ordering but also provides
very crucial information about the nature of low energy excitation. The
absence of any $\lambda$-anomaly (Fig. 3(a)) again confirms absence of long-
range order and/or any structural transition down to 100 mK, consistent with
the magnetic measurement and XRD results, respectively. For an insulating
compound with magnetic spins, $C_{p}$ consists of lattice specific heat
($C_{lat}$) and magnetic specific heat ($C_{m}$). In absence of any analogous
non-magnetic compound, the contribution of $C_{lat}$ has been evaluated by
fitting $C_{p}$ in 30 K - 200 K range by a Debye-Einstein equation with one
Debye term and two Einstein terms (details are in SM sup ) and extrapolating
the fitted curve down to 100 mK. A broad peak is observed around 7 K in
$C_{m}$ vs. $T$ plot (Fig. 3(a)). We can not capture this feature by Schottky
anomaly, arising due to energy level splitting (see SM sup ). On the other
hand, such feature is commonly observed in spin liquid materials and thus,
could be considered as a signature of crossover from thermally disordered
paramagnet to quantum disordered spin liquid state Okamoto  _et al._ (2007b);
Balents (2010); Li  _et al._ (2015); Dey  _et al._ (2017). The position of
this broad peak shows negligible shifts with the application of magnetic field
(shifts $\sim$ 1 K for applied field of 12 T).

At low temperature, $C_{m}$ follows power-law behavior $C_{m}$ = $\gamma
T^{\alpha}$ (Fig. 3(c)). For zero field, the magnitude of coefficient $\gamma$
is 45 mJ/mol K2 and the exponent $\alpha$ is 1.0$\pm$0.05 within 0.1 K - 0.6 K
range. The value of $\gamma$ is very similar to other gapless spin liquid
candidates: like Ba3CuSb2O9 (43.4 mJ/mol K2) Zhou  _et al._ (2011), Ba2YIrO6
(44 mJ/mol K2) Nag  _et al._ (2018b), Ba3ZnIr2O9 (25.9 mJ/mol K2) Nag  _et
al._ (2016), Sr2Cu(Te0.5W0.5)O6 (54.2 mJ/mol K2) Mustonen  _et al._ (2018).
Also, the linear $T$ behavior with nonzero $\gamma$ in an insulator comes due
to gapless spinon excitations with a Fermi surface and has been reported in
several organic and inorganic spin liquid candidates Yamashita  _et al._
(2008, 2011); Zhou  _et al._ (2011); Cheng  _et al._ (2011); Mustonen  _et
al._ (2018); Clark  _et al._ (2014); Uematsu and Kawamura (2019). Otherwise,
any gapped excitation would result in an exponential dependence of $C_{m}$ on
$T$. $\alpha$ becomes 2.6$\pm$0.05 within 0.8 K - 2.1 K. Interestingly, the
application of an external field destroys the linear $T$ behavior of $C_{m}$.
For $\mu_{0}H$ = 4 T, $\alpha$ becomes 2 for 0.15 K $\leq T\leq$ 0.50 K and
2.9 for 0.5 K $\leq T\leq$ 2.4 K. We note that $\alpha$ is found to be between
2 to 3 for several spin nematic phase Nakatsuji  _et al._ (2005, 2007);
Povarov  _et al._ (2019); Kohama  _et al._ (2019). Further studies are
necessary to investigate the possibility of transition from spin liquid to
spin nematic phase by the application of a magnetic field. The amount of
released magnetic entropy ($S_{m}$) is evaluated by integrating $C_{m}/T$
w.r.t. $T$ and is shown in Fig. 3(d). For BNIO, the entropy saturates at 6.9
J/mol K for zero field measurement, which is only 75% of the total entropy
expected for even a $S=1$ system [$R$ln(2$S$+1), where $R$ is universal gas
constant]. The retention of a large amount of entropy at low temperature is
another signature of the spin-liquid nature of BNIO, which has been reported
as well for many other QSL Zhou  _et al._ (2011); Cheng  _et al._ (2011);
Mustonen  _et al._ (2018); Yamashita  _et al._ (2008).

To have a further understanding of the magnetic behavior at low temperature,
we have performed $\mu$SR measurements, which is a very sensitive local probe
to detect a weak magnetic field, even of the order of 0.1 Oe Blundell (1999).
Fig. 4(a) shows asymmetry vs time curves for zero-field (ZF) measurements at
selected temperatures. Neither any oscillation nor the characteristic 1/3rd
recovery of the asymmetry is observed down to 60 mK, strongly implying the
absence of long-range magnetic ordering or spin freezing. For a system with
two interacting spin networks, the local magnetic field, felt by a muon at a
stopping site is contributed by both magnetic sublattices. In such cases, the
muon relaxation function is generally described by a product of two response
functions, representing local fields from two spin networks Uemura  _et al._
(1985). However, our such attempts considering different possible combinations
of relaxation functions, including a spin glass like relaxation Morenzoni  _et
al._ (2008); Uemura  _et al._ (1985), did not provide a satisfactory fitting
of the experimental observed data (see SM sup ). This further supports the
absence of spin glass freezing in present case.

Figure 4:  (a) Asymmetry vs time curves at various temperatures taken at zero
magnetic field and fitted curve (solid line) using equation 1. (b) Variation
of $\nu$ to temperature (shaded area is a guide to the eye). Inset shows
variation of $\nu$ with applied LF at 100 mK

Interestingly, similar to the other hexagonal Ba${}_{3}M$Ir2O9 Nag  _et al._
(2016, 2018a), these asymmetry curves consist of one fast relaxing, one
moderately relaxing, and one hardly relaxing components. We have fitted these
curves using a combination of two dynamical relaxation functions with a common
fluctuation rate $\nu$ and a Kubo-Toyabe function (KT) Hayano  _et al._
(1979),

$A(t)=A_{1}G(t,\Delta H_{1},\nu)+A_{2}G(t,\Delta H_{2},\nu)+A_{3}KT(t,\delta)$
(1)

where $A_{1}$, $A_{2}$, $A_{3}$ are amplitudes. The static KT function,
corresponding to the hardly relaxing component, accounts for the muons
stopping at the silver sample holder as we find that the relaxation curve from
the bare sample holder can also be described by a KT function with similar
$\delta$. The dynamical relaxation, arising due to the presence of a field
distribution ($\Delta H$) with a fluctuation rate ($\nu$) is represented by
the Keren function $G(t,\Delta H,\nu)$ Keren (1994). The presence of two
dynamical relaxations implies two inequivalent muon stopping sites, which are
likely to be related with the two types of crystallographically inequivalent
oxygen in hexagonal Ba${}_{3}M$Ir2O9 Nag  _et al._ (2016, 2018a). The
asymmetry data over a large temperature range (60 mK - 150 K) have been fitted
by allowing $\nu$ to vary with $T$ and, the extracted values of $\nu$ has been
shown as a function of $T$ in Fig. 4(b). The background contribution is
different between measurements in dilution refrigerator and helium cryostat
and, has been kept fixed for our analysis within the corresponding temperature
range. The inequality $\nu>\gamma\Delta H$ ( $\gamma$ = muon gyromagnetic
ratio = 2$\pi\times$135.5 Mrad s-1 T-1) holds for both relaxing components as
$\gamma\Delta H_{1}\sim$0.425 MHz and $\gamma\Delta H_{2}\sim$ 0.09 MHz for
the lowest temperature of our measurement (60 mK). This justifies the use of
dynamical relaxation functions and establishes spin liquid nature of BNIO. We
note that the value of $\nu$ ($\sim$ 4 MHz) at low temperature for BNIO is one
order of magnitude smaller than Ba3ZnIr2O9 Nag  _et al._ (2016) and is likely
to be related with the involvement of large spins on the Ni sublattice.

$\mu$-SR spectra, recorded at 100 mK in presence of an applied longitudinal
field (LF) have further corroborated QSL nature of BNIO. In case of relaxation
arising from a static internal field distribution with width $\Delta H_{i}$,
an applied LF $\sim$ 5-10$\Delta H_{i}$ would completely suppress the
relaxation. From the analysis of ZF $\mu$SR data, we found $\Delta H_{1}\sim$
5 Gauss and $\Delta H_{2}\sim$ 1 Gauss. No such decoupling is observed in the
present case in measurement up to 200 Gauss (see inset of Fig. 4(b) and SM sup
), establishing the dynamic nature of spins in BNIO down to atleast 100 mK.

## Discussion

To understand the underlying mechanism of the observed QSL state, we estimated
the inter-atomic magnetic exchange interactions from the converged
LSDA+$U$+SOC calculations using the formalism of Ref. Kvashnin  _et al._ ,
2015a (see Method section and SM sup  for details). As shown in Table-I, the
strongest interaction is antiferromagnetic, which is between the Ir ions of
the structural dimer. The strong Ir-Ni interaction further testifies three-
dimensional nature of BNIO. Most importantly, Ir-Ir exchange in the a-b plane
($J_{4}$) is found to be antiferromagnetic, resulting in-plane magnetic
frustration and explains the origin of the QSL behavior of the present system.
However, the presence of strong Ir-Ni and Ni-Ni ferromagnetic exchange reduces
the net antiferromagnetic exchange of this system, resulting a relatively
lower value of negative $\theta_{CW}$. We further note that the ferromagnetic
Ni-Ni exchange has been observed also in antiferromagnetic phase of analogous
compound Ba3NiRu2O9 Lightfoot and Battle (1990).

Table 1: Exchange couplings obtained from $ab$-initio calculations. Exchange pathways have been shown in Fig. 1. AFM and FM refers to antiferromagnetic and ferromagnetic interaction, respectively.  Exchange | Interacting | Number | Magnitude | Type |  $|$z${}_{i}J_{i}$/$J_{1}|$
---|---|---|---|---|---
($J_{i}$) | pair | of neighbor ($z_{i}$) | (meV) |  |
$J_{1}$ | Ir-Ir | 1 | -8.91 | AFM | 1
$J_{2}$ | Ir-Ni | 3 | 0.96 | FM | 0.32
$J_{3}$ | Ir-Ir | 3 | 0.10 | FM | 0.03
$J_{4}$ | Ir-Ir | 6 | -0.17 | AFM | 0.11
$J_{5}$ | Ni-Ni | 6 | 0.09 | FM | 0.06

To summarize, our detailed measurements reveal that 6$H$ BNIO containing $S$=1
hosts a gapless spin liquid phase below 2 K. The involvement of Ir5+ and Ni2+
in the magnetic properties of BNIO is revealed by dc magnetic measurements,
$\mu$SR experiments, and electronic structure calculation. The
antiferromagnetic interaction between Ir2O9 dimers in a-b plane facilitates
geometrical frustration driven QSL phase of BNIO. Since many
Ba3$MM_{2}^{\prime}$O9 compounds can be stabilized in 6$H$ structure, we
believe this work will lead to the realization of many 3-D QSLs with large
spin by judicial choice of $M$ and $M^{\prime}$.

### Method

Stoichiometric amount of BaCO3, NiO and Ir metal power were used as starting
materials for the solid state synthesis of BNIO. The mixture was heated
multiple times at 1175∘ C with intermediate grindings till the desired phase
is formed. Powder XRD was carried out using a lab based Rigaku Smartlab
diffractometer and also in the Indian beamline (BL-18B) at the Photon Factory,
KEK, Japan. The diffraction pattern of the final phase was refined by
Reitveild method using FULLPROF Rodríguez-Carvajal (1993).

XAS spectra of Ni $L_{3,2}$-edges were recorded in bulk sensitive total
fluorescence yield mode in 4-ID-C beam line of Advanced Photon Source, USA. DC
magnetic measurements were carried using a Quantum Design (QD) SQUID
magnetometer. Heat capacity measurements ($C_{p}$) were done in a dilution
refrigerator insert coupled with a 16T QD-PPMS system using relaxation
calorimetry. $\mu$SR experiments down to 60 mK were performed using pulsed
muon beam at MuSR spectrometer of ISIS Neutron and Muon Source, UK. A dilution
fridge was used to record $\mu$SR data from 60 mK to 4 K and a cryostat was
used for temperatures above 1.5 K.

The density functional theory (DFT) calculations have been performed in the
local spin-density approximation + Hubbard $U$ (LSDA+U) approach with and
without including spin-orbit coupling (SOC) by means of a full potential
linearized muffin-tin orbital method (FP-LMTO) Andersen (1975); Wills and
Cooper (1987) as implemented in the RSPt code Wills  _et al._ (2000). The
Brillouin-zone (BZ) integration is carried out by using the thermal smearing
method with 10 $\times$ 10 $\times$ 4 k-mesh. For the charge density and
potential angular decomposition inside the muffin-tin (MT) spheres, the value
of maximum angular momentum was taken equal to $l_{max}=8$. To describe the
electron-electron correlation within LSDA+U approach, we have taken $U$ = 6
eV, $J$ =0.8 eV for Ni-$d$ states and $U$ = 2 eV, $J$ =0.6 eV for the Ir-$d$
states. The set of the correlated orbitals located on Ni and Ir sites were
obtained by projecting the electron density onto the corresponding MT sphere
with a certain angular character (so-called “MT-heads” projectionGrechnev  _et
al._ (2007)).

After obtaining the self-consistent fully converged LSDA+$U$+SOC calculations,
the magnetic force theorem Liechtenstein  _et al._ (1987); Katsnelson and
Lichtenstein (2000) was used to extract the effective inter-site magnetic-
interaction parameters ($J_{ij}$). In this approach the magnetic system is
mapped onto the Heisenberg Hamiltonian:

$\hat{H}=-\sum_{i\neq j}J_{ij}\vec{S_{i}}\cdot\vec{S_{j}}.$ (2)

Further, $J_{ij}$ are extracted in a linear-response manner via Green’s
function technique. A detailed discussion of the implementation of the
magnetic force theorem in RSPt is provided in Ref. Kvashnin  _et al._ , 2015b.
This method is considered to be one of the most accurate techniques for the
estimation of exchange interactions and also been successfully employed for
many transition metal compounds Panda  _et al._ (2016).

### Acknowledgement

S.M. acknowledges financial support from ISRO-IISc Space Technology Cell and
Infosys Foundation, Bangalore. S.M. and S.K. acknowledge insightful
discussions with Dr. Subhro Bhattacharjee, Dr. Yogesh Singh, Dr. Pabitra
Biswas. Authors thank the Department of Science and Technology, India for the
financial support, Saha Institute of Nuclear Physics and Jawaharlal Nehru
Centre for Advanced Scientific Research, India for facilitating the
experiments at the Indian Beamline, Photon Factory, KEK, Japan. The authors
also acknowledge the experimental support from UGC-DAE CSR, Indore, staffs of
LT & Cryogenics especially Er. P. Saravanan for their technical support. This
research used resources of the Advanced Photon Source, a U.S. Department of
Energy Office of Science User Facility operated by Argonne National Laboratory
under Contract No. DE-AC02-06CH11357. Experiments at the ISIS Neutron and Muon
Source were supported by a beamtime allocation RB1920396 from the Science and
Technology Facilities Council.

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