# Impact of correlations on topology in Kane-Mele model decorated with
impurities

Jan Skolimowski ID<EMAIL_ADDRESS>International Research
Centre MagTop, Institute of Physics, Polish Academy of Sciences,
Aleja Lotników 32/46, PL-02668 Warsaw, Poland  Wojciech Brzezicki ID
<EMAIL_ADDRESS>Institute of Theoretical Physics, Jagiellonian
University, Prof. S. Łojasiewicza 11, PL-30348 Kraków, Poland  International
Research Centre MagTop, Institute of Physics, Polish Academy of Sciences,
Aleja Lotników 32/46, PL-02668 Warsaw, Poland  Carmine Autieri ID
<EMAIL_ADDRESS>International Research Centre MagTop, Institute of
Physics, Polish Academy of Sciences,
Aleja Lotników 32/46, PL-02668 Warsaw, Poland

###### Abstract

We propose an effective model for the study of the interplay between
correlation and topology by decorating the Kane-Mele model with a set of
localized interacting orbitals hybridized to just one sublattice, breaking the
inversion symmetry. We show that in the time-reversal symmetric case, the
interplay between interactions and hybridization extends the stability of the
topological phase and depending on the driving mechanism very different
behaviors are observed after the topological phase transition (TPT). We
discuss the fate of the TPT in presence of weak ferromagnetic order, by
introducing a weak local magnetic field at the localized orbitals, which
splits the two band inversion points. One of the platforms to apply this model
to are ferrovalley compounds, which are characterized by two independent band
inversion points. Understanding this family of materials is crucial for the
development of the valleytronics. An alternative to spintronics, which uses
valley polarization as opposed to spin degrees of freedom as the building
block, promises great opportunities for the development of information
storage.

##  I Introduction

In the last two decades, the topological phase transitions (TPT) have
attracted considerable attentionHuang and Liu (2020); Continentino (2017).
They can be driven either by applying external factors to the system or by
tuning the interplay of intrinsic factors. In the first category are TPTs
caused by pressureMutch  _et al._ (2019); Kim  _et al._ (2019),
temperatureAguado-Puente and Chudzinski (2022) and magneticSatake  _et al._
(2020); Wysokiński and Brzezicki (2023) or electric fieldIslam  _et al._
(2021). Latter category includes studies exploring the impact of
correlationsRachel (2018); Dzero  _et al._ (2010), long-range orderAmaricci
_et al._ (2018); Skolimowski (2022) or lattice geometries Brzezicki  _et al._
(2019). In case of correlations an additional complication arises from the
fact that most of the topological classifications were obtained for non-
interacting systemsKane and Mele (2005a); Shiozaki  _et al._ (2016). Proper
extension of topological invariant to many-body systems is still under
debateWang and Zhang (2012); Zhao  _et al._ (2023); Blason and Fabrizio
(2023). Nonetheless, the interest in studying the interplay between electronic
correlations and topology has been staidly growing for many years and is still
an active fieldWieckowski  _et al._ (2018); Wysokiński and Brzezicki (2023);
Dzsaber  _et al._ (2022); Pizarro  _et al._ (2020); Barbarino  _et al._
(2019); Wagner  _et al._ (2023). In particular, we should mention the field of
the flat bands in two-dimensional and three-dimensionalLau  _et al._ (2021)
systems. More specifically, the study of the magic-angle twisted bilayer
graphene was addressed as a topological heavy-fermion problem with a relevant
role of the electronic correlationsSong and Bernevig (2022).

A compound with a single band inversion usually becomes more trivial as a
function of the Coulomb repulsion since the Coulomb repulsion promotes the
opening of a trivial band gap. This was shown in model studies using both
mean-fieldZheng  _et al._ (2015); Shao  _et al._ (2021) and many-body Vanhala
_et al._ (2016) approaches. The same happens in the density functional theory
approach if we use an exchange-correlation functional that has more electronic
correlations such as modified Becke-JohnsonHussain  _et al._ (2022), HSE or
strongly constrained and appropriately normedCuono  _et al._ (2023).

Additional complexity in the topological phase diagram of a system can arise
if it has two band inversions at inequivalent $k$-points. In the case of two
band inversion at different $k$-points with different gap closures as a
function of the Coulomb repulsion a two-stage correlation-driven topological
transition was reportedHussain  _et al._ (2023). A more complex topological
phase diagram can be obtained under time-dependent light irradiationZhan  _et
al._ . More in general, if we have a complex Fermi surface the electronic
correlations are able to induce topologyXu  _et al._ (2020); Ma  _et al._
(2022). Going beyond the single-particle picture, several topological Mott
insulators have been proposedRachel (2018); Wysokiński and Brzezicki (2023).

One of the possible platforms to validate these results is provided by the
family of two-dimensional (2D) materials called ferrovalley compoundsTong  _et
al._ (2016). These compounds have 2 inequivalent band inversions at $K$ and
$K^{\prime}$. Due to the presence of two inequivalent band inversions, these
systems can host a more complex correlation-driven TPTHussain  _et al._
(2023); Sheng  _et al._ (2022); Guo and Guo (2023); Guo  _et al._ (2022). One
path to obtain the ferrovalley compounds is through the decoration of a
monolayer with ad-atoms, which is also a promising avenue to achieve new
topological phasesJin  _et al._ (2022) or enhancing certain properties of
already existing phases.Jiang  _et al._ (2023) Decoration of 2D materials
allows for engineering a system that has specific hybridizations and
interactions to achieve different model Hamiltonians, such as the one examined
in this paper.

In this work, we propose and study an effective model describing a hexagonal
lattice, in which spin-orbit coupling (SOC) and inversion symmetry breaking
through hybridization with strongly correlated orbitals coexist. By
construction, it could play the role of a minimal model applicable to the
compounds, which can host ferrovalley physics. There the interplay between
these two factors is believed to be responsible for a wide range of phenomena
such as the Quantum Anomalous Hall (QAH) phase. The hexagonal lattice with SOC
is modeled using the Kane-Mele model (KMM) and the inversion symmetry-breaking
hybridization is modeled through coupling between one of the sublattices of
the KMM and localized correlated orbitals. In order to extend our studies to a
ferrovalley case, in the second part of this paper we will introduce a local
magnetic field acting only on the localized orbitals. This model differs from
the standard model of topological Kondo insulatorsDzero  _et al._ (2010),
where SOC is at the correlated impurity sites enforcing symmetry constraints
on the hybridization term and giving rise to topology. It is also different
from the standard periodic Anderson model on KMMFeng  _et al._ (2013), where
the impurity is always half-filled thus promoting the emergence of Kondo
physics. In Section II we introduce the Hamiltonian and the Cluster
Perturbation Theory method (CPT) used to solve it. In Section III we will
discuss the phase diagram of the model Hamiltonian without time-reversal
breaking local magnetic field. Lastly, in Section IV, we will discuss the
effect of the local magnetic field on the TPT in the KMM model and how it
relates to the two-stage TPT reported in ab initio studies of ferrovalley
materialsHussain  _et al._ (2023).

##  II Model

The model Hamiltonian discussed in this work written in the second
quantization form consists of three terms:

$\mathcal{H}=\mathcal{H}_{latt}+\mathcal{H}_{hyb}+\mathcal{H}_{imp},$ (1)
$\mathcal{H}_{latt}=-t\sum_{<i,j>,\sigma}c^{\dagger}_{i,\sigma}c_{j,\sigma}-\mu\sum_{i,\sigma}c^{\dagger}_{i,\sigma}c_{i,\sigma}+\\\
+it^{\prime}\\!\sum_{\ll
i,j\gg,\sigma,\sigma^{\prime}}\nu_{i,j}c^{\dagger}_{i,\sigma}s^{z}_{\sigma,\sigma^{\prime}}c_{j\sigma^{\prime}}$
(2)

$\mathcal{H}_{hyb}=J\sum_{i\in
A,\sigma}(d^{\dagger}_{i,\sigma}c_{i\sigma}+H.c.),$ (3)
$\mathcal{H}_{imp}=\sum_{i\in
A}Ud^{\dagger}_{i,\uparrow}d_{i,\uparrow}d^{\dagger}_{i,\downarrow}d_{i,\downarrow}+\\\
+\sum_{i\in A,\sigma}(\epsilon_{d}+\sigma
M)d^{\dagger}_{i,\sigma}d_{i,\sigma}-\mu\sum_{i\in
A,\sigma}d^{\dagger}_{i,\sigma}d_{i,\sigma},$ (4)

where $c_{i,\sigma},d_{j,\sigma^{\prime}}$ are the fermionic annihilation
operators of (itinerant) electrons in the orbitals forming the lattice and
localized orbitals, respectively, at sites $i$ and $j$ and with spin
projection $\sigma$ and $\sigma^{\prime}$. The $\mathcal{H}_{latt}$ describes
the standard Kane-Mele modelKane and Mele (2005a, b), with real hopping $t$
between the nearest neighbours in the honeycomb lattice and a complex hopping
$it^{\prime}$ between next-nearest neighbors. The summations $<i,j>$ and $\ll
i,j\gg$ reflect the range of each hooping term. Since the honeycomb lattice is
bipartite we introduce capital Latin letters $A,B$ to distinguish between the
two sublattices. The KMM is a version of the Haldane model in which the time-
reversal symmetry is recovered through the introduction of a spin-dependent
sign of the complex hopping, hence the Pauli matrix $z$
($s^{z}_{\sigma,\sigma^{\prime}}$) in the last term of $\mathcal{H}_{latt}$.
The parameter $\nu_{i,j}=\pm 1$ in this term distinguishes between a clockwise
and anti-clockwise movement of particles within each sublattice. Such hopping
was shown to describe a spin-orbit coupling Kane and Mele (2005b). The
$\mathcal{H}_{hyb}$ combines all terms that connect the lattice with
correlated localized orbitals. In the following we will consider the
decoration of only a single sublattice with localized orbitals, thus summation
is over $i\in A$. Parameter $J$ quantifies the strength of the hybridization
between the electrons in the localized and lattice orbitals. Finally,
$\mathcal{H}_{imp}$ is a set of local Hamiltonians, each having a Hubbard-type
interaction term characterized by the strength $U$, local potential
$\epsilon_{d}$ and Zeeman-like term coupling impurity electrons to an external
magnetic field $M$. The latter is introduced to mimic either intrinsic or
externally driven instability of the system to form magnetic moments. The
parameter $\mu$ in $\mathcal{H}_{latt}$ and $\mathcal{H}_{imp}$ is the
chemical potential. It will be tuned to study the 2/3 and 1/3 filled cases for
which the TPT is allowed, cf. text below for a detailed discussion. In the
following, when $U\neq 0$ then $\epsilon_{d}=-U/2$ will be assumed. To set the
energy scale we use $t=1$.

To solve the interacting Hamiltonian 1 we will use the Cluster Perturbation
Theory method. This method, introduced by Senechal et al. Sénéchal  _et al._
(2000), builds upon the strong coupling expansion of the Hubbard modelPairault
_et al._ (1998, 2000) and allows to study both strong local correlation and
kinetic effectsSénéchal  _et al._ (2002). The $k$-dependent solution of the
full Hamiltonian is built from an exact solution of an isolated cluster
$\mathcal{G}_{0}(\omega)$ and matrix elements describing inter-cluster
coupling in the reciprocal space $V(\vec{k}$) through RPA-like expression

$\mathcal{G}(\omega,\vec{k})=\mathcal{G}_{0}(\omega)\left[1-V(\vec{k})\mathcal{G}_{0}(\omega)\right]^{-1}$
(5)

The solution of the isolated cluster is usually obtained either analytically
for simple subsystems or through the exact diagonalization method in the case
of more complex clusters. The results presented in this work were obtained
using the latter approach, with the isolated cluster being the unit cell of
the lattice in Eq. (1). It consists of three sites: two representing the non-
equivalent sub-lattices of the KMM and the third site representing the
localized orbitals, shown as blue, red and green circles in Fig.
LABEL:lattice:fig. The localized nature of the interacting orbital in this
model makes the inter-cluster coupling matrix $V(\vec{k})$ effectively a
$2\times 2$ matrix, as opposed to a $3$ matrix in a generic case. The $2\times
2$ sub-matrix of $V(\vec{k})$ has almost the same elements as the $H(\vec{k})$
matrix for the bare KMM. The difference is in its off-diagonal elements, where
the hopping between the atoms within the same unit cell is missing. The
effects connected to this matrix element is included in
$\mathcal{G}_{0}(\omega)$. The reduced dimensionality of $V(\vec{k})$ in the
model studied here is responsible for the strong presence of local (intra-
cluster) dynamics in the spectrum of the lattice systems.

The information about the topological state of the system will be read out
from the generalized spin-Chern number ($C_{s}$) defined as a difference of
generalized Chern numbersWang and Zhang (2012) for each spin species. The need
for $C_{s}$ comes from the time-reversal symmetry of this model at $M=0$,
which leads to the cancellation of normal (generalized) Chern number similar
to the standard quantum spin Hall system (QSH)Fu and Kane (2007); Prodan
(2009). The $C_{s}$ invariant is an extension of the standard Chern number to
many-body systems, through the utilization of the eigenstates of the effective
Hamiltonian

$\mathcal{H}_{\rm eff}(\vec{k})=-\mathcal{G}^{-1}(0,\vec{k}).$ (6)

It inherits the spin $S_{z}$ conservation symmetry after the full many-body
Hamiltonian so it can be written as

$\mathcal{H}_{\rm eff}(\vec{k})=\mathcal{H}_{\rm
eff}^{\uparrow}(\vec{k})\oplus\mathcal{H}_{\rm eff}^{\downarrow}(\vec{k}).$
(7)

Now the generalized spin-Chern number $C_{s}$ can be defined as
$C_{s}=C_{s}(\uparrow)-C_{s}(\downarrow)$ where $C_{s}(\sigma)$ is a standard
Chern number of $\mathcal{H}_{eff}^{\sigma}(\vec{k})$. It can be obtained by,
e.g., Kubo formula:

$C({\sigma})=\frac{1}{2\pi}\int_{BZ}d^{2}k\Omega_{\vec{k}}^{\sigma}$ (8)

with

$\Omega_{\vec{k}}^{\sigma}=\\!\\!\sum_{{n\leq n_{F}\atop
m>n_{F}}}\\!\\!\mathfrak{Im}\frac{\left\langle\psi_{\vec{k},\sigma}^{n}\right|\\!\partial_{k_{1}}\mathcal{H}_{\rm
eff}^{\sigma}\\!\left|\psi_{\vec{k},\sigma}^{m}\right\rangle\\!\left\langle\psi_{\vec{k},\sigma}^{m}\right|\\!\partial_{k_{2}}\mathcal{H}_{\rm
eff}^{\sigma}\\!\left|\psi_{\vec{k},\sigma}^{n}\right\rangle}{\left(E_{\vec{k}}^{(n)}-E_{\vec{k}}^{(m)}\right)^{2}}.$
(9)

Here $|\psi_{\vec{k},\sigma}^{n}\rangle$ is an eigenstate of $\mathcal{H}_{\rm
eff}^{\sigma}$ with energy $E_{\vec{k}}^{(n)}$ and $n_{F}$ is the number of
occupied bands. This is a non-zero quantized quantity in the topological phase
of this model due to $S_{z}$ conservation Sheng  _et al._ (2006); Prodan
(2009). In the text below, every time non-trivial topology is mentioned it
will mean $C_{s}(\sigma)\neq 0$. The accuracy of this approach in determining
the topology on a many-body state is currently under debate Zhao  _et al._
(2023); Blason and Fabrizio (2023), yet it provides an estimate for the
qualitative behavior of the topological phase diagram. In addition, it has
been frequently used in similar studies of the interplay between correlations
and topologyWerner and Assaad (2013); Nguyen and Tran (2013); Tupitsyn and
Prokof’ev (2019); Lu and Sénéchal (2022). Using this approach will allow for
qualitative comparison against these results.

We will consider only clusters that are geometrically the same size as the
unit cell of the underlying KMM. Because the additional (localized) orbital
does not have inter-cluster coupling, the effective Hamiltonian will inherit
the same lattice symmetries as the KMM except for the inversion symmetry.

##  III Time-reversal symmetric case

We will start with the analysis of the Hamiltonian 1 in the absence of
magnetic field $M$ ($M$=0) i.e. the time-reversal symmetric case. In the non-
interacting limit, when $U=0$, the presence of localized orbitals can drive a
topological phase transition through the interplay of two factors: (i) the
hybridization strength $J$ and (ii) the position of the localized orbital
level $\epsilon_{d}$. The latter sets the initial position of the flat band,
of localized orbitals, relative to the dispersive bands formed by the
itinerant electrons. The former mixes the two types of orbitals giving rise to
$k$-dependence in the otherwise flat band at $\epsilon_{d}$.

The three orbital unit cell of the model means, that the insulating phase can
only be obtained for an even number of electrons. We will focus mainly on the
$\frac{2}{3}$-filling and only mention the $\frac{1}{3}$-filling in cases,
where it could lead to situations that are physically different. We will also
consider the case of the flat band within the lattice electrons bandwidth, as
later this configuration will allow us to explore the onset of Kondo screening
once the correlations are included.

From the point of view of the TPT the $K/K^{\prime}$ points in the Brillouin
zone (BZ), where the gap inversion in the KMM takes place, are crucial. The
non-interacting Hamiltonian in the orbital basis for one of the spin channels
is given by:

$\mathcal{H}_{non,\sigma}(\vec{k})=\left[\begin{array}[]{ccc}\alpha_{\sigma}(\vec{k})&0&0\\\
0&-\alpha_{\sigma}(\vec{k})&J\\\ 0&J&\epsilon_{d}\end{array}\right].$ (10)

Depending on which $k=K,K^{\prime}$ point we look at the function
$\alpha_{\sigma}(\vec{k})$ can have two values,
$\alpha_{\sigma}(K)=\mathrm{sgn}(\sigma)3\sqrt{3}t^{\prime}$ or
$\alpha_{\sigma}(K^{\prime})=-\mathrm{sgn}(\sigma)3\sqrt{3}t^{\prime}$. At
these two points the inversion symmetry-breaking hybridization $J$ introduces
mixing between only two bands, while the third is effectively decoupled. As a
result, the two hybridized bands repel each other, while the third band
remains unchanged. Upon increasing $J$ one of the hybridized bands is
eventually pushed across the gap to the unaltered band causing the change in
the sign of the band gap and thus a TPT. The role of $\epsilon_{d}$ in this
scenario is to set the initial energy spacing between the in-gap (localized)
band and the conduction band. Hybridization alone is sufficient to drive the
TPT, while $\epsilon_{d}$ is not. Worth stressing is, that in case the lattice
electrons had an inverted gap at $J=0$, switching on the hybridization will
bend the flat band at $K$ and $K^{\prime}$ points in opposite directions. As a
consequence of the hybridization breaking the sub-lattice symmetry of a
lattice with band inversion. On the other hand, if the lattice electrons were
initially in a trivial state, i.e. due to additional sub-lattice splitting
mass term, the band bending will induce curvature with the same sign at either
of the two $k$-points. Thus, from the curvature of the initially localized
band alone, one can deduce the topology of the underlying lattice. In
addition, a hybridization that induces curvature with a different sign at $K$
and $K^{\prime}$ points in the flat band has a greater effect on its overall
bandwidth, further differentiating these two cases.

To illustrate and quantify the interplay between $J$ and $\epsilon_{d}$ in the
non-interacting model we constructed the topological phase diagram, shown in
Fig. LABEL:phase_diag:fig(a). The values of $\epsilon_{d}$ are indicated on
the right-hand side of the figure and the coloured dashed lines show the TPT
lines as a function of $J$ for various $t^{\prime}$. Each dashed corresponds
to a certain value of $t^{\prime}$ displayed in this figure in the same color.
The region below the dashed curves is the stability region of the
topologically non-trivial phase, and above is the trivial insulator. At
exactly $J=0$ there is no TPT since the localized orbitals are disconnected
from the lattice. They are topologically trivial and changing their energy has
no effect on the overall spin-Chern number of the system, which is made non-
trivial only by the lattice electrons. Thus x-axis scale in Fig.
LABEL:phase_diag:fig(a) starts from a very small but non-zero $J$. Outside
this pathological case, as $J$ increases the stability region of the Chern
insulator monotonically decreases irrespective of the $t^{\prime}$. This
monotonic behavior reflects the cooperation of $\epsilon_{d}$ and $J$ in
closing the topological gap.

Armed with a better understanding of the non-interacting limit we will now
proceed to $U\neq 0$ cases, fixing the localized orbital energy level at
$\epsilon_{d}=-U/2$. This choice of $\epsilon_{d}$ fixes the mid-point between
the two levels of the localized orbitals to the center of the bare KMM. The
resulting regions of stability of the topologically non-trivial phases for
various $t^{\prime}$ are indicated in Fig. LABEL:phase_diag:fig(a) by shaded
regions between the axes and continuous lines with circles. These lines define
critical interaction strength $U_{c}(J)$ for which the gap is closed. Their
colors, similarly to the non-interacting case, correspond to the colors of
$t^{\prime}$ indicated in the figure. The $U$ values are indicated on the
left-hand side of the figure. The scale on the left and right y-axis of Fig.
LABEL:phase_diag:fig(a) was chosen in such a way as to allow comparison
between the interacting model and a reference non-interacting model where the
localized orbital contributes to the single energy level $\epsilon_{d}=U/2$.
Their relation will be discussed in the next paragraph.

At small NNN hopping strengths ($t^{\prime}=0.05$) the dashed and solid lines
match quite well. This means that one can draw a parallel between the
interacting model with $\epsilon_{d}=-U/2$ and a non-interacting model with
$\epsilon_{d}=U/2$. It is possible since at $J=0$, $U<6\sqrt{3}t^{\prime}$ and
$\frac{2}{3}$ filling the spectrum of the system consist of a single band
originating from the localized orbitals sandwiched between the bands of the
lattice electrons. The disconnected localized orbitals contribute only one
band due to the particular filling of the system, which makes the localized
orbitals fully occupied. Through the analogy with a Hubbard atom, double
occupancy results in a spectrum with a single level shifted from its bare
value by $U$. The smallness of the gap ($\approx t^{\prime}$) makes a
topologically non-trivial phase unstable already at relatively small values
$J$, due to the limited energy window for the band bending. Weak hybridization
does not allow for correlation effects to spill into the lattice and thus the
system can still be effectively separated into an itinerant (lattice) and
localized electrons.

At larger $t^{\prime}$ the wider KMM gap allows for stronger hybridization
$J$, which leads to the discrepancies between the dashed and solid lines in
the intermediate region ($U\neq 0$ and $J\neq 0$). In the limiting two cases
of $J=0$ or $U=0$ the two lines have to match. At $U=0$ the two models are
simply equivalent. At $J=0$ the (trivial) gap is closed when the upper Hubbard
band crosses the KMM conduction band. Until then it remains below the Fermi
level and the mapping to the effective model with a single flat band at $U/2$
is exact. But lack of hybridization makes this case uninteresting due to the
lack of the TPT. Between the limiting cases (intermediate $J$), an increased
stability of the topologically non-trivial phase is observed in the fully
interacting model, when compared to the effective non-interacting model i.e.
the shaded region extends beyond the dashed lines with the same color. For
$t^{\prime}<0.15$ the shape of the stability region changes only
quantitatively with $t^{\prime}$, keeping the same monotonic decrease with
$J$. When $t^{\prime}>0.1$ a qualitative change starts to appear. The
topological phase first starts to become robust against larger $U$ as $J$
increases. Then the stability region develops a maximum at intermediate $J$
values, before dropping to the $U=0$ limit. As $t^{\prime}$ grows this maximum
grows (moves to larger $U$) and shifts to larger $J$, indicating that stronger
hybridization allows for screening of stronger interactions. But the maximal
allowed $J$ value for the non-trivial phase is bounded by the $U=0$ limit,
which is set by the $J=0$ lattice electrons gap width. Hence the observed
behavior that the larger the $t^{\prime}$ the bigger the area between the
solid and the corresponding dashed lines becomes.

To illustrate this point further, in the two bottom panels of Figs.
LABEL:phase_diag:fig(b,c) we compare the low energy spectrum of the effective
Hamiltonian (Eq. 7) and the corresponding non-interacting model. The former
determines the topology of the interacting system Wang and Zhang (2012) and
the latter plays the role of a reference case in which correlations are
treated on a mean-field level. Fig. LABEL:phase_diag:fig(b) and Fig.
LABEL:phase_diag:fig(c) show three regimes of $U$ for $t^{\prime}=0.1,J=0.4$
and $t^{\prime}=0.18,J=0.8$, respectively. These are weakly interacting (left
column), critical $U$ for the effective non-interacting model (middle column)
and critical $U$ for the fully interacting model (right column), respectively.
The bottom bands (deep below the Fermi level) are not displayed in any of
these cases as they are irrelevant from the point of view of the TPT. At weak
interaction strengths $U=0.5,0.8$, the displayed bands of the two Hamiltonians
behave qualitatively the same. The conduction bands of $\mathcal{H}_{non}$ and
$-\mathcal{G}^{-1}(0,\vec{k})$ follow each other closely around the $K$ point,
and deviate slightly just on the $K\rightarrow K^{\prime}$ path, where
hybridization effects come into play for a given spin channel. The biggest
difference between the spectra is in the location of the valence band.
Crucially, the direct gap at $K$ point in the spectrum of the non-interacting
model is smaller than in the spectrum of $-\mathcal{G}^{-1}(0,\vec{k})$. In
addition, comparing the left and the middle columns one can also see that the
valence band is less prone to move towards the conduction band upon increasing
$U$ once the correlation effects are accounted for. This points towards the
reduction of the bare $U$ value by the hybridization term, through the
delocalization of the charge in the interacting orbitals.

Comparison between the two spectra around the $K^{\prime}$ point, at first
glance, would also suggest an effective increase of $J$, since the gap at this
point in the BZ of $\mathcal{H}_{non}$ is smaller than of the
$-\mathcal{G}^{-1}(0,\vec{k})$. However, the increase in $J$ around
$K^{\prime}$ point is consistent with the reduced effect of $U$. The gap
between the two hybridized levels ($\Delta_{non,\sigma}(\vec{k})$) of the
Hamiltonian in Eq. (10) is given by

$\Delta_{non,\sigma}(\vec{k})=2\sqrt{J^{2}+\left(\frac{\alpha_{\sigma}(\vec{k})+\epsilon_{d}}{2}\right)^{2}}.$
(11)

Since $\alpha_{\sigma=\uparrow}(k=K^{\prime})=-3\sqrt{3}t^{\prime}$ and
$\epsilon_{d}=U/2$ the second term in the square root increases as U
decreases. As a result the fully interacting model, with screened interactions
has a larger gap in the spin-up channel at $K^{\prime}$ point. Contrasting the
$U$ values displayed in the two rightmost columns in the two bottom panels,
which shows the gap closing of $\mathcal{H}_{non}$ and
$-\mathcal{G}^{-1}(0,\vec{k})$, confirms that for larger $t^{\prime}$ this
effect is even stronger extending the stability of the topological phase from
$U=0.9$ at $t^{\prime}=0.1$ to $U=2.9$ at $t^{\prime}=0.18$.

The robustness of the conduction band due to the hybridization effect is not
synonymous with the rise of the Kondo physics Hewson (1993); Riseborough
(2000); Feng  _et al._ (2013). The latter would lead to the formation of a
narrow band pinned to the Fermi level Feng  _et al._ (2013). In this model,
the Kondo screening is, in most cases, suppressed due to near double occupancy
of the interacting orbitals. For the values of $t^{\prime}$ examined in this
work, the interaction strengths for which the topological phase is stable are
below the underlying KMM bandwidth and the system is outside the so-called
Kondo regime Meyer and Nolting (2000), in which the interacting orbitals are
half-filled. As a result, the conduction band has a reasonably large
dispersion and is not pinned to or crossed by the Fermi level. The non-trivial
topology at 2/3 -filling differentiates this from a standard Kane-Mele-Kondo-
lattice model Feng  _et al._ (2013) in which the hybridization term does not
introduce inversion symmetry breaking. As a consequence the TPT takes place
for electron densities in the interacting orbitals closer to half-filling,
bringing the Kondo physics to the forefront.

We now proceed to the analysis of the spectral features of the full
$\mathcal{G}(\omega,\vec{k})$ across the topological phase transition in two
cases: (i) $J$-driven ($U\ll 1$)and (ii) $U$-driven ($J\ll 1$). These two have
to be separated since the trivial states one could reach can be differentiated
through analyzing the distribution of spin-Chern number between each filled
band.

In Fig. LABEL:single_spin_band_TI:fig four $k$-resolved spectral functions for
the spin-up channel along the high symmetry points of the BZ are displayed.
These spectra are for a system with $t^{\prime}=0.15$, weak interactions at
the localized orbitals ($U=0.5$) and different hybridization strengths $J$
across the TPT. The corresponding spin-down channel results can be obtained
through the time-reversal symmetry operator since in the absence of $M$ it is
the symmetry of the system. The top panel shows the spectrum without the
hybridization ($J=0$). As anticipated it consists of two dispersive bands
separated by a gap of $6\sqrt{3}t^{\prime}$ at $K$ and $K^{\prime}$ points,
and a single flat band of the localized orbitals between them. The lack of a
hybridization term allows for the interpretation of the dispersive bands as
originating from the KMM. The middle band comes from the doubly occupied
localized orbitals, underlined by the lack of dispersion. As mentioned above,
it follows the behavior of a doubly occupied Hubbard model in the atomic
limit. The double occupancy is the result of $\frac{2}{3}$ filling, hence the
flat band is $U/2$ away from the KMM gap center. In the case of $\frac{1}{3}$
filling the localized orbitals would have been empty and thus also contribute
a single band to the full spectrum. Its energy is lowered by $U$ with respect
to the one displayed in the top panel of Fig. LABEL:single_spin_band_TI:fig.

After the introduction of a non-zero $J$ the spectrum of
$\mathcal{G}(\omega,\vec{k})$ becomes more complex with features that can be
divided into two sets. The first one consists of bands with orders of
magnitude larger spectral weights, we will refer to them as main bands. These
are the states directly connected to the eigenstates of the effective
Hamiltonian from the Eq. (7). In the other set are excitations with much
smaller spectral weights. We will refer to them as secondary bands. They are
mostly flat and either isolated or have anti-crossings with the main bands. In
the latter case, narrow bands are formed as a result. These excitations appear
only when $U$ and $J$ are non-zero and their spectral weight is directly
related to the hybridization strength. The flatness of these states is a clear
indication of their origin from intra-cluster dynamics.

The evolution with $J$ of the low energy part of the main bands reproduces the
analysis made previously for the eigenstates of the
$\mathcal{H}_{eff}(\vec{k})$. Upon increasing $J$, from zero, the initially
flat conduction band becomes dispersive, due to mixing between mobile non-
interacting lattice electrons and localized interacting electrons. Its
curvature increases with $J$ and has the opposite sign at the $K$ and
$K^{\prime}$ points, as a consequence of the presence of an inverted gap in
the underlying KMM. Hence in Fig. LABEL:single_spin_band_TI:fig the band
repulsion is between the flat valence band and the conduction band at one
$k$-point and between the flat valence band and the main band below it at the
time-reversal symmetric partner of that $k$-point. In the opposite spin
channel, the band bending is in exactly the opposite direction due to the
time-reversal symmetry. The band, which does not participate in the anti-
crossing, does not feel the change in the hybridization strength and remains
at its $J=0$ position. At the critical strength of $J=J_{c}$ the repulsed and
the unaffected bands cross and the TPT takes place (cf. third panel from the
top in Fig. LABEL:single_spin_band_TI:fig).

The non-zero interaction strength plays also an important role in the shape of
the stability region of a topological phase. At $J=0$ it controls how far away
from the center of the inverted gap of the lattice the localized orbital level
is occurring.

The lack of particle-hole symmetry in the topological phase has an interesting
consequence. The off-center (within the KMM gap) placement of the interacting
orbitals means, that as the hybridization strength increases and the band
bending takes place the gap at $K/K^{\prime}$ points is closed for different
$J$ values. For the set of parameters presented in Fig.
LABEL:single_spin_band_TI:fig, it is first closed at $K$ point and later at
$K^{\prime}$ point. After the first gap closing the system already becomes a
trivial insulator and the second gap closing does not change the topology of
the system anymore. But a closer inspection of the ”generalized spin-Chern
number” for each of the bands separately of the effective Hamiltonian in Eq.
(7) reveals how the topological invariant jumps between them after each gap
reopening. At small $J$ the middle band is trivial and the two outer bands
carry an opposite topological invariant. After the first gap reopening the
topological invariants of the two highest energy bands swap. Since the system
is $\frac{2}{3}$ filled the total spin-Chern number of the occupied bands
becomes zero. The second gap reopening, below the chemical potential, leads to
the trivialization of each band separately. In the case of $\frac{1}{3}$
filling the spectra would be particle-hole and $\vec{k}\rightarrow-\vec{k}$
transformed, since the localized orbital would be empty. As a result, the gap
for spin-up electrons will be first closed at $K^{\prime}$ and later for the
$K$ point, but the overall behavior of the spin-Chern number would be
qualitatively the same. As mentioned earlier, the increase in $J$ strength
results also in the appearance of flat band excitations, secondary bands, with
relatively small spectral weight. They are located roughly at multiples of
$\Delta=-\sqrt{J^{2}+t^{2}}$, which is the characteristic energy scale of the
charge fluctuation within the unit cell cluster of the lattice in Fig.
LABEL:lattice:fig. This further confirms their origin from local dynamics
within the unit cell. Due to electron filling (2/3 or 1/3) not being at the
particle-hole symmetry point of this system these flat bands are not visible
in the same way in the particle spectrum ($\omega>\mu$) as they are in the
hole part ($\omega<\mu$) of the spectrum. Best illustrated by the excitation
at $\omega=\Delta$. This flat band crosses the broad dispersive band in the
hole region of the energy spectrum around its upper edge. It has no
counterpart in the electron part of the energy spectrum. This cannot be said
about a pair of flat bands formed at higher energies, which are crossing the
outer band edges of both main bands with large dispersion.

We turn now to the analysis of correlation-driven behaviors. Figure
LABEL:single_spin_band_TI_variousU:fig illustrates the evolution of the
spectral function across the TPT as the interaction strength $U$ is increased,
while a non-zero (weak) hybridization is fixed at $J=0.6$. From the point of
view of the effective non-interacting model, one would expect the gap between
the conduction and valence bands to be closed through a shift of the latter
across the gap. In contrast to the previous case, where it was caused by the
valence band gaining dispersion.

The comparison between the three spectral functions displayed in Fig.
LABEL:single_spin_band_TI_variousU:fig confirms that this different mechanism
driving the gap closure is at play. Due to this fact, the gap is closed only
at either $K$ or $K^{\prime}$ point depending on the spin species, but the
time-reversal symmetry does not allow for spin-dependence in the critical $U$.
The bandwidth of the valence band is not experiencing enhancement with
increasing $U$, thus the gap below the Fermi level, at the TR symmetric
$k$-point, is never closed.

The comparison between the spectra in two top panels of Fig.
LABEL:single_spin_band_TI_variousU:fig shows that initially the valence band
undergoes a reduction in the bandwidth accompanied by only a very small
decrease of the gap at the $K$ point. Eventually, upon increasing $U$ the
topological gap closes and a trivial one opens. The re-opened gap, shown for
$U=3.5$ in two bottom panels of Fig. LABEL:single_spin_band_TI_variousU:fig,
is narrow and indirect. Further increase of $U$ only flattens the conduction
band. This evolution with $U$ is not expected from the effective non-
interacting model, cf. blue curve in the rightmost plot in the bottom row.
There the conduction band undergoes an increase in the bandwidth after the
TPT, as it contains parts of the flat band at $\Gamma$ point and the valence
band of the KMM at $K$ point. The observed behavior in the interacting model
can only be explained as an onset of the Kondo regime. For the top panel weak
interactions ($U<6\sqrt{3}t^{\prime}$) is limiting the hybridization effect to
renormalization of the gap. At $U=2$ the interacting orbital levels move deep
into the KMM bands, and its occupancy drifts away from double occupancy and
towards the half-filling. That is when the simple single-particle picture
starts to slowly break downMeyer and Nolting (2000). The rise of Kondo physics
leads to the creation of a many-body resonant state pinned to the chemical
potentialHewson (1993). This state, in the case of a lattice of localized
interacting orbitals, becomes split by the hybridization gap Pruschke  _et
al._ (2000). Thus after the TPT the spectrum of the system has a narrow gap
and an almost flat valence band. The width of the gap at the chemical
potential is set by the Kondo coherence scale.

On top of the changes around the Fermi level flat bands form at higher
energies, as in the previously discussed case. These secondary bands are also
clustering around multiples of $\Delta$, reflecting their inter-cluster
dynamics origin. The interaction strength, surpassing the hybridization,
enhances their anti-crossing with the dispersive main bands and reveals the
internal structure of the high-energy flat bands. The internal structure of
the flat bands is an aftermath of the charge fluctuations at the interacting
orbitals caused by $J$, doping it away from double occupancy, combined with
their two-level spectra due to a charge gap. The spectral weight of each level
is connected to the density of holes and electrons in the interacting
orbitals, thus the internal structure reveals itself in the hole part of the
spectrum. These effects were probably also present in the $U=0.5$ case, shown
in Fig. LABEL:single_spin_band_TI:fig, but the smallness of $U$ and the
broadening factor used in calculating $\mathcal{G}(\omega,\vec{k})$ did not
allow to resolve them. From the comparison between the results for $U<J$ in
Fig. LABEL:single_spin_band_TI:fig and $U>J$ shown in Fig.
LABEL:single_spin_band_TI_variousU:fig, one can see that the spectral weight
in the high energy flat bands remain small in the former while increases in
the latter. These bands are the precursors of the charge excitation peaks of
the interacting orbitals characteristic in the spectrum of the periodic
Anderson model Pruschke  _et al._ (2000); Meyer and Nolting (2000). As $U$
becomes the dominating energy scale, the spacing between them approaches $U$
and their interpretation becomes clear.

A TPT in an opposite direction, from a band insulator to a topological
insulator, can be also observed in systems upon adding a sufficiently strong
Semenoff-type sub-lattice splitting mass term Semenoff (1984); Haldane (1988).
In this geometry, the lack of an inverted gap makes the action of increasing
$J$ or $U$ similar. In either case, the valence band is pushed towards the
Fermi level, leading to an eventual gap closing and reopening. There still is
some increase in the bandwidth of the valence band due to increased
hybridization, but because the band repulsion is between the two fully
occupied bands at any point of the BZ this effect much weaker. The absence of
a large increase in the bandwidth of the valence band caused by $J$ means that
a hybridization-driven second TPT, back to a trivial insulator, is not
allowed. That would be the case if the $J$ term would allow for gap inversion
between the two occupied bands with different Chern numbers.

##  IV  Local magnetic field

Lastly, we will analyze the impact of the local magnetic field acting on the
electrons in the localized orbitals in the scenarios described above. This
orbital selective field is introduced to mimic a possible ferromagnetic
instability in the system. It can be either an intrinsic instability due to
the system’s dynamics, as reported for a similar model of decorated graphene
Seki  _et al._ (2015), or driven by an external field generated by the lack of
inversion symmetry as in Janus systems Zheng  _et al._ (2023).

In this section we will focus on the $U$-driven transition since this case
could show a similar transition to the reported in Ref. Hussain  _et al._ ,
2023, from trivial to non-trivial and back to the trivial phase upon
increasing interaction strength. In addition, the stability of the non-trivial
phase around the transition point for the paramagnetic case relied on the
emergence of the local Kondo physics. The introduction of a spin-symmetry
breaking term, in theory, should strongly influence the stability of the low-
energy physicsCosti (2000). From that point of view, the $J$-driven transition
is less interesting since the interacting orbitals levels are within the bare
KMM gap and the system overall is less susceptible to display many-body
behavior.

Figure LABEL:mag_field_spin_comp shows the evolution of spectral functions for
similar parameters as used in Fig. LABEL:single_spin_band_TI_variousU:fig, but
in the presence of a small local magnetic field $M=0.1$. Since the time-
reversal symmetry no longer holds Fig. LABEL:mag_field_spin_comp displays the
difference in the spectral weights between the two spin channels along high
symmetry points of the BZ, to show results for both spin channels in a concise
way.

Blue color indicates the spin-down majority and red color the spin-up majority
excitations. The lack of time-reversal symmetry is signaled by the different
gap widths around the Fermi level at $K$ and $K^{\prime}$ points for the two
spin species. Compared with the results for similar $U$ values without a
magnetic field before the TPT, top panels of Fig.
LABEL:single_spin_band_TI_variousU:fig and Fig. LABEL:mag_field_spin_comp, the
gap around $K$-point is reduced for spin-up and increased at $K^{\prime}$ for
spin-down electrons due to the Zeeman term. The magnetic field orientation is
such that the valence band for spin-up electrons is shifted upwards and for
spin-down electrons downwards. As a result, upon increasing $U$ first the gap
at $K$-point in the spin-up channel closes and later the gap closer takes
place at $K^{\prime}$ in the spin-down channel. Nonetheless, the system
transitions only from a TI to a metallic state. The trivial gap is never
opened, as the gap in one spin channel opens within the opposite spin channel
band. Similar behavior was reported in Ref. Hussain  _et al._ , 2023, where
the reopening of a gap around one $k$-point was initially overshadowed by the
metallic behavior at its time-reversal symmetric partner in the BZ. In this
model system, it is caused by the bottom of the spin-down conduction band at
$K^{\prime}$ crossing the Fermi level at the same time the spin-up channel
undergoes a TPT, as shown in the middle panel of Fig.
LABEL:mag_field_spin_comp. This is not coincidental, but a consequence of the
symmetries of the lattice and the sub-lattice symmetry-breaking hybridization.
To understand that it is best to look at the spectrum of the effective non-
interacting model, which was showing qualitatively similar behaviours to the
full model for $U<6\sqrt{3}t^{\prime}$. In the previous analysis, we showed
that around $K$ and $K^{\prime}$ points, there exists a level that is
decoupled from the other two, thus unaffected by $U$ nor $J$. It is also not
affected by $M$, which only acts directly at the interacting orbital and
indirectly at the lattice orbital coupled to it. These decoupled levels before
the TPT are the bottoms of the conduction band at $K$ and $K^{\prime}$ points
for the spin-up and spin-down channels, respectively. As shown in the two top
panels of Fig. LABEL:mag_field_spin_comp they are at the same energy. Because
they do not feel the magnetic field this analysis holds true irrespective of
the strength and sign of $M$. After a gap closing in each of the spin channels
these fixed points jump to the valence band. The existence of these
constraints on the energy levels around the $K/K^{\prime}$ points means that
after the gap closing in the spin-up channel, the total spectral function
cannot have a true gap. Since the bottom of the spin-down conduction band at
$K^{\prime}$ point has to have the same energy as the top of the spin-up
valence band at $K$ point. In a special case of an almost flat valence band, a
pseudo gap is possible. The gap closing at $K^{\prime}$ is shown in the bottom
panel of Fig. LABEL:mag_field_spin_comp. At this stage, the band touching at
$K^{\prime}$ is fully within the spin-up metallic band. From this panel, one
can also see that the Kondo physics is no longer present in the system. The
gap at $K$-point in the spin-up channel is much larger than for the same
parameters and larger $U$ in the paramagnetic case, cf. bottom panel in Fig.
LABEL:single_spin_band_TI_variousU:fig. The discrepancy cannot be explained
simply by the Zeeman field shifting the interacting orbital level. For the
parameters displayed in Fig. LABEL:single_spin_band_TI_variousU:fig and Fig.
LABEL:mag_field_spin_comp, that would account for increasing $U$ by $M/2$
which is still smaller than $U=3.5$ shown in the former figure. This means
that the flat bands seen in the presence of the magnetic field originate from
local moments forming within each spin channel and recovering the effective
single-particle picture.

To test this we analyze the influence of varying the chemical potential on the
spectral function. If the Kondo physics would still be at play, the low energy
excitations should adjust to the position of the chemical potential. In case
the flat bands are simply from hybridizing the local moments with the lattice,
changing $\mu$ should eventually drive spin-polarization and the vanishing of
some flat bands in the spectrum. As shown in Fig. LABEL:mag_field_mu_change,
for $U=4$ and the same parameters as in Fig. LABEL:mag_field_spin_comp,
adjusting the chemical potential in the narrow window of $\delta\mu=0.3$
results in a drastic change of the band structure. This supports the non-Kondo
origin of low-energy flat bands.

The top panel shows the spectrum of the system in case of the chemical
potential pinned to the band crossing point at $K^{\prime}$ point, as a
reference point. At low energies, the band structure in both spin channels
behaves differently due to the trivialization of the gap in the spin-up
channel (red color). However, the high-energy features show the same
qualitative behavior, irrespective of spin direction, with only slight
splitting due to the presence of $M$. Introducing a relatively small shift in
the chemical potential, in any direction, leads to the breakdown of this
resemblance at high energies due to changes in the orbital spin composition of
the system. As shown in the two bottom panels of Fig.
LABEL:mag_field_mu_change, reducing $\mu$ from $\mu=0.8\rightarrow\mu=0.7$
(middle panel) as well as increasing it from $\mu=0.8\rightarrow\mu=1.1$
(bottom panel) results in a formation of the continuous spin-polarized band
with dispersion from the decoupled KMM. The lack of localized levels crossing
these bands is signaling the blocking of the intra-cluster charge fluctuation
which was formed through mixing with one of the two local moment levels in the
interacting orbital, as a result of its (partial) spin polarization. The spin-
up band is located in the hole-part of the spectrum while the spin-down band
is in the electron part of the spectrum, reflecting the direction of $M$.

At $\mu=0.7$ the relative depletion of the spin-down electrons, compared to
$\mu=0.8$, leads to disappearance of the the low energy excitations below the
Fermi level in that spin channel. The system enters spin-polarized interacting
orbital limit as the spectra in the spin-up and spin-down channels become
connected by a reflection symmetry with regard to the gap center of the bare
KMM. Above the KMM gap in the spin-down spectrum, only one continuous band
survives, which is the upper bands of the KMM. Below the gap avoided crossings
are clearly visible, which form due to the hybridization of the local moment
level with the lattice. For $\mu=1.1$ the chemical potential is moved into the
upper band of the underlying KMM introducing more lattice electrons into the
system, thus the two spin-resolved spectra become very distinct. In the spin-
up channel it has only three bands while in the spin-down channel, the band
structure is more complex. The reduction of the number of bands to only three
and the absence of any flat bands means that the local charge fluctuations are
frozen and the system can be directly mapped onto an effective non-interacting
model with the localized orbitals level at $\epsilon_{d}=U/2$. The spin-down
sector, on the other hand, has multiple avoided crossings in the occupied part
of the spectrum but the electron part of the spectrum remains similar to spin-
polarized case. This indicates that the inter-cluster dynamics is not yet
trivialized.

##  V Conclusions

We have studied the interplay between topology and electronic correlations in
the decorated Kane-Mele model where we have two in-equivalent band inversions
at $K$ and $K^{\prime}$ points. We demonstrate that in the presence of the
time-reversal symmetry the interplay between the possible band inversion at
two $k$-points, the electronic correlation and the sub-lattice symmetry
breaking hybridization extends the topological region of the phase diagram
beyond $U$ values predicted by a simple mean-field treatment. Despite the
transition being at 2/3 (1/3) filling the Kondo physics arises in the system
only at large $U$, making the topologically trivial phase a narrow gap
insulator.

By introducing an artificial magnetic field, mimicking ferromagnetic ordering
at localized orbitals, we showed that a TPT at different $U$ values for each
spin channel is not allowed in this model due reminiscence of time-reversal
symmetry around $K$ and $K^{\prime}$ points forcing the position of one of the
original KMM bands for the two spin species to be at the same energy. In
addition, such a magnetic field would immediately destroy the Kondo physics
making the system susceptible to small variation in chemical potential. Our
results reveal that a non-trivial topology can survive in a system after
decorating with correlated orbitals. Ferromagnetic ordering of these orbitals
could block the opening of a trivial gap.

We expect that the inclusion of hybridization to the other sublattice could
allow for observation of separate TPT at the two non-equivalent band inversion
points. Exploration of this topic combined with similar studies using methods
that more accurately account for embedding local charge fluctuation in the
lattice Potthoff  _et al._ (2003); Park  _et al._ (2008); Mejuto-Zaera and
Fabrizio (2023), could help better understand the properties of this model and
explore its relevance to the ferrovalley compounds.

###### Acknowledgements.

We thank M. Wysokiński and C. Mejuto Zaera for the useful discussions. The
work is supported by the Foundation for Polish Science through the
International Research Agendas Programme co-financed by the European Union
within the Smart Growth Operational Programme (Grant No. MAB/2017/1). J. S. is
supported by the National Science Centre (Poland) OPUS 2021/41/B/ST3/04475.
J.S. and W.B. acknowledge support by Narodowe Centrum Nauki (NCN, National
Science Centre, Poland) Project No. 2019/34/E/ST3/00404. We acknowledge the
access to the computing facilities of the Interdisciplinary Center of Modeling
at the University of Warsaw, Grant g91-1418, g91-1419 and g91-1426 for the
availability of high-performance computing resources and support. We
acknowledge the CINECA award under the ISCRA initiative IsC99 ”SILENTS”,
IsC105 ”SILENTSG”, IsB26 ”SHINY” and IsB27 ”SLAM” grants for the availability
of high-performance computing resources and support. We acknowledge the access
to the computing facilities of the Poznan Supercomputing and Networking Center
Grant No. 609.

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