# Rings whose non-invertible elements are
uniquely strongly clean

Peter Danchev  Institute of Mathematics and Informatics, Bulgarian Academy of
Sciences, 1113 Sofia, Bulgaria<EMAIL_ADDRESS><EMAIL_ADDRESS>,
Omid Hasanzadeh  Department of Mathematics, Tarbiat Modares University,
14115-111 Tehran Jalal AleAhmad Nasr, Iran<EMAIL_ADDRESS>and
Ahmad Moussavi  Department of Mathematics, Tarbiat Modares University,
14115-111 Tehran Jalal AleAhmad Nasr, Iran<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We define and explore in details the class of GUSC rings, that are those rings
whose non-invertible elements are uniquely strongly clean. These rings are a
common generalization of the so-called USC rings, introduced by Chen-Wang-Zhou
in J. Pure & Appl. Algebra (2009), which are rings whose elements are uniquely
strongly clean. These rings also generalize the so-called GUC rings, defined
by Guo-Jiang in Bull. Transilvania Univ. Braşov (2023), which are rings whose
non-invertible elements are uniquely clean.

###### Key words and phrases:

clean element (ring), uniquely clean element (ring), strongly clean element
(ring), uniquely strongly clean element (ring)

###### 2010 Mathematics Subject Classification:

16S34, 16U60

##  1\. Introduction and Basic Concepts

Everywhere in the current paper, let $R$ be an associative but not necessarily
commutative ring having identity element, usually stated as $1$. Standardly,
for such a ring $R$, the letters $U(R)$, $\rm{Nil}(R)$ and ${\rm Id}(R)$ are
designed for the set of invertible elements (also termed as the unit group of
$R$), the set of nilpotent elements and the set of idempotent elements in $R$,
respectively. Likewise, $J(R)$ denotes the Jacobson radical of $R$, and ${\rm
Z}(R)$ denotes the center of $R$. We also write $ucn(R)$ for the set of all
uniquely clean elements of $R$. The ring of $n\times n$ matrices over $R$ and
the ring of $n\times n$ upper triangular matrices over $R$ are denoted by
${\rm M}_{n}(R)$ and ${\rm T}_{n}(R)$, respectively. Standardly, a ring is
said to be abelian if each of its idempotents is central, that is, ${\rm
Id}(R)\subseteq{\rm Z}(R)$.

For all other unexplained explicitly notions and notations, we refer to the
classical source [6] or to the cited in the bibliography research sources. For
instance, consulting with [5, Definition 3.4], we set

$ucn_{0}(R)=\\{e+j:e^{2}=e\in{\rm Z}(R),j\in J(R)\\}.$

In order to present our achievements here, we now need the necessary
background material as follows: Mimicking [4], an element $a$ from a ring $R$
is called clean if there exists $e\in{\rm Id}(R)$ such that $a-e\in U(R)$.
Then, $R$ is said to be clean if each element of $R$ is clean. In addition,
$a$ is called strongly clean provided $ae=ea$ and, if each element of $R$ are
strongly clean, then $R$ is said to strongly clean too. On the other hand,
imitating [10], $a\in R$ is called uniquely clean if there exists a unique
$e\in{\rm Id}(R)$ such that $a-e\in U(R)$. In particular, a ring $R$ is said
to be uniquely clean (or just UC for short) if every element in $R$ is
uniquely clean.

Generalizing these notions, in [1] was defined an element $a\in R$ to be
uniquely strongly clean if there exists a unique $e\in{\rm Id}(R)$ such that
$a-e\in U(R)$ and $ae=ea$. In particular, a ring $R$ is uniquely strongly
clean (or just USC for short) if each element in $R$ is uniquely strongly
clean.

In a similar vein, expanding the first part of the above concepts, in [3] a
ring is called a generalized uniquely clean ring (or just GUC for short) if
any non-invertible element is uniquely clean.

Our work targets to extend the last definition by defining and exploring the
following key instruments: A ring is called a generalized uniquely strongly
clean ring (or just GUSC for short) if every non-invertible element is
uniquely strongly clean.

The relationships between all of these notions are showed by the next diagram:

USC GUSC GUC Strongly clean Clean uniquely clean

We are now planning to give a brief program of our results established in the
sequel: In the next second section, we establish some fundamental
characterization properties of GUSC rings – e.g., showing that the class of
such rings coincides with the union of the classes of local rings and USC
rings (see Lemma 2.9). We also prove that a ring $R$ with $J(R)=\\{0\\}$ is
GUSC exactly when it is GUC (see Corollary ). Besides, we positively settle
[3, Question 4.8] (see Proposition 2.23). Some other related assertions are
also obtained. In the final third section, we examine GUSC group rings by
giving a necessary and sufficient condition for a group ring of a locally
finite group to be GUSC (see Theorem 3.3).

##  2\. Examples and Main Results

We begin by marking the following simple but useful observations.

######  Example 2.1.

  1. (i)

Any UC ring is USC, but the converse is not true in general – in fact, the
ring ${\rm T}_{2}(\mathbb{Z}_{2})$ is USC that is not uniquely clean. However,
USC rings are strongly clean.

  2. (ii)

Any UC ring is GUC, but the converse is not true in general – indeed, the ring
$\mathbb{Z}_{5}$ is GUC that is not uniquely clean.

  3. (iii)

Any USC ring is GUSC, but the converse is not true in general – in fact, the
ring
$\mathbb{Z}_{(3)}=\left\\{\dfrac{a}{b}~{}\Big{|}~{}a,b\in\mathbb{Z},(3,b)=1\right\\}$
is GUSC that is not USC.

  4. (iv)

Any GUC ring is GUSC, but the converse is not true in general – indeed, the
ring ${\rm T}_{2}(\mathbb{Z}_{2})$ is GUSC being USC that is not GUC.

  5. (v)

Any GUSC ring is strongly clean, but the converse is not true in general. For
example, for an arbitrary prime number $p$, let $\hat{\mathbb{Z}_{p}}$ be the
ring of $p$-adic integers. In view of [2, Theorem 2.4], ${\rm
M}_{2}(\hat{\mathbb{Z}}_{p})$ is strongly clean that is not USC by virtue of
[1, Corollary 7]. Also, a plain check shows that it is not too GUSC.

It is also clear that a ring $R$ is GUC if, and only if, $R$ is abelian GUSC.

We, furthermore, continue with a series of helpful technicalities.

######  Lemma 2.2.

An element $a$ in a ring $R$ is USC if, and only if, so is $1-a$.

###### Proof.

Straightforward. ∎

######  Lemma 2.3.

If $R$ is a GUSC ring, then $R$ is strongly clean.

###### Proof.

Letting $a\in R$, so $a\in U(R)$ or $a\notin U(R)$. If $a\in U(R)$, then $a$
is known to be a strongly clean element. If, however, $a\notin U(R)$, then by
assumption $a$ is uniquely strongly clean element, and thus $a$ is a strongly
clean element. Finally, in both cases, $R$ is strongly clean, as asserted. ∎

######  Lemma 2.4.

If $\displaystyle\prod_{i\in I}R_{i}$ is GUSC, then each $R_{i}$ is GUSC.

###### Proof.

Let $a_{i}\in R_{i}$, where $a_{i}\notin U(R_{i})$, whence
$(1,1,\ldots,a_{i},1,\ldots)\notin U(\prod R_{i})$. So,
$(1,1,\ldots,a_{i},1,\ldots)$ is USC and hence $a_{i}$ is USC. If, however,
some $a_{i}$ is not USC, then we have

$a_{i}=e_{i}+u_{i}=f_{i}+v_{i},$

where $e_{i}u_{i}=u_{i}e_{i}$, $f_{i}v_{i}=v_{i}f_{i}$, $u_{i},v_{i}\in
U(R_{i})$ and $e_{i},f_{i}\in Id(R_{i})$. Therefore,

$(1,1,\ldots,a_{i},1,\ldots)=(0,0,\ldots,e_{i},0,\ldots)+(1,1,\ldots,u_{i},1,\ldots)=$
$=(0,0,\ldots,f_{i},0,\ldots)+(1,1,\ldots,v_{i},1,\ldots),$

where

$\displaystyle(0,0,\ldots,e_{i},0,\ldots)(1,1,\ldots,u_{i},1,\ldots)=(1,1,\ldots,u_{i},1,\ldots)(0,0,\ldots,e_{i},0,\ldots),$
$\displaystyle(0,0,\ldots,f_{i},0,\ldots)(1,1,\ldots,v_{i},1,\ldots)=(1,1,\ldots,v_{i},1,\ldots)(0,0,\ldots,f_{i},0,\ldots),$
$\displaystyle(0,0,\ldots,e_{i},0,\ldots),(0,0,\ldots,f_{i},0,\ldots)\in
Id(\prod R_{i}),$
$\displaystyle(1,1,\ldots,u_{i},1,\ldots),(1,1,\ldots,v_{i},1,\ldots)\in
U(\prod R_{i}).$

That is why, we have two different strongly clean decompositions for
$(1,1,\ldots,a_{i},1,\ldots)$, and so this is a contradiction, as pursued. ∎

We note that the converse of Lemma 2.4 is manifestly false. For example, both
$\mathbb{Z}_{2}$ and $\mathbb{Z}_{3}$ are GUSC. But the direct product
$\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ is not GUSC, because the element $(0,2)$
is not invertible in $\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ and
$(0,2)=(1,0)+(1,2)=(1,1)+(1,1)$ are two different strongly clean
decompositions of $(0,2)$ in $\mathbb{Z}_{2}\times\mathbb{Z}_{3}$, so that
$(0,2)$ is really not uniquely strongly clean element, as required.

######  Proposition 2.5.

The direct product $\displaystyle\prod_{i\in I}R_{i}$ is GUSC if, and only if,
each $R_{i}$ is uniquely strongly clean.

###### Proof.

Letting each $R_{i}$ be USC, then $\prod R_{i}$ is USC employing [1, Example
2] and hence $\prod R_{i}$ is trivially GUSC.

Conversely, assume that $\prod R_{i}$ is GUSC and that $R_{j}$ is not USC for
some index $j\in I$. Then, there exists $a\in R_{j}$ which is not USC. Hence,
$(0,\ldots,0,a,0,\ldots)$ is not USC in $\prod R_{i}$. But, it is readily seen
that $(0,\ldots,0,a,0,\ldots)$ is not invertible in $\prod R_{i}$ and thus, by
hypothesis, $(0,\ldots,0,a,0,\ldots)$ is not USC, a contradiction. Therefore,
each $R_{i}$ is USC, as needed. ∎

As a consequence, we derive:

######  Corollary 2.6.

Let $R$ be a ring and $0\neq e\in{\rm Id}(R)\cap{\rm Z}(R)$. If $R$ is GUSC,
then so is $eRe$. In addition, if $e$ is non-trivial, then $eRe$ is USC.

###### Proof.

Let $R$ be GUSC. We have $R=eRe\oplus(1-e)R(1-e)$. Hence, $eRe$ is GUSC
applying Lemma 2.4. If $e$ is not trivial, then $eRe$ is USC utilizing
Proposition 2.5, as claimed. ∎

######  Example 2.7.

If $R$ is a local ring, then $R$ is GUSC.

###### Proof.

Let $a\in R$ and $a\notin U(R)$. So, $a=1+(a-1)$ is the only strongly clean
decomposition for $a$, and thus $a$ is uniquely strongly clean, as stated. ∎

The converse of the Example 2.7 is false. For example, it is clear that ${\rm
T}_{2}(\mathbb{Z}_{2})$ is GUSC but is obviously not local.

######  Proposition 2.8.

Let $R$ be a ring. Then, the following are equivalent:

  1. (i)

$R$ is a local ring.

  2. (ii)

$R$ is a GUSC ring and ${\rm Id}(R)=\\{0,1\\}$.

  3. (iii)

$R$ is a strongly clean ring and ${\rm Id}(R)=\\{0,1\\}$.

  4. (iv)

$R$ is a clean ring and ${\rm Id}(R)=\\{0,1\\}$.

  5. (v)

$R$ is an exchange ring and ${\rm Id}(R)=\\{0,1\\}$.

  6. (vi)

$R$ is a GUC ring and ${\rm Id}(R)=\\{0,1\\}$.

###### Proof.

$(1)\Rightarrow(2)$. It is obvious owing to Example 2.7.
$(2)\Rightarrow(3)$. It is clear according to Lemma 2.3.
$(3)\Rightarrow(4)\Rightarrow(5)\Rightarrow(1)$. Evident.
$(1)\Rightarrow(6)$. It is apparent that a similar proof to that of Example
2.7 works.
$(6)\Rightarrow(1)$. Let $a\in R$, where $a\notin U(R)$. It suffices to show
that $a-1\in U(R)$. To this goal, by hypothesis, $a$ is uniquely clean, so we
have $a=e+u$, where $e\in{rmId}(R)=\\{0,1\\}$ and $u\in U(R)$. If $e=0$, it
must be that $a=u\in U(R)$, a contradiction. If $e=1$, we obtain $a=1+u$ and
hence $a-1=u\in U(R)$, as desired. ∎

The next two statements are crucial.

######  Lemma 2.9.

Let $R$ be a ring. Then, the following are equivalent:

  1. (i)

$R$ is a GUSC ring.

  2. (ii)

$R$ is either a local ring or a USC ring.

###### Proof.

An analogous proof to that of [3, Theorem 2.10] is applicable to get the
claim, as wanted. ∎

######  Corollary 2.10.

If $R$ is a GUSC ring, then $\dfrac{R}{J(R)}$ is either a Boolean ring or a
division ring.

###### Proof.

A combination of [3, Corollary 2.11] and [1, Corollary 18] applies to get the
assertion. ∎

######  Proposition 2.11.

Suppose $R$ is an abelian ring and $I$ is an ideal of $R$, where $I\subseteq
J(R)$:

  1. (i)

$\dfrac{R}{I}$ is a GUSC ring if, and only if, $R$ is a GUSC ring.

  2. (ii)

$\dfrac{R}{J(R)}$ is a GUSC ring if, and only if, $R$ is a GUSC ring.

###### Proof.

  1. (i)

Let $\dfrac{R}{I}$ be a GUSC ring and $a\in R$, where $a\notin U(R)$. Write
$a=e+u=f+v$, where $eu=ue$, $fv=vf$, $e,f\in{\rm Id}(R)$ and $u,v\in U(R)$. We
have to show that $e=f$. On the other hand, we write

$a+I=(e+I)+(u+I)=(f+I)+\quad(v+I),$

where

$(e+I)(u+I)=eu+I=ue+I=(u+I)(e+I),$ $(f+I)(v+I)=fv+I=vf+I=(v+I)(f+I),$

$e+I,f+I\in{\rm Id}\left(\dfrac{R}{I}\right)$ and $u+I,v+I\in
U\left(\dfrac{R}{I}\right)$. By hypothesis, $a+I$ is a USC element (as $a\in
U(R)$, so $a+I\notin U(\dfrac{R}{I})$). Therefore, $e+I=f+I$, so $e-f\in
I\subseteq J(R)$. Since $R$ is abelian, we have $(e-f)^{3}=(e-f)$, so $e-f=0$
and hence $e=f$.

Conversely, let $R$ be a GUSC ring and $a+I\in\dfrac{R}{I}$, where $a+I\notin
U\left(\dfrac{R}{I}\right)$. Write

$a+I=(e+I)+(u+I)=\quad(f+I)+(v+I),$

where

$(e+I)(u+I)=(u+I)(e+I),(f+I)(v+I)=\quad(v+I)(f+I),$

$e+I,f+I\in{\rm Id}\left(\dfrac{R}{I}\right)$ and $u+I,v+I\in
U\left(\dfrac{R}{I}\right)$. Then, we have $a-(e+u)\in I\subseteq J(R)$, so
$a=e+(u+j)$ for $j\in J(R)$. Similarly, we have $a=f+(v+j^{\prime})$ for
$j^{\prime}\in J(R)$ and know that $e,f\in{\rm Id}(R)$ (because every GUSC
ring is clean and hence idempotents lifts modulo $I\subseteq J(R)$) and $u+j$,
$v+j^{\prime}\in U(R)$ (because $U(R)+J(R)=U(R)$ holds always for every ring
$R$). Also, $e(u+j)=(u+j)e$ and $f(v+j^{\prime})=(v+j^{\prime})f$ hold, since
$R$ is abelian. And as $a$ is a USC element, we deduce $e=f$. Consequently,
$e+I=f+I$, as expected.

  2. (ii)

The proof is similar to (1).

∎

######  Lemma 2.12.

Let $R$ be a GUSC ring and $n=1+1+\ldots+1$. Then, either $n\in J(R)$ or $n\in
U(R)$.

###### Proof.

An application of the same arguments as those from [3, Proposition 2.17]
assures that if $R$ is USC, then $2\in J(R)$ by [1, Lemma 16]. The remaining
part of the proof is now immediate. ∎

######  Example 2.13.

The ring ${\rm M}_{n}(R)$ is never GUSC for any $n\geq 2$.

###### Proof.

Assuming in a way of contradiction that ${\rm M}_{n}(R)$ is GUSC, then in
virtue of Lemma 2.9 it is either local or USC. First, if ${\rm M}_{n}(R)$ is
local, then ${\rm Id}\left(M_{n}(R)\right)=\\{0,1\\}$ which is a
contradiction, because the matrix $\begin{pmatrix}1&0&\ldots&0\\\
0&0&\ldots&0\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&0&\ldots&0\end{pmatrix}$ is
an idempotent in ${\rm M}_{n}(R)$ that is not equal to $0,1$.

If, however, ${\rm M}_{n}(R)$ is USC, then $1_{M_{n}(R)}$ is not $2$-good in
accordance with [1, Lemma 6] and this is again a contradiction, because we
have always

$\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}=\begin{pmatrix}1&1\\\
1&0\end{pmatrix}+\begin{pmatrix}0&-1\\\ -1&1\end{pmatrix}\quad\text{in}\quad
M_{2}(R).$

Thus, ${\rm M}_{n}(R)$ is not GUSC for every $n\geq 2$, as formulated. ∎

######  Example 2.14.

Let $R$ be a ring. If ${\rm T}_{n}(R)$ is a GUSC ring, then $R$ is also GUSC.
The converse holds, provided $R$ is commutative and USC (or just UC).

###### Proof.

Put $e=diag(1,0,\ldots,0)\in{\rm T}_{n}(R)$. Then, $R\cong e{\rm T}_{n}(R)e$,
so $R$ is a GUSC ring using Corollary 2.6. The converse is clear utilizing [1,
Theorem 10]. ∎

We proceed by proving the following assertions.

######  Proposition 2.15.

If $R$ is a commutative ring, then the polynomial ring $R[x]$ is not GUSC.

###### Proof.

We know that $R[x]$ is not clean, so it is not strongly clean and thus is
definitely not GUSC. ∎

A ring $R$ is called left quasi-duo (resp., right quasi-duo) if every maximal
left ideal (resp., maximal right ideal) of $R$ is an ideal.

######  Proposition 2.16.

If $R$ is a GUSC ring, then $R$ is simultaneously left and right quasi-duo.

###### Proof.

If $R$ is a GUSC ring, then $R$ is either local or USC by Lemma 2.9. Now, if
$R$ is local, then $R$ has a unique maximal ideal. Thus, $R$ is left and right
quasi-duo.

Let us now $R$ be USC and $I$ a maximal left ideal in $R$. Since
$\bar{R}=\dfrac{R}{J(R)}$ is Boolean by [1, Corollary 18], we have
$\dfrac{\bar{R}}{\bar{I}}\cong\mathbb{Z}_{2}$. But then
$\dfrac{R}{I}\cong\dfrac{\bar{R}}{\bar{I}}$ has two elements, so that
$\dfrac{R}{I}=\\{I,1+I\\}$, whence we have $R=I\cup(1+I)$. Now, given $x\in I$
and $r\in R$, we must show that $xr\in I$. In fact, this is clear if $r\in I$;
otherwise, $r=1+y$ and $y\in I$. So, $xr=x+xy\in I$, as required. ∎

A ring $R$ is called semi-potent if every one-sided ideal not contained in
$J(R)$ contains a non-zero idempotent.

######  Proposition 2.17.

Let $R$ be a ring. Then, $\dfrac{R}{J(R)}$ is GUSC if, and only if,
$\dfrac{R}{J(R)}$ is GUC.

###### Proof.

The sufficiency is quite easy.

For the necessity, suppose $\dfrac{R}{J(R)}$ is GUSC. So,
$\bar{R}=\dfrac{R}{J(R)}$ is semi-potent. Then, [7, Theorem 2.1] applies to
get that there exists $\bar{e}\in{\rm Id}(\bar{R})$ such that
$\bar{e}\bar{R}\bar{e}\cong{\rm M}_{2}(S)$ for a non-trivial ring $S$. We,
however, know that by Example 2.13 ${\rm M}_{2}(S)$ is not GUSC, so
$\bar{e}\bar{R}\bar{e}$ is not GUSC, but this is a contradiction taking into
account Corollary 2.6. Hence, $\bar{R}$ is reduced, so it is abelian and,
therefore, $\bar{R}$ is GUC, as required. ∎

As a direct consequence, we extract:

######  Corollary 2.18.

Let $R$ be a ring with $J(R)=\\{0\\}$. Then, $R$ is GUSC if, and only if, $R$
is GUC.

The next three statements are worthwhile.

######  Lemma 2.19.

If $R$ is a local ring, then $ucn(R)=ucn_{0}(R)$.

###### Proof.

It is similar to [3, Lemma 3.2]. ∎

######  Lemma 2.20.

If $R$ is a USC ring, then $ucn(R)=ucn_{0}(R)$.

###### Proof.

For a ring $R$, we know that

$ucn_{0}(R)=\left\\{e+j\mid e^{2}=e\in{\rm Z}(R),j\in J(R)\right\\}.$

We also know that $R$ is a clean ring, thus idempotents lift modulo $J(R)$.
Moreover, $\dfrac{R}{J(R)}$ is boolean bearing in mind [1, Corollary 18], so
that

$ucn\left(\dfrac{R}{J(R)}\right)=ucn_{0}\left(\dfrac{R}{J(R)}\right)$

implying $ucn(R)=ucn_{0}(R)$ in view of [3, Proposition 3.6]. ∎

######  Lemma 2.21.

If $R$ is a GUSC ring, then $ucn(R)=ucn_{0}(R)$.

###### Proof.

If $R$ is GUSC, then Lemma 2.9 tells us that $R$ is either local or USC. So,
the result follows from a combination of Lemma 2.19 and Lemma 2.20. ∎

######  Proposition 2.22.

If $R$ is a GUSC ring, then for every non-invertible element $a\in R$, there
exists a unique idempotent $e\in R$ such that $ea=ae$ and $a-e\in J(R)$.

###### Proof.

One sees from Corollary 2.10 that $\dfrac{R}{J(R)}$ is either a boolean ring
or a division ring. Assume first that $\dfrac{R}{J(R)}$ is boolean. Letting
$a\in R$, where $a\notin U(R)$, there exists a unique idempotent $e\in R$ such
that $u:=a-e\in U(R)$ and $ae=ea$. As $\dfrac{R}{J(R)}$ is boolean, it follows
that $\bar{u}=\bar{1}$. This shows that $u\in 1+J(R)$. Hence,

$a=(1-e)+(2e-1+u)\quad\text{with}\quad 2e-1+u\in J(R).$

Write $a=f+j$ with $f=f^{2}$, $j\in J(R)$ and $af=fa$. Then,

$a=(1-f)+(2f-1+j)$

with $2f-1+j\in U(R)$. By the uniqueness, we have $e=1-f$, and so $f=1-e$, as
desired.

Now, supposing $\dfrac{R}{J(R)}$ is a division ring, we infer that $R$ is
local, so we can take $e=0$, as expected. ∎

We are now ready to attack and resolve [3, Question 4.8]: Let $R$ be a ring.
Is it true that $R$ is a GUC ring if, for every non-invertible element $a\in
R$, there exists an unique idempotent $e\in R$ such that $a-e\in J(R)$?

######  Proposition 2.23.

The answer to this question is in the affirmative.

###### Proof.

First, we show that ${\rm Nil}(R)\subseteq J(R)$. Let $a\in{\rm Nil}(R)$. If
$a\in U(R)$, then $a=0$ and hence $a\in J(R)$. If, however, $a\notin U(R)$, so
there exists an unique idempotent $e\in R$ such that $a-e\in J(R)$. Then, we
have $a+J(R)=e+J(R)$, and thus $a+J(R)\in{\rm Id}(\dfrac{R}{J(R)})$ but
$a+J(R)\in N(\dfrac{R}{J(R)})$. So, $a+J(R)=J(R)$ and hence $a\in J(R)$. Now,
we show that, for every non-invertible element $a\in R$, there exists an
unique central idempotent $e\in R$ such that $a-e\in J(R)$. Let $a\in R$,
where $a\notin U(R)$, so by hypothesis there exists an unique idempotent $e\in
R$ such that $a-e\in J(R)$. It suffices to show that $e\in{\rm Z}(R)$. To this
aim, for every $r\in R$,

$er(1-e),(1-e)re\in{\rm Nil}(R)\subseteq J(R)$

and

$e+(1-e)re,e+er(1-e)\in{\rm Id}(R),$

so we have

$a-(e+er(1-e)),a-(e+(1-e)re)\in J(R).$

However, by the uniqueness, we deduce

$e+(1-e)re=e+er(1-e)=e.$

Therefore, $er=ere=re$ and hence $e\in{\rm Z}(R)$, as wanted. Finally, we
conclude that $R$ is GUC by [3, Theorem 3.5], as promised. ∎

Let $R$ be a ring and $M$ a bimodule over $R$. The trivial extension of $R$
and $M$ is

$T(R,M)=\\{(r,m):r\in R,m\in M\\}$

with the addition defined component-wise and the multiplication defined by the
equality

$(r,m)(s,n)=(rs,rn+ms).$

######  Proposition 2.24.

If the trivial extension $T(R,M)$ is a GUSC ring, then $R$ is a GUSC ring. The
converse holds, provided $ex=xe$ for all $x\in M$ and $e\in{\rm Id}(R)$.

###### Proof.

Choose $a\in R$, where $a\notin U(R)$, so that $(a,0)\notin U(T(R,M))$ and
thus $(a,0)$ is USC. Then, $a$ is USC, because otherwise, if $a$ has two
different strongly clean decompositions, then $(a,0)$ has two different
strongly clean decompositions, which contradicts to our assumption.

For the converse, choose $(a,m)\in T(R,M)$, where $(a,m)\notin U(T(R,M))$, so
that $a\notin U(R)$ (as for otherwise, if $a\in U(R)$, then $(a,m)\in
U(T(R,M))$ with inverse $(a^{-1},-a^{-1}ma^{-1})$, a contradiction), and thus
by hypothesis $a$ is USC. Therefore, $(a,m)$ is USC, because otherwise, if
$(a,m)=(e,0)+(u,m)=(f,0)+(v,m)$, where $(e,0),(f,0)\in{\rm Id}(T(R,M))$,
$(u,m),(v,m)\in U(T(R,M))$ and $(e,0)(u,m)=(u,m)(e,0)$,
$(f,0)(v,m)=(v,m)(f,0)$, then we will have

$a=e+u=f+v,$

where $e,f\in{\rm Id}(R)$, $u,v\in U(R)$ and $eu=ue$, $fv=vf$. Hence, $a$ has
two strongly clean decompositions, a contradiction.

Claim. If $(e,m)^{2}=(e,m)\in T(R,M)$, then $e^{2}=e$ and $m=0$.

In fact, $(e,m)^{2}=(e,m)$ automatically gives by comparison $e^{2}=e$ and
$em=me=m$. So, by hypothesis, we have $2em=m$, whence $2em=em$, and hence
$em=0$. So, $2em=0$ and then $m=0$, proving the claim and thus the entire
result. ∎

Let $R$ be a ring and let $M$ be an bimodule over $R$ which is a non-unital
ring in which the equalities $(mn)r=m(nr)$, $(mr)n=m(rn)$ and $(rm)n=r(mn)$
hold for all $m,n\in M$ and $r\in R$. Then, the ideal-extension $I(R,M)$ of
$R$ by $M$ is defined to be the additive abelian group $I(R,M)=R\oplus M$ with
multiplication given by

$(r,m)(s,n)=(rs,rn+ms+mn).$

We are now in a position to establish the validity of the following statement.

######  Proposition 2.25.

If the ideal extension $I(R,M)$ is a GUSC ring, then $R$ is a GUSC ring. The
converse holds if the following conditions are satisfied:

  1. (a)

If $e\in{\rm Id}(R)$, then $em=me$ for all $m\in M$.

  2. (b)

If $m\in M$, then $m+n+mn=0$ for some $n\in M$.

###### Proof.

The necessity is analogical to the arguments of Proposition 2.24.

To prove the sufficiency, first we know with the aid of (a), (b) and [10,
Proposition7] that every idempotent of $I(R,M)$ is of the type $(e,0)$, where
$e^{2}=e$ as well as that every unit of $I(R,M)$ is of the type $(u,m)$, where
$u\in U(R)$, $m\in M$. Now, let $(a,m)\in I(R,M)$, where $(a,m)\notin
U(I(R,M))$, so that $a\notin U(R)$. Then, by hypothesis $a$ is USC, and thus
$(a,m)$ is USC, because otherwise, if $(a,m)$ has two different strongly clean
decompositions, then arguing as in the proof of Proposition 2.24, the element
$a$ has two different strongly clean decompositions, a contradiction. ∎

Let $A$ be a ring, let $B$ be a unital subring of $A$, and let $R[A,B]$ stand
for the set

$\\{(a_{1},a_{2},\ldots,a_{n},b,b,\ldots):a_{i}\in A,b\in B,1\leq i\leq n\\}$

. Then, $R[A,B]$ is a ring under the component-wise addition and
multiplication.

We now state the following.

######  Proposition 2.26.

Let $A$ be a ring and let $B$ be a unital subring of $A$. Then, $R[A,B]$ is a
GUSC ring if, and only if, $A$, $B$ are both GUSC ring.

###### Proof.

Straightforward. ∎

##  3\. Group Rings

We know with the help of Lemma 2.9 that GUSC rings include USC rings and local
rings, and vice versa. In addition, there are some results on USC group rings
and local group rings proved in [11] and [8], respectively. Correspondingly,
we can also obtain some achievements about GUSC group rings that could be of
some interest and importance. To this purpose, we recall that a group $G$ is a
$p$-group if every element of $G$ is a power of $p$, where $p$ is a prime.
Likewise, a group $G$ is called locally finite if every finitely generated
subgroup is finite. As usual, for an arbitrary ring $R$ and an arbitrary group
$G$, the symbol $RG$ designates the group ring of $G$ over $R$.

We can now offer the following two statements.

######  Proposition 3.1.

Let $R$ be a ring and let $G$ be a group. If $RG$ is a GUSC ring, then $R$ is
a GUSC ring and $G$ is a $p$-group such that $p\in J(R)$.

###### Proof.

We can argue in the same manner as in [3, Proposition 4.2] and in [1,
Proposition 24] to finish the argumentation. ∎

######  Proposition 3.2.

Let $R$ be a ring with $\dfrac{R}{J(R)}\cong\mathbb{Z}_{2}$ and let $G$ be a
locally finite group. Then, $RG$ is a GUSC ring if, and only if, $G$ is a
$2$-group.

###### Proof.

Assuming $RG$ is GUSC, we apply Lemma 2.9 to get that $RG$ is either local or
USC. Firstly, if $RG$ is USC, then $G$ is a $2$-group by [11, Corollary 3.1].
Secondly, if $RG$ is local, then $G$ is a $p$-group and $p\in J(R)$ referring
to [8]. Since $\dfrac{R}{J(R)}\cong\mathbb{Z}_{2}$, we conclude that $R$ is
uniquely clean by [10, Theorem 15], so $R$ is USC. Now, if $p\neq 2$, then
$p\notin J(R)$ using Lemma 2.12. So, $p=2$ and hence $G$ is a $2$-group.

As for the converse implication, if $G$ is a $2$-group, then $RG$ is USC
utilizing [11, Theorem 3.1], and thus $RG$ is GUSC, as stated. ∎

We now arrive at our chief result in this section which gives a criterion when
a group ring of a locally finite group is GUSC.

######  Theorem 3.3.

Let $R$ be a ring and let $G$ be a locally finite group. Then, $RG$ is a GUSC
ring if, and only if, $R$ is a GUSC ring and $G$ is a $p$-group with $p\in
J(R)$.

###### Proof.

Assuming $RG$ is a GUSC ring, then the result follows directly from
Proposition 3.1.

Conversely, assume $R$ is a GUSC ring and $G$ is a $p$-group such that $p\in
J(R)$. In view of Lemma 2.9, we differ two cases as follows: if $R$ is local,
then $RG$ is local by the usage of [8]. If, however, $R$ is USC, then we apply
Lemma 2.12 and [1, Lemma 16] to have $p=2$. So, $G$ is a locally finite
$2$-group and $R$ is USC; thus, we infer that $RG$ is USC in virtue of [11,
Theorem 3.1]. Therefore, $RG$ is a GUSC ring consulting with Lemma 2.9, as
expected. ∎

As an automatic consequence in the case when $p=2$, we yield:

######  Corollary 3.4.

Let $R$ be a ring and let $G$ be a locally finite $2$-group. Then, $RG$ is a
GUSC ring if, and only if, $R$ is a GUSC ring and $2\in J(R)$.

We close our work with the following construction.

######  Example 3.5.

Let $C_{n}$ denote the cyclic group of order $n$. If $n\geq 3$ is odd and $R$
is a boolean ring, then $RC_{n}$ is not a GUSC ring.

###### Proof.

One finds with [3, Example 4.7] in hand that $1+g^{n-1}\notin U(RC_{n})$, both
$g^{n-1},x+g^{n-1}\in U(RC_{n})$ and

$1+g+\cdots+g^{n-1}\in{\rm Id}(RC_{n}),$

where $G=\\{1,g,\ldots,g^{n-1}\\}$ and $x=g+g^{2}+\ldots+g^{n-1}$.

Thus, we receive

$1+g^{n-1}=1+(g^{n-1})=(1+g+\cdots+g^{n-1})+(x+g^{n-1}).$

Consequently, we have two strongly clean decompositions for $1+g^{n-1}$ in
$RC_{n}$. Therefore, $RC_{n}$ is not GUSC, as claimed. ∎

In closing, we pose the following question:

Problem. If $R$ is a semi-potent ring, does it follows that $R$ is GUSC if,
and only if, $R$ is GUC?

Funding: The first-named author, P.V. Danchev, of this research paper was
partially supported by the Junta de Andalucía under Grant FQM 264, and by the
BIDEB 2221 of TÜBÍTAK.

## References

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# Zero magnetic-field orbital vortices in s-wave spin-singlet superconductors

Maria Teresa Mercaldo  Dipartimento di Fisica “E. R. Caianiello”, Università
di Salerno, IT-84084 Fisciano (SA), Italy  Carmine Ortix  Dipartimento di
Fisica “E. R. Caianiello”, Università di Salerno, IT-84084 Fisciano (SA),
Italy  Institute for Theoretical Physics, Center for Extreme Matter and
Emergent Phenomena, Utrecht University, Princetonplein 5, 3584 CC Utrecht,
Netherlands  Francesco Giazotto  NEST, Istituto Nanoscienze-CNR and Scuola
Normale Superiore, Piazza San Silvestro 12, I-56127 Pisa, Italy  Mario Cuoco
SPIN-CNR, IT-84084 Fisciano (SA), Italy, c/o Università di Salerno, IT-84084
Fisciano (SA), Italy

###### Abstract

Breaking of time-reversal and point-group spatial symmetries can have a
profound impact on superconductivity. One of the most extraordinary effects,
due to the application of a magnetic field, is represented by the Abrikosov
vortices with charged supercurrents circulating around their cores. Whether a
similar phenomenon can be obtained by exploiting spatial symmetry breaking,
e.g. through electric fields or mechanical strain, is a fundamentally relevant
but not yet fully settled problem. Here, we show that in two-dimensional spin-
singlet superconductors with unusually low degree of spatial symmetry content,
vortices with supercurrents carrying angular momentum around the core can form
and be energetically stable. The vortex has zero net magnetic flux since it is
made up of counter-propagating Cooper pairs with opposite orbital moments. By
solving self-consistently the Bogoliubov- de Gennes equations in real space,
we demonstrate that the orbital vortex is stable and we unveil the spatial
distribution of the superconducting order parameter around its core. The
resulting amplitude has a characteristic pattern with a pronounced angular
anisotropy that deviates from the profile of conventional magnetic vortices.
These hallmarks guide predictions and proposals for the experimental
detection.

A vortex in a superfluid or superconductor is a point-like hole which around
its core is marked by a phase gradient velocity field of the particles forming
the bosonic or fermionic condensate. Since its original prediction Onsager1949
and investigation Feynman1955  in superfluid helium, vortices have been
successfully observed in a broad range of systems including liquid
heliumVinen1961 ; Bewley2008 ; Gomez2014 , ultracold atomic gases
Hadzibabic2006 , photon fields Allen1992  and exciton-polariton superfluids
Lagoudakis2008 ; Roumpos2011 . In superconductors, vortices carry quantized
magnetic fluxes and typically arrange themselves in regular structures. This
prediction Abrikosov1957  has been firmly demonstrated by imaging and
spectroscopic techniques Tonomura2001 ; Roditchev2015 ; Bonevich1994  in a
large variety of so-called type-II superconductors, thus paving the way to a
sparkling development of novel effects and phenomena with vortex quantum
matter.

While in superconductors the connection between vortices and magnetic field or
time-reversal symmetry breaking is well settled, whether vortex structures can
be induced by breaking spatial symmetries is an open and challenging problem.
In this context, structural and bulk inversion symmetry breaking, leading to
Rashba Rashba1960  and Dresselhaus Dresselhaus1955  spin-orbit coupling
respectively, or lack of rotational or mirror symmetries, e.g. by built-in
electric or strain fields, can play a relevant role. Non-centrosymmetric
superconductors have been proposed to host anomalous Abrikosov vortices
Hayashi2006 ; Garaud2020  but in the presence of applied magnetic fields,
whose unconventional character can also arise from parity mixing of spin-
singlet and spin-triplet pairs Gorkov2001 ; Frigeri2004 , magnetoelectric
effects Levitov1985 ; Edelstein1995 , nonstandard Andreev states Vorontsov2008
; Tanaka2009 , and topological phases Sato2009 ; Schnyder2010 ; Ying2017 . As
it concerns electrical manipulation, the recent observations DeSimoni2018  of
critical supercurrent suppression in metallic superconductors by gating and of
other anomalous effects Bours2020 ; Rocci2020 ; Paolucci2019 ; DeSimoni2019 ;
Paolucci2018 ; Ritter2021 ; Alegria2021 ; Rocci2021 ; Golokolenov2021 ;
Desimoni2021 , further underline a profound complexity, not yet fully
uncovered, behind the coupling between superconductivity, point group symmetry
breaking, and electric or strain fields.

In this Letter, we tackle this fundamental problem and demonstrate the
formation in 2D superconductors with very low crystalline symmetry of a vortex
phase with zero net magnetic flux and spin-singlet pairs having non-vanishing
orbital moments. This phenomenon relies on the direct coupling between the
atomic orbital angular momentum ${\bf L}$ and the crystal wave-vector ${\bf
k}$ via the orbital-driven Rashba coupling Park2011 ; Park2012 ; Kim2013 ;
Mercaldo2020  due to crystalline potentials or externally applied electric
fields that break spatial inversion. We find that the orbital Rashba coupling
($\alpha_{{\text{OR}}}$) can induce and stabilize the nucleation of a vortex
state made of counter-propagating Cooper pairs having total spin zero and
opposite angular momentum associated with atomic orbital moments. Hallmarks of
the orbital vortex state are represented by an inhomogeneous pattern of
charge-neutral supercurrents with three-dimensional orbital moment textures,
and a distinctive angular profile of the spatial amplitude of the order
parameter around the vortex core. On the basis of symmetry constraints,
electric field and/or strain turn out to be the most prospective means to
generate and manipulate orbital vortices.

Figure 1: (a) Illustration of the electron distribution associated with atomic
orbitals having zero (a) and non-vanishing (b,c) projections of the angular
momentum components $L_{\alpha}$ with $\alpha=(a,b,c)$ indicating the
$(d_{xz},d_{yz},d_{xy})$ states. Sketch of $+1$ (b) and $-1$ (c)
configurations with collinear orbital moments for various $\hat{L}$
components. Since the orbital density is the same for the $\pm$ states, we use
a clockwise or anticlockwise arrow to indicate the opposite angular momentum
along a given direction. Here, $|\pm\rangle_{c}=\frac{1}{\sqrt{2}}(\mp
i|0\rangle_{a}+|0\rangle_{b})$ and similar superpositions apply for
$|\pm\rangle_{a}$,$|\pm\rangle_{b}$. Taking for example the $L_{c}$
projection, single electrons (d) or electron pairs (e) with opposite
velocities and angular momentum lead to a current with non-zero orbital
momentum. For an orbital vortex we have opposite sign in the circular velocity
and orbital momentum (f).

Model. In order to address the role of point-group symmetries and orbital
degrees of freedom in setting out a vortex phase, we consider a multi-orbital
2D electronic system that includes both mirror and rotation symmetry breaking
interactions. A minimal description can be based on three bands arising from
atomic orbitals spanning an $L=1$ angular momentum subspace, such as $p$ or
$d$ orbitals. For convenience we refer to orbitals localized at the site of a
square lattice (Fig. 1(a)) with $|0\rangle_{l}$ ($l=(a,b,c)$) representing
Wannier states with zero angular momentum configurations, i.e.
${{}_{\alpha}}\langle 0|\hat{L}|0\rangle_{\alpha}=0$ (Fig. 1a). In this basis,
the angular momentum components are given by $\hat{L}_{k}=i\varepsilon_{klm}$,
with $\varepsilon_{klm}$ the Levi-Civita tensor. The breaking of inversion and
of the horizontal mirror symmetry $\emph{M}_{z}$, e.g. due to structural
inversion asymmetry and tunable by externally applied electric fields, sets
out an orbital Rashba interaction that couples the atomic angular momentum
${\bf L}$ with the crystal wave-vector ${\bf k}$. The residual vertical mirror
symmetries ($\emph{M}_{x}$ and $\emph{M}_{y}$) as well as the $C_{4}$ rotation
around the $\hat{z}$ axis can be broken for instance by local built-in elecric
fields or inhomogeneous strains. This additional symmetry lowering can be
included by terms that involve the product of distinct orbital angular
momentum components. Hence, assuming a conventional s-wave spin-singlet
pairing due to a local attraction, the Hamiltonian in real space can be
expressed as Mercaldo2020 ; Bours2020 ; Mercaldo2021_a :

$\displaystyle\mathcal{H}=\sum_{\langle
i,j\rangle}\Psi^{\dagger}(i)\hat{H}(i,j)\Psi(j)\,,$ (1)

with

$\displaystyle\hat{H}(i,j)=(\hat{t}_{i,j}-\mu\delta_{i,j})\tau_{z}+\hat{\Delta}\tau_{x}\delta_{i,j}$
(2)

within the spinorial basis
$\Psi^{\dagger}(i)=(\Psi_{\uparrow}^{\dagger}(i),\Psi_{\downarrow}(i))$
assuming that
$\Psi_{\sigma}^{\dagger}(i)=(c^{\dagger}_{i,0_{a},\sigma},c^{\dagger}_{i,0_{b},\sigma},c^{\dagger}_{i,0_{c},\sigma})$,
$\tau_{i}$ are the Pauli matrices in the particle-hole space, $\delta_{i,j}$
is the Kronecker delta function. Here, $c_{r,0_{\alpha}\sigma}$ is the
canonical electron annihilation operator at the position $r$, with spin
$\sigma$ and zero orbital polarization components ($\alpha=(a,b,c)$). The
generalized hopping matrix is expressed as
$\hat{t}_{i,j}=-t\hat{L}_{a}^{2}-i\alpha_{\text{OR}}\hat{L}_{b}+\frac{\gamma}{2}(\hat{L}_{c}^{2}+\hat{L}_{b}^{2}-\hat{L}_{a}^{2})+t_{m}[\\{\hat{L}_{a},\hat{L}_{c}\\}+\\{\hat{L}_{b},\hat{L}_{c}\\}]$
if the bond $<i,j>$ is along $x$, while
$\hat{t}_{i,j}=-t\hat{L}_{b}^{2}-i\alpha_{\text{OR}}\hat{L}_{a}+\frac{\gamma}{2}(\hat{L}_{c}^{2}+\hat{L}_{a}^{2}-\hat{L}_{b}^{2})+t_{m}[\\{\hat{L}_{a},\hat{L}_{c}\\}+\\{\hat{L}_{b},\hat{L}_{c}\\}]$
for $y$ direction, with $\\{\hat{A},\hat{B}\\}$ indicating the anticommutator.
$\alpha_{\text{OR}}$ is the strength of the orbital Rashba coupling whose
structure in momentum space reads as
$\hat{O}_{or}=\sin(k_{x})\hat{L}_{b}-\sin(k_{y})\hat{L}_{a}$ SupMat . We also
notice that upon a $C_{4}$ rotation $k_{x}\rightarrow k_{y}$,
$k_{y}\rightarrow-k_{x}$ and the same applies to the angular momentum
components, i.e. $\hat{L}_{a}\rightarrow\hat{L}_{b}$,
$\hat{L}_{b}\rightarrow-\hat{L}_{a}$, $\hat{L}_{c}\rightarrow\hat{L}_{c}$.
Furthermore, for the mirror symmetry $\emph{M}_{y}$ we have that
$k_{y}\rightarrow-k_{y}$ ($k_{x}$ and $k_{z}$ stay unchanged) while the
angular momentum is a pseudovector and hence the component perpendicular to
the mirror plane does not transform, i.e. $\hat{L}_{b}\rightarrow\hat{L}_{b}$,
while the other are inverted, i.e. $L_{a}\rightarrow-\hat{L}_{a}$ and
$L_{c}\rightarrow-\hat{L}_{c}$. On the basis of these symmetry
transformations, we have that the orbital Rashba interaction term breaks only
the horizontal mirror symmetry $\emph{M}_{z}$ while $t_{m}$ breaks the
vertical mirrors and rotations. Note that $\gamma$ is an orbital dependent
anisotropic term that is instead compatible with a $C_{4v}$ point group.
Concerning the superconducting order parameter, we consider a conventional
scenario with only local s-wave spin-singlet pairs with intra- and inter-
orbital components. The general structure of $\hat{\Delta}$ in the orbital
space can be expressed in the following form,
$\hat{\Delta}=\frac{1}{2}\sum_{l,m}g_{\alpha\beta}\Delta_{\alpha\beta}(\hat{L}_{\alpha}\cdot\hat{L}_{\beta}+\hat{L}_{\beta}\cdot\hat{L}_{\alpha}$)
with $g_{\alpha\beta}$ being the orbital dependent pairing interaction and
$\Delta_{\alpha\beta}(r)=\langle
c_{r,0_{\alpha}\uparrow}c_{r,0_{\beta}\downarrow}\rangle$ the related
superconducting order parameters.

Here, $\langle...\rangle$ stands for the ground-state expectation value. Due
to the structure of $\hat{\Delta}$ we observe that the intra- and inter-
orbital components of the order parameter (OP) have different symmetry
properties. The intra-orbital terms $\Delta_{\alpha\alpha}$ are generally non-
vanishing with the point-group symmetries that only pose constraints on their
specific amplitudes. On the other hand, since $\Delta_{\alpha\beta}$ is local
and transforms as
$(\hat{L}_{\alpha}\cdot\hat{L}_{\beta}+\hat{L}_{\beta}\cdot\hat{L}_{\alpha})$
then, it can change sign upon rotation and vertical mirror symmetry
transformation, thus resulting into a vanishing amplitude. A lack of these
symmetries is thus required to have an amplitude $\Delta_{\alpha\beta}$
different from zero. More details about the symmetry property of the
superconducting order parameters are reported in the supplemental material
SupMat . Such low symmetry content can be encountered at the surface or
interface due to, e.g., atomic termination profiles, strain or external
electric field gradients. Alternatively, the point group symmetries can be
intrinsically broken by an unusually low crystalline symmetry arrangement.

Now, we show that, if these order parameters have a non-trivial spatial
dependence, akin to pair density waves with specific relations among their
amplitudes and relative phases, a vortex state with electron pairs having non-
vanishing and collinear orbital moments can be obtained. This outcome is
directly accessible by firstly observing that for any component of the orbital
angular momentum one can construct pairs with $\pm 2$ total projection.
Indeed, for electron pairs with parallel orbital moments oriented along
$\alpha$ we have that $\eta_{+,\alpha}(r)=\langle
c_{r,+_{\alpha}\uparrow}c_{r,+_{\alpha}\downarrow}\rangle$ and
$\eta_{-,\alpha}(r)=\langle
c_{r,-_{\alpha},\uparrow}c_{r,-_{\alpha},\downarrow}\rangle\,$. Then, we
notice that from the structure of the single particle states having $\pm 1$
orbital moment, $\eta_{\pm}$ is expressed as a combination of intra- and
inter-orbital pairing amplitudes. For instance, $\eta_{\pm,c}$ is given by

$\displaystyle\eta_{\pm,c}(r)=\left[\frac{1}{2}\left(-\Delta_{aa}(r)+\Delta_{bb}(r)\right)\pm
i\Delta_{ab}(r)\right]\,,$ (3)

and similar expressions can be obtained for the other components
$\eta_{\pm,a}$ and $\eta_{\pm,b}$ SupMat . Hence, to describe a vortex phase
with electron pairs having equal orbital moments (i.e. orbitally polarized),
the order parameter has to be expressed as $\eta_{\pm}(r)=f(r)\exp[\pm
i\theta(r)]$, whose non-trivial phase winding has opposite sign for the
corresponding $\pm$ orbital angular momentum, while the spatial dependent
amplitude is encoded in $f(r)$. Here, the phase winding is an orbital coherent
effect which is evident in the structure of $\eta_{\pm}$ through its real and
imaginary components, with $\eta_{\pm}$ being time-reversal partners. With the
above elements, the vortex state can be directly obtained by considering the
angular variable $\theta(r)$ for the vector ${\bf(r-r_{0})}=(x-x_{0},y-y_{0})$
in the $x$-$y$ plane with respect to the center ${\bf r}_{0}$, and let the
intra- and inter-orbital order parameters have a pair density modulation in
real space such as
$\Delta_{\alpha\alpha}(r)-\Delta_{\beta\beta}(r)=2f(r)\cos\theta(r)$ and
$\Delta_{\alpha\beta}(r)=f(r)\sin\theta(r)$.

Next, we present one specific hallmark of the vortex state, which is directly
related to its orbital moment character and the presence of orbital currents.

Figure 2: (a) Illustration of electron pairs with non-zero orbital moment
having reversed sign of the angular momentum projections and opposite velocity
thus leading to a net orbital supercurrent without charge flow. (b) Spatial
distribution of the orbital supercurrent around the vortex core (black dot)
for the current $J^{a}$ carrying orbital moment oriented along the $x$
direction. The arrow indicates the direction of the orbital current on each
bond, and the length its amplitude. The vortex state is marked by a non-
trivial phase winding for $\eta_{\pm,b}(r)$ and $\eta_{\pm,a}(r)$ order
parameters. The order parameter $\eta_{\pm,c}(r)$ with $c$-oriented orbital
moment has zero amplitude. The simulation refers to a system size $N_{x}\times
N_{y}$ with $N_{x}=N_{y}=40$ while the following parameters of the 2D model
Hamiltonian have been used:
$\alpha_{\text{OR}}=1.0t$,$\gamma=0.1t$,$t_{m}=0.2t$,$g\Delta_{0}=0.1t$,
$\mu=-0.4t$ and for the order parameters $f_{0}=0.3$,$f_{1}=f_{2}=0.5$, and
$g_{\alpha\beta}=g=2t$.

As for the spin-current Nikolic2006 , one can introduce orbital-current
operators on a bond $\langle i,j\rangle$ for the various orbital moment
components. We focus on the leading term for the transfer of orbital angular
momentum, i.e.
$\hat{J}^{\alpha}_{ij}=i(\,c^{\dagger}_{i,0_{m}\sigma}\hat{L}_{\alpha}^{m,m^{\prime}}c_{j,0_{m^{\prime}}\sigma}-\text{h.c.})$
SupMat . We then evaluate the distribution of the orbital current in the
superconducting state around the vortex core. We have selected a
representative set for the local superconducting order parameters assuming
that $\Delta_{aa}=\Delta_{bb}\neq\Delta_{cc}$ and $\Delta_{ac}=\Delta_{bc}$,
while $\Delta_{ab}=0$. For convenience, the parameterization in real space for
$\hat{\Delta}$ at a given position, $(x_{i},y_{i})$ is expressed as
$[\Delta_{aa(bb)}(x_{i},y_{i})-\Delta_{cc}(x_{i},y_{i})]=f_{1}\Delta_{0}\cos(\theta(x_{i},y_{i}))$,
$\Delta_{cc}(x_{i},y_{i})=f_{0}\Delta_{0}$ and
$\Delta_{ac(bc)}=f_{2}\Delta_{0}\sin(\theta(x_{i},y_{i}))$, where $\Delta_{0}$
sets the overall scale and $f_{0}$, $f_{1}$, $f_{2}$ are coefficients that
take into account the possible orbital anisotropy of the superconducting
state. Variations in the $f$ values do not affect the qualitative outcome of
the orbital current profile.

In the absence of vortices, we have a uniform equilibrium orbital supercurrent
with in-plane directional character, i.e. it substantially transfers Cooper
pairs with angular momentum parallel to $b(c)$ when propagating along $x(y)$,
respectively. The current with an out-of-plane orbital moment is vanishing for
the uniform solution. On the contrary, the pattern of orbital supercurrents
exhibits a characteristic winding of the angular momentum flow due to the
presence of the vortex (Figs. 2b). The vorticity is regular and of the same
type for $J^{a}$ and $J^{b}$ while it has a sort of more turbulent aspect for
$J^{c}$ SupMat . The circulation of orbital moments are fingerprints of the
superconducting orbital vortex. We notice that the flow of the orbital
supercurrent does not show a complete winding around the vortex core. This is
due to the fact that in the ground state there are configurations which are
reminiscent of the uniform pattern of orbital currents in the normal state.
Then, these contributions tend to give a cancellation in some regions
amplifying the amplitude asymmetry around the core (Figs. 2b).

Let us move to the energetics of the orbital vortex state. The strategy here
is to firstly analyze the energy competition between the vortex and uniform
states by assuming a homogeneous profile for the order parameters. Then, we
determine the vortex configuration solving self-consistently the equations in
real space for the local amplitudes of all the order parameters. In Fig. 3a we
present the phase diagram obtained by comparing the free energy of the uniform
phase with that of an orbital vortex state having uniform amplitude of the
order parameter. We set an amplitude $\Delta_{0}$ to fix the overall scale for
intra- and inter-orbital order parameters. Since the coherence length
decreases when $\Delta_{0}$ grows one can effectively span different regimes
of $\xi_{S}/L$, as for BCS superconductors $\Delta_{0}\sim 1/\xi_{S}$ (in the
performed simulation for $g\Delta_{0}/t\sim 0.05$ we have $\xi_{S}/L\sim 1$).
The simulation is performed for a system size $N_{x}\times N_{y}$, where
$N_{x}=N_{y}=L$ and $L=70$ in units of the interatomic distance. We consider
the orbital Rashba coupling as the main varying parameter and assume a given
amplitude of the coupling ($t_{m}=0.2t)$ that breaks mirrors and rotational
symmetries. The outcome indicates that there is a boundary line of critical
$\alpha_{\text{OR}}$ separating the region of phase space where the vortex is
energetically more stable than the uniform state. The $\alpha_{\text{OR}}$
coupling can lead to a transition into the vortex phase with uniform
amplitude. For sufficiently low values of the superconducting interaction and
superconducting gap, corresponding to $\xi_{S}/L\sim 1$, we find that the
orbital Rashba interaction becomes substantially ineffective in sustaining the
orbital vortex phase. This observation qualitatively implies that in
superconductors with low critical temperature the nucleation of orbital
vortices might be less favorable.

To demonstrate the stability of the vortex state and to corroborate this trend
obtained by making an ansatz on the amplitude of the superconducting order
parameters, we performed a full self-consistent analysis of the spatial
dependent order parameters. In Fig. 3b we compare the energy of the solutions
obtained by self-consistently solving the gap equations at each lattice site
for the $\eta_{\alpha}$ assuming either a vortex phase winding or a spatially
uniform phase for the superconducting order parameters. The presented
simulation is for a system size $N_{x}=N_{y}=20$. However, we have verified
for few values of $\alpha_{\text{OR}}$ that for larger size, up to
$N_{x}=N_{y}=70$, the stability is preserved SupMat . We observe that above a
critical amplitude of $\alpha_{\text{OR}}$ the vortex state becomes unstable.
This behavior can be understood by noticing that $\alpha_{\text{OR}}$ is
suppressing the amplitude of the SC-OPs and thus brings the system into a
regime where the orbital Rashba coupling is unable to stabilize the vortex
state. Such self-consistent trajectory of the OP is schematically sketched in
Fig. 3a. The analysis of the amplitude of $\eta$ in the vortex state indicates
a peculiar angular anisotropy (Figs. 3c,d) with a modulation of intensity from
small to large values as one moves around the center of the vortex. The
pattern of $\eta$ is consistent with mirror and rotational symmetry breaking.
The position of maximal amplitude of the order parameter is typically pinned
to the high symmetry lattice directions thus having possible implications
concerning the formation of domains.

Figure 3: Phase diagram with uniform and orbital vortex configurations as a
function of the energy scale ($g\Delta_{0}$), setting the amplitudes of
superconducting order parameters $\Delta_{\alpha\beta}$, and the orbital
Rashba interaction ($\alpha_{\text{OR}}$). The simulation refers to a system
size $N_{x}\times N_{y}$ with $N_{x}=N_{y}=70$, while the model parameters are
the same as in Fig.2. (b) Energy difference between the vortex solution
($E_{v}$) and the uniform one ($E_{u}$) as a function of the orbital Rashba
coupling $\alpha_{\text{OR}}$. Both solutions have been obtained evaluating
the SC OP components $\Delta_{\alpha\beta}$ with an iterative self-consistent
procedure in real space for a system of size $N_{x}=N_{y}=20$. In the self-
consistent analysis the components of the SC OP decrease by increasing
$\alpha_{\text{OR}}$ (see SupMat ). This trajectory is approximately given by
the purple dashed line drawn in (a). In the insets we have a graphical
representation of the order parameters ($\eta_{\pm}$) with opposite winding in
the vortex, assuming that the vector components are the corresponding real and
imaginary parts. (c-d) Density plot of the amplitude $\eta_{b}$ for two values
of $\alpha_{\text{OR}}$, obtained self-consistently for a system of size
$70\times 70$. $|\eta_{b}|$ shows a non-uniform spatial behavior, with a
distribution which changes varying $\alpha_{\text{OR}}$. The white arrows give
a representation of the winding of $\eta_{+,b}$, where the vector components
are the real and imaginary part of $\eta_{+,b}/|\eta_{b}|$. For clarity of
visualization the arrows are not drawn at all the sites. The spatial profile
for $\eta_{\pm,a}$ is the same of $\eta_{\pm,b}$ while $\eta_{\pm,c}$ has a
negligible amplitude. The parameters are $\gamma=0.1t$,$t_{m}=0.2t$,
$\mu=-0.4t$, $g_{ab}=0$, and for the other bands $g_{\alpha\beta}=2t$. As
supplemental material we provide movies about the evolution of $\eta_{\pm,b}$
by varying $\alpha_{\text{OR}}$.

Let us finally discuss how to detect this type of vortex and few direct
consequences of their presence. The orbital vortex has zero net magnetic flux.
However since neutral currents of magnetic dipoles can produce electric fields
Sun2004 , we predict that the vortex state of orbital moments can exhibit a
weak electric field, at a given distance $R$ from the orbital current sources,
according to the following relation ${\bf E}=\int
d^{3}r\frac{\mu_{0}}{4\pi}{\bf
J}_{\alpha}\times\frac{1}{R^{3}}[\hat{n}_{\alpha}-\frac{3{\bf R}({\bf
R}\cdot\hat{n}_{\alpha})}{R^{2}}]$ Sun2004 , with ${\bf J}_{\alpha}$ and
$\hat{n}_{\alpha}$ the orbital current and orbital moment associated with the
$\alpha$ orientation, respectively. This electric field would be typically
screened on a distance of the order of the Thomas-Fermi length. On the basis
of the patterns in Fig. 2, especially outward to surface or along directions
of poor screening inside the material, around the core of the vortex there
would be the most prominent features of an electric field distribution with
in-plane and out-of-plane components.

Another path to unveil the presence of vortices with orbital moments is to
make use of weak perturbations that break time-reversal symmetry such as
external currents or small magnetic fields. In this case, the lack of time
reversal symmetry would lead to uncompensating effects in the vortex orbital
flow and weak orbital moments of the Cooper pairs that would result into
anomalous magnetic flux response. The flux is not quantized and can be
directly probed by means of SQUID or electron microscopy techniques.
Furthermore, to detect the presence of orbital vortices one could also employ
scanning tunneling microscopy (STM). On the basis of the achieved self-
consistent patterns (Figs. 3c,d), we expect that a variation of the orbital
Rashba coupling has an impact on the spatial distribution of the
superconducting order parameter nearby the vortex core. This implies that
spatially resolved STM measurements would detect an enhanced population of in-
gap quasiparticles inside the vortex with different characteristic lengths
even in the absence of external magnetic field.

If the crystalline environment does not provide symmetry breaking interactions
that are sufficiently strong to allow for the nucleation of the orbital
vortex, we expect that the application of external electric fields or strains
can be the most impactful means to yield the orbital vortex phase. Hence, we
foresee the achievement of an electric- or strain-driven dual of the magnetic
vortex phase that can open a wide avenue of explorations of quantum vortex
phases in ultrathin superconductors.

###### Acknowledgements.

M.C., M.T.M and F.G. acknowledge support by the EU’s Horizon 2020 research and
innovation program under Grant Agreement nr. 964398 (SUPERGATE). C.O.
acknowledges support from a VIDI grant (Project 680-47-543) financed by the
Netherlands Organization for Scientific Research (NWO). F.G. acknowledges the
European Research Council under Grant Agreement No. 899315-TERASEC, and the
EU’s Horizon 2020 research and innovation program under Grant Agreement No.
800923 (SUPERTED) for partial financial support. We acknowledge valuable
discussions with A. Caviglia and A. di Bernardo.

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