Persistent low-temperature spin dynamics in the mixed-valence iridate Ba$_{3}$InIr$_{2}$O$_{9}$

TL;DR

This study provides evidence for a quantum spin-liquid state in Ba₃InIr₂O₉, characterized by persistent low-temperature spin dynamics, absence of magnetic order, and gapless excitations, driven by its unique mixed-valence and lattice geometry.

Contribution

It reports the discovery of a putative quantum spin-liquid in a mixed-valence iridate with unique lattice geometry, combining multiple experimental techniques to characterize its dynamic, gapless ground state.

Findings

No magnetic order or spin freezing down to 20 mK.

Gapless ground state indicated by power-law behavior.

Persistent spin dynamics over a wide temperature range.

Abstract

Using thermodynamic measurements, neutron diffraction, nuclear magnetic resonance, and muon spin relaxation, we establish putative quantum spin-liquid behavior in Ba$_3$InIr$_2$O$_9$, where unpaired electrons are localized on mixed-valence Ir$_2$O$_9$ dimers with Ir$^{4.5+}$ ions. Despite the antiferromagnetic Curie-Weiss temperature on the order of 10 K, neither long-range magnetic order nor spin freezing are observed down to at least 20 mK, such that spins are short-range correlated and dynamic over nearly three decades in temperature. Quadratic power-law behavior of both the spin-lattice relaxation rate and specific heat indicates the gapless nature of the ground state. We envisage that this exotic behavior may be related to an unprecedented combination of the triangular and buckled honeycomb geometries of nearest-neighbor exchange couplings in the mixed-valence setting.

Tables

Table 1. Table 1: Refined atomic positions for Ba 3 InIr 2 O 9 at 300 K. The U iso subscript 𝑈 iso U_{\rm iso} are isotropic atomic displacement parameters (in 10 -2  Å 2 ). Lattice parameters: a = 5.8316 ​ ( 1 ) 𝑎 5.8316 1 a=5.8316(1)  Å, c = 14.4877 ​ ( 4 ) 𝑐 14.4877 4 c=14.4877(4)  Å, P ​ 6 3 / m ​ m ​ c 𝑃 subscript 6 3 𝑚 𝑚 𝑐 P6_{3}/mmc , R I = 0.039 subscript 𝑅 𝐼 0.039 R_{I}=0.039 , R p = 0.051 subscript 𝑅 𝑝 0.051 R_{p}=0.051 .

		$x/a$	$y/b$	$z/c$	$U_{\mathrm{iso}}$
Ba1	$2b$	0	0	0.25	1.3(1)
Ba2	$4f$	$\frac{1}{3}$	$\frac{2}{3}$	0.9103(2)	0.9(1)
In	$2a$	0	0	0	0.4(1)
Ir	$4f$	$\frac{1}{3}$	$\frac{2}{3}$	0.1590(1)	1.29(3)
O1	$6h$	0.4867(2)	0.5133(2)	0.25	1.40(5)
O2	$12k$	0.1715(2)	0.3430(4)	0.0841(1)	2.1(1)

Table 2. Table 2: Refined atomic positions for Ba 3 InIr 2 O 9 at 3.4 K. The U iso subscript 𝑈 iso U_{\rm iso} are isotropic atomic displacement parameters (in 10 -2  Å 2 ). Lattice parameters: a = 5.8152 ​ ( 3 ) 𝑎 5.8152 3 a=5.8152(3)  Å, b = 10.0680 ​ ( 5 ) 𝑏 10.0680 5 b=10.0680(5)  Å, c = 14.4619 ​ ( 6 ) 𝑐 14.4619 6 c=14.4619(6)  Å, β = 90.854 ​ ( 3 ) ∘ 𝛽 90.854 superscript 3 \beta=90.854(3)^{\circ} , C ​ 2 / c 𝐶 2 𝑐 C2/c , R I = 0.039 subscript 𝑅 𝐼 0.039 R_{I}=0.039 , R p = 0.061 subscript 𝑅 𝑝 0.061 R_{p}=0.061 . Atomic displacement parameters for oxygen were constrained in the refinement.

		$x/a$	$y/b$	$z/c$	$U_{\mathrm{iso}}$
Ba1	$4e$	0	0.0010(15)	0.25	1.0(1)
Ba2	$8f$	0.0049(9)	0.3360(11)	0.0889(3)	0.1(1)
In	$4a$	0	0	0	0.2(1)
Ir	$8f$	$-0.0081\left(5\right)$	0.3334(6)	0.8397(2)	1.0(1)
O1	$4e$	0	0.4918(10)	0.75	1.2(1)
O2	$8f$	0.230(2)	0.2573(8)	0.7541(4)	1.2(1)
O3	$8f$	$-0.018\left(2\right)$	0.1717(8)	0.9140(6)	1.2(1)
O4	$8f$	0.226(1)	0.4158(9)	0.9251(4)	1.2(1)
O5	$8f$	$-0.263\left(1\right)$	0.4144(9)	0.9079(5)	1.2(1)

Equations

$$\chi (T)=$$

$$\times \frac{g_0^2+5g_1^2{e}^{-{\Delta }_1/k_{B}T}+5g_2^2{e}^{-{\Delta }_2/k_{B}T}}{1+e^{-{\Delta }_1/k_{B}T}+e^{-{\Delta }_2/k_{B}T}},$$

$$P(t)=f{e}^{-({\lambda }_{\mathrm{F}}t{)}^{\beta }}+(1-f)\left(\frac{1}{3}+\frac{2}{3}\left(1-(\sigma t{)}^2\right)e^{-\frac{(\sigma t{)}^2}{2}}\right)$$

$$K=K_0+(A_{\mathrm{hf}}/N_A{\mu }_B)\chi (T),$$

$$1-M_t/M_{\infty }=$$

$$+0.0925e^{-(30Wt{)}^{\beta }}+0.215e^{-(56Wt{)}^{\beta }}$$

$$+0.653e^{-(90Wt{)}^{\beta }}),$$

$$(1/T_{1}T)\sim \sum _q{A}_{\mathrm{hf}}(q{)}^2\times {\chi }^{\prime \prime }(q,\omega \to 0),$$

Full text

I Introduction

Frustrated magnets host multiple exotic states, including quantum spin liquids (QSLs). In a QSL, spins are strongly correlated, but quantum fluctuations prevent them from long-range ordering Balents (2010); Savary and Balents (2017). The initial (and subsequently rebutted) proposal of the QSL resonating-valence-bond state on the triangular lattice of Heisenberg spins Anderson (1973) was followed by similar proposals for several other isotropic (Heisenberg) spin-$\frac{1}{2}$ models, where the formation of QSLs is now established Yan et al. (2011); Mishmash et al. (2013); Kaneko et al. (2014); Hu et al. (2015); Iqbal et al. (2016). A few candidate QSL materials proposed over the last decade bear key experimental signatures of this exotic state, including persistent spin dynamics and the absence of long-range order within the experimentally accessible temperature range Norman (2016); Fåk et al. (2012); Clark et al. (2013); Balz et al. (2016).

More recently, QSL states in anisotropic magnets have been explored. Here, Kitaev model with anisotropic interactions on the honeycomb lattice Kitaev (2006) offers an exact solution for the QSL. Real-world manifestations of the Kitaev physics are found in compounds of $4d$ and $5d$ transition metals Jackeli and Khaliullin (2009), where large spin-orbit coupling triggers strong intersite magnetic anisotropy. However, none of the Kitaev materials reported to date host the QSL ground state in zero field, and long-range order typically sets in at low temperatures, owing to substantial interactions beyond the Kitaev terms Winter et al. (2016, ).

In this paper, we propose an alternative strategy and search for QSL states in the family of mixed-valence $5d$ oxides, where unpaired electrons are localized on dimers of the Ir atoms. This should facilitate access to hitherto unexplored local electronic states Streltsov and Khomskii (2016) and new regimes of anisotropic exchange interactions. Specifically, we report low-temperature magnetic behavior of the mixed-valence iridate Ba3InIr2O9 as a QSL candidate, and confirm its persistent spin dynamics as well as the absence of long-range magnetic order down to at least 20 mK. We further identify quadratic-like power-law behavior of both specific heat and spin-lattice relaxation rate, and compare these observations to existing theoretical results on QSLs.

Ba3InIr2O9 belongs to the family of hexagonal perovskites A3BM2O9. Their structures comprise single BO6 octahedra and M2O9 dimers of two face-sharing MO6 octahedra (Fig. 1a). When magnetic ion occupies the B site, triangular interaction geometry is formed, as in Ba3CoSb2O9, which is arguably the best model spin-$\frac{1}{2}$ antiferromagnet on the triangular lattice Shirata et al. (2012); Zhou et al. (2012); Ma et al. (2016). Placing a $4d$ or $5d$ ion into the B site or into one of the M sites could give rise to a triangular system with leading Kitaev interactions Becker et al. (2015); Catuneanu et al. (2015), but experimental implementation of this idea is hindered by the strong B/M site mixing that occurs, e.g., in Ba3IrTi2O9 Dey et al. (2012); Kumar et al. (2016); Lee et al. (2017). Alternatively, Ir could be introduced into both M sites, while keeping the B site non-magnetic, but for integer valence of M such a spin dimer would simply condense into a non-magnetic singlet Darriet et al. (1976, 1983). Mixed-valence systems with both M sites occupied by a magnetic $5d$ ion are possible too Doi and Hinatsu (2004); Sakamoto et al. (2006); Dey et al. (2013, 2014), and seem to be more promising for finding a QSL, because unpaired electrons localized on the dimers appear. Note that such mixed-valence dimers with an unpaired electron delocalized between the two Ir4.5+ ions are very different from more conventional spin dimers formed by two magnetic ions holding one unpaired electron each.

Guided by this idea, we synthesized polycrystalline samples of Ba3InIr2O9. From our detailed study using neutron diffraction, magnetization and specific heat measurements, muon spin relaxation ($\mu$SR), and nuclear magnetic resonance (NMR) we establish a gapless and, potentially, spin-liquid ground state in Ba3InIr2O9.

II Experimental Details

Polycrystalline samples of Ba3InIr2O9 were prepared by a conventional solid-state reaction method  Sakamoto et al. (2006). Stoichiometric amounts of high-purity BaCO3, In2O3, and Ir metal powder were mixed thoroughly, pressed into pellets, and calcined at $900$ *∘*C for $12$ h. Further, the pellet was crushed into powder, mixed well, pelletized, and fired at $1300$ *∘*C for $4$ days with several intermediate grindings. Neutron diffraction data were collected at the high-resolution instrument SPODI  Hoelzel et al. (2012) at FRM-II (TU Munich) using the wavelength of 1.55 Å. Jana2006 software  Petr̆íc̆ek et al. (2014) was used for structure refinement.

Magnetization measurements were carried out in a Quantum Design $5$ T SQUID magnetometer and in a Quantum Design PPMS $14$ T equipped with the vibrating sample magnetometer in the temperature range $2-350$ K. Additional high-temperature data extending up to 650 K were collected in the SQUID magnetometer using powder sample enclosed in a thin-walled quartz tube.

Heat-capacity measurements in the temperature range $0.4-200$ K were performed in a Quantum Design PPMS using the 3He insert. Low-temperature measurements in the $0.08-1$ K range were performed using a quasi-adiabatic heat pulse method, adapted to a dilution refrigerator.

$\mu$SR experiments were done on two different spectrometers at the Paul Scherrer Institute (Switzerland), LTF for temperatures from 20 mK up to 750 mK and Dolly for temperatures from 250 mK up to 200 K. For the Dolly experiment, about 300 mg of the polycrystalline sample was mounted on a thin copper plate inside the 3He cryostat. In order to ensure good thermal contact, we glued the sample with GE varnish. We used the Veto mode, which allowed to get rid of the background signal from the sample holder. Therefore, the acquired signal is due to muons that stopped inside the sample. For the LTF experiment we used the same sample, again glued with GE on a silver plate. By comparing the results obtained between 250 mK and 750 mK on both the spectrometers, we were able to get rid of the experimental background on LTF.

115In nuclear magnetic resonance (NMR) experiments were carried out with our home-built spectrometer with the dilution-fridge insert. The measurements are performed in the field-sweep mode at a fixed frequency of $70$ MHz down to $24$ mK. The spin echo intensity was obtained by integrating over the spin echo in the time domain. The final spectrum is constructed by plotting the spin-echo intensity as a function of the applied field.

III Results

III.1 Crystal structure

Rietveld refinement of room-temperature neutron diffraction data (Fig. 2) confirms hexagonal crystal structure ($P6_{3}/mmc$). However, at 3.4 K peak splitting of the 203 and 204 reflections, as well as a visible broadening of other peaks, indicated that the symmetry is reduced to monoclinic. The 3.4 K data were refined in the $C2/c$ space group similar to other hexagonal perovskites Doi and Hinatsu (2004); Sakamoto et al. (2006); Ling et al. (2010). Both hexagonal and monoclinic structures feature a single crystallographic position of Ir (Tables 1 and 2) suggesting the true intermediate-valence Ir4.5+ state. This is different from, e.g., Ba5AlIr2O11, where two sites of the dimer belong to two different crystallographic positions, thus making possible charge re-distribution within the dimer Terzic et al. (2015).

In contrast to Ba3IrTi2O9 with its $35-40$% site mixing Dey et al. (2012); Lee et al. (2017), our mixed-valence Ba3InIr2O9 shows high degree of structural order. We were able to obtain reasonable atomic displacement parameters in the fully ordered models of both hexagonal and monoclinic structures (Tables 1 and 2). On the other hand, deliberate admixing of In into the Ir position and vice versa leads to a marginal reduction in the refinement residuals and 2.8(5)% site mixing 111We introduced the site mixing and constrained the overall composition to Ba3InIr2O9. This leads to $R_{I}=0.038$ and $R_{p}=0.050$ compared to $R_{I}=0.039$ and $R_{p}=0.051$ for the fully ordered model (room-temperature data).. We believe that the diffraction data alone may not distinguish between the fully ordered structure and the weak site mixing scenario. Further studies, such as direct imaging with high-resolution electron microscopy, would be useful to resolve this issue.

The low-temperature monoclinic distortion is primarily related to the tilting of the IrO6 and InO6 octahedra. It has nearly no effect on relative positions of the Ir atoms. For example, the Ir–Ir distance within the dimer shrinks from 2.637(2) Å at 300 K to 2.599(4) Å at 3.4 K, presumably due to thermal expansion. The Ir–Ir distances between the dimers do not change at all, compare 5.832(1) Å at 300 K to 5.813(8) Å and 5.815(4) Å at 3.4 K for the nearest-neighbor Ir–Ir distances in the $ab$ plane. Given this negligible structural effect and the absence of any signatures in thermodynamic measurements, we conclude that the hexagonal-to-monoclinic phase transition should be an effect of structural (geometrical) origin and bears no relation to the magnetism of Ba3InIr2O9. Determination of the exact transition temperature requires further dedicated diffraction experiments at intermediate temperatures and lies beyond the scope of our present study.

III.2 Magnetic susceptibility

Temperature dependence of the magnetic susceptibility ($\chi=M/H$) measured in various applied fields is shown in the inset (i) of Fig. 3. Down to 2 K, we did not observe any anomaly or divergence of field-cooled and zero-field-cooled data, suggesting the absence of long-range ordering and spin freezing.

The susceptibility follows the Curie-Weiss (CW) behavior between 10 and 70 K. At higher temperatures (see Fig. 3), the data deviate from the CW law due to thermal changes in the mixed-valence Ir2O9 dimers that can adopt different electronic configurations  Streltsov and Khomskii (2016). A tentative van Vleck fit with the three-level model

[TABLE]

where $N_{A}$ is Avogadro’s number, $g_{0}$ is electronic $g$-factor for the ground state, and $g_{1}$ and $g_{2}$ are $g$-factors for the excited states, which are separated from the ground state by energy gaps of $\Delta_{1}$ and $\Delta_{2}$, respectively, yields decent description of the magnetic susceptibility up to at least 650 K. This fitting function can be understood as follows. In the absence of electronic correlations and spin-orbit coupling, 9 electrons occupying 6 $t_{2g}$ orbitals of two Ir atoms give rise to $S=\frac{1}{2}$ and $S=\frac{3}{2}$ states depending on the filling of the molecular orbitals of the dimer Ziat et al. (2017). The effect of spin-orbit coupling is taken into account by introducing electronic $g$-factors as fitting parameters. Additionally, we assumed that the $S=\frac{3}{2}$ state splits into two, because fitting with one excited state was not successful, whereas two distinct excited states provide a good description of the susceptibility in the broad temperature range.

Fitting the susceptibility with Eq. (1) yields $\chi_{0}=-6.8\times 10^{-5}$ cm3/mol, $\theta=-6.8$ K, $\Delta_{1}=107$ K, $\Delta_{2}=472$ K, $g_{0}=0.872$, $g_{1}=0.678$, and $g_{2}=1.810$. The relatively low value of $\Delta_{1}$ explains the deviation from the Curie-Weiss behavior already above 70 K. The obtained $\Delta_{1}$ and $\Delta_{2}$ are of the same order of magnitude as in the mixed-valence ruthenates isostructural to Ba3InIr2O9 Ziat et al. (2017). The paramagnetic effective moments are $\mu_{\rm eff}=0.76$ $\mu_{B}$ in the ground state, $\mu_{\rm eff,1}=1.31$ $\mu_{B}$ in the first excited state, and $\mu_{\rm eff,2}=3.51$ $\mu_{B}$ in the second excited state.

For the rest of this paper, we focus on the low-temperature regime below 70 K with the paramagnetic effective moment of $\mu_{\rm eff}=0.76$ $\mu_{B}$/dimer, which is comparable to the values reported for mixed-valence iridates earlier Miiller et al. (2012); Terzic et al. (2015). Antiferromagnetic couplings between magnetic moments localized on the mixed-valent dimers are confirmed by the negative Curie-Weiss temperature $\theta_{\chi}=-7$ K. Isothermal magnetization curve at 2 K (see inset (ii) of Fig. 3) does not show any sign of saturation up to 14 T. This weak sensitivity to the field may be due to the strongly reduced $g$-factor (only 44% of its spin-only value) that lessens the effect of the external field.

III.3 Specific heat

First insight into the low-temperature magnetism is obtained from the specific-heat data. A broad peak in $C_{p}/T$ is observed around 1.6 K (Fig. 4a) indicative of a crossover between the paramagnetic (thermally disordered) and spin-liquid (quantum disordered) regimes Li et al. (2015a, 2016), further confirmed by the increase in the muon zero-field relaxation rate around the same temperature (Sec. III.4). The peak shifts toward lower temperatures in the applied field, although the changes are relatively small even at 14 T, and no transition anomaly is observed.

Heat capacity was further measured at temperatures well below the broad maximum. For an insulating material, total heat capacity is a sum of the magnetic, lattice, and nuclear contributions, $C_{p}=C_{m}+C_{\rm lat}+C_{\rm nuc}$. To extract the magnetic specific heat of the material, we need to subtract the lattice part and nuclear part from the total specific heat $C_{p}$. In the absence of a suitable non-magnetic analog, we fitted the measured specific heat with $C_{p}=\beta T^{3}$ in the range $14-20$ K sup . The fitting of the lattice part yields $\beta=1.45$ mJ mol*-1* K*-4* and the Debye temperature $\Theta_{D}=272$ K. The fitted curve is extrapolated to low temperatures and taken as the lattice part ($C_{\rm lat}$) that was subtracted from the experimental data.

At low temperatures, the nuclear contribution becomes prominent. To extract the magnetic heat capacity, we adopted the following procedure:

(1) We fitted the $C_{p}$ data for each field ($B$) from lowest $T$ up to $300-400$ mK with $C_{p}=\alpha/T^{2}+C_{0}\,T^{\gamma}$ sup , where $\alpha/T^{2}$ stands for $C_{\rm nuc}$, and power-law behavior of $C_{\rm mag}$ is assumed.

(2) For each field, the magnetic heat capacity $C_{m}$ is obtained as $C_{m}=C_{p}-C_{\rm nuc}$, whereas $C_{\rm lat}$ is negligible in this temperature range.

(3) Reliability of the $\alpha$ values is verified by plotting the field dependence $\alpha(B^{2})$, which was linear, as expected sup .

After subtracting the nuclear contribution sup , we arrive at the robust power-law behavior $C_{m}=C_{0}T^{\gamma}$ with $C_{0}=832$ mJ mol*-1* K*-2.83* and $\gamma=1.83$ (Fig. 4b). This behavior gives first indication of a gapless ground state, because otherwise low-energy excitations over a spin gap would give rise to the exponential decay of $C_{m}$ at low temperatures. The power-law behavior persists up to at least 14 T, but above 9 T the exponent $\gamma$ is slightly reduced sup .

By integrating $C_{m}/T$ in zero field (see Fig. 4c) within the temperature range from 0.08 to 20 K, we estimated magnetic entropy change of 3.4 J mol*-1* K*-1* (see Fig. 4d), which is about $60\%$ of the entropy expected for spin-$\frac{1}{2}$. Part of the magnetic entropy should be then released at higher temperatures, where the magnetic contribution is concealed behind a much larger lattice term. We note here that a two-step release of the magnetic entropy (and, consequently, two well-separated peaks of the magnetic specific heat $C_{m}$) are not uncommon in frustrated magnets Sindzingre et al. (2000); Nasu et al. (2015).

III.4 $\mu$SR

Whereas thermodynamic measurements provide first hints towards the absence of long-range magnetic order in Ba3InIr2O9, experimental evidence for the QSL formation is not complete without a local probe. To this end, we use muon spin relaxation ($\mu$SR), which is a very sensitive technique to detect static local fields arising from weak long-range order or spin freezing.

The relaxation curves of the muon polarization in zero field (ZF) are shown in Fig. 5. The absence of oscillations in the zero-field (ZF) signal and the lack of the polarization recovery to $1/3$ indicate the absence of any frozen moments in Ba3InIr2O9 in the temperature range from $200$ K down to $20$ mK. The polarization curves can be fitted with Eq. 2 which is a combination of the depolarization due to the muon coupled to the In nuclear magnetism (Kubo-Toyabe Gaussian function) and depolarization due to the electronic magnetism evolving with temperature.

[TABLE]

Here, $f=0.786(2)$ is the fraction of muons coupled to the electronic magnetism, $\sigma=0.094(2)$ $\mu$s*-1* is the nuclear depolarization rate, $\beta=1.308(7)$ is the stretched exponent, and $\lambda$ is the electronic depolarization rate. The stretched exponent $\beta$ is independent of field and temperature. The small deviation from unity could be due to a small distribution of the muon sites close to the In ions.

The ZF $\mu^{+}$ relaxation rate ($\lambda_{\rm ZF}$) obtained from fitting the ZF muon depolarization curves is shown as function of temperature in Fig. 6b. At high temperatures ($20$ K to $3$ K), $\lambda_{\rm ZF}$ remains constant at $\sim 0.12\thinspace\mu$s*-1*, which is consistent with the paramagnetic fluctuations of Ir moments according to the Bloembergen Purcell, and Pound theory Bloembergen et al. (1948).

From $3$ K down to $1$ K, $\lambda_{\rm ZF}$ increases with decreasing temperature. The enhancement of $\lambda_{\rm ZF}$ in a narrow temperature window indicates a slowing down of Ir spin fluctuations due to the development of strong short-range correlations, a common feature seen in other QSL candidates Fåk et al. (2012); Clark et al. (2013); Li et al. (2016). Upon further cooling, $\lambda_{\rm ZF}$ shows temperature-independent plateau-like behavior between $1$ K and $20$ mK. The plateau-like behavior in $\lambda_{\rm ZF}$ vs $T$ (see Fig. 6b) has also been observed in several other QSL compounds.

To verify if the plateau-like behavior is originating from the muons directly coupled to the frustrated spins Colman et al. (2011) or the muons coupled to defects Kermarrec et al. (2011); Gomils̆ek et al. (2016), we have performed $\mu$SR experiments under transverse field (TF) of 0.4 T and estimated the $\mu$SR line shift ($K^{\mu}$) as a function of temperature  Orain et al. (2014) as shown in Fig. 8a. $K^{\mu}$ increases with decreasing temperature and saturates below $3$ K, from where $\lambda$ (see Fig. 6b) starts increasing. This indicates that antiferromagnetic (AFM) spin fluctuations are dominating at low temperature in Ba3InIr2O9. The similarity between the temperature dependence of $K^{\mu}$ and NMR shift ($K_{\rm NMR}$) (see Fig. 8a) proves that the muons are directly coupled to the Ir moments.

In order to identify the origin of the depolarization rate being static or dynamic, we performed decoupling experiments at $250$ mK (shown in Fig. 6a) applying longitudinal fields (LF) from $5$ mT up to $100$ mT. Should the low-temperature plateau in $\lambda_{\rm ZF}$ arise from a static field, the size of this field can be estimated as $B_{\rm loc}=\lambda/\gamma_{\mu}\approx 1.4$ mT, where $\gamma_{\mu}=135.5\times 2\pi$ s*-1*$\mu$T*-1* is the gyromagnetic ratio for muons. The static field can then be decoupled by applying an external magnetic field $\simeq 14$ mT, which is one order of magnitude higher than $B_{\rm loc}$. The polarization curve under a field of $100$ mT does not show the signs of a full polarization, suggesting that the plateau does not originate from a static internal field and the spins are dynamic in nature even at the lowest temperature of $20$ mK, as expected in a QSL. The plateau-like behavior of $\lambda_{\rm ZF}$ toward lowest temperature indicates much slower spin dynamics of the material compared to the $\mu$SR time window. It is another characteristics of QSL candidates Fåk et al. (2012); Clark et al. (2013); Li et al. (2016).

III.5 NMR

Nuclear magnetic resonance (NMR) is a concurrent probe for local magnetic fields and spin dynamics. 115In NMR spectra measured at $70$ MHz in the temperature range $0.024-138$ K are shown in Fig. 7(a). At high temperatures (for example, at $138$ K), the spectra exhibit one single isotropic line which indicates the unique crystallographic site for indium, consistent with our structural analysis. The spectra can be described satisfactorily assuming the $I=9/2$ nuclei with the $380$ kHz quadrupolar coupling constant ($\nu_{Q}$).

Below about $100$ K, we observed a second line, which remained unshifted within the entire temperature range. This second line presumably originates from trace amounts (0.9 %) of a non-magnetic impurity phase present in the sample. For estimating the impurity contribution in the spectral line shape, we measured the spectra at several temperatures with $\tau=50$ $\mu$s and $\tau=200$ $\mu$s, where $\tau$ is the pulse separation between the spin-echo sequence $\pi/2-\tau-\pi$. The intrinsic line has much shorter spin-lattice relaxation time $T_{1}$ as well as the spin-spin relaxation time $T_{2}$ compared to the impurity line sup . Hence it is expected that with longer $\tau$ the intrinsic part of the spectra is already relaxed, whereas the impurity contribution sustains. The comparison of the spectra corresponding to different $\tau$ measured at $T=25$ K is shown in Fig. 7c.

The spectra are described at all temperatures below 100 K with two Gaussian lines, one for magnetic (intrinsic) and the other one for a non-magnetic (impurity) contribution as shown in Fig. 7(a)). At low temperatures, the two lines (intrinsic line and the impurity line) merge together. For estimating the Knight shift of the intrinsic line, we have used the peak of the intrinsic contribution marked with the green arrows in Fig. 7a. This peak shifts towards higher fields with decreasing temperature yielding the local spin susceptibility $K$. In general,

[TABLE]

where $K_{0}$ is the temperature-independent part of the line shift $K$, and $N_{A}$ is the Avogadro’s number. The hyperfine coupling constant is estimated at $A_{\rm hf}=-1.675$ T/$\mu_{B}$ sup . The temperature dependence of the 115In line shift ($K$) is shown in Fig. 8a. A continuous increase in the line shift $K(T)$ with decreasing temperature from $100$ K down to $\sim 4$ K indicates the development of spin correlations. The inverse of the temperature-dependent part of the line shift $1/(K-K_{0})$ is shown as a function of temperature in the inset of Fig. 8a. A CW fit of the data yields the CW temperature $\theta_{\rm NMR}=-14$ K. This value should be more reliable than $\theta_{\chi}=-7$ K estimated from the bulk susceptibility, since the NMR line shift does not contain any impurity contributions.

At low temperature the total spectra remain unchanged as shown in Fig  7b. This indicates that both the line shift $K$ and line width $\Delta H$ of the spectra remain constant down to the base temperature of $25$ mK. Saturation of both these quantities suggests saturation of spin correlations below $\sim 4$ K. The finite and temperature-independent value of $K$ below 4 K gives strong evidence for the gapless behavior. From $\Delta H$, we estimated the saturated magnetic moment $\mu_{s}=0.89$ $\mu_{B}$/f.u. close to 1 $\mu_{B}$ expected for one unpaired electron per the Ir2O9 dimer.

To probe the low-energy spin dynamics, we have studied the NMR spin-lattice relaxation rate ($1/T_{1}$) at various temperatures. ($1/T_{1}$) was measured at the peak position of the intrinsic line using long pulses to avoid any interference from the impurity line. $1/T_{1}$ at different temperatures are obtained by fitting (shown in the inset of Fig. 8b) the longitudinal nuclear magnetization recovery curves with the equation

[TABLE]

where $1/T_{1}=2W$. Here, $M$ is the nuclear magnetization, $\beta$ is a stretched exponent, which accounts for a distribution of the $T_{1}$ values due to disorder, and $C$ is a pre-factor.

In general,

[TABLE]

where $\chi^{\prime\prime}(q,\omega)$ is the imaginary part of the dynamical spin susceptibility. The temperature dependence of $1/T_{1}$ is shown in Fig. 8b. Between $30$ K and $4$ K, $1/T_{1}$ is almost temperature-independent, indicating the paramagnetic behavior. A weak hump-like feature is seen around $1.6$ K, which coincides with the hump in the $C_{p}/T$ data (Fig. 4a) and with the rapid increase in $\lambda_{\rm ZF}$ (Fig. 6b). No long-range order occurs in Ba3InIr2O9. Therefore, this hump is not a broadened signature of a magnetic transition. Instead, it may indicate some intrinsic physics, as recently proposed for the honeycomb iridates Yoshitake et al. (2016). The applicability of this scenario to Ba3InIr2O9 may be an interesting venue for future research.

Below 0.3 K, ($1/T_{1}$) follows the $\sim T^{2.2}$ power law similar to the $\sim T^{1.83}$ behavior of $C_{m}(T)$.

IV Discussion and Summary

Our results establish gapless ground state and persistent spin dynamics down to at least 20 mK, rendering Ba3InIr2O9 a potential QSL material. However, its low-temperature behavior is rather different from theoretical expectations for a gapless QSL, where magnetic excitations are usually described in terms of a spinon Fermi surface and should give rise to the linear behavior of both NMR spin-lattice relaxation rate and magnetic specific heat Lawler et al. (2008); Yamashita et al. (2008, 2011) at low temperatures. Our data for Ba3InIr2O9 clearly deviate from this scenario.

Microscopically, we expect two superexchange pathways. The mixed-valence Ir2O9 dimers may interact in the $ab$ plane via the coupling $J_{T}$ (the Ir–Ir distance of 5.83 Å) and along the $c$ direction via the coupling $J_{H}$ (the Ir–Ir distance of 5.71 Å), see Fig. 1. Note that we consider Ir–Ir distances, because magnetic electrons are expected to occupy a molecular orbital of the dimer with equal contributions of both Ir sites Streltsov and Khomskii (2016). While the coupling $J_{T}$ forms a triangular lattice in the $ab$ plane, as already anticipated in Ba3IrTi2O9 Dey et al. (2012); Kumar et al. (2016); Lee et al. (2017); Becker et al. (2015); Catuneanu et al. (2015), the coupling $J_{H}$ leads to a buckled honeycomb geometry. Depending on the ratio between $J_{T}$ and $J_{H}$, the system can interpolate between the purely two-dimensional regime ($|J_{T}|\gg|J_{H}|$, or vice versa) and a 3D behavior when both $|J_{H}|$ and $|J_{T}|$ are of the same size.

On the phenomenological level, quadratic behavior of the zero-field specific heat is expected in an algebraic spin liquid on the kagome lattice Ran et al. (2007); Hermele et al. (2008). Another manifestation of the $T^{2}$ behavior was proposed for a spin model combining the triangular and honeycomb geometries Chen et al. (2012), as in Ba3InIr2O9, but its applicability to our system is obscured by the large single-ion anisotropy term that is central to the model but seems unlikely in iridates Winter et al. (2016). Intriguingly, the robust quadratic behavior of the zero-field specific heat has been reported in frustrated magnets with slow dynamics Ramirez et al. (1992); Nakatsuji et al. (2005) and interpreted in the framework of Halperin-Saslow modes for two-dimensional spin glasses Sachdev (1992); Podolsky and Kim (2009). It was also observed in Li2RhO3, the Kitaev antiferromagnet, where spins freeze below 6 K Khuntia et al. (2017). To what extent these scenarios apply to Ba3InIr2O9 with its clear signatures of persistent spin dynamics should be a subject of future investigations.

Mixed-valence dimers are also reported in ruthenates with the hexagonal perovskite structure. However, these compounds show long-range magnetic order or disordered static magnetism at low temperatures Ziat et al. (2017). This is different from our case, where persistent spin dynamics is observed. From magnetism point of view, mixed-valence hexagonal perovskites remain a largely uncharted territory. The nature of local moments within the mixed-valence dimers requires further investigation, and peculiarities of magnetic interactions between the dimers remain to be explored. Experiments reveal instances of non-trivial magnetic states, including the potential QSL state in Ba3InIr2O9, and call for a better understanding from theoretical and microscopic perspectives.

Altogether, we demonstrated that Ba3InIr2O9 is a structurally well-ordered QSL candidate showing the $\sim T^{1.83}$ behavior of zero-field specific heat and the $\sim T^{2.2}$ power-law for the spin-lattice relaxation rate at low temperatures. Not only would it be interesting for further experimental (probe of the spin excitations) and theoretical (microscopic analysis of the magnetic model) research, it also puts forward mixed-valence iridates as a promising playground for finding new QSL materials. The unique combination of the triangular and buckled honeycomb geometries bears strong connections to current theoretical models of frustrated magnetism Kitaev (2006); Rousochatzakis et al. (2015); Jackeli and Avella (2015); Becker et al. (2015); Li et al. (2015b); Rousochatzakis et al. (2016), and may lead to novel manifestations of the QSL physics when different lattice geometries concur.

Acknowledgements.

We gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG) within the collaborative research center TRR 80 (Augsburg/Munich). TD, MM, and AT were funded by the Federal Ministry for Education and Research through the Sofja Kovalevskaya Award of Alexander von Humboldt Foundation. FB acknowledges financial support from the project SOCRATE (ANR-15-CE30-0009-01) of the ANR French agency. PK acknowledges support from the European Commission through Marie Curie International Incoming Fellowship (PIIF-GA-2013-627322). Part of the work was performed at the LTF and DOLLY spectrometers of the Swiss Muon Source (Paul Scherrer Institute, Villigen, Switzerland). We thank C. Baines for his support on LTF, and Dana Vieweg for her help with high-temperature susceptibility measurements.