On the Irreducibility of Some Quiver Varieties

On the Irreducibility of Some Quiver Varieties††This paper is a contribution
to the Special Issue on Noncommutative Manifolds and their Symmetries in
honour of Giovanni Landi. The full collection is available at
https://www.emis.de/journals/SIGMA/Landi.html

Claudio BARTOCCI ${}^{{\dagger}^{1}{\dagger}^{2}}\\!$, Ugo BRUZZO
${}^{{\dagger}^{3}{\dagger}^{4}{\dagger}^{5}{\dagger}^{6}{\dagger}^{7}}\\!$,
Valeriano LANZA ${}^{{\dagger}^{8}}\\!$ and Claudio L.S. RAVA
${}^{{\dagger}^{1}}\\!\\!\\!$

C. Bartocci, U. Bruzzo, V. Lanza and C.L.S. Rava

${}^{{\dagger}^{1}}$ Dipartimento di Matematica, Università di Genova, Via
Dodecaneso 35, 16146 Genova, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>http://www.dima.unige.it/~bartocci/index.htm

${}^{{\dagger}^{2}}$ Laboratoire SPHERE, CNRS, Université Paris Diderot (Paris
7), 75013 Paris, France

${}^{{\dagger}^{3}}$ SISSA (Scuola Internazionale Superiore di Studi
Avanzati),
${}^{{\dagger}^{3}}$ Via Bonomea 265, 34136 Trieste, Italy<EMAIL_ADDRESS>http://www.people.sissa.it/~bruzzo/webpage/

${}^{{\dagger}^{4}}$ Departamento de Matemática, Universidade Federal da
Paraíba,
${}^{{\dagger}^{4}}$ Campus I, João Pessoa, PB, Brasil

${}^{{\dagger}^{5}}$ IGAP (Institute for Geometry and Physics), Trieste, Italy

${}^{{\dagger}^{6}}$ INFN (Istituto Nazionale di Fisica Nucleare), Sezione di
Trieste, Italy

${}^{{\dagger}^{7}}$ Arnold-Regge Center for Algebra, Geometry and Theoretical
Physics, Torino, Italy

${}^{{\dagger}^{8}}$ Departamento de Análise, IME, Universidade Federal
Fluminense,
${}^{{\dagger}^{8}}$ Rua Professor Marcos Waldemar de Freitas Reis, Niterói,
RJ, Brazil<EMAIL_ADDRESS>
Received March 13, 2020, in final form July 10, 2020; Published online July
26, 2020

We prove that certain quiver varieties are irreducible and therefore are
isomorphic to Hilbert schemes of points of the total spaces of the bundles
$\mathcal{O}_{\mathbb{P}^{1}}(-n)$ for $n\geq 1$.

quiver representations; Hilbert schemes of points

14D20; 14D21; 14J60; 16G20

We thank Gianni Landi for the long friendship and for all
collaborations that took place over the last 35 years

##  1 Introduction

Nakajima’s quiver varieties were introduced by Hiraku Nakajima in [11] to
study the moduli spaces of instantons on ALE spaces, and have been extensively
studied since then, see, e.g., [8, 10, 12, 13]. They provide a modern and
significant example of how algebra and geometry can be sometimes so deeply,
yet surprisingly connected: in fact, their main feature is that they allow one
to put in relation some moduli spaces of bundles (or torsion-free sheaves)
over certain smooth projective varieties with some moduli spaces of
representations of suitable algebras (the so-called _path algebras_ of a
quiver and quotients of them). A major example of this bridge is given by the
moduli space of framed sheaves on $\mathbb{P}^{2}$, which can be identified
with the moduli space of semistable representations of the ADHM quiver (see
[12] for details).

The way this relation is usually looked at is the one that inspired Nakajima’s
first pioneering work: the philosophy is to use the algebraic data we get on
one side (usually called _ADHM data_) to parameterize the geometric moduli
spaces we have on the other side, i.e., the objects we are actually interested
in (see for example [4, 6, 14]). But sometimes it may be useful to switch
roles and use the geometric interpretation as a “tool” to prove something
interesting per se on the algebraic side. For instance, this is the case when
one deals with irreducibility problems: to determine whether a variety of
matrices is irreducible is known to be a challenging problem (see [15] and
references therein), and in the specific case of Nakajima’s quiver varieties
the conclusive result by Crawley-Boevey stating that _all of them_ are indeed
irreducible has been achieved only by using hyperkähler geometry techniques
[5].

In [1] we introduced a collection of new quiver varieties
$\mathcal{M}(\Lambda_{n},\vec{v}_{c},w_{c},\vartheta_{c})$, ${n\geq 1}$ (see
below for the notation); for $n\neq 2$ they are not Nakajima’s quiver
varieties, as the quivers involved are not doubles. We proved that
$\mathcal{M}(\Lambda_{1},\vec{v}_{c},w_{c},\vartheta_{c})$ is isomorphic to
the Hilbert scheme of points of the total space of
$\mathcal{O}_{\mathbb{P}^{1}}(-1)$, and, in particular, that it is therefore
irreducible (as the Hilbert scheme is so [7]). For $n\geq 2$ we only proved a
weaker result, i.e., that only a certain connected component of
$\mathcal{M}(\Lambda_{n},\vec{v}_{c},w_{c},\vartheta_{c})$ can be identified
with
$\operatorname{Hilb}^{c}(\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^{1}}(-n)))$.
However, as $\mathcal{M}(\Lambda_{2},\vec{v}_{c},w_{c},\vartheta_{c})$ is a
Nakajima quiver variety, its irreducibility follows from Crawley-Boevey’s
result, so that one only has to determine whether the varieties
$\mathcal{M}(\Lambda_{n},\vec{v}_{c},w_{c},\vartheta_{c})$ are irreducible for
$n\geq 3$. In this paper we prove this fact, completing the work of [1],
actually showing directly that
$\operatorname{Hilb}^{c}(\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^{1}}(-n)))$
is isomorphic to the whole
$\mathcal{M}(\Lambda_{n},\vec{v}_{c},w_{c},\vartheta_{c})$. As this technique
also works for the case $n=2$ we include it as well.

##  2 Some background

The quivers we are going to consider are extracted from the ADHM data for the
Hilbert schemes of points of the varieties $\operatorname{Hilb}^{c}(X_{n})$,
where $X_{n}$ is the total space of the line bundle
$\mathcal{O}_{\mathbb{P}^{1}}(-n)$, and, in turn, the construction of the ADHM
data is based on the description of the moduli spaces of framed sheaves on the
Hirzebruch surfaces $\Sigma_{n}$ in terms of monads that was given in [2]. We
denote by $H$ and $E$ the classes in $\operatorname{Pic}(\Sigma_{n})$ of the
sections of the natural ruling $\Sigma_{n}\to\mathbb{P}^{1}$ that square to
$n$ and $-n$, respectively. We fix a curve $\ell_{\infty}$ in $\Sigma_{n}$
belonging to the class $H$ (the “line at infinity”). A framed sheaf on
$\Sigma_{n}$ is a pair $(\mathcal{E},\theta)$, where $\mathcal{E}$ is a rank
$r$ torsion-free sheaf which is trivial along $\ell_{\infty}$, and
$\theta\colon\mathcal{E}|_{\ell_{\infty}}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{O}_{\ell_{\infty}}^{\oplus
r}$ is an isomorphism. A morphism between framed sheaves
$(\mathcal{E},\theta)$, $(\mathcal{E}^{\prime},\theta^{\prime})$ is by
definition a morphism
$\Lambda\colon\mathcal{E}\longrightarrow\mathcal{E}^{\prime}$ such that
$\theta^{\prime}\circ\Lambda|_{\ell_{\infty}}=\theta$. The moduli space
parameterizing isomorphism classes of framed sheaves $(\mathcal{E},\theta)$ on
$\Sigma_{n}$ with Chern character
$\textrm{ch}(\mathcal{E})=\big{(}r,aE,-c-\frac{1}{2}na^{2}\big{)}$, where
$r,a,c\in\mathbb{Z}$ and $r\geq 1$, will be denoted $\mathcal{M}^{n}(r,a,c)$.
We normalize the framed sheaves so that $0\leq a\leq r-1$.

A monad $M$ on a scheme $X$ is a three-term complex of locally free
$\mathcal{O}_{X}$-modules of finite rank, having nontrivial cohomology only at
the middle term (cf. [16, Definition II.3.1.1]). It was proved in [2] that a
framed sheaf $(\mathcal{E},\theta)$ on $\Sigma_{n}$ with invariants $(r,a,c)$
is the cohomology of a monad

$M(\alpha,\beta)\colon\ \lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
5.5pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
29.5pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{U}_{\vec{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern 72.23056pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
72.23056pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{V}_{\vec{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
97.98143pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\beta}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 114.96112pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
114.96112pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathcal{W}_{\vec{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
160.46948pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
160.46948pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}\ignorespaces}}}}\ignorespaces,$
(2.1)

where $\vec{k}$ is the quadruple $(n,r,a,c)$, and

$\mathcal{U}_{\vec{k}}:=\mathcal{O}_{\Sigma_{n}}(0,-1)^{\oplus
k_{1}},\qquad\mathcal{V}_{\vec{k}}:=\mathcal{O}_{\Sigma_{n}}(1,-1)^{\oplus
k_{2}}\oplus\mathcal{O}_{\Sigma_{n}}^{\oplus
k_{4}},\qquad\mathcal{W}_{\vec{k}}:=\mathcal{O}_{\Sigma_{n}}(1,0)^{\oplus
k_{3}},$

with

$k_{1}=c+\dfrac{1}{2}na(a-1),\qquad k_{2}=k_{1}+na,\qquad
k_{3}=k_{1}+(n-1)a,\qquad k_{4}=k_{1}+r-a.$

The space $L_{\vec{k}}$ of pairs in
$\operatorname{Hom}(\mathcal{U}_{\vec{k}},\mathcal{V}_{\vec{k}})\oplus\operatorname{Hom}(\mathcal{V}_{\vec{k}},\mathcal{W}_{\vec{k}})$
fitting into (2.1), such that the cohomology of the complex is torsion-free
and trivial at infinity, is a smooth algebraic variety. There is a principal
$\operatorname{GL}(r,\mathbb{C})$-bundle $P_{\vec{k}}$ over $L_{\vec{k}}$
whose fibre at a point $(\alpha,\beta)$ is the space of framings for the
corresponding cohomology of (2.1). The algebraic group

$G_{\vec{k}}=\operatorname{Aut}(\mathcal{U}_{\vec{k}})\times\operatorname{Aut}(\mathcal{V}_{\vec{k}})\times\operatorname{Aut}(\mathcal{W}_{\vec{k}})$

acts freely on $P_{\vec{k}}$, and the moduli space $\mathcal{M}^{n}(r,a,c)$ is
the quotient $P_{\vec{k}}/G_{\vec{k}}$ [2, Theorem 3.4]. This is nonempty if
and only if $c+\frac{1}{2}na(a-1)\geq 0$, and when nonempty, it is a smooth
algebraic variety of dimension $2rc+(r-1)na^{2}$.

When $r=1$ we can assume $a=0$, and there is an identification

$\mathcal{M}^{n}(1,0,c)\simeq\operatorname{Hilb}^{c}(\Sigma_{n}\setminus\ell_{\infty})=\operatorname{Hilb}^{c}(X_{n}).$

A first step to construct ADHM data for the Hilbert schemes of points of the
varieties $X_{n}$ is to show that the Hilbert schemes can be covered by open
subsets that are isomorphic to the Hilbert scheme of $\mathbb{C}^{2}$, and
therefore have an ADHM description, according to Nakajima. Then one proves
that these “local ADHM data” can be glued to provide ADHM data for the Hilbert
schemes of $X_{n}$.

Let $P^{n}(c)$ be the set of collections $(A_{1},A_{2};C_{1},\dots,C_{n};e)$
in $\operatorname{End}(\mathbb{C}^{c})^{\oplus
n+2}\oplus\operatorname{Hom}(\mathbb{C}^{c},\mathbb{C})$ satisfying the
conditions
(P1)
$\displaystyle\begin{cases}A_{1}C_{1}A_{2}=A_{2}C_{1}A_{1},&\qquad\text{when
$n=1$},\\\\[2.0pt] \begin{aligned} A_{1}C_{q}&=A_{2}C_{q+1},\\\
C_{q}A_{1}&=C_{q+1}A_{2}\end{aligned}\qquad\text{for}\quad
q=1,\dots,n-1,&\qquad\text{when $n>1$;}\end{cases}$
(P2) $A_{1}+\lambda A_{2}$ is a _regular pencil_ of matrices, i.e., there
exists $[\nu_{1},\nu_{2}]\in\mathbb{P}^{1}$ such that
$\det(\nu_{1}A_{1}+\nu_{2}A_{2})\neq 0$;
(P3) for all values of the parameters
$([\lambda_{1},\lambda_{2}],(\mu_{1},\mu_{2}))\in\mathbb{P}^{1}\times\mathbb{C}^{2}$
satisfying

$\lambda_{1}^{n}\mu_{1}+\lambda_{2}^{n}\mu_{2}=0$

there is no nonzero vector $v\in\mathbb{C}^{c}$ such that

$\begin{cases}C_{1}A_{2}v=-\mu_{1}v,\\\ C_{n}A_{1}v=(-1)^{n}\mu_{2}v,\\\
v\in\ker
e\end{cases}\qquad\text{and}\qquad\left(\lambda_{2}{A_{1}}+\lambda_{1}{A_{2}}\right)v=0.$

The group
$\operatorname{GL}(c,\mathbb{C})\times\operatorname{GL}(c,\mathbb{C})$ acts on
$P^{n}(c)$ according to

$(A_{i},C_{j},e)\mapsto\big{(}\phi_{2}A_{i}\phi_{1}^{-1},\phi_{1}C_{j}\phi_{2}^{-1},e\phi_{1}^{-1}\big{)}$

for $i=1,2$, $j=1,\dots,n$,
$(\phi_{1},\phi_{2})\in\operatorname{GL}(c,\mathbb{C})\times\operatorname{GL}(c,\mathbb{C})$.

The following result expresses the fact that the collections
$(A_{1},A_{2};C_{1},\dots,C_{n};e)$ satisfying conditions (P1) to (P3) are
ADHM data for the varieties $\operatorname{Hilb}^{c}(X_{n})$ (this is Theorem
3.1 in [2]).

######  Theorem 2.1.

$P^{n}(c)$ is a principal
$\operatorname{GL}(c,\mathbb{C})\times\operatorname{GL}(c,\mathbb{C})$-bundle
over $\operatorname{Hilb}^{c}(X_{n})$.

##  3 The main result

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$0$$1$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\scriptstyle{a_{1}}$$\scriptstyle{a_{2}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{c_{1}}$$\scriptstyle{c_{2}}$$\scriptstyle{c_{n}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{2}}$$\scriptstyle{i_{n-1}}$$\infty$

Figure 1: The quivers $Q_{n}$.

Now we turn to the purpose of this paper, namely, proving that the Hilbert
schemes of points of the varieties $X_{n}$ are isomorphic to moduli spaces of
representations of suitable quivers. For any $n\geq 2$ let $Q_{n}$ be the
framed quiver in Fig. 1, where $\infty$ is the framing vertex. Let $J_{n}$ be
the two sided ideal of $\mathbb{C}Q_{n}$ generated by the relations

$\begin{cases}a_{2}c_{q+1}-a_{1}c_{q}=0,\\\
c_{q+1}a_{2}-c_{q}a_{1}-i_{q}j=0\end{cases}\qquad\text{for}\quad
q=1,\dots,n-1.$ (3.1)

Our purpose is to describe the spaces of representations of the quiver $Q_{n}$
with relations $J_{n}$, i.e., the spaces of representations of the quotient
algebra $\Lambda_{n}=\mathbb{C}Q_{n}/J_{n}$.

We recall some basic definitions. Given
$\vec{v}=(v_{0},v_{1})\in\mathbb{N}^{2}$ and $w\in\mathbb{N}$, a
$(\vec{v},w)$-dimensional representation of $\Lambda_{n}$ is the datum of a
triple of $\mathbb{C}$-vector spaces $V_{0}$, $V_{1}$, $W$, with $\dim
V_{i}=v_{i}$, $\dim W=w$, and of an element
$(A_{1},A_{2};C_{1},\dots,C_{n};e;f_{1},\dots,f_{n-1})$ in

$\operatorname{Hom}_{\mathbb{C}}(V_{0},V_{1})^{\oplus
2}\oplus\operatorname{Hom}_{\mathbb{C}}(V_{1},V_{0})^{\oplus
n}\oplus\operatorname{Hom}_{\mathbb{C}}(V_{0},W)\oplus\operatorname{Hom}_{\mathbb{C}}(W,V_{0})^{\oplus
n-1}$

satisfying the relations determined by equations (3.1), namely

$\begin{cases}A_{2}C_{q+1}-A_{1}C_{q}=0,\\\
C_{q+1}A_{2}-C_{q}A_{1}-f_{q}e=0\end{cases}\qquad\text{for}\quad
q=1,\dots,n-1.$ (Q1)

The space $\operatorname{Rep}(\Lambda_{n},\vec{v},w)$ of all
$(\vec{v},w)$-dimensional representations of $\Lambda_{n}$ is an affine
variety, on which the group
$G_{\vec{v}}=\operatorname{GL}(v_{0},\mathbb{C})\times\operatorname{GL}(v_{1},\mathbb{C})$
acts by basis change. Indeed, we ignore the action of
$\operatorname{GL}(w,\mathbb{C})$ on the vector space $W$ attached to the
framing vertex. As usual, to get a well behaved quotient space one has to
perform a GIT construction by introducing a suitable notion of stability. This
was done by A. King [9] and, in a slightly different way, by A. Rudakov [17].
In the case of a quiver with a framing vertex, the following definition can be
shown to be equivalent to the King–Rudakov one [3, 5].

######  Definition 3.1.

Fix $\vartheta\in\mathbb{R}^{2}$. A $(\vec{v},w)$-dimensional representation
$(V_{0},V_{1},W)$ of $\Lambda_{n}$ is said to be _$\vartheta$ -semistable_ if,
for any subrepresentation $S=(S_{0},S_{1})$ $\subseteq(V_{0},V_{1})$, one has:

$\displaystyle\text{if $S_{0}\subseteq\ker e$, then $\vartheta\cdot(\dim
S_{0},\dim S_{1})\leq 0$};$ (3.2) $\displaystyle\text{if
$S_{0}\supseteq\operatorname{Im}f_{i}$ for $i=1,\dots,n-1$, then
$\vartheta\cdot(\dim S_{0},\dim S_{1})\leq\vartheta\cdot(v_{0},v_{1})$}.$
(3.3)

A $\vartheta$-semistable representation is _$\vartheta$ -stable_ if a strict
inequality holds in (3.2) whenever $S\neq 0$ and in (3.3) whenever
$S\neq(V_{0},V_{1})$.

Let $\operatorname{Rep}(\Lambda_{n},\vec{v},w)^{\rm ss}_{\vartheta}$ be the
subset of $\operatorname{Rep}(\Lambda_{n},\vec{v},w)$ consisting of
$\vartheta$-semistable representations. By [9, Proposition 5.2], the coarse
moduli space of $(\vec{v},w)$-dimensional $\vartheta$-semistable
representations of $\Lambda_{n}$ is the GIT quotient

$\mathcal{M}(\Lambda_{n},\vec{v},w,\vartheta)=\operatorname{Rep}(\Lambda_{n},\vec{v},w)^{\rm
ss}_{\vartheta}/\\!/G_{\vec{v}}.$

It can be proved that the open subset $\mathcal{M}^{\rm
s}(\Lambda_{n},\vec{v},w,\vartheta)\subset\mathcal{M}(\Lambda_{n},\vec{v},w,\vartheta)$
consisting of stable representations makes up a fine moduli space. Notice
that, for quivers without a framing, this holds only when the dimension vector
is primitive [9, Proposition 5.3], whilst this requirement is not necessary in
the case of framed quivers [5]. Theorem 4.5 of [1] states that the Hilbert
scheme of points $\operatorname{Hilb}^{c}(X_{n})$ can be embedded into
$\mathcal{M}(\Lambda_{n},\vec{v},w,\vartheta)$ for suitable choices of
$\vec{v}$, $w$, and $\vartheta$. Precisely, one has the following result:

######  Theorem 3.2.

For every $n\geq 2$ and $c\geq 1$ let

$\vec{v}_{c}=(c,c),\qquad w_{c}=1,\qquad\vartheta_{c}=(2c,1-2c),$

and let $\mathcal{H}(n,c)$ be the irreducible component of
$\mathcal{M}(\Lambda_{n},\vec{v}_{c},1,\vartheta_{c})$ given by the equations

$f_{1}=f_{2}=\cdots=f_{n-1}=0.$ (3.4)

Then $\operatorname{Hilb}^{c}(X_{n})\simeq\mathcal{H}(n,c)$.

Let $\mbox{pr}\colon\operatorname{Rep}(\Lambda_{n},\vec{v}_{c},1)^{\rm
ss}_{\vartheta_{c}}\to\mathcal{M}(\Lambda_{n},\vec{v}_{c},1,\vartheta_{c})$ be
the quotient map. The proof of Theorem 3.2 basically consists in proving that
the counterimage $\operatorname{pr}^{-1}(\mathcal{H}(n,c))=:Z_{n}(c)$
coincides with the total space of the principal fibration $P^{n}(c)$ we
introduced in Section 2. As it is quite involved and requires a few
intermediate Lemmas and Propositions, we refer the reader to [1] for further
details. Here we only note that the starting point is given by the stability
conditions in Definition 3.1.

######  Remark 3.3.

The set of $(\vec{v}_{c},w_{c})$-dimensional representations of $\Lambda_{n}$
which are semistable according to Definition 3.1 does not change if we let the
stability parameter vary inside the open cone

$\Gamma_{c}=\left\\{\vartheta=(\vartheta_{0},\vartheta_{1})\in\mathbb{R}^{2}\,|\,\vartheta_{0}>0,\,-\vartheta_{0}<\vartheta_{1}<-\frac{c-1}{c}\vartheta_{0}\right\\}.$

It can be shown that for any stability parameter $\bar{\vartheta}$ on the open
rays

$\displaystyle
R_{1}=\big{\\{}(\vartheta_{0},\vartheta_{1})\in\mathbb{R}^{2}\,|\,\vartheta_{0}>0,\,\vartheta_{0}+\vartheta_{1}=0\big{\\}},$
$\displaystyle
R_{2}=\big{\\{}(\vartheta_{0},\vartheta_{1})\in\mathbb{R}^{2}\,|\,\vartheta_{0}>0,\,(c-1)\vartheta_{0}+c\vartheta_{1}=0\big{\\}}$

there exist representations which are $\bar{\vartheta}$-semistable, but not
$\vartheta_{c}$-semistable. So, $\Gamma_{c}$ is a chamber in the space
$\mathbb{R}^{2}_{(\vartheta_{0},\vartheta_{1})}$ of stability parameters and
the closed rays $\overline{R_{1}}$, $\overline{R_{2}}$ are its walls.
Furthermore, inside $\Gamma_{c}$ semistability and stability are equivalent
(cf. [1, Lemma 4.7]): in particular, points in
$\mathcal{M}(\Lambda_{n},\vec{v}_{c},1,\vartheta_{c})$ can be thought of as
$G_{\vec{v}_{c}}$-orbits of representations in
$\operatorname{Rep}(\Lambda_{n},\vec{v}_{c},1)$.

A full description of the chamber/wall decomposition of the space
$\mathbb{R}^{2}_{(\vartheta_{0},\vartheta_{1})}$ will be the object of a
future work.

We wish to prove that the component $\mathcal{H}(n,c)$ of
$\mathcal{M}(\Lambda_{n},\vec{v}_{c},1,\vartheta_{c})$ introduced in Theorem
3.2 coincides with the whole moduli space
$\mathcal{M}(\Lambda_{n},\vec{v}_{c},1,\vartheta_{c})$ (this will be Theorem
3.8). Let us introduce the following notation

$\mathcal{R}(\Lambda_{n},c)=\operatorname{Rep}(\Lambda_{n},\vec{v}_{c},1);\qquad\mathcal{R}^{\rm
ss}(\Lambda_{n},c)=\operatorname{Rep}(\Lambda_{n},\vec{v}_{c},1)_{\vartheta_{c}}^{\rm
ss}.$

Given a representation
$(A_{1},A_{2};C_{1},\dots,C_{n};e;f_{1},\dots,f_{n-1})\in\mathcal{R}(\Lambda_{n},c)$,
we form the pencil $A_{1}+\lambda A_{2}$, with $\lambda\in\mathbb{C}$. We
recall that a pencil $A_{1}+\lambda A_{2}$ is regular if there is a point
$[\nu_{1},\nu_{2}]\in\mathbb{P}^{1}$ such that
$\det(\nu_{1}A_{1}+\nu_{2}A_{2})\neq 0$.

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$0$$1$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\scriptstyle{a_{1}}$$\scriptstyle{a_{2}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{b_{1}}$$\scriptstyle{b_{2}}$$\scriptstyle{b_{n-1}}$$\scriptstyle{d_{2}}$$\scriptstyle{d_{3}}$$\scriptstyle{d_{n}}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i_{1}}$$\scriptstyle{i_{2}}$$\scriptstyle{i_{n-1}}$$\infty$

Figure 2: The quivers $Q^{\prime}_{n}$ for $n\geq 3$.

To prove Theorem 3.8 it is convenient to introduce “augmented” framed quivers
defined as follows: let $Q^{\prime}_{2}=Q_{2}$, and, for every $n\geq 3$, let
$Q^{\prime}_{n}$ be the framed quiver in Fig. 2. Let $J^{\prime}_{2}=J_{2}$
and, for all $n\geq 3$, let $J^{\prime}_{n}$ be the two sided ideal of
$\mathbb{C}Q^{\prime}_{n}$ generated by the relations

$\begin{cases}a_{2}d_{q+1}-a_{1}b_{q}=0,\\\
d_{q+1}a_{2}-b_{q}a_{1}-i_{q}j=0\end{cases}\qquad\text{for}\quad
q=1,\dots,n-1.$ (3.5)

We set $\Lambda^{\prime}_{n}=\mathbb{C}Q^{\prime}_{n}/J^{\prime}_{n}$ for all
$n\geq 2$. Notice that $\Lambda^{\prime}_{2}=\Lambda_{2}$; for $n\geq 3$, the
algebra $\Lambda_{n}$ can be obtained by taking the quotient of
$\Lambda^{\prime}_{n}$ by a suitable ideal. Indeed, let $K_{n}$ be the two
sided ideal of $\Lambda^{\prime}_{n}$ generated by the relations

$\bar{b}_{q}=\bar{d}_{q}\qquad\text{for}\quad q=2,\dots,n-1,$ (3.6)

where $\bar{x}$ is the class in $\Lambda^{\prime}_{n}$ of the element
$x\in\mathbb{C}Q^{\prime}_{n}$. Let
$\tilde{p}_{n}\colon\mathbb{C}Q^{\prime}_{n}\longrightarrow\mathbb{C}Q_{n}$ be
the $\mathbb{C}$-algebra morphism determined by the assignments

$\tilde{p}_{n}(a_{q})=a_{q},\qquad\tilde{p}_{n}(b_{q})=c_{q},\qquad\tilde{p}_{n}(d_{q})=c_{q},\qquad\tilde{p}_{n}(j)=j,\qquad\tilde{p}_{n}(i_{q})=i_{q}.$
(3.7)

It is straightforward that $\tilde{p}_{n}$ is surjective and that its kernel
is the two sided ideal $L_{n}\subset\mathbb{C}Q^{\prime}_{n}$ generated by the
relations

$b_{q}=d_{q}\qquad\text{for}\quad q=2,\dots,n-1.$ (3.8)

It follows directly from equation (3.7) that $\tilde{p}_{n}$ maps the set of
generators of $J^{\prime}_{n}$ (see equation (3.5)) onto the set of generators
of $J_{n}$ (see equation (3.1)), so that

$\tilde{p}_{n}(J^{\prime}_{n})=J_{n}.$

Then it is not hard to check that $\tilde{p}_{n}$ induces a surjective
morphism $p_{n}\colon\Lambda^{\prime}_{n}\to\Lambda_{n}$, whose kernel, by
equations (3.6) and (3.8), is

$\ker p_{n}=L_{n}/(L_{n}\cap J^{\prime}_{n})=K_{n}.$

In conclusion, we have proved the following lemma.

######  Lemma 3.4.

There is an isomorphism of $\mathbb{C}$-algebras
$\Lambda^{\prime}_{n}/K_{n}\simeq\Lambda_{n}$.

One of the reasons to introduce the augmented quivers $Q^{\prime}_{n}$ is that
their path algebras carry an action of the group ${\rm SO}(2,\mathbb{C})$
which descends to the quotient algebra $\Lambda^{\prime}_{n}$. This action
will be instrumental in proving the regularity of the pencil $A_{1}+\lambda
A_{2}$.

Elements of ${\rm SO}(2,\mathbb{C})$ will be denoted by
$\nu=\left(\begin{smallmatrix}\nu_{1}&\nu_{2}\\\
-\nu_{2}&\nu_{1}\end{smallmatrix}\right)$. Given arrows

$(a_{1},a_{2};b_{1},\dots,b_{n-1};d_{2},\dots,d_{n};j;i_{1},\dots,i_{n-1})$

as above and $\nu\in{\rm SO}(2,\mathbb{C})$, we set

$\begin{pmatrix}a^{\prime}_{1}\\\
a^{\prime}_{2}\end{pmatrix}=\nu\begin{pmatrix}a_{1}\\\
a_{2}\end{pmatrix},\qquad\begin{pmatrix}b^{\prime}_{q}\\\
d^{\prime}_{q+1}\end{pmatrix}=\nu^{-1}\begin{pmatrix}b_{q}\\\
d_{q+1}\end{pmatrix}\qquad\text{for}\quad q=1,\dots,n-1.$

The assignment

$\displaystyle(a_{1},a_{2};b_{1},\dots,b_{n-1};d_{2},\dots,d_{n};j;i_{1},\dots,i_{n-1})$
$\displaystyle\qquad{}\longmapsto(a^{\prime}_{1},a^{\prime}_{2};b^{\prime}_{1},\dots,b^{\prime}_{n-1};d^{\prime}_{2},\dots,d^{\prime}_{n};j;i_{1},\dots,i_{n-1}),$

induces an action

$\widetilde{\Phi}_{n}\colon\ {\rm
SO}(2,\mathbb{C})\to\operatorname{Aut}_{\mathbb{C}\hbox{-alg}}(\mathbb{C}Q^{\prime}_{n}),$

which leaves invariant the generators of the ideal $J^{\prime}_{n}$, that is,

$\widetilde{\Phi}_{n}(\nu)\bigl{(}J^{\prime}_{n}\bigr{)}=J^{\prime}_{n}.$

So one has an induced action

$\Phi_{n}\colon\ {\rm
SO}(2,\mathbb{C})\to\operatorname{Aut}_{\mathbb{C}\hbox{-alg}}(\Lambda^{\prime}_{n}).$

We wish now to study the space
$\operatorname{Rep}(\Lambda^{\prime}_{n},\vec{v}_{c},1)=\mathcal{R}(\Lambda^{\prime}_{n},c)$
of $(c,c)$-dimensional framed representations of $\Lambda^{\prime}_{n}$ and
its open subset
$\operatorname{Rep}(\Lambda^{\prime}_{n},\vec{v}_{c},1)_{\vartheta_{c}}^{\rm
ss}=\mathcal{R}^{\rm ss}(\Lambda^{\prime}_{n},c)$ of
$\vartheta_{c}$-semistable representations (defined analogously to Definition
3.1). For $n=2$ there is nothing new, since
$\mathcal{R}(\Lambda^{\prime}_{2},c)=\mathcal{R}(\Lambda_{2},c)$ and
$\mathcal{R}^{\rm ss}(\Lambda^{\prime}_{2},c)=\mathcal{R}^{\rm
ss}(\Lambda_{2},c)$. For $n\geq 3$, $\mathcal{R}(\Lambda^{\prime}_{n},c)$ is
the affine subvariety of the vector space

$\operatorname{Hom}_{\mathbb{C}}(V_{0},V_{1})^{\oplus
2}\oplus\operatorname{Hom}_{\mathbb{C}}(V_{1},V_{0})^{\oplus
2n-2}\oplus\operatorname{Hom}_{\mathbb{C}}(V_{0},W)\oplus\operatorname{Hom}_{\mathbb{C}}(W,V_{0})^{\oplus
n-1}$

whose points
$(A_{1},A_{2};B_{1},\dots,B_{n-1};D_{2},\dots,D_{n},e;f_{1},\dots,f_{n-1})$
satisfy the relations determined by equations (3.5), namely,

$\begin{cases}A_{2}D_{q+1}=A_{1}B_{q},\\\
D_{q+1}A_{2}=B_{q}A_{1}+f_{q}e\end{cases}\qquad\text{for}\quad q=1,\dots,n-1.$
(Q$1^{\prime}$)

######  Lemma 3.5.

$\mathcal{R}^{\rm ss}(\Lambda^{\prime}_{n},c)$ is the open subset of
$\mathcal{R}(\Lambda^{\prime}_{n},c)$ determined by the conditions:

  1. (Q$2^{\prime}$)

for all subrepresentations $S=(S_{0},S_{1})$ such that $S_{0}\subseteq\ker e$,
one has $\dim S_{0}\leq\dim S_{1}$, and, if $\dim S_{0}=\dim S_{1}$, then
$S=0$;

  2. (Q$3^{\prime}$)

for all subrepresentations $S=(S_{0},S_{1})$ such that
$S_{0}\supseteq\operatorname{Im}f_{i}$, for $i=1,\dots,n-1$, one has $\dim
S_{0}\leq\dim S_{1}$.

###### Proof.

Given a subrepresentation $(S_{0},S_{1})$, we set $s_{i}=\dim S_{i}$, $i=0,1$.
By substituting the definitions of $\vec{v}_{c}$ and $\vartheta_{c}$ given in
Theorem 3.2 into equations (3.2) and (3.3) one gets

$\displaystyle\text{if $S_{0}\subseteq\ker e$, then $s_{0}\leq
s_{1}-\dfrac{s_{1}}{2c}$};$ (3.9) $\displaystyle\text{if
$S_{0}\supseteq\operatorname{Im}f_{i}$ for $i=1,\dots,n-1$, then $s_{0}\leq
s_{1}+\dfrac{1}{2}-\dfrac{s_{1}}{2c}$}.$ (3.10)

Whenever $s_{1}>0$, one has $0<\frac{s_{1}}{2c}<1$; hence, equation (3.9) is
equivalent to condition (Q$2^{\prime}$). On the other hand, as
$0\leq\frac{1}{2}-\frac{s_{1}}{2c}<\frac{1}{2}$, equation (3.10) is equivalent
to condition (Q$3^{\prime}$). ∎

######  Proposition 3.6.

For each point of $\mathcal{R}^{\rm ss}(\Lambda^{\prime}_{n},c)$ the
associated matrix pencil $A_{1}+\lambda A_{2}$ is regular.

###### Proof.

Let
$(A_{1},A_{2};B_{1},\dots,B_{n-1};D_{2},\dots,D_{n},e;f_{1},\dots,f_{n-1})$ be
a point of $\mathcal{R}^{\rm ss}(\Lambda^{\prime}_{n},c)$, and assume that
$A_{1}+\lambda A_{2}$ is singular. If $c=1$, then $A_{1}+\lambda A_{2}$ is
singular if and only if $A_{1}=A_{2}=0$. But this implies the
subrepresentation $(V_{0},0)$ does not satisfy condition (Q$3^{\prime}$).
Hence we can assume $c\geq 2$. The fact that the pencil $A_{1}+\lambda A_{2}$
is singular implies that there is a nontrivial element

$v(\lambda)=\sum_{\alpha=0}^{\varepsilon}(-\lambda)^{\alpha}v_{\alpha}\in
V_{0}\otimes_{\mathbb{C}}\mathbb{C}[\lambda]$ (3.11)

such that

$(A_{1}+\lambda A_{2})v(\lambda)=0\qquad\text{for all
$\lambda\in\mathbb{C}$.}$ (3.12)

By arguing as in the proof of [1, Lemma 4.11], one can show that the minimal
degree polynomial solution $v(\lambda)$ for the pencil $A_{1}+\lambda A_{2}$
has necessarily degree $\varepsilon>0$. Let us inductively define the vector
spaces $\\{U_{i}\\}_{i\in\mathbb{N}}$ as follows:

$\begin{cases}U_{0}=\langle v_{0},\dots,v_{\varepsilon}\rangle,&\\\
U_{2k+1}=A_{1}(U_{2k})+A_{2}(U_{2k})&\text{for}\quad k\geq 0,\\\ \displaystyle
U_{2k}=\sum\limits_{q=1}^{n-1}B_{q}(U_{2k-1})+\sum\limits_{q=2}^{n}D_{q}(U_{2k-1})&\text{for}\quad
k\geq 1.\end{cases}$

Note that each $U_{j}$, with $j$ even, is a subspace of $V_{0}$, while each
$U_{j}$, with $j$ odd, is a subspace of $V_{1}$. So, if we introduce the
subspaces

$S_{0}=\sum_{k=0}^{\infty}U_{2k}\subset V_{0},\qquad
S_{1}=\sum_{k=0}^{\infty}U_{2k+1}\subset V_{1},$

it follows that $(S_{0},S_{1})$ is a subrepresentation of $(V_{0},V_{1})$. We
will show that this subrepresentation fails to satisfy either condition
(Q$2^{\prime}$) or condition (Q$3^{\prime}$) of Lemma 3.5, so that one gets a
contradiction.

By substituting equation (3.11) into equation (3.12) one finds out that

$\begin{cases}A_{1}v_{0}=0,\\\
A_{1}v_{\alpha}=A_{2}v_{\alpha-1},\qquad\alpha=1,\dots,\varepsilon,\\\
A_{2}v_{\varepsilon}=0,\end{cases}$ (3.13)

so that

$U_{1}=\langle A_{1}v_{1},\dots,A_{1}v_{\varepsilon}\rangle=A_{1}(U_{0}).$
(3.14)

There are two possible cases, either i) $U_{0}\subseteq\ker e$, or ii)
$U_{0}\not\subseteq\ker e$.

i) If we suppose that $U_{0}\subseteq\ker e$, equation (3.13) and condition
(Q$1^{\prime}$) imply that

$U_{2}=\sum_{q=2}^{n}D_{q}A_{1}(U_{0}).$

By letting $w_{q,\alpha}=D_{q}A_{1}v_{\alpha}$, $\alpha=1,\dots,\varepsilon$,
for each $q=2,\dots,n$ we obtain an element

$(w_{q,1},\dots,w_{q,\varepsilon})\in U_{2}^{\oplus\varepsilon}$

such that
$\sum\limits_{\alpha=0}^{\varepsilon-1}(-\lambda)^{\alpha}w_{q,\alpha+1}$ is a
polynomial solution for the pencil $A_{1}+\lambda A_{2}$ of degree
$\varepsilon-1$. Since we have supposed $\varepsilon$ to be minimal, one has
$(w_{q,1},\dots,w_{q,\varepsilon})=0$. From that it is easy to deduce that
$U_{2}=0$ and that $(S_{0},S_{1})=(U_{0},U_{1})$. So, since $\ker A_{1}\cap
U_{0}\neq 0$ by equation (3.13), then equation (3.14) entails that
$(S_{0},S_{1})$ is a subrepresentation violating condition (Q$2^{\prime}$).

ii) Suppose now that $U_{0}$ is not contained in $\ker e$. So, there is at
least one $\gamma\in\\{0,\dots,\varepsilon\\}$ such that $e(v_{\gamma})\neq
0$. Condition (Q$1^{\prime}$) implies that

$\operatorname{Im}f_{q}=\langle f_{q}e(v_{\gamma})\rangle\subseteq
U_{2}\qquad\text{for all}\quad q=1,\dots,n-1.$ (3.15)

To simplify computations, we may assume $\gamma=0$ and $e(v_{0})=1$. Actually,
one checks that the ${\rm SO}(2,\mathbb{C})$ action on $\Lambda^{\prime}_{n}$
induces an action on $\mathcal{R}(\Lambda^{\prime}_{n},c)$, which commutes
with the $G_{\vec{v}_{c}}$ action defined on the same space, and therefore it
restricts to an ${\rm SO}(2,\mathbb{C})$ action on $\mathcal{R}^{\rm
ss}(\Lambda^{\prime}_{n},c)$. Moreover, this action preserves the regularity
of the matrix pencil $A_{1}+\lambda A_{2}$. An element
$\nu=\left(\begin{smallmatrix}\nu_{1}&\nu_{2}\\\
-\nu_{2}&\nu_{1}\end{smallmatrix}\right)\in{\rm SO}(2,\mathbb{C})$ produces a
change of basis

$(v_{0},\dots,v_{\varepsilon})\mapsto\nu\cdot(v_{0},\dots,v_{\varepsilon})=(v^{\prime}_{0},\dots,v^{\prime}_{\varepsilon}),$

so that

$e(v^{\prime}_{0})=\sum_{\alpha=0}^{\varepsilon}(-\nu_{2})^{\alpha}\nu_{1}^{\varepsilon-\alpha}e(v_{\alpha}).$

Since $(e(v_{0}),\dots,e(v_{\varepsilon}))\neq(0,\dots,0)$, there is
$\nu\in{\rm SO}(2,\mathbb{C})$ so that $e(v^{\prime}_{0})\neq 0$. Moreover,
$e(v^{\prime}_{0})$ can be assumed to be $1$.

Next, by using condition (Q$1^{\prime}$) and equation (3.13), along with the
identity $A_{2}(\operatorname{Im}f_{q})=\langle A_{2}f_{q}(1)\rangle$
$=\langle A_{2}f_{q}(e(v_{0}))\rangle$, it is not hard to show that

$A_{2}(\operatorname{Im}f_{q})\subseteq A_{1}(U_{2})\qquad\text{for all}\quad
q=1,\dots,n-1.$ (3.16)

Now we show that

$U_{2k+1}\subseteq\sum_{l=1}^{k}A_{1}(U_{2l})$ (3.17)

for all $k\geq 1$. Assume $k=1$. By using equations (3.13), (Q$1^{\prime}$)
and condition $e(v_{0})=1$, one gets

$f_{q}e(v_{\alpha})=e(v_{\alpha})D_{q+1}A_{1}v_{1}$

for $q=1,\dots,n-1$. Hence, by using equations (3.13) and (Q$1^{\prime}$)
again one shows that

$U_{2}=\sum_{q=2}^{n}D_{q}A_{1}(U_{0}).$

Then $U_{3}$ is spanned by the sets of vectors

$\\{A_{1}D_{q}A_{1}v_{\alpha}\\}_{\begin{subarray}{l}q=2,\dots,n\\\
\alpha=1,\dots,\varepsilon\end{subarray}}\subseteq
A_{1}(U_{2}),\qquad\\{A_{2}D_{q}A_{1}v_{\alpha}\\}_{\begin{subarray}{l}q=2,\dots,n\\\
\alpha=1,\dots,\varepsilon\end{subarray}}\subseteq A_{2}(U_{2})$

and it follows directly from equations (Q$1^{\prime}$) that
$A_{2}D_{q}A_{1}v_{\alpha}\in A_{1}(U_{2})$, for $q=2,\dots,n$. So
$U_{3}\subseteq A_{1}(U_{2})$.

Let us now suppose that equation (3.17) holds true for $1\leq k\leq m$, with
$m\geq 1$. This means that $U_{2m+1}$ is spanned by vectors of the form
$A_{1}w$ with $w\in U_{2l}$, $l=1,\dots,m$. By noticing that $U_{2m+2}$ is
spanned by vectors of the form $B_{p}A_{1}w$ and $D_{q}A_{1}w^{\prime}$, with
$w\in U_{2l}$ and $w^{\prime}\in U_{2l^{\prime}}$ for
$l,l^{\prime}=1,\dots,m$, and by using equation (Q$1^{\prime}$) and the
inductive hypothesis one finds out that

$U_{2m+2}\subseteq\sum_{l=1}^{m}\sum_{q=2}^{n}D_{q}A_{1}(U_{2l})+\sum_{q=1}^{n-1}\operatorname{Im}f_{q}.$

From this it follows

$\displaystyle A_{1}(U_{2m+2})+A_{2}(U_{2m+2})$ $\displaystyle=U_{2m+3}$
$\displaystyle\subseteq
A_{1}(U_{2(m+1)})+\sum_{l=1}^{m}\sum_{q=2}^{n}A_{2}D_{q}A_{1}(U_{2l})+\sum_{q=1}^{n-1}A_{2}(\operatorname{Im}f_{q})$
$\displaystyle\subseteq
A_{1}(U_{2(m+1)})+A_{1}(U_{2})+\sum_{l=1}^{m}\sum_{q=2}^{n}A_{2}D_{q}A_{1}(U_{2l}),$
(3.18)

where in the last step equation (3.16) has been used. For $q=2,\dots,n$,
equation (Q$1^{\prime}$) implies that

$A_{2}D_{q}A_{1}(U_{2l})\subseteq A_{1}(U_{2(l+1)}).$

Thus, from equation (3.18) we may conclude that

$U_{2(m+1)+1}\subseteq\sum_{l=1}^{m+1}A_{1}(U_{2l}),$

so that the inclusion (3.17) is proved.

This and equation (3.14) imply that

$S_{1}=\sum_{k=0}^{\infty}U_{2k+1}\subseteq
A_{1}(U_{0})+\sum_{k=1}^{\infty}\sum_{l=1}^{k}A_{1}(U_{2l})=\sum_{k=0}^{\infty}A_{1}(U_{2k})=A_{1}\left(\sum_{k=0}^{\infty}U_{2k}\right)=A_{1}(S_{0}).$

But $A_{1}(S_{0})\subseteq S_{1}$, so that

$S_{1}=A_{1}(S_{0}).$

By equation (3.13), one has $\ker A_{1}\cap S_{0}\neq 0$, and therefore $\dim
S_{1}<\dim S_{0}$. Finally, equation (3.15) implies that the subrepresentation
$(S_{0},S_{1})$ violates condition (Q$3^{\prime}$). ∎

When $n\geq 3$, there is a map
$\mathcal{R}(\Lambda_{n},c)\longrightarrow\mathcal{R}(\Lambda^{\prime}_{n},c)$
given by

$(A_{1},A_{2};C_{1},\dots,C_{n};e;f_{1},\dots,f_{n-1})\mapsto(A_{1},A_{2};C_{1},\dots,C_{n-1};C_{2},\dots,C_{n},e;f_{1},\dots,f_{n-1}).$

This map provides a $G_{\vec{v}_{c}}$-equivariant isomorphism of
$\mathcal{R}(\Lambda_{n},c)$ onto the subvariety of
$\mathcal{R}(\Lambda^{\prime}_{n},c)$ cut by the equations

$B_{q}=D_{q}\qquad\text{for}\quad q=2,\dots,n-1$

(cf. equations (3.6)). Through this isomorphism $\mathcal{R}(\Lambda_{n},c)$
may be regarded as a closed subvariety of
$\mathcal{R}(\Lambda^{\prime}_{n},c)$.

######  Lemma 3.7.

When $n\geq 3$, one has that

$\mathcal{R}^{\rm ss}(\Lambda_{n},c)=\mathcal{R}^{\rm
ss}(\Lambda^{\prime}_{n},c)\cap\mathcal{R}(\Lambda_{n},c).$

###### Proof.

Semistability is a numerical condition which is to be checked on the set of
all submodules of a given representation. Hence, it is enough to show that for
any left $\Lambda_{n}$-module $M$, an abelian subgroup $N\subset M$ is a left
$\Lambda_{n}$-submodule if and only if it is a left
$\Lambda^{\prime}_{n}$-submodule (notice that $M$ has also a natural structure
of left $\Lambda^{\prime}_{n}$-module, induced by restriction of scalars; cf.
Lemma 3.4). However, precisely because the algebra $\Lambda_{n}$ is a quotient
of $\Lambda^{\prime}_{n}$, the category $\Lambda_{n}$-mod is a full
subcategory of $\Lambda^{\prime}_{n}$-mod, and this implies in particular that
the set of all subobjects of a given $\Lambda_{n}$-module is the same in the
two categories. ∎

######  Theorem 3.8.

The component $\mathcal{H}(n,c)$ of the moduli space
$\mathcal{M}(\Lambda_{n},\vec{v}_{c},1,\vartheta_{c})$ defined by equations
(3.4) coincides with the whole of
$\mathcal{M}(\Lambda_{n},\vec{v}_{c},1,\vartheta_{c})$.

###### Proof.

For each representation
$(A_{1},A_{2};C_{1},\dots,C_{n};e;f_{1},\dots,f_{n-1})\\!\in\\!\mathcal{R}^{\rm
ss}(\Lambda_{n},c)$, equations (3.4) hold if and only if the pencil
$A_{1}+\lambda A_{2}$ is regular (condition (P2) in Section 2 and in [1]):

  * •

in the proof of Proposition 4.9 of [1] it has been shown that condition (P2)
holds in $Z_{n}(c)=\operatorname{pr}^{-1}(\mathcal{H}(n,c))$; i.e., equations
(3.4) imply the regularity of the pencil;

  * •

further on, in the proof of Theorem 4.5 of [1] it has been shown that
$Z_{n}(c)$ actually coincides with the open subset of $\mathcal{R}^{\rm
ss}(\Lambda_{n},c)$ (denoted $\mathcal{R}_{n}(c)$ in [1]) where condition (P2)
is satisfied; i.e., the regularity of the pencil implies equations (3.4).

So, the orbit of a $\vartheta_{c}$-semistable representation

$(A_{1},A_{2};C_{1},\dots,C_{n};e;f_{1},\dots,f_{n-1})\in\mathcal{R}^{\rm
ss}(\Lambda_{n},c)$

lies in $\mathcal{H}(n,c)$ if and only if the pencil $A_{1}+\lambda A_{2}$ is
regular. Then the conclusion follows from Proposition 3.6 and Lemma 3.7. ∎

##  4 A remark involving the 2-Kronecker quiver

We want to rephrase Proposition 3.6 is a slightly different way which involves
the Kronecker quiver with two arrows $Q_{K}$

$0$$1$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{a_{1}}$$\scriptstyle{a_{2}}$$\textstyle{\bullet}$

The new claim, Proposition 4.2, may be regarded as a statement in relative
Geometric Invariant Theory.

The vector space of $\vec{v}_{c}=(c,c)$-dimensional representations of $Q_{K}$
is the space
$\operatorname{Rep}(Q_{K},\vec{v}_{c})=\operatorname{Hom}_{\mathbb{C}}(V_{0},V_{1})^{\oplus
2}$. Since Definition 3.1 only applies to framed quivers, we need a slightly
different notion of semistability. So we recall from [9, 17] that, given
$\vartheta\in\mathbb{R}^{2}$, a $\vec{v}_{c}$-dimensional representation of
$Q_{K}$ is said to be _$\vartheta$ -semistable_ if, for any proper nontrivial
subrepresentation supported by $(S_{0},S_{1})$ $\subseteq(V_{0},V_{1})$, one
has

$\frac{\vartheta\cdot(\dim S_{0},\dim S_{1})}{\dim S_{0}+\dim
S_{1}}\leq\frac{\vartheta\cdot\vec{v}_{c}}{2c}.$ (4.1)

A $\vartheta$-semistable representation is _$\vartheta$ -stable_ if strict
inequality holds in (4.1).

As in Section 3, we set $\vartheta_{c}=(2c,1-2c)$.

######  Lemma 4.1.

A point $(A_{1},A_{2})\in\operatorname{Rep}(Q_{K},\vec{v}_{c})$ is
$\vartheta_{c}$-semistable if and only if the matrix pencil $A_{1}+\lambda
A_{2}$ is regular.

###### Proof.

Let $(A_{1},A_{2})$ be a representation of $Q_{K}$ supported by the pair of
vector spaces $(V_{0},V_{1})$, and consider a proper subrepresentation
supported by $(S_{0},S_{1})$. If the stability parameter is
$\vartheta_{c}=(2c,1-2c)$, the inequality (4.1) is equivalent to

$\frac{2c\dim S_{0}+(1-2c)\dim S_{1}}{\dim S_{0}+\dim S_{1}}\leq\frac{1}{2},$

which is in turn equivalent to

$\dim S_{0}\leq\dim S_{1}.$ (4.2)

It is not hard to show that (4.2) implies

$\dim(A_{1}(S)+A_{2}(S))\geq\dim S\quad\text{for all vector subspaces
$S\subseteq V_{0}$.}$ (4.3)

Conversely, if condition (4.3) is satisfied, then, given any subrepresentation
supported by $S=(S_{0},S_{1})$, one has

$\dim S_{1}\geq\dim(A_{1}(S_{0})+A_{2}(S_{0}))\geq\dim S_{0}.$

Finally, by [1, Lemma 4.10] condition (4.3) is equivalent to the fact that the
matrix pencil $A_{1}+\lambda A_{2}$ is regular. ∎

Recall that $\mathcal{R}(\Lambda_{n},c)$ is the affine subvariety of

$\operatorname{Rep}(Q_{n},\vec{v}_{c},1)=\operatorname{Hom}_{\mathbb{C}}(V_{0},V_{1})^{\oplus
2}\oplus\operatorname{Hom}_{\mathbb{C}}(V_{1},V_{0})^{\oplus
n}\oplus\operatorname{Hom}_{\mathbb{C}}(V_{0},W)\oplus\operatorname{Hom}_{\mathbb{C}}(W,V_{0})^{\oplus
n-1}$

defined by equations (Q1). Let us denote by
$\pi_{n}\colon\mathcal{R}(\Lambda_{n},c)\to\operatorname{Rep}(Q_{K},\vec{v}_{c})$
the restriction of the natural projection
$\operatorname{Rep}(Q_{n},\vec{v}_{c},1)\to\operatorname{Hom}_{\mathbb{C}}(V_{0},V_{1})^{\oplus
2}=\operatorname{Rep}(Q_{K},\vec{v}_{c})$.

As a straightforward consequence of Lemma 4.1, Proposition 3.6 may be
rephrased in the following terms.

######  Proposition 4.2.

Each $(\vec{v}_{c},1)$-dimensional $\vartheta_{c}$-semistable representation
of $\Lambda_{n}$ is mapped by $\pi_{n}$ to a $\vec{v}_{c}$-dimensional
$\vartheta_{c}$-semistable representation of $Q_{K}$:

$\pi_{n}\bigl{(}\mathcal{R}^{\rm
ss}(\Lambda_{n},c)\bigr{)}\subseteq\operatorname{Rep}(Q_{K},\vec{v}_{c})_{\vartheta_{c}}^{\rm
ss}.$

### Acknowledgements

This research was partly supported by INdAM-GNSAGA and PRIN “Geometria delle
varietà algebriche”. V.L. was supported by the FAPESP Postdoctoral Grants No.
2015/07766-4 (Bolsa no País) and 2017/22052-9 (Bolsa Estágio de Pesquisa no
Exterior). This research was partly carried out while he was visiting the
Department of Mathematics of the Northeastern University (Boston), and he
wishes to thank that institution for hospitality. We thank the referees for
their valuable suggestions which allowed us to substantially improve the
presentation.

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# Giant asymmetric proximity-induced spin-orbit coupling in twisted
graphene/SnTe heterostructure

Marko Milivojević  Institute of Informatics, Slovak Academy of Sciences, 84507
Bratislava, Slovakia  Institute for Theoretical Physics, University of
Regensburg, 93053 Regensburg, Germany  Faculty of Physics, University of
Belgrade, 11001 Belgrade, Serbia  Martin Gmitra  Institute of Physics, Pavol
Jozef Šafárik University in Košice, 04001 Košice, Slovakia  Institute of
Experimental Physics, Slovak Academy of Sciences, 04001 Košice, Slovakia
Marcin Kurpas  Institute of Physics, University of Silesia in Katowice, 41‑500
Chorzów, Poland  Ivan Štich  Institute of Informatics, Slovak Academy of
Sciences, 84507 Bratislava, Slovakia  Department of Natural Sciences,
University of Saints Cyril and Methodius, 917 01 Trnava, Slovakia  Jaroslav
Fabian  Institute for Theoretical Physics, University of Regensburg, 93053
Regensburg, Germany

###### Abstract

We analyze the spin-orbit coupling effects in a three-degree twisted bilayer
heterostructure made of graphene and an in-plane ferroelectric SnTe, with the
goal of transferring the spin-orbit coupling from SnTe to graphene, via the
proximity effect. Our results indicate that the point-symmetry breaking due to
the incompatible mutual symmetry of the twisted monolayers and a strong
hybridization has a massive impact on the spin splitting in graphene close to
the Dirac point, with the spin splitting values greater than 20 meV. The band
structure and spin expectation values of graphene close to the Dirac point can
be described using a symmetry-free model, triggering different types of
interaction with respect to the threefold symmetric graphene/transition-metal
dichalcogenide heterostructure. We show that the strong hybridization of the
Dirac cone’s right movers with the SnTe band gives rise to a large asymmetric
spin splitting in the momentum space. Furthermore, we discover that the
ferroelectricity-induced Rashba spin-orbit coupling in graphene is the
dominant contribution to the overall Rashba field, with the effective in-plane
electric field that is almost aligned with the (in-plane) ferroelectricity
direction of the SnTe monolayer. We also predict an anisotropy of the in-plane
spin relaxation rates. Our results demonstrate that the group-IV
monochalcogenides MX (M=Sn, Ge; X=S, Se, Te) are a viable alternative to
transition-metal dichalcogenides for inducing strong spin-orbit coupling in
graphene.

††preprint: AIP/123-QED

##  I Introduction

The field of spintronics ZFS04 ; FME+07  has been providing fundamental
science and technology with new functionalities derived from the electron spin
as the fundamental degree of freedom. Two important interactions enable the
efficient transfer of the information stored in the electron spin to
electrically or magnetically readable one: spin-orbit and exchange coupling.

Its long spin-relaxation times SKX+16 ; DFP+16  and high electronic mobility
BSJ+08 ; DSB+08  make graphene an excellent material for spin transport.
However, to enable the manipulation of the spins in graphene and therefore
truly unlock its full potential as a platform for spintronics, enhancing and
controlling graphene’s spin-orbit coupling (SOC) and inducing magnetism is
necessary. This can be achieved via the van der Waals proximity effects GG13 ;
ZMS+19 ; HMM+20 ; SFK+21 ; HZL+22 ; SBB+22 ; ZZN+22 ; ZRM+23 , which can
successfully manipulate electronic, optical, magnetic, and spin properties of
the target material.

It has already been demonstrated that graphene can be successfully
proximitized so that it can become magnetic HKG+14 . Similarly, strong SOC
materials, such as transition metal dichalcogenides (TMDC) ZHU ; KZD+13 ;
SYZ+13 ; KGR13 ; ABX+14 ; KBG+15 , can induce giant SOC into another two-
dimensional material, without destroying its essential electronic properties.
In graphene/TMDC heterostructures, giant spin lifetime anisotropy is observed
GAK+17 ; CGF+17 ; BST+18 ; ZCR+21 , defined as the ratio of lifetimes for
spins pointing out of the graphene plane to those pointing in the plane.
However, the preserved three-fold symmetry of graphene/TMDC heterostructures
yields in-plane isotropy of spin relaxation rates. In contrast,
heterostructures with no common symmetry are expected to induce fully three-
dimensional spin relaxation anisotropy. This is a strong motivation to explore
beyond the well-investigated graphene/TMDC stacks GF15 ; GF17 ; NZG+21 ;
LSK+22 ; PDR+22 ; SMK+23  in the search for novel materials IHS+19 ; TF23
that could be used as an alternative to induce both strong spin-orbit coupling
and exotic spin textures in graphene. Particularly interesting are mixed-
lattice heterostructures with incompatible symmetries, expected to give rise
to novel spin textures.

Figure 1: Atomic structural model of the graphene/SnTe heterostructure: (a)
side view; (b) top view, with the arrow indicating electrical polarization.
(c) the first Brillouin zone of graphene with high symmetry points and
$k$-path used for band structure calculations indicated by the blue line.

In this paper, we investigate an example of a mixed-lattice heterostructure:
hexagonal graphene and rectangular monolayer SeTe, which is a representative
of group-IV monochalcogenides GC15 ; FLL+15 ; GCN15 ; PVL19 . Monolayer SnTe
has already been recognized as a material that can advance spintronics
applications SCG+20 ; WGP+20 . It has strong SOC AI19  and in-plane
ferroelectricity AMS+19 ; JOS+21  that is locked to the out-of-plane
persistent spin texture. We consider a small twist angle of three degrees
between the bilayer constituents, giving rise to the symmetry-free environment
of the proximitized graphene. The performed DFT calculations and effective
Hamiltonian modeling with no symmetry restrictions serve as proof that a Dirac
cone nature of graphene is not destroyed due to the strong hybridization of
graphene and SnTe bands, responsible for the observed giant asymmetric
($\propto 20\,$meV) spin splitting of graphene bands. Furthermore, analysis of
the effective model reveals that the ferroelectricity-induced Rashba SOC is
much stronger than the usual in-plane Rashba field, present due to the broken
inversion symmetry. Finally, we predict that the in-plane spins of the
graphene relax anisotropically due to the symmetry-free environment set up by
the SnTe monolayer.

This paper is organized as follows. In Section II we describe the
graphene/SnTe heterostructure, alongside the necessary details regarding the
density-functional theory (DFT) calculation of the corresponding band
structure. In Section III, we analyze the obtained band structure with and
without SOC and describe the Hamiltonian model that can satisfactorily explain
the DFT data. Furthermore, the impact of the obtained spin texture on spin
relaxation is given in Section III.3, while the short conclusions are given in
Section IV.

##  II First-principles calculations details

An atomic model of the graphene/SnTe heterostructure investigated in the paper
is given in FIG. 1. The lattice parameter of graphene is equal to
$a_{0}=2.46\text{\AA}$ HLS65 , while for the SnTe monolayer the values are
$a=4.58\text{\AA}$ and $b=4.56\text{\AA}$, in line with the literature AI19
(see Appendix A for more details).

The commensurate heterostructure was obtained using the CellMatch code L15  by
straining the SnTe monolayer for $3.32$%, containing 8 C, 2 Sn, and 2 Te
atoms. The supercell lattice vectors ${\bm{b}}_{1}$ and ${\bm{b}}_{2}$ can be
defined in terms of the graphene ones, ${\bm{a}}_{1}=a_{0}{\bm{e}}_{x}$ and
${\bm{a}}_{2}=a_{0}(-1/2{\bm{e}}_{x}+\sqrt{3}/2{\bm{e}}_{y})$, as
${\bm{b}}_{1}=-3{\bm{a}}_{1}+10{\bm{a}}_{2}$ and
${\bm{b}}_{2}=-{\bm{a}}_{1}+2{\bm{a}}_{2}$. The relative twist angle of the
armchair direction of the SnTe monolayer with respect to graphene is three
degrees. The nonzero twist angle between the bilayer constituents breaks all
the common symmetries between the graphene and SnTe monolayers and triggers
the symmetry-free environment for graphene electrons. The applied strain could
have a quantitative impact on the obtained results; nevertheless, our goal is
to investigate to which level mixed-lattice heterostructures with no common
symmetry could give rise to new physical phenomena.

We perform the electronic structure calculation of the twisted graphene/SnTe
heterostructure by means of plane wave DFT suite Quantum ESPRESSO (QE) QE1 ;
QE2 , using the revised Perdew-Burke-Ernzerhof generalized gradient
approximation for the exchange-correlation functional PBEsol . The kinetic
energy cut-offs for the wave function and charge density were chosen to be 47
Ry and 325 Ry, respectively, assuming small Methfessel–Paxton energy level
smearing MP89  of 1 mRy. Additionally, $18\times 80$ $k$-points mesh for
Brillouin zone integration was used. and a vacuum of 18 Å in the $z$-direction
was assumed. The interaction between the monolayers was modeled using the
semiempirical Grimme-D2 van der Waals correction G06 ; BCF+08 . The positions
of atoms were relaxed using the quasi-Newton scheme using scalar-relativistic
pseudopotentials, keeping the force and energy convergence thresholds for
ionic minimization to $7\times 10^{-4}$ Ry/bohr and $10^{-7}$ Ry,
respectively. The average distance between the graphene plane and the closest
SnTe planes (in $z$-direction) of the relaxed structure is equal to 3.35 Å.
For the self-consistent calculation including the spin-orbit coupling, the
fully relativistic pseudopotential was used, keeping the same convergence
threshold and $k$-mesh as in the case of the orbital relaxation. Also, dipole
correction B99  was applied to properly determine Dirac point energy offset
due to dipole electric field effects.

We note that for convenience reasons the illustration of the band structure
unfolded to the Brillouin zone of graphene is done with the DFT Vienna ab-
initio simulation package VASP 6.2 KF96 ; KF99 , using as the input the
relaxed structure from the QE code.

##  III Band structure analysis, effective model, and spin relaxation

In FIG. 2 we present the band structure of the graphene/SnTe heterostructure
unfolded to the $\Gamma$KM path of the Brillouin zone (see FIG. 1c), with
spectral weights of graphene and SnTe presented in a) and b), respectively.
First, we note that the Dirac cone is electron-doped and is slightly shifted
with respect to the K point. The K-point shift is an expected consequence of
the broken three-fold rotational symmetry SSL16 . Furthermore, the spectral
weights of heterostructure atomic function reveal strong hybridization of the
right movers of the Dirac cone with the ferroelectric SnTe bands, suggesting a
very strong transfer of spin-orbit coupling from SnTe to graphene via the
proximity effect. This assumption is consistent with DFT calculation, for
which we derive an effective model and extract the relevant parameters that
match the calculated band and spin structure.

Figure 2: Calculated band structure for graphene/SnTe heterostructure unfolded
to graphene Brillouin zone path in the vicinity of the K point. The $k$-path
is indicated in FIG. 1 and corresponds to 25% of corresponding full widths.
The spectral weights of unfolded bands calculated for (a) graphene and (b)
SnTe are presented. The band structure calculation is performed with VASP.

###  III.1 Effective model Hamiltonian

Figure 3: Comparison between the DFT data and the fit obtained using the model
Hamiltonian (1) (parameters can be found in Table 1) in the vicinity of the
$K$ point, $K=\frac{4\pi}{3a}(1,0)$, along different directions in the
Brillouin zone: in a) the band structure of the Dirac bands (labeled ad C1,
C2, V1, and V2) along the $\Gamma$KM path in the $x$-direction (see the inset)
is given; the corresponding spin expectation values (in $\hbar$ units) are
given in FIGS. b)-d). Furthermore, in FIGS. e)-h) the k-path is chosen to be
in the y-direction with the center at the K point.

To model the proximity-induced spin-orbit effects in graphene, we derive an
effective Hamiltonian for the Dirac cone of graphene based on the symmetry of
the studied heterostructure KIF17 ; LK19 ; DRK+19 . As there is no restriction
due to symmetry, we employ a general Hamiltonian form, spanning the graphene
sublattice and spin spaces, restricted to terms linear in momenta (relative to
K) ${\bf k}$.

Specifically, we model the Hamiltonian around the Dirac point as

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{\rm orb}+H_{\rm I}+H_{\rm
R}+H_{\rm HI},$ (1)

where $H_{\rm orb}$ represents the orbital Hamiltonian, $H_{\rm I}$ is the
intrinsic spin-orbit coupling Hamiltonian, $H_{\rm R}$ is the Rashba spin-
orbit coupling term always present when there is a structural asymmetry in the
system. Finally, the hybridization-induced SOC term $H_{\rm HI}$ accounts for
the SOC acquired due to the hybridization of the right movers of the Dirac
cone with the SnTe bands.

In more detail, the orbital Hamiltonian can be described as

$\displaystyle H_{\rm orb}$ $\displaystyle=$ $\displaystyle\mu+\hbar v_{\rm
F}(\tau k_{x}\sigma_{x}+k_{y}\sigma_{y})+\Delta\sigma_{z}+\Delta_{\rm
s}\sigma_{x},$ (2)

where $\mu$ represents the chemical potential, $v_{\rm F}$ is the Fermi
velocity, $\Delta$ represents the staggered potential, while $\Delta_{\rm
s}\sigma_{x}$ is the term responsible for the shift of the Dirac cone in the
x-direction due to the broken out-of-plane rotational symmetry of the
graphene/SnTe heterostructure, giving rise to the uniaxial deformation of the
graphene lattice SSL16 . Finally, $\tau=1$ (-1) corresponds to the K (K’)
point and $k_{x}$ and $k_{y}$ are the Cartesian components of the electron
wave vector measured from K (K’).

Next, we assume the presence of the sublattice-dependent intrinsic SOC which
can be written as GF15

$\displaystyle H_{\rm I}$ $\displaystyle=$ $\displaystyle\lambda_{\rm
I}^{A}[(\sigma_{z}+\sigma_{0})/2]\tau S_{z}+\lambda_{\rm
I}^{B}[(\sigma_{z}-\sigma_{0})/2]\tau S_{z},$ (3)

where $\lambda_{\rm I}^{A/B}$ represent the strength of the intrinsic SOC
parameters on the sublattice $A/B$, while $S_{z}$ is the Pauli z-matrix acting
in the spin-subspace.

Next, we discuss the effects of momentum-dependent Rashba spin-orbit coupling.
In the general case, the gradient of the crystal potential $\nabla V({\bf r})$
can have all three components. The z-component gives rise to the usual Rashba
spin-orbit field $\propto k_{x}S_{y}-k_{y}S_{x}$, while x- and y- components
of the crystal potential gradient can generate the novel terms $k_{y}S_{z}$
and $k_{x}S_{z}$, respectively, which can be interpreted as the x- and y-
component of the effective electric field. The appearance of the novel Rashba
spin-orbit terms is to be expected since SnTe is a strong in-plane
ferroelectric material (in the armchair AMS+19 ; JOS+21  direction of the SnTe
monolayer), suggesting a strong gradient of the crystal potential in the
polarization direction. The three-degree twist angle between the polarization
(armchair) direction of the SnTe monolayer and the x-axis of the
heterostructure’s coordinate frame indicates that both in-plane components of
the crystal potential gradient should be present. Thus, we can model $H_{\rm
R}$ as

$\displaystyle H_{\rm R}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{3}\tau^{\delta_{i,2}}\sigma_{i}\Big{[}\beta_{1}^{i}k_{x}S_{z}+\beta_{2}^{i}k_{y}S_{z}+\lambda^{i}(k_{x}S_{y}-k_{y}S_{x})\Big{]}$

additionally assuming the general pseudospin dependence of the Rashba SOC
Hamiltonian, justified due to the absence of common symmetry between the
constituents in the studied heterostructure. Finally, let us mention that
interaction $\propto k_{x}\sigma_{z}$ and $k_{y}\sigma_{z}$ (in-plane Rashba
field terms) give rise to the out-of-plane spin texture. On the other hand,
the terms $\propto k_{x}\sigma_{y}-k_{y}\sigma_{x}$ are called the out-of-
plane Rashba field terms, giving rise to the in-plane spin texture.

Finally, to model the momentum-independent hybridization of the right movers
of the Dirac cone with the SnTe bands, we assume the presence of the
momentumless hybridization-induced SOC term

$\displaystyle H_{\rm HI}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{3}\tau^{1+\delta_{i,2}}\sigma_{i}(\alpha^{i}_{1}S_{x}+\alpha^{i}_{2}S_{y}),$
(5)

where the parameters $\alpha^{i}_{1/2}$, $i=0,...3,$ represent the pseudospin-
dependent components of the spin-orbit field, while $\delta_{i,j}$ is the
Kronecker delta function, being equal to 1 when $i=j$, and 0 otherwise.

###  III.2 Model parameters

In order to extract the model parameters, we have performed DFT calculations
with QE focusing on the Dirac cone structure of graphene bands. Comparison
between the DFT data and the fit obtained using the model Hamiltonian (1) can
be found in FIG. 3. More precise, in FIG. 3 a)-d) band structure and spin
expectation values of the graphene bands around the Dirac cone along the
$\Gamma$KM path in the $x$-direction is given ($k_{y}=0$), using the values of
$k_{x}$ in the range $k_{x}\in(-\kappa,\kappa)$, measured from the K point,
where $\kappa=0.026\,1/\text{\AA}$. In FIG. 3 e)-h), the band structure and
spin expectation values are given by assuming the k-path in the y-direction of
the graphene Brillouin zone ($k_{x}=0$), where $k_{y}\in(-\kappa,\kappa)$ is
the distance with respect to the K point (see inset).

The obtained parameters are gathered in Table 1. The Fermi velocity, in this
case, is equal to $0.76\times 10^{6}$ m/s, which is of the same order of
magnitude as in other proximitized graphene heterostructures YSM+12 ; JZ21 ;
TJU+22 . Other spin-independent terms consist of the chemical potential $\mu$,
staggered field potential $\Delta\sigma_{z}$, followed by the Dirac cone
direction shift in the x-direction, $\Delta_{\rm s}\sigma_{x}$.

When discussing the sublattice-resolved spin conserving SOC parameters
$\lambda_{\rm I}^{A}$ and $\lambda_{\rm I}^{B}$, it is to be mentioned that it
can be written as $\lambda_{\rm I}^{A/B}=\lambda_{I}\pm\delta\lambda_{\rm I}$,
where $\lambda_{I}=14.46$ meV is called the intrinsic or Kane-Mele SOC term
CY62 ; KM05 , while $\delta\lambda_{\rm I}=-0.05$ meV is the valley-Zeeman
term GF15 ; SMK+23 . Whereas the intrinsic SOC Hamiltonian gives rise to the
out-of-plane spin texture, the in-plane HI field induces the in-plane spin
texture. Within the pseudo-spin dependent terms, we can identify the dominant
ones, proportional to $(\sigma_{0}+\sigma_{1})S_{y}$, since
$\alpha_{2}^{0}=\alpha_{2}^{2}$. A simple numerical analysis of the model
Hamiltonian shows that the term $\alpha_{2}^{0}(\sigma_{0}+\sigma_{1})S_{y}$
affects only the right movers of the Dirac cone and orients the spin in the
y-direction, with the size of the spin splitting proportional to the parameter
$\alpha_{2}^{0}$. The appearance and the dominant influence of such a term can
be traced to the strong hybridization of the right movers of the Dirac cone
with SnTe bands, crucially affecting the shape of the Dirac bands. Moreover,
since the SnTe band is parallel to the right movers of the Dirac cone over a
wide range of ${\bf k}$, it was natural to assume that the k-independent term
was to describe such an interaction.

Finally, we discuss the Rashba SOC. As mentioned in Sec. III.1, the gradient
of the crystal potential in the $x/y/z$ direction gives rise to the Rashba
terms $\propto k_{y}S_{z}/k_{x}S_{y}/(k_{x}S_{y}-k_{y}S_{x})$, whose influence
is quantified by the pseudospin-dependent parameters
$\beta_{2}^{i}/\beta_{1}^{i}/\lambda^{i}$, $i=0,..,3$. To get a deeper insight
into the nature of the effective electric field generating the Rashba
interaction, we define a generalized pseudospin-independent Rashba coupling
strength terms as $|\beta_{1/2}|=\sqrt{\sum_{i=0}^{3}(\beta_{1/2}^{i})^{2}}$
and $|\lambda|=\sqrt{\sum_{i=0}^{3}(\lambda^{i})^{2}}$. Using this formula and
the fitted parameters, we obtain $\beta_{1}=47.61$ meV Å, $\beta_{2}=368.73$
meV Å, $\lambda=155.60$ meV Å. We see that the Rashba coupling parameters are
significant, but still much smaller than the Rashba parameter in SnTe (1.23 eV
Å AI19 ). Furthermore, the main contribution to the overall Rashba spin-orbit
coupling comes from the in-plane components, suggesting the ferroelectric
origin of the Rashba SOC. By comparing the ratio of $\beta_{1}$ and
$\beta_{2}$, the 7.35 angle of the in-plane electric field generating Rashba
is determined, being almost aligned (3 degrees) with the polarization
direction of the SnTe monolayer. This confirms the assumption of the
ferroelectric nature of the in-plane Rashba SOC field.

Table 1: Fitting parameters of the graphene bands in the vicinity of the K point. Chemical potential is given as $\mu$, Fermi velocity as $v_{\rm F}$, staggered potential as $\Delta$, while the strain-induced K point shift is equal to $\Delta_{\rm s}$. Sublattice-dependent intrinsic spin-orbit coupling is represented as $\lambda_{\rm I}^{A/B}$; also, we label the pseudo-spin dependent Rashba parameters as $\beta_{1/2}$ and $\lambda$, with components $\beta_{1/2}^{i}$ and $\lambda^{i}$ ($i=0,...,3$) spanning the pseudo-spin space. Finally, $\alpha_{1/2}$ represent the in-plane HI field, with pseudo-spin components $\alpha_{1/2}^{i}$, $i=0,...,3$. $v_{\rm f}$[$10^{6}$m/s] | 0.76
---|---
$(\mu,\Delta,\Delta_{\rm s})$ [meV] | $(-416.03,0.18,-48.54)$
$(\lambda_{\rm I}^{A},\lambda_{\rm I}^{B})$ [meV] | $(14.41,14.51)$
$\beta_{1}$ [meV Å] | $(0.97,1.51,-27.30,-38.97)$
$\beta_{2}$ [meV Å] | $(-116.50,-349.00,1.94,-24.17)$
$\lambda$ [meV Å] | $(-42.77,-5.85,-4.48,149.42)$
$\alpha_{1}$ [meV] | $(0.05,0.08,-1.52,-2.17)$
$\alpha_{2}$ [meV] | $(5.67,5.67,0.11,-1.35)$
Figure 4: Spin texture of the spin split Dirac cone bands around the
$K=4\pi/3a(1,0)$ point is given for each of the four bands. The values of
${k_{x}/k_{y}}\in(-0.03,0.03)\text{\AA}^{-1}$ are measured with respect to the
K point. The arrows represent the in-plane spin texture, while the out-of-
plane spin texture is represented by the color scale.

The spin texture of the proximitized graphene of Dirac bands (C1, C2, V1, and
V2, see FIG. 3a) is given in FIG. 4, assuming that the
$k_{x}/k_{y}\in(-0.03,0.03)1/\text{\AA}$ is the distance measured from the
$K=4\pi/3a(1,0)$ point. First, one notices that the in-plane texture,
represented in FIG. 4 by arrows, is almost fully polarized in the y-direction,
representing a signature of the hybridization-induced SOC term
$\propto(\sigma_{0}+\sigma_{1})S_{y}$, responsible for the highly asymmetric
band structure and the giant (bigger than 20 meV) spin splitting of the Dirac
bands. On the other hand, the z component of spin is influenced by both the
intrinsic SOC $H_{\rm I}$ and the in-plane Rashba SOC Hamiltonian terms. This
was concluded indirectly by comparing the averaged z component of the spin-
orbit field (see below); however, a similar conclusion stems from the
comparison of the DFT data for $S_{z}$ with the Hamiltonian model (1) that
uses the parameters from Table 1 while manually turning off the in-plane
Rashba coupling parameters $\beta_{1/2}^{i}$ in one case and intrinsic SOC
parameters $\lambda_{\rm I}^{A/B}$ in the second case.

###  III.3 Proximity-induced spin relaxation

Finally, we show that the novel spin texture induced into graphene by the
interaction with the SnTe ML has a genuine fingerprint on the spin relaxation
rate anisotropy, not observed before. In the case of twisted graphene/TMDC
heterostructure, the common ${\bf C}_{3}$ symmetry of the heterostructure
gives rise to the in-plane isotropy of spin relaxation rates and the giant in-
plane/out-of-plane spin relaxation rate anisotropy CGF+17 ; GVC+18 ; ZGF19 .
In the case of the graphene/SnTe heterostructure, the absence of common
symmetry between the heterostructure constituents triggers new SOC terms and
as a consequence, new channels for spin relaxation. Assuming the Dyakonov-
Perel DP72  mechanism of spin relaxation, which should be dominant in
proximitized graphene ZCR+21 , spin relaxation rates can be described as

$\displaystyle\tau_{x}^{-1}$ $\displaystyle=$
$\displaystyle\tau_{p}(\langle{\Omega^{2}_{{\bf k},y}}\rangle_{\rm
p}+\langle{\Omega^{2}_{{\bf k},z}}\rangle_{\rm p})+\tau_{\rm
iv}(\langle{\Omega^{2}_{y}}\rangle_{\rm
iv}+\langle{\Omega^{2}_{z}}\rangle_{\rm iv}),$ $\displaystyle\tau_{y}^{-1}$
$\displaystyle=$ $\displaystyle\tau_{\rm p}(\langle{\Omega^{2}_{{\bf
k},x}}\rangle_{\rm p}+\langle{\Omega^{2}_{{\bf k},z}}\rangle_{p})+\tau_{\rm
iv}(\langle{\Omega^{2}_{x}}\rangle_{\rm
iv}+\langle{\Omega^{2}_{z}}\rangle_{\rm iv}),$ $\displaystyle\tau_{z}^{-1}$
$\displaystyle=$ $\displaystyle\tau_{\rm p}(\langle{\Omega^{2}_{{\bf
k},x}}\rangle_{\rm p}+\langle{\Omega^{2}_{{\bf k},y}}\rangle_{\rm
p})+\tau_{\rm iv}(\langle{\Omega^{2}_{x}}\rangle_{\rm
iv}+\langle{\Omega^{2}_{y}}\rangle_{\rm iv}),$

where $\tau_{i}$, $i=x,y,z$, represents a lifetime of spins pointing in the
$i$ direction; $\tau_{\rm p}$ is the momentum relaxation time, $\tau_{\rm iv}$
is the intervalley scattering time, while $\langle{\Omega_{i}^{2}}\rangle_{\rm
iv}$ and $\langle{\Omega_{{\bf k},i}^{2}}\rangle_{\rm p}$ are the Fermi
contour averages of the momentum-independent (index iv) or momentum-dependent
(index p) effective spin-orbit field components. By definition, the average
$\langle{\Omega^{2}}\rangle$ of an arbitrary spin-orbit field component
$\Omega^{2}$ is defined as

$\langle{\Omega^{2}}\rangle=\frac{1}{\rho(E_{F})S_{BZ}}\int_{FC}\frac{\Omega^{2}}{\hbar|v_{F}(k)|}dk,$
(7)

where $\rho(E_{F})$ represents the density of states per spin at the Fermi
level, $v_{F}(k)$ is the Fermi velocity, while $S_{BZ}$ is the area of the
Fermi surface.

To calculate the momentum-dependent and momentum-independent spin-orbit
fields, one has to turn on/off the corresponding interactions of the graphene
Hamiltonian (1). Detailed, in the case of the z-components of the spin-orbit
field, terms proportional to $\lambda_{I}^{A/B}$ are momentum-independent,
while the terms proportional to $\beta_{1/2}^{i}$, $i=0,...,3$, are momentum-
dependent; for the in-plane spin-orbit field components, one has to
distinguish between contributions of the momentum-independent terms
proportional to $\alpha_{1/2}^{i}$, $i=0,...,3$, and the momentum-independent
terms proportional to $\lambda^{i}$, $i=0,...,3$. The procedure to calculate
the spin-orbit fields for valence and conductance band can be found in
Appendix B. Assuming the Fermi level $E_{\rm F}$ in a range from -170 meV to
-50 meV below the valence band maximum, the spin-orbit field averages
$\langle{\Omega^{2}_{x/y/z}}\rangle_{\rm iv}$ and $\langle{\Omega^{2}_{{\bf
k},x/y/z}}\rangle_{\rm p}$ are calculated for the valence bands. The results
that can be found in Appendix B show that the dominant contribution to the
spin relaxation rate comes from $\langle{\Omega^{2}_{{\bf k},y}}\rangle_{\rm
iv}$. When compared to $\langle{\Omega^{2}_{{\bf k},y}}\rangle_{\rm iv}$, the
z-components $\langle{\Omega^{2}_{{\bf k},z}}\rangle_{\rm p}$ and
$\langle{\Omega^{2}_{z}}\rangle_{\rm iv}$ are also sizable for bigger $|E_{\rm
F}|$, while for smaller $|E_{\rm F}|$ $\langle{\Omega^{2}_{x}}\rangle_{\rm
iv}$ should be taken into the account. To estimate the impact of the averaged
spin-orbit field components on the spin relaxation rates, the relative ratio
$\tau_{\rm iv}/\tau_{\rm p}$ has to be known. Although this value is hard to
determine from the experiments, it can be estimated to be within the range
$\tau_{\rm iv}/\tau_{\rm p}\in(10-60)$ ZCR+21 .

In FIG. 5, we analyze the ratio of spin relaxation rates $\tau_{y}/\tau_{x}$,
$\tau_{z}/\tau_{x}$, and $\tau_{y}/\tau_{z}$ on the doping level, being in the
range from -170 meV to -50 meV below the valence band maximum, assuming
$\tau_{\rm iv}=10\tau_{\rm p}$. Our calculations have shown that the results
are independent of the ratio $\tau_{\rm iv}/\tau_{\rm p}\in(10-60)$. This is a
clear consequence of the two facts, the dominant roles of $\tau_{\rm iv}$ time
scale and averaged momentum-independent spin-orbit fields appearing in
(III.3). Focusing on the spin relaxation rates, the results demonstrate the
prominent in-plane spin relaxation anisotropy, while the ratio
$\tau_{z}/\tau_{x}$ is proportional to 1, suggesting that the dominant time
scale is given by $\tau_{y}$. This is dramatically different from the
calculated CGF+17  and observed GAK+17  spin relaxation rates in graphene/TMDC
heterostructures, where $\tau_{x}=\tau_{y}\ll\tau_{z}$, suggesting the
importance of substrate in inducing and manipulating different spin properties
in graphene.

##  IV Conclusions

We analyzed the proximity-induced spin-orbit effect in a heterostructure made
of graphene and SnTe monolayer, assuming a relative twist angle of three
degrees between the monolayers. Focusing on the band structure of graphene
close to the Dirac cone, we show that, due to the strong hybridization with
SnTe bands, giant asymmetric spin splitting of the graphene bands occurs. We
show that the data from the ab-initio calculations can be fitted to the
symmetry-free model and discuss the novel terms present in the effective
Hamiltonian due to the broken symmetry. We show that the hybridization-induced
spin-orbit coupling induces strong spin-splitting asymmetry, having its origin
in the hybridization of the right movers of the Dirac cone with SnTe bands.
Moreover, the strong in-plane ferroelectricity of the SnTe monolayer is
transferred to the graphene bands via the Rashba effect, whose in-plane field
is much stronger than the more common out-of-plane field, also present in
graphene/TMDC heterostructures. Finally, we report large in-plane anisotropy
of spin relaxation rates in graphene bands, present due to the broken symmetry
of the studied heterostructure. Thus, we show that group-IV monochalcogenide
MX (M=Sn, Ge; X=S, Se, Te) monolayers can be an efficient alternative to
transition-metal dichalcogenides for inducing strong spin-orbit coupling into
a desired material and novel patterns of spin relaxation rates.

Figure 5: Dependence of anisotropies of Dyakonov-Perel spin relaxation times
$\tau_{y}/\tau_{x}$, $\tau_{z}/\tau_{x}$, and $\tau_{y}/\tau_{z}$ on the
position of the Fermi level $E_{F}$. In the calculations, the relation between
the momentum relaxation time $\tau_{\rm p}$ and the intervalley scattering
time $\tau_{\rm iv}$ is set to be $\tau_{\rm iv}=10\tau_{\rm p}$.

###### Acknowledgements.

M.M. acknowledges the financial support provided by the Ministry of Education,
Science, and Technological Development of the Republic of Serbia and DAAD
Research Grant 57552336. This project has received funding from the European
Union’s Horizon 2020 Research and Innovation Programme under the Programme
SASPRO 2 COFUND Marie Sklodowska-Curie grant agreement No. 945478. M.G.
acknowledges financial support provided by Slovak Research and Development
Agency provided under Contract No. APVV-SK-CZ-RD-21-0114 and by the Ministry
of Education, Science, Research and Sport of the Slovak Republic provided
under Grant No. VEGA 1/0105/20 and Slovak Academy of Sciences project IMPULZ
IM-2021-42 and project FLAG ERA JTC 2021 2DSOTECH. M.K. acknowledges financial
support provided by the National Center for Research and Development (NCBR)
under the V4-Japan project BGapEng V4-JAPAN/2/46/BGapEng/2022. I.Š
acknowledges financial support by APVV-21-0272, VEGA 2/0070/21, VEGA
2/0131/23, and by H2020 TREX GA No. 952165 project. J.F. acknowledges support
from EU project 101135853 (2DSPIN-TECH) and FLAG ERA JTC 2021 2DSOTECH. The
authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. for
funding this project by providing computing time on the GCS Supercomputer
SuperMUC-NG at Leibniz Supercomputing Centre.

##  Appendix A Crystal parameters of SnTe monolayer

The crystal parameter determination was obtained by minimizing the total
energy of the system in the absence of the spin-orbit coupling within the
plane wave QE package, using the SG15 Optimized Norm-Conserving Vanderbilt
(ONCV) pseudopotentials H13 ; SG15 , with the kinetic energy cut-offs for the
wave function and charge density 80 Ry and 320 Ry, respectively. The force and
energy convergence thresholds were set to $10^{-5}$ Ry/bohr and $10^{-8}$ Ry,
respectively. For the Brillouin zone integration, $10\times 10\times 1$
$k$-points mesh was considered using the Monkhorst-Pack scheme; a vacuum of 20
Å in the $z$-direction was used. The obtained crystal parameters are
$a=4.58\text{\AA}$ and $b=4.56\text{\AA}$, in line with AI19 .

Since an important characteristic of a SnTe monolayer is its ferroelectricity
CLL+16 ; SCG+20 ; WGP+20 , we also calculate this property of a system using
the determined crystal parameters. In this case, the fully relativistic SG15
ONCV pseudopotentials with the same energy cut-offs for the wave function and
charge density as in the non-relativistic case were used, with the $k$ point
mesh $25\times 25\times 1$. By performing the Berry phase calculation KV93 ,
the polarization value of 23.39 $\mu C/{\rm cm}^{2}$ was obtained when the
effective thickness 1nm for the monolayer is used, consistent with WLX+14 .

##  Appendix B Derivation of the spin-orbit field in proximitized graphene

Figure 6: Fermi contour average of the momentum-dependent
$\langle{\Omega^{2}_{{\bf k},x/y/z}}\rangle_{\rm p}$ and momentum-independent
$\langle{\Omega^{2}_{x/y/z}}\rangle_{\rm iv}$ spin-orbit fields versus the
position of the Fermi level calculated.

Here we show how the spin-orbit field ${\bm{\Omega}}_{\bf k}$ in proximitized
graphene ZCR+21  can be derived straightforwardly from the effective
Hamiltonian H (1). As a first step, the orbital part of the Hamiltonian,
$H_{\rm orb}$ (2), is diagonalized,

$H_{\rm orb}\Psi=E\Psi,$ (8)

where
$\Psi=\\{|{\Psi_{CB1}}\rangle,|{\Psi_{CB2}}\rangle,|{\Psi_{VB1}}\rangle,|{\Psi_{VB2}}\rangle\\}$
describe eigenvectors of conductance and the valence bands of graphene. As the
next step, the effective Hamiltonian (1) is written in the basis $\Psi$ and
has the following form

$\mathcal{H}=\begin{pmatrix}\mathcal{H}_{CC}&\mathcal{H}_{CV}\\\
\mathcal{H}_{VC}&\mathcal{H}_{VV}\end{pmatrix},$ (9)

where $\mathcal{H}_{CV/VC}$ represents the interaction between the conductance
and the valence bands. Finally, the Hamiltonian $\mathcal{H}$ is projected
onto the conductance or valence band only, taking into account the interaction
between them,

$\displaystyle\mathcal{H}_{CB}$ $\displaystyle\equiv$
$\displaystyle\Big{[}\mathcal{H}_{CC}+\mathcal{H}_{CV}(E-\mathcal{H}_{VV})^{-1}\mathcal{H}_{VC}\Big{]},$
$\displaystyle\mathcal{H}_{VB}$ $\displaystyle\equiv$
$\displaystyle\Big{[}\mathcal{H}_{VV}+\mathcal{H}_{VC}(E-\mathcal{H}_{CC})^{-1}\mathcal{H}_{CV}\Big{]}.$
(10)

The obtained conductance/valence band Hamiltonians $\mathcal{H}_{CB/VB}$ can
be written in the following form

$\mathcal{H}_{CB/VB}=\epsilon_{0}\mathcal{I}_{2}+\frac{\hbar}{2}\begin{pmatrix}\Omega_{Z}&\Omega_{X}-\rm{i}\Omega_{Y}\\\
\Omega_{X}+\rm{i}\Omega_{Y}&-\Omega_{Z}\end{pmatrix},$ (11)

where $\mathcal{I}$ is the identity 2$\times$2 matrix, $\epsilon_{0}$ is the
orbital energy, while ${\bm{\Omega}}=(\Omega_{X},\Omega_{Y},\Omega_{Z})$
corresponds to the spin-orbit field, representing the system dependent
quantity that crucially affects the spin relaxation times of valence and
conductance bands of graphene.

Using the third-order perturbation theory we have taken into account the
interaction between the conductance and valence bands and calculated the Fermi
contour averages of the momentum-dependent and momentum-independent spin-orbit
fields $\langle{\Omega^{2}_{x/y/z}}\rangle_{\rm iv}$ and
$\langle{\Omega^{2}_{{\bf k},x/y/z}}\rangle_{\rm p}$ as in (7). In our
calculations, we have assumed the Fermi level $E_{\rm F}$ values from -170 meV
to -50 meV below the valence band maximum. This justifies the usage of the
perturbation approach since there is a sizable gap between the valence and
conductance bands. In FIG. 6, the dependence of the Fermi contour averaged
spin-orbit field on $E_{\rm F}$ is given, showing that the dominant spin
relaxation channel is $\langle{\Omega^{2}_{{\bf k},y}}\rangle_{\rm iv}$, while
$\langle{\Omega^{2}_{{\bf k},z}}\rangle_{\rm p}$ and
$\langle{\Omega^{2}_{z}}\rangle_{\rm iv}$ are comparable for bigger $|E_{\rm
F}|$; for smaller $|E_{\rm F}|$ $\langle{\Omega^{2}_{x}}\rangle_{\rm iv}$
should also be taken into the account.

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