# Floquet engineering of Kitaev quantum magnets

Umesh Kumar  Theoretical Division, T-4, Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA  Saikat Banerjee  Theoretical Division, T-4, Los
Alamos National Laboratory, Los Alamos, New Mexico 87545, USA  Shi-Zeng Lin
Theoretical Division, T-4, Los Alamos National Laboratory, Los Alamos, New
Mexico 87545, USA  Theoretical Division, T-4 and CNLS, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA

###### Abstract

In recent years, there has been an intense search for materials realizing the
Kitaev quantum spin liquid model. A number of edge-shared compounds with
strong spin-orbit coupling, such as RuCl3 and iridates, have been proposed to
realize this model. Nevertheless, an effective spin Hamiltonian derived from
the microscopic model relevant to these compounds generally contains terms
that are antagonistic toward the quantum spin liquid. This is consistent with
the fact the zero magnetic field ground state of these materials is generally
magnetically ordered. It is a pressing issue to identify protocols to drive
the system to the limit of the Kitaev quantum spin model. In this work, we
propose Floquet engineering of these Kitaev quantum magnets by coupling
materials to a circularly polarized laser. We demonstrate that all the
magnetic interactions can be tuned in situ by the amplitude and frequency of
the laser, hence providing a route to stabilize the Kitaev quantum spin liquid
phase.

## Introduction

Light-matter interaction not only provides an important way to probe
materials’ properties but also offers exciting opportunities to control the
physical properties of quantum materials [1, 2, 3]. It has been demonstrated
experimentally that light can induce superconductivity [4, 5, 6], magnetism
[7, 8], topological states [9, 10, 11], and other novel quantum states of
matter [12]. One particular fruitful protocol is the so-called Floquet
engineering by driving materials periodically with a continuous laser. In this
Floquet approach, the coupling between light and matter can tune the
microscopic parameters, generate new interactions, and even stabilize entirely
new states and excitations that do not exist in equilibrium [13, 14]. Inspired
by the success of light control of quantum states in the past, we investigate
the light-driven transition into the quantum spin liquid in strongly
correlated magnetic materials.

Quantum spin liquid (QSL) is one class of highly entangled quantum states of
matter, which hosts fractionalized excitations [15, 16]. The elegant exact
solution of the Kitaev spin model unambiguously shows the existence of a
quantum spin liquid [17]. The presence of an external magnetic field can open
a gap and the model supports the Majorana fermions which have the potential
for applications in robust topological quantum computation [18, 19]. Kitaev’s
work has motivated tremendous efforts both theoretically and experimentally to
materialize the Kitaev model in quantum magnets [20, 21, 22]. Jackeli and
Khaliullin showed that one can realize the Kitaev spin interaction in edge-
shared octahedral structure with strong spin-orbit coupling [23]. Several
candidate materials, including RuCl3 [24] and iridates have been identified
and have been extensively studied. Encouraging signs of QSL has been reported
[25], but unambiguous identification of the QSL remains a challenge. One
obstacle from a theoretical perspective is that there exist competing magnetic
interactions originated from the different correlation induced exchange
processes in materials, which drives the model Hamiltonian away from the ideal
Kitaev QSL model [26, 27]. Numerical study shows that the Kitaev QSL is only
stable in a small region of the model parameter space. [26] Therefore, it is
highly desirable to design a protocol for continuously in situ tuning the
magnetic interactions in materials to drive the system into QSL phases.

In this work, we propose to use light to tune magnetic interactions in spin-
orbit coupled Mott insulators to favor QSL. We consider the multi-orbital
strongly spin-orbit coupled Hubbard model, which describes several candidate
materials including RuCl3 and iridates. In a simple picture, the coupling of
electrons to light has two effects. First, the periodic modulation of the
hopping of electrons between different sites effectively normalizes the
hopping parameters. Secondly, electrons can absorb or emit photos during the
virtual hopping process, and as a consequence, the energy barrier due to
Coulomb repulsion also gets modified. Therefore, the strength and even the
sign of magnetic interactions can be controlled by light. In addition, an
effective Zeeman field can be generated by circularly polarized light through
the process called inverse Faraday effect, which provides a new handle to
control the quantum states [28, 29]. Here, we derive a low energy spin
Hamiltonian from the Hubbard model, both using the numerical exact
diagonalization (ED) method and the analytical perturbation theory, and
explore in detail how the magnetic interactions can be tuned by the frequency
and amplitude of the light.

## Results

### Multi-orbital system

We focus on the edge-shared iridates (5d) and ruthenates (4d) as our target
materials. In the octahedral crystal field, these materials have been proposed
to be described by the $J$-$K$-$\Gamma$ Hamiltonian [26] and as possible
candidates to realize Kitaev QSL phase. Here, the $t_{2g}$ manifold of the
transition metal (TM) is known to host a hole in the $J_{\text{eff}}=1/2$
sector, which is separated from $J_{\text{eff}}=3/2$ states due to a large
spin-orbit coupling ($\lambda$) in the system. The $J_{\mathrm{eff}}=1/2$
multiplet is composed of $d_{xy},\leavevmode\nobreak\ d_{yz}$ and $d_{zx}$
orbitals of the TM atom. On the other hand, the TM atoms form a two-
dimensional (2D) honeycomb lattice perpendicular to [111] direction composed
of the standard $x,\leavevmode\nobreak\ y,\leavevmode\nobreak\ z$ \- bonds as
shown in Fig. 1(a). For subsequent discussion, we constrain our analysis for
the $z$-bond which is oriented along the [110] direction. The $x$ and $y$
bonds can be recovered by $C_{3}$-rotation of this bond. Along the $z$-bond,
only $p_{z}$-orbital of the oxygen/chloride ligand is active and have a finite
hopping between the $d_{yz}$ and $d_{zx}$ orbitals as shown in panel (b) and
(c) of Fig. 1, respectively.

Figure 1: Schematics for the edge-shared iridates/ruthenates. Panel (a) shows
the full edge-shared hexagon realized in materials. Panel (b) and (c) shows
the hopping channels for the $z$-bond mediated by the oxide/chloride [ligand
atoms] $p$ orbitals. Note that only $d_{yz}$ and $d_{zx}$ orbitals can hop via
the ligand $p_{z}$ orbitals at (1,0,0) and (0,1,0), respectively.
Additionally, only the upper and lower hopping paths in panels (b) and (c),
respectively, are non-vanishing.

These multi-orbital systems are captured well by Hubbard-Kanamori Hamiltonian
and can be mapped to effective spin Hamiltonians using exact diagonalization
(ED) [30] or perturbation theory [26, 31]. The corresponding Hamiltonian is
written as

$\mathcal{H}_{i}^{C}=\frac{U}{2}\sum_{\alpha,\sigma}n_{i\alpha\sigma}n_{i\alpha\bar{\sigma}}+\frac{U^{\prime}}{2}\sum_{\begin{subarray}{c}\alpha\neq\beta,\\\
\sigma,\sigma^{\prime}\end{subarray}}n_{i,\alpha,\sigma}n_{i\beta\sigma^{\prime}}-\frac{J_{\mathrm{H}}}{2}\sum_{\begin{subarray}{c}\alpha\neq\beta,\\\
\sigma,\sigma^{\prime}\end{subarray}}d_{i\alpha\sigma}^{\dagger}d_{i\alpha\sigma^{\prime}}^{\phantom{\dagger}}d_{i\beta,\sigma^{\prime}}^{\dagger}d_{i\beta\sigma}^{\phantom{\dagger}}+\frac{J_{\mathrm{P}}}{2}\sum_{\begin{subarray}{c}\alpha\neq\beta,\\\
\sigma\end{subarray}}d_{i\alpha\sigma}^{\dagger}d_{i\alpha\bar{\sigma}}^{\dagger}d_{i\beta\sigma}^{\phantom{\dagger}}d_{i\beta\bar{\sigma}}^{\phantom{\dagger}}+\Delta\sum_{i\sigma}p^{\dagger}_{i\sigma}p_{i\sigma},$
(1)

where $U$ and $U^{\prime}$ are the strength of the intra-orbital and inter-
orbital Coulomb repulsion and $J_{\mathrm{H}}$ is the Hund’s coupling for the
orbitals $\alpha,\beta\in\\{d_{xy},d_{yz},d_{zx}\\}$, and
$J_{\mathrm{P}}=J_{\mathrm{H}}$. $\Delta$ parametrizes the ligand charge-
transfer energy. Note that we have ignored the atomic spin-orbit coupling
(SOC) term
($\lambda/2\sum_{i\sigma}d^{\dagger}_{i\sigma}(\mathbf{L}\cdot\mathbf{S})d_{i\sigma}$),
assuming $\lambda\ll U,U^{\prime},J_{\mathrm{H}},\Delta$.

Focusing on the four-site cluster [two TM atoms oriented along the $z$-bond
with their two edge-shared ligand sites], we now write down the tight-binding
Hamiltonian in presence of a circularly polarized light (CPL) as

$\mathcal{H}_{K}(t)=\sum_{\langle\alpha,\beta\rangle\sigma}e^{i\delta
F(t)}t_{\alpha\beta}d_{1\alpha\sigma}^{\dagger}d_{2\beta\sigma}^{\phantom{\dagger}}+t_{pd}\sum_{\sigma}\Big{(}e^{i\delta{F}_{x}(t)}d_{1zx\sigma}^{\dagger}p_{1\sigma}^{\phantom{\dagger}}-e^{i{F}_{y}(t)}p_{1\sigma}^{\dagger}d_{2yz\sigma}^{\phantom{\dagger}}+e^{i\delta{F}_{y}(t)}d_{1yz}^{\dagger}p_{2\sigma}^{\phantom{\dagger}}-e^{i{F}_{x}(t)}p_{2\sigma}^{\dagger}d_{2zx\sigma}^{\phantom{\dagger}}\Big{)}+\text{h.c.},$
(2)

where $t_{pd}$ is the hopping amplitude between the $p$\- and $d$-orbitals.
The hopping matrix between the three $d$-orbitals $t_{\alpha\beta}$ is
obtained through Slater-Koster [32] interatomic matrix elements as

$t_{\alpha\beta}d_{1\alpha\sigma}^{\dagger}d_{2\beta\sigma}^{\phantom{\dagger}}=\begin{bmatrix}d_{1yz\sigma}^{\dagger}&d_{1zx\sigma}^{\dagger}&d_{1xy\sigma}^{\dagger}\end{bmatrix}\begin{bmatrix}t_{3}&t_{2}&0\\\
t_{2}&t_{3}&0\\\ 0&0&t_{1}\\\ \end{bmatrix}\begin{bmatrix}d_{2yz\sigma}\\\
d_{2zx\sigma}\\\ d_{2xy\sigma}\end{bmatrix}.$ (3)

The time-dependence of the drive appears as a Peierls phase in these multi-
orbital Hamiltonian [30, 33]. The total Hamiltonian for the two TM-sites is
given by $\mathcal{H}(t)=\mathcal{H}_{K}(t)+\sum_{i=1,2}\mathcal{H}_{i}^{C}$.
For the sake of simplicity, we consider the light along [001] direction with
circular polarization, ${\bf F}(t)=E_{0}(\hat{\mathbf{x}}\sin\omega
t+\hat{\mathbf{y}}\cos\omega t)/\omega$. The phase along the TM-TM bond
$\left(\tfrac{1}{\sqrt{2}},\tfrac{1}{\sqrt{2}},0\right)$ is given by $\delta
F(t)=\zeta\sin(\omega t-\pi/4)$ and the phases along TM-ligand bonds are given
by $\delta F_{x}(t)=\zeta\sin\omega t/\sqrt{2}$ and $\delta
F_{y}(t)=\zeta\cos\omega t/\sqrt{2}$. Here, the dimensionless parameter
$\zeta$ is defined as $\zeta=R_{ij}E_{0}/\omega$ [$R_{ij}$ is the bond-length
between the sites for the associated hopping]. The values of the parameters
entering Eq. (1)-(3) are adapted from the recent ab-initio [34] and
photoemission [35] studies relevant for $\alpha$-RuCl3: $t_{1}=0.044$,
$t_{2}=0.08$, $t_{3}=0.109$, $t_{pd}=-0.8$, $U=3.0$, $J_{\mathrm{H}}=0.45$,
$\Delta=5$.

Here, we assumed a reduced value of $t_{2}$ so as to account for finite
$t_{pd}$. All the values are in units of eV, unless stated otherwise. We also
use relations, $U^{\prime}=U-2J_{\mathrm{H}}$ and $J_{p}=J_{\mathrm{H}}$ in
the Hubbard-Kanamori Hamiltonian, which preserves the rotational invariant
form. Consequently, the Hubbard-Kanamori term in Eq. (1) can be mapped to

$\mathcal{H}_{i}^{C}=\sum_{i}\bigg{[}\frac{U-3J_{\mathrm{H}}}{2}\left(N_{i}-5\right)^{2}-2J_{\mathrm{H}}\mathbf{S}_{i}^{2}-\frac{J_{\mathrm{H}}}{2}\mathbf{L}_{i}^{2}\bigg{]}+\Delta\sum_{i}n^{p}_{i\sigma},$
(4)

where $N_{i}$ is electron number, $S_{i}$ is total spin, and $L_{i}$ is total
orbital angular momentum at site $i$ [36, 26]. The above simplified form of
the Hubbard-Kanamori Hamiltonian [Eq. (4)] allows us to evaluate the energy of
the doubly occupied states at the TM sites. Since, the SOC $\lambda$ is
ignored, we label the doubly occupied sites in terms of the orbital and spin
angular momentum $\mathbf{L}$, $\mathbf{S}$, respectively, as
$|^{2S+1}L,L_{z},S_{z},0\rangle$, where $L,S\leavevmode\nobreak\
[=(0,0),(1,1),(2,0)]$ are the orbital and the spin angular momentum of the
doublet, respectively, with $L_{z}$ and $S_{z}$ being their respective
$z$-components. The allowed values of $L$ and $S$ provides three different
energies for the doublets (two particles on TM site) given by:
$E_{S}=U+2J_{\mathrm{H}}$, $E_{P}=U-3J_{\mathrm{H}}$, $E_{D}=U-J_{\mathrm{H}}$
and singly occupied case (one particle in each TM site), $E_{0}=0$.

### Effective spin-exchange model

Since the tight-binding parameters are small compared to interaction strengths
[$t_{\alpha\beta},t_{pd}\ll U,U^{\prime},J_{\mathrm{H}},\Delta$], the model
can be analyzed by a low-energy effective spin-exchange Hamiltonian by
ignoring the high-energy charge degrees of freedom. The static limit of our
model has been extensively studied previously leading to the well-known
$J-K-\Gamma$ spin Hamiltonian [26, 37]. Here, we utilize the Floquet approach
to derive a time-independent effective Hamiltonian from Eq. (1)-Eq. (2) and
subsequently perform ED calculations to estimate the various spin-exchange
parameters of the concomitant spin Hamiltonian. On the other hand, the time-
independent version of the latter can be naturally captured by a minimal
magnetic model [based on the mirror reflection, inversion and broken time-
reversal symmetry of TM-ligand-TM atomic cluster associated with the $z$-bond]
as

$\mathcal{H}_{\mathrm{eff}}=\begin{bmatrix}S_{1}^{x}&S_{1}^{y}&S_{1}^{z}\end{bmatrix}\begin{bmatrix}J&\Gamma&\Gamma^{\prime}\\\
\Gamma&J&\Gamma^{\prime}\\\ \Gamma^{\prime}&\Gamma^{\prime}&J+K\\\
\end{bmatrix}\begin{bmatrix}S_{2}^{x}\\\ S_{2}^{y}\\\
S_{2}^{z}\end{bmatrix}+\textbf{h}\cdot(\textbf{S}_{1}+\textbf{S}_{2}).$ (5)

Since the trigonal distortion is ignored in our model, we have
$\Gamma^{\prime}=0$ [26]. The emergent Zeeman magnetic field $\mathbf{h}$ is
an effect of the broken time-reversal symmetry due to the applied CPL and is
not accounted for in the static version of the associated model.

We now briefly discuss how the mapping of the multi-orbital model to the spin
model is carried out. In the method section, we discuss how the time-dependent
Hamiltonian can be mapped to a time-independent form. The time-independent
matrix in Bloch structure form is given by,
$(\epsilon_{\alpha}+m\omega)|\psi_{\alpha,m}\rangle=\sum_{m^{\prime}}H_{m-m^{\prime}}|\psi_{\alpha,m^{\prime}}\rangle$.
Here, integer $m$ is the Floquet sector index and $\alpha$ consists of: (i) a
single hole in each TM site and an empty ligand site, (ii) two holes [double
occupancy] on a TM site while the ligand and the other TM site is empty and
(iii) one particle each on a ligand and a TM site [see Supplement [38] for the
exact forms]. The singly occupied states are a product of
$J_{\mathrm{eff}}=1/2$ states on the two TM sites, the doublets consists of
states $|^{2S+1}L,L_{z},S_{z},0\rangle$, where $L,S\leavevmode\nobreak\
[=(0,0),(1,1),(2,0)]$, as mentioned earlier.

The full Floquet Hamiltonian ($H$) can be written in the eigen-decomposed form
as,
$H=\sum_{\mathfrak{n}}E_{\alpha,\mathfrak{n}}|\phi_{\mathfrak{n}}\rangle\langle\phi_{\mathfrak{n}}|$,
where,
$|\phi_{\mathfrak{n}}\rangle=\sum_{\alpha,m}a_{\alpha,m}^{\mathfrak{n}}|\psi_{\alpha,m}\rangle$.
Notice that $|\psi_{\alpha^{\prime},n}\rangle$ contains all the configuration
for the two-site problem. To restrict the basis to $m=0$ and singly occupied
basis, one can use a projector; $P_{s,0}=P_{\alpha\in s}P_{m=0}$. We then have
the projected spin Hamiltonian given by; $H_{\text{PG}}=P_{s,0}HP_{s,0}$. The
spin model is valid only when the Floquet band for the singly occupied is well
separated from the other Floquet band and the upper Hubbard band. We define
$\Delta_{l}=\big{(}E_{\text{min}}(m=0,s)-E_{\text{min-1}}\big{)}/W$ and
$\Delta_{u}=\big{(}E_{\text{max+1}}(m=0,s)-E_{\text{max}}(m=0,s)\big{)}/W$ for
each $\zeta$ $i.e.\leavevmode\nobreak\ $ $\Delta_{u},\Delta_{l}\ll 1$. Here
$E_{\text{min}}(m=0,s)$ and e $E_{\text{max}}(m=0,s)$ are the minimum and
maximum energy for the singly occupied states in the $m=0$ Floquet sector, and
$W=E_{\text{max}}(m=0,s)-E_{\text{min}}(m=0,s)$ for a given $\zeta$.

Fig. 2 and 3 plot the estimates using ED for the system without and with
ligand, respectively. Inset in the panels of these figures plots the energy
states for the $m=0$ Floquet states with singly occupied configuration (in
black) and its nearby states. The singly occupied states (in black) are
separated from other states, which validates our calculation for the drive
frequencies presented in our work.

Figure 2: Tuning $J$, $K$, $\Gamma$, and $h$ parameters for the model after
integrating out the ligand sites. Panels (a) – (d) show the result for drive
frequency $\omega=1.2,2.1,3.2$ and $7$, respectively. Inset, in each panel,
shows the energy levels in the vicinity of $m=0$ Floquet singly occupied
states. The black curve (inset) indicates the states with the largest weights
on the singly occupied configurations. Note that the vertical axis is plotted
in different intervals to highlight the relative strength between different
parameters.

#### Spin-exchange model: ED without ligands

The ligand degrees of freedom (oxide/chloride $p$ orbitals), in the edge-
shared ruthenetes or iridates, are usually integrated out leading to an
effective description of the model in terms of the TM $d$-orbitals only [26].
In this section, we perform ED on the Floquet Hamiltonian obtained from
$\mathcal{H}(t)=\mathcal{H}_{K}(t)+\sum_{i=1,2}\mathcal{H}_{i}^{C}$ and
analyze the results by integrating out the ligand [38]. The system without the
ligand is simulated by turning off the direct hopping $t_{pd}$ between TM and
ligand sites and rescaling the TM-TM hopping $t_{2}$ to
$t_{2}+t_{pd}^{2}/\Delta$ in the Eq. (2). Consequently, the four-site cluster
effectively becomes a two-site system. In this case, the emergent Zeeman
magnetic field $\mathbf{h}$ term is absent as there is no residual orbital
current along the TM-TM bond. However, the multi-orbital nature leads to
finite non-vanishing spin-exchange couplings $J$, $K$, and $\Gamma$.

The corresponding results are shown in Fig. 2. We choose four distinct light
frequencies avoiding the three critical resonant frequencies
$\omega=\\{E_{P},E_{D},E_{S}\\}[=\\{1.65,2.45,3.9\\}]$ of the doublet states
and illustrate the relative variation of $J$, $K$ and $\Gamma$ with applied
laser strength in the four panels [Fig. 2]. Inset in each panel shows the
separation of the $m=0$ Floquet singly occupied states from the adjacent
$m\neq 0$ states. Panel (a) shows the variation of the couplings at drive
frequency $\omega=1.2$, below the doublet $E_{P}$ energy. In this case, we
notice that $J,K,\Gamma$ can all change their sign and enhance the magnitude,
but the relative tunability is missing. Panel (b) shows the result for
$\omega=2.1$, in between doublet $E_{P}$ and $E_{D}$ energies and retains the
similar feature as in the prior case. Panel (c) and (d) show the results for
$\omega=3.2$ [$E_{D}<\omega<E_{S}$] and $\omega=7$ [$\omega>E_{S}$],
respectively. In these two cases, however, we notice that the sign of the
parameters cannot be changed, and also the magnitudes of the couplings can be
enhanced only mildly. As will be shown by perturbation calculations below, the
ratio of $\Gamma/K$ is fixed, independent of the amplitude and frequency of
the light.

#### Spin-exchange model: ED with ligands

We now come to the main part of our paper. Here, we present the results using
the full Hamiltonian given by Eq. (1) and Eq. (2) for the parameters discussed
previously and consider the effect of ligands in full glory. Recently, the
inclusion of ligands has been investigated in a few studies [39]. The
inclusion of ligand degree of freedom in the multi-orbital description is
crucial primarily for its two-fold significance. Firstly, in real materials
[iridates/ruthenates] the ligand effect is unavoidably present and a thorough
quantitative material-specific analysis needs to incorporate it properly.
Secondly and most importantly, the inclusion of ligand $p$-orbital along with
the TM $d$-orbital offers the desired relative tunability of the spin-exchange
couplings - $J$, $K$ and $\Gamma$ [see Fig. 3], which was absent as we saw in
the previous section. Finally, the inclusion of ligand sites constitutes the
full TM-ligand-TM atomic cluster with its associated orbital current which
leads to a finite Zeeman magnetic field through inverse Faraday effect [28,
29]. In Fig. 3 we show the variation of all the parameters defined in Eq. (5)
with the driving strength ($\zeta$) evaluated using ED calculations.

As before, we again choose six distinct light frequencies avoiding the four
critical resonant frequencies $\omega=\\{E_{P},E_{D},E_{S},\Delta\\}$
[$=\\{1.65,2.45,3.9,5\\}$] of the doublet states and illustrate the relative
variation of $J$, $K$ and $\Gamma$ with applied laser strength in the six
panels (a)-(f) [Fig. 3]. The ED analysis is always performed away from the
Floquet resonances in order to avoid the breakdown of the spin picture. Inset
in each panel shows the separation of the $m=0$ Floquet singly occupied states
from the adjacent $m\neq 0$ Floquet states. For field strength $\zeta=0$
[static case], we have $J=0.0007$, $K=0.005$, and $\Gamma=0.003$ in each
panel, whereas the Zeeman magnetic field $\mathbf{h}$ is zero. The vanishing
of the magnetic field is due to the preserved time-reversal symmetry of the
multi-orbital system in the absence of CPL.

Figure 3: Tuning J, K, $\Gamma$ and $h$ parameters in a system with ligands as
a function of drive strength ($\zeta$). Panel (a)-(f) show the tuning for
drive frequency $\omega=1.2,2.3,3.2,4.5,7$ and 12 respectively. Inset in each
panel shows the energy levels in the vicinity of $m=0$ Floquet singly occupied
states. The black line indicates the states with the largest weights on the
singly occupied configurations. Notice that the $y$-axis are plotted in the
different intervals to highlight the relative strength between different
parameters.

Panel (a) shows the variation of the various parameters defined in Eq. (5)
with the driving amplitude $\zeta$ at the drive frequency $\omega=1.2$ eV,
below the lowest doublet energy $E_{P}$. Indeed for the reasons mentioned
earlier, we find a finite Zeeman magnetic field $\mathbf{h}$ in the driven
system, in contrast to vanishing $\mathbf{h}$ in the case without ligand. In
this case, the parameters $J$, $K$, $\Gamma$, and $\mathbf{h}$ can all change
their sign and their magnitudes can be enhanced. Additionally, these
parameters vanish at different values of $\zeta$ [compared to the situation in
Fig. 2] allowing for their relative tuning. We also note that the magnitude of
the Kitaev term $K$ and anisotropy $\Gamma$ can be enhanced by a few orders of
magnitude compared to the Heisenberg exchange term $J$. Panel (b) shows the
variation of the parameters with $\zeta$ for $\omega=2.3$ in between the
$E_{P}$ and $E_{D}$ doublet. In addition to our findings in panel (a), we
notice that $K$ at $\zeta=3.7$ can be tuned to a few orders of magnitudes
larger compared to all the other parameters, making it ideal for realizing the
Kitaev QSL phase. The results for $\omega=3.2$ [$E_{D}<\omega<E_{S}$] are
shown in panel (c). In this case, the magnetic field $\mathbf{h}$ can achieve
the largest magnitude compared to the other parameters. Moreover, we notice
limited tunability at larger drive strength $\zeta$. Panel (d) shows the
result for $\omega=4.5$, in the regime between $E_{S}$ and $\Delta$. In this
case, the system can be tuned to large $J$ and $K$, leading to a realization
of the Kitaev-Heisenberg model. In addition, the almost vanishing Zeeman field
provides a promising route to realize gapless Kitaev QSL. For $\omega=7$ above
the largest energy scale $\Delta$, in our model [see panel (e)], we notice a
significant enhancement of $K$ at low drive strength, making it another
frequency regime suited for realizing the Kitaev phase. Panel (f) shows the
result for $\omega=12$, large frequency, well above $\Delta$. In this case, we
observe that all the parameters values decrease with the driving strength
similar to the case without ligand [see Fig. 2(d)]. In contrast, the
tunability of the signs of these parameters persists.

Figure 4: Schematics for evaluating spin Hamiltonian parameters at the third-
order perturbation theory. We start with one particle each at the TM site. In
the intermediates states, the particle moves to a ligand and then to the other
TM site creating a doublet. Finally, it returns to the original site, with or
without a spin-flip. We also consider a time-reversal partner of this hopping
path. Figure 5: Comparison of the ED and perturbation theory results. Panel
(a) shows the result for the case without ligand at drive frequency
$\omega=7$. Panel b) shows the comparison in the model with ligands at
$\omega=12$.

### Spin-exchange model: Perturbation theory

So far we analyzed our model in the Floquet approximation utilizing the ED
calculations and discussed the results in various regimes of the model-based
parameter values. In this section, we derive the low-energy spin-exchange
Hamiltonian based on perturbation theory and compare the analytical
expressions of the couplings $J$, $K$, $\Gamma$, and $\mathbf{h}$ with the
numerical estimates from the ED.

In the absence of ligand degree of freedom, we perform a second-order
perturbation theory [38] and time-dependent Schrieffer-Wolff transformation
[28] to obtain the various couplings defined in Eq. (5) as

$\displaystyle J$
$\displaystyle=\frac{4}{27}\sum_{m=-\infty}^{\infty}\mathcal{J}_{m}^{2}(\zeta)\Bigg{[}\frac{6t_{1}(t_{1}+2t_{3})}{E_{P}+m\omega}+\frac{2(t_{1}-t_{3})^{2}}{E_{D}+m\omega}+\frac{(2t_{1}+t_{3})^{2}}{E_{S}+m\omega}\Bigg{]},$
(6a) $\displaystyle K$
$\displaystyle=\frac{8}{9}\sum_{m=-\infty}^{\infty}\mathcal{J}_{m}^{2}(\zeta)\frac{J_{\mathrm{H}}[(t_{1}-t_{3})^{2}-3t_{2}^{2}]}{(E_{P}+m\omega)(E_{D}+m\omega)},$
(6b) $\displaystyle\Gamma$
$\displaystyle=\frac{16}{9}\sum_{m=-\infty}^{\infty}\mathcal{J}_{m}^{2}(\zeta)\frac{J_{\mathrm{H}}t_{2}(t_{1}-t_{3})}{(E_{P}+m\omega)(E_{D}+m\omega)},$
(6c)

where $\mathcal{J}_{m}(\zeta)$ is the Bessel function of the first kind of
order $m$, $\zeta=R_{ij}E_{0}/\omega$ is the dimensionless amplitude of the
applied laser and $R_{ij}$ is the bond length between the two TM sites. Notice
that the expressions for $K$ and $\Gamma$ have nodes at the identical values
of the drive strength $\zeta$ dictated by the combination of the Bessel
functions and the denominator only and also are found to follow a constant
relative strength as illustrated in Fig. 2. Therefore, it forbids the relative
tunability of these parameters in the system. The comparison of our ED results
with the analytical expressions for the couplings $J$, $K$, and $\Gamma$ is
shown in Fig. 5(a). We find a very good agreement between the two in this
case.

We now focus on the final part of our work and obtain the spin-exchange model
using perturbation theory [38] and time-dependent Schrieffer-Wolff
transformation [28]. In this case, the ligand degrees of freedom are taken
into consideration and we have to rely on a third-order perturbation theory to
properly capture the ligand effects. The perturbation theory is performed
utilizing the processes illustrated in Fig. 4 and we obtain the couplings
defined in Eq. (5) as

$\displaystyle J$
$\displaystyle=\frac{8t_{pd}^{2}}{27}\sum_{n,m=-\infty}^{\infty}\mathfrak{J}_{m,n}(\zeta)\Bigg{[}\frac{\sin[(m-n)\psi_{0}]}{\Delta+m\omega}\left(\frac{2t_{1}+t_{3}}{E_{S}+l\omega}+\frac{t_{1}-t_{3}}{E_{D}+l\omega}+3\frac{t_{1}+t_{3}}{E_{P}+l\omega}\right)\Bigg{]},$
(7a) $\displaystyle K$
$\displaystyle=\frac{8t_{pd}^{2}}{9}\sum_{n,m=-\infty}^{\infty}\mathfrak{J}_{m,n}(\zeta)\bigg{[}\frac{J_{\mathrm{H}}}{(\Delta+m\omega)}\frac{\sin[(m-n)\psi_{0}](t_{1}-t_{3})-3\cos[(m-n)\psi_{0}]t_{2}}{(E_{P}+l\omega)(E_{D}+l\omega)}\bigg{]},$
(7b) $\displaystyle\Gamma$
$\displaystyle=\frac{8t_{pd}^{2}}{9}\sum_{n,m=-\infty}^{\infty}\mathfrak{J}_{m,n}(\zeta)\bigg{[}\frac{J_{\mathrm{H}}}{(\Delta+m\omega)}\frac{\cos[(m-n)\psi_{0}](t_{1}-t_{3})+\sin[(m-n)\psi_{0}]t_{2}}{(E_{P}+l\omega)(E_{D}+l\omega)}\bigg{]},$
(7c) $\displaystyle h$
$\displaystyle=\frac{4t_{pd}^{2}}{9}\sum_{n,m=-\infty}^{\infty}\mathfrak{J}_{m,n}(\zeta)\frac{\sin[(m-n)\psi_{0}]}{\Delta+m\omega}\bigg{[}\frac{t_{1}-t_{3}}{E_{P}+l\omega}+\frac{t_{1}-t_{3}}{E_{D}+l\omega}\bigg{]},$
(7d)

where
$\mathfrak{J}_{m,n}(\zeta)=\mathcal{J}_{m+n}(\zeta)\mathcal{J}_{-m}(\zeta
r_{ij}/R_{ij})\mathcal{J}_{-n}(\zeta r_{ij}/R_{ij})$, $r_{ij}$ is the bond-
length between the TM and ligand sites, $l=m+n$ and $\psi_{0}$ is the angle
between the TM-TM and TM-ligand bond. Note that previously $\psi_{0}$ was
considered to be equal to $\pi/4$. We observe that in the static case
$\zeta=0$ the super-exchange coupling $J$ and the Zeeman magnetic field
$\mathbf{h}$ vanish. This vanishing of super-exchange coupling is consistent
with the Jackeli-Khaliluin formalism [23] and the absence of $\mathbf{h}$ at
$\zeta=0$ is consistent with the intact time-reversal symmetry of the multi-
orbital system.

We further observe the distinct analytical structure of the Kitaev term $K$
and the anisotropy $\Gamma$ [Eqs. (7b,7c)] which dictates that the zeros of
these parameters occur at different drive strength $\zeta$. This is in stark
contrast to the results discussed without ligand in Eqs. (6b,6c) and is
consistent with the variation of these parameters as shown in Fig. 3. Fig.
5(b) shows the results for the relative comparison between the perturbation
[adding the contributions from Eq. (6a-6c) to Eq. (7a-7d)] and ED
calculations. In this case, we find an excellent agreement for the case of the
magnetic field $\mathbf{h}$, but have a relatively poor agreement in the case
of the other parameters. Note that with the inclusion of ligand, there are
many more paths possible for the super-exchange in the high order processes.
Therefore, this deviation can possibly be accounted for by higher-order
contributions, which cannot be neglected due to larger values of $t_{pd}$,
whereas our perturbation term accounts only till third order. On the other
hand, the better agreement of the perturbation results and the ED calculations
for the Zeeman magnetic field $\mathbf{h}$ is justified. In this case, there
is only one process, in the next order perturbation, which contributes and is
roughly proportional to $t^{4}_{pd}/\Delta^{2}U$. Within our parameter choice,
this term is much smaller than the second or third-order contributions and can
safely be neglected.

## Discussion

In our discussion above, we have not taken into account the role of heating
due to laser irritation. When the laser frequency is close resonances of the
system, electrons can be excited efficiently resulting in severe heating.
Under the drive, the model Hamiltonian can be divided into different Floquet
Hamiltonian sectors according to the number of the photon, $m$, the system
absorbs/emits. Initially, electrons occupy the $m=0$ sector. The laser drives
the electrons to other Floquet sectors, which causes heating. Meanwhile, the
effective Floquet Hamiltonian at $m=0$ sector is modified through
hybridization with other sectors. There are resonances within the same Floquet
sector. In our system, at the single-ion level, we have crystal field
splitting between $t_{2g}$ and $e_{g}$ orbitals, and splitting effective
$J_{\mathrm{eff}}=3/2$ and $J_{\mathrm{eff}}=1/2$ manifold due to the spin-
orbit coupling. The frequency of the laser should be tuned away from these
resonances. In the Mott insulator, one dominant mechanism of heating is the
excitation of the doublons by laser. This resonance occurs when the photon
energy is close to the energy difference between different Hubbard bands. The
population of the doublons increases continuously with time in the presence of
laser irritation, which eventually invalidates the effective spin Hamiltonian
description. It is calculated in Ref. [40] that the rate of doublon generation
can be very low when the laser frequency is away from resonance set by the
lower and upper Hubbard band. The effective spin Hamiltonian we derived here
is valid in a long time window, where the effect of doublons on magnetic
interactions is also negligible. This long time window is known as the Floquet
prethermal region, which can be exponentially long in time before the system
enters into a featureless infinite temperature region due to heating [41, 42,
40, 43, 44, 45, 46, 47].

We remark that not all the magnetic interactions can be tuned independently
because we only have two tuning parameters, i.e. amplitude and frequency, of
the incident circularly polarized light. As demonstrated explicitly in Fig. 3,
we can achieve the region where $K$ is dominant over all other magnetic
interactions, thus realizing a favorite condition for the Kitaev QSL.
Furthermore, the Kitaev interaction $K$ can be tuned to be ferromagnetic or
antiferromagnetic by light in a single material. It is believed that RuCl3 and
irritates are described by a ferromagnetic Kiteav interaction without a light
drive. The antiferromagnetic Kitaev model has attracted considerable interest
recently because it may host a new gapless QSL in the intermediate magnetic
field region [48, 49, 50, 51, 52, 53]. Both the antiferromagnetic Kitaev
interaction and magnetic field can be induced by light, thus the driven system
allows us to access this gapless QSL. We can also tune into a parameter region
where $\Gamma$ interaction is dominant, where a new type of QSL has been
suggested recently [54].

So far, we have focused on the circularly polarized light, our results can be
readily generalized to the case with a linearly polarized light. In this case,
the effective Zeeman field due to the inverse Faraday effect is absent because
the time-reversal symmetry is preserved. It is natural to expect one can tune
the system into an anisotropic spin Hamiltonian by linearly polarized light,
which can also support QSL [33].

While we are working on the current project, there appear two nice theoretical
works [29, 55, 56] on the tuning of magnetic interactions in Kitaev quantum
magnets by light recently, which have some overlap with our current work. In
Refs. [55, 56], the authors treat the ligands by an effective hopping between
two transition metal sites, and then employ Floquet theory for the two site
problem. In this treatment, the ratio between K and $\Gamma$ interaction is
fixed and cannot be tuned by laser. The effects of time reversal symmetry
breaking and hence the inverse Faraday effect is absent because the direct
hopping of electrons between two sites does not produce a net hopping phase
associated with the circularly polarized light. In Ref. [29], the effective
magnetic interactions and the Zeeman field due to the inverse Faraday effect
were calculated numerically. The authors also determined the magnetic phase
diagram by exact diagonalization of the effective spin Hamiltonian
numerically. In our work, we have derived analytical expressions of these
magnetic interactions and effective Zeeman field, which provides new insights
on how the magnetic interactions are tuned by light.

In conclusion, we show that light can tune magnetic interactions in Kitaev
quantum magnets by explicitly calculating an effective spin Hamiltonian from
the multi-orbit spin-orbit coupled Hubbard model in the presence of a
circularly polarized light. We employ both the exact diagonalization of the
Floquet Hamiltonian and analytical perturbation theory. We demonstrate that
magnetic interactions favorite for QSL can be achieved by tuning the frequency
and amplitude of the light. Our work points to a promising route to stabilize
quantum spin liquid by coupling quantum magnets to light.

## Methods

### Floquet Theory

We use Bloch wave theory,
$|\psi(t)\rangle=e^{-i\epsilon_{\alpha}t}|\psi_{\alpha}(t)\rangle$ and solve
the Schrodinger equation given by
$H(t)|\psi(t)\rangle=i\frac{\partial}{\partial t}|\psi(t)\rangle$. Using the
Fourier transform given by $H_{l}=\frac{1}{T}\int_{0}^{T}e^{il\omega t}H(t)dt$
and $|\psi_{\alpha,m}\rangle=\frac{1}{T}\int_{0}^{T}e^{im\omega
t}|\psi_{\alpha}(t)\rangle$, we can solve the Schrodinger equation, where the
solution is given by,
$\big{(}\epsilon_{\alpha}+m\omega\big{)}|\psi_{\alpha,m}\rangle=\sum_{m^{\prime}}H_{m-m^{\prime}}|\psi_{\alpha,m^{\prime}}\rangle$.
Here, $\alpha$ are the basis states for the singly occupied, doubly occupied
states on the TM and the singly occupied states on the ligand. The results for
the quasi-energies converge quickly for off-resonance and in our case $|m|<8$,
is sufficient to get converged results. We use $m=0$-Floquet sector to
estimate the the parameters in the spin model.

### Formalism for Perturbation results

We evaluate the perturbation results for the system without ligands using the
second order perturbation theory [26, 33]. The effective Hamiltonian is given
by

$\mathcal{H}^{(2)}_{\mathrm{eff}}=-\sum_{\alpha,\beta}\sum_{n\neq
0}\langle\beta|\mathcal{T}_{\text{dd}}\frac{|n\rangle\langle
n|}{E_{n}-E_{0}}\mathcal{T}_{\text{dd}}|\alpha\rangle|\beta\rangle\langle\alpha|.$
(8)

Here,
$\alpha,\beta\in\Big{\\{}|+\tfrac{1}{2},+\tfrac{1}{2}\rangle,|+\tfrac{1}{2},-\tfrac{1}{2}\rangle,|-\tfrac{1}{2},+\tfrac{1}{2}\rangle,|-\tfrac{1}{2},-\tfrac{1}{2}\rangle\Big{\\}}$.
The $t_{2g}$-orbitals have $l_{\text{eff}}=1$. The $J_{\mathrm{eff}}=1/2$
states for a site in the orbital basis are given by
$|+\tfrac{1}{2}\rangle=\tfrac{1}{\sqrt{3}}\big{(}d_{yz,\downarrow}^{\dagger}+id_{zx,\downarrow}^{\dagger}+d_{xy,\uparrow}^{\dagger}\big{)}|0\rangle$,
$|-\tfrac{1}{2}\rangle=\tfrac{1}{\sqrt{3}}\big{(}d_{a,yz,\uparrow}^{\dagger}-id_{a,zx,\uparrow}^{\dagger}-d_{a,xy,\downarrow}^{\dagger}\big{)}|0\rangle$.
$|n\rangle$ are the intermediates states, which consists of double occupancy
on a TM site, are $(L,S)=[(0,0),(1,1),(2,0)]$, here, $L$ and $S$ are the total
spin and angular momentum of the doubly occupied site. Here $\mathcal{T}_{dd}$
is the hopping between the TM sites.

For the system with ligands, the second order results are the same as Eq. (8).
The third order corrections are evaluated using

$\mathcal{H}^{(3)}_{\mathrm{eff}}=\sum_{\alpha,\beta}\sum_{n\neq
0}\langle\beta|\mathcal{T}_{\text{dd}}\frac{|n\rangle\langle
n|}{E_{n}-E_{0}}\mathcal{T}_{\text{pd}}\frac{|L\rangle\langle
L|}{E_{L}-E_{0}}\mathcal{T}_{\text{pd}}|\alpha\rangle|\beta\rangle\langle\alpha|.$
(9)

Here, $|L\rangle$ (${E_{L}}$) are the eigen-states (energies) for the
occupancy on the ligand sites. $\mathcal{T}_{pd}$ is the hopping between the
TM and the ligand sites. The parameters, $J,K,\Gamma$ and $h$ evaluated using
this formalism are given in Eqs. (7a-7d).

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## Acknowledgements

This work was carried out under the auspices of the US DOE NNSA under Contract
No. 89233218CNA000001 through the LDRD Program. SZL was also supported by the
US Department of Energy, Office of Science, Basic Energy Sciences, Materials
Sciences and Engineering Division, Condensed Matter Theory Program.

## Author contributions statement

U.K. and S.Z.L. planned the project. U.K. developed the Floquet Exact
Diagonalization code and performed all of the numerical simulations. U.K.
together with S.B. carried out perturbation calculations. U.K., S.B. and
S.Z.L. discussed the results and wrote the manuscript.

## Additional information

See supplement.pdf