# Disordered magnetic ground state in a quasi-1-D d 4 columnar iridate Sr3LiIrO6

Abhisek Bandyopadhyay, Debu Das, A Chakraborty, S Bhowal, Vinod Kumar, G B G Stenning, C Ritter, D T Adroja, M Moretti Sala, A Efimenko, et al.

► To cite this version:

Abhisek Bandyopadhyay, Debu Das, A Chakraborty, S Bhowal, Vinod Kumar, et al. Disordered magnetic ground state in a quasi-1-D d 4 columnar iridate Sr3LiIrO6. Journal of Physics: Condensed Matter, 2024, 36 (42), pp.425804. 10.1088/1361-648x/ad63eb. hal-04815049

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# Disordered magnetic ground state in a quasi-1-D $d^4$ columnar iridate $Sr_3LilrO_6$

To cite this article: Abhisek Bandyopadhyay et al 2024 *J. Phys.: Condens. Matter* **36** 425804

View the [article online](#) for updates and enhancements.

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# Disordered magnetic ground state in a quasi-1-D $d^4$ columnar iridate Sr$_3$LiIrO$_6$

Abhisek Bandyopadhyay<fnref n="1" />,<fnref n="2" />,* ID, Debu Das<fnref n="2" />, A Chakraborty<fnref n="3" />,<fnref n="4" />, S Bhowal<fnref n="3" />,<fnref n="5" />,<fnref n="14" />, Vinod Kumar<fnref n="6" />, G B G Stenning<fnref n="1" />, C Ritter<fnref n="7" /> ID, D T Adroja<fnref n="1" /> ID, M Moretti Sala<fnref n="8" />, A Efimenko<fnref n="9" />, C Meneghini<fnref n="10" /> ID, F Bert<fnref n="11" />, P K Biswas<fnref n="12" /> ID, I Dasgupta<fnref n="3" /> ID, T Saha Dasgupta<fnref n="13" />, A V Mahajan<fnref n="6" /> and Sugata Ray<fnref n="2" /> ID

E-mail: abhisek.ban2011@gmail.com and abhisek.bandyopadhyay@stfc.ac.uk

Received 3 March 2024, revised 27 June 2024

Accepted for publication 16 July 2024

Published 25 July 2024

<figure><img src="image_2.png" /></figure>

CrossMark

## Abstract

Spin-orbit coupling offers a large variety of novel and extraordinary magnetic and electronic properties in otherwise ‘ordinary pool’ of heavy ion oxides. Here we present a detailed study on an apparently isolated hexagonal 2H spin-chain $d^4$ iridate Sr$_3$LiIrO$_6$ with geometric frustration. Our structural studies reveal Li-Ir chemical order with desired stoichiometry in this compound, while x-ray absorption together with x-ray photoemission spectroscopic characterizations establish pure 5+ valence of Ir. We have established a magnetic ground state with finite Ir$^{5+}$ magnetic moments in this compound, contrary to the anticipated nonmagnetic $J_{eff} = 0$ state, through combined dc susceptibility, $^7$Li nuclear magnetic resonance (NMR), muon spin relaxation ($\mu$SR) and *ab-initio* electronic structure studies. These investigations together with ac magnetic susceptibility and specific heat measurements reveal that despite having noticeable antiferromagnetic correlation among the Ir$^{5+}$ local moments, this system does not magnetically

<figure><img src="image_3.png" /></figure>

<footnote n="1">¹ ISIS Neutron and Muon Source, STFC, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, United Kingdom</footnote>

<footnote n="2">² School of Materials Sciences, Indian Association for the Cultivation of Science, 2A & 2B Raja S. C. Mullick Road, Jadavpur, Kolkata 700 032, India</footnote>

<footnote n="3">³ School of Physical Sciences, Indian Association for the Cultivation of Science, 2A & 2B Raja S. C. Mullick Road, Jadavpur, Kolkata 700 032, India</footnote>

<footnote n="4">⁴ Department of Physics, Indian Institute of Technology, Kanpur 208016, India</footnote>

<footnote n="5">⁵ Materials Theory, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Zurich, Switzerland</footnote>

<footnote n="6">⁶ Department of Physics, Indian Institute Of Technology Bombay, Powai, Mumbai 400076, India</footnote>

<footnote n="7">⁷ Institut Laue-Langevin, 71 Avenue des Martyrs, CS 20156, 38042 Grenoble Cedex 9, France</footnote>

<footnote n="8">⁸ Dipartimento di Fisica, Politecnico di Milano, P.zza Leonardo da Vinci 32, I-20133 Milano, Italy</footnote>

<footnote n="9">⁹ ESRF–The European Synchrotron, 71 Avenue des Martyrs, 38000 Grenoble, France</footnote>

<footnote n="10">¹⁰ Dipartimento di Scienze, Università Roma Tre, Via della Vasca Navale, 84 I-00146 Roma, Italy</footnote>

<footnote n="11">¹¹ Laboratoire de Physique des Solides, Université Paris-Saclay, CNRS, 91405 Orsay, France</footnote>

<footnote n="12">¹² ISIS Facility, Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX110QX, United Kingdom</footnote>

<footnote n="13">¹³ Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for Basic Sciences, Block JD, Sector 3, Saltlake, Kolkata 700106, India</footnote>

<footnote n="14">¹⁴ Present address: Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India</footnote>

<footnote>* Author to whom any correspondence should be addressed.</footnote>
order down to at least 0.05 K, possibly due to geometrical exchange frustration, arising from the comparable nearest- and next-nearest-neighbor interchain Ir–O–O–Ir superexchange interaction strengths with opposite signs. However, the zero-field µSR analysis shows emergence of a considerable proportion of spin-freezing on top of a spin-fluctuating dynamic magnetic background down to the lowest measured temperature of 1.7 K, possibly due to some inhomogeneity and/or the much stronger intra-column Ir–Ir magnetic exchange interaction strength relative to the inter-column Ir–Ir ones. The linear temperature dependence of the magnetic specific heat ($C_m$) in both zero and applied magnetic fields, plus the power-law behavior of the NMR spin-lattice relaxation rate suggest a gapless spinon density of states in this charge gapped disordered magnetic ground state of $Sr_3LiIrO_6$.

Supplementary material for this article is available online

Keywords: 5d iridates, geometric frustration, spin-orbit coupling, disordered magnetic state

# 1. Introduction

Magnetic frustration disrupts conventional long-range magnetic order which can establish several exciting highly entangled ground states [1–5]. Materials supporting strongly quantum-entangled states are potential for data storage and memory applications, and can be important in the context of quantum computation.

In this context, it is of interest to explore the enhanced quantum fluctuations in the geometrically frustrated heavier 5d Ir-oxides. The strong spin-orbit entanglement of the Ir 5d valence electrons together with large crystal field promotes the formation of quantum singlets, thus softening conventional magnetic order and hence diverse interesting quantum disordered magnetic ground states could be realized. Here another most discussed prevailing controversy in the context of pentavalent $d^4$ (Ir$^{5+}$:5$d^4$) iridates is the experimental deviation from the strongly spin-orbit coupled ideal nonmagnetic $J_{eff} = 0$ state, and the concomitant development of finite Ir-magnetic moments [6–11]. Furthermore, these spontaneously generated finite Ir$^{5+}$ moments are susceptible to quantum fluctuations due to geometrical exchange frustration arising in the given geometrically frustrated magnetic lattice [8–10, 12–14].

In such a backdrop, the 2H-prototype $A_3A'MO_6$ ($A = \text{Sr Ca}; A'$, any transition metal or nonmagnetic cation; $M = \text{transition metal}$) family of hexagonal oxides appear interesting because of their unconventional magnetic properties due to the interplay between low dimensionality, magnetic frustration, and magnetocrystalline anisotropy [15]. More recently the exploration of this family of compounds containing heavier 4d/5d transition metal ions has been extended with the aim of maximizing the spin-orbit coupling (SOC) effect and studying the resulting novel ground state magnetic behaviors in the strong SOC regime [15–19]. Additionally, due to spatially extended 4d/5d electronic orbitals, the extended superexchange interactions via multiple oxygen ions between the neighboring spin-chains effectively influence their respective magnetic ground states [20–23]. It would therefore be an interesting proposition to check if, by introducing Ir$^{5+}$ at the M-site along with a nonmagnetic monovalent alkali metal cation at the A'-site, the

effective Ir SOC could be maximized through increased isolation of Ir atoms in the structure, which would allow testing the atomic-like strong SOC driven nonmagnetic $J = 0$ ground state or alternately, if there is emergence of spin-orbit coupled unconventional Ir$^{5+}$-magnetism in the limit of low dimensionality and geometrical frustration, paving pathways towards the highly entangled quantum magnetic phases in such an apparently reduced dimensional structure.

Here we present a detailed structural, chemical, electronic, magnetic and thermodynamic characterizations of a $d^4$ columnar spin-chain iridate $Sr_3LiIrO_6$ (SLIO). By means of systematic structural and chemical investigations we infer that this system is largely free from structural disorder between the Li- and Ir-sites as well as the Sr- and Li-sites, and also there is nearly no trace of detectable cation-nonstoichiometry and/or oxygen-deficiency in this material. The combined x-ray absorption (XAS) and x-ray photoemission spectroscopy (XPS) studies confirm pure single 5+ oxidation state of Ir. The bulk dc magnetization study together with spin-polarized electronic structure calculations suggest a noticeable effective paramagnetic moment ($\sim 0.45 \mu_B$) on individual Ir$^{5+}$ ion, thus refuting the strongly spin-orbit coupled nonmagnetic $J_{eff} = 0$ ground state in SLIO. Further, the $^7$Li NMR spectroscopy together with the low-T bulk dc susceptibility, the 2 K M vs H isotherm and the ‘two-level Schottky anomaly’ analysis of the 0.05–4 K magnetic heat capacity ($C_m$) data reveal the presence of $\sim 1\%–2\%$ Ir$^{4+}$ defect centers on top of the intrinsic local Ir$^{5+}$ moments in this material. Finally, our combined dc, ac magnetic susceptibilities ($\chi_{dc}$ and $\chi'_{ac}$, respectively), NMR µSR, heat capacity ($C_p$), and theoretical studies unveil that despite having significant antiferromagnetic (AFM) interaction ($\Theta_{CW} \approx -49$ K) between the sizeable Ir$^{5+}$ local moments, the SLIO possesses neither any magnetic long-range ordering nor a spin-glass frozen magnetic ground state down to 0.05 K at least. However, our zero-field (ZF) µSR characterization reveals development of two-component muon depolarization in the temperature range of 1.7–200 K with slow- and fast-relaxing components corresponding to distinct proportions of dynamic and frozen magnetism, which might be coexisting in this material. The persistent spin dynamics could be ascribed
to the geometric exchange frustration in the edge-shared Ir-
triangular network (figure 6(e)) as a result of the comparable
interchain Ir-Ir magnetic exchange interactions. On the other
hand, the much stronger intrachain Ir-Ir magnetic interaction
acts against the geometric frustration, and probably favoring
partial spin-freezing alongside the continually fluctuating Ir<sup>5+</sup>
moments in this system. Furthermore, the magnetic specific
heat, C<sub>m</sub>, data in the 0.05-4 K range display a combined &gamma;T +
&alpha;T<sup>2</sup> form, which together with the power-law behavior of <sup>7</sup>Li
NMR spin-lattice relaxation rate (below 80 K) with T, point
towards the gapless nature of spin excitations from a metal-
like spinon Fermi surface of the disordered magnetic ground
state of SLIO. Lastly, considering our earlier report on colum-
nar d<sup>4</sup> iridate, Sr<sub>3</sub>NaIrO<sub>6</sub> (SNIO) [12], and also by inspecting
the bandwidth vs. moment trend for three different materials of
the d<sup>4</sup> columnar iridate family and a 6H-hexagonal perovskite
iridate Ba<sub>3</sub>ZnIr<sub>2</sub>O<sub>9</sub> (BZIO) we conclude that only a moder-
ate SOC, far from atomic j &minus; j coupling regime, is present in
SLIO too.

**2. Methodology**

**2.1. *Experimental techniques***

Polycrystalline SLIO sample was prepared using standard
solid state reaction route. The high purity (>99.9%) SrCO3,
Li2CO3 and IrO2 powders, as starting precursors, were stoichiometrically weighted and homogeneously mixed in an
agate mortar. The mixture was initially calcined at 600 °C in
air for 10 hours, and then at several higher temperatures in
steps of 100 °C up to 800 °C (12 hours dwell time at each
of these set temperatures) in air. The purity of the sample
was checked using the x-ray powder diffraction (XRD) with
the Bruker AXS: D8 Advance x-ray diffractometer with Cu
Kα radiation at room temperature (RT). The collected RT
XRD data were analyzed following the Rietveld technique
within FULLPROF program [24] to get the crystal structure.
The high-resolution neutron powder diffraction (NPD) data
were collected from the phase pure polycrystalline sample
of SLIO at 300 K on D2B, Institut Laue-Langevin (ILL),
Grenoble, France, using a wavelength of λ = 1.594 Å in a
6 mm V-can in an annular geometry. Further, the 4 K and
100 K NPD data have been measured on the high intensity powder diffractometer D20 of ILL as well using the
90° take off angle option giving high resolution and λ =
1.54 Å in a 9 mm V-can in an annular geometry. The elemental composition and chemical homogeneity of this sample
were tested through energy dispersive x-ray (EDX) analysis
using a field emission scanning electron microscope (JEOL,
JSM-7500 F) and a ZEISS Supra 55VP scanning electron
microscope with an Oxford Instruments energy-dispersive x-ray (EDX) spectrometer. Our EDX results are supported by
the inductively coupled plasma-optical emission spectroscopy
(ICP-OES) using a Perkin Elmer Optima 2100 DV instrument. The Ir L3-edge (EIr = 11215 eV) XAS spectra were
collected at the XAFS beamline [25] of the Elettra synchrotron radiation facility, Italy, at RT in standard transmission

geometry. The XAS from pure Ir metal foil placed in the refer-
ence sample holder chamber after the sample, was measured
simultaneously to get precise x-ray beam energy calibration.
Three spectra have been collected, checked for energy calib-
ration and averaged up giving high quality data with low stat-
istical noise less than 1 × 10⁻⁴ (see figure S3 in supplement-
ary information(SI)). The experimental XAS data have been
treated along the standard procedures for background subtrac-
tion, normalization and extraction of structural XAFS signal
[26]. The Data were quantitatively analyzed in the near edge
(XANES) and extended (EXAFS) regions to acquire precise
information about the average valence state and local atomic
structure around Ir. A well-established [27, 28] multi-shell
procedure is used for structural EXAFS data refinement to
obtain information about the intermediate range order and in
particular the Li/Ir anti-site disorder. The *k*-weighted experi-
mental data (χ(*k*)) in the 3–20 Å⁻¹ *k*-range were fitted to the
theoretical EXAFS formula calculated assuming a Gaussian
disorder model, where amplitude and phase functions were
calculated using the program FEFF8 [29]. The main struc-
tural signals were individuated on the basis of the crystal-
lographic structure by assessing their statistical significance
for the refinement (F-test) exploiting the MINUIT statistical
tools [30] implemented in the EXAFS data analysis program
Fitexa [26]. Physical constraints on the fitting parameters were
applied to reduce the correlations and improve the reliability
of the results (see below). The core level and valence band
XPS measurements were carried out using an Omicron elec-
tron spectrometer, equipped with a Scienta Omicron sphera
analyzer and an Al Kα monochromatic source with an energy
resolution of 0.5 eV. Before collecting spectra, the sample
surface was cleaned by in-situ Ar sputtering to negate sur-
face oxidation effect. The temperature dependent electrical
resistivity was measured using standard four-probe technique
in the 100–400 K range in a lab-based resistivity setup. The
temperature dependence of dc magnetic susceptibility was
measured in the 2–400 K range and in magnetic fields up to
±70 kOe using a Quantum Design superconducting quantum
interference device (SQUID) magnetometer. The heat capa-
city (Cₚ) was further investigated as a function of temperature
(2–300 K) in both the zero and applied several magnetic fields
in a physical property measurement system (PPMS, Quantum
Design) at IIT Bombay, India. Moreover, the Cₚ versus *T* as
well as the *T*-dependence of ac magnetic susceptibility χ<sub>ac</sub>(*T*)
were measured down to 0.05 K in a physical property meas-
urement system (Dynacool PPMS, Quantum Design Inc) at
the MCL of ISIS facility, UK. In addition, the local mag-
netic probe muon-spin-rotation/relaxation (μSR) was imple-
mented on this sample (few drops of GE Varnish were spread
all over the powder sample to allow for better thermaliza-
tion) within the EMU spectrometer of the ISIS facility in the
1.7–300 K temperature range in a helium flow cryostat. The
<sup>7</sup>Li nuclear magnetic resonance (NMR) was also studied in
a fixed applied magnetic field of *H* = 93.954 kOe with a
RT bore using a standard pulse NMR technique in the 4–
300 K range using an Oxford continuous flow cryostat. Finally,
the high-resolution low-energy and low-resolution higher
energy resonant inelastic x-ray scattering (RIXS) spectra were
measured at the Ir L₃-edge of the polycrystalline SLIO sample (in powder form), along with the isostructural isovalent columnar iridate SNIO and an 6 H-hexagonal iridate BZIO (both are in powder forms) for comparison, at the ID20 beamline of the European Synchrotron Radiation Facility (ESRF) [31] within exactly the same technical specifications and at the same temperatures as those clearly described in our previous work on the Na-member of this d⁴ columnar iridate Sr₃AlrO₆ (A = Na, Li, K) family of systems [12].

## 2.2. *Theoretical methods*

The electronic structure calculations based on density functional theory (DFT) presented in this paper are carried out in the plane-wave basis within generalized gradient approximation (GGA) [32] of the Perdew-Burke-Ernzerhof exchange correlation supplemented with Hubbard *U* as encoded in the Vienna *ab-initio* simulation package (VASP) [33, 34] with projector augmented wave potentials [35, 36]. The calculations are done with usual value of *U* and Hund's coupling (*J*<sub>H</sub>) chosen for Ir with *U*<sub>eff</sub> (≡ *U* − *J*<sub>H</sub>) = 1.5 eV [12, 37] in the Dudarev scheme [38]. In order to achieve convergence of energy eigenvalues, the kinetic energy cut off of the plane wave basis is chosen to be 550 eV. The Brillouin-Zone integrations are performed with 8 × 8 × 8 Monkhorst grid of *k*-points. The symmetry protected ionic relaxation of the experimentally obtained crystal structure has been carried out within VASP calculation using the conjugate-gradient algorithm until the Hellman-Feynman forces on each atom were less than the tolerance value of 0.01 eV Å<sup>-1</sup>. The optimized structure has been used for calculations.

To get a quantitative estimate of the noncubic crystal distortion mediated Ir-t<sub>2g</sub> crystal field splitting and realistic hopping parameters of our low-energy Hamiltonian, we employed the muffin-tin orbital (MTO) based Nth order MTO (NMTO) method [39, 40], as implemented in Stuttgart code, retaining Ir-t<sub>2g</sub> orbitals within the basis set and down-folding higher degrees of freedom. The NMTO method relies on the self-consistent potentials borrowed from the linear MTO (LMTO) calculations [41]. For the self-consistent LMTO calculations within the atomic sphere approximation (ASA), the space filling in the ASA is obtained by inserting appropriate empty spheres in the interstitial regions.

# 3. Results and discussion

## 3.1. *Structure from x-ray & NPD and composition verification*

Rietveld refinement of the powder x-ray diffraction (PXRD) pattern (figure 1(a)), collected from the SLIO sample at 300 K, confirms pure single phase with rhombohedral R3c space group. Considering the fact that the x-rays are quite weakly sensitive to the accurate determination of the site occupancies and the atomic positions of the lighter cations as well as oxygen atoms in a given crystal structure, we therefore confirmed the precise crystallographic structural details by performing Rietveld refinement on the room-temperature (RT) as well as the 4 K and 100 K NPD data (see figures 1(b) and

(c)). Our NPD refinements also confirm the phase-pure R3c crystal symmetry in the sample, without any unusual change in the refined structural parameters. The RT refined crystal structure (see figure 1(d)) consists of alternately arranged face-sharing LiO₆ trigonal prisms and IrO₆ octahedral units, leading to the infinite one dimensional (1D) chains along the *c*-axis. These chains are separated from each other by nonmagnetic Sr cations. Attempts to include Ir/Li anti-site defects do not statistically improve the refinement, rather attempts to fit the data with the Ir/Li site-disordered model failed to confirm any Ir on the Li-site (see figure S1 and table S1 in the SI), and hence, such a possibility of Ir/Li site-mixing can be convincingly excluded, even more when taking into account the EXAFS results (see below). Further, any cation nonstoichiometry and/or oxygen vacancy could not be convincingly confirmed from the NPD results. Any site-exchange between the Sr (18e) and Li (6a) atoms has also been discarded from the NPD refinement. The structural parameters obtained from the 300 K NPD analysis are indicated in table 1. In summary, our combined XRD & NPD establish a perfectly ordered SLIO sample with desired stoichiometry, in agreement with reference [42]. In addition, the IrO₆ octahedra undergo weak trigonal distortion in terms of the little deviation of O-Ir-O bond angles (~90.8°) from ideal 90° expected in cubic symmetry. It is noteworthy to state at this point that the intrachain Ir-Ir distance (~5.5 Å) is relatively smaller in SLIO compared to that of the SNIO (~5.8 Å) [12], thus indicating possibility of stronger intracolumn Ir-Ir magnetic interactions in SLIO.

The stoichiometry of this sample has further been checked by SEM-EDX, which indicates more or less chemical homogeneity of SLIO. Further, our EDX results showed that the cation stoichiometry is retained nearly at the target composition, i.e. Sr : Li : Ir = 27.76(2):8.61(1):9.42(1) ≡ 3 : 1 : 1 within the given accuracy of the measurement (see the SEM images and the results obtained from elemental mapping of the selected different regions in the supplementary information). However, these data cannot completely rule out the possibility of some oxygen off-stoichiometry or Li-vacancy in the form of Sr<sub>3</sub>LiIrO<sub>6</sub>±δ or Sr<sub>3</sub>Li<sub>1-δ</sub>IrO<sub>6</sub> in this system, because the lighter Li cation has very small x-ray characteristic energy to be accurately probed and the oxygen has low x-ray fluorescence yields, self-absorption, and the issue of detector efficiency. The cation-stoichiometry was also quantified through ICP-OES analysis which again reveals the actual stoichiometry to be **close to** the desired level.

## 3.2. *Ir local atomic structure from Ir L<sub>3</sub>-edge EXAFS analysis*

Analysis of combined XRD and NPD data provides precise information on crystallographic symmetry and the average arrangement of atoms in the unit cell (long-range order, LRO). However, it is well known that specific electronic and/or steric effects between atoms can cause the structure and composition to deviate locally from the global crystallographic structure (short range order, SRO). EXAFS is a specific and chemically selective probe to describe the local order properties around a selected atomic species in the sample, and is particularly relevant to the study of complex systems where
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Rietveld refined room temperature (a) XRD pattern, the Rietveld refined (b) room temperature and (c) 4 K neutron powder diffraction (NPD) patterns for $Sr_3LiIrO_6$. (d) The refined crystal structure showing three different exchange interactions by blue dotted lines/curves.</figcaption></figure>

<table><caption>**Table 1.** Structural information extracted from the Rietveld refinement of the room temperature NPD data of $Sr_3LiIrO_6$ sample. Space group: $R\bar{3}c$, $a = b = 9.6419(10)$ Å, $c = 11.1450(11)$ Å, $\gamma = 120^{\circ}$, $V = 897.29(16)$ Å$^3$; $R_p = 4.23$, $R_{wp} = 5.33$, $R_{exp} = 4.29$, $\chi^2 = 1.55$, $\chi^2_{Bragg} = 1.71$.</caption><thead><tr><th>Atoms</th><th>Site</th><th>Occupancy</th><th>x</th><th>y</th><th>z</th><th>B (Å<sup>2</sup>)</th></tr></thead><tbody><tr><td>Sr</td><td>18e</td><td>1.000</td><td>0.35 819(19)</td><td>0</td><td>0.25</td><td>0.623(30)</td></tr><tr><td>Li</td><td>6a</td><td>1.000</td><td>0</td><td>0</td><td>0.25</td><td>1.649(275)</td></tr><tr><td>Ir</td><td>6b</td><td>1.000</td><td>0</td><td>0</td><td>0</td><td>0.528(35)</td></tr><tr><td>O</td><td>36f</td><td>1.000</td><td>0.17 477(16)</td><td>0.02 155(15)</td><td>0.10 525(11)</td><td>0.712(25)</td></tr></tbody></table>

details of local order can decisively affect the macroscopic properties [27, 28].

In our specific case, anti-site defects can be excluded at the crystallographic scale, but may still be active at the local scale (not long range coherent) and may affect the ground state magnetic responses, [43] as the Li/Ir anti-site disorder would eventually increase the face-sharing Ir-Ir connectivity in the system. Such effects may remain unexplained if only the long range crystallographic structure (LRO) is considered. Therefore we carefully analyzed the Ir L$_3$-edge EXAFS spectra specifically looking for anti-site Ir/Li or Ir/Sr defects. The k$^2$-weighted experimental data and the best-fit curve obtained assuming no antisite defects, are shown in figure 2, together with the contributions we found statistically relevant for the analysis, whose atomic geometry is schematized in the diagram in the figure, where S$_1$ corresponds to the 6 oxygen atoms

bonded at about 2.0 Å from the Ir absorber, S$_2$ and S$_3$ correspond to 2 Li and 6 Sr next neighbors, at about 2.8 Å and 3.3 Å from the absorber, respectively. The data analysis also had to include the multiple scattering contribution from 24 triangular paths Ir–O–O (see diagram), labeled as S$_4$ in the figure. Finally, we included the single and multiple scattering contributions from nearly collinear Ir–O–Sr (∠Ir–O–Sr ≃ 175°, not shown in the figure 2) around 4.4 Å. The signal multiplicities were fixed to the crystallographic values, so we used 13 independent parameters (including the energy shift) to reproduce the main EXAFS features that account for the local Ir atomic structure up to about 4.5 Å. It is noteworthy that the photoelectron scattering amplitude and phase functions of Ir, Sr and Li are quite different, so they can be easily distinguished, allowing high sensitivity to even small chemical changes, especially in the search for the anti-site defects. The
<figure><img src="image_5.png" /><figcaption>**Figure 2.** (Left Panel) $k^2$ weighted Ir L$_3$-edge experimental EXAFS data (open sky blue circles) and respective best fit curve (brown solid line) for Sr$_3$LiIrO$_6$ sample. The partial contributions from different paths used for the analysis are shown, vertically shifted for clarity. (Right Panel) Modulus (|FT|) and imaginary part (Imm(FT)), vertically shifted for clarity, of the $k^2$ weighted Fourier transform of experimental (open sky blue circles) and best fit curve (brown solid line).</figcaption></figure>

fitted structural parameters are shown in table 2, which are in good agreement with the crystallographic structure. Attempts to include a fraction of Ir(Sr)/Li antisite defects was realized by adding an Ir-Ir(Sr) shell at around 2.8 ($S'_2$) with multiplicity $x$ and simultaneously reducing the $S_2$ multiplicity by (1-x), the parameter $x$ being refined together with the distance and mean square relative displacement parameters ($\sigma^2$) for the $S'_2$. The best-fit is obtained for $x = 0$, which demonstrates that, just as for the XRD and NPD, there is no significant antisite disorder effect at short range. In order to set a limit to the fraction of the detectable defects we fixed to $x = 0.03$, while the distance and mean square relative displacement parameters were refined. The reduced $\chi_L^2$ raises respect to the model $x = 0$ either considering Ir-Ir or Ir-Sr defects demonstrating that even a small fraction of antisite defects is not statistically significant, thus further emphasizing our claim of large degree of chemical order in both the local and global levels.

### 3.3. Electronic characterization

Estimation of Ir-oxidation state has the central importance in the context of ground state magnetism of 5d iridates, as Ir$^{4+}$ and Ir$^{6+}$ species are strongly magnetic [44–47], while the Ir$^{5+}$ ion is ideally nonmagnetic $J_{eff} = 0$ [48]. On the other hand, deviation from this ideal state in pentavalent $d^4$ iridates is often attributed to the presence of magnetic Ir$^{4+}$/Ir$^{6+}$ impurities [49]. Looking at the Ir L$_3$-edge XANES features help to precisely assess its average valence state (figure 3(a)). The asymmetric structure of the spectral line, in the form of a weak shoulder in the lower energy side (shown by the solid blue arrow in figure 3(a)), resembles quite well with the higher oxidation state (>4+) of Ir [10, 50, 51]. The corresponding second derivative curve (inset to figure 3(a)), representative of the white-line feature, clearly reveals well-resolved doublet

<table><caption>**Table 2.** Local atomic structure around the average Ir atom as obtained from the multi-shell EXAFS data refinement (see text). The multiplicities of the contributions (N) were fixed to their crystallographic values. The statistical uncertainties on the last digit of the refined parameters are reported in parenthesis, the crystallography distances ($R_{XRD}$) are shown for sake of comparison. The $S_5$ signal accounts for Ir-Sr$_2$ (single scattering, SS) path, and Ir-O-Sr$_2$ 3-legs (3S), and Ir-O-Sr$_2$-O 4-legs (4S) paths. In order to take into account for the not collinear arrangement a free parameter $dr$ is introduced so that $R_{3S} = R_{SS} + dr/2$ and $R_{4S} = R_{SS} + dr$ and $\theta_{IrOsr} = \pi - \arccos(R_{SS}/R_{4S} \approx 173(3)°)$ deg. The normalized best fit residual is $R^2 = 4.2 \times 10^{-2}$.</caption><thead><tr><th>Shell</th><th>N</th><th>R (Å)</th><th>σ<sup>2</sup> (×10<sup>2</sup> Å<sup>2</sup>)</th><th>R<sub>XRD</sub> (Å)</th></tr></thead><tbody><tr><td>S<sub>1</sub>: Ir-O</td><td>6</td><td>1.979(5)</td><td>0.18(2)</td><td>1.99</td></tr><tr><td>S<sub>2</sub>: Ir-Li</td><td>2</td><td>2.87(1)</td><td>0.11(2)</td><td>2.79</td></tr><tr><td>S<sub>3</sub>: Ir-Sr<sub>1</sub></td><td>6</td><td>3.24(2)</td><td>0.05(1)</td><td>3.24</td></tr><tr><td>S<sub>4</sub>: Ir-O-O (MS)</td><td>24</td><td>3.35(2)</td><td>0.44(3)</td><td>3.40</td></tr><tr><td>S<sub>5</sub>: Ir-O-Sr<sub>2</sub> (SS+MS)</td><td>6</td><td>4.42(3)</td><td>0.45(3)</td><td>4.43</td></tr></tbody></table>

feature, identifying the $2p \rightarrow t_{2g}$ (lower energy shoulder) and $2p \rightarrow e_g$ (higher energy peak) transitions. Further, we have compared the Ir L$_3$-edge measured in our SLIO sample and in the 10% Ir/Na antisite disordered SNIO sample [12], and as shown in the figure S3 of SI, the observed red-shift (~0.15 eV) of the raising of the absorption edge for SNIO with respect to SLIO is likely related to the Ir/Na antisite defect of SNIO as already demonstrated in our previous study. As the very accurate energy reproducibility of the setup at the XAFS beamline is able to probe even much smaller changes, and experimentally, the Ir-L$_3$ edge shift is expected to be about 1.5 eV per valence charge unit (see [52]), we therefore conclude that the Ir XANES shape coherently indicates the pure 5+ oxidation state of Ir [10, 50, 51] in SLIO. To definitively assess the pure 5+ valence state of Ir in this compound the Ir 4f core level XPS spectrum was measured and fitted using a single spin-orbit split doublet (figure 3(b)) within the KOLXPD software program. The energy positions of the $4f_{7/2}$ and $4f_{5/2}$ doublets along with their spin-orbit separation of 3.02 eV confirm the pure Ir(5+) valence state [8–10, 13, 14].

### 3.4. Electrical resistivity, XPS valence band, non-spin polarized electronic structure calculation, and the Ir-L$_3$ edge RIXS studies

*Electrical resistivity and XPS valence band*

Similar to several other 5d iridates [8, 10, 13, 14, 44, 45, 53], SLIO also shows insulating behavior throughout the temperature range (see figure 4(a)). In addition, XPS valence band spectrum (inset to figure 4(a)) clearly reveals absence of finite density of states at the Fermi level (shown by vertical green dashed line in the inset of figure 4(a)), affirming the charge-gapped electronic state for SLIO. Such a gapped electronic behavior establishes importance of SOC in SLIO too.

*Non-spin polarized electronic structure*

To further apprehend the electronic structure, we have carried out non-spin polarized GGA + SOC electronic structure calculations of SLIO. The GGA + SOC Ir-d projected
<figure><img src="image_6.png" /><figcaption>**Figure 3.** (a) Ir-L₃ edge XANES spectrum (shaded orange circles), with the solid blue arrow indicating asymmetric shoulder; Inset: corresponding second derivative curve (shaded dark cyan circles); (b) Ir-4f core-level XPS spectra (shaded black circles) along with the fitting (red solid line).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 4.** (a) Temperature dependence of electrical resistivity (shaded black circles); Inset: XPS valance band spectrum (shaded violet circles), with the vertical green dashed line representing the Fermi energy position *E*<sub>F</sub>; (b) The orbital resolved density of states of Ir-d orbital within GGA+SOC; Comparison of the Ir-L<sub>3</sub> edge RIXS spectra in (c) low-resolution high-energy features and (d) high-resolution low-energy excitations.</figcaption></figure>

density of states plot in figure 4(b) reveal strong crystal field splitting of ~3.9 eV within the *t*<sub>2g</sub> and *e*<sub>*g*</sub> of Ir-*d* states due to the local octahedral environment of the Ir atoms, which is consistent with the low-resolution high energy RIXS results. Presence of trigonal distortion within the IrO<sub>6</sub> octahedra further lifts the degeneracy of the *t*<sub>2g</sub> states into low lying singly degenerate *a*<sub>1g</sub> and higher lying doubly degenerate *e*<sub>*g*</sub><sup>π</sup> states

with small non-cubic crystal field splitting of value 0.03 eV. Inclusion of SOC breaks the Ir-*t*<sub>2g</sub> states into *J*<sub>eff</sub> = 3/2 (Γ<sub>8</sub>) and *J*<sub>eff</sub> = 1/2 (Γ<sub>7</sub>) states as seen in the density of states plot of figure 4(b). The four electrons of the valence Ir-*d* orbital completely occupy the pseudo-spin *J*<sub>eff</sub> = 3/2 states keeping the *J*<sub>eff</sub> = 1/2 state completely empty, resulting in an insulating state.
### Low-resolution high-energy Ir-L$_3$ edge RIXS

The near isolation of Ir atoms in SLIO is supported by the presence of narrow crystal field excitation (around 3.9 eV) and charge transfer peaks (around 6 and 9 eV) in the low-resolution Ir-L$_3$ RIXS spectrum of SLIO compared to the 6H-hexagonal BZIO [54] (see figure 4(c)). The band-width of Ir-t$_{2g}$ states for SLIO is found to be ~0.8 eV which is slightly higher than that of it is sister compound SNIO (band-width ~0.7 eV)[12], but substantially smaller than BZIO [54]. Another notable feature to address is that the t$_{2g}$-e$_g$ crystal field splitting energy for SLIO (~3.9 eV) in the present study appears at relatively higher compared to that of SNIO (~3.5-3.6 eV) [12] and 6H-BZIO (~3.2-3.3 eV) [54], illustrated in figure 4(c), which could be assigned to the shorter Ir-O bond lengths (1.98 Å) in SLIO with respect to those found on SNIO and BZIO (≈2.03 and 1.9-2.0 Å respectively) [8, 12], and the lower level of Ir-t$_{2g}$ splitting due to much lesser IrO$_6$ octahedral distortion in SLIO with respect to the SNIO and BZIO [8, 12]. So, it would now be quite natural to assume that a combination of higher isolation among Ir$^{5+}$ ions, resulting narrow bandwidth and weaker trigonal distortion, can help the realization of highest SOC, i.e. near atomic limit and consequently can help to achieve the coveted $J_{eff} = 0$ nonmagnetic ground state.

### Low-energy high-resolution Ir-L$_3$ RIXS

Next we compare the low-energy, high-resolution Ir-L$_3$ edge RIXS spectra between the two isovalent isostructural columnar iridates Sr$_3$(Na,Li)IrO$_6$ and a 6H hexagonal iridate BZIO, measured within the same technical specifications [54], and the results are resumed in figure 4(d). It is evident that there are substantial differences in the spectral width, intensity, energy positions as well as number of spectral features (peak at 0.18 eV energy loss and double peak feature in the 0.3-0.4 eV range of the energy loss in BZIO, shown by the bicolored arrow and ellipse respectively, which are missing in both the columnar systems) between the columnar and 6H family of d$^4$ iridates [12, 54] which could certainly be ascribed to the enhanced degree of intersite Ir-Ir hopping due to face-sharing IrO$_6$ octahedral connectivity, as well as greater extent of Ir t$_{2g}$ trigonal crystal field splitting in BZIO [8] compared to the columnar SNIO [12] and SLIO. On the other hand, like in SNIO [12], we observe three similar inelastic peaks below 1.5 eV in SLIO. Although the shape and energy positions of these three peaks overall appear similar in both of them, subtle changes in these inelastic RIXS features are clearly visible, as demonstrated by the intensity enhancement and slight peak width broadening, as well as development of weak but noticeable shoulders in the higher-energy sides of the respective first and second inelastic peaks (marked by two bicolored arrows for SNIO in figure 4(d)). Clearly these differences in spectral features between the two columnar iridates lie in the difference between their respective Ir-O octahedral distortions, Ir-Ir hopping pathways and extent of Na/Li-Ir anti-site disorder. Moreover, there are more numbers of interchain Ir-Ir connectivity and the respective hopping networks

(via Ir-O-O-Ir) as a consequence of larger Na/Ir site-disorder, in contrary to only two different Ir-Ir interchain distances (schematically presented in figure 1(d) by $J_2^{ex}$ and $J_3^{ex}$ at ~5.87 and 6.7 Å respectively) and the corresponding Ir-O-O-Ir hopping pathways in the fully ordered SLIO. Also, there would be bimodal distributions of local noncubic crystal fields, Ir-O covalent interactions, Ir-Ir superexchange interactions $J_{SE}(\equiv 4t^2/U)$ and the effective SOC strengths in SNIO due to the presence of ~90% Ir in the ordered sites along with the ~10% Ir in the antisite positions [12], which is naturally absent in SLIO. All these issues give rise to different extent of lifting of Ir-t$_{2g}$ degeneracy, and accordingly, rearranging the spin-orbit derived Ir-$J_{eff}$ states in these two systems. In such a scenario, we must infer that the aforementioned changes in low-energy Ir-L$_3$ RIXS features of SLIO relative to the SNIO and BZIO should be due to the dissimilar Ir-Ir hopping connectivity and also the influence of different trigonal crystal field splitting on the Ir t$_{2g}$ orbitals in this SLIO. Infact, both hopping and noncubic crystal field effect impart a strong impact on the effective SOC strength in d$^4$ iridate systems [54]. Therefore, demonstrating the spin-orbit-coupled Ir energy levels from the perspective of atomic J picture solely, as has been the widely accepted description until recently [55], becomes insufficient, as revealed by Nag et al [54] and Revelli et al [56] very recently. So, as estimation of the effective SOC strength $\lambda_{eff}$ on Ir within only a purely atomic limit is not at all a reasonable approach, we refrain from doing such an estimation here in SLIO.

Next we have constructed a single site (atomic) many-body multiplet model Hamiltonian for the Ir t$_{2g}$ states of SLIO, [13, 37, 57-59] in order to estimate the strength of SOC,

$$H_{tot} = H_{ncf} + H_{int} + H_{so}. \qquad (1)$$

Here $H_{ncf}$, $H_{so}$ and $H_{int}$ represent the respective Hamiltonians for local non-cubic crystal field, SOC and the Coulomb interaction, which form the effective many-body atomic Hamiltonian. $H_{int}$ comprises intra-orbital Coulomb correlation (U), Hund's coupling strength ($J_H$) and inter-orbital Coulomb correlation ($U' \equiv U - J_H$). The strength of non-cubic crystal field turns out to be 0.03 eV from our single-particle mean-field calculations using the MTO based N th-order MTO method as implemented in the Stuttgart code [39]. In order to extract the strength of SOC ($\lambda$) and $J_H$, the energy losses in experimental RIXS spectrum are mapped with the differences between the eigenstates of the effective atomic many-body Hamiltonian (see figure 5(a)). Such an analysis estimates the strength of Hund's coupling and SOC strength for SLIO to be 0.25-0.26 eV and 0.39 eV, respectively, and are expected to be renormalized upon inclusion of intersite hopping. Further due to the presence of trigonal crystal field a small though finite moment exists ($\langle J^2 \rangle = 0.01$) at the Ir site even within our atomic level calculation, which will be explored in details in the next section.
<figure><img src="image_8.png" /><figcaption>**Figure 5.** (a) Calculated excitation energies as a function of $\lambda$ from model calculation for Hund's coupling strength 0.25 eV (black) and 0.26 eV (cyan); (b) Variation of total moment $\langle J^2 \rangle$ with diagonal hopping $\gamma$ within the two-site spin polarized $t_{2g}$-only model, Inset: projection of the ground state onto pure $J=0$ state to indicate the deviation with increasing relative hopping strength.</figcaption></figure>

### 3.5. dc and ac magnetic susceptibilities

Now in order to unfold the crucial question of whether the elusive $J_{\text{eff}} = 0$ state is realized in SLIO, or not, the temperature dependence of the zero-field-cooled (ZFC) and field-cooled (FC) dc magnetic susceptibilities $\chi(T)$ have been measured under different applied magnetic fields, and the results are summarized in figures 6(a)–(c). As observed, the featureless paramagnetic-like bulk dc susceptibility curves without any ZFC/FC bifurcation for $H \ge 2000$ Oe clearly indicates absence of long-range magnetic ordering transition down to 2 K. For the low applied fields (see figure 6(a) and its inset), a noticeable irreversibility between the ZFC and FC magnetizations appears from a relatively higher temperature (~200–225 K), and gets completely suppressed upon increasing applied fields, possibly indicating some proportions of spin-freezing [3, 60, 61]. As such a spin-freezing does not produce a peak/cusp-like feature in either of the T-dependent high-field dc magnetic susceptibility and the T-dependence of the real part of ac magnetic susceptibility curves (see inset to figure 6(f) for $\chi'_{ac}$ versus T variations at selected different frequencies) throughout the measured 2–300 K range, it evidently refutes a conventional spin-glass magnetic ground state in SLIO. Interestingly, unlike our previous work on the columnar SNIO counterpart [12], the development of a higher-T magnetic feature in the form of ZFC/FC splitting in low applied fields could be due to the much stronger intrachain Ir–Ir exchange relative to the interchain Ir–Ir magnetic interactions (see latter in the spin-polarized DFT calculations in section 3.9) and/or to the inhomogeneity induced by some Li-vacancy/oxygen-off-stoichiometry in this sample (not reliably detected from our structural characterizations data due to instrumental sensitivity limit). The field-dependent isothermal magnetization ($M-H$) curves, as presented in figure S5 of SI, clearly discard coercivity and remanent magnetization even at the lowest measured 2 K, thus negating hysteresis-like behavior. Further, the 2 K Arrot plot ($M^2$ versus $H/M$ in the inset of figure 6(d)) renders an interception on the negative $M^2$ axis, refuting any spontaneous magnetization

vis-a-vis FM components [62] in SLIO. The observed nonlinearity in the $2K M-H$ data, however, does not resemble the standard paramagnetic behavior, rather it is qualitatively analogous to the reported several $d^4$ iridate systems [10, 14, 49, 63, 64].

Now in such disordered magnetic system, it is noteworthy to state that within weak temperature-dependence of the measured dc susceptibility, the Curie–Weiss (CW) fitting parameters always carry some uncertainties which critically depend on both the temperature range of CW fitting and the applied magnetic field [65]. Considering all the possibilities, the C–W analysis ($\chi = \chi_0 + \frac{C_W}{T-\Theta_{CW}}$ with $\chi_0$ being the temperature independent paramagnetic susceptibility, and $C_W$, $\Theta_{CW}$, the Curie constant and the Curie–Weiss temperature, respectively) on the 10 and 70 kOe field-cooled $\chi(T)$ data seems to be most reliable in the given temperature range (see bottom left insets and main panels of figures 6(b) and (c)). The resulting fits yield a $\Theta_{CW} \approx -45$ to $-49$ K, suggesting even stronger (relative to SNIO [12]) AFM Ir–Ir interaction, which is in agreement with the stronger intrachain Ir–Ir exchange strength (see discussion in section 3.9 latter), and an effective paramagnetic moment, $\mu_{\text{eff}}$, to be ~0.45 $\mu_B$/Ir$^{5+}$, thus violating the $J_{\text{eff}} = 0$ nonmagnetic ground state description in SLIO. At this point, it is of utmost importance to check if the nature of the low-$T$ susceptibility rise has an intrinsic origin or if such a $\chi(T)$ upturn is originating from extrinsic paramagnetic impurity. To confirm this, we carefully investigated our low-$T$ (below 10 K) dc susceptibility data using both power-law scaling and CW law. Notably, the 10 and 70 kOe $\chi_{dc}(T)$ data show deviation from the power-law dependence at lowest temperatures. Rather, as illustrated by red solid line in the main panel and top right inset of figure 6(b) and the solid blue line in the main panel and top right inset of figure 6(c), a CW fit satisfactorily describes the respective 2–10 K $\chi$ versus $T$ data. These low-temperature CW analysis results in an effective paramagnetic moment, $\mu_{\text{eff}}^{\text{LT}}$ of ~0.025 $\mu_B$/Ir and a small negative $\Theta_{CW} \sim -1.4$ to $-1.7$ K. The small $\mu_{\text{eff}}^{\text{LT}}$ of SLIO could stem from the presence of a low fraction of Ir$^{4+}$ defect spins (~1.5% of $J_{\text{eff}} = 1/2$). The initially negative Weiss temperature ($\Theta_{CW} \sim -1.35$ to $-1.7$ K) points towards nearly uncorrelated paramagnetic Ir$^{4+}$ defects at the lowest temperature. However, such a small fraction of magnetic Ir$^{4+}$ defects cannot explain the large effective paramagnetic moments of ~0.45 $\mu_B$/Ir estimated from the high-$T$ CW fittings, reflecting the magnetic moments to be intrinsic to system’s Ir$^{5+}$ and not due to the isolated paramagnetic centers. To understand the origin of $\mu_{\text{eff}}^{\text{LT}}$, we further analyzed the $2K M$ versus $H$ isotherm as shown in figure 6(d). We used a modified Brillouin function, $M(H) = \chi_0 H + f N_A g \mu_B J B_J(g \mu_B J H/k_B T)$ [49], where $f$, $N_A$, $g$, and $k_B$ represent a scaling factor to account for the finite number of paramagnetic impurities, the Avogadro constant, the Lande $g$-factor, and the Boltzman constant, respectively. The first term, $\chi_0 H$ represents the temperature independent contribution, while the second term, the Brillouin function $B_J$, takes care of the behavior of paramagnetic spins as a function of magnetic field at a particular temperature [49].
<figure><img src="image_9.png" /><figcaption>**Figure 6.** (a) Temperature dependence of ZFC and FC dc magnetic susceptibility data for 2000 Oe (main panel) and 100 Oe (Inset) applied magnetic fields; (b) *T*-dependence of the 10 kOe ZFC and FC dc magnetic susceptibility curves in log-log scale along with the Curie–Weiss (CW) fittings in 2–10 K (solid red line) and 100–300 K (solid blue line) region. Insets: Corresponding inverse susceptibility versus temperature variations together with linear CW fits; (c) *T*-dependence of the 70 kOe ZFC and FC dc magnetic susceptibility curves in log-log scale along with the Curie–Weiss (CW) fittings in 2–10 K (solid blue line) and 150–400 K (solid green line) ranges. Insets: Corresponding inverse susceptibility versus temperature variations together with linear CW fits; (d) Field dependent isothermal magnetization (*M* versus *H*) curve for the first quadrant at 2 K along with a modified Brillouin fit. Inset: the 2 K Arrot plot (*M*² versus *H*/*M*); (e) the edge-shared frustrated Ir-triangular network formed by interchain Ir–Ir exchange connectivity *J*₂ᶜₓ and *J*₃ᶜₓ; (f) *T*-dependence of the real part of ac magnetic susceptibility *χ*<sub>ac</sub> (0.05–4 K) at some selected frequencies and in excitation field *H*<sub>ac</sub> = 1 Oe with no dc bias, Inset: *χ*'<sub>ac</sub> versus *T* curves in 2–300 K range at three different frequencies.</figcaption></figure>
Our 2 K *M* vs *H* data analysis (see in figure 6(d)) reveals ~2% of *J*<sub>eff</sub> = 1/2 spins and a *g*-factor of 1.97. These findings match very well with the estimates from the low-*T* (2–10 K) CW fittings assuming *g* = 2. It is important to note that while the temperature-independent susceptibility contribution from the low-*T* CW fitting yields χ<sub>0</sub> = 7.65 × 10<sup>-4</sup> emu mol<sup>-1</sup>, we obtain a larger value of χ<sub>0</sub> = 16.5 × 10<sup>-4</sup> emu mol<sup>-1</sup> from the analysis of field dependent isothermal magnetization data at 2 K. Such a mismatch probably arises from non-negligible AFM spin correlations which have not been taken into account in the modified Brillouin function [49]. So, we may elucidate that the low-*T* bulk magnetic response, in particular, the susceptibility upturn in the χ vs *T* data, and the nonlinearity in the 2 K *M* vs *H* isotherm is contributed by the small amount (~1.5%–2%) of defect Ir<sup>4+</sup> spins, on top of the strongly anti-ferromagnetically interacting intrinsic Ir<sup>5+</sup> moments in this material.

Lastly, we measured the ac magnetic susceptibility down to
50 mK at some selected frequencies, and the results for the in-
phase component of the ac magnetic susceptibility as a func-
tion of temperature at different frequencies with an ac excit-
ation field of 1 G amplitude and zero dc field are shown in
figure 6(f). Absence of any anomaly till down to 50 mK clearly
rejects a spin-glass frozen magnetic ground state in SLIO.

3.6. Spin polarized DFT calculation, and applicability of the
two-Ir-site model in support of magnetism

To explore the possibility of magnetism, the spin polarized
GGA + SOC + U calculation keeping the spin quantiza-
tion axis along [001] direction, has been carried out of the
AFM configuration having anti-parallel spin orientations at
the neighboring Ir sites within the linear chain. Total energy
calculated from spin-polarized density functional theory in
presence of SOC suggests AFM configuration to be lower in
energy than both of the non-magnetic and ferromagnetic con-
figuration by energy values Δ*E*/f.u.~6.62 meV and 6.54 meV
respectively. This suggests the presence of AFM interactions
in SLIO, which is consistent with the negative Θ<sub>CW</sub> extrac-
ted from the high-temperature CW fit of the dc susceptibility
data. In summary, our calculations reveal a magnetic ground
state with AFM spin configurations and large values of spin
& orbital moments on individual Ir<sup>5+</sup> ion, in contrast to the
coveted nonmagnetic *J*<sub>eff</sub> = 0 state as suggested by Ming *et al*
in a recent theoretical study [66].

Our results are further corroborated with a two site mul-
tiplet model Hamiltonian, where incorporation of nearest
neighbor hopping *H*<sub>*t*</sub> in the single site atomic Hamiltonian
(equation (1)) results in Ir moments of about 0.31 μ<sub>B</sub>. In
figure 5(b) we have plotted the variation of <*J*<sup>2</sup>> with rela-
tive hopping parameter γ, where the scale factor γ is defined as
*t*<sub>cal</sub> = γ *t*<sub>slio</sub>. Here *t*<sub>cal</sub> and *t*<sub>slio</sub> are respectively the hopping mat-
rix used in our two site model calculation and nearest neigh-
bor hopping matrix within the *t*<sub>2g</sub> basis of SLIO obtained from
NMTO calculations. Evidently γ = 1 represents the <*J*<sup>2</sup>>
value corresponding to the realistic hopping strength of the
system. To explore the effect of hopping, we vary the hop-
ping matrix *t*<sub>cal</sub> in the multiplet model calculation for a range

of γ = 0 to 2. As per our expectation the higher hopping
strengths promotes the larger Ir moments. Further, our calcu-
lated <J²> value for three different SOC strength viz. λ =
0.29 meV (blue), 0.39 meV (violet) and 0.49 meV (red); sug-
gest that moment generation is higher for the low values of λ.
In the inset of figure 5(b), we have plotted the projection of
the SLIO ground state wave-function |ψg⟩ onto pure J<sub>eff</sub> = 0
state (|ψ<sub>j=0</sub>⟩) for λ = 0.39 eV. It is evident from the plot that
for realistic values of hopping the system substantially devi-
ates from non-magnetic J<sub>eff</sub> = 0 state. Our calculation of
figure 5(b) suggests although the SOC suppresses the value
of the moment significantly but finite non-cubic crystal field
and emergence of intermediate d³ - d⁵ and d² - d⁶ config-
urations owing to nearest neighbor hopping promote gener-
ation of Ir⁵⁺ moment in SLIO. Therefore, we argue that sim-
ilar to other analogous d⁴ systems, the ground state of SLIO
deviates from the pure atomic J<sub>eff</sub> = 0 state due to non-
cubic crystal field together with hopping-induced delocaliza-
tion of holes. Moreover, in line with our observation of a mag-
netic ground state in terms of the finite local Ir⁵⁺ moments
in SLIO, Kaushal *et al* also recently confirmed the gener-
ation of magnetic moments with existing short-range AFM
interactions among them in one-dimensional t²g iridate chains
using the density matrix renormalization group (DMRG)
technique [67].

3.7. ⁷Li NMR

Bulk magnetization in many of the *d*⁴ iridates has often been
attributed to extrinsic paramagnetic impurities on top of the
nonmagnetic *J*<sub>eff</sub> = 0 ground state [49, 68]. Therefore, to fur-
ther verify the intrinsic nature of the developed Ir<sup>5+</sup> mag-
netic moments, the ⁷Li NMR was measured on the bulk SLIO
sample in the temperature range of 4–300 K in a fixed field of
93.954 kOe. The ⁷Li nucleus has a nuclear spin *I* = 3/2, and
thus, is 1st order quadrupolar active. The observed resonance
lineshape (see for example, at 200 K in figure 7(a)) is simulated
by considering the parameters *K*<sub>iso</sub> = 80 ppm, *K*<sub>ax</sub> = 95 ppm,
*K*<sub>aniso</sub> = 0 ppm, *η* = *V<sub>xx</sub>*−*V<sub>yy</sub>* = 0, and *ν*<sub>Q</sub> = 33 kHz. Here, the
first three parameters *K*<sub>iso</sub>, *K*<sub>ax</sub>, and *K*<sub>aniso</sub> are isotropic, axial,
and anisotropic Knight shifts, respectively, corresponding to
quadrupolar lineshape. *V*<sub>xx</sub>, *V*<sub>yy</sub>, and *V*<sub>zz</sub> are the principal com-
ponents of the electric field gradient tensor with *V*<sub>zz</sub> defined as
the highest among the three. The quadrupole frequency is des-
ignated by *ν*<sub>Q</sub>.

The temperature dependence of the NMR Knight shifts for
both the central and satellite lines, extracted from their meas-
ured resonance frequencies, ν as ω = 2πν = γH₀(1 + K), is
illustrated in figure 7(b). γ/2π is the gyromagnetic ratio of ⁷Li
nucleus and μ₀H₀ is the magnetic field. The Knight shift res-
ults from the hyperfine interaction between the electron spins
and the nuclear spins. As shown in figure 7(b), the nearly tem-
perature independent Knight shifts and their low absolute val-
ues could possibly refer to the weak hyperfine coupling of
the ⁷Li with the Ir<sup>5+</sup> local moments or that the local spin
susceptibility is small and only weakly temperature depend-
ent or perhaps both. In contrary, the NMR line-widths (Full
width at half maximum) of both the central and satellite lines
<figure><img src="image_10.png" /><figcaption>**Figure 7.** (a) The 200 K $^7$Li NMR spectrum (half-filled magenta circles), along with the theoretical fit (solid cyan line); Inset: same at T = 30 K; T-dependence of the NMR Knight shifts (b) and line-widths (c); (d) Linear fit (green solid line) to the NMR central line-width versus 70 kOe dc susceptibility curve; (e) T-dependence of the $^7$Li NMR spin-lattice relaxation rate $T_1^{-1}$ (shaded cyan circles) in logarithmic scale and the corresponding BPP model fit (red solid line); Inset: Power-law dependency of the $T_1^{-1}$ with T at low-T; (f) the exponent extracted from the $T_1$ data fitting with the stretched exponential function as a function of T and the red line shows the average value of the exponent which is ~0.7.</figcaption></figure>

(see figure 7(c)) possess increase with decreasing temperature. Such an increasing trend scales with the variation of bulk dc susceptibility upon temperature lowering, thereby supporting intrinsic nature of the observed Ir$^{5+}$ local moments in SLIO, as such an increase in line-width originates from varying dipolar coupling to the local Ir$^{5+}$ moments as a function of T, similar to that in SNIO [12]. Hence, like in SNIO [12], here in SLIO too, the T-independent NMR Knight shift could possibly be attributed to weak hyperfine interaction of $^7$Li with Ir$^{5+}$ local moments given that the magnetic Ir is not directly probed through our NMR measurement, rather, nonmagnetic Li is the NMR active nucleus, causing the $^7$Li nuclei to probe

the Ir local moments of this system via transferred-hyperfine coupling through an overlap of Ir 5d orbitals with O-2p states. In addition, while the presence of tiny amounts (~1.5–2%) of magnetic Ir$^{4+}$ defect centers could possibly be responsible for the low-T increase in the bulk dc susceptibility, the hyperfine coupling, in contrary, fails to capture the interactions of the isolated paramagnetic centers and hence, the NMR Knight shift possibly does not track any T dependence at low temperatures, unlike the low-T Curie–Weiss-like upturn of $\chi_{dc}$. It should also be noted in this context that there is no signature of spin-freezing at around 200–225 K in our T-dependent NMR Knight shift spectra, in agreement with the high-field dc
susceptibility results and pointing towards the sustainability of the partially frozen phase against only zero or low magnetic fields. Furthermore, from the slope of the linear fit to the central NMR linewidth versus 70 kOe dc susceptibility plot (figure 7(d)) with temperature being an explicit parameter, we find a dipolar coupling constant to be about $A = 3.35 \times 10^{22}$ per cm$^3$atom, and the average distance between the Ir moments and the $^7$Li nucleus to be about $2.59$ Å, in agreement with the Ir–Li interatomic distance of ~2.7 Å obtained from our structural studies. This further validates the intrinsic nature of the Ir$^{5+}$ local moments.

Finally, the $^7$Li NMR spin-lattice relaxation rate $1/T_1$ was measured using a saturation recovery sequence. The recovery was found not to be single exponential. Considering that $^7$Li has $I = 3/2$ and hence a quadrupole moment, one might expect a multi-exponential recovery if the full spectrum is not irradiated with the saturating sequence. However, given that the $\nu_Q$ is small, one expects to irradiate nearly the full spectrum. We fitted the longitudinal nuclear magnetization recovery to a stretched exponential given by $M(t) = M_0[1 - \exp(-(t/T_1)^\beta)]$. The rather large value of spin-lattice relaxation time $T_1$ (see figure 7(e)) indicates a weak coupling to the local magnetic moments. The corresponding relaxation rate $1/T_1$ (shown in figure 7(e)) increases with decreasing temperature and exhibits a broad maximum at around 80 K, which could not be associated with conventional spin glass freezing as critical slowing down of the spin-fluctuations in a spin-glass system would give rise to a diverging spin-lattice relaxation below the freezing temperature [61, 69, 70]. We therefore suggest that this arises due to contribution from the relaxation of the fluctuations of local Ir$^{5+}$ moments as motion of the lithium ions might not be expected at these temperatures. In the Bloembergen-Purcell-Pound formalism [71, 72], the relaxation rate varies as $P(\frac{\tau}{1+\omega^2\tau^2})$ where $\tau$ is the correlation time of the nuclear dipoles and $\omega$ is the NMR frequency. The correlation time (rate) follows an Arrhenius behavior $\tau = \tau_0\exp(E_A/k_BT)$ where $E_A$ is the activation energy. A maximum occurs around $\omega\tau = 1$. From the fit to our data (see figure 7(e)), we obtain $E_A/k_B \approx 200$ K, $\tau_0 \approx 10^{-10}$ s, and the pre-factor to be $P = \gamma^2h_\perp^2 = 3.5 \times 10^8 s^{-2}$. Given that the gyromagnetic ratio for $^7$Li is $2\pi \times 16.546$ MHz/Tesla, we find the transverse fluctuating field $h_\perp \approx 2$ Gauss, which is comparable to Ba$_2$YIrO$_6$, supporting that the local Ir$^{5+}$ moments are present in our SLIO. So, the very weak spin-lattice relaxation rate is rather surprising as local Ir$^{5+}$ moments are present, but it is possibly due to the symmetric location of $^7$Li between the magnetic iridium, filtering out the AFM fluctuations. Notably, from about 80 K, $1/T_1$ decreases with $T$ lowering and maintains a power-law behavior (inset to figure 7(e)) with power 1.5 till down to 10 K. This suggests building up of magnetic correlations and persistent fluctuations of the correlated Ir$^{5+}$-local moments, likely in agreement with the $T$-dependent dc, ac magnetic susceptibilities and NMR line-widths (see in figures 6(a)–(c) and 7(c)). Such a power-law-dependency of the NMR spin-lattice relaxation rate, $1/T_1$, refers to the gapless spin excitation spectrum in

SLIO, which is consistent with the reported many other iridates Na$_4$Ir$_3$O$_8$ [73], Ba$_3$InIr$_2$O$_9$ [74], SNIO [12], and low-dimensional quantum magnets [61, 75, 76]. The $T$-dependence of the exponent $\beta$ extracted out of the $T_1$ data fitting with the stretched exponential function is presented in figure 7(f).

### 3.8. True dimensionality, origin of local moments, irrelevancy of strong SOC, and geometric frustration

The intra and inter-chain Ir–Ir hopping interaction strengths are calculated to be ~130 meV and ~76 meV, respectively. This estimated substantial nearest-neighbor inter-chain Ir–Ir hopping interaction through inter-orbital Ir-$d_{xz}$ and $d_{yz}$ overlap suggests three-dimensional nature of magnetic exchange within apparently one-dimensional structure of SLIO. The observation of finite local magnetic moments on individual Ir$^{5+}$ ion immediately refutes the nonmagnetic $J_{eff} = 0$ ground state proposition, and thus, an isolated atom-like description of Iridium in SLIO, and hence, establishes SLIO as a magnetically three-dimensional (3D) system, consistent with the previously reported isostructural columnar spin-chain compounds [23, 77–79]. Further, this relatively large value of Ir$^{5+}$ magnetic moment (with respect to the strong spin-orbit coupled ideal nonmagnetic $J = 0$ state) strongly endorses only an intermediate effective SOC picture [80, 81] in SLIO, thereby opposing the anticipated higher atomic Ir-SOC picture in this compound.

The origin of such a finite Ir$^{5+}$ local moment in SLIO could be assigned to the intersite Ir–Ir hopping via extended Ir–O–O–Ir superexchange pathways within the possible intra and interchain Ir–Ir magnetic exchange connectivity [represented by three different exchange interactions in figure 1(c)], causing delocalization of intra-site Ir$^{5+}$ holes, and therefore, creating deviation from a perfect atomic $J_{eff} = 0$ arrangement [10, 12–14, 54]. In addition, the trigonal distortion of the IrO$_6$ octahedra and the subsequent Ir-$t_2g$ splitting, as well as the hybridization of extended Ir-5d orbitals together with the stronger Ir–O covalent interactions might cause rearrangement in the spin-orbit coupled Ir energy levels in SLIO, which naturally leads to the possible breakdown of atomic $J$ limit [81–84]. But surprisingly, despite having significant AFM correlation (by means of large negative $\Theta_{CW}$) between the local Ir$^{5+}$ moments, the SLIO compound fails to trigger magnetic ordering down to 0.05 K at least.

To corroborate our prediction of magnetism within SLIO, we have calculated the magnetic exchange interactions between the Ir atoms. For a quantitative estimation of the Ir–Ir exchange couplings, we have calculated the symmetric exchange interactions by mapping total energies of several spin configurations within GGA + SOC + U scheme to the Heisenberg model $H_{spin} = \sum_{ij} J_{ij} \vec{S}_i \cdot \vec{S}_j$. Our calculations show that the nearest-neighbor intra-chain Ir–Ir interaction $J_1^{ex}$ is dominant and AFM in nature with a magnitude of 4.26 meV. The second ($J_2^{ex}$) and third ($J_3^{ex}$) nearest-neighbor interactions, representing the interchain Ir–Ir exchanges, are relatively weaker with magnitude of 0.07 meV and 0.05 meV, and
<figure><img src="image_11.png" /><figcaption>**Figure 8.** (a) Time evolution of the zero-field muon asymmetry spectra along with relevant fittings for some selected temperatures from 300 to 1.7 K. Inset: crossover in the muon asymmetry curves between 300 and 1.7 K; (b) T-dependence of the (b) muon depolarization rate subjected to frozen magnetism, (c) slow ($\lambda'_1$) and fast ($\lambda'_2$) relaxation rates corresponding to 1/3 polarization tail and the dynamical component, respectively, and (d) stretched exponent $\beta$ of the dynamical component; (e) T-evolution of the frozen fraction.</figcaption></figure>

respectively FM and AFM in nature. Interestingly, the twelve inter-chain $J_2^\text{ex}$ and $J_3^\text{ex}$ bonds with comparable and oppositely signed exchange interaction strengths therefore form an edge-sharing frustrated triangular network (see figure 6(e)), prohibiting magnetic ordering in SLIO.

Although, the absence of any magnetic long-range ordering despite significant AFM correlation might probably indicate persistent spin fluctuations in SLIO until 2 K and is likely consistent with the existing several candidate QSL systems [8, 9, 12–14, 65], the low-field ZFC/FC branching from a rather high-T could be related to finite spin-freezing, resulting from the weakening of exchange frustration in the Ir triangular network.

### 3.9. Muon spin relaxation/rotation (μSR)

At this point, experimental verification for a true magnetic ground state is incomplete without employing a local probe. Muon spin relaxation/rotation (μSR) is an extremely sensitive microscopic technique, in particular, for the disordered and/or weakly magnetic materials to ensure the actual magnetic ground state as well as to detect the nature of local spin dynamics and internal magnetic fields. Based on the fact that positrons are most likely emitted in the direction of muon spin polarization when it decays, the muon spin polarization $P(t)$ is measured by recording the asymmetry $A(t)$ [$P(t) \propto A(t)$] between the positron count rates in detectors placed on opposite sides of the sample. The time-evolution of the ZF muon decay asymmetry $A(t)$ for the SLIO sample has been measured with decreasing temperature down to 1.7 K and the results are presented in figure 8(a). Strikingly the relaxation is rather strong and shows a marked temperature dependence. This is at variance with the expectation of a weak relaxation from dilute impurities (see, for example, [85]). The absence of coherent spontaneous oscillations in either of the muon polarization curves refute conventional long-range magnetic ordering of SLIO down to 1.7 K, although, since the initial part of the signal is lost due to the limited time resolution at the ISIS pulsed muon source, we cannot definitively rule

out the possibility of fast and strongly damped oscillations. Nonetheless, as shown in the inset to figure 8(a), there is a prominent crossing of the muon polarization curves at low ($T < 150$ K) and high ($T > 200$ K) temperatures, giving evidence for the development of spin-freezing in SLIO, in line with the ZFC/FC divergence of the low-field dc magnetization data. Beside this spin-freezing phenomenon, even at 1.7 K, at long time, the persistent non-zero relaxation reveals slow persisting spin fluctuations (i.e. decaying 1/3rd tail) thus pointing towards an overall fluctuating liquid-like magnetic ground state in SLIO. Furthermore, a nearly perfect overlap of the 10 K and 1.7 K polarization curves corroborate the enduring spin fluctuations, and hence, pertinent spin dynamics in the ground state of SLIO. In order to gain a quantitative understanding of the spin dynamics in this compound we have analyzed the muon polarization curves in the temperature range of 300–1.7 K using a two-component relaxation with the form,

$$A(t) = A_0 + A_{\text{rel}} \left[ f\left(\frac{2}{3}e^{-(\sigma t)^2/2} + \frac{1}{3}e^{-\lambda'_1 t}\right) + (1-f)e^{-(\lambda'_2 t)^\beta} \right]. \quad (2)$$

The first part of equation (2), $f(\frac{2}{3}e^{-(\sigma t)^2/2} + \frac{1}{3}e^{-\lambda'_1 t})$, is representative of the frozen magnetism and is characterized by rapid depolarization of 2/3 of the initial polarization, reflected in an apparent loss of the initial asymmetry, due to muons coupled to a broad distribution of quasi-static internal magnetic fields, followed by slow exponentially relaxing remaining 1/3 tail due to fluctuating component of the internal field. The additional stretched exponential relaxation, $(1-f)e^{-(\lambda'_2 t)^\beta}$ stands for dynamic paramagnetism. $A_0$ and $A_{\text{rel}}$ be the background accounting for the fraction of muons falling in the sample holder and the asymmetry amplitude of the muons stopping at the sample sites, respectively, and are fixed at ~0.05 and ~0.22 during fitting throughout the temperature range of measurements; $f$ represents the fraction of the frozen spins. $\lambda'_1$ and $\lambda'_2$ represent the slow and fast muon spin relaxation rates corresponding to the frozen and dynamic magnetism, respectively; $\beta$ is the stretched exponent associated with the
dynamical component; $\sigma$ is the muon depolarization rate due to quasi-static internal field distribution. The T-dependence of the $\mu$SR fitting parameters found from satisfactory fit to each of the $\mu$SR polarization curves using equation (2) are summarized in figures 8(b)–(e). As shown in figure 8(c), the relaxation rate $\lambda'_2$ reveals a gradual slow increase with decreasing temperature below 300 K, suggesting continuous slowing down of the Ir moment fluctuations, and hence, development of magnetic correlations among the local Ir$^{5+}$ moments in SLIO. The $\lambda'_2$ reaches a weak maximum of ~0.92 $\mu$s$^{-1}$ at about 150 K, which could not be accounted for by any magnetic ordering transition as a magnetically ordered state (even for a spin-glass ground state) would naturally give rise to a diverging muon spin relaxation rate at the transition temperature. Rather, following the 150 K maximum the $\lambda'_2$ consistently decreases upon lowering down the temperature till to 10 K and then below 10 K there is no further decrease in relaxation along with the presence of a noticeable finite relaxation rate (~0.4 $\mu$s$^{-1}$) at the lowest measured T of 1.7 K, elucidating persistent spin fluctuations in SLIO as commonly seen in many geometrically frustrated magnets accommodating dynamic magnetic ground state [86–88]. The associated stretched exponent $\beta$ of this dynamical component of relaxation possesses (figure 8(d)) a monotonic decrease from ~1 for T = 300 K to ~0.55 for T = 150 K, and then with a small increase finally attaining a nearly level-off of $\beta$ ~ 0.7 in the T-range of 1.7–50 K, indicating a gradual crossover from a homogeneous paramagnetic to an inhomogeneous magnetic environment in this material. The obtained value of $\beta$ at the base temperature is larger than the $\beta$ = 1/3 of the canonical spin glass [89], thus negating a frozen magnetic ground state and supporting further a fluctuating disordered magnetic ground state in SLIO. Interestingly, at $T \le 200$ K (the same temperature at which crossing of the $\mu$SR polarization curves starts) there is a clear development of finite proportion of spin freezing along with the dynamical part, and upon further cooling till down the lowest measured T of 1.7 K a coexisting frozen and fluctuating spin dynamics of ~40% and 60%, respectively, is evidenced from our $\mu$SR fitting results (see figure 8(e)). In corroboration with the development of frozen magnetism, a rapid increase in the muon depolarization rate $\sigma$ with temperature lowering below 250 K (figure 8(b)) suggests emergence of quasi-static Ir moments and consequently, a gradual enhancement of the Ir spin-freezing-assisted static internal magnetic field at low temperature. Considering the nearly unchanged $\sigma$-values for $T \le 50$ K and that this $\sigma$ to be due to static internal field distribution from the frozen Ir-spins, the magnitude of that static field would be approximated by $B_\sigma = \frac{\sigma}{\gamma_\mu} \approx 200$ Gauss ($\gamma_\mu/2\pi = 135.5$ MHz/Tesla is the muon gyromagnetic ratio). Assuming that positively charged muons sit at the negative oxygen sites in the structure (i.e. ~2 Å away from the Ir ions) and that $B_\sigma$ be the dipolar interaction field experienced by the muons due to Ir moments, the frozen moments producing this dipolar static field, $B_\sigma$, could be roughly estimated to be $\approx 0.1 \mu_B$. This frozen moment cannot be explained only by tiny amounts (~1.5%–2%) of Ir$^{4+}$ magnetic defects, so it is quite obvious to infer that such a large static internal field

arises from the frozen Ir$^{5+}$ local moments out of an overall fluctuating magnetic background. In summary, we conclude that the magnetic ground state of SLIO is inhomogeneous and consists of coexisting fluctuating and frozen dynamics of Ir local moments. Such a dynamic/frozen phase coexistence in SLIO without perceptible site disorder, unlike the previously studied antisite disordered SNIO where an inhomogeneous quantum spin liquid ground state was identified due to comparable intra- and interchain Ir–Ir exchange interactions [12], could be attributed to much larger intrachain Ir–Ir exchange coupling strength compared to the interchain Ir–Ir magnetic exchange interactions. Despite the presence of slowly fluctuating iridium spins down to the lowest measured 1.7 K, such an appreciable spin-freezing at a rather high T would eventually reduce the extent of frustration in SLIO as is also discernible from the frustration index, $f = |\frac{\Theta_{cw}}{T_{freezing}}| \approx 0.22$, which can be further corroborated by the orders of magnitude apart intra and interchain Ir–Ir exchange interaction strengths shown earlier in our spin-polarized DFT calculations. Furthermore, it is worth to mention here that this spin-freezing mechanism at a reasonably higher temperature (<250 K) is observed only in zero or low fields. The frozen phase is apparently fragile with respect to the applied field as shown in dc magnetization measurements and from the absence of such a frozen phase in the $^7$Li NMR data at 93.954 kOe field, supporting possible presence of some inhomogeneity as the origin of spin freezing in SLIO.

A more comprehensive argument can therefore be posed that the ideal $J=0$ state is never stabilized in this material and spontaneously generated finite iridium moments exist unambiguously while the system homogeneously behaves as a paramagnet at higher temperature ($T > 250$ K). As the temperature is lowered through 200 K, ~40% of iridium moments undergo spin-freezing while the remaining ~60% of iridium spins continue to fluctuate, and upon cooling through 50 K the fluctuating Ir moments start to pair up to form singlets with vanishing magnetization [13] at low $T$ (<10 K) while a tiny proportion of isolated Ir$^{4+}$ magnetic defects [due to Li vacancy, oxygen off-stoichiometry, site-disorder, etc below the sensitivity of our structural and chemical characterizations] talk to each other with a very low residual AFM interaction. It is therefore quite plausible to consider the small fluctuating internal fields to arise from diluted magnetic centers out of a nonmagnetic background. This is consistent with the scenario proposed for $Ba_2YIrO_6$ [13] and $Ba_2YMoO_6$ [90]. Our $\mu$SR results further reinforce the existent tiny amounts of isolated magnetic impurity centers, which are a cause for non-saturating bulk susceptibility at low T and the nearly temperature-independent NMR Knight shift in SLIO, common in frustrated magnetic systems [13, 49, 91, 92].

### 3.10. Specific heat

The magnetic frustration of any magnetically disordered material is commonly expressed by the amount of magnetic entropy retained within the system at very low temperatures. To further check the magnetic ground state and also in order
<figure><img src="image_12.png" /><figcaption>**Figure 9.** (a) *T*-dependence of total specific heat *C*<sub>*p*</sub> in the low-*T* region for zero field as well as different applied magnetic fields, Upper Inset: Lattice part extraction (solid red line) of the zero field *C*<sub>*p*</sub> data (open black circles) using Debye–Einstein model, Lower Inset: Expanded view of *C*<sub>*p*</sub> versus *T* data at low temperatures (<15 K); (b) *T*-dependence of the magnetic specific heat *C*<sub>*m*</sub> at zero and applied magnetic fields, Inset: *T*-dependence of the magnetic entropy *S*<sub>*M*</sub> in zero and applied fields; (c) *T*-dependent magnetic specific heat *C*<sub>*m*</sub> data in the 0.05–4 K range for *H* = 0 and several applied magnetic fields, along with the respective fittings (solid colored lines), Inset: log–log plot of the *C*<sub>*m*</sub>*T* versus *T* data at each case along with applicable linear fitting (solid colored lines) for *H* = 0 and 3 Tesla fields; (d) the variation of the coefficients of the linear (green squares; left y-axis) and quadratic (red squares; right y-axis) terms is shown as a function of *H*; (e) magnetic field dependence of the two-level Schottky energy gap (open blue circles) and the linear fitting (solid red line).</figcaption></figure>

to probe the nature of low-energy magnetic excitations in this
d⁴ columnar iridate SLIO, the heat capacity Cₚ was measured
as a function of temperature in both zero field and several
applied higher magnetic fields down to 0.05 K. Considering
the fact that the lattice contribution is only negligible in the
measured 0.05–4 K Cₚ data, we can approximate the total Cₚ
as the Cₘ, the magnetic specific heat, in this low-temperature
region. The collected Cₚ (and also Cₘ in the 0.05–4 K range)
versus T data (see figures 9(a) and (c) corresponding to the 2–
300 K and 0.05–4 K range, respectively) do not show any sharp
λ-like anomaly, hallmark of thermodynamic phase transition
into a long-range magnetically ordered state, in any of the
applied magnetic fields, further affirming absence of magnetic
ordering and/or structural phase transition in SLIO till down
to 0.05 K. But, a very weak hump-like broad feature develops
below ~10 K (see bottom inset to figure 9(a)) which gradu-
ally shifts towards higher temperatures with the increase of
applied fields. In contrary to our previous study on antisite

disordered SNIO [12], an attempt to model this low-*T* weakly
field-dependent Cₚ(*T*) behavior with the 'two-level Schottky
anomaly' description fails (see figure S6 in SI), in agreement
with the disorder-free sample in the present case. This eventu-
ally refutes the paramagnetic impurity centers to be at the ori-
gin of observed weak field dependence in the low-*T* Cₚ versus
*T* data in SLIO [3, 64, 93]. So, the total Cₚ (2–300 K) can be
modeled by the sum of the lattice (C<sub>lattice</sub>), and magnetic (C<sub>m</sub>)
contributions. The C<sub>lattice</sub> was obtained after fitting the high-*T*
(180–300 K) Cₚ data using Debye–Einstein model (see SI for
details), yielding a Debye temperature Θ<sub>D</sub> ≈ 150 K. This fit is
then extrapolated down to the lowest measuring temperature
(see inset to figure 9(a)) and taken as the C<sub>lattice</sub>, which was
then subtracted from the total Cₚ, so that the sample is now
only left out with magnetic contribution to the heat capacity
due to local Ir<sup>5+</sup> moments.

Consequently, the *T*-dependence of the magnetic specific
heat *C*<sub>*m*</sub> is displayed in figure 9(b), which reveals a noticeable
field-dependence in the 2–10 K range, indicating short-range magnetic correlations in SLIO. As displayed in figure 9(b), the $C_m$ attains a broad maximum at ~32–42 K region, which does not show any change in applied fields up to 9 Tesla, thus reflecting its origin to be from intrinsic excitations and not any paramagnetic defects [13, 94, 95]. It is important to point out here in this context that the temperature region of this broad field-independent peak in the $C_m$ data closely resembles the exchange energy scale, $k_B\Theta_{CW}$. Moreover, such a broad field-independent behavior of the magnetic heat capacity in SLIO is quite common in reported several frustrated magnets [2, 60, 96].

The release in zero field magnetic entropy, $S_M$ [inset to figure 9(b)], obtained by integrating the zero field $C_M/T$ with $T$, achieves a value of ~6.12 J mol·K⁻¹ till 150 K, which is ≈68% of the maximum of $R\ln(2S + 1)$, with $S = 1$ corresponding to the number of unoccupied states accessed by the spin-orbit coupled 5d electrons, for a completely magnetically ordered state. This points towards a larger release of magnetic entropy (~30% entropy is retained within the system), implying lesser extent of geometrical exchange frustration in SLIO compared to the columnar SNIO counterpart [12]. This scenario of weakening in frustration in SLIO could again be accounted for by the small frustration index, $f$ (=$|\Theta_{cw}/T_{freezing}|$) ≈ 0.22, as well as the much stronger intra-column Ir–Ir exchange coupling over the inter-column Ir–Ir magnetic interaction strengths (see discussion in section 3.8). However, the SLIO still preserves a moderate percentage of magnetic entropy (≈30%), which together with slightly decreasing magnetic entropy release (i.e. a small enhancement in the retained entropy) in applied magnetic fields endorse persistence of spin-fluctuations and also low-energy spin-excitations in this material [5, 97–99].

We now critically look into the 0.05–4 K $C_m$ versus $T$ data. Interestingly, as displayed in the inset to figure 9(c), the ZF $C_m/T$ data has a negative log($T$) dependence with decreasing temperature, followed by a weak peak-like feature at ~0.1 K. As our combined dc, ac magnetizations, magnetic specific heat, NPD, and µSR data deny magnetic long-range ordering down to 0.05 K at least, the 0.1 K anomaly of the ZF $C_m/T$ data cannot be assigned to magnetic phase transition, rather it could be due to the dominance of magnetic fluctuations, resulting in a weak peak-like feature. Interestingly, the 0.1 K peak of the ZF data is gradually broadened in fields and evolved into a ‘two-level Schottky anomaly’-like feature (inset to figure 9(c)) which is gradually moved towards the higher temperatures with increasing field, a common occurrence in many frustrated magnetic materials [12, 100–103]. To have an idea about the nature of spin-excitations, We have now analyzed the 0.05–4 K $C_m$ versus $T$ data for both zero and applied fields (see main panel of figure 9(c)) using the following equation

$$C_m = \gamma \cdot T + \alpha \cdot T^2 + f \cdot C_{Sch} + D \cdot T \cdot (-\log T) \quad (3)$$

'f' represents paramagnetic impurity centers, while $C_{Sch}$ denotes the standard 'two-level Schottky anomaly' contribution, originating from that $f$% of free spins, with an expression $C_{Sch} = R \times (\frac{\Delta}{k_B \cdot T})^2 \times \frac{\exp(\Delta/k_B \cdot T)}{[1+\exp(\Delta/k_B \cdot T)]^2}$; where, $\Delta/k_B$ defines

the two-level Schottky energy gap, $\gamma$ being Sommerfeld coefficient typically observed in metals. The negative logarithmic temperature dependence of $C_m/T$ in both the zero and applied $H \le 30$ kOe (see inset to figure 9(c)), as defined in equation (3), is indicative of the presence of quantum criticality in the system [104, 105]. Notably, this -log $T$ dependence becomes insignificant for the higher applied fields (≥5 Tesla). Our fitting results (figure 9(c)) give an 'f' value varying between 0.5%–1.1%, implying presence of extremely low proportions of the impurity free spin centers in SLIO, which is in agreement with the CW fitting of the low-$T$ dc $\chi$ data. We obtained $f \sim 0.01$ from the 9 Tesla $C_m$ versus $T$ data fit, and kept this $f$ value fixed in all the applied fields. As shown in figure 9(e), the Schottky energy gap, $\Delta/k_B$, maintains a linear relation with $H$, and the estimated 'g' value of 1.9 closely matches with the ideal value of 'Lande-g factor', $g = 2$, for a free $S = 1/2$ spin of $5d^5$ Ir⁴⁺ ion. Furthermore, $C_m$ reveals a linear $T$-dependence, which together with finite $T$ linear term [$\gamma_{avg} \approx 10$ J mol·mol⁻¹·K⁻²], shown in the left y-axis of figure 9(d)], signify low-lying gapless spin excitations or a metal-like spinon Fermi surface of SLIO [2, 8, 13, 90, 94, 106–108]. Significantly, the $C_m$ data at very low $T$ unveil a $T^2$-dependency along with finite $T^2$-term [$C_m \sim \alpha \cdot T^2$, with $\alpha_{avg} \approx 5$ J mol·K⁻³ for $H \le 5$ Tesla and $\alpha_{avg} \approx 15$ J mol·K⁻³ for $H > 5$ Tesla, as shown in the right y-axis of figure 9(d)]. Most importantly, only a combined $\gamma T + \alpha T^2$ term gives rise to satisfactory fit to data. The quadratic $T$-dependence of $C_m$ is common in the spin-orbit coupled heavier 4d and 5d transition metal based frustrated quantum magnets as a signature of gapless spin excitation spectrum [61, 109, 110].

### 3.11. Comparative bandwidth versus magnetic moments study

Ultimately, with the conclusion of magnetic ground state in SLIO having intrinsic Ir⁵⁺ local moments (~0.45 $\mu_B$) we now critically investigate the trend of the theoretically predicted bandwidth $v_S$ magnetic moments (obtained from both the spin-polarized DFT calculations and the CW fittings on the experimental data) for three different $d^4$ columnar iridate systems Sr₃AlR₀₆ ($A$ = Li, Na, K) and a 6H-hexagonal perovskite iridate BZIO. The results are summarized in figure 10. It is to be noted here that the general trend of the bandwidth versus moment curves compliments each other very well. The higher values of the theoretical moments relative to the experimentally observed moments is quite natural. We infer that the $d^4$ iridates invariably possess finite magnetic moments, contrary to the expected nonmagnetic $J = 0$ ground state, and there is gradual systematic decrease in the spin-orbit coupled Ir⁵⁺ magnetic moments with increasing bandwidth, thus denying the pertaining standard notion of more atomic like SOC (i.e. isolated Ir) ensuring the realization of vanishing magnetization. Such an opposite trend could only be explained within the framework of moderate effective SOC strength for the $d^4$ iridates, which therefore naturally introduces a serious challenge to the general perception of strong SOC in iridates. Further, in support of the deviation from strong SOC picture in particular for $d^4$ iridates we highlight the fact that in a solid the
<figure><img src="image_13.png" /><figcaption>**Figure 10.** Bandwidth dependence of effective Ir⁵⁺-magnetic moments, obtained from experiments (shaded red circles with error bars) and calculations (shaded blue circles).</figcaption></figure>

strength of the SOC is renormalized as $\lambda_{\text{eff}} = \lambda_{\text{atomic}}/S$, where S is the total spin [111]. Consequently, for $d^4$ iridate systems with S = 1, SOC is much weaker than in $d^5$ iridates with S = 1/2, providing further credence of intermediate SOC regime for the $d^4$ iridates.

## 4. Summary and conclusion

In summary, we have investigated a 2H-hexagonal columnar spin chain $d^4$ iridate SLIO through combined structural, chemical, electronic, transport, magnetic and thermodynamic characterizations. Our bulk dc susceptibility, together with local probes $^7$Li NMR $\mu$SR and the spin polarized electronic structure calculations refute the coveted nonmagnetic $J_{\text{eff}} = 0$ ground state in a nearly stoichiometric structurally ordered SLIO, rather a large intrinsic local magnetic moment is developed on individual Ir$^{5+}$ ion of SLIO likely due to intersite Ir-Ir hopping (via extended Ir-O-O-Ir superexchange pathways within the intra- and inter-chain Ir-Ir exchange connectivity), trigonal distortion of the IrO$_6$ octahedra, hybridization of extended Ir-5d orbitals, Ir-O covalent interactions, and the bandwidth effect. Despite having significant AFM interaction between the local Ir$^{5+}$ moments, our combined heat capacity, dc, ac magnetic susceptibilities, $^7$Li NMR and $\mu$SR measurements discern absence of magnetic long-range ordering in SLIO down to 50 mK at least as a result of spin fluctuations

due to geometrical exchange frustration. On top of it, our ZF $\mu$SR investigation infer the coexistence of frozen and dynamic magnetism without altering their individual fractions in SLIO. The magnetic heat capacity $C_m$ at very low-T includes both the linear and quadratic T-dependence, conveying reminiscence to gapless spin excitations from a metal-like spinon Fermi surface. In summary, our combined experimental and theoretical studies point towards a gapless disordered magnetic ground state in SLIO comprising coexistent frozen and fluctuating spin dynamics. It appears that the real strength of SOC in the $d^4$ iridate systems may not be as strong to be solely described within the $j-j$ coupling limit.

## Data availability statement

All data that support the findings of this study are included within the article (and any supplementary files).

## Acknowledgments

A B thanks EPSRC UK for the funding (Grant No. EP/W00562X/1). A B also acknowledges CSIR, India and IACS for supporting fellowship. A C acknowledges Indian Institute of Technology, Kanpur and Science and Engineering Research Board (SERB) National Postdoctoral Fellowship (PDF/2021/000346), India for funding. S R acknowledges the Department of Science and Technology (DST) [Project No. CRG/2023/001060] for support. A B and S R thank Jawaharlal Nehru Centre for Advanced Scientific Research from DST-Synchrotron-Neutron project, for performing experiments at ESRF (Proposal No. HC-2872) synchrotron radiation facility, Grenoble, France. The authors further acknowledge the European Synchrotron Radiation Facility (ESRF) for provision of synchrotron radiation facilities and we would like to thank Dr M Moretti Sala and Dr A Efimenko for assistance and support in using beamline ID20. A B and S R thank ISIS muon facility, UK for providing facilities in performing the muon-spin-relaxation measurements. ILL is acknowledged for beamtime allocation under the experiment code EASY-1107, doi: 10.5291/ILL-DATA.EASY-1107. The Ir-$L_3$ XAFS measurements have been carried out in Elettra, Italy. The authors also thank the Indian Institute of Technology, Bombay, and the Technical Research Center of Indian Association for the Cultivation of Science, Kolkata, for support in research.

## ORCID iDs

Abhisek Bandyopadhyay https://orcid.org/0000-0001-9332-8985

C Ritter https://orcid.org/0000-0003-3674-3378

D T Adroja https://orcid.org/0000-0003-2280-079X

C Meneghini https://orcid.org/0000-0003-4846-8422

P K Biswas https://orcid.org/0000-0002-7367-5960

I Dasgupta https://orcid.org/0000-0002-6012-9530

Sugata Ray https://orcid.org/0000-0001-7180-1241
References

[1] Savary L and Balents L 2017 *Rep. Prog. Phys.* **80** 016502

[2] Balents L 2010 *Nature* **464** 199–208

[3] Dey T, Mahajan A V, Khuntia P, Baenitz M, Koteswararao B and Chou F C 2012 *Phys. Rev.* B **86** 140405

[4] Quilliam J A, Bert F, Kermarrec E, Payen C, Guillot-Deudon C, Bonville P, Baines C, Luetkens H and Mendels P 2012 *Phys. Rev. Lett.* **109** 117203

[5] Kitagawa K, Takayama T, Matsumoto Y, Kato A, Takano R, Kishimoto Y, Bette S, Dinnebier R, Jackeli G and Takagi H 2018 *Nature* **554** 341–5

[6] Kayser P, Kennedy B J, Ranjbar B, Kimpton J A and Avdeev M 2017 *Inorg. Chem.* **56** 2204

[7] Bremholm M, Yim C K, Hirai D, Climent-Pascual E, Xu Q, Zandbergen H W, Ali M N and Cava R J 2012 *J. Mater. Chem.* **22** 16431

[8] Nag A *et al* 2016 *Phys. Rev. Lett.* **116** 097205

[9] Nag A *et al* 2018 *Phys. Rev.* B **97** 064408

[10] Bandyopadhyay A, Carlomagno I, Simonelli L, Sala M M, Efimenko A, Meneghini C and Ray S 2019 *Phys. Rev.* B **100** 064416

[11] Laguna-Marco M A *et al* 2020 *Phys. Rev.* B **101** 014449

[12] Bandyopadhyay A *et al* 2022 *Phys. Rev.* B **105** 104431

[13] Nag A *et al* 2018 *Phys. Rev.* B **98** 014431

[14] Khan M S, Bandyopadhyay A, Nag A, Kumar V, Mahajan A V and Ray S 2019 *Phys. Rev.* B **100** 064423

[15] Lefrançois E, Chapon L C, Simonet V, Lejay P, Khalyavin D, Rayaprol S, Sampathkumaran E V, Ballou R and Adroja D T 2014 *Phys. Rev.* B **90** 014408

[16] Niitaka S, Kageyama H, Kato M, Yoshimura K and Kosuge K 1999 *J. Solid State Chem.* **146** 137

[17] Mikhailova D *et al* 2012 *Phys. Rev.* B **86** 134409

[18] Mohapatra N, Iyer K K, Rayaprol S and Sampathkumaran E V 2007 *Phys. Rev.* B **75** 214422

[19] Flahaut D, Hébert S, Maignan A, Hardy V, Martin C, Hervieu M, Costes M, Raquet B and Broto J M 2003 *Eur. Phys. J.* B **35** 317

[20] Shi Y, Guo Y, Yu S, Arai M, Sato A, Belik A A, Yamaura K and Takayama-Muromachi E 2010 *J. Am. Chem. Soc.* **132** 8474

[21] Kan E, Wu F, Lee C, Kang J and Whangbo M-H 2011 *Inorg. Chem.* **50** 4182

[22] Calder S, Lumsden M D, Garlea V O, Kim J W, Shi Y G, Feng H L, Yamaura K and Christianson A D 2012 *Phys. Rev.* B **86** 054403

[23] Porter D G, Senn M S, Khalyavin D D, Cortese A, Waterfield-Price N, Radaelli P G, Manuel P, zur-Loye H-C, Mazzoli C and Bombardi A 2016 *Phys. Rev.* B **94** 134404

[24] Carvajal J R 1993 *Physica B* **192** 55

[25] Cicco A D, Aquilanti G, Minicucci M, Principi E, Novello N, Cognigni A and Olivi L 2009 *J. Phys.: Conf. Ser.* **190** 012043

[26] Meneghini C, Bardelli F and Mobilio S 2012 *ESTRA-FitEXA: A software package for EXAFS data analysis* *Nucl. Instrum. Methods Phys. Res.* B **285** 153–7

[27] Jana S, Singh V, Kaushik S D, Meneghini C, Pal P, Knut R, Karis O, Dasgupta I, Siruguri V and Ray S 2010 *Phys. Rev.* B **82** 180407(R)

[28] Bandyopadhyay A, Neogi S K, Paul A, Meneghini C, Dasgupta I, Bandyopadhyay S and Ray S 2017 *Phys. Rev.* B **95** 024432

[29] Ankudinov A L, Ravel B, Rehr J J and Conradson S D 1998 Real-space multiple-scattering calculation and interpretation of x-ray-absorption near-edge structure *Phys. Rev.* B **58** 7565–76

[30] James F MINUIT—function minimization and error analysis *Reference Manual (Report)*. Version 94.1 (CERN Program Library Long Writeup. D506) Archived from the original on 2008-05-26

[31] Sala M M *et al* 2018 A high-energy-resolution resonant inelastic x-ray scattering spectrometer at ID20 of the European Synchrotron Radiation Facility *J. Synchrotron Radiat.* **25** 580–91

[32] Perdew J P, Burke K and Ernzerhof M 1996 *Phys. Rev. Lett.* **77** 3865

[33] Kresse G and Hafner J 1993 *Phys. Rev.* B **47** 558

[34] Kresse G and Furthmüller J 1996 *Phys. Rev.* B **54** 11169

[35] Blöchl P E 1994 *Phys. Rev.* B **50** 17953

[36] Kresse G and Joubert D 1999 *Phys. Rev.* B **59** 1758

[37] Chakraborty A and Dasgupta I 2019 *J. Magn. Magn. Mater.* **492** 165798

[38] Dudarev S L, Botton G A, Savrasov S Y, Humphreys C J and Sutton A P 1998 *Phys. Rev.* B **57** 1505–9

[39] Andersen O K and Saha-Dasgupta T 2000 *Phys. Rev.* B **62** R16219

[40] Andersen O K, Saha-Dasgupta T and Ezhov S 2003 *Bull. Mater. Sci.* **26** 19

[41] Andersen O K and Jepsen O 1984 *Phys. Rev. Lett.* **53** 2571

[42] Segal N, Vente J F, Bush T S and Battle P D 1996 *J. Mater. Chem.* **6** 395

[43] Chen Q *et al* 2017 *Phys. Rev.* B **96** 144423

[44] Kim B J *et al* 2008 *Phys. Rev. Lett.* **101** 076402

[45] Cao G *et al* 2013 *Phys. Rev.* B **87** 155136

[46] Cao G, Durairaj V, Chikara S, DeLong L E, Parkin S and Schlottmann P 2007 *Phys. Rev.* B **76** 100402

[47] Lampe-Önnerud C, Sigrist M and zur Loye H C 1996 *J. Solid St. Chem.* **127** 25

[48] Chen G and Balents L 2011 *Phys. Rev.* B **84** 094420

[49] Dey T *et al* 2016 *Phys. Rev.* B **93** 014434

[50] Choy J-H, Kim D-K, Demazeau G and Jung D-Y 1994 *J. Phys. Chem.* **98** 6258

[51] Choy J-H, Kim D-K, Hwang S-H, Demazeau G and Jung D-Y 1995 *J. Am. Chem. Soc.* **117** 8557

[52] Sardar K, Petrucco E, Hiley C I, Sharman J D B, Wells P P, Russell A E, Kashtiban R J, Sloan J and Walton R I 2014 *Angew. Chem., Int. Ed.* **53** 10960–4

[53] Zhu W K, Lu C-K, Tong W, Wang J M, Zhou H D and Zhang S X 2015 *Phys. Rev.* B **91** 144408

[54] Nag A, Bhowal S, Sala M M, Efimenko A, Dasgupta I and Ray S 2019 *Phys. Rev. Lett.* **123** 017201

[55] Kusch M *et al* 2018 *Phys. Rev.* B **97** 064421

[56] Revelli A *et al* 2019 *Sci. Adv.* **5** eaav4020

[57] Das S, Halder A, Chakraborty A, Dasgupta I and Saha-Dasgupta T 2020 *Phys. Rev.* B **101** 184422

[58] Bhowal S and Dasgupta I 2020 *J. Magn. Magn. Mater.* **507** 166827

[59] Bhowal S and Dasgupta I 2021 *J. Phys. Cond. Mater.* **33** 45

[60] Okamoto Y, Nohara M, Aruga-Katori H and Takagi H 2007 *Phys. Rev. Lett.* **99** 137207

[61] Khuntia P, Manni S, Foronda F R, Lancaster T, Blundell S J, Gegenwart P and Baenitz M 2017 *Phys. Rev.* B **96** 094432

[62] Kumar D and Pramanik A K 2016 *J. Magn. Magn. Mater.* **409** 20

[63] Laguna-Marco M A, Kayser P, Alonso J A, Martínez-Lope M J, van V M, Choi Y and Haskel D 2015 *Phys. Rev.* B **91** 214433

[64] Corredor L T *et al* 2017 *Phys. Rev.* B **95** 064418

[65] Nag A and Ray S 2017 *J. Magn. Magn. Mater.* **424** 93

[66] Ming X, Wan X, Autieri C, Wen J and Zheng X 2018 *Phys. Rev.* B **98** 245123

[67] Kaushal N, Herbrych J, Alvarez G and Dagotto E 2021 *Phys. Rev.* B **104** 235135

[68] Hammerath F *et al* 2017 *Phys. Rev.* B **96** 165108

[69] Quilliam J A, Bert F, Colman R H, Boldrin D, Wills A S and Mendels P 2011 *Phys. Rev.* B **84** 180401
[70] Shimizu Y, Miyagawa K, Kanoda K, Maesato M and Saito G 2003 *Phys. Rev. Lett.* **91** 107001

[71] Bloembergen N, Purcell E M and Pound R V 1947 *Nature* **160** 475

[72] Bloembergen N, Purcell E M and Pound R V 1948 *Phys. Rev.* **73** 679

[73] Shockley A C, Bert F, Orain J-C, Okamoto Y and Mendels P 2015 *Phys. Rev. Lett.* **115** 047201

[74] Dey T *et al* 2017 *Phys. Rev. B* **96** 174411

[75] Yoshitake J, Nasu J and Motome Y 2016 *Phys. Rev. Lett.* **117** 157203

[76] Mendels P and Bert F 2010 *J. Phys. Soc. Japan* **79** 011001

[77] Lee K-S, Koo H-J and Whangbo M-H 1999 *Inorg. Chem.* **38** 9

[78] Hillier A D, Adroja D T, Kockelmann W, Chapon L C, Rayaprol S, Manuel P, Michor H and Sampathkumaran E V 2011 *Phys. Rev. B* **83** 024414

[79] McClarty P A, Hillier A D, Adroja D T, Khalyavin D D, Rayaprol S, Manuel P, Kockelmann W and Sampathkumaran E V 2020 *J. Phys. Soc. Japan* **89** 064703

[80] Calder S, Singh D J, Garlea V O, Lumsden M D, Shi Y G, Yamaura K and Christianson A D 2027 *Phys. Rev. B* **96** 184426

[81] Liu X *et al* 2012 *Phys. Rev. Lett.* **109** 157401

[82] Chen G, Balents L and Schnyder A P 2009 *Phys. Rev. Lett.* **102** 096406

[83] Takegami D *et al* 2020 *Phys. Rev. B* **102** 045119

[84] Paramekanti A, Singh D J, Yuan B, Casa D, Said A, Kim Y-J and Christianson A D 2018 *Phys. Rev. B* **97** 235119

[85] Aczel A A *et al* 2022 *Phys. Rev. Mater.* **6** 094409

[86] Li Y, Adroja D T, Biswas P K, Baker P J, Zhang Q, Liu J, Tsirlin A A, Gegenwart P and Zhang Q 2016 *Phys. Rev. Lett.* **117** 097201

[87] Mendels P, Bert F, de Vries M A, Olariu A, Harrison A, Duc F, Trombe J C, Lord J S, Amato A and Baines C 2007 *Phys. Rev. Lett.* **98** 077204

[88] Gardner J S *et al* 1999 *Phys. Rev. Lett.* **82** 1012

[89] Campbell I A, Amato A, Gygax F N, Herlach D, Schenck A, Cywinski R and Kilcoyne S H 1994 *Phys. Rev. Lett.* **72** 1291-4

[90] Vries M A de, Mclaughlin A C and Bos J-W G 2010 *Phys. Rev. Lett.* **104** 177202

[91] Olariu A, Mendels P, Bert F, Duc F, Trombe J C, Vries M A de and Harrison A 2008 *Phys. Rev. Lett.* **100** 087202

[92] Gomilsek M, Klanjsek M, Pregelj M, Luetkens H, Li Y, Zhang Q M and Zorko A 2016 *Phys. Rev. B* **94** 024438

[93] Kumar R, Sheptyakov D, Khuntia P, Rolfs K, Freeman P G, Rønnow H M, Dey T, Baenitz M and Mahajan A V 2016 *Phys. Rev. B* **94** 174410

[94] Clark L *et al* 2013 *Phys. Rev. Lett.* **110** 207208

[95] Yamashita S, Nakazawa Y, Oguni M, Oshima Y, Nojiri H, Shimizu Y, Miyagawa K, Kanoda K 2008 *Nat. Phys.* **4** 459

[96] Cheng J G, Li G, Balicas L, Zhou J S, Goodenough J B, Xu C and Zhou H D 2011 *Phys. Rev. Lett.* **107** 197204

[97] Kelly Z A, Gallagher M J and Mcqueen T M 2016 *Phys. Rev. X* **6** 041007

[98] Mustonen O, Vasala S, Sadrollahi E, Schmidt K P, Baines C, Walker H C, Terasaki I, Litterst F J, Baggio-Saitovitch E and Karppinen M 2018 *Nat. Commun.* **9** 1085

[99] Nguyen L T and Cava R J 2019 *Phys. Rev. Mater.* **3** 014412

[100] Kundu S *et al* 2020 *Phys. Rev. Lett.* **125** 267202

[101] Zhou H D, Choi E S, Li G, Balicas L, Wiebe C R, Qiu Y, Copley J R D and Gardner J S 2011 *Phys. Rev. Lett.* **106** 147204

[102] Han T-H, Chisnell R, Bonnoit C J, Freedman D E, Zapf V S, Harrison N, Nocera D G, Takano Y and Lee Y S arXiv: 1402.2693v1

[103] de Vries M A, Kamenev K V, Kockelmann W A, Sanchez-Benitez J and Harrison A 2008 *Phys. Rev. Lett.* **100** 157205

[104] Tripathi R *et al* 2022 *Phys. Rev. B* **106** 064436

[105] Löhneysen H v, Rosch A, Vojta M and Wölfle P 2007 *Rev. Mod. Phys.* **79** 1015

[106] Norman M R and Micklitz T 2009 *Phys. Rev. Lett.* **102** 067204

[107] Shen Y *et al* 2016 *Nature* **540** 559

[108] Ortiz B R *et al* 2023 *Nat. Phys.* **19** 943-9

[109] Zhou Y, Lee P A, Ng T-K and Zhang F-C 2008 *Phys. Rev. Lett.* **101** 197201

[110] Lawler M J, Paramekanti A, Kim Y B and Balents L 2008 *Phys. Rev. Lett.* **101** 197202

[111] Khomskii D I 2014 *Transition Metal Compounds* (Cambridge University Press)