# Some identities for the Narayana polynomial and its derivatives

Nguyen Tien Dung 111Department of Mathematics, FPT University, Hoa Lac High
Tech Park, Hanoi, Vietnam. Department of Mathematics, VNU University of
Science, Vietnam National University, Hanoi, 334 Nguyen Trai, Thanh Xuan,
Hanoi, 084 Vietnam. Email<EMAIL_ADDRESS>
(June 18, 2022)

###### Abstract

In this note, by employing a nice property of semicircular distributions, we
derive some identities for the Narayana polynomial and its derivatives.

_Keywords:_ Narayana polynomials, Semicircular distribution.
2020 Mathematics Subject Classification: 11B83, 60C05.

##  1 Introduction

The Narayana numbers $N(r,k)$ and the Narayana polynomials
$\mathcal{N}_{r}(z)$ are defined by

$N(r,k)=\frac{1}{r}\binom{r}{k-1}\binom{r}{k},$ (1.1)
$\mathcal{N}_{r}(z)=\sum_{k=1}^{r}N(r,k)z^{k-1}.$ (1.2)

Because of their applications in Combinatorics theory, the Narayana numbers
and the Narayana polynomials have been investigated by several authors. There
is a large number of combinatorial properties can be found in the literature,
see [1, 2, 3, 4] and references therein.

The main result of this paper can be formulated as follows.

######  Theorem 1.1.

For every $r\geq 3,$ we have

$\mathcal{N}_{r}(z)=\frac{2r-1}{r+1}(1+z)\mathcal{N}_{r-1}(z)-\frac{r-2}{r+1}(1-z)^{2}\mathcal{N}_{r-2}(z),$
(1.3)
$\mathcal{N}^{\prime}_{r}(z)=\frac{r-1}{2z}\mathcal{N}_{r}(z)+\frac{(r-1)(z-1)}{2z}\mathcal{N}_{r-1}(z),$
(1.4)
$\mathcal{N}^{\prime\prime}_{r}(z)=-\frac{r-1}{z^{2}}\mathcal{N}_{r}(z)+\left(\frac{(r-1)^{2}}{z}+\frac{r-1}{z^{2}}\right)\mathcal{N}_{r-1}(z).$
(1.5)

The proof of the above theorem is deferred to Section 2. Let us end up this
Section with some remarks.

######  Remark 1.1.

In his paper [3], Lasalle established the following recurrence

$(z+1)\mathcal{N}_{r}(z)-\mathcal{N}_{r+1}(z)=\sum_{n\geq
1}(-z)^{n}\binom{r-1}{2n-1}A_{n}\mathcal{N}_{r-2n+1}(z),$ (1.6)

where the numbers $A_{n}$ satisfies the recurrence

$(-1)^{n-1}A_{n}=C_{n}+\sum_{j=1}^{n-1}(-1)^{j}\binom{2n-1}{2j-1}A_{j}C_{n-j},$

with $A_{1}=1$ and $C_{n}=\frac{1}{n+1}\binom{2n}{n}$ the Catalan number.
Interestingly, our relation (1.3) is simpler than (1.6).

######  Remark 1.2.

From (1.3) and (1.4) we obtain the following recurrence for the first order
derivative

$\mathcal{N}^{\prime}_{r}(z)=\frac{r-1}{2r+2}\frac{r-2+3rz}{z}\mathcal{N}_{r-1}(z)-\frac{(r-1)(r-2)}{2r+2}\frac{(1-z)^{2}}{z}\mathcal{N}_{r-2}(z).$

Similarly, we can express $\mathcal{N}^{\prime\prime}_{r}(z)$ in term of
$\mathcal{N}_{r-1}(z)$ and $\mathcal{N}_{r-2}(z).$

######  Remark 1.3.

It is known that $C_{r}:=\mathcal{N}_{r}(1),r\geq 1$ is the sequence of the
Catalan numbers. Hence, by choosing $z=1,$ we obtain from (1.3), (1.4) and
(1.5) the following identities

$C_{r}=\frac{4r-2}{r+1}C_{r-1},$
$\sum_{k=1}^{r}(k-1)N(r,k)=\frac{r-1}{2}C_{r},$
$\sum_{k=1}^{r}(k-1)(k-2)N(r,k)=-(r-1)C_{r}+r(r-1)C_{r-1}.$

Similarly, by choosing $z=2,$ we also obtain the identities for the sequence
of the large Schröder numbers $S_{r}:=\frac{1}{2}\mathcal{N}_{r}(2),r\geq 1.$

##  2 Proofs

Let $X$ be a random variable with the semicircular distribution, i.e. $X$ has
a density function $f(x)$ given by

$f(x)=\begin{cases}\frac{2}{\pi}\sqrt{1-x^{2}}&\quad\text{ if }-1\leq x\leq
1\\\ 0&\quad\text{ otherwise}.\end{cases}$ (2.1)

It is known from [1] that the Narayana polynomial can be expressed as follows

$\mathcal{N}_{r}(z)=\mathbb{E}[(1+z+2\sqrt{z}X)^{r-1}],$ (2.2)

where $\mathbb{E}$ denotes the expectation operator.

In order to prove Theorem 1.1, we need to use the following property of
semicircular distributions.

######  Lemma 2.1.

For any continuously differentiable function $h,$ we have

$\mathbb{E}[h(X)X]=\frac{1}{3}\mathbb{E}[h^{\prime}(X)(1-X^{2})].$

###### Proof.

We observe that

$F(x):=\int_{x}^{1}yf(y)dy=\frac{2}{\pi}\int_{x}^{1}y\sqrt{1-y^{2}}dy=\frac{2}{3\pi}(1-x^{2})\sqrt{1-x^{2}},\,\,-1\leq
x\leq 1.$

Hence, by integrating by parts, we obtain

$\displaystyle\mathbb{E}[h(X)X]$
$\displaystyle=\int_{-1}^{1}h(x)xf(x)dx=-\int_{-1}^{1}h(x)dF(x)$
$\displaystyle=\int_{-1}^{1}F(x)h^{\prime}(x)dx=\int_{-1}^{1}\frac{1}{3}(1-x^{2})h^{\prime}(x)f(x)dx$
$\displaystyle=\frac{1}{3}\mathbb{E}[h^{\prime}(X)(1-X^{2})].$

The proof of the lemma is complete. ∎

We now are ready to present the proof of Theorem 1.1.

Proof of (1.3). Using the representation (2.2), we have

$\displaystyle\mathcal{N}_{r}(z)$
$\displaystyle=\mathbb{E}[(1+z+2\sqrt{z}X)^{r-2}(1+z+2\sqrt{z}X)]$
$\displaystyle=(1+z)\mathcal{N}_{r-1}(z)+2\sqrt{z}\mathbb{E}[(1+z+2\sqrt{z}X)^{r-2}X],$

and hence,

$\mathcal{N}_{r}(z)-(1+z)\mathcal{N}_{r-1}(z)=2\sqrt{z}\mathbb{E}[(1+z+2\sqrt{z}X)^{r-2}X].$
(2.3)

Furthermore, by using Lemma 2.1 and the relation (2.3), we deduce

$\displaystyle\mathcal{N}_{r}(z)-(1+z)\mathcal{N}_{r-1}(z)$
$\displaystyle=\frac{4(r-2)}{3}z\mathbb{E}[(1+z+2\sqrt{z}X)^{r-3}(1-X^{2})].$

This gives us the following

$z\mathbb{E}[(1+z+2\sqrt{z}X)^{r-3}X^{2}]=-\frac{3}{4(r-2)}\left(\mathcal{N}_{r}(z)-(1+z)\mathcal{N}_{r-1}(z)\right)+z\mathcal{N}_{r-2}(z).$
(2.4)

Once again we use the representation (2.2) to get

$\displaystyle\mathcal{N}_{r}(z)$
$\displaystyle=\mathbb{E}[(1+z+2\sqrt{z}X)^{r-3}(1+z+2\sqrt{z}X)^{2}]$
$\displaystyle=(1+z)^{2}\mathcal{N}_{r-2}(z)+4(1+z)\sqrt{z}\mathbb{E}[(1+z+2\sqrt{z}X)^{r-3}X]+4z\mathbb{E}[(1+z+2\sqrt{z}X)^{r-3}X^{2}],$

which, together with (2.3) and (2.4), yields

$\displaystyle\mathcal{N}_{r}(z)$
$\displaystyle=(1+z)^{2}\mathcal{N}_{r-2}(z)+2(1+z)\left(\mathcal{N}_{r-1}(z)-(1+z)\mathcal{N}_{r-2}(z)\right)$
$\displaystyle-\frac{3}{r-2}\left(\mathcal{N}_{r}(z)-(1+z)\mathcal{N}_{r-1}(z)\right)+4z\mathcal{N}_{r-2}(z).$

So it holds that

$\frac{r+1}{r-2}\mathcal{N}_{r}(z)=\frac{2r-1}{r-2}(1+z)\mathcal{N}_{r-1}(z)-(1-z)^{2}\mathcal{N}_{r-2}(z)$

and (1.3) follows.

Proof of (1.4). It follows from the representation (2.2) that

$\mathcal{N}^{\prime}_{r}(z)=(r-1)\mathbb{E}\left[(1+z+2\sqrt{z}X)^{r-2}\left(1+\frac{X}{\sqrt{z}}\right)\right],$

which implies that

$\mathcal{N}^{\prime}_{r}(z)-(r-1)\mathcal{N}_{r-1}(z)=\frac{r-1}{\sqrt{z}}\mathbb{E}\left[(1+z+2\sqrt{z}X)^{r-2}X\right].$
(2.5)

Combining (2.3) and (2.5) yields

$\mathcal{N}^{\prime}_{r}(z)-(r-1)\mathcal{N}_{r-1}(z)=\frac{r-1}{2z}\left(\mathcal{N}_{r}(z)-(1+z)\mathcal{N}_{r-1}(z)\right).$

So (1.4) follows.

Proof of (1.5). We have

$\displaystyle N^{\prime\prime}_{r}(z)$
$\displaystyle=(r-1)(r-2)\mathbb{E}\left[(1+z+2\sqrt{z}X)^{r-3}\left(1+\frac{X}{\sqrt{z}}\right)^{2}\right]$
$\displaystyle-\frac{r-1}{2}\mathbb{E}\left[(1+z+2\sqrt{z}X)^{r-2}\frac{X}{z\sqrt{z}}\right]$
$\displaystyle=(r-1)(r-2)\mathbb{E}\left[(1+z+2\sqrt{z}X)^{r-3}\left(1+\frac{2X}{\sqrt{z}}+\frac{X^{2}}{z}\right)\right]$
$\displaystyle-\frac{r-1}{2}\mathbb{E}\left[(1+z+2\sqrt{z}X)^{r-2}\frac{X}{z\sqrt{z}}\right].$

Hence, recalling the relations (2.3) and (2.4), we obtain

$\displaystyle\mathcal{N}^{\prime\prime}_{r}(z)$
$\displaystyle=(r-1)(r-2)\bigg{(}\mathcal{N}_{r-2}(z)+\frac{1}{z}\left(\mathcal{N}_{r-1}(z)-(1+z)\mathcal{N}_{r-2}(z)\right)$
$\displaystyle-\frac{3}{4(r-2)z^{2}}\left(\mathcal{N}_{r}(z)-(1+z)\mathcal{N}_{r-1}(z)\right)+\frac{1}{z}\mathcal{N}_{r-2}(z)\bigg{)}$
$\displaystyle-\frac{r-1}{4z^{2}}\left(\mathcal{N}_{r}(z)-(1+z)\mathcal{N}_{r-1}(z)\right).$

Rearranging the terms we get

$\mathcal{N}^{\prime\prime}_{r}(z)=-\frac{r-1}{z^{2}}\mathcal{N}_{r}(z)+\left(\frac{(r-1)^{2}}{z}+\frac{r-1}{z^{2}}\right)\mathcal{N}_{r-1}(z).$

Thus the proof of (1.5) is complete.

## References

  * [1] T. Amdeberhan, V. H. Moll, C. Vignat, A probabilistic interpretation of a sequence related to Narayana polynomials. Online J. Anal. Comb. No. 8 (2013), 25 pp.
  * [2] W. Y. C. Chen, L. X. W. Wang, A. L. B. Yang, Schur positivity and the $q$-log-convexity of the Narayana polynomials. J. Algebraic Combin. 32 (2010), no. 3, 303–338.
  * [3] M. Lassalle, Two integer sequences related to Catalan numbers. J. Combin. Theory Ser. A 119 (2012), no. 4, 923–935.
  * [4] T. Mansour, Y. Sun, Identities involving Narayana polynomials and Catalan numbers. Discrete Math. 309 (2009), no. 12, 4079–4088.


††institutetext: Physics department, Syracuse University,
NY 13244-1130, USA

# Spin $\frac{1}{2}$ from Gluons

A. P. Balachandran<EMAIL_ADDRESS>
###### Abstract

The theta vacuum in $QCD$ is the standard vacuum, twisted by the exponential
of the Chern-Simons term. But what is the quantum operator $U(g)$ for winding
number $1$?

We construct $U(g)$ in this note. The Poincaré rotation generators commute
with it only if they are augmented by the spin $\frac{1}{2}$ representation of
the Lorentz group coming from large gauge transformations. This result is
analogous to the ‘spin-isopin’ mixing result due to Jackiw and Rebbi Jackiw ,
and Hasenfratz and ’t Hooft hasenfratz  and similar results in fuzzy physics
Fuzzy  and quantum gravity due to Friedman and Sorkin sorkin .

Hence states can drastically affect repreentations of observables. This fact
is further shown by charged states dressed by infrared clouds. Following Mund,
Rehren and Schroer MRS , we find that Lorentz invariance is spontaneously
broken in these sectors. This result has been extended earlier to $QCD$
(references nair  given in the Final Remarks) where even the global $QCD$
group is shown to be broken.

It is argued that the escort fields of MRS  are the Higgs fields for Lorentz
and colour breaking. They are string-localised fields where the strings live
in a union of de Sitter spaces. Their oscillations and those of the infrared
clouds generate the associated Goldstone modes.

##  1 Introduction

In quantum physics, there are always two related aspects. The first is the
algebra $\mathcal{A}$ of observables which represent elements subject to
experimental measurements. The second is the state $\omega$ which represents
the quantum ensemble which will be subject to measurements. $\omega$ is a
positive linear functional on $\mathcal{A}$, so that if $a\in\mathcal{A}$ ,
$\omega(a)$ is a complex number. Also $\omega(a^{*}a)\geq 0$ and
$\omega(\mathbb{I})=1$, the two properties needed for a probability measure.

In this view, the Hilbert space $H$ and the representation of $\mathcal{A}$ on
$H$ are emergent concepts which can be found using the GNS construction. The
abstract algebra $\mathcal{A}$ is always the same, but the representations of
$\mathcal{A}$ depend on $\omega$.

It can happen that two $\omega$’s give equivalent representations, but matrix
elements of observables between vectors in these representations vanish: this
vanishing theorem requires taking the direct sum of these representations.

It can also happen that the emergent representations are inequivalent. Here
too, no observable can excite a vector in one to a vector in the other.

In either case, we say that the representations are superselected. If a
Lagrangian symmetry changes the superselection sector, it is said to be
spontaneously broken.

In an infinite ferromagnet, the vector states in an irreducible representation
can be all those with the same direction of asymptotic spins. Observables can
be those which affect the local spins without changing the asymptotic value.
That defines an irreducible representation of observables.

Another Hilbert space will have vectors with asymptotic direction of spins
being in a different direction, but still observables causing only local
disturbances of spins. These two irreducible representations are equivalent,
but no observable has a non-zero matrix element between vectors of the two
representations.

In the case of a charged Higgs field $\phi$ breaking say $U(1)$ gauged
symmetry, it can happen that we have two families of states defining their
Hilbert spaces, the expectation values of $\phi(x)$ as $|\vec{x}|$ goes to
infinity differing in magnitude. This difference can be caused by the Higgs
potential. In this case, the $U(1)$ gauge field has different masses in the
two cases so that the representations are inequivalent. But still the local
observables define the same algebra .

These remarks illustrate that we need both the abstract algebra $\mathcal{A}$
and a state on it to realise $\mathcal{A}$ as operators on a Hilbert space.In
this paper, we elaborate on this idea for $SU(N)$ theta vacua in non-abelian
gauge theories. These vacua are based on the fact that the homotopy group
$\pi_{3}(SU(N))=Z$. The quantum states are classified by representations of
this group. If $g_{n}$ is a winding number $n$ transformation, then $n$ has
the image $\exp(in\theta)$ on the theta states. They define a representation
of observables on these states. If $U(g_{n})$ is the quantum operator
implementing the winding number $n$ transformation on theta states, and $g$
denotes $g_{1}$, then $U(g)$ acting on a theta state must have eigenvalue
$e^{i\theta}$. We will find $U(g)$ explicitly. It is a ‘large’ gauge
transformation so that all observables must commute with it. The $g$ in
question is the configuration that occurs for Skyrme solitons.

##  2 Remarks on Gauge Transformations

We will work with an $SU(N)$ gauge theory with the Gell-Mann matrices
$\lambda_{\alpha}$ as its Lie algebra generators in its defining
$N$-dimensional representation. We also fix an $SU(2)$ subgroup with Pauli
matrices $\tau_{i}$ as its generators.

On a spatial slice $\mathbb{R}^{3}$, the gauge group $\mathcal{G}$ is the
group of smooth maps

$g:\mathbb{R}^{3}\rightarrow SU(N)$

with $g(x)$ having a definite limit as the spatial coordinate goes to
infinity, that is as $|\vec{x}|\rightarrow\infty$. It has been called the sky
group by Balachandran and Vaidya sky . It is the analogue of the spi group for
asymptotically flat spaces introduced by Ashtekar and Hanson abhay .

Let $\lambda_{\alpha}$ be the $SU(N)$ Gell-Mann matrices. Then if $\Xi$ is a
Lie algebra valued test function,

$\Xi(\vec{x})=\Xi^{\alpha}(\vec{x})\lambda_{\alpha},$

with $\Xi^{\alpha}(\vec{x})$ approaching definite limits as
$|\vec{x}|\rightarrow\infty$ ,the Lie algebra generators of the sky group are

$Q(\Xi)=\int
d^{3}xtr~{}(D_{i}\Xi(\vec{x})E_{i}(\vec{x})+\Xi(\vec{x})J_{0}(\vec{x}))$

where $D_{i}$ is the covariant derivative, $E_{i}$ is the ( Lie algebra
valued) electric field, $J_{0}$ is the $SU(N)$ charge density from matter
sources and the trace is in the Lie algebra representation.

If the test functions are compactly supported or vanish at infinity, $Q(\Xi)$
represents the smeared Gauss law as one can see by partial integration. So all
observables are required to commute with it. In addition, $Q(\Xi)$ is required
to vanish on quantum states. These are called ‘small gauge transformations’.

If $\Xi^{\alpha}$ do not all vanish at infinity, considerations based on
locality show that observables still commute with them andres . But $Q(\Xi)$
need no longer vanish on quantum states. For example , in $QED$, if $\Xi$ goes
to a constant at infinity and does not vanish on quantum states, then it means
that we are working in a charged sector.

So an operator which does not commute with such $Q(\Xi)$ is not an observable
, it changes the superselection sector. It is an intertwiner between two
representations of the observables. We will see that generic elements of the
Lorentz or $SU(N)$ groups do precisely that.Hence they are spontaneously
broken.

##  3 The Theta Vacua

The theta vacua are quantum vector states which respond to $U(g)$ with
eigenvalue $exp(i\theta)$ and which are invariant under small gauge
transformations.They can be inferred from instanton physics and are given by

$|\theta\rangle=exp(i\theta\int K(A))|0\rangle$

where $K(A)$ is the Chern-Simons term ,

$K(A)=\frac{1}{8\pi^{2}}tr~{}(A\wedge dA+\frac{2}{3}A\wedge A\wedge A),$

and $|0\rangle$ is the Poincare’-invariant vacuum.

Under a winding number 1 transformation $g$ of $A$, $A\rightarrow gDg^{-1}$,
$\int K(A)$ acquires the additional term

$\frac{1}{24\pi^{2}}\int tr~{}(dg~{}g^{-1})^{3}=1$

so that the above Chern-Simons twisted vacuum is indeed the theta vacuum
vector.

Note that $\int K(A)$ is invariant under small gauge transformations.

##  4 The Operator Implementing Winding Number Transformations

We can guess that it is a large gauge transformation. We propose to show that
it is the finite gauge transforrmation generated by $Q(h)$ where
$h(\vec{x})=(\vec{\tau}\cdot\hat{\vec{x}})\tilde{h}(r)$ with

$\tilde{h}(0)=0,\qquad\tilde{h}(\infty)=-\pi.$

This test function is well-defined at $r$ equals 0 as $\tilde{h}$ vanishes
there. But as $\tilde{h}$ is not zero as $r$ becomes $\infty$, it generates a
large gauge transformation. It will be recognised that $h$ is the winding
number 1 Skyrmion configuration. ( see for example topology_book  ). An
important feature of $h$ is that it is invariant under simultaneous rotation
of $\hat{\vec{x}}$ and $\vec{\tau}$. This plays a crucial role in describing
spin $\mathit{\frac{1}{2}}$ nucleons using the chiral model of pions.

##  5 More on Superselection Sectors : Many Theta Vacua

Let us note a striking feature of theta vacua .When one writes
$\vec{\tau}.\hat{\vec{x}}\in h$, there is an identification of directions such
as the third direction in $\vec{\tau}$ and $\vec{x}$ spaces, or an
identification of angular momentum generators in the two spaces. We can also
write $h’$ equals $(\vec{\tau’}.\hat{x})\tilde{h}$ where $\tau^{\prime}_{i}$s
are any rotated Pauli matrices. That too will give a theta sector from $h$.
But $(h’-h)$ does not vanish at infinity and so $Q(h’-h)$ is a large gauge
transformation. Hence $Q(h’)$ and $Q(h)$ define different superselection
sectors even though their eigenvalues on the Chern-Simons-twisted vacua are
the same ! Superposition of such theta vacua produces a mixed state for the
observables. If $CP$ violation from instantons is found, we can ask which
mixed state is responsibe for it.

In this discussion, we have kept the Chern-Simons term fixed and changed the
superselection operator defining the $2\pi$ rotation. But we can equally well
keep the superselection operator fixed and rotate the components $A^{\alpha}$
of $A=A^{\alpha}\tau_{\alpha}$ by a global colour rotation generated by
$Q(\vec{\tau}\cdot\vec{\beta}/2)$, $\beta\neq\alpha$. This will change the
Chern-Simons term leading to new theta states as before. The response
$exp(i\theta)$ of the two Chern-Simons terms to the winding number one
transformation is still the same.

##  6  Spin 1/2 from Gluons

There is a 1980 paper by Friedman and Sorkin sorkin  with a similar title and
we have adapted our title from theirs. There are also papers with similar
results by Jackiw and Rebbi Jackiw  and Hasenfratz and ’t Hooft hasenfratz  in
the theory of non-abelian monopoles.

As emphasised in the introduction, quantum theory requires both an algebra of
observables and a state .( That is the case also in classical theory.) In
functional integral approaches, the latter is defined by the Lagrangian. It
can happen that the latter is defined entirely by bosonic variables, but still
quantum theory contains spinorial states. There are plenty of examples. The
books topology_book  and bundle_book  describe many instances, both from
soliton physics( eg.Skyrmions ) and from molecular physics ( such as the
ethylene molecule). The theta states are other examples. A vector state in
this case is defined by the vacuum twisted by a Chern-Simons term. The algebra
of observables is gauge invariant.

A superselecton sector contains a large gauge transformation $U(g)$. We
claimed above that this $U(g)$ for us generates a winding number $1$
transformation. We also claimed that this $U(g)$ is given by the Skyrmion
configuration for $g$. Let us prove this result.

Let $\Psi$ be a coloured field in the $N$-dimensional $SU(N)$ representation.
A finite transformation on $\Psi$ is then given by

$\displaystyle e^{iQ(h)}\Psi(x)e^{-iQ(h)}$ $\displaystyle=$
$\displaystyle\sum_{n}\frac{i^{n}}{n!}[Q(h),[Q(h),\cdots[Q(h),\Psi]\cdots]]$
(1) $\displaystyle=$
$\displaystyle\sum_{n}\frac{i^{n}}{n!}((\vec{\tau}\cdot\hat{x})\tilde{h}(r))^{n}\Psi((x)=exp(i(\tau\cdot\hat{x})\tilde{h}(r))\Psi(x)$
$\displaystyle\equiv$ $\displaystyle g(h)\Psi(x)$

Here $g$ is a Skyrmion configuration which is well-defined :

$g(h)=\cos h(r)+i(\tau\cdot\hat{x})\sin h(r)$

## A Remark

Let $h’$ be defined using $\tau’$. The $g(h)$ above and a $g(h’)$ are both
$-\mathbb{I}$ at $r\rightarrow\infty$ although $Q(h-h’)$ does not come from
the Gauss law and need not vanish on quantum states. Thus when restricted to
the sphere at $\infty$, the map from the Lie algebra to the Lie group level is
not injective. This result has played a role in the above analysis.

Back to the main theme. The expression (1) is valid also in the pure gluon
sector when the state is given by the Chern-Simons-twisted vacuum. The latter
involves the connection $A=A^{\alpha}\lambda_{\alpha}$ and $U(g)$ gauge
transforms it with $g(h)$ as in (1).

Now the gluons rotate only with tensorial angular momentum. This operator
rotates just $\hat{x}$ in $\hat{h}$. But that changes $Q(h)$, changing also
the superselection sector. We can conclude that the canonical angular momentum
$L_{i}$ for the gluon sector is spontaneously broken.

But consider adding the gauge rotation $Q(\tau_{i}/2)$ to $L_{i}$. The added
term rotates $\tau_{i}$ as well in $Q(h)$ so that $L_{i}+Q(\tau_{i}/2)$
commutes with $Q(h)$ : it does not change the supeselection sector. That is,
the total angular momentum $J_{i}=L_{i}+Q(\tau_{i}/2)$ does not change the
superselection sector.

The $2\pi$ rotation from $J_{i}$ acting on the twisted vacuum changes its sign
: the Chern-Simons twisted vacuum is spinorial. In this way we get spinorial
states in the gluon sector.

##  7 The Lorentz Group

Let $K_{i}$ be the canonical boost associated to $L_{i}$. Then
$K_{i}+Q(i\tau_{i}/2)$ and $L_{i}+Q(\tau_{i}/2)$ fulfil the $SL(2,C)$ algebra
and are appropriate generators for a Majorana field.

( We can also consider $L_{i}+Q(-i\tau_{i}/2)$ ). A Majorana field
transforming unitarily by these operators can also be constructed using
Weinberg’s methods weinberg .)

Unfortunately this choice of boosts does not seem to preserve the
superselection sector. For example , in $Q(h)$, $\tau_{i}$ will transform by
the non-unitary $(1/2,0)$ representation of $SL(2,C)$ and that does not seem
to be compensated by the transformation of $x$. So the Lorentz group is
spontaneously broken, a result known from other papers. But the spinorial
cover of the Euclidean group with $L_{i}+Q(\tau_{i}/2)$ and spacetime
translations seem implementable in the theta sectors.

In a subsequent paper, we show that infrared effects canonically induce fields
on the two-sphere at ‘infinity’ with $SL(2,C)$ conformal dimension $1$. Acting
on the vacuum, they create states on the local algebra which under $SL(2,C)$
intertwine inequivalent irreducible representations.

##  8 The Chern-Simons term for $SO(3)\subset SU(N)$

When $N\geq 3$, there is an $SO(3)$ subgroup in $SU(N)$ acting say on the
first three components of the $N$-dimensional vector space. This group had a
prominent role in our work on dibaryons lizzi  as solitons.

Now we can construct the Chern-Simons term $K(A)$ and the associated twisted
vacuum with $A$ valued in the $SO(3)$ algebra. The image of $\tau_{i}/2$ are
the $3\times 3$ angular matrices $l_{i}$. ( These are conventionally called
$\theta_{i}$ as in our group theory book group_book  , but we will use $l_{i}$
instead to avoid confusion with the theta of theta vacua.) Accordingly, the
Skyrmion configuration is changed to

$h^{\prime}(\vec{x})=(\vec{\mathbf{l}}\cdot\hat{x})\tilde{h}(r).$

Its finite transformation equals

$\hat{g}(\hat{x})=e^{i\vec{l}\cdot\hat{x}}h(r)$

with winding number 4 and so the eigenvalue of winding number transformation
on the Chern-Simons twisted vacuum is $e^{4i\theta}$. The periodicity in theta
now is accordingly $\frac{2\pi}{4}$.

The angular momentum $J_{i}=L_{i}+l_{i}$ is now tensorial. The boost
generators are $K_{i}+Q(il_{i})$ , but the associated Lorentz group changes
the superselection sector.

##  9 Brief Remarks on Escort Fields

In the abstract, we remarked that the Goldstone modes of Lorentz symmetry
breaking are incorporated in the escort fields of MRS .That is the case :
these fields incorporate a ’string’ from the direction of the Wilson line, and
it can locally fluctuate creating quantised Goldstone modes. But to keep this
paper focused, we will discuss such points in later worknext .

##  10 Final Remarks

There is more to be said on superselection sectors and their relation for
example to Wilson lines and the Rindler space. They will be discussed in later
work.

An older result discussed in nair  concerns $QCD$ :As it is non-abelian, its
generators do not generically commute with $Q(\Xi)$ : only the stability group
of $Q(\Xi)$ does so.

A particular result among others with direct application is the calculation of
the Landau-Yang process, strctly forbidden by Lorentz invariance and allowed
by its breaking. This calculation with Asorey, Balachandran and Momenz2gamma
is complete and will soon be reported.

It is striking that theta vacua convert the gluon sector to spinorial states
and that the theta states are infinitely degenerate. These results will have
an impact on axion phenomenology, which is yet to be explored.

## Acknowledgments

I thank Manolo Asorey, Andres Reyes-Lega, V.P.Nair. Sasha Pinzul and Sachin
Vaidya for many discussions. I am also grateful to Arshad Momen for critical
comments and help with the preparation of this article.

## References

  * (1) R. Jackiw and C. Rebbi, Phys. Rev. Lett. 36 (1976) 1116.
  * (2) P. Hasenfratz and G. ’t Hooft, Phys.Rev.Lett. 36 (1976) 1119.
  * (3) A. Balachandran, S.Kurkcuoglu and S. Vaidya, Lectures on Fuzzy and Fuzzy Susy Physics, World Scientific Publishing Company (2007).
  * (4) J. L. Friedman and R. D. Sorkin, Phys. Rev. Lett. 44 (1980) 1100.
  * (5) J. Mund, K. H. Rehren and B. Schroer, arXiv:hep-th/2109.10342.
  * (6) A.P. Balachandran, V. P. Nair, A. Pinzul, A. F. Reyes-Lega, S. Vaidya”Superselection, Boundary Algebras and Duality in Gauge Theories” ( arXiv:2112.08631 [hep-th] ) and refs. 1, 12 and 13 therein.
  * (7) A. P. Balachandran and S. Vaidya, Eur. Phys. J. Plus 128 (2013) 118.
  * (8) A. Ashtekar and R. O. Hanson , J. Math. Phys. 19(1978)1534.
  * (9) A. P. Balachandran, G Marmo, B S Skagerstam and A Stern , Classical Topology and Quantum States World Scientific Publishing Company (1991).
  * (10) A. P. Balachandran, G Marmo, B S Skagerstam and A Stern, Gauge Symmetries and Fibre Bundles - Applications to Particle Dynamics”, Lecture Notes in Physics 188 , Springer-Verlag(1983), for an updated version see arXive: quant-ph/1702.08910v2.
  * (11) A. P. Balachandran and A. F. Reyes-Lega in G. Marmo, D. Martin de Diego, M. Muñoz Lecanda (eds.) Classical and Quantum Physics, Springer Proceedings in Physics. Vol. 229. Springer, Cham.(2019) ( https://arxiv.org/abs/1807.05161v2).
  * (12) S. Weinberg, The Quantum Theory of Fields, Vol. 1. , Cambridge Univ. Press. (2005).
  * (13) A.P. Balachandran, A. Barducci, F. Lizzi, V.G.J. Rodgers and A. Stern, Phys.Rev.Lett. 52 (1984) 887, A.P. Balachandran, F. Lizzi, V.G.J. Rodgers and A. Stern, Nucl.Phys.B 256 (1985) 525, see also topology_book .
  * (14) A. P. Balachandran, S. G. Jo and G. Marmo, Group Theory and Hopf Algebras : Lectures for Physicists, World Scientific Publishing Co Pte Ltd ( 2010).
  * (15) A.P. Balachandran and colleagues , in preparation.
  * (16) M. Asorey, A. P. Balachandran and A. Momen, in preparation.


#  Dependence of the magnetic interactions in MoS2 monolayer on Mn-doping
configurations

Adlen Smiri Faculté des Sciences de Bizerte, Laboratoire de Physique des
Matériaux: Structure et Propriétés,
Université de Carthage, 7021 Jarzouna, Tunisia LPCNO, Université Fédérale de
Toulouse Midi-Pyrénées,
INSA-CNRS-UPS, 135 Av. de Rangueil, 31077 Toulouse, France  Iann C. Gerber
LPCNO, Université Fédérale de Toulouse Midi-Pyrénées,
INSA-CNRS-UPS, 135 Av. de Rangueil, 31077 Toulouse, France  Samir Lounis Peter
Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum
Jülich and JARA, 52425 Jülich, Germany  Sihem Jaziri Faculté des Sciences de
Bizerte, Laboratoire de Physique des Matériaux: Structure et Propriétés,
Université de Carthage, 7021 Jarzouna, Tunisia  Faculté des Sciences de Tunis,
Laboratoire de Physique de la Matiére Condensée,
Département de Physique, Université Tunis el Manar, Campus Universitaire 2092
Tunis, Tunisia

###### Abstract

Understanding the magnetic properties of the various Mn doping configurations
that can be encountered in $2H$-MoS2 monolayer could be beneficial for its use
in spintronics. Using density functional theory plus Hubbard term (DFT$+$U)
approach, we study how a single isolated, double- and triple-substitution
configurations of Mn atoms within a MoS2 monolayer could contribute to its
total magnetization. We find that the doping-configuration plays a critical
role in stabilizing a ferromagnetic state in a Mn-doped MoS2 monolayer.
Indeed, the Mn-Mn magnetic interaction is found to be ferromagnetic and strong
for Mn in equidistant substitution positions where the separation average
range of 6-11 $\angstrom$. The strongest ferromagnetic interaction is found
when substitutions are in second NN Mo-sites of the armchair chain. Clustering
is energetically favorable but it strongly reduces the ferromagnetic exchange
energies. Our results suggest that ordering the Mn dopants on MoS2 monolayer
is needed to increase its potential ferromagnetism.

##  I Introduction

On the grounds of their special structural and electronic properties Najmaei
et al. (2013); Tan and Zhang (2015); Li et al. (2016); Liu et al. (2013);
Lebègue and Eriksson (2009), the two-dimensional transition metal (M)
dichalcogenide (X) semiconductors (2D-TMDs) have shown peculiar optical
characteristics Zhu et al. (2015); Kidd et al. (2016); Kylänpää and Komsa
(2015); Raja et al. (2017); Stier et al. (2016) leading to several
applications, such as optoelectronics Wang et al. (2012); Choi et al. (2014),
including lasers and light-emitting diodes Zhang et al. (2015); Sundaram et
al. (2013); Ye et al. (2014, 2015). In the past few years, doping-induced
magnetism in nonmagnetic 2D-TMDs, such as MoS2 Cheng et al. (2013); Mishra et
al. (2013); Fang et al. (2018); Ramasubramaniam and Naveh (2013); Fan et al.
(2016); Wang et al. (2016); Zhang et al. (2015); Cong et al. (2015); Yue et
al. (2013), WS2 Song et al. (2017), WSe2 Song et al. (2017) or SnS2 Sun et al.
(2016) systems has deserved considerable attention. The magnetic properties of
doped 2D-TMDs, such as strong ferromagnetism (FM) Cheng et al. (2013); Mishra
et al. (2013); Fang et al. (2018); Ramasubramaniam and Naveh (2013); Fan et
al. (2016); Wang et al. (2016); Gao et al. (2016); Sun et al. (2016); Seixas
et al. (2015); Yue et al. (2013) and large magnetic anisotropy, are sought to
be used in the ultimately small magnetic devices. To achieve this purpose,
nonmetal (H, B, Cr, etc.) Yue et al. (2013) and transition-metal (Mn, Fe, Nb,
etc.) Cheng et al. (2013); Mishra et al. (2013); Fang et al. (2018);
Ramasubramaniam and Naveh (2013); Fan et al. (2016); Wang et al. (2016); Zhang
et al. (2015); Cong et al. (2015); Yue et al. (2013) dopants through various
doping strategies, such as substitution at M or X-sites Cheng et al. (2013);
Mishra et al. (2013); Fang et al. (2018); Ramasubramaniam and Naveh (2013);
Fan et al. (2016); Wang et al. (2016); Gao et al. (2016); Sun et al. (2016);
Seixas et al. (2015); Song et al. (2017) and adsorption Cong et al. (2015),
have been used to tune magnetism in 2D-TMDs Cheng et al. (2013); Mishra et al.
(2013); Fang et al. (2018); Ramasubramaniam and Naveh (2013); Fan et al.
(2016); Wang et al. (2016); Gao et al. (2016); Sun et al. (2016); Seixas et
al. (2015); Cong et al. (2015); Song et al. (2017).

Among 2D-TMDs, $2H$-MoS2 monolayer (ML) has shown specific electronic
transport properties, like considerable electron mobility (up to 1000
cm${}^{2}/Vs$ at low temperature) Cai et al. (2014); Baugher et al. (2013);
Lembke and Kis (2012); Schmidt et al. (2014) or low power dissipation Lembke
and Kis (2012); Radisavljevic et al. (2011); Zhang et al. (2012). These
features make this material a promising 2D-TMDs candidate for electronic
transport devices, essentially for next-generation transistors Lembke and Kis
(2012); Radisavljevic et al. (2011); Zhang et al. (2012); Lin et al. (2013).
These transport abilities has led to extensive efforts to induce magnetism in
MoS2 ML Mishra et al. (2013); Fang et al. (2018); Ramasubramaniam and Naveh
(2013); Wang et al. (2016); Zhang et al. (2015) in order to control the
electron spin and thus reach spintronic applications. To this end, one of the
effective tools is to substitute some Mo atoms in ML by Manganese ones to
produce Mn-doped MoS2, which has attracted wide interest, either theoretically
Mishra et al. (2013); Fang et al. (2018); Ramasubramaniam and Naveh (2013) or
experimentally Wang et al. (2016); Zhang et al. (2015). Indeed, using first-
principles calculations, it was demonstrated that this substitution is
energetically favorable under S-rich regime, which is common in reaction
medium for MoS2 nanosheets’ synthesis Fang et al. (2018). Moreover,
substitutional Mn atoms was found to prefer clustering in MoS2 ML Fan et al.
(2016). Experimentally, doped Mn-MoS2 sheets have been successfully
synthesized through different methods. In particular, Kehao et al. Zhu et al.
(2015), succeeded to incorporate Mn in MoS2 ML via vapor phase deposition
techniques. More recently, a hydrothermal method for Mn-doped MoS2 ML
synthesis has been proposed by Jieqiong et al. Wang et al. (2016).

According to several works Ramasubramaniam and Naveh (2013); Mishra et al.
(2013); Fan et al. (2016); Gao et al. (2016), the Mn impurities within MoS2 ML
are coupled ferromagnetically. In the earliest study of Mn-doped MoS2 ML,
Ramasubramaniam and Naveh Ramasubramaniam and Naveh (2013) attributed the
origin of this FM coupling to the double-exchange magnetic interaction Sato et
al. (2010). This type of interaction is due to the presence of delocalized
carriers between Mn impurities. However, the double-exchange mechanism was
ruled out by Mishra et al. Mishra et al. (2013) based on the fact that there
exists an antiferromagnetic (AFM) coupling between Mn atoms and their closest
Sulfur atoms. More recently, another origin of magnetic interaction among Mn
impurities in MoS2 ML has been proposed and called successive spin
polarizations (SSP) Andriotis and Menon (2014); Andriotis et al. (2015). The
SSP magnetic coupling model is based on the spin-polarization induced by
impurities in the host material to their nearest environment, i.e the closest
atoms mainly Andriotis and Menon (2014); Andriotis et al. (2015); Andriotis
and Menon (2012, 2013). In particular, a specific Mn impurity dictates the
spin polarization of its first Next Nearest neighbors (NN), namely Mo and S,
which in return dictate the spin polarization of the NN possible dopant
Andriotis and Menon (2014); Andriotis et al. (2015). Unlike the double-
exchange mechanism, the SSP FM coupling is based on localized electronic
processes that take place between Mn impurities Andriotis and Menon (2014).
Therefore, the SSP FM coupling can take place at low magnetic dopant
concentrations which can be below the percolation threshold Andriotis and
Menon (2014); Andriotis et al. (2015). Hence, a local enhancement of FM
coupling by manipulating Mn-doping configurations may lead to avoid the need
of high-doping concentration and still get strong ferromagnetism. Since the
doping concentration cannot be easily controlled in experiments Wang et al.
(2016), the risk is high to lose the semi-conducting property of MoS2 ML at
large dopant concentration Wang et al. (2016). To do this, one must first
understand the role of doping configurations in stabilizing the FM state of
MoS2 ML.

In both studies of Ramasubramaniam and Naveh Ramasubramaniam and Naveh (2013)
and Mishra et al. Mishra et al. (2013), the FM coupling strength between two
Mn impurities was found to decrease with respect to Mn-Mn distance’s
increasing. Additionally, according to the SSP model, one can expect that the
strength of the FM coupling can also depend drastically of the very local
configuration between the two Mn atoms Andriotis and Menon (2014); Andriotis
et al. (2015); Andriotis and Menon (2012, 2013). For instance, in the case of
1T${}^{{}^{\prime}}-$MoS2 ML doped by substitution with Mn atoms, the strength
of their magnetic interaction was found to be highly dependent on their
relative positions Lin and Ni (2016). Indeed, the FM coupling was found more
pronounced when two Mn dopants were separated by 6.38 $\angstrom$ than by 3.81
$\angstrom$ Lin and Ni (2016). Mind that similar conclusions have been drawn
already in the case of FM coupling between Co atoms embedded in a single
graphene sheet Lisenkov et al. (2012).

In Ref. Ramasubramaniam and Naveh (2013) it was shown that the ferromagnetism
in Mn-doped MoS2 ML becomes important when the Mn-doping concentration
increases. In particular, in 10-15% Mn-doping range, the Curie temperature was
found to be above room temperature. Motivated by this result, Jieqiong et al.
Wang et al. (2016) succeeded to elaborate a MoS2 ML heavily doped with Mn
impurities which gives rise to robust ferromagnetism. However, they also
demonstrated that the different resulting doping configurations contribute
differently, and even not, to the overall ML’s ferromagnetism Wang et al.
(2016). On the one hand, those Mn with NN forming Mn clusters are typically
antiferromagnetic and thus do not contribute to the overall magnetization. On
the other hand, only those Mn dopants that are at suitable distances can order
ferromagnetically Wang et al. (2016). The diversity of magnetic behaviour of
the different Mn doping configurations in MoS2 ML results on two different FM
phases in this material Wang et al. (2016). Therefore, distinguishing these
different contributions is of high interest in order to potentially control
magnetism in Mn-doped MoS2 ML.

Using Density Functional Theory plus Hubbard term (DFT$+$U), we perform a
comprehensive investigation of structural stability and magnetic properties,
namely magnetic exchange interaction and magnetic moments, of few near Mn-
dopants embedded in MoS2 ML. By placing Mn atoms in armchair- and/or zigzag-
substitution Mo-sites with different Mn-Mn distances, we generate several
doping configurations. Our aim is to explore the effect of these doping
configurations on the magnetic coupling nature and strength among Mn
impurities. To this end, the outline of this paper is as follows: we start by
a description of our computational details and methods in section II. In
section III we present and discuss our results for MoS2 ML with multiple Mn
dopings: in III. A, we validate our computational parameters and approachs by
comparing our results of the isolated Mn-induced magnetic and electronic
properties to that of literature. In III. B, we study the structural
stability, pairwise exchange interaction of two Mn-dopants placed on armchair
or zigzag chains as a function of Mn-Mn separations. In III. C, to broaden our
understanding of the magnetic exchange interaction behavior versus the doping
configurations, we add a third Mn-dopant. Indeed, by manipulating the three
dopant positions, we are able to determine the effect of the doping
clustering, doping shape (tiangle- or line- like) and equidistant or non-
equidistant doping on the magnetic properties. We also discuss the dependence
of the magnetic exchange interaction on inter-dopant distances. Our results
are compared to previous calculations and to the experiment of Jieqiong et al.
Wang et al. (2016). Finally, we conclude our results in section IV.

##  II Methods and Computational details

Our work was based on spin-polarized DFT implemented in Vienna ab initio
simulation package (VASP) Kresse and Hafner (1993, 1994). The exchange-
correlation interaction was described using the Perdew-Burke-Ernzerhof (PBE)
formulation of generalized gradient approximation (GGA) Perdew et al. (1996).
In addition, for 3d Mn orbitals, the Hubbard term correction (GGA+U) Dudarev
et al. (1998), was adopted. An on-site U parameter of 5 eV, assigned to Mn
impurity in Ref. (Wang et al., 2017), was considered. The core potential was
approximated by the projected augmented wave (PAW) scheme Blöchl (1994). A
cutoff energy of 400 eV for the plane-wave basis set, was found sufficient to
achieve a few meV convergence in energy in conjunction with a Brillouin zone
sampling of 4$\times$4$\times$1 Gamma-centered Monkhorst-Pack grids. Finer
grids (8$\times$8$\times$1) were used for density of states investigations.
The criteria of atom force convergence, used for the structure relaxations,
was fixed to 0.02 eV/$\angstrom$.

A distance of 20 $\angstrom$ between adjacent MoS2 MLs in perpendicular
direction was considered to eliminate spurious interactions resulting from the
periodic boundary conditions. Three cases of Mn doping were adopted: an
isolated Mn atom per supercell, two Mn atoms per supercell and three Mn atoms
per supercell. In order to significantly reduce long range magnetic
interaction between dopants in neighboring cells, a supercell of size
5$\times$5$\times$1 were used to contain one and two Mn dopants while for
three Mn dopants, we have considered a 7$\times$7$\times$1 supercell.

The Mn impurities were placed in different positions inside the supercells.
The exchange interaction among them was evaluated by the exchange energy,
$\Delta E=E_{\textrm{FM}}-E_{\textrm{AFM}}$. $\Delta E$ is the energy
difference between the parallel and antiparallel impurity spin orientations.
$E_{\textrm{FM}}$ and $E_{\textrm{AFM}}$ are the DFT total energies of self
consistent calculations for the FM and AFM configurations, respectively. The
magnetic coupling nature (FM or AFM) and its strength was determined by the
sign and amount of the exchange energy, respectively. It should be noted that
our aim is to evaluate the exchange interaction through $\Delta E$ between a
few dopants inside the supercell. A large negative $\Delta E$ indicates a
large FM coupling with a relatively high Curie temperature Kudrnovský et al.
(2004); Kan et al. (2013).

##  III Results and discussion

###  III.1 Single Mn atoms in MoS2 monolayer

We begin our work by examining the electronic properties of an individual Mn
dopant in a MoS2 ML at a doping concentration of 4%, see figure 1. In this
case, Mn impurities are well separated by a distance equal to 16.25
$\angstrom$. Therefore, one can assume that Mn impurities do not interact with
each others.

Figure 1:  A 4%-Mn atom doped ML is represented by $5\times 5\times 1$
supercell, the yellow atoms are the Sulfur (S), the blue atoms are the
Molybdenum (Mo) and the Manganese atom is denoted by purple color.

(a)

(b)

Figure 2:  a) The spin-resolved total electron density of states (DOS) and b)
the projected electron density of states (PDOS) of Mn-doped MoS2. The blue and
green lines denote the spin-up and spin-down channels, respectively. The Fermi
level is indicated by the red line.

The splitting of 3d orbital of Mn impurity in the MoS2 ML are first
investigated. The spin-resolved total electron density of states (DOS) and the
projected electron density of states (PDOS), are shown in 2a and 2b figures,
respectively. As clearly seen in figure 2a, highly localized states occur in
the band gap near the conduction band minimum. The origin of these states is
the 3d Mn orbitals, figure 2b. The latter split into three groups the in-plane
(d${}_{xy/x^{2}-y^{2}}$) orbitals, the out-of-plane (dxz/yz) orbitals and the
perpendicular d${}_{z^{2}}$ orbital. The only occupied states are the spin up
d${}_{z^{2}}$ orbital which gives rise to a total magnetic moment (MM) of
1$\mu_{B}$. All these results are in good agreement with previous studies
Cheng et al. (2013); Mishra et al. (2013); Fang et al. (2018); Ramasubramaniam
and Naveh (2013).

###  III.2 Double-substitution configurations of Mn atoms in MoS2 monolayer

In this section, two Mo atoms, from a 5$\times$5$\times$1 supercell, are
replaced by two Mn atoms (noted Mn1 and Mn2 in figure 3), which represents 8%
doping concentration in MoS2 ML. In this case, the available substitution
positions suggest four inequivalent configurations as plotted in figure 3. We
have the following: (i) the configuration a (figure 3a), in which the Mn pairs
are placed on 2nd NN positions of an armchair chain, with a Mn-Mn separation
of $5.7$ $\angstrom$; (ii) the configuration b (figure 3b) in which, the Mn
pairs are placed on 4nd NN positions of an armchair chain, with a Mn-Mn
separation of $11.3$ $\angstrom$; (iii) the configuration c (figure 3c) in
which the two Mn atoms are placed on two consecutive positions of a zigzag
chain where the separation is equal to $3.5$ $\angstrom$; (iv) the
configuration d (figure 3d) in which the Mn pairs are placed on 2nd NN
substitution positions of a zigzag chain and separated by $6.5$ $\angstrom$.

(a) -0.18 eV, 5.7 $\angstrom$

(b) -0.05 eV, 11.3 $\angstrom$

(c) -0.14 eV, 3.5 $\angstrom$

(d) -0.15 eV, 6.5 $\angstrom$

Figure 3: Spin density isosurface distributions for 8%-Mn-doped MoS2 in
5$\times$5$\times$1 supercells. The upper supercells contain each two Mn atoms
in two armchair positions while the lower supercells contain each two Mn atoms
in two zigzag positions. The energy differences between the AFM and FM
coupling states ($\Delta E$) and Mn-Mn distances are listed for each
configuration. The green and red isosurfaces represent positive and negative
spin density, respectively. The iso-surface value is $0.015$
e$\angstrom^{-3}$.

To get an idea about the stability of different configurations, their relative
energies are listed in Table 1. The lowest energy is corresponding to
configuration c which contains the closest Mn impurities. This is followed by
the configurations d, b and a. One can notice that the configurations with
armchair position substitutions are less stable than those with zigzag
position substitutions. Furthermore, we find that all double-doping
configurations have a common total MM of $\sim$ 2$\mu_{B}$. This value is
similar to that found by many previous reports (Mishra et al., 2013; Fang et
al., 2018; Ramasubramaniam and Naveh, 2013; Fan et al., 2016).

Table 1:  Distance between impurities
$L_{\text{Mn}_{1}-\text{Mn}_{2}}(\angstrom)$, total MMs, local MMs (for Mn1
and Mn2) and magnetic energies of various configurations which are shown in
figure 3.

Double doping-configurations |  | Separation distances |  | Relative energy |  | Total magnetic |  | Local magnetic | moment ($\mu_{B}$) |  | $\Delta E$
---|---|---|---|---|---|---|---|---|---|---|---
|  | $L_{\text{Mn}_{1}-\text{Mn}_{2}}(\angstrom)$ |  | (eV) |  | moment ($\mu_{B}$) |  | Mn1 | Mn2 |  | (eV)
a |  | 5.7 |  | 0.51 |  | 2.00 |  | 3.18 | 3.21 |  | -0.18
b |  | 11.3 |  | 0.51 |  | 1.99 |  | 3.19 | 3.21 |  | -0.05
c |  | 3.5 |  | 0.00 |  | 2.00 |  | 3.22 | 3.22 |  | -0.14
d |  | 6.5 |  | 0.40 |  | 2.00 |  | 3.20 | 3.20 |  | -0.15

In order to figure out the magnetic coupling nature in each doping
configuration, their exchange energies are presented in table 1. The exchange
energy is found negative for the four configurations which means that they
have stable FM states. In particular, for Mn-Mn separations equal or less than
$5.7$ $\angstrom$, the double-doping configurations have large $\Delta E$,
above 0.10 eV, which should stabilize their FM nature at high temperature. The
FM exchange interactions depend on the doping configuration as well as Mn-Mn
separations. More specifically, the configuration a shows the strongest FM
exchange coupling even greater than configuration c which has the closest Mn
pair. This dominance of the FM interaction between Mn impurities in the
configuration a is also found in Ref. Fan et al. (2016).

Using the SSP model, we investigate the magnetic coupling of two Mn dopants in
different positions. To this end, the spin-polarized charge density isosurface
distributions of the FM state for the different cases are shown the figure 3.
According to the SSP model the Mn-Mn FM coupling is based on the interaction
with the spin-polarized neighboring atoms, namely S and Mo atoms Andriotis and
Menon (2014). The Mn-Mn FM coupling is justified by the fact that the two
dopants are identical and both induce the same type of polarization on nearby
atoms Andriotis and Menon (2014). For all doping configurations, the induced
spins on the nearby host atoms are antiparallel to those of the Mn dopants.
The same spin density behavior has been found in previous studies Mishra et
al. (2013); Fang et al. (2018); Ramasubramaniam and Naveh (2013); Fan et al.
(2016). Furthermore, as shown in the table 1, the local MMs of the dopants Mn
are larger than the total MM of the doped ML. In fact, the AFM coupling
between the impurities of Mn and their host NN atoms is at the origin of the
reduction of the total MM. The latter result is in agreement with Ref. Fan et
al. (2016).

As we mentioned before, the FM coupling depends on the strength of the induced
polarization on the NN anions mediating the two dopants. Indeed, the strength
of the induced antiparallel spin density that resides on the Mn-Mn mediating S
and Mo, differs from one configuration to another, see figure 3. For the
different configurations in figure 3, we classify the induced polarization
between Mn dopants from the most important to the weakest as c than d, a and b
configurations. For configuration b (figure 3b), the distance between two Mn
dopants is $11.3$ $\angstrom$ which is so large that the SSP processes between
them is weak and therefore one can consider them to be almost isolated. For
the rest of cases, we notice that the FM coupling is inversely proportional to
the spin density magnitude that mediates the two Mn impurities. In particular,
for the case of configuration c (figure 3c) there is a strong AFM coupling
between Mn and the mediated S and Mo atoms, which weakens the FM interaction
between the two dopants. For configuration d (figure 3d), Mn1 and the mediated
host material atoms are less AFM coupled compared to configuration c. Thus,
configuration d shows more stable Mn1-Mn2 FM coupling than c configuration.
Unlike the latter two configurations c and d, a has the lowest AFM coupling
between the dopants and the mediated atoms, which promotes its FM stability.
It can be said that the antiparallel spin density in the middle of the dopants
filters the FM interaction between the two dopants, more than it is important,
more than the FM exchange is weak. In general, it can be said that the Mn-Mn
mediating antiparallel spin density screens the FM interaction between the two
dopants, more than it is important, more than the Mn-Mn FM exchange is weak.

###  III.3 Triple-substitution configurations of Mn atoms in MoS2 monolayer

In this section, we study the magnetic properties of three substitutional Mn
impurities that replace three close Mo atoms of MoS2 ML. Unlike the previous
section, we expand the supercell to 7$\times$7$\times$1, which means a
$6.12\%$ doping concentration. A large number of configurations are considered
in which the Mn atoms are placed in various relative Mo-sites. In figure 4, we
summarize the resulting triple-doping configurations (TDCs). In each case, for
clarity, we show the relevant portion of the 7$\times$7$\times$1 supercell
that contains the three dopants (figure 4).

Beginning by treating the TDCs stabilities, their relative energies are listed
in table 2. Our calculations indicate that the configurations where the Mn
impurities are placed at NN positions, namely configuration d, g and j, are
more energetically favored compared to the rest of configurations. In other
words, the three Mn dopants prefer to stay close to each other. In particular,
the TDC lowest energy is associated to the configuration j, in which the
impurities are bonded to the same S atom, see figure 4j. These results suggest
that high concentration doping can lead to the clustering of Mn impurities in
the MoS2 ML.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 4: Schematics showing various configurations of triple-dopant
configurations (TDCs) that were considered. Only the relevant portions of the
7$\times$7$\times$1 supercells are shown, for clarity. Blue, yellow, and
purple balls represent Mo, S, and Mn atoms, respectively. In each case, the Mn
atoms are denoted by $(1)$, $(2)$ and $(3)$. The energy differences between
the FM state and the energetically-closest AFM state ($\Delta E^{*}$) are
listed for each doping configuration.

(a)

(b)

(c)

Figure 5:  The local spin magnetic moments in Bohr magneton of Mn impurities
and of their surrounding neighboring atoms as obtained from the FM state for
a) configuration a, b) configuration j, and c) configuration h depicted in
Fig. 4. Figure 6: a) The gain in energy of the FM ordering over AFM ordering
of spins for the three Mn impurities as a function of the average separation
distance, $<\mathrm{L}>$, between the impurities. the negative (positive)
energy corresponds to FM (AFM) ordering being more stable.

It should be noted also, as in table 2, that while the NN TDCs, d g and j,
have a large total MMs of $\sim$ 5 $\mu_{B}$, the rest of configurations have
a total MMs of just 3 $\mu_{B}$. This difference of total MM originates from
the different environments of dopants. The total MMs result from the
competition between the Mn positive local MMs and the negative local MMs of
the host atoms. For instance, local spin magnetic moments of Mn impurities and
their surrounding atoms for configurations a, j and h are depicted in figure
5. For the NN TDCs, local MMs of $\sim$ 10 $\mu_{B}$ and $\sim$ -3.8 $\mu_{B}$
were found for the Mn impurities and the host atoms, respectively. However,
for the rest of configurations, local MMs of $\sim$ 9.7 $\mu_{B}$ and $\sim$
-5 $\mu_{B}$ were found for the Mn impurities and the host atoms,
respectively. The reduced MM of the environment of dopants in case of NN TDCs
causes the increasing of their overall MMs.

The energy of the different considered magnetic states is denoted
$E(\sigma_{1},\sigma_{2},\sigma_{3})$, where $\sigma_{i}$ represents the spin
orientation of Mni impurities. Obviously, $E(\uparrow,\uparrow,\uparrow)$ is
the energy of the FM state. The enumeration of Mni is shown in figure 4 for
all TDCs. For the sake of comparison, we denote the closest Mn neighbors Mn1
and Mn2.

Table 2: Relative energies, total magnetic moments (MMs) and the energy
differences between FM and non-FM spin configurations, $\Delta E$, of various
triple doping configurations. The asterisk indicates the energetically closest
spin configuration to the ground state.

Configurations | Relative energy (eV) | MM($\mu_{B}$) | Spin configurations |  $\Delta E$ (eV) |
---|---|---|---|---|---
a | 2.11 | 2.99 | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\downarrow,\uparrow)$ | -0.24 |
|  |  | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\uparrow,\downarrow)^{*}$ | -0.14 |
b | 2.10 | 2.99 | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\downarrow,\uparrow)$ | -0.13 |
|  |  | $E(\uparrow,\uparrow,\uparrow)-E(\downarrow,\uparrow,\uparrow)$ | -0.12 |
|  |  | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\uparrow,\downarrow)^{*}$ | -0.02 |
c | 2.26 | 2.99 | $E(\uparrow,\uparrow,\uparrow)-E(\downarrow,\uparrow,\uparrow)$ | -0.04 |
|  |  | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\downarrow,\uparrow)^{*}$ | -0.02 |
d | 0.71 | 5.00 | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\downarrow,\uparrow)$ | -0.14 |
|  |  | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\uparrow,\downarrow)^{*}$ | -0.07 |
e | 2.45 | 1.00 | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\downarrow,\uparrow)$ | -0.07 |
|  |  | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\downarrow,\uparrow)$ | -0.07 |
|  |  | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\uparrow,\downarrow)^{*}$ | 0.01 |
f | 1.88 | 2.99 | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\downarrow,\uparrow)$ | -0.25 |
|  |  | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\uparrow,\downarrow)^{*}$ | -0.11 |
g | 0.45 | 4.97 | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\downarrow,\uparrow)^{*}$ | -0.05 |
h | 1.30 | 3.00 | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\downarrow,\uparrow)$ | -0.14 |
|  |  | $E(\uparrow,\uparrow,\uparrow)-E(\downarrow,\uparrow,\uparrow)$ | -0.09 |
|  |  | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\uparrow,\downarrow)^{*}$ | -0.00 |
i | 2.90 | 3.00 | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\downarrow,\uparrow)$ | -0.22 |
|  |  | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\uparrow,\downarrow)^{*}$ | -0.10 |
j | 0.00 | 5.00 | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\downarrow,\uparrow)^{*}$ | -0.01 |
k | 1.34 | 4.99 | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\downarrow,\uparrow)$ | -0.10 |
|  |  | $E(\uparrow,\uparrow,\uparrow)-E(\downarrow,\uparrow,\uparrow)$ | -0.11 |
|  |  | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\uparrow,\downarrow)^{*}$ | -0.03 |
l | 1.83 | 3.00 | $E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\uparrow,\downarrow)^{*}$ | -0.12 |

In Table 2, we present the energy differences, $\Delta E$, between FM and non-
FM configurations for the various TDCs. In each TDC, the energy difference
between the ground state and the energetically-closest magnetic state is
asterisked $\Delta E^{*}$ in table 2. The FM state is found to be stable for
all TDCs except configuration e which has an AFM ground state. Furthermore,
for each TDC the energy differences, $\Delta E$, where the AFM energies have
the spin configurations ($\downarrow,\uparrow,\uparrow$) or
($\uparrow,\downarrow,\uparrow$), are the most important. This result proves
that the NN impurities, Mn1 and Mn2 as shown in figure 4, have a strong FM
coupling. However, the presence of a third close Mn impurity tends in general
to destabilise the FM state. For configuration e, the ground state becomes
even AFM ($\uparrow,\uparrow,\downarrow$). Clearly, modifying the doping
configurations alters the stability of the FM state and with that certainly
the strength and nature of the inter-impurities magnetic interactions.

In particular, depending on the FM stability and the geometric similarities,
three TDC groups stand out. (i) The first TDC group is formed by
configurations a, f, i, and l. Here, in each configuration, at least one Mn
dopant is placed at the mediator of the segment formed by the other two Mn
dopants. At least two pairs of Mn atoms are 2nd NNs, see figures 4a, 4f, 4i,
4l. The TDCs in this group have the most stable FM states compared to the rest
of configurations. Indeed, $\Delta E^{*}$ are of the order of hundreds meV
which is comparable to the a, c, and d double-doping configuration energies.
Similar to the case of double-doping configurations, Mn dopants on the 2nd NN
positions of an armchair chain, configuration a, exhibits the most stable FM-
state. Ordering the Mn dopants on the MoS2 ML in this particular set of
configurations can increase the temperature stability of the FM state. (ii)
The second group includes configurations d, g and j. In contrast to the first
group, here the Mn atoms are placed in the NN positions (see figures 4d, 4g,
4j). In this case, the FM interaction of the TDC is weak since $\Delta E^{*}$
is of the order of tens meV. In this group, the lowest $\Delta E^{*}$ is found
for configuration j. The origin of the reduction of the overall FM state of
configuration j is attributed to the strong AFM coupling of the three dopants
with the mediating atoms, see figure 5b. Therefore, clustering of Mn
impurities in the ML is not preferable if we want to get strong
ferromagnetism. This means that although clustering is energetically
favorable, we need to avoid it. This result is consistent with an observation
of Jieqiong et al. Wang et al. (2016) in which those Mn with NN form Mn
clusters are typically AFM. (iii) A weak FM interaction is also found in the
third group which includes the configurations b, e, h, and k. Here, the three
dopants are placed at different distances from each others (see figures 4b,
4e, 4h, 4k). In other words, one Mn dopant is far from the other two Mn which
are close to each other. As shown in figure 5c for configuration h, a strong
negative local MM of $\sim$ $-0.4$ $\mu_{B}$ resides on the mediating Mo
between the two close Mn and the far Mn dopant. This explains why the third
far Mn reduces the FM state of configuration i. In Ref. Andriotis et al.
(2015), according to SSP, if one dopes with two different magnetic impurities,
the spin polarization on the mediating host atom will take the characteristics
of the strongest polarization induced on it by the two neighboring impurities.
In our case, the situation is quite similar, the two close Mn atoms act as one
atom that induces a strong AFM polarization on the host atoms which dictates
the spin polarization of the third Mn atom. This favors the tendency towards a
weak FM coupling or even for an AFM coupling, as obtained for configuration e,
between the two close Mn impurities and the far Mn impurity.

To get a better insight on the ferromagnetic stability of the TDCs, we plot in
6, $\Delta E^{*}$ as a function of the average separation distances between
the impurities. One notices that the most stable FM state is realized for the
average separation distance ranging from 6 to 9 $\angstrom$. Outside these
inter-impurity distances, the FM state is weakened but maintained up to large
distances. Our findings are in agreement with the experiment reported in Ref.
Wang et al. (2016), which indicates that only Mn impurities that are at
suitable distances can order ferromagnetically. Figure 6 shows furthermore
that the doping configuration plays a criticial role in the magnetic stability
of the impurities complexes since for comparable averaged inter-impurity
distances the energy differences can be very different. For instance, $\Delta
E^{*}$ for configuration j (-0.01 eV) is about 5 times smaller than that of
configuration g (-0.05 eV) although they are both characterized by the same
averaged Mn-Mn distance.

After our discussion of the magnetic stability of the various complexes, we
complete our study by evaluating the magnetic interactions among the Mn atoms.
To this end, we map the energy differences obtained from first-principles for
the various studied magnetic states to those of the classical Heiseberg model,
$H=-\frac{1}{2}\sum_{i\neq j}J_{ij}\bm{e}_{i}\bm{e}_{j}$. Here $\bm{e}_{i}$ is
the unit vector defining the direction of Mn atomic MM at site $i$ and
$J_{ij}$ are the magnetic exchange coupling constants between the local
moments at Mn-sites $i$ and $j$. For the double-doping configuration,
$J_{12}^{double}=\displaystyle-\frac{\Delta E}{2}$. For the triple-doping
configurations, we have different cases. When Mn-complex form an equilateral
triangle, the three possible AFM states are degenerate, i.e.
$(\downarrow,\uparrow,\uparrow)\equiv(\uparrow,\downarrow,\uparrow)\equiv(\uparrow,\uparrow,\downarrow)$
and
$J_{12}^{triple}=J_{23}^{triple}=J_{13}^{triple}=\displaystyle-\frac{\Delta
E}{4}$. In the case of two inequivalent AFM states,
$J_{12}^{triple}=J_{23}^{triple}=\displaystyle-\frac{\Delta E_{1}}{4}$ and
$J_{13}^{triple}=\displaystyle\frac{\Delta E_{1}-2\Delta E_{2}}{4}$ where
$\Delta E_{1}=E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\downarrow,\uparrow)$
and $\Delta
E_{2}=E(\uparrow,\uparrow,\uparrow)-E(\uparrow,\uparrow,\downarrow)$. Finally,
when the three AFM states are inequivalent,
$J_{12}^{triple}=\displaystyle\frac{-\Delta E_{1}-\Delta E_{2}+\Delta
E_{3}}{4}$, $J_{23}^{triple}=\displaystyle\frac{\Delta E_{1}-\Delta
E_{2}-\Delta E_{3}}{4}$ and $J_{13}^{triple}=\displaystyle\frac{-\Delta
E_{1}+\Delta E_{2}-\Delta E_{3}}{4}$ where $\Delta
E_{3}=E(\uparrow,\uparrow,\uparrow)-E(\downarrow,\uparrow,\uparrow)$.

The estimated values of $J_{ij}$ are listed in table 3. Comparing the
constants $J_{ij}^{triple}$ to $J_{ij}^{double}$, we see that some of them
remain almost unaltered and some others change significantly. For instance,
the first $J_{ij}$ line in table 3 shows that except for the doping
configurations h and k, the exchange coupling constants $J_{ij}^{double}$ of
Mn pairs placed on the 2nd NN Mo-site, are comparable to $J_{ij}^{double}$.
Furthermore, for the other two lines, it is clear that there is a large
fluctuation of $J_{ij}^{triple}$ with respect to $J_{ij}^{double}$. This
reveals the dependence of pairwise magnetic coupling on the pair environment
which contains a third dopant as expected from our previous discussion.
Interestingly, we note one case where the magnetic interaction becomes AFM:
configuration h while for configurations d, k and e the magnetic interactions
between the furthest apart Mn atoms is negligible.

Table 3:  Estimated magnetic exchange coupling constants $J_{ij}$ of triple-
and double-doping configurations. Each $J_{ij}$ row corresponds to the Mn-
pairs with common separation distance L${}_{Mn_{i}-Mn_{j}}$.

| L${}_{Mn_{i}-Mn_{j}}$ ($\angstrom$) | Triple doping-configurations | Double doping-configurations
---|---|---|---
|  | a | b | h | i | k |  | a
$J_{ij}$(eV) |  $\sim$ 5.7 |  $J_{12(23)}=$ 0.06 |  $J_{12}=$ 0.05 |  $J_{23}=$ 0.01 |  $J_{12(23)}=$ 0.05 |  $J_{23}=$ 0.00 |  |  $J_{12}=$ 0.09
|  | d | e | g | j | h | k | c
$J_{ij}$(eV) |  $\sim$ 3.5 |  $J_{12(23)}=$ 0.03 |  $J_{12}=$ 0.04 |  $J_{12(23,13)}=$ 0.01 |  $J_{12(23,13)}=$ 0.02 |  $J_{12}=$ 0.06 |  $J_{12}=$ 0.04 |  $J_{12}=$ 0.07
|  | d | e | f | h | k | l | d
$J_{ij}$(eV) |  $\sim$ 6.5 |  $J_{13}=$ 0.00 |  $J_{23}=$ -0.00 |  $J_{12(23)}=$ 0.06 |  $J_{13}=$ -0.01 |  $J_{13}=$ 0.01 |  $J_{12(23,13)}=$ 0.03 |  $J_{12}=$ 0.07

##  IV Conclusion

Performing DFT+U calculations, we show that the magnetic stability and the
magnetic exchange interaction between neighboring dopants are very sensitive
to the doping-configuration geometry and the dopant separation distances. Our
calculations suggest on the one hand that placing Mn dopants at equidistant
Mo-sites where the average dopants separation is 6-9 $\angstrom$, enhances the
ferromagnetism of Mn-doped MoS2 ML. The Mn impurities that are placed on the
2nd NN Mo-sites of an armchair chain have the strongest FM coupling. On the
other hand, the FM exchange interaction is found to be reduced dramatically
when we have a Mn impurity close to a Mn-cluster. Interestingly, the
ferromagnetic interactions are in general finite for large inter-impurity
distances. In addition, the Mn impurities in the closest Mo-sites, clusters,
show weak FM coupling. The diversity in the FM coupling strength for the
various doping configuration is due to the strength of antiparallel spin
polarized Mo and S atoms that are mediating the interactions among Mn
impurities. When the Mn impurities approach each other the anti-parallel
mediating MMs increase, which reduces the FM exchange interaction. It should
be noted that, the doping configuration in which the FM exchange is low are
found energetically favorable indicating that Mn impurities have the tendency
to clustering within the MoS2 ML. Our results show that doping control is very
necessary to take advantage of magnetic properties of this material. This is
achievable with atomic manipulation using scanning tunneling microscopy, which
in its spin-polarized version allows even to extract magnetic exchange
interactions at the atomic scale Zhou et al. (2010); Loth et al. (2012). This
offers the possibility of confirming our predictions.

## ACKNOWLEDGMENT

I.C.G. thanks the CALMIP initiative for the generous allocation of
computational time through project p0812, as well as GENCI-CINES and GENCI-
IDRIS for Grant No. 2018-A004096649.
S.L. acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (ERC-
consolidator Grant No. 681405 DYNASORE).

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