# Light Induced Magnetism in Metals via Inverse Faraday Effect

Priya Sharma  Department of Physics and Institute for Materials Science,
University of Connecticut, Storrs, CT 06269, USA  Alexander V. Balatsky
Department of Physics and Institute for Materials Science, University of
Connecticut, Storrs, CT 06269, USA  Nordita, KTH Royal Institute of Technology
and Stockholm University, Hannes Alfvéns väg 12, SE-106 91 Stockholm, Sweden

###### Abstract

We present a microscopic calculation of the inverse Faraday effect in metals.
We derive a static local magnetic moment induced on the application of high-
frequency light, using the Eilenberger formulation of quasiclassical theory.
We include the effect of disorder and formulate a theory applicable across the
entire temperature range, in the absence of external applied fields. For
light-induced electric fields of amplitude $\sim 100kV/cm$, the induced fields
are large, $\sim 0.1T$ for metallic Nb! The predictions of our theory agree
with recent experimental and theoretical results [1, 2]. An extension of this
approach to superconductors would open a new route of inducing orbital
magnetic field and potentially vortices in superconductors.

The ultrafast manipulation of magnetic states is of technological importance
in non-volatile magnetic storage, ubiquitous in modern day device applications
[3]. Light-induced magnetism [4], light-induced superconductivity [5], light-
induced translational and hidden orders [6] have been investigated as examples
of optical manipulation of quantum states. The demands on pushing up the read
and write speeds in magnetic storage media are ever-increasing. The state-of-
the-art NVMe technology boasts up to subnanosecond read/write speeds ($\sim
7.5GB/sec$ for Gen 4 NVMe drives) with intrinsic limitations set by the
generation of magnetic fields by current. The search for technologies to
control magnetization without the use of magnetic fields is accelerating with
optical means emerging as a promising leader [7]. The Nobel Prize in Physics
2023 highlights the emergence of attosecond pump-probe methods to study
electron dynamics in quantum matter. With the advent of THz laser technology,
the ability to manipulate magnetic order on ultrafast sub-picosecond
timescales is being explored for magnetic switching/reversal [8, 9]. The
physics that governs light-matter interactions on these ultrafast timescales
makes available a route to quantum states inaccessible in equilibrium, that
can be technologically transformative. Historically, the understanding of
magneto-optic phenomena was associated with magnetization dynamics involving
spin degrees of freedom, including spin-orbit(SO) interactions. The SO
interaction set the time-scale for this dynamics.

Figure 1: Schematic of geometry to observe induced magnetization in metals via
the IFE. Circularly polarized light is shone on the metal surface. (Image
generated with help from OpenAI’s DALL-E tool)

Here, we evaluate a spin-independent and purely orbital method to instantiate
and further manipulate magnetic order in metals dynamically by interaction
with THz light via the Inverse Faraday Effect [10]. This non-dissipative
magneto-optical effect is known to induce static magnetic moments in various
materials on the application of high frequency light.

The Inverse Faraday Effect(IFE) was first proposed by Pitaevskii [10] as an
effect of a time-varying electric field on the stress tensor in a dispersive
medium. [11] coined the name and first reported an observed induced
magnetization in various liquids and glasses. This suggested the interesting
approach of transduction viz., transferring the orbital momentum of light to
the electronic degrees of freedom in matter. Further, it offers a means for
rectification of quantum order to generate quantum features that did not exist
in the absence of dynamical fields. The IFE has been theoretically discussed
[12, 13, 14] and observed experimentally in garnets and magnetic materials
where magnetic moments are induced on the application of high-frequency light
[15]. Ab initio methods are the leading theoretical techniques to investigate
optically induced magnetization dyamics [15]. A semiclassical theory for the
IFE in metals was derived by Hertel [16] which has been further developed
recently [17] and applied to superconductors, where the creation of vortices
by light has been proposed phenomenologically. Buzdin and coworkers suggest
vortex generation via the Kibble-Zurek mechanism by quenching into the
superconducting state [18]. Ginzburg Landau(GL) theory has been used to
calculate the induced moments in superconductors and proposed for the
manipulation of vortices using the IFE [19, 20]. Floquet theory has been
applied to study the IFE in Mott insulators [21].

In this paper, we present a microscopic calculation of the IFE applicable to
metals and extensible to superconductors. We use quantum many-body theory to
address the extent to which we can controllably produce magnetization in a
metal using circularly polarized light. We find the effect to be significant
and the induced moments predicted by our theory are large. Our microscopic
theory offers better degree of control via material parameters such as the
plasma frequency of the material $\omega_{p}$; it is applicable over the
entire temperature range, unlike phenomenological and GL theories. It applies
on quasiclassical length scales that are microscopic on the scale of the spot
size of incident light, but long compared to atomic length scales. In our
knowledge, the IFE has not been formulated in terms of the Green’s functions
for electrons in metals thus far. We propose this comprehensive approach for
light manipulation of quantum materials, emphasizing the potential emergence
of novel quantum ordered states and topological excitations via a light-
specific mechanism.

We consider a metal exposed to circularly polarized electromagnetic radiation
of frequency $\omega$, as shown in Fig.1. The primary response of
electrons(quasiparticles) is to the oscillating electric field given by
$\bf{E}=\bf{E}_{0}e^{{{i\bf{K}.\bf{R}}}-i\omega t}$. We find the induced
magnetization density via the IFE is given by,

${\bf{M}}_{ind}=\mu_{B}\frac{\omega_{p}^{2}}{2\omega(\omega^{2}+16\Gamma^{2})}\,(i\epsilon_{0}\,{\bf{E}}_{0}\times{\bf{E}}_{0}^{\star})\zeta^{2}(\frac{\omega}{2k_{B}T})\,\,\,,$
(1)

where $\Gamma$ is the rate at which quasiparticles scatter from
impurities/disorder and $\epsilon_{0}$ and $\mu_{B}$ are the free-space
permittivity and the Bohr magneton, respectively. The function
$\zeta^{2}(x)\leq 1$ over the temperature range and is discussed in Section
II. ${\bf{M}}_{ind}$ is maximal in the clean limit ($\Gamma\rightarrow 0$) and
at low temperatures when $\zeta$ is maximal. For high-frequency THz light, the
induced magnetization density is given by equation(1). For typical metals we
find the effective magnetic field to be on the scale of $0.001-0.1$T, and the
specific material estimates are given in Table 1. dc magnetization induction,
discussed here, epitomises a rectified quantum order where the dc order is
induced via a nonlinear effect upon application of the ac external field.

##  I Theoretical Framework

We consider high-frequency circularly polarized light of frequency, $\omega$
shining on a metal. The electrodynamic response has been considered by [17],
where the effect is described by the hydrodynamics of the electron density
given by the London equation. Here we consider the corrections to the Green’s
functions for electrons in response to the light-induced electric field. To
simplify the analysis, we neglect the crystal structure (and band structure)
of electrons and consider an isotropic Fermi surface. The crystal point group
dependent tensors would need to be used for other symmetries.

We employ the Keldysh formulation [22] of quasiclassical theory [23, 24] which
describes phenomena that occur on characteristic length scales much larger
than the Fermi wavelength, $k_{F}^{-1}$ and characteristic time scales much
longer than the inverse Fermi energy $\varepsilon_{F}^{-1}$ (Here, $\hbar=1$).
It is established as a robust formulation for the theoretical description of
degenerate fermionic systems and superconductors, in particular. We consider
the response of electrons to the oscillating electric field of the
electromagnetic wave, $\bf{E}=\bf{E}_{0}e^{i\bf{K}.\bf{R}-i\omega t}$. The
driving term corresponding to the response to the oscillating magnetic field
of the electromagnetic wave is much smaller, viz., $\propto v_{F}/c\ll 1$,
$v_{F}$ being the Fermi velocity, and we ignore it. We use the Eilenberger
equations to formulate and derive the second-order current response via the
IFE.

The quasiclassical Green’s function, $\hat{g}$ is the propagator for
quasiparticles with effective mass $m^{\star}$, energy $\varepsilon$ and Fermi
momentum ${\bf{p}}_{F}=m^{\star}{\bf{v}}_{F}$, given by solutions to the
Eilenberger equation [23]. Observables such as the quasiparticle density are
calculated from the Keldysh components of the Green’s function, $\hat{g}^{K}$,
viz.,
$n=\int\frac{d\varepsilon}{2\pi}\frac{d^{3}p}{(2\pi\hbar)^{3}}\mathscr{T}(\hat{g}^{K})$;
here, $\mathscr{T}$ refers to a trace over Nambu and spin indices. We expand
$\hat{g}^{K}$ in the external field $\bf{E}$,
$\hat{g}^{K}=\hat{g}_{0}^{K}+\hat{g}_{1}^{K}+\hat{g}_{2}^{K}$, with
$\hat{g}_{i}^{K}\propto\mathcal{O}(\bf{E}^{i})$ being the $i$-th order
correction to $\hat{g}^{K}$ and evaluate the current response. We are
interested in the second-order $\mathcal{O}(\bf{E}^{2})$ corrections to the
current density, $\bf{j}_{2}$, averaged over the period of the oscillating
field. We show (details in Appendix) that

${\bf{j}}_{2}=e\,\langle\,\bar{n}_{1}({\bf{p}})\,\,\bar{\bf{v}}_{1}({\bf{p}})\,\rangle_{{\bf{{p}}},(2\pi/\omega)}\,\,\,,$
(2)

where
$\langle...\rangle_{{\bf{p}},(2\pi/\omega)}\equiv(\frac{\omega}{2\pi})\int
dt\int\frac{d^{3}p}{(2\pi)^{3}}$ refers to an average over the Fermi surface
and an average over a time-period of the field; $n_{1}$ and $\bf{v}_{1}$ are
the first-order corrections to the quasiparticle density and velocity,
respectively viz.,

$\bar{n}_{1}({\bf{p}})\equiv\int{d\varepsilon}\mathscr{T}\hat{g}_{1}^{K}({\bf{p}},\varepsilon)\,\,\,\,;\,\,\,\,\bar{\bf{v}}({\bf{p}})\equiv{\bf{v}}_{F}\int{d\varepsilon^{\prime}}\mathscr{T}\hat{\tau}_{3}\hat{g}_{1}^{K}({\bf{p}},\varepsilon^{\prime})\,\,\,,$
(3)

where $\hat{\tau}$ are Pauli matrices in Nambu space. We calculate the first-
order corrections $\hat{g}_{1}^{K}$ from the Eilenberger equations and obtain
the time-averaged $\bf{j}_{2}$, where we average over a two-dimensional
isotropic Fermi surface (details in Appendix). $\bf{j}_{2}$ represents a
static current response to an applied oscillating field via the IFE. It
contains curl-less terms and a term that is $\propto\nabla\times\bf{M}$. We
identify the latter as a magnetization density induced by the IFE. The
irrotational terms describe the so-called ponderomotive forces on electrons.

##  II Results and Discussion

The induced magnetization density is given by the expression(1) with
$\zeta(\beta\omega)=\frac{tanh(\beta\omega)}{\sqrt{2-tanh^{2}(\beta\omega)}}$,
where $\beta=(2k_{B}T)^{-1}$. For $\beta\omega\gtrsim 1$, $\zeta\sim 1$ and

$\beta\omega\gtrsim
1\,\,:\,\,{\bf{M}}_{ind}\,\,\rightarrow\,\,\mu_{B}\frac{\omega_{p}^{2}}{2\omega(\omega^{2}+16\Gamma^{2})}\,{(i\epsilon_{0}\,{\bf{E}}\times{\bf{E}}^{\star})}\,\,\,.$
(4)

In the clean limit($\Gamma=0$), this agrees with the semiclassical expression
derived by Hertel[16]. For $\beta\omega\ll 1$, $\zeta\sim\beta\omega$ and

$\beta\omega\ll
1\,\,:\,\,{\bf{M}}_{ind}\,\,\rightarrow\,\,\mu_{B}\frac{\omega_{p}^{2}}{2(\omega^{2}+16\Gamma^{2})}\,{(i\epsilon_{0}\,{\bf{E}}\times{\bf{E}}^{\star})}\frac{\omega}{4(k_{B}T)^{2}}$
(5)

Therefore, for fixed disorder at a given temperature, the induced
magnetization goes through a peak as the frequency $\omega$ is varied. For
weak disorder set by $\beta\Gamma\lesssim 1$,

$\displaystyle{\bf{M}}_{ind}=\mu_{B}\frac{\omega_{p}^{2}}{2(\omega^{2}+16\Gamma^{2})}\,{(i\epsilon_{0}\,{\bf{E}}\times{\bf{E}}^{\star})}\frac{\omega}{4(k_{B}T)^{2}}$
$\displaystyle\propto$
$\displaystyle\omega\,\,\,\,\,\,\,\,\,\,;\,\,\,\,\,\beta\omega<\beta\Gamma\ll
1$ (6)
$\displaystyle=\mu_{B}\frac{\omega_{p}^{2}}{2(\omega^{2}+16\Gamma^{2})}\,{(i\epsilon_{0}\,{\bf{E}}\times{\bf{E}}^{\star})}\frac{\omega}{4(k_{B}T)^{2}}$
$\displaystyle\propto$
$\displaystyle\omega^{-1}\,\,\,\,\,;\,\,\,\,\,\beta\Gamma<\beta\omega\ll 1$
$\displaystyle=\mu_{B}\frac{\omega_{p}^{2}}{2\omega(\omega^{2}+16\Gamma^{2})}\,{(i\epsilon_{0}\,{\bf{E}}\times{\bf{E}}^{\star})}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle\propto$
$\displaystyle\omega^{-3}\,\,\,\,\,;\,\,\,\,\,\beta\Gamma\ll
1\lesssim\beta\omega\,\,\,.$

For strong disorder set by $\beta\Gamma\gtrsim 1$,

$\displaystyle{\bf{M}}_{ind}=\mu_{B}\frac{\omega_{p}^{2}}{2(\omega^{2}+16\Gamma^{2})}\,{(i\epsilon_{0}\,{\bf{E}}\times{\bf{E}}^{\star})}\frac{\omega}{4(k_{B}T)^{2}}$
$\displaystyle\propto$
$\displaystyle\omega\,\,\,\,\,\,\,\,\,\,;\,\,\,\,\,\beta\omega\ll
1\lesssim\beta\Gamma$ (7)
$\displaystyle=\mu_{B}\frac{\omega_{p}^{2}}{2\omega(\omega^{2}+16\Gamma^{2})}\,{(i\epsilon_{0}\,{\bf{E}}\times{\bf{E}}^{\star})}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle\propto$
$\displaystyle\omega^{-1}\,\,\,\,\,;\,\,\,\,\,1\ll\beta\omega\lesssim\beta\Gamma$
$\displaystyle=\mu_{B}\frac{\omega_{p}^{2}}{2\omega(\omega^{2}+16\Gamma^{2})}\,{(i\epsilon_{0}\,{\bf{E}}\times{\bf{E}}^{\star})}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$
$\displaystyle\propto$
$\displaystyle\omega^{-3}\,\,\,\,\,;\,\,\,\,\,1\lesssim\beta\Gamma\ll\beta\omega\,\,\,.$

The induced magnetization peaks at $\omega\sim\Gamma$ for weak disorder
($\Gamma\lesssim k_{B}T$) and at $\omega\sim k_{B}T$ for strong
disorder($\Gamma\gtrsim k_{B}T$). In the low-frequency limit, screening
effects become important. We have not included these effects in our
calculation as we focus on the optical response. The effective induced field
is given by ${\bf{B}}_{eff}=\mu_{0}\,{\bf{M}}_{ind}$. For metallic $Nb$, the
induced fields via the inverse Faraday effect are plotted in Fig 2. Table
LABEL:tab:M-ind lists the effective induced fields at room temperature for
some materials.

Figure 2: Effective induced magnetic fields $\bf{B}_{eff}$ in units of Tesla for elemental Nb. Plasma frequency $\omega_{p}=20eV$ for Nb and applied electric fields of $E=100kV/cm$ [25] are used here. For reference, $\Gamma=0.01eV$ corresponds to impurity mean free paths $\sim 0.1\mu m$.  Material | $\tau(\times 10^{-12}sec)$ |  $\omega_{p}$ (eV) |  $B_{eff}$(T) | $\omega(THz)$ | T(K)
---|---|---|---|---|---
Nb | 0.01 | 20 | 0.16 | 1 | 10
Pb | 0.02 | 1.8 | 0.04 | 1 | 10
Au NP | 0.0273 | 5.8 | 0.04 | $10^{3}$ | 300
Table 1: Effective induced magnetic fields at $T\sim 10K$ for various
materials in units of Tesla. $\tau$ is the relaxation time for electrons.
$E=100kV/cm$ for $\omega=1THz$ was used [26]. The values of the entries for Au
nanoparticles are taken from [1] with laser peak intensity $9.1\times
10^{13}W/m^{3}$.

The metallic skin depth of metals is in the range of $\lesssim 100nm$ and the
induced magnetic moments in metals can be measured in finite size geometries.
Indeed, this effect has been calculated using ab initio methods [2] and
observed in Au nanoparticles [1]. Our calculation yields a result(Table
LABEL:tab:M-ind) consistent with the measured values of effective fields
induced via the IFE in Au nanoparticles reported in [1], and consistent with
numerical ab initio predictions reported in [2].

##  III Summary

We consider a Keldysh Green’s function approach to formulate the pure optical
generation of magnetic fields in metals via the IFE. We perform a fully
microscopic calculation that apllies to the entire temperature range and
captures the effects of disorder scattering with no phenomenological fitting
parameters. We give the estimated effective magnetic fields induced for a
range of materials in Table LABEL:tab:M-ind. Our theory would be formally
applicable to the superconducting state where, with proper modifications of
the equations of motion, we can explore the induction of vortex states by
locally induced magnetic fields via the IFE. The proposed effect highlights
the utility of so-called ”rectification” for static manipulation of the phase
of quantum coherent states.

##  IV Acknowledgements

We are grateful to G. Aeppli, S. Bonetti, I. Khaymovich, P. Oppeneer, B.
Spivak, O. Tjernberg, J. Wiesenrieder and J. A. Sauls for useful discussions.
We acknowledge support from European Research Council under the European Union
Seventh Framework ERS-2018-SYG 810451 HERO, the Knut and Alice Wallenberg
Foundation KAW 2019.0068 and the University of Connecticut.

##  Appendix A Quasiclassical Theory : Definitions

The central object of the theory is the quasiclassical Green’s function,
$\check{g}$ which is the propagator for quasiparticles of effective mass
$m^{\star}$ with energy $\varepsilon$ and Fermi momentum
${\bf{p}}_{F}=m^{\star}{\bf{v}}_{F}$, given by the solutions to the equation,

$[\varepsilon\check{\tau}_{3}-\check{\sigma},\check{g}]_{\circ}+i{\bf{v}}_{F}\cdot\partial_{\vec{R}}\check{g}=\check{0}\,\,\,,$
(8)

where

$\check{g}=(\begin{array}[]{c c}\hat{g}^{R}&\hat{g}^{K}\\\
\hat{0}&\hat{g}^{A}\end{array})\,\,\,;\,\,\,\check{\tau}_{3}=(\begin{array}[]{c
c}\hat{\tau}_{3}&\hat{0}\\\
\hat{0}&\hat{\tau}_{3}\end{array})\,\,\,;\,\,\,\check{0}=(\begin{array}[]{c
c}\hat{0}&\hat{0}\\\ \hat{0}&\hat{0}\end{array})\,\,\,.$ (9)

Here, $\hat{g}^{R,A,K}$ are the Retarded, Advanced and Keldysh propagators and
the $\hat{\,}$ refers to matrices in Nambu space. $\hat{\tau}_{j}$ $(j=1,2,3)$
are the Pauli matrices in Nambu space. $\check{\sigma}$ is the self-energy
with corresponding Retarded, Advanced and Keldysh components. The circle
product is defined as :

$\hat{a}\circ\hat{b}({\bf{p}},{\bf{R}},\varepsilon,t)=\hat{a}({\bf{p}},{\bf{R}},\varepsilon-\frac{1}{2i}\frac{\partial}{\partial
t_{2}},t_{1})\,\hat{b}({\bf{p}},{\bf{R}},\varepsilon+\frac{1}{2i}\frac{\partial}{\partial
t_{1}},t_{2})\mid_{t_{1}=t_{2}=t}\,\,\,,$ (10)

where the $\partial/\partial t_{1}$ should be understood as acting on the left
[27]. This formulation is applicable for arbitrary external frequencies,
$\omega$ much smaller than the Fermi energy, $\omega\ll\varepsilon_{F}$.
Putting equation(9) in equation(8), we get separate equations for the $R,A,K$
propagators,

$\displaystyle[\varepsilon\hat{\tau}_{3}-\hat{\sigma}^{R,A},\hat{g}^{R,A}]_{\circ}+i{\bf{v}}_{F}\cdot\nabla\hat{g}^{R,A}$
$\displaystyle=$ $\displaystyle\hat{0}\,\,\,;$ (11)
$\displaystyle(\varepsilon\hat{\tau}_{3}-\hat{\sigma}^{R})\circ\hat{g}^{K}-\hat{\sigma}^{K}\circ\hat{g}^{A}+\hat{g}^{R}\circ\hat{\sigma}^{K}-\hat{g}^{K}\circ(\varepsilon\hat{\tau}_{3}-\hat{\sigma}^{A})+i{\bf{v}}_{F}\cdot\nabla\hat{g}^{K}$
$\displaystyle=$ $\displaystyle\hat{0}\,\,\,.$

Consider electromagnetic radiation of frequency $\omega$ shone on a metal. For
high-frequency light, the primary response of electrons(quasiparticles) is to
the oscillating electric field of the electromagnetic wave. For the purpose of
this calculation, we neglect the response of electrons to the oscillating
magnetic field of light. Consider the electric field to be given by
${\bf{E}}={\bf{E}}_{0}e^{i{\bf{K}}.{\bf{R}}-i\omega t}$. The response of
quasiparticles to this field can be calculated using equations(11). We expand
$\hat{g}$ in the external field ${\bf{E}}$,

$\hat{g}^{X}=\hat{g}_{0}^{X}+\hat{g}_{1}^{X}+\hat{g}_{2}^{X}\,\,\,\,\,,(X=R,A,K)$
(12)

with $\hat{g}_{i}^{X}\propto\mathcal{O}({\bf{E}}^{i})$ being the $i$-th order
correction to $\hat{g}^{X}$ and evaluate the current response. The current
density operator at a space-time point $({\bf{R}},t)$ is given by

${\bf{j}}({\bf{R}},t)=e\,n({\bf{R}},t)\,{\bf{v}}({\bf{R}},t)\,\,\,,$ (13)

where $n({\bf{R}},t)$ and ${\bf{v}}({\bf{R}},t)$ are the local density and
velocity of charge carriers operators, respectively. (${\bf{R}}$ and $t$ are
the centre of mass space and time coordinates).

$\displaystyle n({\bf{R}},t)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}}n({\bf{p}},t)e^{i{\bf{p}}\cdot{\bf{R}}}$
(14) $\displaystyle{\bf{v}}({\bf{R}},t)$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p^{\prime}}{(2\pi)^{3}}{\bf{v}}({\bf{p}}\,^{\prime},t)e^{i{\bf{p}}\,^{\prime}\cdot{\bf{R}}}$

Using equations(14) in equation(13), we get ${\bf{j}}$,

${\bf{j}}({\bf{R}},t)=e\langle\,\bar{n}({\bf{p}})\,\,\bar{v}({\bf{p}})\,\rangle_{FS}\,\,\,,$
(15)

where $\langle...\rangle_{FS}\equiv\int\frac{d^{3}p}{(2\pi)^{3}}$ refers to an
average over the Fermi surface and

$\displaystyle\bar{n}({\bf{p}})$ $\displaystyle\equiv$ $\displaystyle
N_{f}\int d\varepsilon\,\mathscr{T}\hat{g}^{K}({\bf{p}},\varepsilon,t)$ (16)
$\displaystyle\bar{v}({\bf{p}})$ $\displaystyle\equiv$
$\displaystyle\frac{N_{f}}{n_{0}}{\bf{v}}_{F}\int{d\varepsilon^{\prime}}\mathscr{T}\hat{\tau}_{3}\hat{g}^{K}({\bf{p}},\varepsilon^{\prime},t)\,\,\,.$

Here, $n_{0}$ is the equilibrium density of charge carriers, $N_{f}$ is the
density of states at the Fermi level and $\mathscr{T}$ refers to the trace
over the Nambu and spin degrees of freedom.

At equilibrium, $\hat{g}_{0}$ is given by the solution to the Eilenberger
equation in the absence of external fields,

$\hat{g}_{0}^{R,A}=\pm
i\,\hat{\tau}_{3}\,\,\,,\,\,\,\hat{g}_{0}^{K}=-2i\,tanh(\beta\varepsilon)\,\hat{\tau}_{3}\,\,\,,$
(17)

where $\beta\equiv(2k_{B}T)^{-1}$ and $\varepsilon^{R,A}=\varepsilon\pm i0+$.
We use the notation for the Nambu elements,
$\hat{g}^{X}\equiv\Big{(}\begin{array}[]{c c}g^{X}&f^{X}\\\
\underline{f}^{X}&\underline{g}^{K}\end{array}\Big{)}$. $g^{X}$ and
$\underline{g}^{X}$ (and analogously $f^{X}$ and $\underline{f}^{X}$) are
related by symmetries,
$[\hat{g}^{R}]^{\dagger}=-\hat{\tau}_{3}\hat{g}^{A}\hat{\tau}_{3}$ and
$[\hat{g}^{K}]^{\dagger}=\hat{\tau}_{3}\hat{g}^{K}\hat{\tau}_{3}$.

##  Appendix B Calculation of IFE

We take into account the effect of impurities that scatter quasiparticles at a
rate $\Gamma=\frac{\hbar}{\tau}$ where $\tau=\frac{l_{mfp}}{v_{F}}$ is the
relaxation time and $l_{mfp}$ is the mean free path for scattering between
point impurities. The dynamical response of electrons to light of a given
frequency $\omega$ crosses over from the ballistic ($\Gamma\ll\omega$) to the
diffusive ($\Gamma\gg\omega$) regime as the rate of impurity scattering
increases. In the hydrodynamic limit, diffusive motion of electrons renders
the propagators and transport equation isotropic over the Fermi surface. The
response of electrons in this diffusive limit is given by solutions to the
Usadel equation for Fermi-surface averaged propagators. In the ”clean” or
ballistic limit, the propagators retain their momentum-space structure and are
given by solutions to the Eilenberger equation(11). We consider the dynamics
of electrons in this ballistic regime. The impurity self-energy is included as
$\sigma^{R}=i\Gamma$.

We are interested in calculating the time-averaged corrections to the current
density ${\bf{j}}$ that are of second order in the field,
${\bf{j}}_{2}\propto\mathcal{O}({\bf{E}}^{2})$,

${\bf{j}}_{2}=e\langle n_{2}{\bf{v}}_{0}\rangle_{FS,(2\pi/\omega)}+e\langle
n_{0}{\bf{v}}_{2}\rangle_{FS,(2\pi/\omega)}+e\langle
n_{1}{\bf{v}}_{1}\rangle_{FS,(2\pi/\omega)}\,\,\,\,\,.$ (18)

$n_{2}({\bf{p}})$ is given by equation(16) using the second-order corrections
$\hat{g}_{2}^{K}$. $\hat{g}_{2}^{K}$ is obtained from equation(11) with the
driving term $\propto({\bf{v}}_{F}\cdot{\bf{E}})\,\hat{g}_{1}^{K}$. As
$\hat{g}_{1}^{K}\propto({\bf{v}}_{F}\cdot{\bf{E}})$ (shown in equation(28)),
this gives $\langle
n_{2}{\bf{v}}_{0}\rangle_{FS}\propto\langle{\bf{v}}_{F}({\bf{v}}_{F}\cdot{\bf{E}})({\bf{v}}_{F}\cdot{\bf{E}})\rangle_{FS}=0$.
Similarly, the second term in equation(18) also gives a zero contribution.
This leaves only the third term non-zero in ${\bf{j}}_{2}$ above. $n_{1}$ and
${\bf{v}}_{1}$ are given by $\hat{g}_{1}^{K}$, as in equation(16).

$\hat{g}_{1}^{X}$ are given by solutions to the equation,

$\displaystyle(\varepsilon^{R,A}\,\hat{\tau}_{3}-\hat{\sigma}^{R,A}-\hat{\sigma}_{ext}^{R,A})\circ\hat{g}^{R,A}-\hat{g}^{R,A}\circ(\varepsilon^{R,A}\,\hat{\tau}_{3}-\hat{\sigma}^{R,A}-\hat{\sigma}_{ext}^{R,A})+i{\bf{v}}_{F}\cdot\nabla\hat{g}^{R,A}=0$
(19)
$\displaystyle(\varepsilon^{R}\,\hat{\tau}_{3}-\hat{\sigma}^{R}-\hat{\sigma}_{ext}^{R})\circ\hat{g}^{K}-\hat{g}^{K}\circ(\varepsilon^{A}\,\hat{\tau}_{3}-\hat{\sigma}^{A}-\hat{\sigma}_{ext}^{A})+\hat{g}^{R}\circ\hat{\sigma}^{K}-\hat{\sigma}^{K}\circ\hat{g}^{A}+i{\bf{v}}_{F}\cdot\nabla\hat{g}^{K}$
$\displaystyle=$ $\displaystyle 0\,,$

where $\hat{\sigma}_{ext}$ is the applied external field. Using the notation
in equation(12), the definition(10), and the self-energies given by
$\hat{\sigma}^{R,A}=\pm i\Gamma\hat{\tau}_{3}$, we evaluate the convolution
products. With

$\hat{\sigma}_{0}^{K}=(\hat{\sigma}_{0}^{R}-\hat{\sigma}_{0}^{A})\,tanh(\beta\varepsilon)=2i\Gamma\hat{\tau}_{3}\,tanh(\beta\varepsilon)\,\,\,,$
(20)

and

$\hat{\sigma}_{1}^{K}=\Gamma\hat{t}^{R}\hat{g}^{K}\hat{t}^{A}=\Gamma\hat{g}^{K}\,\,\,,$
(21)

the driving terms given by
$\hat{\sigma}_{ext}^{R,A}\circ\hat{g}^{R,A}-\hat{g}^{R,A}\circ\hat{\sigma}_{ext}^{R,A}$
for the Retarded/Advanced components and
$\hat{\sigma}_{ext}^{R}\circ\hat{g}^{K}-\hat{g}^{K}\circ\hat{\sigma}_{ext}^{A}$
for the $K$ component, respectively. The applied external field
$({\bf{A}},\Phi)$ gives the associated self-energy,
$\hat{\sigma}_{ext}=e{\bf{v}}_{F}\cdot{\bf{A}}-e\Phi$. The vector potential
${\bf{A}}$ is related to the oscillating electric field of light,
${\bf{E}}=\frac{({\bf{E}}_{0}\,e^{-i\omega
t}+c.c.)}{2}=-\frac{\partial{A}}{\partial t}$. We neglect screening effects
and set $\Phi=0$. Then to $\mathcal{O}({\bf{E}})$, ,

$\displaystyle(\varepsilon-i\Gamma-\frac{1}{2i}\frac{\partial}{\partial
t})\,\hat{\tau}_{3}\hat{g}_{1}^{K}$ $\displaystyle-$
$\displaystyle(\varepsilon+i\Gamma+\frac{1}{2i}\frac{\partial}{\partial
t})\,\hat{g}_{1}^{K}\hat{\tau}_{3}$ (22)
$\displaystyle-2i\Gamma\,tanh\Big{(}\beta(\varepsilon-\frac{1}{2i}\frac{\partial}{\partial
t})\Big{)}\,\hat{\tau}_{3}\hat{g}_{1}^{A}$ $\displaystyle+$ $\displaystyle
2i\Gamma\,tanh\Big{(}\beta(\varepsilon+\frac{1}{2i}\frac{\partial}{\partial
t})\Big{)}\,\hat{g}_{1}^{R}\hat{\tau}_{3}$
$\displaystyle-\Gamma\,\hat{g}_{1}^{K}\,\hat{g}_{0}^{A}(\varepsilon+\frac{1}{2i}\frac{\partial}{\partial
t})$ $\displaystyle+$
$\displaystyle\hat{g}_{0}^{R}(\varepsilon-\frac{1}{2i}\frac{\partial}{\partial
t})\Gamma\,\hat{g}_{1}^{K}$
$\displaystyle+i{\bf{v}}_{F}\cdot\nabla\hat{g}_{1}^{K}+2i\,(tanh\Big{(}\beta(\varepsilon+\frac{\omega}{2})\Big{)}$
$\displaystyle-$ $\displaystyle
tanh\Big{(}\beta(\varepsilon-\frac{\omega}{2})\Big{)})\,\frac{e}{-i\omega}{\bf{v}}_{F}\cdot{\bf{E}}=0\,\,\,,$
$\displaystyle(\varepsilon-i\Gamma-\frac{1}{2i}\frac{\partial}{\partial
t})\,\hat{\tau}_{3}\hat{g}_{1}^{R}-(\varepsilon-i\Gamma$ $\displaystyle+$
$\displaystyle\frac{1}{2i}\frac{\partial}{\partial
t})\,\hat{g}_{1}^{R}\hat{\tau}_{3}+i{\bf{v}}_{F}\cdot\nabla\hat{g}_{1}^{R}=0\,\,\,.$

Examining these equations, we seek solutions of the form

$\hat{g}_{1}^{X}=\hat{g}_{+}^{X}\,e^{i\omega t}+\hat{g}_{-}^{X}\,e^{-i\omega
t}\,\,\,.$ (23)

$\hat{g}_{\pm}^{X}=(\begin{array}[]{c c}g_{\pm}^{X}&0\\\
0&\underline{g}^{X}_{\pm}\end{array})$, using the notation defined in Appendix
A, with the symmetry
$\underline{g}^{R}(\hat{p},\varepsilon)=-g^{R}(-\hat{p},-\varepsilon)^{\star}$
and
$\underline{g}^{K}(\hat{p},\varepsilon)=g^{K}(-\hat{p},-\varepsilon)^{\star}$.
(Here, ${\bf{p}}=p_{F}\hat{p}$). Using the ansatz, equation(23) in the
Retarded equation(22), we get equations for $g_{\pm}^{R}$
($\underline{g}_{\pm}$ can be obtained using symmetries),

$(\varepsilon-i\Gamma\pm\frac{\omega}{2})\,g_{\mp}^{R}-(\varepsilon-i\Gamma\mp\frac{\omega}{2})\,g_{\mp}^{R}+i{\bf{v}}_{F}\cdot\nabla
g_{\mp}^{R}=0\,\,\,.$ (24)

Now ${\bf{v}}_{F}\cdot{\bf{K}}\propto\frac{v_{F}}{c}\omega\ll\omega$. So we
use ${\bf{K}}=0$ solutions. From the equation above,

$g_{\pm}^{R}=\underline{g}_{\mp}^{R}=0\,\,\,.$ (25)

Similarly, $\hat{g}_{\pm}^{A}=0$. Finally, for the Keldysh components,

$\displaystyle(\varepsilon-i\Gamma+\frac{\omega}{2})\,g_{-}^{K}$
$\displaystyle-$
$\displaystyle(\varepsilon+i\Gamma-\frac{\omega}{2})\,g_{-}^{K}-2i\Gamma\,tanh\Big{(}\beta(\varepsilon+\frac{\omega}{2})\Big{)}\,g_{-}^{A}$
$\displaystyle+2i\Gamma
tanh\Big{(}\beta(\varepsilon-\frac{\omega}{2})\Big{)}\,g_{-}^{R}$
$\displaystyle-$ $\displaystyle
i\Gamma\,g_{-}^{K}\,-i\Gamma\,g_{-}^{K}+i{\bf{v}}_{F}\cdot\nabla g_{-}^{K}$
(26) $\displaystyle+2i\,(tanh\Big{(}\beta(\varepsilon$ $\displaystyle+$
$\displaystyle\frac{\omega}{2})\Big{)}-tanh\Big{(}\beta(\varepsilon-\frac{\omega}{2})\Big{)})\,{\frac{e}{-2i\omega}}{\bf{v}}_{F}\cdot{\bf{E}}_{0}=0\,,$
$\displaystyle(\varepsilon-i\Gamma-\frac{\omega}{2})\,g_{+}^{K}$
$\displaystyle-$
$\displaystyle(\varepsilon+i\Gamma+\frac{\omega}{2})\,g_{+}^{K}-2i\Gamma\,tanh\Big{(}\beta(\varepsilon-\frac{\omega}{2})\Big{)}\,g_{+}^{A}$
$\displaystyle+2i\Gamma
tanh\Big{(}\beta(\varepsilon+\frac{\omega}{2})\Big{)}\,g_{+}^{R}$
$\displaystyle-$ $\displaystyle
i\Gamma\,g_{+}^{K}\,-i\Gamma\,g_{+}^{K}+i{\bf{v}}_{F}\cdot\nabla g_{+}^{K}$
(27) $\displaystyle+2i\,(tanh\Big{(}\beta(\varepsilon$ $\displaystyle-$
$\displaystyle\frac{\omega}{2})\Big{)}-tanh\Big{(}\beta(\varepsilon+\frac{\omega}{2})\Big{)})\,{\frac{e}{2i\omega}}{\bf{v}}_{F}\cdot{\bf{E}}_{0}^{\star}=0\,,$

and using the R,A solutions in equations(25), the ${\bf{K}}=0$ solutions are

$\displaystyle g_{-}^{K}$ $\displaystyle=$
$\displaystyle\frac{e}{\omega(\omega-4i\Gamma)}({\bf{v}}_{F}\cdot{\bf{E}}_{0})(tanh\Big{(}\beta(\varepsilon+\frac{\omega}{2})\Big{)}-tanh\Big{(}\beta(\varepsilon-\frac{\omega}{2})\Big{)})$
(28) $\displaystyle\underline{g}_{+}^{K}$ $\displaystyle=$
$\displaystyle\frac{e}{\omega(\omega+4i\Gamma)}({\bf{v}}_{F}\cdot{\bf{E}}_{0}^{\star})(tanh\Big{(}\beta(\varepsilon+\frac{\omega}{2})\Big{)}-tanh\Big{(}\beta(\varepsilon-\frac{\omega}{2})\Big{)})\,\,\,,$
$\displaystyle g_{+}^{K}$ $\displaystyle=$
$\displaystyle-\frac{e}{\omega(\omega+4i\Gamma)}({\bf{v}}_{F}\cdot{\bf{E}}_{0}^{\star})(tanh\Big{(}\beta(\varepsilon+\frac{\omega}{2})\Big{)}-tanh\Big{(}\beta(\varepsilon-\frac{\omega}{2})\Big{)})$
$\displaystyle\underline{g}_{+}^{K}$ $\displaystyle=$
$\displaystyle-\frac{e}{\omega(\omega-4i\Gamma)}({\bf{v}}_{F}\cdot{\bf{E}}_{0})(tanh\Big{(}\beta(\varepsilon+\frac{\omega}{2})\Big{)}-tanh\Big{(}\beta(\varepsilon-\frac{\omega}{2})\Big{)})\,\,\,.$

Using the ansatz in equation(23) to find $n_{1}$ and ${\bf{v}}_{1}$, given by
equation(16),

$\displaystyle n_{1}$ $\displaystyle=$ $\displaystyle
N_{f}\int{d\varepsilon}\,(g^{K}+\underline{g}^{K})$ $\displaystyle=$
$\displaystyle N_{f}\int{d\varepsilon}\,(g_{-}^{K}\,e^{-i\omega
t}+\underline{g}_{+}^{K}\,e^{i\omega t}+g_{+}^{K}\,e^{i\omega
t}+\underline{g}_{-}^{K}\,e^{-i\omega t})$ $\displaystyle{\bf{v}}_{1}$
$\displaystyle=$
$\displaystyle\frac{N_{f}}{n_{0}}{\bf{v}}_{F}\int{d\varepsilon}\,(g_{-}^{K}\,e^{-i\omega
t}+g_{+}^{K}\,e^{i\omega t}-\underline{g}_{+}^{K}\,e^{i\omega
t}-\underline{g}_{-}^{K}\,e^{-i\omega t})\,\,\,.$

In the limit ${\bf{K}}=0$, and in the absence of all self-energy and applied
external field terms, equations(B) give

$\displaystyle\varepsilon\circ{g}_{\pm}^{K}-{g}_{\pm}^{K}\circ\varepsilon+i{\bf{v}}_{F}\cdot\nabla
g_{\pm}^{K}$ $\displaystyle=$ $\displaystyle 0$ (30) $\displaystyle
g_{\pm}^{K}$ $\displaystyle=$ $\displaystyle-\frac{i{\bf{v}}_{F}\cdot\nabla
g_{\pm}^{K}}{\omega}\,\,\,.$

This is an expansion of $g_{\pm}^{K}$ valid to $\mathcal{O}({\bf{E}})$ and to
leading order in the gradient $\mathcal{O}(\nabla)$. $g_{\pm}^{K}$ is of order
$\mathcal{O}({\bf{E}})$ and we have used equation(30) for free electrons
(zeroth order in self-energies) to obtain the $\nabla$ term, which is
therefore of order $\mathcal{O}({\bf{E}}^{0})$, thus giving the expansion(30)
of $\mathcal{O}({\bf{E}})$. Now, define

$\mathcal{G}_{\pm}^{X}\equiv\int
d\varepsilon\,g_{\pm}^{X}\,\,\,;\,\,\,\underline{\mathcal{G}}_{\pm}^{X}\equiv\int
d\varepsilon\,\underline{g}_{\pm}^{X}\,\,\,,$ (31)

and use equations(B-30) in equation(15) to find (to leading order in
$\mathcal{O}(\nabla)$),

$\displaystyle{\bf{j}}_{2}$ $\displaystyle=$ $\displaystyle
e\frac{N_{f}^{2}}{n_{0}}\,\langle\,{\bf{v}}_{F}(\mathcal{G}_{-}^{K}\,e^{-i\omega
t}+\mathcal{G}_{+}^{K}\,e^{i\omega
t}+\underline{\mathcal{G}}_{+}^{K}\,e^{i\omega
t}+\underline{\mathcal{G}}_{-}^{K}\,e^{-i\omega t})\,\,\,$
$\displaystyle\cdot\\{-\frac{i{\bf{v}}_{F}\cdot\nabla}{\omega}(\mathcal{G}_{-}^{K}\,e^{-i\omega
t}+\mathcal{G}_{+}^{K}\,e^{i\omega
t}-\underline{\mathcal{G}}_{+}^{K}\,e^{i\omega
t}-\underline{\mathcal{G}}_{-}^{K}\,e^{-i\omega t})\\}\,\rangle_{FS}\,\,\,.$

Time-averaging over a period of oscillation, we get static terms,

$\langle{\bf{j}}_{2}\rangle_{2\pi/\omega}=e\frac{N_{f}^{2}}{\omega\,n_{0}}\,\langle\,{\bf{v}}_{F}(\mathcal{G}_{-}^{K}(i{\bf{v}}_{F}\cdot\nabla)\underline{\mathcal{G}}_{+}^{K}-\underline{\mathcal{G}}_{+}^{K}(i{\bf{v}}_{F}\cdot\nabla)\mathcal{G}_{-}^{K}+\mathcal{G}_{+}^{K}(i{\bf{v}}_{F}\cdot\nabla)\underline{\mathcal{G}}_{-}^{K}-\underline{\mathcal{G}}_{-}^{K}(i{\bf{v}}_{F}\cdot\nabla)\mathcal{G}_{+}^{K})\rangle_{FS}\,\,\,.$
(33)

Using the definitions(31) and the solutions (28) to evaluate
$\mathcal{G}^{K}$,

$\displaystyle\mathcal{G}_{-}^{K}$ $\displaystyle=$
$\displaystyle\frac{e}{2(\omega-4i\Gamma)}({\bf{v}}_{F}\cdot{\bf{E}}_{0})\zeta(\beta\omega)\,\,\,,$
(34) $\displaystyle\underline{\mathcal{G}}_{+}^{K}$ $\displaystyle=$
$\displaystyle\frac{e}{2(\omega+4i\Gamma)}({\bf{v}}_{F}\cdot{\bf{E}}_{0}^{\star})\zeta(\beta\omega)\,\,\,,$
$\displaystyle\mathcal{G}_{+}^{K}$ $\displaystyle=$
$\displaystyle-\frac{e}{2(\omega+4i\Gamma)}({\bf{v}}_{F}\cdot{\bf{E}}_{0}^{\star})\zeta(\beta\omega)\,\,\,,$
$\displaystyle\underline{\mathcal{G}}_{-}^{K}$ $\displaystyle=$
$\displaystyle-\frac{e}{2(\omega-4i\Gamma)}({\bf{v}}_{F}\cdot{\bf{E}}_{0})\zeta(\beta\omega)\,\,\,,$

where

$\zeta(\beta\omega)=\frac{tanh(\beta\omega)}{\sqrt{2-tanh^{2}(\beta\omega)}}\,.$
(35)

We now use these in the expression(33) to get the static induced current
density.

$\langle{\bf{j}}_{2}\rangle_{2\pi/\omega}=\frac{i\,e^{3}N_{f}^{2}\,\zeta^{2}(\beta\omega)}{\omega(\omega^{2}+16\Gamma^{2})n_{0}}\langle\,{\bf{v}}_{F}({\bf{v}}_{F}\cdot{\bf{E}}_{0})({\bf{v}}_{F}\cdot\nabla)({\bf{v}}_{F}\cdot{\bf{E}}_{0}^{\star})-{\bf{v}}_{F}({\bf{v}}_{F}\cdot{\bf{E}}_{0}^{\star})({\bf{v}}_{F}\cdot\nabla)({\bf{v}}_{F}\cdot{\bf{E}}_{0})\rangle_{FS}\,.$
(36)

Evaluating the Fermi surface averages,

$\langle{\bf{j}}_{2}\rangle_{2\pi/\omega}=\frac{e}{4m}\frac{\omega_{p}^{2}}{(\omega^{2}+16\Gamma^{2})}\frac{\nabla\times(i\epsilon_{0}\,{\bf{E}}\times{\bf{E}}^{\star})}{\omega}\zeta^{2}(\beta\omega)+{\bf{j}}_{pm}\,\,\,,$
(37)

since $N_{F}\varepsilon_{F}=n_{0}$ and the plasma frequency
$\omega_{p}=n_{0}e^{2}/m\epsilon_{0}$. The second term refers to the current
induced by ponderomotive forces on the electrons. The first term above may be
expressed in terms of a magnetization density, ${\bf{M}}_{ind}$ induced via
the IFE viz.,

$\displaystyle{\bf{j}}_{IFE}$ $\displaystyle=$
$\displaystyle\nabla\times{\bf{M}}_{ind}$ $\displaystyle{\bf{M}}_{ind}$
$\displaystyle=$
$\displaystyle\mu_{B}\frac{\omega_{p}^{2}}{2\omega(\omega^{2}+16\Gamma^{2})}\,{(i\epsilon_{0}\,{\bf{E}}\times{\bf{E}}^{\star})}\zeta^{2}(\beta\omega)\,\,\,.$
(38)

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††thanks: These authors contributed equally to this work††thanks: These
authors contributed equally to this work

# Highly Tunable Ru-dimer Molecular Orbital State in 6H-perovskite Ba3MRu2O9

Bo Yuan  Department of Physics and Astronomy, McMaster University, Hamilton,
Ontario L8S 4M1, Canada  Beom Hyun Kim  Center for Theoretical Physics of
Complex Systems, Institute for Basic Science, Daejeon 34126, Republic of Korea
Qiang Chen  Department of Physics and Astronomy, McMaster University,
Hamilton, Ontario L8S 4M1, Canada  Daniel Dobrowolski  Department of Physics
and Astronomy, McMaster University, Hamilton, Ontario L8S 4M1, Canada  Monika
Azmanska  Department of Physics and Astronomy, McMaster University, Hamilton,
Ontario L8S 4M1, Canada  G.M. Luke  Department of Physics and Astronomy,
McMaster University, Hamilton, Ontario L8S 4M1, Canada  Shiyu Fan  National
Synchrotron Light Source II, Brookhaven National Laboratory,Upton, NY, 11973,
USA  Valentina Bisogni  National Synchrotron Light Source II, Brookhaven
National Laboratory,Upton, NY, 11973, USA  Jonathan Pelliciari  National
Synchrotron Light Source II, Brookhaven National Laboratory,Upton, NY, 11973,
USA  J.P. Clancy  Department of Physics and Astronomy, McMaster University,
Hamilton, Ontario L8S 4M1, Canada

###### Abstract

Molecular orbital (MO) systems with clusters of heavy transition metal (TM)
ions are one of the most important classes of model materials for studying the
interplay between local physics and effects of itinerancy. Despite a large
number of candidates identified in the family of 4d TM materials, an
understanding of their physics from competing microscopic energy scales is
still missing. We bridge this gap by reporting the first resonant inelastic
X-ray scattering (RIXS) measurement on a well-known series of Ru dimer systems
with a 6H-perovskite structure, Ba3MRu2O9 (M3+=In3+, Y3+, La3+). Our RIXS
measurements reveal an extremely fragile MO state in these Ru dimer compounds,
evidenced by an abrupt change in the RIXS spectrum accompanying a tiny change
in the local structure tuned by the M-site ion. By modelling the RIXS spectra,
we attribute the enhanced electronic instability in Ba3MRu2O9 to the combined
effect of a large hopping and a small spin-orbit coupling in the Ru dimers.
The unique combination of energy scales uncovered in the present study make Ru
MO systems ideal model systems for studying quantum phase transitions with
molecular orbitals.

“Cluster Mott insulators”, or materials comprised of “molecules” of transition
metal (TM) ions have been one of the most important classes of model systems
in the study of strongly correlated phenomena. Due to strong inter-site
hopping that is comparable or even larger than intra-atomic interactions,
orbitals from all sites within each “molecule” or cluster are strongly
hybridized to form so-called “molecular orbitals” (MO). Here, the competition
between hopping and intra-atomic interactions, combined with an increase in
orbital degeneracy provide the minimal ingredients for many interesting
phenomena such as orbital-selective localization [1, 2] and non-local quantum
entanglement[3] that have no analogue in isolated TM ions. Moreover, the
exactly solvable nature of the finite-sized problem allows detailed comparison
between theory and experiment. These two properties place the MO materials
halfway between the extreme limits of a solid and an isolated ion, making them
unique model systems for studying the interplay between local (e.g. strong
correlation) and non-local physics (e.g. itinerancy).

Most MO candidates have been identified from the family of heavy TM materials
with 4d or 5d ions such as Ru and Ir, where inter-site hopping is enhanced by
the extended nature of the 4d and 5d orbitals 111 Given the large number of
4d/5d MO candidate materials, we cite only a few review articles. For example,
see Ref. [43, 44, 45]. A particularly important class of materials are the
6H-perovskites with chemical formula Ba3MB2O9 [5, 6, 7, 8, 9, 10], where M is
a non-magnetic ion and B=Ir/Ru is magnetic. As shown in Fig. 1, the magnetic
unit is a simple face-sharing Ru or Ir dimer with an extremely short Ru-Ru/Ir-
Ir bond that favours the formation of MO’s. Simplicity and ease of tuning
local structure via M-site substitution make these materials ideal platforms
for studying MO physics.

Despite a wealth of experimental evidence for magnetism driven by MO physics
in these dimer compounds[11, 12, 13, 14, 15, 16, 17], an understanding of
their microscopic physics, as well as that of other MO candidates with more
complex cluster geometry[18, 19, 20] became available only recently from
resonant inelastic X-ray scattering (RIXS). Since the first measurement on
Ba3CeIr2O9[21], Ir L3-edge RIXS has been applied to a large number of Ir dimer
systems[22, 23, 24, 25, 26, 3], allowing a direct determination of their
microscopic electronic Hamiltonian. In contrast, due to the difficulty in
conducting analogous RIXS measurements at the Ru L3 edges, direct measurements
of the electronic excitations in 4d Ru MO systems is completely missing. The
combination of smaller inter-site hopping due to more localized 4d orbitals,
and drastically different intra-atomic interactions, such as the significantly
reduced 4d spin-orbit coupling (SOC), has the potential to realize
qualitatively different microscopic physics in Ru-based MO systems compared to
their Ir counterparts, unlocking completely new regimes of MO physics.

Figure 1:  (a) Crystal structure of Ba3MRu2O9 without the oxygen ions. (b)
Local structure of the Ru2O9 dimer as a function of the ionic radius of the
M3+ ion. Data points with different symbols are taken from: Ref. [27, 6,
7]($\bigtriangleup$), Ref. [28, 29]($\Diamond$), Ref. [30]($\square$) and Ref.
[12, 11]( ), the last of which used the same sample as the present study. The
thick blue trend line in each panel is obtained by a linear fit to all
reported bond lengths. The local structural parameters for M3+ = In3+, Y3+ and
La3+ are denoted by vertical dotted lines. (c) Schematic illustration of the
non-degenerate $a_{1g}$ and doubly degenerate $e_{g}^{\pi}$ orbitals in the
$t_{2g}$ manifold under a trigonal crystal field. The hoppings between them
are denoted as $t_{1}$ and $t_{2}$, respectively

Here, we address this outstanding question by reporting Ru M3-edge RIXS
measurements of the dimer electronic excitations in a well-known series of Ru
6H-perovskites. The choice of Ru M3-edge is motivated by recent works
demonstrating the feasibility of Ru M3-edge RIXS [31, 32, 33], which, compared
to the more traditional Ru L3-edge RIXS, uses soft (Ei=463 eV) rather than
tender X-ray photons (Ei=2838 eV) and is therefore technically less
challenging. The 6H-perovskite series studied here, Ba3MRu2O9, with M3+=In3+,
Y3+, La3+ and 7 electrons in the Ru2O9 dimer, has been intensively studied in
the past due to its anomalous magnetic properties[11, 12, 6] (See Supplemental
Materials [34] for materials characterization and experimental details).

Our measurements reveal RIXS spectra in the Ba3MRu2O9 series that are
qualitatively different from the predictions of a local model, and place them
well in the MO limit with a large inter-site hopping. Remarkably, the measured
RIXS spectra show an abrupt change with only a tiny change in local structure
from Ba3InRu2O9/Ba3YRu2O9 to Ba3LaRu2O9 [at most $1\%-2\%$ change in Ru-O and
Ru-Ru bond lengths - see Fig. 1 (b)], thus providing direct evidence for the
fragility of electronic states in these Ru dimer compounds. Our theoretical
modelling attributes the enhanced electronic instability in the Ru dimer to a
large inter-site hopping and reduced SOC compared to its Ir counterparts. A
direct manifestation of such instability is the prediction of an electronic
transition driven by a very small change in the trigonal distortion of the
dimer, accompanied by an abrupt change in the simulated RIXS spectra which is
in excellent agreement with the experiment. The high degree of tunability for
the electronic states of the Ru dimer uncovered in the present study shows
that Ru MO compounds are ideal candidates for studying quantum phase
transitions involving molecular orbitals.

The overall RIXS features of a Ru dimer are shown by the color map in Fig. 2
(a), where we plot RIXS intensity as a function of incident photon energy,
$\mathrm{E_{i}}$ (horizontal axis), and energy transfer, $\hbar\omega$
(vertical axis) for Ba3YRu2O9. Four excitations can be clearly resolved at (A)
$\hbar\omega\lesssim 2~{}$eV, (B) $\hbar\omega\sim$3.5 eV, (C)
$\hbar\omega\sim$6 eV and (D) $\hbar\omega\sim$9 eV. Aside from a difference
in the absolute $\mathrm{E_{i}}$’s used in our experiment, corresponding to
the $3p_{\frac{3}{2}}\rightarrow 4d$ transition, the main RIXS features shown
in Fig. 2 (a) are qualitatively similar to those observed in Ru L3-edge RIXS
measurements [35, 36, 37], allowing us to assign A-D as (A) intra-$t_{2g}$
excitations, (B) $t_{2g}-e_{g}$ excitations, and charge transfer excitations
from the oxygen $p$ orbitals to (C) the Ru $t_{2g}$ and (D) $e_{g}$ orbitals,
respectively. The difference in resonant $\mathrm{E_{i}}$’s of these features
($\sim$3 eV) corresponds to the different intermediate states where the core
electron is excited to the $t_{2g}$ and $e_{g}$ orbitals for A/C and B/D,
respectively. RIXS spectra for Ba3MRu2O9 with different M-site ions are
compared in Fig. 2(b) and (c). Fig. 2 (c) is an enlarged version of (b) that
focusses on the intra $t_{2g}$ excitations with $\hbar\omega\lesssim$ 2eV. The
$\mathrm{E_{i}}$ for these scans is chosen to resonate with the intra-$t_{2g}$
features. As expected from the small changes in local structure going from
M=In to M=La, the high energy electronic excitations (B,C,D) are qualitatively
similar for the three Ba3MRu2O9 compounds, consistent with minimal changes in
the dominant energy scales such as $t_{2g}-e_{g}$ splitting and Ru-O charge
transfer energy across the Ba3MRu2O9 series.

In contrast to the high energy region, the low energy RIXS spectra of the
intra-$t_{2g}$ excitations [Fig. 2 (c)] are drastically different in these
three compounds. Two broad peaks can be resolved in Ba3InRu2O9 and Ba3YRu2O9
at $\hbar\omega\sim$0.3 eV and $\hbar\omega\sim$0.7 eV, respectively.
Strikingly, these features are very different in Ba3LaRu2O9: the low energy
peak is suppressed, while a sharp peak-like feature gains intensity at
$\sim$0.7 eV. As we show in the Supplemental Materials[34], the suppression of
the low energy RIXS intensity below $\hbar\omega<$0.5 eV relative to the high
energy feature in Ba3LaRu2O9 is a robust observation independent of
$\mathrm{E_{i}}$’s.

Qualitatively different RIXS spectra shown in Fig. 2 (c) provide direct
evidence for distinct electronic states in Ba3LaRu2O9 and
Ba3InRu2O9/Ba3YRu2O9. This is quite surprising considering the small magnitude
of the structural change across the Ba3MRu2O9 series [Fig. 1 (b)]. Our
observations are in stark contrast to analogous RIXS measurements on face
sharing Ir-dimers (Ir5+:Ref [22], Ir4.5+:Ref [25, 26, 24], Ir4+:Ref [23, 21]).
There, qualitatively similar RIXS spectra robust against small changes in
local structures are observed for compounds with identical Ir valences. A
modification of the RIXS spectra by a change in local structure is usually
attributed to a change in energies of the excited states. Here, this
explanation requires one to move up a bulk of the low-lying excited states at
$\hbar\omega\sim 0.3~{}$eV in Ba3InRu2O9 and Ba3YRu2O9 to $\hbar\omega\sim
0.7$ eV in Ba3LaRu2O9. However, such a large shift in the excited states seems
highly unlikely given at most a $1\%-2\%$ change in Ru-O and Ru-Ru bond
lengths from Ba3YRu2O9 to Ba3LaRu2O9 [Fig. 1 (b)] with no change in local
symmetry[11, 12, 6].

Figure 2:  (a) Map of RIXS intensity as a function of incident energy,
$\mathrm{E_{i}}$, and energy transfer, $\hbar\omega$, for Ba3YRu2O9. The four
features in the RIXS map denote intra-$t_{2g}$ excitations (A), $t_{2g}-e_{g}$
excitations (B) and charge transfer excitation between the oxygen $p$-orbital
and the Ru-$t_{2g}$ (C) and $e_{g}$ orbitals (D). (b) RIXS intensity as a
function of $\hbar\omega$ for Ba3MRu2O9 (M=In, Y, La) using
$\mathrm{E_{i}}$=463 eV. $\mathrm{E_{i}}$=463 eV, marked by the vertical
dashed line in (a), is chosen to resonate with the intra-$t_{2g}$ features [A
in (a)]. To directly compare RIXS scans of different compounds, the RIXS
intensity is normalized with respect to the integrated intensity between 2 eV
and 12 eV covering features B-D, which should be relatively material-
independent. (c) RIXS scans focussing on the intra-$t_{2g}$ excitations below
$\sim$ 2eV obtained by zooming into the grey shaded region in (b). All
measurements shown here are carried out at 40 K. Figure 3:  Energy levels and
simulated RIXS spectra (a-d) in the localized limit with $t_{1}\approx$0 and
(e-i) in the limit of large $t_{1}$. (a-d): (a) The energy level diagram for
the $t_{2g}^{3}$ and $t_{2g}^{4}$ sites in a Ru2O9 dimer with 7 electrons (the
degeneracy of each multiplet is proportional to the line thickness) and (b)
the corresponding RIXS spectrum for $t_{1}=t_{2}=\Delta_{tr}=0$. (c,d)
Variation of the RIXS spectrum up to $\hbar\omega<2~{}$eV as a function of
trigonal distortion, $\Delta_{tr}$ for the case of (c) zero coupling
($t_{1}=t_{2}=0$) (d) weak coupling ($t_{1}=0.1$ eV and $t_{2}=0$) between Ru
ions. The RIXS intensity is shown on a linear color scale. The energy
transfer, $\hbar\omega$, and trigonal distortion, $\Delta_{tr}$, are shown on
the vertical and horizontal axes, respectively. (e-i): (e) Variation of the
RIXS spectrum as a function of $t_{1}$ with $t_{2}=\Delta_{tr}=0$. (f) is the
same as (e) but zoomed into the region with $\hbar\omega<0.3~{}$eV marked by
the dotted yellow box. The excitation energies are plotted as a function of
$t_{1}$ on top of the RIXS spectra in (f) as white dotted lines. The solid
yellow line in (f) highlights the evolution of the
$(L=0,S=\frac{3}{2},J=\frac{3}{2})\bigotimes(L=1,S=1,J=1)$ multiplet as a
function of $t_{1}$. (g) Variation of the RIXS spectrum as a function of
$\Delta_{tr}$ for fixed $t_{1}=0.75~{}$eV and $t_{2}=0.05$. For all
calculations in (a-f), $U,J_{H}$ and $\lambda$ are fixed at $U=2.5~{}$eV,
$J_{H}=0.3$ eV and $\lambda=0.15~{}$eV. (h,i) Comparison between simulated
RIXS spectra using (h) $\Delta_{tr}=-0.1$ eV (I) and (i) $\Delta_{tr}=0.14$ eV
(II) with the measured intra-$t_{2g}$ RIXS spectrum of Ba3LaRu2O9 and
Ba3YRu2O9/Ba3InRu2O9, respectively. The two $\Delta_{tr}$ values have been
marked as vertical dashed lines in (g). To compare with experiment, the
calculated spectra in (h) and (i) have been scaled by the same factor. The
rest of the parameters are the same in (h) and (i) with $U=2.5~{}$eV,
$J_{H}=0.3~{}$eV, $\lambda=0.15~{}$eV, $t_{1}=0.75~{}$eV, $t_{2}=0.05~{}$eV.

To resolve this conundrum, we carried out RIXS modelling for the dimer by
taking into account the six $t_{2g}$ orbitals on the two Ru ions (See
Supplemental Materials [34] for details). Given the local $D_{3h}$ symmetry,
there are six independent parameters within this minimal model[38]: Hubbard
interactions ($U$), Hund’s coupling ($J_{H}$), SOC ($\lambda$), trigonal
distortion ($\Delta_{tr}$) and hopping between singly degenerate $a_{1g}$ and
doubly degenerate $e_{g}^{\pi}$ orbitals, denoted by $t_{1}$ and $t_{2}$ [See
Fig. 1 (c)], respectively. To reduce the number of parameters, we fix the
intra-atomic interactions to $U=$ 2.5 eV, $J_{H}=$ 0.3 eV, and $\lambda=$ 0.15
eV, which show little variation across different Ru-compounds[39, 35, 40], and
focus on the effects of $\Delta_{tr}$, $t_{1}$ and $t_{2}$. Due to the short
Ru-Ru bond, $\Delta_{tr}$, $t_{1}$ and $t_{2}$ are expected to be
substantially modified from their corresponding values with well-separated
RuO6 octahedra. In particular, a small separation between transition metal
ions enables direct overlap between the axial $a_{1g}$ orbitals and can
significantly increase $t_{1}$, which has been shown to be comparable or even
larger than the intra-atomic interactions in some face-sharing Ir dimers[21,
24].

Qualitatively, the importance of $t_{1}$ in the present Ru dimer compounds is
clear by comparing the measured RIXS spectra [Fig. 2 (c)] to the prediction of
a local picture by setting $t_{1}=t_{2}=0$ [Fig. 3 (a-d)]. In this limit, the
Ru2O9 dimer can be thought of as two weakly coupled Ru ions with a
$t_{2g}^{3}$ and $t_{2g}^{4}$ electronic configuration, respectively, whose
electronic states consist of well-separated Hund’s multiplets as shown in Fig.
3 (a). Due to the residual orbital degeneracy, the ground $(L=1,S=1)$
multiplet in the $t_{2g}^{4}$ case is further split by SOC into the $J=0,J=1$
and $J=2$ multiplets. As shown in Fig. 3 (b), the RIXS spectrum in the
localized limit consists of an intense low energy feature at
$\hbar\omega\lesssim$0.3 eV and a weaker feature at much higher energy with
$\hbar\omega\sim$0.7 eV-0.9 eV, corresponding to transitions within the ground
state multiplet and to the excited Hund’s multiplets, respectively [red and
blue arrows in Fig. 3 (a)]. Crucially, the large separation between these
excitations, which is a direct consequence of the small $\lambda$ compared to
$J_{H}$ in the ruthenates (in contrast to $\lambda\gtrsim J_{H}$ in
iridates[41]), implies that they are weakly coupled and hence very robust
against small perturbations. This is confirmed by the simulated RIXS spectra
in Fig. 3 (c)-(d), where we show the variation of the RIXS spectrum up to
$\hbar\omega<2~{}$eV (vertical axis) as a function of changes in local
structure parametrized by $\Delta_{tr}$ (horizontal axis) for the case of (c)
$t_{1}=t_{2}=$0 and (d) $t_{1}=0.1$ eV and $t_{2}=$0\. Other than a slight
shift in energy of different RIXS features, the RIXS spectrum varies smoothly
with $\Delta_{tr}$ and resembles that of Fig. 3 (b) for the entire range of
$\Delta_{tr}$. Within a local picture, it is therefore difficult to explain
the abrupt changes in RIXS spectrum observed with minimal changes in local
structure across the Ba3MRu2O9 series. Moreover, the RIXS spectrum of
Ba3LaRu2O9 shows only one peak at $\sim 0.7~{}$eV, completely different from
the simulation in Fig. 3(b)-(d) which always shows two peaks below 1 eV. As we
show below, both of these observations can be explained by the presence of a
large inter-site hopping $t_{1}$.

As shown by the calculated $t_{1}$-dependence of the excitation energies and
the RIXS spectra in Fig. 3 (e), increasing $t_{1}$ moves up a bulk of the low
energy states within the $(L=0,S=\frac{3}{2})\bigotimes(L=1,S=1)$ manifold
together with most of its spectral weight. This part of the spectral weight
eventually merges with the high energy transitions for $t_{1}\gtrsim
0.7~{}$eV, and provides a natural explanation for the presence of only one
peak in the RIXS spectrum of Ba3LaRu2O9.

More importantly, unlike the smooth variation of the RIXS spectrum with
$\Delta_{tr}$ in the local limit [Fig. 3 (c)-(d)], varying $\Delta_{tr}$ in
the limit of large $t_{1}$ as shown in Fig. 3 (g) leads to an abrupt change in
the RIXS spectrum at $\Delta_{tr}^{c}\approx 0.13~{}$eV, across an electronic
transition where the first excited states cross the ground states [See
Supplemental Materials for justification of the choice of $t_{1}$ and $t_{2}$
in Fig. 3 (g)-(i)]. Remarkably, as shown by the direct comparison between
theory and experiment in Fig. 3 (h)-(i), the simulated RIXS spectrum at
$\Delta<\Delta_{tr}^{c}$ (I) and $\Delta>\Delta_{tr}^{c}$ (II) provides an
excellent description of the spectrum of Ba3LaRu2O9 and Ba3InRu2O9/Ba3YRu2O9,
respectively, by capturing the suppression/enhancement of the high/low energy
peaks. It is worth emphasizing that the abrupt changes in the RIXS spectrum at
$\Delta_{tr}^{c}$ [Fig. 3 (g)-(i)] come from a modification of the RIXS matrix
elements due to the abrupt change in ground-state wave-function. Consequently,
only a small tweak in $\Delta_{tr}$ is needed, consistent with a very small
change in local structure. Our model also predicts a large drop in effective
moment from $\sim 3.7~{}\mu_{B}$ to $\sim 1~{}\mu_{B}$ across the electronic
transition [34]. These values are in reasonable agreement with the values from
magnetic susceptibility measurements [34, 12, 11] and provide further support
for our theory.

The ability to drive an electronic transition at very small $\Delta_{tr}^{c}$
[Fig. 3 (g)] is a direct manifestation of the enhanced electronic instability
of the dimer in the limit of large $t_{1}$ compared to the local limit [Fig. 3
(c)-(d)]. This can be understood from the evolution of the low energy states
as a function of $t_{1}$ shown in Fig. 3 (f). Unlike the local model where the
first excited states occur at $\sim\frac{2}{3}\lambda\sim 0.1~{}$eV, there is
an abundance of low energy states with $\hbar\omega<50~{}$meV at large
$t_{1}$, which could cross the ground states and drive an electronic
transition in the dimer with a small change in $\Delta_{tr}$. Further
demonstrating the enhanced electronic instability of the dimer, we found that
an electronic transition can also be induced by other small perturbations such
as $t_{2}$ [34]. However, the predicted change in RIXS spectrum by varying
$t_{2}$ is inconsistent with the experimental observations. We therefore
attribute the observed change in RIXS spectrum to a change in $\Delta_{tr}$.

Of special importance among the low energy states is the doublet originating
from the excited $(L=0,S=\frac{3}{2},J=\frac{3}{2})\bigotimes(L=1,S=1,J=1)$
multiplet [marked by the yellow solid line in Fig. 3 (f)] corresponding to a
scenario where $t_{2g}^{3}$ is in its ground state and $t_{2g}^{4}$ is in the
excited $J=1$ state in the $t_{1}=0$ limit. Due to the reduced SOC in Ru, this
doublet is pushed to a very small energy ($\sim~{}30~{}$meV) in the limit of
large $t_{1}$ and strongly hybridizes with states from the ground multiplet.
As illustrated in the Supplemental Materials[34], such hybridization is
crucial for the electronic transition shown in Fig. 3 (g), and marks a clear
distinction between Ir and Ru dimer systems. The relevance of the $J=1$
excited multiplet is reminiscent of the famous excitonic magnetism proposed
for $t_{2g}^{4}$ ions[42], which is difficult to realize in isolated Ru4+ and
Ir5+ ions with a large spin-orbit gap. Our theory therefore suggests an
interesting mechanism for its realization in MO systems, where the energy of
the J=1 exciton can be substantially lowered by large inter-site hopping.
Although not strictly applicable to the present system with 7 electrons per
dimer, Ru-dimer compounds Ba3MRu2O9 with M=Ce4+, Pr3+ [13, 9] and 8 electrons
per Ru2O9 dimer might be promising candidates for a MO version of excitonic
magnetism.

In conclusion, by carrying out Ru M3-edge RIXS measurements on a well-known
series of Ru dimer compounds with the 6H-perovskite structure, Ba3MRu2O9
(M3+=In3+, Y3+, La3+), we find direct spectroscopic evidence for an extremely
fragile molecular orbital state, manifested as an abrupt change in RIXS
spectrum concomitant with a very subtle change in local structure tuned by the
M-site ion. By modelling our RIXS spectra, we attribute the enhanced
electronic instability in these dimer compounds to a cooperative effect
between large inter-site hopping and small SOC, a unique combination found
only in the Ru but not Ir MO systems. Our study therefore highlights the Ru MO
systems as ideal platforms for studying quantum phase transitions involving
molecular orbitals.

###### Acknowledgements.

BY acknowledges support from the NSERC postdoctoral fellowship. JPC is
supported by the NSERC Discovery Grant program. BHK was supported from the
Institute for Basic Science in the Republic of Korea through the Project No.
IBS-R024-D1. This research used the SIX beamline (2-ID) of the National
Synchrotron Light Source II, a U.S. Department of Energy (DOE) Office of
Science User Facility operated for the DOE Office of Science by Brookhaven
National Laboratory under Contract No. DE-SC0012704.

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