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La Coppa del Mondo di rugby 2023 (; ) è la 10ª edizione della Coppa del Mondo di rugby, massima competizione internazionale di rugby a 15 organizzata da World Rugby.

È in programma dall' al  in Francia e, a differenza della precedente occasione in cui fu organizzata in tale Paese, nel 2007, è destinata a svolgersi senza il sostegno logistico di altre federazioni che misero a disposizione i propri stadi; per contro è la quarta edizione della Coppa in cui la Francia, sia come Paese organizzatore che come fornitore di impianti, è coinvolta, dopo quella citata del 2007 e quelle del 1991 (affidata all'Inghilterra) e del 1999 (Galles).

Sono 9 le città prescelte e l'area di Parigi è rappresentata dallo Stade de France, impianto che sorge nella banlieue della capitale, a Saint-Denis, destinato a ospitare la finale e, nell'occasione, a divenire il terzo stadio, dopo Eden Park ad Auckland e Twickenham a Londra, a essere sede di due finali di Coppa del Mondo.

Storia

L'organizzazione
Il processo di selezione per la scelta del Paese ospitante iniziò il , data dell'invito ufficiale da parte di World Rugby a presentare entro un mese le dichiarazioni di interesse all'organizzazione della Coppa del 2023.
Furono quattro le federazioni che risposero all'appello, , , Italia e .
Dichiarazioni recenti degli  avevano indotto a supporre che anch'essi fossero interessati a concorrere per un'eventuale candidatura, ma nessun passo ufficiale venne da tale federazione.
Le quattro federazioni ebbero un anno di tempo per presentare i propri dossier a partire dal giugno successivo, data ultima di formalizzazione della candidatura.

Il progetto presentato dall'Irish Rugby Football Union prevedeva un torneo articolato su 12 sedi, 9 nella Repubblica d'Irlanda e 3 nell'Irlanda del Nord, con Dublino e Belfast a mettere complessivamente a disposizione 5 stadi; gli organizzatori espressero la previsione che la competizione avrebbe apportato un beneficio di 800 milioni di euro all'economia dell'isola grazie a circa  turisti attesi.
La South African Rugby Union – che, per poter presentare la propria candidatura, ebbe bisogno che il governo le revocasse il divieto a organizzare manifestazioni internazionali, inflittole a causa di insufficienti misure per contrastare discriminazioni razziali nell'accesso alla disciplina – presentò per la competizione un piano che si sviluppava su 7 città e 8 sedi (Johannesburg con due impianti, il Soccer Stadium di Soweto e l'Ellis Park, che ospitò la finale del 1995).
L'impatto stimato sull'economia del Paese fu di circa 27 miliardi di rand (~1,56 miliardi di euro).

Più articolate le candidature di Francia e Italia: la Fédération Française de Rugby era la più recente organizzatrice della competizione, avendola ospitata nel 2007, e nello staff che preparò il dossier figuravano molti elementi legati a quell'evento: Pierre Camou, presidente federale, avviò la pratica poi ereditata dal suo successore Bernard Laporte, già allenatore della nazionale.
Quest'ultimo affidò il comitato organizzatore a Claude Atcher, che già fu direttore del torneo nel 2007, e gli ex giocatori Sébastien Chabal e Frédéric Michalak furono incaricati del ruolo di ambasciatori del progetto che, nella sua stesura originale, prevedeva lo svolgimento del torneo in 12 stadi su altrettante città; in corso d'opera Parigi, Montpellier e Lens furono tagliate e le sedi ridotte a 9.
Delle città rimaste, l'unica a non aver mai ospitato il torneo in precedenza è Nizza; delle altre otto, cinque presentavano impianti già utilizzati in passato per la competizione mondiale, e tre contribuivano con stadi costruiti in occasione del campionato europeo di calcio 2016.
Nella documentazione della candidatura fu messa in rilievo l'importanza dell'evento quale traino per l'incremento dell'elettrificazione dei servizi di trasporto pubblico onde renderlo sostenibile per l'ambiente.
Per quanto riguarda altresì la Federazione Italiana Rugby, la sua candidatura ebbe vita breve perché vittima di veti politici: nel dossier presentato dall'allora presidente federale Alfredo Gavazzi figuravano 12 città candidate ospiti, 11 delle quali in comune con quelle incluse dal sindaco di Roma Ignazio Marino nella candidatura per organizzare i Giochi olimpici del 2024: qualora la Capitale se li fosse visti assegnare, infatti, tali città avrebbero dovuto ospitarne il torneo di calcio e avrebbero usufruito quota parte dei fondi del CIO (1,7 miliardi di euro) e del governo (4 miliardi) per le ristrutturazioni e i restyling necessari.
Tuttavia il nuovo sindaco della Capitale Virginia Raggi, con una decisione discussa dal governo e contestata dal CONI, bocciò il progetto olimpico a fine settembre 2016 e Gavazzi, che aveva legato la proposta della F.I.R. a quella di Roma e del CONI, fu costretto a ritirare la candidatura italiana prima ancora che World Rugby la mettesse ai voti, per sopravvenuta mancanza dei presupposti economici e tecnici per poterla sostenere.

Dopo i controlli di compatibilità e congruenza delle candidature, esse furono messe ai voti dei 39 membri del consiglio di World Rugby il  a Londra: nella prima votazione a tre il favorito Sudafrica giunse secondo con 13 voti alle spalle della Francia, prima a 18; l'Irlanda, ultima con 8, fu bocciata; nel ballottaggio tra le due rimanenti, la Francia ebbe la meglio con 24 voti (61,5%) contro 15.

A maggio 2017 giunse anche l'ufficializzazione delle sedi.
Delle 9 città prescelte l'unica esordiente, come già preannunciato in sede di candidatura, è Nizza con il suo Allianz Riviera; Marsiglia (Vélodrome), Nantes (la Beujoire), e Saint-Étienne (Geoffroy Guichard) furono già sedi della ; Saint-Denis, con il suo Stade de France, fu sede della  e ospitò la finale della Coppa del 2007; Tolosa ospitò in precedenza tre edizioni della Coppa, quella del 1991 con lo stadio Ernest Wallon e quelle del 1999 e 2007 con il Municipal, scelto anche per il 2023.
Per quanto riguarda invece le città già presenti che contribuiscono al torneo con un nuovo impianto, Lione nel 2007 fu sede della manifestazione con lo stadio di Gerland; lo stadio prescelto per il 2023 è il Parc Olympique Lyonnais, struttura di proprietà del club calcistico Olympique di Lione; benché insistente nel territorio della metropoli di Lione, si trova amministrativamente nel comune di Décines-Charpieu.
Stesso discorso per Lilla, in cui l'impianto si trova nella limitrofa Villeneuve-d'Ascq: nel 1991 ospitò la competizione con quello che all'epoca era noto come Stadium Nord, mentre nel 2023 mette a disposizione il nuovo Pierre Mauroy, inaugurato nel 2012.
Infine Bordeaux, il cui stadio Chaban-Delmas servì come sede nelle Coppe del 1999 (con il nome di Parc Lescure) e del 2007, nel 2023 ospita incontri nel nuovo stadio Matmut-Atlantique, inaugurato nel 2015 in previsione del campionato europeo di calcio dell'anno seguente.

Sia l'Allianz Riviera che il Matmut-Atlantique sono noti rispettivamente con i nomi originali di Stadio di Nizza e Stadio di Bordeaux stante il divieto di utilizzo dei nomi commerciali per competizioni internazionali.

Il torneo
Il sorteggio dei gironi si tenne nel parigino Palais Brongniart il , a qualificazioni ancora in corso.
Dodici delle venti squadre del torneo erano automaticamente qualificate grazie al piazzamento nella : nell'ordine  (campione uscente),  (finalista),  e  (semifinalisti),  ,  e  (quartifinalisti), , ,  e  (terze classificate dei propri rispettivi gironi).
Tali squadre furono sorteggiate nelle prime tre fasce di ogni girone: facendo riferimento al piazzamento World Rugby al , le quattro squadre con il ranking più alto occuparono la prima fascia, le quattro a seguire la seconda fascia e quelle con il ranking più basso la terza fascia.
Le ultime due fasce sono riservate alle squadre impegnate nel percorso di qualificazione: nella quarta le prime qualificate delle Americhe e dell'Europa, quella dell'Oceania e la vincitrice della zona Asia/Pacifico, nella quinta la qualificata dall'Africa, le seconde qualificate di Americhe ed Europa e la vincitrice del torneo finale di ripescaggio.

Nel girone A, quello riservato al Paese ospitante, il sorteggio mette di fronte , , , Americhe 1 (posto nel frattempo rivelatosi appannaggio dell') e Africa 1, slot successivamente conquistato dalla ; il girone B si compone nell'ordine di , , , Asia/Pacifico (appannaggio di  a luglio 2022) ed Europa 2 (originariamente , poi penalizzata a favore della  per via di irregolarità nel tesseramento di giocatori equiparati).
Insieme a ,  e , nel girone C furono sorteggiate la qualificata Europa 1 e la vincitrice dei ripescaggi (alfine risultata essere il , tornato alla Coppa dopo 16 anni), mentre nell'ultimo girone terminarono , , , Oceania 1 (slot che a luglio 2021 fu occupato da ) e America 2, a sorpresa rivelatasi essere il  dopo la vittoria nel barrage finale contro gli .

Squadre qualificate

Impianti

Formula
Le 20 squadre sono divise in 4 gironi da 5 squadre ciascuna che si affrontano con il metodo del girone all'italiana.
Il punteggio assegnato è quello dell'Emisfero Sud: 4 punti per la vittoria, 2 ciascuno per il pareggio e zero per la sconfitta e, in aggiunta a ciò, un punto eventuale alla squadra sconfitta con sette o meno punti nonché un ulteriore punto alla squadra autrice di almeno quattro mete nell'incontro, indipendentemente dal risultato.
Le prime due classificate di ogni girone si qualificano ai play-off e la squadra terza classificata di ogni girone, inoltre, è automaticamente qualificata alla  al pari delle otto quartifinaliste.

In ordine di abbinamento dal primo al quarto, gli incontri dei quarti di finale sono la vincitrice del girone C contro la seconda del girone D; la vincitrice del girone B contro la seconda del girone A; la vincitrice del girone D contro la seconda del girone C e, infine, la vincitrice del girone A contro la seconda del girone B.
Gli accoppiamenti di semifinale sono predeterminati: le vincitrici delle prime due partite si incontrano nella prima semifinale, quelle delle altre due partite nella seconda semifinale.
Le squadre vincenti le semifinali si incontrano per il titolo di campione del mondo, quelle sconfitte per il terzo posto.

I quattro quarti di finale sono equalmente suddivisi tra Marsiglia e Saint-Denis; semifinali e le finali per il primo e il terzo posto solo a Saint-Denis.

Gironi

Fase a gironi

Girone A

Classifica

Girone B

Classifica

Girone C

Classifica

Girone D

Classifica

Play-off

Quarti di finale

Semifinali

Finale per il 3º posto

Finale

Note

Collegamenti esterni

redpajama_set_name: RedPajamaWikipedia


text:
\section{Introduction}

The spin-orbit-assisted Mott-insulating character of iridates provides a very active
playground for quantum magnetism \cite{WitczakKrempa14,Rau16,Schaffer16,Trebst17,Winter17,Hermanns18,Cao18,Takagi19,Motome20,Takayama21,Nguyen21,Trebst22}.
For $5d^5$ and $5d^4$ configurations, spin-orbit coupling $\lambda$\,=\,$\zeta/2S$ lifts the
degeneracy of the $t_{2g}$ orbitals and yields electronic bands which in many compounds are
sufficiently narrow for a moderate on-site Coulomb interaction $U$ to result in Mott insulators
with local $j$\,=\,$1/2$ and $j$\,=\,$0$ moments, respectively.
Particular attention has been paid to $j$\,=\,$1/2$ moments in compounds with corner-sharing or edge-sharing
IrO$_6$ octahedra, such as Sr$_2$IrO$_4$ or Na$_2$IrO$_3$ with dominant isotropic Heisenberg exchange or
bond-directional Kitaev exchange couplings, respectively \cite{Jackeli09}.
The experimental realization of Kitaev couplings has triggered an intense quest for novel quantum spin-liquid
states in so-called Kitaev materials \cite{Trebst17,Singh10,Singh12,Kitagawa18,Takagi19,Motome20,Trebst22}.

\begin{figure}[tb]
\centering
\includegraphics[width=0.4\columnwidth]{Revelli_fig01a.png}
\includegraphics[width=0.08\columnwidth]{Revelli_fig01a2.png}
\includegraphics[width=0.44\columnwidth]{Revelli_fig01b.pdf}
   \caption{Left: Sketch of the crystal structure of Ba$_3$InIr$_2$O$_9$ with triangular layers of
   	well-separated Ir$_2$O$_9$ units. Bold blue lines connect Ir ions within a dimer.
   	Olive/gray/red: Ba/In/O.
   	Right: Permittivity $\varepsilon^{\prime}$ (black) and dielectric loss $\varepsilon^{\prime\prime}$ (red)
	at 20\,K in the microwave range. The small $\varepsilon^{\prime\prime}$ demonstrates the
	insulating character of Ba$_3$InIr$_2$O$_9$.
}
\label{fig:struc}
\end{figure}

The rich physics encountered in the iridates is based on the subtle balance of $U$, $\zeta$, and hopping $t$.
Compared to corner- or edge-sharing IrO$_6$ octahedra, the less intensively studied case of
\textit{face-sharing} octahedra features enhanced hopping \cite{KhomskiiZhETF},
opening access to unexplored regions in $U$-$\zeta$-$t$ space.
We focus on the structural motif of two face-sharing octahedra that form Ir$_2$O$_9$ units or dimers,
as realized in the Ba$_3$$X$Ir$_2$O$_9$ family with the layered
(6H)-BaTiO$_3$ structure formed for a wide variety of mono-, di-, tri-, or tetravalent $X$
ions \cite{Doi04,Kim04,Sakamoto06,Nguyen21,Dey12,Dey14,Kumar16,Lee17,Dey17,Khan19,Nag19,Kumar21,Garg21},
see Fig.\ \ref{fig:struc}.
Within the layers, the Ir$_2$O$_9$ units are arranged on a triangular lattice.
In total there are 12 neighbors in three dimensions, analogous to a \textit{hcp} lattice of Ir$_2$O$_9$ units.
In Ba$_3$InIr$_2$O$_9$ with Ir$^{4.5+}$ ions and three holes per dimer, no long-range magnetic order
is observed but persistent spin dynamics were reported down to 20\,mK using thermodynamic measurements,
nuclear magnetic resonance, and muon spin resonance \cite{Dey17}. This promisingly points towards a
possible gapless spin-liquid ground state.

An essential step for the understanding of the magnetism of Ba$_3$$X$Ir$_2$O$_9$
is the determination of the character of the magnetic moments.
To this end, one has to resolve the electronic structure of the Ir$_2$O$_9$ units.
In the limit of dominant $U$, one expects spin-orbit entangled moments localized
on each Ir site with strong exchange interactions \cite{Xu19}.
For mixed-valent compounds such as Ba$_3$InIr$_2$O$_9$, this limit may e.g.\ feature charge ordering of
$5d^5$ Ir$^{4+}$ and $5d^4$ Ir$^{5+}$ sites with $j$\,=\,$1/2$ and $j$\,=\,$0$ moments coexisting
on a dimer. However, depending on the choice of $X$ ions the Ir$_2$O$_9$ units exhibit a
very short Ir-Ir distance of 2.5--2.6\,\AA\ \cite{Doi04,Dey17}, shorter than the value realized
in Ir metal, 2.7\,\AA. This suggests an alternative scenario in which the large intra-dimer
Ir-Ir hopping yields quasi-molecular orbitals with possibly quenched orbital moments \cite{Streltsov16},
as proposed for Na$_2$IrO$_3$ \cite{Mazin12,Foy13}.
For iridate dimers, the formation of quasi-molecular orbitals was claimed in
Ba$_5$AlIr$_2$O$_{11}$, Ba$_3$BiIr$_2$O$_9$, and Ba$_3$CeIr$_2$O$_9$
\cite{Streltsov17,Wang18,Miiller12,Revelli19}.
For the latter compound, it has been shown that RIXS measurements of the dynamical
structure factor are particularly well suited to nail down a quasi-molecular orbital
character of the electronic states as well as to unravel the role of
spin-orbit coupling \cite{Revelli19}.
In Ba$_3$CeIr$_2$O$_9$ with two holes per dimer, the comparison of RIXS and theory established
that the quasi-molecular states are predominantly built from spin-orbit entangled $j$\,=\,$1/2$
states, giving rise to a pseudospin $j_{\rm dim}$\,=\,$0$ character as opposed to a spin-singlet
dimer \cite{Revelli19}.
The compound Ba$_3$InIr$_2$O$_9$ with three holes per dimer has been discussed as being
close to a phase transition \cite{Li20} between a $j_{\rm dim}$\,=\,1/2 state
with dominant spin-orbit coupling and a $j_{\rm dim}$\,=\,3/2 state for larger hopping,
see Fig.\ \ref{fig:sketch}.
This phase transition is related to van-Vleck-type excitonic magnetism \cite{Khaliullin13},
also known as singlet magnetism \cite{KhomskiiBook},
in nominally non-magnetic $d^4$ $j$\,=\,0 compounds in which magnetic
moments arise from the hopping- or exchange-induced condensation of excited states.
Similarly, the condensation of spin-orbit excitons has been discussed in one-dimensional
$t_{2g}$ systems \cite{Kaushal19}.
The quasi-molecular dimer orbitals provide a different flavor of this mechanism.
Condensation of the $j_{\rm dim}$\,=\,3/2 excitation occurs on the
dimer level based on strong intra-dimer hopping, while exchange between the
dimer moments is related to inter-dimer hopping with a smaller energy scale.

\begin{figure}[tb]
	\centering
	\includegraphics[width=0.85\columnwidth]{Revelli_fig02.png}
	\caption{Quasi-molecular orbitals based on local single-hole states with $j$\,=\,1/2
	and 3/2. In the hole picture, bonding (anti-bonding) orbitals are depicted in red (blue).
	In this $j$ basis, hopping is block-diagonal for $t_{e_g^\pi}$\,=\,$-t_{a_{1g}}$
	(see main text). For three holes, increasing hopping yields a phase transition
	from $j_{\rm dim}$\,=\,1/2 (left) to $j_{\rm dim}$\,=\,3/2 (right) \cite{Li20}.
	In Ba$_3$InIr$_2$O$_9$, the hopping parameters deviate from the idealized case
	depicted here, but this sketch still provides a good starting point for an intuitive
	understanding of the electronic structure.
	}
	\label{fig:sketch}
\end{figure}

Here, we study the crystal structure and the electronic structure of Ba$_3$InIr$_2$O$_9$,
which represents a model system for Ir dimers with unpaired electrons. We find a rich RIXS
spectrum of intra-$t_{2g}$ excitations and demonstrate their quasi-molecular nature by the
observation of a sinusoidal RIXS interference pattern, i.e., the dynamical structure factor
of a dimer, similar to the two-hole case of Ba$_3$CeIr$_2$O$_9$ \cite{Revelli19}.
A comparison of RIXS spectra with exact diagonalization calculations establishes a
$j_{\rm dim}$\,=\,$3/2$ ground state,
while the observation of RIXS features below about 0.3\,eV supports that Ba$_3$InIr$_2$O$_9$
is close to the phase transition to a $j_{\rm dim}$\,=\,$1/2$ state.

The existence of quasi-molecular magnetic moments in dimers with an odd number of charge
carriers clearly establishes that Ba$_3$InIr$_2$O$_9$ belongs to the class of cluster Mott
insulators \cite{AbdElmeguid04,Chen14,Khomskii21}.
Due to strong intra-dimer Ir-Ir hopping, charge carriers are delocalized inside the
dimer-type cluster in quasi-molecular orbitals. The Mott gap in this case refers to
\textit{inter}-dimer charge excitations which are suppressed by Coulomb repulsion.
In comparison, the two holes per dimer in the sister compound Ba$_3$CeIr$_2$O$_9$ fully
occupy the lowest-lying molecular orbital, leading to a non-magnetic ground state.
Cluster Mott insulators offer a new route to tailor spin-orbit-entangled quasi-molecular
magnetic moments. In this spirit, our RIXS results reveal the local quasi-molecular orbital
structure of the spin-liquid candidate material Ba$_3$InIr$_2$O$_9$, providing an
essential step towards a thorough understanding of its magnetic properties.

\section{Experimental}
\label{sec:exp}

Single crystals of Ba$_3$InIr$_2$O$_9$ were grown at 1200$^\circ$C in air using a BaCl$_2$ flux,
as described in Ref.\ \cite{Li20}.
We first synthesized polycrystalline material using stoichiometric amounts of In$_2$O$_3$, BaCO$_3$,
and Ir powder \cite{Dey17}. This polycrystalline material was used to grow the single crystals.
Single-crystal x-ray diffraction measurements were carried out using a Bruker AXS Kappa APEX II
four-circle diffractometer that operates with a wave length of $0.7\,$\AA\ (Mo $K_{\alpha}$ radiation).
The studied crystal was shaped like a hexagonal prism and a numerical absorption correction is
based on a model with eight faces at distances from the center of about 100\,$\mu$m.

RIXS measurements were performed at beamline ID20 at the ESRF \cite{Moretti18}.
The crystal studied in RIXS had the shape of a hexagonal prism with a size of about 0.3\,mm
and a thickness of roughly 0.05\,mm.
The RIXS intensity of intra-$t_{2g}$ excitations was resonantly maximized by choosing an incident energy
of 11.215\,keV at the Ir $L_3$ edge.
Data were collected at 10\,K.\@
At low temperature, a mono\-clinic distortion with a monoclinic angle $\beta$\,=\,$90.854(3)^\circ$
was reported based on powder neutron diffraction data \cite{Dey17}.
Orienting the sample on the RIXS spectrometer with a spot size of the incident beam of about
$(20\times 10)\,\mu$m$^2$, we found that the measurement averages over different monoclinic domains.
With $\sin\beta \! \approx \! 0.9999$, we neglect the small distortion for the sake of simplicity
and assume hexagonal $P6_3/mmc$ symmetry for the RIXS measurements.
We studied the (001) surface in horizontal scattering geometry with the \textit{c} axis in the
scattering plane and the \textit{a} axis along the vertical direction.
The incident photons were $\pi$ polarized.
Combining a Si(844) back-scattering monochromator and $R$\,=\,2\,m Si(844) spherical diced crystal
analyzers, an overall energy resolution of 25\,meV was achieved \cite{Moretti13}.
Replacing the Si(844) back-scattering monochromator with a Si(311) channel-cut yields a lower resolution
of 0.36\,eV but a larger signal-to-noise ratio.
RIXS spectra were measured with the high-resolution setup by scanning the energy loss at constant
transferred momentum {\bf q}. Additionally, we collected {\bf q} scans at constant energy loss,
both with the low- and the high-resolution setup.
For comparison, we measured RIXS on Ba$_3$CeIr$_2$O$_9$ using the same setup and the same sample
orientation as for Ba$_3$InIr$_2$O$_9$, as described previously \cite{Revelli19}.

\section{Results}

\subsection{Insulating behavior}

We probe the conductivity of Ba$_3$InIr$_2$O$_9$ by dielectric spectroscopy in the microwave range.
A 4-point measurement of the dc resistivity was unfeasible since the size of our single crystals
does not exceed 0.3\,mm.
We employ a 2-point approach in which case extrinsic contact effects have to be considered.
These can be reduced by measuring at frequencies that are high enough
to short out the contacts.
The crystal was contacted in capacitor geometry on a microstrip waveguide with the electric
field parallel to the $c$ axis.
Using a vector network analyzer (Rhode\&Schwarz ZNB8) in the frequency range from 300\,kHz to 30\,MHz,
the permittivity of Ba$_3$InIr$_2$O$_9$ was measured at 20\,K, see Fig.\ \ref{fig:struc}.
A significant uncertainty in the absolute values is caused by the high relative error
in the determination of the small effective area $A_s$\,=\,0.02\,mm$^2$ and
thickness $d_s$\,=\,0.04\,mm.
However, the experimental result corresponds to a real part of the conductivity
$\sigma \approx 10^{-6}/\Omega$\,cm,
which firmly establishes the insulating character of Ba$_3$InIr$_2$O$_9$.
Independent of the exact absolute values, this result is supported by the observation that
the dielectric loss $\varepsilon^{\prime\prime}$ is much smaller than the permittivity $\varepsilon^{\prime}$,
as expected for an insulator.

\subsection{Single-crystal structure refinement}
\label{sec:struc}

In single-crystal x-ray diffraction at 295\,K, we collected the intensities of $69\,971$ Bragg
reflections, yielding $877$ independent reflections with respect to the hexagonal space group $P 6_3/m m c$.
The integration yields an internal $R$ value \cite{karplus}
of $R^2$(int)\,=\,6.53\,\% after absorption correction.
Using the software JANA \cite{petricek}, three refinements were carried out in space group $P 6_3/m m c$.
One assumes the ordered structure, with In (Ir) atoms located on the Wyckoff sites 2\textit{a} (4\textit{f}).
The second refinement considers a statistical distribution of In and Ir atoms,
with the 2\textit{a} and 4\textit{f} sites both occupied by 1/3 In and 2/3 Ir.
The third refinement assumes partial disorder, allowing free occupations of these sites
with the constraint that each site is fully occupied.
We employ isotropic atomic displacement parameters for all atoms and Gaussian isotropic
Becker-Coppens extinction correction \cite{Becker1974a}. For the refinements with In and Ir
sharing a site, these atoms are constrained to have the same atomic displacement parameters.

The best result was obtained for the partially ordered model. It yields
$R$(obs)\,=\,5.96\,\%, w$R$(obs)\,=\,8.41\,\%, $R$(all)\,=\,5.96\,\%,
and w$R$(all)\,=\,8.41\,\%.
We find that the 2\textit{a} site is occupied by 98.5(1.1)\,\% In atoms and
1.5(1.1)\,\% Ir atoms, while the 4\textit{f} sites are occupied by
92.7(1.4)\,\% Ir atoms and 7.3(1.4)\,\% In atoms.
For comparison, the fully disordered model can be ruled out, w$R$(all)\,=\,19.56\,\%,
while the ordered model yields w$R$(all)\,=\,9.14\,\%.
In total, the single-crystal data point towards a small surplus of In ions with
about 7\% of In ions on the Ir 4\textit{f} dimer sites while the In 2\textit{a} sites
are almost completely occupied by In. This has to be compared with the powder data
which indicate In-Ir disorder of about 2.8(5)\% \cite{Dey17} in a refinement
with fixed In to Ir total occupation ratio.

We have searched for a breaking of the translational symmetry by analyzing half-integer
indexed Bragg reflections with respect to the hexagonal lattice.
In standard operation of the x-ray generator with
50\,kV voltage many of these reflections were observed with statistically significant intensities.
At 30\,kV, which is below the voltage required to generate photons with half wavelength, 0.35\,\AA ,
these signals are suppressed although we significantly enhanced the counting time to 2000\,s/degree.
All half-indexed reflections can thus be attributed to $\lambda/2$ contamination that possesses a greater
impact in x-ray diffraction experiments on highly absorbing materials like iridates.
The comparison of all observed half-indexed reflections with their doubled integer-indexed partners
further corroborates this conclusion.

\begin{table}[t]
	\begin{tabular}[t]{cp{.2mm}cp{.2mm}cp{.2mm}cp{.2mm}cp{.2mm}c}
		& &     x            & &     y       & &     z         & &  U$_{\rm iso}$  & &     occ. \\\hline
		Ba1  & &     0            & &  0.00202(7) & &  1/4          & &  73(1) & &  1 \\
		Ba2  & &    -0.00113(14)  & & 0.33664(5)  & & 0.08965(2)  & &  87(1)  & & 1 \\
		In   & &     0            & & 0           & & 0             & &  39(1)  & & 0.968(3)\\
		Ir'  & &     0            & &  0          & &  0            & &  39    & &  0.032\\
		Ir   & &    -0.00426(7)   & &  0.33345(3) & &  0.84138(1)  & &  40(1) & &  0.897(3)\\
		In'  & &    -0.00426      & &  0.33345    & &  0.84138     & &  40    & &  0.103\\
		O1   & &     0            & &  0.4880(8)  & &  3/4          & &  38(3)  & &  1\\
		O2   & &     0.2217(10)   & & 0.2545(6)   & &  0.7433(3)    & &  38   & & 1\\
		O3   & &     0.0242(15)   & & 0.1716(5)   & &  0.9153(4)    & &  38   & & 1\\
		O4   & &     0.2272(11)   & & 0.4166(7)   & &  0.9263(3)    & &  38  & & 1\\
		O5   & &    -0.2616(11)   & & 0.4113(7)   & &  0.9079(4)    & &  38   & & 1\\
	\end{tabular}
	\caption{Results of the refinements of the crystal structure in Ba$_3$InIr$_2$O$_9$ in
		monoclinic space group $C2/c$ with room-temperature x-ray single-crystal diffraction data.
		The lattice parameters are $a$\,=\,5.829(3)\,\AA, $b$\,=\,10.096(5)\,\AA, and $c$\,=\,14.487(7)\,\AA\
		for $\beta$\,=\,90$^\circ$ (see main text).
		The parameters U$_{\rm iso}$ are given in $10^{-4}$\AA $^2$.
		Errors are given in brackets for the last digits; if no errors are given,
		parameters are fixed by symmetry or were constrained.
		\label{tab:crystal-structure} }
\end{table}

The space group $P 6_3/m m c$ exhibits the selection rule $hh\overline{2h}l$\,$:$\,$l$\,$=$\,$2n$
caused by the $c$ glide plane.
At room temperature, we observed several such forbidden reflections for which
a $\lambda/2$ contamination was excluded by scanning with reduced generator voltage.
Typical reflections are (6 $\overline{3}$ $\overline{3}$), ($\overline{3}$ $\overline{3}$ $\overline{3}$),
or ($\overline{6}$ 3 1).
The breaking of hexagonal symmetry can be described in the mono\-clinic space group $C2/c$
that was proposed for other members of the Ba$_3$$X$$M$$_2$O$_9$ family \cite{Doi04,Sakamoto06}.
The overall structure in this family corresponds to the 6H-BaTiO$_3$ phase, which can
be considered as a mixing of cubic and hexagonal perovskite arrangements.
The $X$O$_6$ octahedra form a single layer and
share a corner with neighboring $M$O$_6$ octahedra as in the cubic perovskite, see Fig.\ \ref{fig:struc}.
In contrast, the $M$ sites in the double-octahedron dimer layer share faces as in hexagonal perovskites.
In the high-symmetry arrangement in space group $P 6_3/m m c$ all oxygen ions in a plane between
two metal planes are on the same height and there are only two distinct O positions
(one on the common face of the double octahedra, one in between $X$ and $M$ ions).
Since the arrangement around the $X$ sites resembles the cubic perovskites it is not surprising
to find the main structural instability of these materials, i.e., the rotation of the $X$O$_6$ octahedra.
This already has been proposed for several Ba$_3$$X$$M$$_2$O$_9$ compounds \cite{Doi04,Sakamoto06}
including Ba$_3$InIr$_2$O$_9$ at low temperature \cite{Dey17}. The observation of the corresponding superstructure
reflection however shows that the distortion is already present at room temperature in our single crystal.

We have refined the crystal structure in space group $C2/c$ by taking the twinning into account, see Table I.\@
The symmetry reduction from hexagonal to monoclinic symmetry results in six different twin orientations,
whose contributions need to be summed up in the analysis. The refinement indicates a nearly equal distribution
of these twins which results in correlations of the parameters describing the distortion.
Furthermore the lattice constants determined on the single crystal cannot show the monoclinic distortion
due to the twinning, so that we fix $\beta$=90$^{\circ}$.
The number of positional parameters is considerably enhanced from 7 in $P 6_3/m m c$ to 20 in $C2/c$.
We have used the low-temperature structure \cite{Dey17} as starting parameters
in the refinement with strong damping. We find a considerable improvement of the fit when applying the lower symmetry: $wR_{all}$ ($R_{all}$) decreases from 8.41\,\%
(5.96\%) to 5.01\% (5.13\%) and the goodness of fit parameter is reduced from 5.39 to 1.87.
There is thus no doubt that the single crystal of Ba$_3$InIr$_2$O$_9$ exhibits the structural distortion
as described in space group $C2/c$ already at room temperature.
The main element of this distortion consists in the rotation of the
InO$_6$ octahedron: the In-O-Ir bond angle amounts to
$178.2^{\circ}$ in the hexagonal refinement and to
$171.1^{\circ}$, $169.8^{\circ}$, and $172.2^{\circ}$ in the distorted structure.
The structural phase transition involves the condensation of a $\Gamma_6^+$ phonon mode \cite{Stokes88} associated
with the rotation of the octahedra around an axis parallel to the layers. Due to the degeneracy of this mode
in the hexagonal high-temperature space group several low-temperature symmetries are possible. In $C2/c$ the octahedra rotate around a [210] direction of the hexagonal lattice, which corresponds to the monoclinic $b$ axis.
There is also some distortion in the InO$_6$ octahedron with the O-In-O bond angle deviating by up to 5$^{\circ}$
from 90$^{\circ}$, which is also frequently observed in cubic perovskites \cite{Cwik2003}.
Furthermore, there is some buckling in the O layers arising from the octahedron rotation.
However there is little impact on the bond distances.
For the intra-dimer Ir-Ir distance we find 2.648\,\AA\ at room temperature.

\subsection{RIXS spectra}

High-resolution RIXS spectra of Ba$_3$InIr$_2$O$_9$ are shown in Fig.\ \ref{fig:rixst2geg} for
selected values of the transferred momentum $\mathbf{q}$.
Note that we use reciprocal lattice units (r.l.u.) for indices $(h,k,l)$
while reciprocal space vectors are given in absolute units.
The RIXS data are analyzed in the hexagonal lattice with $P6_3/mmc$ symmetry,
as discussed in Sect.\ \ref{sec:exp}.
Iridates are known to show a large cubic crystal-field splitting 10\,Dq\,$\approx$\,3\,eV
between $t_{2g}$ and $e_g^\sigma$ orbitals due to the spatially extended character of the $5d$ states.
Accordingly, the RIXS features observed below 1.5\,eV can be attributed to intra-$t_{2g}$ excitations.
Excitations to $e_g^\sigma$ orbitals set in at about 2.5\,eV, see right panel of Fig.\ \ref{fig:rixst2geg},
and the increase above 4.5\,eV can tentatively be identified with charge-transfer excitations
between Ir and O sites.
Around zero energy loss, the data are dominated by the elastic line. For $\pi$ polarization of the
incident photons, the elastic line can be suppressed by choosing a scattering angle of $2\theta$\,=\,$90^\circ$.
Figure \ref{fig:rixst2geg} shows data for $\mathbf{q}$\,=\,(0.3\,\,0\,\,$l$)\,r.l.u.\
that were measured with $2\theta$\,=\,$51^\circ$, $65^\circ$, $80^\circ$, and $97^\circ$, and the
strongest elastic line is observed for $l$\,=\,11.2 with $2\theta$\,=\,$51^\circ$.

The RIXS spectrum of intra-$t_{2g}$ excitations is very rich in Ba$_3$InIr$_2$O$_9$.
Figure \ref{fig:rixst2geg} shows the effect of changing $q_c$, the component of the transferred
momentum parallel to the $c$ axis, and Fig.\ \ref{fig:rixsGMK} addresses a possible dispersion
within the hexagonal plane, perpendicular to $c$, showing data for different
high-symmetry points for two different values of $q_c$.
We find that the intensity of the intra-$t_{2g}$ excitations strongly varies with $q_c$,
while the peak energies do not depend considerably on $\mathbf{q}$.
For constant $q_c$, the data basically fall on top of each other in Fig.\ \ref{fig:rixsGMK}.
The largest effect of $\mathbf{q}$ on the peak energy is observed for the feature at about
0.45\,eV, which sets an upper limit of 10\,meV for a possible dispersion,
see the inset of Fig.\ \ref{fig:rixsGMK}.
This reflects the fact that inter-dimer hopping of the excited states is small.
Since such inter-dimer hopping interactions form the microscopic basis for magnetic
exchange interactions, a dispersion smaller than 10\,meV suggests that also exchange
interactions between dimers are small. This agrees with the small Curie-Weiss temperature
$\Theta_{\rm CW}$\,=\,$-6.8$\,K derived from an analysis of magnetic susceptibility data
and the small temperature scale of about 2\,K for the occurrence of correlated
spin-liquid-like behavior \cite{Dey17}, which suggests inter-dimer exchange interactions
of less than 1\,meV.

\begin{figure}[tb]
	\centering
	\includegraphics[width=\columnwidth]{Revelli_fig03.pdf}
	\caption{High-resolution RIXS spectra of Ba$_3$InIr$_2$O$_9$ at 10\,K
	for $\mathbf{q}$\,=\,(0.3\,\,0\,\,$2.8n$)\,r.l.u.\ with integer $n$ and
	$2.8\cdot 2\pi/c \approx \pi/d$\,=\,$Q_d$, where $d$ denotes the intra-dimer Ir-Ir distance.
	Left: The intra-$t_{2g}$ excitations below about 1.5\,eV show a pronounced
	even/odd behavior with respect to $n$. The intensity is modulated with the period
	$2Q_d$\,=\,$2\pi/d$\,=\,$5.56(2) \cdot 2\pi/c$, cf.\ Fig.\ \ref{fig:rixslscan}.
	We use a finite $h$\,=\,0.3 to avoid fulfilling the Bragg condition and the concomitant
	enhancement of the elastic line.
	Right: Excitations involving $e_g^\sigma$ orbitals are observed above 2.5\,eV.\@
	}
	\label{fig:rixst2geg}
\end{figure}

\begin{figure}[tb]
	\centering
	\includegraphics[width=\columnwidth]{Revelli_fig04.pdf}
	\caption{RIXS spectra of Ba$_3$InIr$_2$O$_9$ at the high-symmetry points $\it{\Gamma}$, $M$,
	and $K$ for two different values of $q_c$. For $\it{\Gamma}$, the data were measured
	with a small offset in $\mathbf{q}$ to avoid strong elastic scattering.
	Within the experimental energy resolution, the RIXS features do not show any considerable
	dispersion. The largest change of up to 10\,meV is found for the peak at 0.45\,eV (see inset).
	The data are dominated by the strong dependence of the intensity on $q_c$,
	very similar to the behavior shown in Fig.\ \ref{fig:rixst2geg}.
	The larger overall intensity stems from the difference in $h$,
	compare Fig.\ \ref{fig:rixslscan}b.
	The integer values $l$\,=\,17 and 20 were chosen since they are close
	to extrema of the intensity modulation, see Fig.\ \ref{fig:rixslscan}a.
	}
	\label{fig:rixsGMK}
\end{figure}

\subsection{RIXS interference patterns and quasi-molecular orbitals}
\label{sec:interference}

The absence of strong dispersion shows that inter-dimer hopping is small. In the following,
we address the role of intra-dimer hopping for the electronic structure.
RIXS provides an unambiguous tool to distinguish different limits.
For dominant $U$ and small intra-dimer hopping $t$, the presence of local $j$\,=\,$1/2$ or
$j$\,=\,$0$ moments can be detected via characteristic narrow features in the RIXS spectra \cite{Gretarsson13,Rossi17,Liu12,Revelli19b,Reig20,Khan21,Moretti14,Yuan17,Kusch18,Nag18}.
For the opposite limit of dominant intra-dimer hopping, the observation of sinusoidal double-slit-type
RIXS interference patterns provides a litmus test for a quasi-molecular electronic structure \cite{Revelli19}.
To explain the occurrence of such interference patterns, we address the RIXS process at the
Ir $L_3$ edge \cite{Ament11}, in which an incident x-ray photon excites an electron out of a
strongly localized $2p$ core level into an empty $5d$ level. In particular, the choice of
11.215\,keV for the incident energy resonantly enhances absorption into a $t_{2g}$ orbital
in the $5d$ shell.  This intermediate state decays by emitting a photon with lower energy,
and the $2p$ core hole is filled by a $5d$ valence electron. The resonance enhancement particularly
boosts the sensitivity to intra-t$_{2g}$ excitations, which may show orbital or magnetic character.
In other words, resonant inelastic scattering proceeds from the ground state via an intermediate state
with a strongly localized $2p$ core hole to a final state with an excitation that may be delocalized,
for instance a magnon.
For a given excitation, the total RIXS amplitude is a coherent sum running over the scattering
processes on all Ir sites over which this final excited state is delocalized \cite{Gelmukhanov94,Ma94}.
If the excited state is a quasi-molecular orbital excitation in which electrons are
delocalized over two Ir sites, the interference between x-ray photons emitted from these
dimer sites yields a sinusoidal interference pattern as a function of the transferred
momentum {\bf q}, in close analogy to the case of a double-slit experiment \cite{Ma95}.
The same applies to, e.g., magnetic excitations if spin correlations are restricted
to two adjacent sites, and the corresponding sinusoidal interference pattern
was observed in RIXS on the honeycomb Kitaev materials Na$_2$IrO$_3$ and
$\alpha$-Li$_2$IrO$_3$ \cite{Revelli20}.

\begin{figure}[tb]
	\centering
	\includegraphics[width=0.9\columnwidth]{Revelli_fig05a.pdf}
	\includegraphics[width=0.9\columnwidth]{Revelli_fig05b.pdf}
	\caption{RIXS intensity $I(\mathbf{q})$ of Ba$_3$InIr$_2$O$_9$ at 10\,K.\@
	a) Data for $\mathbf{q}$\,=\,(-0.3\,\,0\,\,$l$)\,r.l.u.\  (top axis).
	The bottom axis emphasizes the period $2Q_d$\,=\,$2\pi/d$\,=\,$5.56 \cdot 2\pi/c$
	which is incommensurate with the Brillouin zone.
	Solid lines show damped sinusoidal fits, see Eq.\ (\ref{eq:Iqc}).
	Green symbols: Low-resolution data, $\Delta E$\,=\,0.36\,eV,
	covering the range 0.62 -- 0.98\,eV.\@ Integration time was 10\,s per $\mathbf{q}$ point.
	Brown symbols: Data integrated from 0.08 to 0.21\,eV with the high-resolution setup,
	$\Delta E$\,=\,25\,meV, and 20\,s integration time per $\mathbf{q}$ point.
	The advantage of the low-resolution setup is the enhanced signal-to-noise ratio.
	Data were normalized by the incident flux.
	b) As a function of $h$, the RIXS intensity, integrated from 0.3 to 1.1\,eV
	shows monotonic behavior. Data were normalized to the value at $h$\,=\,0.
	The solid red line depicts the expected behavior due to self-absorption.
	}
	\label{fig:rixslscan}
\end{figure}

This explains the strong dependence of the intensity on $\mathbf{q}\cdot\mathbf{d} = q_c\,d$,
cf.\ Fig.\ \ref{fig:rixst2geg}, where $\mathbf{d}$\,=\,(0,0,$d$) is the vector connecting
two Ir sites within a dimer. The sinusoidal character of the intensity modulation,
\begin{equation}
\label{eq:Iqc}
I(q_c) \propto \sin^2(q_c \,d/2) + {\rm const}\, ,
\end{equation}
is revealed by measurements of the RIXS intensity $I(q_c)$ for fixed energy loss,
see Fig.\ \ref{fig:rixslscan}a. The observed period
$2Q_d$\,=\,$2\pi/d$\,=\,$5.56(2) \cdot 2\pi/c$
yields an Ir-Ir distance $d$\,=\,2.601(9)\,\AA\ at 10\,K,
in excellent agreement with the neutron diffraction result 2.599(4)\,\AA\ at 3.4\,K \cite{Dey17}.
The sinusoidal form of the modulation and the value of the period $2Q_d$ unambiguously prove
that the intra-$t_{2g}$ excitations are delocalized over the two sites of an Ir dimer,
hence the electronic structure of Ba$_3$InIr$_2$O$_9$ needs to be described in a quasi-molecular picture.

The $\sin^2(q_c \,d/2)$ interference pattern deviates from the $\cos^2(q_c\,d/2)$ behavior
well known from a conventional elastic double-slit experiment. For RIXS on dimers,
the observation of a $\cos^2(q_c\,d/2)$ or $\sin^2(q_c\,d/2)$ interference pattern
contains information on the symmetry of the investigated states \cite{Revelli19}.
This can be explained by the dipole selection rules for both photon absorption and photon emission.
Consider, e.g., an even ground state.
In the  limit $\mathbf{q}$\,=\,0, RIXS excitations are allowed via an odd intermediate state to an
even final state. For finite $\mathbf{q}$, we additionally have to consider the geometrical path difference
for scattering events on the two adjacent Ir sites. Together, this yields a
$\cos^2(q_c\,d/2)$ ($\sin^2(q_c\,d/2)$) interference pattern for even (odd) final states
in the case of an even ground state.

The Ir$_2$O$_9$ dimers show mirror symmetry but no inversion symmetry,
the two face-sharing octahedra are rotated with respect to each other by $\pi$ around $c$.
Parity therefore is not a good quantum number of the eigenstates. This results in interference patterns
with mixed behavior, $I(q_c) \propto u \sin^2(q_cd/2) + v \cos^2(q_cd/2)$, cf.\ Eq.\ (\ref{eq:Iqc}).
In RIXS on Ba$_3$InIr$_2$O$_9$, the $\sin^2(q_c\,d/2)$ behavior strongly prevails for the intra-$t_{2g}$
excitations below about 1\,eV, similar to the case of Ba$_3$CeIr$_2$O$_9$ \cite{Revelli19}.
Figure \ref{fig:rixslscan}a demonstrates this dominant $\sin^2(\pi\,q_c/2Q_d)$ behavior for
the integrated intensity plotted as a function of $q_c/Q_d$,
and the spectra depicted in Figs.\ \ref{fig:rixst2geg} and \ref{fig:rixsGMK}
corroborate this behavior for energies below about 1.1\,eV.\@
These spectra were recorded for values of $q_c \approx nQ_d$ that correspond to extrema
of the interference pattern.
RIXS spectra for, e.g., odd $n$ agree with each other over almost the entire range of
energy loss. Below about 1.1\,eV, the intensity is higher for odd $n$ and the data
strongly differ from the spectra for even $n$.
Even though parity is not a good quantum number, we find that excitations from
bonding to anti-bonding orbitals, as defined in Sect.\ \ref{sec:discuss}, predominantly show
$\sin^2(q_c\,d/2)$ behavior. The experimental result hence agrees with a $j_{\rm dim}$\,=\,3/2
ground state as depicted in Fig.\ \ref{fig:sketch}, where all three holes occupy bonding orbitals
and most of the excitations correspond to transitions to anti-bonding ones.
This conclusion is supported by the analysis in Sect.\ \ref{sec:discuss}.

Wang \textit{et al.} \cite{Wang18} reported on RIXS data of Ba$_5$AlIr$_2$O$_{11}$
which also shows face-sharing IrO$_6$ octahedra that form Ir dimers with three holes.
However, these Ir dimer sites are crystallographically non-equivalent,
resulting in charge disproportionation \cite{Streltsov17}. The formation of
quasi-molecular orbitals, predicted by Streltsov, Cao, and Khomskii \cite{Streltsov17},
has been derived from the RIXS spectra by arguing that they deviate from those of
local moments and by comparing with cluster calculations.
Data addressing the RIXS intensity as a function of $\mathbf{q}\cdot \mathbf{d}$
were not reported. Measured with an energy resolution
of 80\,meV, the RIXS spectra of Ba$_5$AlIr$_2$O$_{11}$ show five broad peaks at
0.18, 0.32, 0.57, 0.74, and 1.13\,eV.\@ They resemble our results obtained at an
interference minimum assuming a somewhat reduced energy scale.

\subsection{In-plane momentum dependence}

In contrast to the pronounced sinusoidal dependence of $I(q_c)$, the RIXS intensity shows a moderate,
monotonic intensity change as a function of $h$, see Fig.\ \ref{fig:rixslscan}b.
Any change of $\mathbf{q}$ for fixed incident energy requires to change the experimental geometry.
This affects matrix elements and self-absorption effects.
The latter can be described by \cite{Minola15} $1/(1+\sin\theta_{\rm in}/\sin\theta_{\rm out})$,
where $\theta_{\rm in}$ and $\theta_{\rm out}$ denote the angle between the sample surface and the
incident and emitted beams, respectively. For grazing incidence, i.e., small $\theta_{\rm in}$,
the incident photons are absorbed close to the surface. This facilitates the escape of the re-emitted photons
and results in an enhanced signal. The overall $h$ dependence of the intensity agrees with this expectation,
as depicted by the red line in Fig.\ \ref{fig:rixslscan}b.

\begin{figure}[tb]
	\centering
\includegraphics[width=\columnwidth]{Revelli_fig06.pdf}
	\caption{Comparison of RIXS spectra of Ba$_3$InIr$_2$O$_9$ (top) and Ba$_3$CeIr$_2$O$_9$ (bottom).
	The Ce compound has two holes per dimer, i.e., no partially filled quasi-molecular orbitals,
	and exhibits a clear onset of RIXS intensity at about 0.5\,eV.\@
	The different Ir-Ir distance $d_{\rm Ce}$\,=\,$c_{\rm Ce}/5.83$ gives rise to a slightly different
	period $2Q_{\rm Ce}$ of the interference pattern.
}
	\label{fig:rixsInCe}
\end{figure}

Additional matrix-element effects are revealed by comparing spectra for different $h$,
see top panel of Fig.~\ref{fig:rixsInCe}. For the same value of $l$, the intensity is larger for
$h$\,=\,$-2.5$ than for $h$\,=\,$0.3$, but this effect is most pronounced around 0.7\,eV.
Remarkably, a similar behavior is observed from 0.5 to 1.1\,eV in the sister compound
Ba$_3$CeIr$_2$O$_9$ \cite{Revelli19} with two holes per dimer, see bottom panel of Fig.\ \ref{fig:rixsInCe}.
In Ba$_3$CeIr$_2$O$_9$, the two prominent peaks at about 0.7\,eV and 0.9\,eV show $\sin^2(q_cd/2)$
behavior \cite{Revelli19}, very similar to Ba$_3$InIr$_2$O$_9$. In reasonable approximation,
these two peaks of Ba$_3$CeIr$_2$O$_9$ can be described in a single-particle picture as excitations
from the bonding quasi-molecular orbital formed from local $j$\,=\,1/2 states
to bonding quasi-molecular orbitals built from local $j$\,=\,$3/2$ states \cite{Revelli19}.
For the 0.7\,eV peak in Ba$_3$InIr$_2$O$_9$, the similar energy range and the similar dependence
on $h$ suggest a related character. Our analysis supports a contribution of such states
but finds also further excitations in this energy range, see Sect.\ \ref{sec:discuss}.

\subsection{Low-energy behavior}
\label{sec:lowE}

\begin{figure}[tb]
	\centering
\includegraphics[width=\columnwidth]{Revelli_fig07.pdf}
	\caption{Low-energy RIXS spectra of Ba$_3$InIr$_2$O$_9$ at 10\,K.\@
		a) Zoom in on two data sets shown in Fig.\ \ref{fig:rixst2geg}, measured with 10\,meV step size
		for $q_c$ values on extrema of the interference pattern, see Fig.\ \ref{fig:rixslscan}.
		Integration time was 20\,s per detector image.
	b) Data collected with $2\theta$\,=\,$90^\circ$ to suppress the elastic line.
	With the given incident energy, this fixes $|\mathbf{q}|$. With this restriction,
	we can reach $q_c$\,=\,$18.5 \cdot 2\pi/c \! \approx \! 6.7\,Q_d$, reasonably close
	to the interference maximum at $7Q_d$, and
	\mbox{$\mathbf{q}$\,=\,(-2.66\,\,1/3\,\,17)\,r.l.u}.\ with $q_c$\,$\approx$\,$6Q_d$,
	an interference minimum.
	For a quantitative comparison, one has to take into account that the intensity at $6.7Q_d$
	(dark red symbols) is roughly 10\,\% smaller than at $7Q_d$, see Fig.\ \ref{fig:rixslscan}a.
	Self-absorption has been corrected.
	Dashed lines with small symbols show an estimate of the inelastic contribution obtained
	by subtracting the already suppressed elastic line.
	Dark blue (dark red): 5\,meV (10\,meV) step size and 40\,s (30\,s) integration time.
	}
	\label{fig:rixs3}
\end{figure}

In Ba$_3$CeIr$_2$O$_9$, the RIXS intensity vanishes below 0.5\,eV, which in a single-particle
picture reflects the energy difference between filled and empty quasi-molecular
orbitals \cite{Revelli19}.
In contrast, Ba$_3$InIr$_2$O$_9$ exhibits finite RIXS intensity also at low energy, see Fig.\ \ref{fig:rixs3}.
Previously, the existence of excitations at about 10\,meV and 40\,meV has been estimated from
an analysis of the magnetic susceptibility \cite{Dey17}.
To better resolve the low-energy features, we suppressed the elastic line by choosing a scattering angle
$2\theta$\,=\,$90^\circ$ with incident $\pi$ polarization, see Fig.\ \ref{fig:rixs3}b.
Dashed lines with small symbols estimate the inelastic contribution after subtraction of the
remaining elastic line.
With the given energy resolution of 25\,meV, the data in Fig.\ \ref{fig:rixs3}b show that the RIXS intensity
remains finite for any energy loss below 0.3\,eV.\@
The data for $l$\,=\,17 appear continuum-like but the result for $l$\,=\,18.5 points towards peaks
at about 0.03\,eV, 0.10\,eV, and 0.23\,eV.\@

\begin{figure}[t]
	\centering
	\includegraphics[width=\columnwidth]{Revelli_fig08.pdf}
	\caption{Oscillator fit (solid lines) of RIXS data averaged over the spectra measured
	at about $\it{\Gamma}$, $M$, and $K$, cf.\ Fig.~\ref{fig:rixsGMK}. We distinguish
	$l \approx 20$ and $17$
	which roughly correspond to the extrema of the interference pattern at
	$q_c$\,=\,$7Q_d$ and $6Q_d$, cf.\ Fig.\ \ref{fig:rixslscan}a.
	The elastic line has been subtracted.
	Parameters are given in Table \ref{tab:fit}.
	Vertical bars denote the excitation energies found in the fit.
	}
	\label{fig:rixsdiff}
\end{figure}

Within a dimer, low-energy excitations occur in the vicinity of the phase transition from
$j_{\rm dim}$\,=\,1/2 to $j_{\rm dim}$\,=\,3/2, see Sect.\ \ref{sec:discuss}.
The peak energies and the RIXS resolution are much larger than the inter-dimer magnetic
energy scale given by the peak in the specific heat at about 2\,K \cite{Dey17}.
Therefore a direct connection between the observed low-energy excitations and the magnetic
fluctuations of a possible spin-liquid ground state can be ruled out.
However, a small but finite magnetic contribution to the low-energy RIXS intensity
may arise due to In/Ir site mixing. Sakamoto \textit{et al.} \cite{Sakamoto06} studied
powder samples of Ba$_3$$X$Ir$_2$O$_9$ for different metals $X$ including In and found
that $X$-Ir cation disorder is only relevant for $X$\,=\,Ti$^{4+}$ and Sc$^{3+}$.
This can be explained by the similar ionic radii of Ir$^{4+}$, Ti$^{4+}$, and Sc$^{3+}$
\cite{Sakamoto06}. Larger $X$ ions such as Ce$^{4+}$ or In$^{3+}$ as well as the different
valence of In and Ir suppress cation disorder.
For Ba$_3$InIr$_2$O$_9$, Dey \textit{et al.} \cite{Dey17} discussed that the structural data of
powder samples are well described by fully ordered structure models without cation disorder,
while deliberately adding site mixing yields an estimate of 2.8(5)\,\%.
Our single-crystal x-ray diffraction data indicate about 1-2\,\% of Ir ions on the In $2a$ site
\textit{between} dimers, see Sec.\ \ref{sec:struc}.
Even though the estimated density of such Ir moments is small, their contribution
may be relevant since each of them is coupled to six neighboring dimers.
The $2a$ sites are connected with the $4f$ dimer sites in a corner-sharing configuration
with $180^\circ$ degree Ir-O-Ir bonds, and Ir moments on the $2a$ sites are expected to enhance
inter-dimer coupling. The bonding geometry is closely related to the case of Sr$_2$IrO$_4$ which
shows large exchange interactions and magnetic excitations up to about 0.2\,eV \cite{Kim12,Kim14}.
In Ba$_3$InIr$_2$O$_9$, the sinusoidal modulation observed for integration from 0.08 to 0.21\,eV
demonstrates the existence of dimer excitations at low energy, see Fig.\ \ref{fig:rixslscan}a.
However, the $l$ dependence is small between 0.2\,eV and 0.3\,eV, see Fig.\ \ref{fig:rixs3}b.
A small magnetic contribution caused by In-Ir site disorder hence cannot be ruled out.

\section{Discussion}
\label{sec:discuss}

\begin{table}[t]
	\begin{tabular}[t]{cp{2mm}cp{2mm}cp{2mm}cp{2mm}cp{2mm}c}
		energy [eV] && $I_{6Q_d}$ && $I_{7Q_d}$ && $\frac{I_{7Q_d}-I_{6Q_d}}{I_{7Q_d}+I_{6Q_d}}$ && width [meV] && $\mu$ \\ \hline
		0.03   &&      &&      &&       &&     &&  \\
		0.10   && 2.4  && 3.1  &&  0.14 &&  81 && 2.0  \\
		0.23   && 3.7  && 3.9  &&  0.02 && 106 && 4.8  \\
		0.34   && 6.8  && 13.9 &&  0.34 &&  76 && 2.0 \\
		0.42   && 0.3  && 10.7 &&  0.94 &&  99 && 1.6  \\
		0.45   && 7.2  && 5.1  && -0.17 &&  82 && 1.9 \\
		0.50   && 0.5  && 4.7  &&  0.79 &&  90 && 1.9 \\
		0.58   && 7.9  && 9.4  &&  0.09 &&  80 && 1.5 \\
		0.63   && 7.6  && 8.5  &&  0.06 &&  93 && 1.9 \\
		0.69   && 3.0  && 16.8 &&  0.70 &&  97 && 1.8 \\
		0.73   && 1.4  && 6.6  &&  0.66 &&  83 && 1.6 \\
		0.81   && 9.1  && 14.2 &&  0.22 && 148 && 1.8 \\
		0.98   && 2.1  && 7.4  &&  0.56 && 137 && 1.3 \\
	\end{tabular}
	\caption{Parameters of the fit depicted in Fig.\ \ref{fig:rixsdiff}, where only the
	peak intensities are allowed to vary with $q_c$.
	The Pearson VII oscillator describes a Lorentzian (Gaussian) line shape for
	$\mu$\,=\,1 ($\infty$) \cite{Wang05Pearson}.
	Above about 1\,eV it is not possible to distinguish different features.
	The lowest excitation energy of 0.03\,eV is estimated from the data in Fig.\ \ref{fig:rixs3}b.
\label{tab:fit} }
\end{table}

Having established the quasi-molecular nature of the electronic states, we aim to achieve a
microscopic understanding of the electronic structure of Ba$_3$InIr$_2$O$_9$. To this end,
we compare our RIXS data with exact diagonalization results. Typical for iridates and other
$5d$ compounds, the electronic structure is governed by the interplay of Coulomb interactions,
spin-orbit coupling, hopping interactions, and the non-cubic part of the crystal field.
However, for the cluster Mott insulator Ba$_3$InIr$_2$O$_9$ the focus is in particular on the
competition between hopping and spin-orbit coupling. Is hopping strong enough to counteract
the effects of spin-orbit coupling?
In general, individual $5d^5$ Ir$^{4+}$ sites show local moments with $j$\,=\,1/2 and 3/2.
In the case of two holes per dimer as in Ba$_3$CeIr$_2$O$_9$ with Ir$^{4+}$, (anti-) bonding
states of these spin-orbit entangled $j$ states provide a good starting point for the
description \cite{Revelli19}.
Below we show that the same applies to Ba$_3$InIr$_2$O$_9$, as sketched in Fig.\ \ref{fig:sketch},
giving rise to the rich excitation spectrum observed in RIXS.\@

First, we determine the peak energies by fitting the RIXS data using Pearson VII oscillators \cite{footnotePearson}.
For the fit, we considered the dispersion as negligible, as discussed above, and averaged
the data of Fig.\ \ref{fig:rixsGMK} over $\it{\Gamma}$, $M$, and $K$ for constant $q_c$
to improve statistics.
Motivated by the intensity modulation depicted in Fig.\ \ref{fig:rixslscan}a, we focus on
$q_c/(2\pi/c) \approx 20$ and 17, i.e., close to a maximum and a minimum of interference.
In the fit, only the peak intensities are allowed to change as a function of $q_c$,
all other parameters are treated as independent of $q_c$.
The fit describes the RIXS spectra very well, see Fig.\ \ref{fig:rixsdiff}.
The corresponding parameters are summarized in Table \ref{tab:fit},
and the peak energies are plotted as symbols in the middle panel of Fig.\ \ref{fig:3holeQMOs}.

The interaction Hamiltonian with on-site Coulomb
repulsion $U$ and Hund's coupling $J_H$ reads
\cite{Perkins14}
\begin{eqnarray}
\nonumber
H_C & = & U \, \sum_{i,\alpha} n_{i\alpha \uparrow} n_{i\alpha \downarrow}
+  \frac{1}{2} (U-3J_H) \sum_{i,\sigma,\alpha\neq\alpha^\prime} n_{i\alpha \sigma} n_{i\alpha^\prime \sigma} \\
\nonumber
& + & (U-2J_H) \sum_{i,\alpha\neq\alpha^\prime} n_{i\alpha \uparrow} n_{i\alpha^\prime \downarrow}
\\
\nonumber
& + & (U-2J_H)\sum_{i} \Big( 15-5\sum_{\alpha,\sigma} n_{i\alpha \sigma} \Big)
\\
\nonumber
& + & J_H \sum_{\alpha \neq \alpha'} \left( c^\dagger_{\alpha \uparrow} c^\dagger_{\alpha \downarrow} c_{\alpha'\downarrow} c_{\alpha'\uparrow}
- c^\dagger_{\alpha\uparrow} c_{\alpha\downarrow} c^\dagger_{\alpha'\downarrow} c_{\alpha'\uparrow}\right)
\, ,
\end{eqnarray}
with $n_{i\alpha\sigma}$ being the number operator for the $t_{2g}$ orbitals
$\alpha \! \in \! \{a_{1g},e_g^{\pi+},e_g^{\pi-}\}$ \cite{KhomskiiZhETF}
at site $i,j \! \in \! \{1,2\}$ and \mbox{$\sigma \! \in \! \{\uparrow,\downarrow\}$},
and the operators $c^\dagger_{\alpha \sigma}$ create holes.
We stick to $P6_3/mmc$ symmetry for the analysis of our RIXS data and consider
the trigonal crystal field via $\Delta_{\rm CF} L_z^2$, splitting the $t_{2g}$ level into
$a_{1g}$ and $e_g^\pi$ orbitals, and the inter-site hopping amplitudes $t_{a_{1g}}$ and
$t_{e_g^\pi}$. For $D_{3h}$ symmetry of the Ir$_2$O$_9$ dimer, hopping is diagonal
in the $a_{1g}$--$e_g^\pi$ basis.
Finally, we employ spin-orbit coupling $\zeta \, \mathbf{l}\cdot \mathbf{s}$.

\begin{figure}[t]
	\centering
\includegraphics[width=\columnwidth]{Revelli_fig09.pdf}
	\caption{Excitation energies of the many-body states calculated by exact
	diagonalization for $U$\,=\,1.5\,eV.\@
	Left: Dependence on $\zeta$ for $t$\,=\,$\Delta_{\rm CF}$\,=\,0 and $J_H$\,=\,0.33\,eV.\@
	For $\zeta$\,=\,0.43\,eV, the lowest excitation energies
	of about 0.4\,eV and 0.7\,eV agree with the dominant characteristic RIXS features of
	$t_{2g}^4$ iridates \cite{Yuan17,Kusch18,Nag18} and with the spin-orbit exciton of
	$t_{2g}^5$ iridates at $3/2\zeta$.
	Right: Effect of hopping in the simplified case	$t$\,=\,$t_{a_{1g}}$\,=\,$-t_{e_g^\pi}$
	for $\zeta$\,=\,0.43\,eV and $J_H$\,=\,$\Delta_{\rm CF}$\,=\,0, where hopping is
	block-diagonal in the $j$ basis. Notably, several excitation energies are independent of $t$.
	Middle: Excitation energies as a function of hopping $t_{a_{1g}}$ for the realistic
	parameter set with $\zeta$\,=\,0.43\,eV, $J_H$\,=\,0.33\,eV, $\Delta_{\rm CF}$\,=\,0.1\,eV,
	and $t_{e_g^\pi}/t_{a_{1g}}$=\,$-0.6$.
	For these parameters, the ground state changes at about 0.52\,eV, and good agreement
	with the experimental peak energies (symbols) is obtained for $t_{a_{1g}}$=\,0.53\,eV.\@
	The weak RIXS feature at 0.23\,eV (open blue symbol) is
	tentatively attributed to Ir/In site disorder, see Sec.\ \ref{sec:lowE}.
	}
	\label{fig:3holeQMOs}
\end{figure}

Discussing three $t_{2g}$ holes on two Ir sites,
we start from vanishing hopping $t$\,=\,$t_{a_{1g}}$\,=\,$t_{e_g^\pi}$\,=\,0 and a cubic crystal field,
i.e., $\Delta_{\rm CF}$\,=\,$0$.
States with all three holes on the same site are about 2\,$U$ above the ground state and can be neglected.
For $t$\,=\,0 we hence are facing simple product states $|t_{2g}^5\rangle_i\,|t_{2g}^4\rangle_{j}$
of single-site multiplets with one and two holes.
Their joint excitation energies as function of $\zeta$ are depicted in the left panel of
Fig.\ \ref{fig:3holeQMOs} for $U$\,=\,1.5\,eV and $J_H$\,=\,0.33\,eV.\@
Energies are plotted up to $\zeta$\,=\,0.43\,eV, a realistic value for
iridates \cite{KimKhalNa14}.
While $t_{2g}^5$ systems in cubic approximation show a single intra-$t_{2g}$ excitation
at $3/2\zeta \approx 0.7$\,eV \cite{Gretarsson13,Rossi17,Liu12,Revelli19b,Reig20,Khan21,Moretti14}
the dominant RIXS features in $t_{2g}^4$ compounds are observed
at roughly 0.4 and 0.7\,eV \cite{Yuan17,Kusch18,Nag18}, see dashed lines in top panel of
Fig.\ \ref{fig:theoRIXS}.

Hopping is diagonal in the single-particle $a_{1g}$--$e_g^\pi$ basis and mixes the local multiplets.
We therefore skip a more detailed discussion of the $t_{2g}^4$ multiplets but mention in passing
that the local $t_{2g}^4$ ground state shows $j$\,=\,0.
In contrast to the $t_{2g}^4$ states, the single-hole $t_{2g}^5$ states with $j$\,=\,1/2 and 3/2
are still important for an intuitive picture of the electronic states of a dimer.
In particular, most of the characteristic features of the electronic structure already
become apparent when considering an idealized system with $\Delta_{\rm CF}$\,=\,$J_H$\,=\,0 and
$t_{e_g^\pi}/t_{a_{1g}}$\,=\,$-1$. The latter makes hopping block-diagonal in the $j$\,=\,1/2
and 3/2 basis, so that $j$ (but not $j_z$) remains a good quantum number.
In this case, the eigenstates can be built from bonding and anti-bonding states
\begin{align}\label{eq:BAB}
|j^m_\pm\rangle &\equiv \frac{1}{\sqrt{2}} \big( |j,m\rangle_1  \pm |j,m\rangle_2 \big)
\end{align}
where $m$ denotes suitable linear combinations of
$j_z$\,=\,$\pm 1/2$ or $j_z$\,=\,$\pm 3/2, \pm 1/2$,
and the definition of states on sites 1 and 2 takes into account that
the IrO$_6$ are rotated with respect to each other \cite{KhomskiiZhETF}.
Product states of these simple states form a very good starting point for the
discussion of the many-body eigenstates, even though $t_{e_g^\pi}/t_{a_{1g}}$
is closer to $-1/2$ in Ba$_3$InIr$_2$O$_9$ and Coulomb interaction is strong.

For the idealized case with $t$\,=\,$t_{a_{1g}}$\,=\,$-t_{e_g^\pi}$,
the right panel of Fig.\ \ref{fig:3holeQMOs} depicts the effect of hopping
on the excitation energies. We employ $\zeta$\,=\,0.43\,eV and $U$\,=\,1.5\,eV,
keeping $\Delta_{\rm CF}$\,=\,$J_H$\,=\,0 for simplicity.
For not too large $t$, the doubly degenerate ground state shows two holes in the bonding
$j$\,=\,1/2 orbital $|\sfrac{1}{2}^\sigma_+\rangle$ and the third hole in the anti-bonding orbital,
resulting in $j_{\rm dim}$\,=\,1/2,
\begin{align}\label{eq:0}
|\sfrac{1}{2}_{\rm dim}\rangle_\sigma
& =  |\sfrac{1}{2}^\uparrow_+\rangle \, |\sfrac{1}{2}^\downarrow_+\rangle \,
 |\sfrac{1}{2}^\sigma_-\rangle\, ,
\end{align}
as sketched in Fig.\ \ref{fig:sketch}.
In contrast, the first excited state at small $t$ has two holes in the anti-bonding orbital,
\begin{align}\label{eq:1}
|1\rangle_\sigma & =  |\sfrac{1}{2}^\sigma_+\rangle \, |\sfrac{1}{2}^\uparrow_-\rangle \,
|\sfrac{1}{2}^\downarrow_-\rangle\, .
\end{align}
The absolute energies of
$|\sfrac{1}{2}_{\rm dim}\rangle_\sigma$
and $|1\rangle_\sigma$ vary like $\pm t$, and hence the
excitation energy of $|1\rangle_\sigma$ rises with a slope $2t$. Even though $U$ is large,
this perfectly agrees with a simple picture of non-interacting holes since hopping for
these three-particle states does not affect double occupancy.

The further low-energy states show one hole in an (anti-)bonding $j$\,=\,3/2 orbital,
\begin{align}\label{eq:2}
|\sfrac{3}{2}_{\rm dim}\rangle_m
& =  |\sfrac{1}{2}^\uparrow_+\rangle \, |\sfrac{1}{2}^\downarrow_+\rangle \,
|\sfrac{3}{2}^m_+\rangle \,
\\ \label{eq:3}
|2\rangle_m
& =  |\sfrac{1}{2}^\uparrow_+\rangle \, |\sfrac{1}{2}^\downarrow_+\rangle \,
|\sfrac{3}{2}^m_-\rangle \,
\\ \label{eq:4}
|3\rangle_{\sigma\sigma^\prime m}
& =  |\sfrac{1}{2}^\sigma_+\rangle \, |\sfrac{1}{2}^{\sigma^\prime}_-\rangle \,
|\sfrac{3}{2}^m_+\rangle \, .
\end{align}
For $t$\,=\,0, all of these states lie at $\sfrac{3}{2}\, \zeta$. For $U$\,=\,0, their energies
are given in the simple non-interacting picture described above, i.e., each bonding (anti-bonding)
state contributes $-t$ ($+t$) to the excitation energy. In this limit, the energy of state
$|\sfrac{3}{2}_{\rm dim}\rangle_m$ decreases with a slope $-3t$, which corresponds to an
excitation energy $\sfrac{3}{2}\, \zeta -2t$. Accordingly, $|\sfrac{3}{2}_{\rm dim}\rangle_m$
becomes the ground state for $t$ larger than a critical value.
For finite $U$, the slope is reduced and the behavior is not perfectly linear anymore,
see right panel of Fig.\ \ref{fig:3holeQMOs}.
The reduced slope reflects the Coulomb interaction between holes in the bonding orbitals
for $j$\,=\,1/2 and 3/2 which compete for the same space between the two Ir sites.

In contrast to the behavior of $|\sfrac{3}{2}_{\rm dim}\rangle_m$, the energies of
$|2\rangle_m$ and $|3\rangle_{\sigma\sigma^\prime m}$ vary like $-t$, independent of $U$.
Therefore, their excitation energy remains constant, $\sfrac{3}{2}\, \zeta$.
With this we collected the main ingredients to describe the general properties observed
in the RIXS spectra of Ba$_3$InIr$_2$O$_9$. For realistic parameters, the simple states
described above are split into a multitude of levels, see middle panel of Fig.\ \ref{fig:3holeQMOs},
but the main features of the RIXS spectra can still be explained in an intuitive picture,
as described in the following and depicted in Fig.\ \ref{fig:sketch}.

First of all, the insensitivity of the excitation energy to hopping roughly remains valid
for many states. This applies to the phase observed for small hopping but still is valid
above the phase transition. This explains that the main RIXS intensity in Ba$_3$InIr$_2$O$_9$
is found in the energy range where single-site-like $d^4$ and $d^5$ iridates show RIXS features
based on $\zeta$ and $J_H$
\cite{Gretarsson13,Rossi17,Liu12,Revelli19b,Reig20,Khan21,Moretti14,Yuan17,Kusch18,Nag18},
i.e., around 0.4\,eV and 0.7\,eV.\@
Note that this does not imply that the character of these states is the same as for vanishing hopping.
It rather signifies that these states
in terms of bonding/anti-bonding have the same character as the ground state.
For instance for small hopping, they are mainly built  from two bonding and one anti-bonding orbital
and hence show a very similar dependence on hopping as the state $|\sfrac{1}{2}_{\rm dim}\rangle_\sigma$.

Second, the observation of distinct low-energy RIXS features below 0.3\,eV suggests that
Ba$_3$InIr$_2$O$_9$ is close to the phase transition \cite{Li20} where
-- in the simple picture given in the right panel of Fig.\ \ref{fig:3holeQMOs}
and sketched in Fig.\  \ref{fig:sketch}  -- the ground state changes from
$|\sfrac{1}{2}_{\rm dim}\rangle_\sigma$ to $|\sfrac{3}{2}_{\rm dim}\rangle_m$.
The energies of the low-energy features strongly disperse as a function of hopping and
hence restrict the possible range of $t_{a_{1g}}$  and $t_{e_g^\pi}$. However, the slope
of these states is renormalized by Coulomb interaction, as discussed above.
Therefore the relevant range of hopping parameters depends on $U$ \cite{Li20}.
Deviations from the idealized case $t_{e_g^\pi}/t_{a_{1g}}$\,=\,$-1$
mix states with different $j$ and thereby lift the four-fold degeneracy in
Eq.\ \eqref{eq:2}, giving rise to two Kramers doublets.
For the parameters used in Fig.\ \ref{fig:3holeQMOs}, this splitting is of the same order
of magnitude as the non-cubic crystal-field splitting  $\Delta_{\rm CF}$.

\begin{figure}[t]
	\centering
	\includegraphics[width=\columnwidth]{Revelli_fig10.pdf}
	\caption{Exact diagonalization results for RIXS spectra at (0\,\,0\,\,$nQ_d$).
	Top and middle panels show data on opposite sides of the phase transition for
	$t_{a_{1g}}$\,=\,0.52\,eV and 0.53\,eV, respectively.
	All other parameters are identical, i.e., $\zeta$\,=\,0.43\,eV,
	$U$\,=\,1.5\,eV, $J_H$\,=\,0.33\,eV, $\Delta_{\rm CF}$\,=\,0.1\,eV,
	and $t_{e_g^\pi}/t_{a_{1g}}$\,=\,$-0.6$, as used in the middle panel of Fig.\ \ref{fig:3holeQMOs}.
	Bottom: result for larger $t_{a_{1g}}$\,=\,0.62\,eV and $U$\,=\,3.0\,eV.\@
	To take into account the finite In-Ir site disorder, all spectra contain the small contribution
	of individual $t_{2g}^4$ sites depicted by dashed lines in the top panel (see main text).
	}
	\label{fig:theoRIXS}
\end{figure}

With Ba$_3$InIr$_2$O$_9$ being close to the phase transition, the central question to be answered
concerns the ground state. We address this issue via the RIXS matrix elements and calculate
RIXS spectra in the dipole approximation as described in Ref.\ \cite{Revelli19}.
In the top and middle panels of Fig.\  \ref{fig:theoRIXS}, we compare results for
$t_{a_{1g}}$\,=\,0.52\,eV and 0.53\,eV, located on opposite sides of the phase transition.
All other parameters are kept fixed. In contrast to the excitation energies that are very similar
in the two cases, the RIXS matrix elements sensitively depend on the ground state, which is
reflected in the RIXS intensity of different features as well as in their dependence on $q_c$.
In the experiment, we find that the RIXS intensity is larger for odd values of $Q_d$ (red and
orange curves in Figs.\ \ref{fig:rixst2geg} and \ref{fig:theoRIXS}). This is reproduced for
hopping larger than the critical value, i.e., a ground state related to
$|\sfrac{3}{2}_{\rm dim}\rangle_m$.
The corresponding middle panel of Fig.\ \ref{fig:theoRIXS} shows reasonable agreement with
the experimental data concerning the main features of the spectrum. In contrast, the overall
RIXS intensity is very similar for even and odd $Q_d$ for $t_{a_{1g}}$\,=\,0.52\,eV,
i.e., smaller than the critical value where the ground state is based on
$|\sfrac{1}{2}_{\rm dim}\rangle_\sigma$.
We hence conclude that hopping exceeds the critical value while the presence of low-energy
features demonstrates that Ba$_3$InIr$_2$O$_9$ is still close to the phase transition,
see Fig.\ \ref{fig:sketch}.
Concerning the character of the ground state, our result agrees with an analysis of the
magnetic susceptibility of Ba$_3$InIr$_2$O$_9$ \cite{Li20}, and the corresponding sequence
with the lowest bonding $j$\,=\,3/2 orbital lying below the anti-bonding $j$\,=\,1/2 orbital
agrees with a RIXS study on Ba$_3$CeIr$_2$O$_9$ with two holes per dimer \cite{Revelli19}.

Concerning the electronic parameters, we use $\zeta$\,=\,0.43\,eV and $J_H$\,=\,0.33\,eV,
which are typical for iridates \cite{KimKhalNa14,Revelli19}.
Determination of the hopping parameters is more subtle. For given $\zeta$, the critical hopping for
the phase transition depends on $U$ \cite{Li20}. Using $t_{e_g^\pi}/t_{a_{1g}}$\,=\,$-0.6$,
we obtain very similar results for the RIXS spectra for $U$\,=\,1.5\,eV and 3.0\,eV
if $t_{a_{1g}}$ is adapted from 0.53\,eV to 0.62\,eV, see Fig.\ \ref{fig:theoRIXS}.
These values roughly agree with first-principles density-functional theory calculations \cite{YingLi}
which yield $t_{a_{1g}}$\,=\,$0.47$\,eV and $t_{e_g^\pi}/t_{a_{1g}}$=\,$-0.53$.
Finally, the effect of a small crystal-field splitting $\Delta_{\rm CF}$ is not pronounced,
which prevents an accurate determination. We employ $\Delta_{\rm CF}$\,=\,0.1\,eV.\@

For these parameters, the many-body ground state still is well approximated by
$|\sfrac{3}{2}_{\rm dim}\rangle_m$ which describes about 86\,\% of the weight.
The main admixture, carrying about 8\,\%, stems from
\begin{align}\label{eq:2prime}
|4\rangle_m
& =  |\sfrac{1}{2}^\uparrow_-\rangle \, |\sfrac{1}{2}^\downarrow_-\rangle \,
|\sfrac{3}{2}^{m}_+\rangle \,
\\
|5\rangle_{\sigma\sigma^\prime m}
& =  |\sfrac{1}{2}^\sigma_+\rangle \, |\sfrac{1}{2}^{\sigma^\prime}_-\rangle \,
|\sfrac{3}{2}^{m}_-\rangle \, ,
\end{align}
and a further 3\,\% are contributed by
\begin{align}\label{eq:6}
|6\rangle_{\sigma m m^\prime}
	& =  |\sfrac{1}{2}^\sigma_+\rangle \, |\sfrac{3}{2}^m_+\rangle \,
	|\sfrac{3}{2}^{m^\prime}_+\rangle \, .
\end{align}
Similarly, $|\sfrac{1}{2}_{\rm dim}\rangle_\sigma$ carries about 83\,\% of the weight
of the lowest excited many-body state, and $|2\rangle_m$ and
$|3\rangle_{\sigma\sigma^\prime m}$ together contribute 8\,\%.
We hence conclude that the spin-orbit entangled bonding and anti-bonding orbitals capture
the dominant physics for realistic parameters.

Finally, we address the possible role of In-Ir site disorder on the RIXS spectra.
With one exception, the exact diagonalization calculations describe
the experimentally observed peak energies very well, see red symbols in middle panel
of Fig.\ \ref{fig:3holeQMOs}. At low energy, however, theory predicts two excitation energies,
and our calculations do not yield any RIXS peak around 0.2\,eV.\@
This can be reconciled with the experimental data assuming that the weak RIXS intensity
around 0.23\,eV, cf.\ Fig.\ \ref{fig:rixs3} and blue symbol in Fig.\ \ref{fig:3holeQMOs},
is caused by a magnetic contribution of a few \% of Ir ions on In sites
that form 180$^\circ$ Ir-O-Ir bonds with Ir on dimer sites,
as discussed in Sec.\ \ref{sec:lowE}.
Furthermore, the calculated spectra contain a small contribution of individual
$t_{2g}^4$ Ir$^{5+}$ sites, see dashed lines in top panel of Fig.\ \ref{fig:theoRIXS}.
These reflect the presence of 7.3\,\% In$^{3+}$ ions on dimer sites, as derived
from our analysis of the crystal structure, see Sect.\ \ref{sec:struc}.
For simplicity, we assume that an In$^{3+}$ ion on a dimer site is accompanied by
an Ir$^{5+}$ ion. This contribution, however, is small and overlaps with dimer features,
it is hence not important for our analysis.
Around 0.4\,eV, the small $\mathbf{q}$ dependence of the intensity of this single-site
contribution reflects a corresponding change of the scattering geometry. Such effects
are typically much smaller than the $\mathbf{q}$ dependence observed for the
quasi-molecular dimer contribution.

\section{Conclusions}

In spin-orbit entangled iridates, experimental realizations of quantum spin liquids are very limited
and hotly debated. The honeycomb iridate H$_3$LiIr$_2$O$_6$ for instance does not show
magnetic order down to 50\,mK \cite{Kitagawa18} but the role of disorder related to the H ions
has been pointed out \cite{Yadav18,Li18,Knolle19b}.
The compound Ba$_3$InIr$_2$O$_9$ was discussed as an outstanding case with clear experimental evidence
for persistent spin dynamics down to 20\,mK from thermodynamic and spectroscopic data \cite{Dey17}.
The dimer structure based on face-sharing IrO$_6$ octahedra requires to solve the issue of the
microscopic character of the local moments on a quantitative level. Understanding these local moments
is a prerequisite for a successful description of magnetism in this compound.

Overall, our results establish that Ba$_3$InIr$_2$O$_9$ is a spin-orbit-entangled cluster Mott insulator.
We prove the insulating character via dielectric spectroscopy and demonstrate
that the crystal structure shows a monoclinic distortion already at room temperature.
Based on the observation of a double-slit type sinusoidal interference pattern, our RIXS data
establish the quasi-molecular orbital character of the electronic structure in
face-sharing geometry. The three holes are fully delocalized over the two sites of a dimer.
The physics is related to the case of the sister compound Ba$_3$CeIr$_2$O$_9$ in which two holes
per dimer form a spin-orbit entangled non-magnetic $j_{\rm dim}$\,=\,0 ground state \cite{Revelli19}.
In Ba$_3$InIr$_2$O$_9$, the many-body dimer ground state can be approximated as a spin-orbit entangled
Kramers doublet based on the $j_{\rm dim}$\,=\,3/2 state in which two holes occupy the
bonding orbital built from local $j\,$=\,1/2 moments, as in Ba$_3$CeIr$_2$O$_9$, while the third hole
is in a bonding $j$\,=\,3/2 state. This character of the ground state explains the dominant
$\sin^2(q_c\,d/2)$ behavior of the RIXS intensity.
The simple picture of bonding orbitals of local $j$ states works best in the idealized case
$t_{e_g^\pi}/t_{a_{1g}}$\,=\,$-1$, where hopping is block-diagonal in the $j$ basis,
but it still yields a reasonable description for realistic parameters.
Furthermore, the observation of low-energy RIXS features demonstrates that Ba$_3$InIr$_2$O$_9$
is close to the phase transition to a $j_{\rm dim}$\,=\,1/2 state \cite{Li20}.

The realization of a new type of a gapless spin liquid based on spin-orbit entangled
moments in quasi-molecular orbitals would open up a different and very promising perspective
on iridates and quantum spin liquids in general. In Ba$_3$InIr$_2$O$_9$, the proposed
unconventional spin-liquid character is supported by a quadratic behavior found in the specific heat
at low temperature, which deviates from the typical spin-liquid scenario of fermionic
spinons \cite{Dey17}. However, this has to be taken with a grain of salt since the
possible role of a small percentage of cation disorder needs to be further elucidated.

The magnetism in Ba$_3$InIr$_2$O$_9$ is based on inter-dimer exchange interactions which
occur on an energy scale that challenges the state-of-the-art energy resolution of RIXS at
the Ir $L_3$ edge, calling for thorough theoretical studies of exchange interactions between
quasi-molecular states on neighboring dimers and further spectroscopic investigations
with other techniques.

\acknowledgments

We thank Y. Li, S. M. Winter, and R. Valent{\'\i} for fruitful discussions
and for sharing their DFT results with us.
We acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
-- Project numbers 277146847 and 247310070 -- CRC 1238 (projects A02, B02, B03, and C03)
and CRC 1143 (project A05), respectively.
The work in Augsburg was supported by the DFG through project 107745057 (TRR 80) and
by the Federal Ministry for Education and Research
through the Sofja Kovalevskaya Award of the Alexander von Humboldt Foundation.
M.H.\@ acknowledges partial funding by the Knut and Alice Wallenberg Foundation and the Swedish Research Council.

redpajama_set_name: RedPajamaArXiv


text:
include $(ELEMVARS)

# locations of source files separated by ':'
VPATH=src:../common/src

SRC = \
	main.cpp                  \
	command_line.cpp          \
	spooky_v2.cpp             \
	matrix_market_file.cpp    \
	utils.cpp                 \
	$(SYSTEM_FILE)            \
	constants.cpp             \
	file_loader.cpp           \
	nmf_options.cpp           \
	nmf.cpp                   \
	delimited_file.cpp        \
	error.cpp                 \
	bit_matrix.cpp            \
	bit_matrix_ops.cpp        \
	nnls.cpp                  \
	assignments.cpp           \
	node.cpp                  \
	clust_options.cpp         \
	clust.cpp                 \
	tree.cpp                  \
	flat_clust_output.cpp     \
	hierclust_writer_factory.cpp \
	hierclust_xml_writer.cpp  \
	hierclust_json_writer.cpp \
	flatclust_xml_writer.cpp  \
	flatclust_json_writer.cpp

# object files replace the .cpp extension with .o
OBJS = $(patsubst %.cpp, objs/%.o, $(SRC))

TARGET=hierclust

# # paths to all include folders
INCLUDES = \
	-Iinclude \
	-I../common/include \
	-I$(ELEMENTAL_INCLUDES)

all: hierclust_bin

hierclust_bin: bin/$(TARGET)

# link the target binary
bin/$(TARGET) : $(OBJS)
	@mkdir -p $(dir $@)
	$(CXX) $(CXXFLAGS) -o $@ $^ $(LDFLAGS) $(LIBS)

# The -MMD switch causes several things to happen.  First, a makefile rule is
# generated that contains the the compiler to generate dependency files for
# each file being compiled.  The dependency files are given a '.d' extension
# and are written to the default location for preprocessed output.  The
# 'include' statement below includes them in the makefile.  The -MP option
# causes a phony target to be generated for each dependency other than the
# main CPP file being compiled (such as for header files).  This feature
# prevents the problem of having to include headers explicitly in the makefile
# and keeping them updated.
objs/%.o: %.cpp
	@mkdir -p $(dir $@)
	$(CXX) -c $(CXXFLAGS) $(INCLUDES) -MMD -MP -o $@ $<

clean:
	@rm -rf bin* objs*

install:
	install -m 0755 bin/$(TARGET) $(DESTDIR)$(bindir)

uninstall:
	-rm $(DESTDIR)$(bindir)/$(TARGET) >/dev/null 2>&1

# include all dependency files as individual targets
ifneq ($(MAKECMDGOALS),clean)
-include $(wildcard objs/*.d)
endif

.PHONY: clean all hierclust_bin install uninstall

redpajama_set_name: RedPajamaGithub


My fabulous, gorgeous, talented, youthful, vibrant, funny, caring and all-around wonderful sister, Cathy, gave me this recipe, which she makes for her husband, Gregg, who just happens to be an authority on anything chocolate.
I asked if I could post it because, as Cathy says "It's killer, and you will never smell such a heavenly aroma in your whole life as this cake baking in the oven. Not kidding."
Heat oven to 350F. Grease & flour tube pan.
Beat sugar, margarine, eggs and vanilla in large mixer bowl on medium speed, scraping bowl occasionally, 2 minutes. Beat in flour, baking powder, baking soda and salt alternately with sour cream on low speed.
Spread 1/3 of batter in pan and sprinkle with 1/3 of cinnamon sugar and 1/3 of chips; do this two more times.
Bake until wood pick inserted in center comes out clean - about 1 hour.
This recipe also works very well in a 9x13 pan or two 9x5x3 loaf pans (you would only do the layering procedure twice instead of 3 times).
Oh, and if you like bigger chunks of chocolatey-goodness in your cake, use regular sized chocolate chips instead of mini chips. Your call!


text:
215-260-4881 gail@gailcummings.com
I hope you have had time this holiday season to recharge, renew and refresh – and get ready for a productive, enthusiastic, and joyful 2023.
A bucket list is comprised of the experiences one wants to have during their lifetime. It seems that many people hold on to their lists until they are older and have "more time."
Inspiration Indeed!
I recently went to a concert of one of my favorite musicians: Keith Urban. At age 54, with more than 30 years in the music industry, he rocked a sold-out 25,000-person outdoor venue for 2+ hours on a Sunday evening in 90-degree heat. What an inspiration!
People Supporting People
We shouldn't have to pretend that everything is okay when it isn't.
Jan 5, 2023 | Emotions, Family, Personal Growth, Relationships
Wishing you a great start to the New Year! I hope you have had time this holiday season to recharge, renew and refresh...
Oct 20, 2022 | Emotions, Personal Growth
Why Wait? A bucket list is comprised of the experiences one wants to have during their lifetime. It seems that many...
Aug 18, 2022 | Emotions, Personal Growth, Relationships
Inspiration Indeed! Music creates community. It moves us, touching our emotions of joy, sadness, love, and loss.I...
Emotions, Family, Personal Growth, Relationships
Emotions, Personal Growth
Emotions, Personal Growth, Relationships
People Supporting People We shouldn't have to pretend that everything is okay when it isn't.Two days ago, on July 4th,...
One Heart at a Time
One Heart at a Time On a recent road trip, I stopped in Charlotte, North Carolina and discovered the city loves public...
The Slap Heard Round the World
Emotions, Therapy
The Slap Heard Round the World Everyone has heard, seen and dissected the "slap" Will Smith bestowed on Chris Rock. On...
Courage Under Pressure
Courage Under Pressure Who among us hasn't had a moment, a day or a week where we felt the pressure in our lives –...
Toxic Positivity
Toxic Positivity "My comfort is more important than your reality..."Getting into a good public high school in NYC,...
Our 2021 Superpower
Our 2021 Superpower is . . . After having our world suddenly turned upside down in 2020, it seems that 2021 could not...
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redpajama_set_name: RedPajamaCommonCrawl


text:
Robert George 'Out of Danger' After Being Hospitalized With Life-Threatening Heart Complication
CP Current Page: U.S. | Friday, December 11, 2015 | Coronavirus →
By Samuel Smith, U.S. Editor Follow | Friday, December 11, 2015
Princeton professor Robert George tells those in attendance at the In Defense of Christians press conference in Washington, Sept. 10, 2014, that more voices need to be heard of American and European Christians to urge their governments to remedy the atrocities facing Middle East Christians. | (PHOTO: THE CHRISTIAN POST/SONNY HONG)
Princeton professor and prominent religious freedom and traditional marriage advocate Robert George has been hospitalized with a serious heart ccomplication that now appears as though it is no longer life threatening.
George, who is the chair of the State Department's United States Commission on International Religious Freedoms and the McCormick Professor of Jurisprudence at Princeton, began feeling abdominal pain on Sunday and checked himself into the University Medical Center of Princeton at Plainsboro in New Jersey, where he was diagnosed with a life-threatening tear on the inner layer of his aorta.
After he disclosed his condition, a number of George's friends, colleagues and peers asked followers on social media to pray for his health and a speedy recovery.
"After a day of observation and testing, his doctors concluded that he was suffering from an aortic arterial dissection. This is a very rare and quite serious condition," Ryan Anderson, one of George's former students and senior fellow at the Heritage Foundation, posted on Facebook Thursday night.
Anderson explained that George was in the hospital's critical care unit recovering from an aortic angiography, which appeared to have confirmed the original diagnosis. In an email from one of George's close friends, Anderson was told that George was later transported to the Weill Cornell Medical Center in Manhattan where an expert in these kinds of heart conditions is available.
"Robby sounded pretty good and quite lucid. Surgery is a last resort, because it would be risky," the undisclosed friend's email to Anderson states. "Right now he is being pumped with blood thinners, so that circulation to the whole body is restored (by making it so that the blood can pass through the aorta notwithstanding the anomaly there). Robby says that he is told that this can be a stable long-term solution, and thus that surgery might not be required at all."
Sherif Girgis, one of George's former students who co-authored the book What Is Marriage?: Man and Woman: A Defense along with George and Anderson, posted on Facebook Friday morning that he was told that surgery is no longer required and George's condition is past the point of being life-threatening.
"The latest, from a friend of Robert P. George 'Deo Gratias! [thanks be to God]," Girgis wrote. "I just spoke to him and he just spoke to the doctor. He is going to be okay (no surgery). He is past the life-endangering point. Keep praying for his recovery."
Bradford Wilson, the executive director of the James Madison Program at Princeton University, also confirmed that George is "out of danger."
"Although the condition can be life-threatening, we are pleased to report that Professor George now appears to be out of danger, thanks to swift and astute diagnostic work done by the staff of University Medical Center of Princeton before his transfer to Weill Cornell," Wilson said in a statement posted online. "Although the Superior Mesenteric Artery is a branch of the aorta, the condition is not a form of heart disease. It threatens the digestive organs by impeding blood flow to them. We are grateful for the prayers and good wishes of Professor George's many friends and admirers here at Princeton and around the nation and world. We look forward to providing additional reports on Professor George's recovery."
Many prominent conservatives took to social media to offer their support to George when they heard of his condition, including Russell Moore, Eric Metaxas and Republican presidential candidate and former Florida Gov. Jeb Bush.
"Please join me in praying for Robert P. George @McCormickProf this evening as he's been hospitalized in serious condition," Bush tweeted.
Please join me in praying for Robert P. George @McCormickProf this evening as he's been hospitalized in serious condition.
— Jeb Bush (@JebBush) December 11, 2015
Author and radio host Eric Metaxas issued a statement to the The Christian Post on Friday calling George "a gift to the country."
"Robby George is one of the most brilliant minds of our time, and his willingness to winsomely engage those on the other side of the fiercest hot-button social issues is nothing less than a gift to the country," Metaxas said. "He's a dear friend and it's my honor to ask faith-filled prayers for his recovery."
Contact: <ahref="mailto:samuel.smith@christianpost.com">samuel.smith@christianpost.com, @IamSamSmith (Twitter)