††thanks: Equal contribution authors††thanks: Equal contribution
authors††thanks: corresponding author

#  Unusual spin dynamics in the low-temperature magnetically ordered state of
Ag3LiIr2O6

Atasi Chakraborty  School of Physical Sciences, Indian Association for the
Cultivation of Science, Jadavpur, Kolkata 700032, India  Vinod Kumar
Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai
400076, India  Sanjay Bachhar  Department of Physics, Indian Institute of
Technology Bombay, Powai, Mumbai 400076, India  N. Büttgen  Experimentalphysik
V, Elektronische Korrelationen und Magnetismus, Institut für Physik,
Universität Augsburg, 86135 Augsburg, Germany  K. Yokoyama  ISIS Pulsed
Neutron and Muon Source, STFC Rutherford Appleton Laboratory, Harwell Campus,
Didcot, Oxfordshire OX110QX, UK  P. K. Biswas  ISIS Pulsed Neutron and Muon
Source, STFC Rutherford Appleton Laboratory, Harwell Campus, Didcot,
Oxfordshire OX110QX, UK  V. Siruguri  UGC-DAE-Consortium for Scientific
Research Mumbai Centre, Bhabha Atomic Research Centre, Mumbai 400085, India
Sumiran Pujari  Department of Physics, Indian Institute of Technology Bombay,
Powai, Mumbai 400076, India  I. Dasgupta  School of Physical Sciences, Indian
Association for the Cultivation of Science, Jadavpur, Kolkata 700032, India
A.V. Mahajan  Department of Physics, Indian Institute of Technology Bombay,
Powai, Mumbai 400076, India

###### Abstract

Recently, there have been contrary claims of Kitaev spin-liquid behaviour and
ordered behavior in the honeycomb compound Ag3LiIr2O6 based on various
experimental signatures. Our investigations on this system reveal a low-
temperature ordered state with persistent dynamics down to the lowest
temperatures. Magnetic order is confirmed by clear oscillations in the muon
spin relaxation ($\mu$SR) time spectrum below 9 K till 52 mK. Coincidentally
in 7Li nuclear magnetic resonance, a wipe-out of the signal is observed below
$\sim$ 10 K which again strongly indicates magnetic order in the low
temperature regime. This is supported by our density functional theory
calculations which show an appreciable Heisenberg exchange term in the spin
Hamiltonian that favors magnetic ordering. The 7Li shift and spin-lattice
relaxation rate also show anomalies at $\sim$ 50 K. They are likely related to
the onset of dynamic magnetic correlations, but their origin is not completely
clear. Detailed analysis of our $\mu$SR data is consistent with a co-existence
of incommensurate Néel and striped environments. A significant and
undiminished dynamical relaxation rate ($\sim 5$ MHz) as seen in $\mu$SR deep
into the ordered phase indicates enhanced quantum fluctuations in the ordered
state.

##  I Introduction

Kitaev’s seminal proposal of bond-dependent magnetic interactions stabilizing
a novel $Z_{2}$ spin-liquid ground state with Majorana excitations, followed
by the important material-specific advance of Jackeli and Khaliullin [1] has
triggered significant experimental effort to synthesize such materials. They
advocated honeycomb lattice structures of $4d$/$5d$ element based oxides with
edge-sharing oxygen octahedra and strong spin-orbit coupling as having the
necessary ingredients to host the Kitaev model [2]. Several promising
candidates with such layered honeycomb structure have since been investigated:
Na2IrO3 [3, 4, 5], $\alpha$-Li2IrO3 (as also its three dimensional polymorphs)
[6, 7], and $\alpha$-RuCl3 [8, 9]. However, it has been revealed that these
materials order magnetically [10, 11, 12, 13, 14, 15, 9] due to the presence
of Heisenberg and other non-Kitaev terms, and the fingerprint of the Kitaev
interactions may only be realised either at higher temperatures or under
application of a magnetic field.

In this family of materials a new addition has been H3LiIr2O6 [16], where all
of the interlayer Li+ ions of $\alpha$-Li2IrO3 (LIO) are replaced by H+ ions,
retaining the LiIr2O6 planes. Various measurements have confirmed the absence
of magnetic ordering down to 0.05 K in H3LiIr2O6 [17] which has been argued to
be a spin-orbit entangled quantum spin-liquid [17]. To complicate matters
further, x-ray diffraction (XRD) has revealed the presence of stacking faults
between the honeycomb planes [16], and the low temperature behavior was
attributed to local moments induced by these defects. Theoretical ab-initio
calculations have also shown for these systems that although the bond-
dependent Kitaev interactions are significant, the Heisenberg and other non-
Kitaev terms are not negligible. It has been suggested that these systems lie
close to the tricritical point between ferromagnetic, zigzag and
incommensurate spiral order resulting in the absence of magnetic order [18].
Calculations further reveal that the interlayer O-H-O geometry as well as lack
of hydrogen order also strongly influence the Kitaev and other exchange
interactions having strong impact on its magnetic properties.[18, 19]

Very recently, the compound Ag3LiIr2O6 (ALIO) has been synthesized [20], where
proximate Kitaev spin-liquid physics has been claimed based on scaling
behaviour of various thermodynamic quantities in the presence of quenched
disorder and a two step release of magnetic entropy. Replacement of the
lighter H+ ions by the heavier Ag+ ions leads to an increase in the inter-
layer separation which can significantly influence the various magnetic
exchange interactions. Estimates of the magnetic interactions using ab-initio
study are currently lacking for this system.

We have been working on the hexagonal Ag3LiM2O6 (M = Mn, Ru, Ir) system with
the intention of developing a comprehensive understanding of this honeycomb
system. The Mn-based ($3d^{3}$ or $S=3/2$) system exhibits long-range order
below about 50 K. There is also evidence of Berezinskii-Kosterlitz-Thouless
behavior from an analysis of the electron spin resonance line broadening data
near the transition temperature [21]. The Ru-based system ($4d^{4}$ which
might be $S=1$ or $J=0$) was expected to be a possible candidate for excitonic
magnetism [22]. Our investigations however revealed that spin-orbit coupling
might not be significant in this case. But the expected long-range ordered
state was not seen in spite of the apparently unfrustrated geometry of the
honeycomb lattice. Rather, the local moments were found to be on the
borderline of being dynamic and static at low temperatures, based on muon spin
relaxation ($\mu$SR) data [23]. Such behavior is likely driven by higher order
bi-quadratic or ring exchange terms in the Hamiltonian over and above the
usual Heisenberg couplings.

Continuing our investigations in this series of systems, we focus in this
paper on the Ir analog containing Ir4+ ions ($J_{\text{eff}}=1/2$) with the
purpose of looking for possible Kitaev spin-liquid physics due to the enhanced
spin-orbit coupling of the Ir moments. Our experimental and theoretical
results as itemized below are however quite far from such expectations:

  * •

We observe clear oscillations in $\mu$SR relaxation data below $\sim 9$K
providing strong evidence for magnetic order below this temperature. Analysis
of the muon data in the ordered state, complemented with density function
theory (DFT) simulations of the muon stopping site points towards the co-
existence of incommensurate Néel and stripe ordered magnetic domains.

  * •

We also provide theoretical estimates for the various magnetic interactions in
ALIO via DFT-based computations. We find the ratio of the nearest neighbor
Kitaev exchange ($K_{1}$) to the Heisenberg exchange ($J_{1}$) to be in the
range $|\frac{K_{1}}{J_{1}}|$ $\sim$ 2.0-3.5, placing the ALIO system far from
the pure Kitaev limit, and closer to the phase boundaries between stripe, Néel
and 120∘ order of the phase diagram [24].

  * •

We observe two anomalies in the 7Li nuclear magnetic resonance (NMR) shift
variation with temperature. The first one that is present at $T\sim$ 10 K
clearly signifies the onset of magnetic long-range order as seen from the so-
called wipe-out of the NMR signal, which agrees with our conclusions from muon
data.

  * •

The second anomaly is a broad maximum at about 50 K in the temperature
variation of the 7Li NMR line shift (which tracks the intrinsic spin
susceptibility) and is reminiscent of such features in quasi-one and quasi-two
dimensional Heisenberg antiferromagnetic systems due to short-range magnetic
correlations. The 7Li NMR 1/$T_{1}$ also has a maximum at $T\sim$ 50 K.
Together with the observed progressive loss of the NMR intensity at this
temperature, this could be signifying dynamic short-range magnetic
correlations similar to those seen in CeCu2Si2 [25, 26].

  * •

Finally, we also find a large value of the muon relaxation rate ($\sim 5$ MHz)
that remains essentially flat and undiminished deep into the ordered phase,
i.e. down to 52 mK which is about $1/200^{\text{th}}$ of the ordering
temperature of $\sim 10$ K. This is quite striking and noteworthy, rather
reminiscent of spin-liquid behavior [27] in spite of the unambiguous evidence
for magnetic order mentioned above (i.e., clear oscillations in $\mu$SR
relaxation data.) We interpret this as a signature of persistent spatio-
temporal fluctuations of the Néel and stripe ordered domains, possibly driven
by quantum effects given our theoretical estimates of the various magnetic
exchange energy scales in ALIO.

Our results are in sharp contrast with a recent report on the same compound
[28]. This report was based on the low-temperature scaling behaviors as stated
before, and in particular, on an apparent two step release of magnetic entropy
suggesting ALIO may be a proximate Kitaev spin-liquid. We argue that the
(extracted) magnetic specific heat in the high-temperature regime is very
uncertain. This is simply due to the overwhelming contribution of the lattice
to the total specific heat, especially in the high-temperature region
($T\gtrsim$ 30 K). Consequently, the inference of the high-temperature peak
(position and magnitude) and that of a two-step entropy release is rather
insecure, let alone ascribing it to Majorana excitations. We note here that an
even more recent report [29] has found evidence for magnetic ordering in a
cleaner batch of samples in line with our observations.

The remainder of the paper is organised as follows: we start by giving the
details on the structure of ALIO and relevant technical details on
measurements and theoretical methods in Sec. II. We next present our main
pieces of experimental evidence that establish a low-temperature magnetic
ordered state coming from $\mu$SR and NMR data in Sec. III. This is followed
by a presentation of our theoretical estimates for the various magnetic
exchange couplings based on DFT in Sec. IV. Sec. V is devoted to a detailed
discussion of our observations using bulk probes (heat capacity and
susceptibility) and the high temperature anomaly in NMR vis-a-vis long-range
magnetic order below 10 K as inferred from our observations and computation,
versus Kitaev spin-liquid scenario as in Ref. [28]. Concluding remarks are
given in Sec. VI.

##  II Structure and technical details

Figure 1: (a) shows the unit cell of Ag3LiIr2O$6$. The edge shared IrO6
honeycomb network is shown (b). The $x,y,z$ local axes point toward the
transition metal (Ir) to ligand (O) direction. X (red dotted line), Y (green
dotted line) and Z (blue dotted line) type Ir-Ir nearest-neighbor bonds are
perpendicular to the chosen $x,y,z$ local axes respectively.

ALIO crystallizes in base centered monoclinic symmetry having space group
C2/m. The crystal structure is shown in Fig. 1(a). The magnetic building block
consists of Ir in an octahedral environment with nearest neighbor ($nn$)
oxygen ligands. The IrO6 octahedra form an edge shared honeycomb geometry in
the a-b plane containing Li ions at the center (see Fig. 1(b)). The honeycomb
layers in ALIO are identical to its parent compound, $\alpha-$LIO but the
chemical bonds between the layers are modified. The interlayer Li atoms in
$\alpha-$LIO are octahedrally coordinated with six oxygens in the two adjacent
O6 honeycomb layers, whereas the Ag atoms in ALIO are linearly connected to
two oxygens in neighboring layers making a 1800 O-Ag-O bond-angle.

Polycrystalline samples of ALIO were prepared by a two-step process as
described in the supplementary information (SM) [30]. The resulting product
was Ag3LiIr2O6 as verified by lab x-ray diffraction measurements using a
PANalytical X’Pert PRO diffractometer using Cu-K${\alpha}$ radiation
($\lambda=1.54182$Å). Small amounts of residual Ag and $\alpha$-Li2IrO3 were
detected in the x-ray diffraction pattern.

The magnetization measurements have been performed in a Quantum Design SQUID
Vibrating Sample Magnetometer in the temperature range 2-400 K and in applied
fields ranging from 0 Oe to 70 kOe. The heat capacity measurements have been
done in a Quantum Design PPMS in the temperature range 2-250 K, in various
field values in the range 0-90 kOe. $\mu$SR measurements were carried out
using the MUSR spectrometer at the ISIS Neutron and Muon Source at the STFC
Rutherford Appleton Laboratory in the UK. The powder sample was loaded on a
silver sample holder to minimize the background signal. The holder was then
mounted on a dilution refrigerator insert and a standard cryostat stick for
measuring temperatures ranging from 50 mK up to 150 K. 7Li NMR measurements
have been performed in a fixed field of 93.954 kOe, using a Tecmag
spectrometer in a continuous flow cryostat in the temperature range of 4-300
K. Measurements have also been performed in a swept field magnet down to 1.5 K
at various frequencies (and therefore, fields). From our measurements, we
obtained 7Li NMR spectra, spin-lattice (1/$T_{1}$) and spin-spin (1/$T_{2}$)
relaxation rates as a function of temperature in various fields.

The first-principles electronic structure calculations in the framework of
density functional theory (DFT) are carried out within the generalized
gradient approximation (GGA) for the exchange-correlation functional following
the Perdew-Burke-Ernzerhof prescription. We have employed the plane-wave basis
as implemented within the Vienna Ab initio Simulation Package (VASP) [31, 32]
with projector augmented wave potentials [33, 34] as well as in the
$N^{\text{th}}$-order muffin-tin orbital (NMTO) and linear muffin-tin orbital
(LMTO) basis sets as implemented in the STUTTGART code [35]. The consistency
between the two sets of calculations in two choices of basis sets is cross-
checked in terms of band structure, density of states etc. The VASP
calculations are done with usual values of Coulomb correlation $U$ [36] and
Hund’s coupling ($J_{H}$) chosen for Ir with $U_{eff}(\equiv U-J_{H})=1.5$ eV
in Dudarev scheme [37]. The details of the VASP, LMTO and NMTO calculations
are described in the SM [30].

##  III The low temperature ordered state of ALIO

###  III.1 $\mu$SR

The depolarisation of the muons as a function of temperature is shown in Fig.
2. At high temperatures, a slow (Gaussian-like) decay of the muon polarisation
is seen whereas below about 20 K, a faster (exponential-like) decay is
discernible which gets even faster with a decrease in temperature. Finally,
below about 9 K, clear oscillations in the muon asymmetry as a function of
time are seen. Fits of the time decay of the muon asymmetry at various
temperatures have then been carried out to obtain the variation of the local
moment dynamics with temperature.

Figure 2: Variation of the muon asymmetry with time is shown at selected
temperatures. Clear oscillations are seen below about 9 K indicative of long-
range magnetic order. Fits at some representative temperatures are shown as
explained in the text.

We find that at temperatures above 15 K, the data are well fit to a product of
a static Kubo-Toyabe function with an exponential in addition to a constant
background $A_{0}$ i.e.,
$\mathit{A\mathrm{(}t\mathrm{)}}=A_{2}G_{KT}(\Delta,t)e^{-\lambda_{2}(T)t}+A_{0}$.
Here, $G_{KT}(\Delta,t)$ is the Kubo-Toyabe function which models the
relaxation of muons in a Gaussian distribution of magnetic fields from nuclear
moments. From these fits we obtain the field distribution $\Delta$ to be about
1.6 Oe. This value is typical of nuclear dipolar fields at the muon site, in
the present case arising from 107,109Ag, 6,7Li, and 191,193Ir nuclei. The
exponential term $e^{-\lambda_{2}(T)t}$ arises from the relaxation due to
fluctuations of the electronic local moments. This relaxation rate
$\lambda_{2}(T)$ is small at high temperatures and gradually increases as the
local moment fluctuation rate gets smaller (see Fig. 4). We notice a sharper
increase of $\lambda_{2}(T)$ below about 20 K.

The intermediate region of 13 K to 9 K shows a sharply falling muon asymmetry
with $\lambda_{2}(T)$ showing a sharp increase with decreasing temperature as
an approach to long-range order. Going further down in temperature, we find
that below 9 K, there are clear oscillations in the muon asymmetry as a
function of time. This is a classic signature of the presence of long-range
magnetic order. The temporal decay of the muon asymmetry is nearly unchanged
from 7 K down to 52 mK. The data in the range 52 mK-7 K were well fit (see
Fig. 3) by the following equation:

$\begin{split}A(t)&=A_{0}+A_{1}e^{-\lambda_{1}(T)t}J_{0}(\gamma H_{1}t)\\\
&+A_{2}e^{-\lambda_{2}(T)t}J_{0}(\gamma H_{2}t)+A_{3}e^{-\lambda_{3}(T)t}\\\
\end{split}$ (1)

where $\gamma$ is the muon gyromagnetic ratio ($\gamma=2\pi\times 135.539$
MHz/Tesla).

Figure 3: Variation of the muon asymmetry with time is shown at 52 mK together
with a fit to Eq. 1. The fit parameters are nearly unchanged up to 7 K. Figure
4: Variation of the muon relaxation rate (main figure) and the local field at
the muon site (inset) as a function of temperature for ALIO is shown.

The $A_{3}e^{-\lambda_{3}(T)t}$ term is ascribed to muons which are initially
parallel to the internal field components and hence do not precess. As for the
other significant terms, $J_{0}(\gamma H_{1}t)$ and $J_{0}(\gamma H_{2}t)$ are
zeroth order Bessel functions. In case of ordering that is commensurate with
the lattice, one expects an exponentially damped sinusoidal variation of
$A(t)$ in the ordered state. The Bessel function variation observed here is
indicative of magnetic order incommensurate with the lattice [38] (such as for
a spin density wave), where the muon experiences fields upto a maximum of
$H_{1}$ or $H_{2}$, in the present case. We found that the fit was better with
two Bessel functions rather than one (see SM [30] for a comparison),
suggesting the presence of two types of magnetic environments for the muons.
These could arise either from the presence of two kinds of regions with
different spin order, or possibly from crystallographically inequivalent muon
stopping sites.

From our DFT calculations (see Sec. IV for details), we conclude that while
the Néel type order has the lowest energy, the stripy phase is not much higher
in energy, which lends credence to the first possibility. Thus assuming this
scenario of a single muon stopping site, the site was determined from
calculations of the electrostatic energy and was found to be about 1 Å from
the oxygen ion (see SM [30] for details). This is similar to that found in
cuprates and other oxide materials [39]. We then calculated the dipolar
magnetic fields at the muon stopping site in Néel and stripy environments
respectively. The calculated field values of 139 Oe for the Néel phase and 266
Oe for the stripy phase (assuming a moment of 0.5 $\mu_{B}$/Ir which is
typical for Ir4+) are in reasonable register with the values of 129 Oe and 232
Oe obtained as the averages of fit parameters $H_{1}$ and $H_{2}$ respectively
between 52 mK to 1.4 K (see inset of Fig. 4).

Finally, we look at the variation of the muon relaxation rates $\lambda_{1}$
and $\lambda_{2}$ vs $T$. It shows a peak at about 9 K, but does not fall to
low values even at 52 mK as is expected to happen for progressively slower
dynamics as we go deeper into the ordered state (Fig. 4). It rather stays
almost flat and undiminshed at a value of about 5 MHz in the low temperature
side as seen in Fig. 4. We speculate that persistent spatio-temporal
fluctuations of the stripy and Néel regions are responsible for this. In such
a scenario, this would present an interesting example where these fluctuations
persist even at temperatures more than two orders of magnitude lower than the
transition temperature ($\sim$ 9 K) till at least a thermal energy scale of
$\sim$ 50 mK (4.3 $\mu$eV). Could these then be quantum mechanical in origin?

From a quantitative point of view, the rather large value of 5 MHz for the
muon depolarisation rate is quite remarkable, comparable to (or even larger
than) those seen in a variety of spin liquids (for example, see the review
Ref. [27] and the references therein; also see recent Refs. [40, 41] not
covered in this review). Even in $\alpha$-RuCl3 which has been established to
be magnetically ordered, the muon relaxation rate is appreciable ($\sim 0.5$
MHz in a single crystal [42] and $\sim 2$ MHz/$\sim 4$ MHz in a
polycrystalline sample [43]) though smaller compared to ALIO. However, these
observations on $\alpha$-RuCl3 are limited respectively to roughly $1/2$ the
ordering temperature in the single crystal work [42], and about
$1/5^{\text{th}}$ the ordering temperature in the polycrystalline sample study
[43]. Whereas our observations on ALIO go down to $1/200^{\text{th}}$ of the
ordering temperature and can be considered well-representative of the ground
state physics. This suggests that the persistent dynamics are really a feature
of the magnetically ordered many-body ground state.

###  III.2 NMR

Figure 5: The 7Li NMR shift increases with decreasing temperature and then
shows a broad plateau below 50 K.  Figure 6: The variation of he 7Li NMR spin-
lattice relaxation rate is shown with $T$. This also shows a peak at about 50
K where there is a susceptibility anomaly.

Having established the presence of magnetic order in ALIO from our zero field
$\mu$SR data at low temperatures, we now move over to the local probe
technique of NMR to examine the variation of the intrinsic spin susceptibility
in the paramagnetic state, as well as to look for complementary evidence of
ordering. The bulk susceptibility on the other hand can have a low-$T$ upturn
arising from extrinsic contributions or orphan spins which may not be
reflective of the intrinsic properties. We have therefore performed 7Li local
probe NMR measurements to determine the shift of the 7Li resonance (7K) with
respect to a diamagnetic reference as a function of temperature. The results
are shown in Fig. 5 where it is seen that the intrinsic susceptibility (in the
form of a 7K line shift) increases with decreasing temperature and then
exhibits a broad plateau region below about 50 K.

Furthermore, as NMR is a good probe of low-energy excitations, we have
performed 7Li NMR spin-lattice relaxation rate measurements as a function of
temperature. The recovery of the longitudinal nuclear magnetisation after a
saturating pulse sequence was well fitted with a single exponential. As shown
in Fig. 6, we find that 1/$T_{1}$ increases with decreasing temperature and
has a peak around 50 K similar to systems which show ordering. The 7K and
1/$T_{1}$ results are found independent of the applied field between 18-94
kOe.

We now contrast our observations with those reported in a very recent Ref.
[44] performed on two different batches of samples (A and B) of ALIO. Wang et
al. found that the cleaner sample A showed a single peak in the NMR spectrum
as opposed to two peaks for sample B. The longitudinal nuclear magnetisation
recovery for sample A was found to be single exponential in contrast to a
stretched exponential variation for sample B. Also, the values of the
relaxation rate were higher for sample A compared to sample B. The near
absence of a second peak in the NMR lineshape of our sample of ALIO, together
with a single exponential recovery in $T_{1}$ and with the absolute value of
the 7Li NMR relaxation rate 1/$T_{1}$ on the higher side (similar to sample A
of Ref. [44]) suggests that our sample is of high quality.

Figure 7: The variation of the 7Li NMR spin-spin relaxation rate (left
$y$-axis) is shown with $T$. The right $y$-axis shows the product of the
nuclear magnetisation and temperature normalized to the maximum, as a function
of $T$.

We also monitored the total NMR spectral intensity as a function of
temperature which naturally involves the measurement of the spin-spin
relaxation rate (1/$T_{2}$). The total spectral intensity should normally vary
as a Curie law due to a Curie variation of the nuclear magnetisation. Hence,
the product of the nuclear magnetisation $M_{0}$ and $T$ should remain
constant with temperature in case the same number of nuclei contribute to the
signal at all temperatures. Our results for 1/$T_{2}$ and $M_{0}T$ are shown
in Fig. 7. We observe an onset of a decrease in intensity already around 50 K
and a near complete wipe-out below 10 K. The observed wipe-out is a classic
signature of the onset of long-range magnetic ordering. An anomaly is also
seen in the 1/$T_{2}$ data around 10 K. We thus conclude that there is an
onset of short-range correlations around 50 K and eventually long-range
magnetic ordering sets in at about 10 K.

##  IV  First Principles Electronic Structure Calculations

In order to gain a microscopic understanding of the electronic and magnetic
behavior, we first optimised the crystal structure parameters using first-
principles density functional theory calculations. The crystal structure of
ALIO has been shown in Fig. 1, and we recall that the various exchange
parameters, in particular the Kitaev exchange interaction are strongly
dependent on the nearest-neighbor ($nn$) bond-length and bond-angles. To
cross-check the experimental refinement of the position of the light atoms
(such as O) based on X-ray diffraction, we have independently determined the
structural parameters by carrying out an ionic relaxation simulation for ALIO
while maintaining the crystal symmetries of space group C2/m (see SM [30] for
details). A comparison of the structural data for the experimental and the
relaxed structures are presented in Table 1 along with the structural data of
H3LiIr2O6 (HLIO) and the parent compound $\alpha$-LIO with similar
stoichiometries. We find for the relaxed structure of ALIO, the Ag-O bond
lengths along the O-Ag-O are nearly equal and substantially larger in
comparison to its H counterpart while the Ir-O-Ir angle for the Z-bond are
nearly identical for both the systems. All the subsequent calculations are
performed with this relaxed structure of ALIO.

Table 1: Bond lengths are given in Å and bond-angles are in
degrees$(^{\circ})$.

Parameters |  $\alpha$-Li2IrO3 | H3LiIr2O6 | Ag3LiIr2O6 | Ag3LiIr2O6
---|---|---|---|---
| (exp) [45] | (exp) [18] | (exp) [20] | (relax)
Ir-Ir distance |  |  |  |
(X/Y-bond) | 2.98 | 3.10 | 3.06 | 3.04
Ir-Ir distance |  |  |  |
(Z-bond) | 2.98 | 3.05 | 3.04 | 3.08
Ir-O-Ir angle |  |  |  |
(X/Y-bond) | 94.74 | 99.77 | 98.55 | 97.76
Ir-O-Ir angle |  |  |  |
(Z-bond) | 95.42 | 99.03 | 92.43 | 100.08
(Li,H,Ag)-O |  |  |  |
distance | 1.88, 2.13 | 1.23, 1.27 | 1.94, 2.09 | 2.08, 2.10

We begin with an investigation of the electronic structure of ALIO without
magnetic order. The results of our calculations are summarized in the top
panel of Fig. 8, where we have plotted the total as well as the Ir projected
density of states (DOS). We find that the octahedral environment of Ir splits
its $d$ states into $t_{\mathrm{2g}}$ and $e_{\mathrm{g}}$ states by a large
crystal field splitting ($\Delta_{CF}$ $\sim$ 4.3 eV) characteristic of
iridates. The $t_{\mathrm{2g}}$ states are further split due to monoclinic
distortion and host the Fermi level. The oxygen states are completely
occupied, while the Ir-$e_{\mathrm{g}}$, Li-$s$ and Ag-$d$ states are
completely empty consistent with the nominal ionic formula
Ag${}^{1+}_{3}$Li+1Ir${}^{4+}_{2}$O${}^{-2}_{6}$.

In view of the large $\Delta_{CF}$, the entire physics of these systems is
essentially governed by the $t_{\mathrm{2g}}$ states. We have therefore
constructed a low energy tight-binding model retaining only the
$t_{\mathrm{2g}}$ states in a local basis (where the local $x$, $y$, and $z$
axes point towards the ligands) and downfolded all other higher degrees of
freedom using the NMTO downfolding method [35]. The C2/m group splits the
$t_{\mathrm{2g}}$ states retaining two-fold symmetry at each metal site. The
various hopping interactions between the Ir atoms reveal that the hopping
corresponding to the first $nn$ is stronger compared to the other
interactions. The C2/m space group provides two types of symmetry inequivalent
nearest-neighbor bonds: (i) Z bonds, parallel to the crystallographic
$b$-axis, are of local C2h symmetry; (ii) X or Y bonds with lower symmetry.
The nearest neighbor $d$-$d$ hopping integrals for the Z-bond expected for the
ideal structure are extracted from NMTO downfolding calculations by suitable
averaging and the values are $t_{1}\equiv t_{xz,xz}=t_{yz,yz}$ = -49.1 meV,
$t_{2}\equiv t_{xz,yz}=t_{yz,xz}$ = 204.0 meV, $t_{3}\equiv t_{xy,xy}$ =-1.4
meV, $t_{4}\equiv t_{xz,xy}=t_{xy,xz}=t_{yz,xy}=t_{xy,yz}$ = 46.2 meV. While
$t_{2}$ is strongest as expected, $t_{3}$ is however strongly suppressed. This
will have a profound impact on the exchange intercations to be discussed later
in the section.

Figure 8: Non-magnetic total density of states (DOS) (brown shaded region) and
partial Ir-$d$ DOS (cyan shaded region) are shown. Top and bottom panels show
DOS for GGA calculations without and with spin-orbit coupling.

We now present the effect of spin-orbit coupling (SOC) on the electronic
structure of ALIO. The total and the Ir projected DOS are displayed in Fig. 8.
We find that the SOC further splits the $t_{2g}$ states into low-lying, four-
fold degenerate $J_{\text{eff}}=3/2$ and high-lying two-fold degenerate
$J_{\text{eff}}=1/2$ states. The five electrons of nominal Ir4+ completely
fill the $J_{\text{eff}}=3/2$ states, while the $J_{\text{eff}}=1/2$ state is
half-filled which upon inclusion of a moderate Hubbard interaction $U$ makes
the system insulating. In the limit $U>>t$, the holes occupying the
$J_{\text{eff}}=1/2$ states are nearly localized on the metal sites and the
low energy degrees of freedom are pseudo-spin-1/2 variables $\mathbf{S_{i}}$
connected to the $J_{\text{eff}}=1/2$ states.

This spin-orbit entangled pseudo-spin state of Ir atom on a honeycomb lattice
in the strong coupling limit hosts bond dependent anisotropic Kitaev exchanges
in addition to the usual isotropic Heisenberg exchange terms. The nearest
neighbor spin Hamiltonian may be written as $H_{\text{spin}}=\sum_{\langle
i,j\rangle}\mathbf{S}_{i}\cdot\mathbf{J}_{ij}\cdot\mathbf{S}_{j}$, where
$\mathbf{J}_{ij}$ is a 3 $\times$ 3 symmetric matrix due to the presence of
local inversion symmetry and is given by:

$\begin{bmatrix}J_{1}&\Gamma_{1}&\Gamma^{\prime}_{1}\\\
\Gamma_{1}&J_{1}&\Gamma^{\prime}_{1}\\\
\Gamma^{\prime}_{1}&\Gamma^{\prime}_{1}&J_{1}+K_{1}\\\ \end{bmatrix}\quad$ (2)

The various parameters of the spin Hamiltonian (Eq. 2) are calculated
following Refs. [15, 24] using the hoppings obtained from the NMTO downfolding
method mentioned earlier and neglecting the crystal field terms. For $U$ =1.7
eV, $J_{H}$ = 0.3 eV and $\lambda$ = 0.4 eV suitable for ALIO [15], we obtain
for the spin Hamiltonian ($J_{1}$, $K_{1}$, $\Gamma_{1}$
,$\Gamma^{\prime}_{1}$) in Eq. 2 as (+3.19, -11.4, -1.3, -2.99) meV and (2.86,
-5.85, -0.67, -1.54) meV by using the strong coupling expressions from Ref.
[15] and Ref. [24] respectively. The reported values of ($J_{1}$, $K_{1}$,
$\Gamma_{1}$,$\Gamma^{\prime}_{1}$) from two different studies on similarly
stoichiometric HLIO structure that has been suggested to be a Kitaev quantum
spin-liquid [17] are ($1.8$ meV, $-12.0$ meV, $-0.2$ meV, -3.2meV) [19] and
($-1.3$ meV, $-15.4$ meV, $+1.5$ meV, $-5.1$ meV) [18] respectively. While the
ferromagnetic nature of $K_{\mathrm{z}}$ obtained for ALIO is similar to that
of HLIO, the magnitude of $|\frac{K_{1}}{J_{1}}|$ is 2-3.5. This is much
smaller in comparison to HLIO (8.5-12.5). It is therefore likelier that ALIO
will order like the parent compound $\alpha$-LIO [7], for which the (two)
reported estimates of ($J_{1}$, $K_{1}$, $\Gamma_{1}$, $\Gamma^{\prime}_{1}$)
are ($-4.6$ meV, $-4.2$ meV, $+11.6$ meV, $-4.3$ meV) and ($-3.1$ meV, $-6.3$
meV, $+9.4$ meV, $-0.1$ meV) [15].

Perturbative analysis [24] estimates that $J\sim\frac{t_{dd}^{2}}{U}$ and
$K\sim\frac{t_{pd}^{4}}{\Delta_{pd}^{2}}\frac{J_{H}}{U^{2}}$ with the
Heisenberg term predominantly governed by direct exchange, while the Kitaev
interaction is mostly due to superexchange processes along the Ir-O-Ir paths.
Here, $t_{dd}$ and $t_{pd}$ stand for the hopping amplitudes between $d$
orbitals of neighboring Ir ions and between Ir-$d$ and O-$p$ states
respectively. $\Delta_{pd}$ is the charge-transfer energy. Unlike Li-$s$ in
honeycomb $\alpha$-LIO, the Ag-$d$ orbitals form strong covalent bonds with
ligand O-$p$ orbitals (seen from the crystal orbital Hamiltonian population
(COHP) plot in SM [30]) resulting in strong $d$-$p$ mixing. This modifies the
Ir-O-Ir superexchange interaction and enhances the Kitaev term ($K$). In
comparison to $\alpha$-LIO ($\Gamma_{1}$= 11.6 meV) [15], the increased Ir-O-
Ir bond angle in ALIO suppresses $t_{3}$ and leads to the reduced $\Gamma_{1}$
similar to HLIO. We have also calculated the second and third neighbor
interaction strengths and these are found to be much weaker,
($J_{\mathrm{2nd}}<6\%\text{ of }J_{\mathrm{nn}}$; $J_{\mathrm{3rd}}<1\%\text{
of }J_{\mathrm{nn}}$). Our calculated parameters place ALIO in the vicinity of
the striped phase and the Neel (AFM) phase in the phase diagram reported in
Ref. [24]. The 120∘ magnetic phase is, as well, in close proximity for these
parameters.

Figure 9:  Comparison of different magnetic configurations within GGA+SOC+U
calculation

To determine the likely ground state of ALIO, we have considered several
magnetic configurations whose spin and orbital moments are consistent with the
$J_{\text{eff}}=1/2$ state of Ir, and calculated their energies within
GGA+SOC+U scheme (see Fig. 9). Apart from the zigzag AFM order that has been
observed in the parent compound $\alpha$-LIO [7], we have also examined three
other representative magnetic orders: ferromagnet (FM), Néel AFM, and stripe
AFM which have been observed in honeycomb materials for the $q=0$ magnetic
structure [9, 4]. Whereas the Néel AFM order has the lowest energy among the
chosen configurations, the stripy phase is not much higher in energy. The
$q\neq 0$ solutions and the incommensurate phases are not taken into
consideration which may also dictate the ground state magnetism, especially
the 120∘ ordered phase.

##  V Discussion on Bulk Probes and high-temperature anomaly

Figure 10: The magnetic heat capacity (Cm) and the entropy change (Sm) are
plotted in units of Rln2 as a function of temperature. Representative error
bars are given at a few temperatures. Anomalies are observed at
T${}_{H}\simeq$ 50 K and T${}_{L}\simeq$ 13 K.

In this section, we discuss bulk probes — heat capacity (Fig. 10) and magnetic
susceptibility (Fig. 11) —, and the high temperature anomaly in the NMR data
in light of the low-temperature ordered state established in the previous
sections. We start with heat capacity data. As is well-known, the measured
heat capacity contains a contribution from the crystal lattice degrees of
freedom in addition to magnetic contributions. At higher temperatures, it is
the lattice constribution which generally dominates the heat capacity. To
determine the lattice part, one can make use of structurally analogous non-
magnetic variants of the given compound if available. In such cases,
corrections are necessary in inferring the lattice contribution of the
magnetic compound from that of the non-magnetic analogue to account for the
differences in their effective Debye temperatures of the two compounds due to
the different ionic masses and unit cell volumes. Following such a correction,
the heat capacities of the two compounds must coincide in a high-temperature
region where the magnetic contribution is negligible. In Ref. [28, 29], the
authors have made use of Ag3LiSn2O6 (ALSO) as the nonmagnetic analog. However,
it appears that no correction has been applied before subtracting these data
from the measured heat capacity of ALIO. This can be seen in Fig. 3(a) of Ref.
[28] and Fig. 2(a) of Ref. [29] where the data for of ALIO and ALSO do not
appear to overlap at high-temperatures.

The magnetic specific heat of our sample, using the data for the structurally
analogous nonmagnetic Ag3LiTi2O6 (ALTO) for the lattice contribution (see SM
[30] for details), is given in Fig. 10. Anomalies are seen in our data at
about 13 K and 50 K. Based on our NMR and $\mu$SR data described in Sec. III,
the lower-temperature anomaly is from long-range order. The higher-temperature
50 K anomaly could be from short-range correlations (recall the broad plateau
seen in the 7Li NMR shift with temperature, Fig. 5). However, it should be
noted that at high temperatures, the magnetic heat capacity is quite small
compared to the lattice contribution, and hence a small error in the lattice
heat capacity gives rise to a large error in the magnetic heat capacity. For
instance, at about 50 K, the inferred magnetic specific heat is only about 13
% of the total specific heat. Therefore, a 5 % error in the lattice heat
capacity will result in nearly 50 % error in magnetic heat capacity at 50 K.
Consequently, the magnetic entropy change as well suffers from high
uncertainty, especially in the high-temperature region. As a result, the
inference of a two-stage entropy release is suspect in our opinion which calls
into question an interpretation in terms of localised and itinerant Majorana
excitations.

Figure 11: Variation of $\chi$ with temperature is shown together with the
inverse susceptibility. The inset shows that the susceptibility measured in a
field of 100 Oe has a clear bifurcation below 100 K suggesting static moment
formation.

The DC magnetic susceptibility $\chi(T)=M/H$ of Ag3LiIr2O6 is shown in Fig.
11. It varies in a Curie-Weiss manner at high-temperature but has an anomaly
at about 50 K. In particular, as shown in the inset of Fig. 11, in a low field
of 100 Oe, there is a clear bifurcation between the zero-field-cooled ZFC and
field-cooled FC curves a little below 100 K. Such a measurement is currently
not available in the literature for comparison.

Given the above bulk probe data, we revisit the high temperature anomaly seen
in NMR data which one might want to optimistically interpret as having to do
with Kitaev physics. Broad anomalies, such as the one in the 7Li NMR line
shift of ALIO at about 50 K (Fig. 5), are expected in low-dimensional quantum
magnets with antiferromagnetic couplings at temperatures on the scale of the
dominant exchange coupling. In such a case, an anomaly is also expected in the
temperature dependence of 1/$T_{1}$. It should be noted that 7Li nuclei are
located symmetrically with respect to the Ir moments on the lattice, which
will filter out Néel-type antiferromagnetic fluctuations. However, given the
evidence from $\mu$SR in Sec. III.1 of the eventual incommensuration below 9
K, as also the possibility of the (incommensurate) stripy phase as a competing
phase for which the fluctuations are not cancelled at the 7Li site, one
expects now incomplete filtering at the 7Li site. The appearance of a peak at
50 K (Fig. 6) then likely reflects a build-up of short-range dynamic magnetic
correlations at the dominant exchange scale as concluded in Sec. III.2. This
50 K scale is indeed in consistent with our DFT estimates (Sec. IV) for the
various exchange couplings (also, see SM [30]). On the other hand, as
discussed in the NMR section (Sec. III.2), the absence of a sharp divergence
in 1/$T_{1}$ at the ordering temperature ($\sim$ 10 K) is likely due to a
wipe-out effect where a large fraction of the 7Li nuclei (nearly 90 % as seen
from Fig. 7) are already out of the window of observation by 10 K.

We now return to the bifurcation seen in the low-field ZFC/FC susceptibility
data at about 100 K suggesting a freezing of magnetic moments (inset of Fig.
11). In higher fields, this anomaly moves to about 50 K. If this arose from
intrinsic regions, it would result in a sharp loss of the NMR spectral
intensity at this temperature. On the other hand, the observed loss of NMR
intensity is rather gradual as the temperature decreases from 100 K (Fig. 7).
Further, one should have seen a reflection of the freezing of moments in the
$\mu$SR relaxation rate $\lambda_{2}$ which is a zero-field measurement.
However, $\lambda_{2}$ shows a rather gentle variation with temperature with a
sharper increase only below 20 K (Fig. 4). This then suggests that the ZFC/FC
bifurcation is extrinsic in origin, possibly related to moments localized at
stacking faults that naturally occur these systems, and one has to look for a
different cause for the decrease in the NMR intensity.

We note that such a decrease of the NMR intensity has been seen in the heavy
Fermion superconductor CeCu2Ge2 _above_ the superconducting transition
temperature as well as in its non-superconducting variants [25, 26]. In these
cases, it was ascribed to dynamical magnetic correlations arising from the
itinerant physics of the “heavy-fermion band magnetism”. In the present case,
our DFT estimates of the Heisenberg and Kitaev terms tell us that they are
comparable ($|\frac{K_{1}}{J_{1}}|$ is about 2-3.5) and in the range of the
energy scale $\sim$ 50 K. As discussed earlier in this section, these
competing interactions can be driving the development of short-range dynamical
magnetic correlations in ALIO. Whether there is some form of itinerancy – by
which, we have in mind some form of a quasi-particle excitation continuum [46]
– in the ALIO system driven by the Kitaev terms leading to the above decrease
of the NMR intensity is an open question.

##  VI Conclusions

In summary, ALIO is shown to exhibit magnetic long-range order below 9 K. This
observation is consistent with our DFT calculations which find comparable
Heisenberg and Kitaev exchange couplings. Though $5d$ Ir4+ possesses strong
spin-orbit coupling that lie at the origin of substantial bond-dependent
anisotropic Kitaev exchange terms, however they are tamed by the Heisenberg
exchange term in this system which leads to magnetic long-range order. More
specifically, the introduction of the heavy Ag atom in ALIO in place of Li in
$\alpha-$Li2IrO3 or H in HLIO strongly affects the local structure and
enhances the interlayer Ag-$d$ and O-$p$ hybridization along linear O-Ag-O
bonds. The increased bond angle originating due to Ag-O electronic repulsion
essentially contributes to the nearest-neighbor Kitaev exchange. The nearest-
neighbor Kitaev and Heisenberg exchanges are found to be ferromagnetic and
antiferromagnetic respectively. Although a comparison of the energy among
various magnetic configurations within the unit cell finds the Néel state to
be the lowest in energy, the stripy ordered state is not very far in energy
(Fig. 9). Both $J$ and $K$ parameters strongly depend on the local geometry,
an interesting aspect that can be explored in future studies by the tuning of
bond-length and bond-angles driven by pressure or the application of a
magnetic field. This can open up the possibility of suppressing $J$ to
negligible values so that the Kitaev term $K$ dominates the physics of the
ground state which might then be a Kitaev spin-liquid.

We end with the most unusual finding from our study: the continued presence of
a high muon relaxation rate ($\sim 5$ MHz) deep into the ordered state as
exemplified by Fig. 4. Our observations go down to temperatures as low as 52
mK which is $1/200^{\text{th}}$ of the ordering temperature as highlighted
earlier in the introduction and the text surrounding Fig. 4. This points to
considerable dynamics of the ordered moments even in the ground state. The
$\alpha$-RuCl3 system also shows a significant muon depolarisation rate on the
ordered side [42, 43], however as mentioned in Sec. III.1, these datasets are
limited to about $1/5^{\text{th}}$ of the ordering temperature. A somewhat
similar conclusion of persistent dynamics has been drawn [44, 47] based on an
analysis of the 7Li NMR 1/$T_{1}$ in the ordered state of ALIO. The natural
question is what is the source of this dynamics. Our data taken together with
our computations of the muon stopping site and their dipolar fields are
consistent with the co-existence of Néel and stripe ordered domains. We
conjecture that the persistent dynamics are due to spatio-temporal
fluctuations of these two kinds of ordered domains that are likely driven by
quantum effects in presence of non-neglible Kitaev terms. This is a well-
motivated question for theory, i.e. what is the effect of quantum fluctuations
coming from the spin-flip Kitaev terms in the ordered regions of the phase
diagram [24]. Ultimately, an explanation for the persistent dynamics whether
driven by Kitaev terms or not is needed.

##  VII acknowledgments

We thank Nandini Trivedi and Vikram Tripathi for discussions. We thank MoE and
Department of Science and Technology, Govt. of India for financial support.
A.V.M. kindly acknowledges the Alexander von Humboldt foundation for the
financial support for the renewed research stay at the University of Augsburg
in 2018. N.B. and A.V.M. kindly acknowledge support from the German Research
Society (DFG) via TRR80 (Augsburg, Munich). Author group from IIT Bombay
acknowledges support of measurement facilities at their institution.
Experiments at the ISIS Neutron and Muon Source were supported by a beam-time
allocation RB2068009 from the Science and Technology Facilities Council [48].
S.P. acknowledges financial support from IRCC, IIT Bombay (17IRCCSG011) and
SERB, DST, India (SRG/2019/001419). I.D. thanks Department of Science and
Technology, Technical Research Centre (DST-TRC) and Science and Engineering
Research Board (SERB) India (Project No. EMR/2016/005925) for support.

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