# Small cocircuits in minimally vertically $4$-connected matroids

James Oxley  Mathematics Department, Louisiana State University, Baton Rouge,
Louisiana, USA<EMAIL_ADDRESS>and  Zach Walsh  Mathematics Department,
Louisiana State University, Baton Rouge, Louisiana, USA<EMAIL_ADDRESS>
###### Abstract.

Halin proved that every minimally $k$-connected graph has a vertex of degree
$k$. More generally, does every minimally vertically $k$-connected matroid
have a $k$-element cocircuit? Results of Murty and Wong give an affirmative
answer when $k\leq 3$. We show that every minimally vertically $4$-connected
matroid with at least six elements has a $4$-element cocircuit, or a
$5$-element cocircuit that contains a triangle, with the exception of a
specific non-binary $9$-element matroid. Consequently, every minimally
vertically $4$-connected binary matroid with at least six elements has a
$4$-element cocircuit.

##  1\. Introduction

A graph $G$ is _minimally $k$-connected_ if it is $k$-connected, and
$G\\!\setminus\\!e$ is not $k$-connected for all edges $e$ of $G$. While
$k$-connected graphs have no vertices of degree less than $k$, Halin [1]
proved that every minimally $k$-connected graph has a vertex of degree exactly
$k$. Since vertices of small degree are useful, for example, for facilitating
inductive arguments, Halin’s result was strengthened several times until
finally Mader [2] proved a tight lower bound on the number of vertices of
degree $k$ in a minimally $k$-connected graph.

Although matroids in general do not have vertices, it has been common to use
cocircuits as matroid analogues of vertices. When we seek analogues of Halin’s
theorem, we find that there are two widely used notions of matroid
connectivity, namely (Tutte) connectivity, and vertical connectivity.

The analogue of Halin’s result for $k$-connectivity was studied by Reid, Wu,
and Zhou [6]. A matroid $M$ is _minimally $k$-connected_ if it is
$k$-connected, and $M\\!\setminus\\!e$ is not $k$-connected for all elements
$e$ of $M$. Reid et al. sought to prove the following (see [5, Problem
14.4.9]).

######  Conjecture 1.1.

If $M$ is a minimally $k$-connected matroid with $|E(M)|\geq 2(k-1)$, then $M$
has a cocircuit of size $k$.

Results of Murty [3] and Wong [8] prove the conjecture when $k\leq 3$. Reid et
al. [6] proved that the conjecture holds when $k=4$, with a unique exception,
the $9$-element rank-$4$ matroid $N_{9}$ that is represented over
$\operatorname{GF}(3)$ by the matrix in Figure 1.

######  Theorem 1.2 (Reid, Wu, Zhou).

Let $M$ be a minimally $4$-connected matroid with at least six elements such
that $M\ncong N_{9}$. Then $M$ has a $4$-element cocircuit.

$\begin{bmatrix}1&0&0&0&0&1&1&-1&1\\\ 0&1&0&0&1&0&1&1&-1\\\
0&0&1&0&1&1&0&1&1\\\ 0&0&0&1&-1&1&1&0&-1\end{bmatrix}.$

Figure 1. A ternary representation for $N_{9}$.

Reid et al. constructed $N_{9}$ from a $2$-$(9,4,3)$-design, so it is highly
structured. It has a transitive automorphism group, and each element $x$
satisfies $N_{9}\\!\setminus\\!x\cong P_{8}$ and
$N_{9}/x\cong\operatorname{AG}(2,3)\\!\setminus\\!e$. Since
$\operatorname{AG}(2,3)\\!\setminus\\!e$ is not binary, $N_{9}$ is not binary.

In the same paper, Reid et al. disprove Conjecture 1.1 for each $k\geq 5$, by
finding a minimally $k$-connected matroid with $2k+1$ elements and no
$k$-cocircuits. They conjecture that Conjecture 1.1 holds when $|E(M)|\neq
2k+1$, so the analogue of Halin’s theorem for $k$-connectivity may still hold
for $k\geq 5$.

In this paper, we focus on the analogue of Halin’s theorem for vertical
$k$-connectivity. This is a weaker connectivity property than
$k$-connectivity, in the sense that every $k$-connected matroid is also
vertically $k$-connected. A matroid $M$ is _minimally vertically
$k$-connected_ if it is vertically $k$-connected, and $M\\!\setminus\\!e$ is
not vertically $k$-connected for all elements $e$ of $M$. Thus Halin’s theorem
prompts the following natural question for vertical $k$-connectivity.

######  Problem 1.3.

Does every minimally vertically $k$-connected matroid with at least $2k+2$
elements have a $k$-cocircuit?

Since a graph $G$ is $k$-connected if and only if the graphic matroid $M(G)$
is vertically $k$-connected, this problem seeks a direct generalization of
Halin’s result. The condition that the matroid has at least $2k+2$ elements is
necessary, due to the construction of Reid et al. [6].

When $k\leq 3$, minimal vertical $k$-connectivity and minimal $k$-connectivity
coincide, so the results of Murty [3] and Wong [8] affirmatively answer
Problem 1.3 when $k\leq 3$. However, for $k\geq 4$, minimal vertical
$k$-connectivity is a strictly weaker condition than $k$-connectivity. We take
a step towards resolving Problem 1.3 for $k=4$ by showing that every minimally
vertically $4$-connected matroid, except for the $9$-element matroid in
Theorem 1.2, has a small cocircuit with special structure.

######  Theorem 1.4.

Let $M$ be a minimally vertically $4$-connected matroid with at least six
elements such that $M\ncong N_{9}$. Then $M$ has a $4$-cocircuit, or a
$5$-cocircuit that contains a triangle.

Since no binary matroid has a $5$-cocircuit that contains a triangle, Theorem
1.4 shows that Problem 1.3 has an affirmative answer for binary matroids when
$k=4$.

######  Theorem 1.5.

Every minimally vertically $4$-connected binary matroid with at least six
elements has a $4$-cocircuit.

Despite these positive results, we conjecture that Problem 1.3 has a negative
answer when $k=4$ in the following strong sense.

######  Conjecture 1.6.

There is an infinite family of minimally vertically $4$-connected matroids
with no $4$-cocircuits.

In a previous version of this paper we claimed to prove this conjecture, and
we are grateful to the anonymous referee who pointed out an error in the
proof. While we could not fix this error, we expect that there is a clever
construction that proves Conjecture 1.6.

There is a key relationship between $4$-connectivity and vertical
$4$-connectivity that allows us to use Theorem 1.2 in the proof of Theorem
1.4. Specifically, a non-uniform matroid is minimally $4$-connected if and
only if it is minimally vertically $4$-connected, and has no triangles. Thus,
by Theorem 1.2, it suffices to prove Theorem 1.4 for matroids with a triangle.
In fact, we show that every triangle of a minimally vertically $4$-connected
matroid intersects a small cocircuit with special structure.

######  Theorem 1.7.

Let $M$ be a minimally vertically $4$-connected matroid with a triangle $T$.
Then

  1. $(1)$

$M$ has a $4$-cocircuit that contains exactly two elements of $T$, or

  2. $(2)$

$M$ has a $5$-cocircuit that contains a triangle and exactly two elements of
$T$, or

  3. $(3)$

$|E(M)|\leq 11$.

We can even relax the condition that $M$ is minimally vertically
$4$-connected, and still find a cocircuit with specific structure.

######  Theorem 1.8.

Let $M$ be a vertically $4$-connected matroid with a triangle $T=\\{e,f,g\\}$
so that none of $M\\!\setminus\\!e$, $M\\!\setminus\\!f$, $M\\!\setminus\\!g$
is vertically $4$-connected. Then either

  1. $(1)$

$M$ has a cocircuit $C^{*}$ so that $|C^{*}\cap T|=2$ and $M|C^{*}\cong
U_{2,k}\oplus U_{2,2}$ for some $k\geq 2$, or

  2. $(2)$

$r(M)\leq 6$, and, if $M$ has no $U_{2,4}$-restrictions, then $|E(M)|\leq 11$.

This is an analogue of a result of Tutte [7] called Tutte’s Triangle Lemma
(see [4, Lemma 8.7.7]), which finds a $3$-cocircuit, or _triad_ , that
intersects a given triangle of a $3$-connected matroid.

######  Theorem 1.9 (Tutte).

Let $M$ be a $3$-connected matroid with a triangle $\\{e,f,g\\}$ so that
neither $M\\!\setminus\\!e$ nor $M\\!\setminus\\!f$ is $3$-connected. Then $M$
has a triad that contains $e$ and exactly one of $f$ and $g$.

Our proof of Theorem 1.8 follows the proof of Theorem 1.9. Before proving
Theorems 1.4, 1.5, 1.7, and 1.8 in Section 3, we discuss some preliminaries in
Section 2. We close by discussing several related open problems in Section 4.

##  2\. Preliminaries

We follow the notation of Oxley [4]. Given a matroid $M$ with ground set $E$
and rank function $r$, the function $\lambda_{M}$ defined by

$\displaystyle\lambda_{M}(X)=r(X)+r(E-X)-r(M)$

is the _connectivity function_ of $M$. Tutte [7] proved that this function is
_submodular_ , which means that all $X,Y\subseteq E(M)$ satisfy

$\displaystyle\lambda_{M}(X)+\lambda_{M}(Y)\geq\lambda_{M}(X\cup
Y)+\lambda_{M}(X\cap Y).$

For a positive integer $j$, a partition $(X,Y)$ of the ground set of a matroid
$M$ is a _vertical j-separation_ of $M$ if $\lambda_{M}(X)<j$ and
$\min\\{r(X),r(Y)\\}\geq j$. The vertical $j$-separation is _exact_ if
$\lambda_{M}(X)=j-1$. For an integer $k$ exceeding one, a matroid $M$ is
_vertically k-connected_ if it has no vertical $j$-separations with $j<k$.

While this is the main notion of connectivity in this paper, we also make some
use of (Tutte) $k$-connectivity. For $j\geq 1$, a partition $(X,Y)$ of the
ground set of a matroid $M$ is a _j-separation_ of $M$ if $\lambda_{M}(X)<j$
and $\min\\{|X|,|Y|\\}\geq j$. The $j$-separation is _exact_ if
$\lambda_{M}(X)=j-1$. For $k\geq 2$, a matroid $M$ is _k-connected_ if it has
no $j$-separations with $j<k$. Every vertical $j$-separation is a
$j$-separation, so every $k$-connected matroid is vertically $k$-connected.

There are two natural relationships between $k$-connectivity and vertical
$k$-connectivity for non-uniform matroids. First, a non-uniform matroid is
$k$-connected if and only if it is vertically $k$-connected, and has no
circuits with fewer than $k$ elements. In particular, a non-uniform simple
matroid is $4$-connected if and only if it is vertically $4$-connected and has
no triangles. Second, a non-uniform matroid $M$ is $k$-connected if and only
if $M$ and $M^{*}$ are both vertically $k$-connected. These two relationships
combine to show that a non-uniform vertically $k$-connected matroid has no
cocircuits with fewer than $k$ elements. In particular, a non-uniform
vertically $4$-connected matroid has no triads. We need one more useful
property of vertically $4$-connected matroids.

######  Lemma 2.1.

Let $M$ be a vertically $4$-connected matroid. For a subset $X$ of $E(M)$ and
an element $x$ of $X$, if $M|X$ is isomorphic to $U_{2,4}$ or
$U_{2,4}\oplus_{2}U_{2,4}$, then $M\\!\setminus\\!x$ is vertically
$4$-connected.

###### Proof.

Suppose that $M|X\cong U_{2,4}$. Let $(A,B)$ be a vertical $3$-separation of
$M\\!\setminus\\!x$. Then $A$ or $B$ contains at least two elements of $X-x$
and thus spans $x$, so either $(A\cup x,B)$ or $(A,B\cup x)$ is a vertical
$3$-separation of $M$, a contradiction.

Suppose that $M|X\cong U_{2,4}\oplus_{2}U_{2,4}$. Then $X$ is the union of
disjoint triangles $T_{1}$ and $T_{2}$. Let $x\in T_{1}$, and let $(A,B)$ be a
vertical $3$-separation of $M\\!\setminus\\!x$. Then, without loss of
generality, $A$ contains at least three elements of $(T_{1}\cup T_{2})-x$. As
$A$ does not span $x$, we deduce that $A\cap X=T_{2}$, so $B$ spans $x$, a
contradiction. ∎

##  3\. The proofs of the main results

In this section, we prove Theorems 1.4, 1.5, 1.7, and 1.8. We first prove a
lemma about the interaction of vertical $3$-separations of $M\\!\setminus\\!a$
and $M\\!\setminus\\!b$, where $a$ and $b$ are in a common triangle.

######  Lemma 3.1.

Let $M$ be a vertically $4$-connected simple matroid with a triangle
$T=\\{a,b,c\\}$. Let $(X_{a},Y_{a})$ and $(X_{b},Y_{b})$ be exact vertical
$3$-separations of $M\\!\setminus\\!a$ and $M\\!\setminus\\!b$, respectively,
so that $b\in X_{a}$ and $a\in X_{b}$. Then either

  1. $(i)$

$M$ has a cocircuit $C^{*}$ so that $|C^{*}\cap T|=2$ and $M|C^{*}\cong
U_{2,k}\oplus U_{2,2}$ for some $k\geq 2$, or

  2. $(ii)$

$r(X_{a}\cap Y_{b})=2$ and $r(X_{b}\cap Y_{a})=2$, and either $|X_{a}\cap
X_{b}|=1$, or $r(Y_{a}\cap Y_{b})\leq 2$ and $X_{a}\cap X_{b}\neq\varnothing$.

###### Proof.

Suppose that neither $(i)$ nor $(ii)$ holds. Because $M$ has a triangle but
has rank at least three, $M$ is not uniform. Thus, it has no cocircuits of
size less than four. Observe that $c\in Y_{a}\cap Y_{b}$ otherwise $(X_{a}\cup
a,Y_{a})$ or $(X_{b}\cup b,Y_{b})$ is a vertical $3$-separation of $M$.

######  3.1.1.

$M\\!\setminus\\!a,b$ is $3$-connected.

For some $j\in\\{1,2\\}$, let $(A,B)$ be a $j$-separation of
$M\\!\setminus\\!a,b$ with $c\in A$ where $A$ is closed. Then
$\lambda_{M}(A\cup\\{a,b\\})\leq j$, and so either $r(A\cup\\{a,b\\})\leq j$
or $r(B)\leq j$. As $M$ is simple and has no triads, $r(A\cup\\{a,b\\})\geq 2$
and $r(B)\geq 2$. Thus $j=2$, and $r(A\cup\\{a,b\\})=2$ or $r(B)=2$. Suppose
that $r(B)=2$. Then $(i)$ holds with $C^{*}=B\cup\\{a,b\\}$. If
$r(A\cup\\{a,b\\})=2$, then $M|(A^{\prime}\cup\\{a,b\\})\cong U_{2,4}$ for a
$2$-element subset $A^{\prime}$ of $A$, so, by Lemma 2.1, $M\\!\setminus\\!a$
is vertically $4$-connected, a contradiction. Thus 3.1.1 holds.

We make repeated use of the following.

######  3.1.2.

$a\in\operatorname{cl}(X_{b}-a)$ and $b\in\operatorname{cl}(X_{a}-b)$.

If $a\notin\operatorname{cl}(X_{b}-a)$, then
$\lambda_{M\\!\setminus\\!b}(X_{b}-a)\leq 2$. But then the complement of
$X_{b}-a$ in $M\\!\setminus\\!b$ contains $a$ and $c$ and thus spans $b$, so
$\lambda_{M}(X_{b}-a)\leq 2$. This implies that $r_{M}(X_{b}-a)=2$, so $(i)$
holds with $C^{*}=(X_{b}\cup b)-\operatorname{cl}(Y_{b})$. Thus 3.1.2 holds.

The following is due to the submodularity of $\lambda_{M\\!\setminus\\!a,b}$.

######  3.1.3.

  1. $(i)$

$\lambda_{M\\!\setminus\\!a,b}(Y_{a}\cap
Y_{b})+\lambda_{M\\!\setminus\\!a,b}(X_{a}\cap X_{b})\leq 4$, and

  2. $(ii)$

$\lambda_{M\\!\setminus\\!a,b}(X_{a}\cap
Y_{b})+\lambda_{M\\!\setminus\\!a,b}(X_{b}\cap Y_{a})\leq 4$.

We prove 3.1.3$(ii)$. The proof of 3.1.3$(i)$ is similar. We have
$\lambda_{M\\!\setminus\\!a,b}(X_{a}-b)\leq\lambda_{M\\!\setminus\\!a}(X_{a})=2$.
Similarly, $\lambda_{M\\!\setminus\\!a,b}(Y_{b})\leq 2$. Then

(1) $\displaystyle 4$
$\displaystyle\geq\lambda_{M\\!\setminus\\!a,b}(X_{a}-b)+\lambda_{M\\!\setminus\\!a,b}(Y_{b})$
(2) $\displaystyle\geq\lambda_{M\\!\setminus\\!a,b}(X_{a}\cap
Y_{b})+\lambda_{M\\!\setminus\\!a,b}((X_{a}\cup Y_{b})-b)$ (3)
$\displaystyle=\lambda_{M\\!\setminus\\!a,b}(X_{a}\cap
Y_{b})+\lambda_{M\\!\setminus\\!a,b}(X_{b}\cap Y_{a}),$

where (2) holds by the submodularity of $\lambda_{M\\!\setminus\\!a,b}$, and
(3) holds because the complement of $(X_{a}\cup Y_{b})-b$ in
$M\\!\setminus\\!a,b$ is $X_{b}\cap Y_{a}$. Thus 3.1.3$(ii)$ holds.

We now determine the ranks of $X_{a}\cap Y_{b}$ and $X_{b}\cap Y_{a}$.

######  3.1.4.

$r(X_{a}\cap Y_{b})=r(X_{b}\cap Y_{a})=2$.

If $X_{a}\cap Y_{b}=\varnothing$, then $X_{a}-b\subseteq X_{b}$. But then
$X_{b}$ spans $b$, so $(X_{b}\cup b,Y_{b})$ is a vertical $3$-separation of
$M$, a contradiction. Suppose that $X_{a}\cap Y_{b}=\\{d\\}$. Then $|X_{a}\cap
X_{b}|\geq 2$, or else $(i)$ holds with $C^{*}=X_{a}\cup a$. Similarly,
$|Y_{a}\cap Y_{b}|\geq 2$, or else $\\{d,c,b\\}$ is a triad of $M$. Then 3.1.1
implies that $\lambda_{M\\!\setminus\\!a,b}(X_{a}\cap X_{b})\geq 2$ and
$\lambda_{M\\!\setminus\\!a,b}(Y_{a}\cap Y_{b})\geq 2$. By 3.1.3$(i)$, it
follows that $\lambda_{M\\!\setminus\\!a,b}(Y_{a}\cap Y_{b})=2$. The
complement of $Y_{a}\cap Y_{b}$ in $M\\!\setminus\\!a,b$ contains $X_{a}-b$
and $X_{b}-a$, and thus spans $a$ and $b$, by 3.1.2. This implies that
$\lambda_{M}(Y_{a}\cap Y_{b})=2$, and so $r(Y_{a}\cap Y_{b})=2$, since $M$ is
vertically $4$-connected. But then $(i)$ holds with $C^{*}=(Y_{b}\cup
b)-\operatorname{cl}(X_{b})$, a contradiction. Thus, $|X_{a}\cap Y_{b}|\geq
2$, and by symmetry, $|X_{b}\cap Y_{a}|\geq 2$.

Now $\lambda_{M\\!\setminus\\!a,b}(X_{a}\cap Y_{b})\geq 2$ and
$\lambda_{M\\!\setminus\\!a,b}(X_{b}\cap Y_{a})\geq 2$, by 3.1.1. Then
3.1.3$(ii)$ implies that $\lambda_{M\\!\setminus\\!a,b}(X_{a}\cap Y_{b})=2$.
The complement of $X_{a}\cap Y_{b}$ in $M\\!\setminus\\!a,b$ contains
$X_{b}-a$, and thus spans $a$, by 3.1.2. But then it also spans $b$ since
$\\{a,b,c\\}$ is a triangle, and so $\lambda_{M}(X_{a}\cap Y_{b})=2$. Since
$M$ is vertically $4$-connected and $|X_{a}\cap Y_{b}|\geq 2$, it follows that
$r(X_{a}\cap Y_{b})=2$. By symmetry, $r(X_{b}\cap Y_{a})=2$ as well. Thus
3.1.4 holds.

Suppose that $X_{a}\cap X_{b}=\varnothing$. Then since $r(X_{a}\cap Y_{b})=2$,
outcome $(i)$ holds with $C^{*}=(X_{a}\cup a)-\operatorname{cl}(Y_{a})$, a
contradiction. Thus $|X_{a}\cap X_{b}|\geq 1$. Since $(ii)$ does not hold, we
have $r(X_{a}\cap X_{b})\geq 2$ and $r(Y_{a}\cap Y_{b})\geq 3$. Then 3.1.1
implies that $\lambda_{M\\!\setminus\\!a,b}(X_{a}\cap X_{b})\geq 2$ and
$\lambda_{M\\!\setminus\\!a,b}(Y_{a}\cap Y_{b})\geq 2$. By 3.1.3$(i)$, it
follows that $\lambda_{M\\!\setminus\\!a,b}(Y_{a}\cap Y_{b})=2$. The
complement of $Y_{a}\cap Y_{b}$ in $M\\!\setminus\\!a,b$ contains $X_{b}-a$
and $X_{a}-b$, and thus spans $a$ and $b$, by 3.1.2. Thus,
$\lambda_{M}(Y_{a}\cap Y_{b})=2$, and so $r(Y_{a}\cap Y_{b})=2$, a
contradiction. ∎

The following easily implies Theorem 1.8. We add an extra condition to outcome
(2) to help deal with matroids on at most $11$ elements.

######  Theorem 3.2.

Let $M$ be a vertically $4$-connected matroid with a triangle $T=\\{e,f,g\\}$
so that none of $M\\!\setminus\\!e$, $M\\!\setminus\\!f$, $M\\!\setminus\\!g$
is vertically $4$-connected. Then either

  1. $(1)$

$M$ has a cocircuit $C^{*}$ so that $|C^{*}\cap T|=2$ and $M|C^{*}\cong
U_{2,k}\oplus U_{2,2}$ for some $k\geq 2$; or

  2. $(2)$

$r(M)\leq 6$, and if $M$ has no $U_{2,4}$-restrictions, then $|E(M)|\leq 11$,
while if $|E(M)|=11$, then $M$ has disjoint triangles $T_{1}$ and $T_{2}$,
neither of which is $T$.

###### Proof.

Suppose that (1) does not hold for $M$. If $M$ is uniform, then $r(M)=2$ and
outcome (2) holds, so we may assume that $M$ is non-uniform.

######  3.2.1.

If $M\\!\setminus\\!e$ has an exact vertical $j$-separation $(X,Y)$ with
$j\in\\{1,2\\}$, then $(2)$ holds.

It is easy to show that if $j=1$, then $M\cong U_{2,3}$. Suppose $j=2$. Then
$r(X)\leq 2$, otherwise $(X,Y\cup e)$ is a vertical $3$-separation of $M$, a
contradiction. Similarly, $r(Y)\leq 2$. Since
$\lambda_{M\\!\setminus\\!e}(X)=1$, this implies that $r(M)=3$. Also, if $M$
has no $U_{2,4}$-restrictions, then $|E(M)|\leq 7$. Thus 3.2.1 holds.

Let $(A_{e},B_{e})$, $(A_{f},B_{f})$, and $(A_{g},B_{g})$ be vertical
$3$-separations of $M\\!\setminus\\!e$, $M\\!\setminus\\!f$, and
$M\\!\setminus\\!g$, respectively, so that $f\in A_{e}\cap A_{g}$ and $e\in
A_{f}$. Then $g\in B_{e}\cap B_{f}$ and $e\in B_{g}$. By 3.2.1, we may assume
that each of the designated vertical $3$-separations is exact, or else (2)
holds. We apply Lemma 3.1 with

  * •

$(X_{a},Y_{a})=(A_{e},B_{e})$ and $(X_{b},Y_{b})=(A_{f},B_{f})$,

  * •

$(X_{a},Y_{a})=(B_{e},A_{e})$ and $(X_{b},Y_{b})=(B_{g},A_{g})$, and

  * •

$(X_{a},Y_{a})=(B_{f},A_{f})$ and $(X_{b},Y_{b})=(A_{g},B_{g})$.

In each case, outcome $(ii)$ of Lemma 3.1 holds. First, suppose that at least
two of $|A_{e}\cap A_{f}|$, $|B_{e}\cap B_{g}|$, and $|B_{f}\cap A_{g}|$ are
equal to one. Up to relabeling $e$, $f$, and $g$, we may assume that
$|A_{e}\cap A_{f}|=1$ and $|B_{e}\cap B_{g}|=1$. Note that $r(A_{e}\cap
B_{f})=2$ and $r(B_{e}\cap A_{g})=2$. Then $A_{e}-f$ and $B_{e}-g$ are each a
union of a rank-$2$ set and a rank-$1$ set, so $r(A_{e})\leq 4$ and
$r(B_{e})\leq 4$. Since $\lambda_{M\\!\setminus\\!e}(A_{e})=2$, this implies
that $r(M)\leq 6$. Also, if $M$ has no $U_{2,4}$-restrictions, then
$|A_{e}|\leq 5$ and $|B_{e}|\leq 5$, so $|E(M)|\leq 11$. If $|E(M)|=11$, then
$A_{e}$ and $B_{e}$ each contain a triangle, so (2) holds.

Second, suppose that fewer than two of $|A_{e}\cap A_{f}|$, $|B_{e}\cap
B_{g}|$, and $|B_{f}\cap B_{g}|$ are equal to one. Then, by Lemma 3.1, at
least two of $r(B_{e}\cap B_{f})$, $r(A_{e}\cap A_{g})$, $r(A_{f}\cap B_{g})$
are at most two. Up to relabeling $e$, $f$, and $g$, we may assume that
$r(B_{e}\cap B_{f})\leq 2$ and $r(A_{e}\cap A_{g})\leq 2$. But then $A_{e}$
and $B_{e}$ are each the union of two sets of rank at most two, so
$r(A_{e})\leq 4$ and $r(B_{e})\leq 4$. Since
$\lambda_{M\\!\setminus\\!e}(A_{e})=2$, this implies that $r(M)\leq 6$.

Now assume that $M$ has no $U_{2,4}$-restrictions. Then $|A_{e}-f|\leq 5$ and
$|B_{e}-g|\leq 5$, and so $|E(M)|\leq|A_{e}-f|+|B_{e}-g|+|T|\leq 13$. We show
that $|E(M)|\leq 11$. Suppose that $|A_{e}|=6$. Since $r(A_{e}\cap B_{g})=2$
and $r(A_{e}\cap A_{g})\leq 2$, each of these sets is a triangle. So, $A_{e}$
is the disjoint union of triangles $T_{1}$ and $T_{2}$, where $f\in T_{2}$.
Each of $A_{f}$ and $B_{f}$ contains an element of $T_{2}$, or else $A_{f}$ or
$B_{f}$ spans $f$. As $|A_{e}\cap B_{f}|\geq 2$, it follows that $|T_{1}\cap
B_{f}|\geq 1$. If $|T_{1}\cap B_{f}|\geq 2$, then $M$ has a
$U_{2,4}$-restriction consisting of $T_{1}$ and the element in $T_{2}\cap
B_{f}$, since $r(A_{e}\cap B_{f})=2$. Thus, $|T_{1}\cap B_{f}|=1$. Let $s\in
T_{1}\cap B_{f}$. Then $A_{f}$ spans $s$, and so (1) holds with
$C^{*}=(B_{f}\cup f)-\operatorname{cl}(A_{f})$, a contradiction. Thus,
$|A_{e}|\leq 5$, and a similar argument shows that $|B_{e}|\leq 5$, so
$|E(M)|\leq 11$. It is clear from the argument that if $|A_{e}|=|B_{e}|=5$,
then each contains a triangle, and so (2) holds. ∎

###### Proof of Theorem 1.7.

As $M$ is minimally vertically $4$-connected, Lemma 2.1 implies that $M$ has
no $U_{2,4}$-restriction. Thus, in Theorem 3.2, we see that $k\leq 3$ in
outcome $(1)$, while $|E(M)|\leq 11$ in outcome $(2)$. The theorem follows. ∎

Theorem 1.4 directly implies Theorem 1.5, since a circuit and a cocircuit in a
binary matroid meet in an even number of elements. Thus we need only prove
Theorem 1.4. To do this, we must investigate matroids with at most $11$
elements.

###### Proof of Theorem 1.4.

Let $M$ be a minimally vertically $4$-connected matroid with at least six
elements. Suppose that $M$ has no $4$-cocircuits and no $5$-cocircuits
containing a triangle. By Theorem 1.7, we must have that $|E(M)|\leq 11$.
Theorem 1.2 implies that $M$ has a triangle $T$.

######  3.2.2.

$M$ has no element $a$ for which $M\\!\setminus\\!a$ has an exact vertical
$j$-separation for $j\in\\{1,2\\}$.

Let $(X,Y)$ be an exact vertical $j$-separation of $M\\!\setminus\\!a$ with
$j\in\\{1,2\\}$. It is easy to show that if $j=1$, then $M\cong U_{2,3}$, a
contradiction. Thus $j=2$. If $r(X)\geq 3$, then $(X,Y\cup a)$ is a vertical
$2$\- or $3$-separation of $M$, a contradiction. So $r(X)=2$, and, similarly,
$r(Y)=2$. By Lemma 2.1, this implies that $|E(M)|\leq 7$. Since
$\lambda_{M\\!\setminus\\!a}(X)=1$, it also implies that $r(M)=3$. However, by
Lemma 2.1, $M$ has no disjoint triangles in a common plane, so $|E(M)|=6$.
Since $M$ is a $6$-element rank-$3$ matroid that is $3$-connected, it is
isomorphic to $M(K_{4})$, $\mathcal{W}^{3}$, $Q_{6}$, $P_{6}$, or $U_{3,6}$
[4, Corollary 12.2.19], so $M$ has a $4$-cocircuit, a contradiction. Thus
3.2.2 holds.

A consequence of 3.2.2 is that $M$ has no cocircuits with fewer than four
elements. For all $a$ in $E(M)$, the deletion $M\\!\setminus\\!a$ has an exact
$3$-separation $(A_{a},B_{a})$. Since each of $A_{a}$ and $B_{a}$ must contain
a cocircuit of $M\\!\setminus\\!a$ but $M$ has no cocircuits of size less than
five, $|E(M)|\geq 9$. As $M\\!\setminus\\!a$ has no exact $2$-separations, we
may assume that $T\subseteq B_{a}$. Then $|E(M)|\neq 9$, otherwise
$|A_{a}|=4=|B_{a}|$ and $B_{a}\cup a$ is a $5$-cocircuit containing a
triangle, a contradiction.

Next suppose that $|E(M)|=10$. Then $|B_{a}|=5$ and $|A_{a}|=4$. Also, $A_{a}$
does not contain a triangle, or else $A_{a}\cup a$ is a $5$-cocircuit of $M$
that contains a triangle. Then $M|A_{a}\cong U_{3,4}$ otherwise $M|A_{a}$ has
a coloop $x$ and $(A_{a}-x,B_{a}\cup x)$ is a vertical $3$-separation of
$M\\!\setminus\\!a$, so $(A_{a}-x)\cup a$ is $4$-cocircuit of $M$, a
contradiction. Similarly, each element $x\in B_{a}-T$ is in
$\operatorname{cl}(B_{a}-x)$, or else $(B_{a}-x)\cup a$ is a $5$-cocircuit of
$M$ that contains $T$. This implies that $r(B_{a})=3$. As $r(A_{a})=3$, it
follows that $r(M)=4$. So, for each $a\in E(M)-T$, there is a pair $P_{a}$ of
elements so that $r(T\cup P_{a})=3$. Since $M|A_{a}\cong U_{3,4}$ and
$a\notin\operatorname{cl}(B_{a})$, it follows that $P_{a}\neq P_{b}$ if $a\neq
b$ otherwise $A_{a}-b$ spans $A_{a}\cup a$, a contradiction. Let $\mathcal{P}$
be the set of pairs $P$ for which $r(P\cup T)=3$. Then
$|\mathcal{P}|\geq|E(M)-T|=7$. However, the sets in $\mathcal{P}$ are pairwise
disjoint, or else $M$ has a $6$-element plane and thus has a $4$-cocircuit.
Since $|\mathcal{P}|\geq 7$, this implies that $|E(M)|\geq 17$, a
contradiction.

Finally, suppose that $|E(M)|=11$. By Theorem 3.2, $M$ has disjoint triangles
$T_{1}$ and $T_{2}$. Let $X=E(M)-(T_{1}\cup T_{2})$. For each $a\in X$, let
$(A_{a},B_{a})$ be a vertical $3$-separation of $M\\!\setminus\\!a$ so that
$T_{1}\subseteq A_{a}$, while $T_{2}$ is contained in $A_{a}$ or $B_{a}$. If
$T_{2}\subseteq A_{a}$ for some $a\in X$, then, by Lemma 2.1, $r(A_{a})\geq
4$, so $r(M)\geq 5$. If $T_{2}\subseteq B_{a}$ for some $a\in X$, then
$|A_{a}|=|B_{a}|=5$. Then $M|A_{a}$ and $M|B_{a}$ each have no coloops, or
else $M$ has a $5$-cocircuit containing a triangle. This implies that
$r(M)=4$. Thus, either $T_{2}\subseteq A_{a}$ for all $a\in X$, or
$T_{2}\subseteq B_{a}$ for all $a\in X$. If the former holds, then
$M|B_{a}\cong U_{3,4}$ for each $a\in X$. It follows that $r(X)=3$, so
$r(B_{a}\cup a)=r(B_{a})$, a contradiction. We deduce that $T_{2}\subseteq
B_{a}$ for all $a\in X$. This implies that, for each $a\in X$, there is a pair
$P_{a}$ for which $r(T_{2}\cup P_{a})=3$. If $P_{a}=P_{b}$ for distinct
$a,b\in X$, then $T_{2}\cup P_{a}$ and its complement in $M$ each have rank
$3$, so $M$ is not vertically $4$-connected, a contradiction. Let
$\mathcal{P}$ be the set of pairs $P$ for which $r(T_{2}\cup P)=3$. Then
$|\mathcal{P}|\geq|X|=5$. However, the sets in $\mathcal{P}$ are pairwise
disjoint, or else $T_{2}$ is contained in a $6$-element plane that is disjoint
from $T_{1}$, so $T_{1}$ is contained in a $5$-cocircuit of $M$. Since
$|\mathcal{P}|\geq 5$, this implies that $|E(M)|\geq 16$, a contradiction. ∎

##  4\. Open Problems

Our results lead to some natural open problems in several different
directions. First, we conjecture that Theorem 1.4 can be strengthened so that
the cocircuit contains a specific element of $T$, as in Theorem 1.9.

######  Conjecture 4.1.

Let $M$ be a minimally vertically $4$-connected matroid with a triangle
$T=\\{e,f,g\\}$. Then

  1. $(1)$

$M$ has a $4$-cocircuit that contains $e$ and exactly one of $f$ and $g$, or

  2. $(2)$

$M$ has a $5$-cocircuit that contains a triangle and $e$, and exactly one of
$f$ and $g$, or

  3. $(3)$

$|E(M)|\leq 11$.

This would be helpful for trying to find a tight lower bound on the number of
$4$-cocircuits and $5$-cocircuits containing a triangle in a minimally
vertically $4$-connected matroid.

Second, we conjecture that Theorem 1.5 extends to vertical $k$-connectivity
with $k\geq 5$ due to restrictions on the interaction between circuits and
cocircuits in binary matroids.

######  Conjecture 4.2.

For each integer $k\geq 5$, every minimally vertically $k$-connected binary
matroid with at least $2k+2$ elements has a $k$-cocircuit.

We close with the following extension of Conjecture 1.6, which would show that
Problem 1.3 has a negative answer when $k\geq 4$.

######  Conjecture 4.3.

For each integer $k\geq 4$, there is an infinite family of minimally
vertically $k$-connected matroids with no $k$-cocircuits.

## Acknowledgements

We thank an anonymous referee for pointing out an error in our construction of
an infinite family of minimally vertically $4$-connected matroids with no
$4$-cocircuits in a previous version of this paper.

## References

  * [1] R. Halin. Untersuchungen über minimale n-fach zusammenhängende Graphen. _Math. Ann._ 182: 175–188, 1969.
  * [2] W. Mader. Zur Struktur minimal n-fach zusammenhängender Graphen. _Abh. Math. Sem. Univ. Hamburg_ 49: 49–69, 1979.
  * [3] U. S. R. Murty. Extremal critically connected matroids. _Discrete Math._ 8: 49–58, 1974.
  * [4] J. Oxley. _Matroid Theory_ , Second Edition. Oxford Graduate Texts in Mathematics. Oxford University Press, 2011.
  * [5] J. G. Oxley. _Matroid Theory_. Oxford Sci. Publ. The Clarendon Press, Oxford University Press, New York, 1992.
  * [6] T. J. Reid, H. Wu, X. Zhou. On minimally $k$-connected matroids. _J. Combin. Theory Ser. B_ , 98: 1311–1324, 2008.
  * [7] W. T. Tutte. Connectivity in matroids. _Canad. J. Math._ 18: 1301–1324, 1966.
  * [8] P. K. Wong. On certain n-connected matroids. _J. Reine Angew. Math._ , 299/300: 1–6, 1978.


# Correlated carrier dynamics in a superconducting van der Waals
heterostructure

Prakiran Baidya  Department of Physics, Indian Institute of Science, Bangalore
560012, India  Vivas Bagwe  Pratap Raychaudhuri  Tata Institute of Fundamental
Research, Mumbai 400005, India  Aveek Bid<EMAIL_ADDRESS>Department of
Physics, Indian Institute of Science, Bangalore-560012, India

###### Abstract

Study of Berezinskii-Kosterlitz-Thouless transitions in clean, layered two-
dimensional superconductors promises to provide insight into a host of novel
phenomena like re-entrant vortex-dynamics, underlying unconventional metallic
phases, and topological superconductivity. In this letter, we report the study
of charge carrier dynamics in a novel 2-dimensional superconducting van der
Waals heterostructure comprising of monolayer MoS2 and few-layer NbSe2 ($\sim
15$ nm). Using low-frequency conductance fluctuation spectroscopy, we show
that the superconducting transition in the system is percolative. We present a
phenomenological picture of different phases across the transition correlating
with the evaluated noise. The analysis of the higher-order statistics of
fluctuation reveals non-Gaussian components around the transition indicative
of long-range correlation in the system.

##  I Introduction

With the practical realization of graphene [1], the past decade has seen an
extensive exploration of layered systems. The van der Waals heterostacking of
these crystalline layered materials promises to exhibit parameter-driven
exotic phenomena including topologically non-trivial states [2, 3, 4, 5] and
strongly correlated phases [6, 7]. An example is atomically thin
superconductors in the ‘true’ 2-dimensional (2D) limit. Contrary to the
3-dimensional superconductors, for a superconductor in the 2D limit, the
transition occurs through Berezinskii-Kosterlitz-Thouless (BKT) mechanism [8,
9]. Below a characteristic critical temperature $T_{BKT}$, the vortices and
antivortices are bound – the thermal unbinding of these pairs at $T>T_{BKT}$
gives rise to the transition from the dissipationless to a finite resistive
state in the system.

There are two distinct experimental strategies employed to identify a BKT
transition as one approaches it from above – (i) measurement of the superfluid
density $n_{S}$ which is expected to go to zero discontinuously at the
transition [31, 11] and (2) extrapolation of the temperature dependence of the
resistivity measured at $T>T_{BKT}$ to lower $T$ using the formalism developed
by Halperin and Nelson [12]. The first technique of looking for discontinuity
in the superfluid density as a signature of BKT physics does not work well for
superconductors buried inside heterostructures. The second method fails for
disordered superconductors due to two reasons: (i) the presence of impurities
tends to broaden the transition [13, 14, 15] and (ii) the inhomogeneities
change the value of the vortex-core energy from that predicted within the 2D
XY model [16].

The study of carrier dynamics through resistance fluctuation spectroscopy has
emerged as a powerful alternative probe to identify BKT transitions [14, 17].
Though this technique has been employed to probe the BKT physics in thin-film
superconductors, the study of fluctuation statistics is not well explored in
layered systems, specifically in van der Waals heterostructures. With
increasing interest in such systems as platforms for realizing low dimensional
superconductivity in the clean limit and topological superconductivity, there
is an urgent need for a detailed study of such systems.

In a previous study, we have reported the observation of two-dimensional Ising
superconductivity in a van der Waals heterostructure comprising of single-
layer-MoS2 (SL-MoS2) and bulk NbSe2 [18]. We established that the reduced
dimensionality comes from an effective thinning of NbSe2 due to the coupling
with the MoS2 layer making it a perfect example of a ‘buried’ van der Waals
superconductor. Thus, the conductance fluctuation spectroscopy technique
becomes very relevant to probe the BKT physics in this system.

In this letter, we report a detailed study of the carrier dynamics of this
heterostructure through low-frequency conductance fluctuation spectroscopy
around the BKT transition. Through systematic measurements, we establish that
superconductivity has a percolative nature. We also find proof of correlated
dynamics arising from long-range interaction of vortices-antivortices near the
$T_{BKT}$ establishing universal BKT nature of the superconducting transition
in this system.

##  II Device Fabrication

To fabricate the device, we mechanically exfoliated single-layer flakes of
MoS2 from a bulk crystal [18]. The thickness of the flake was confirmed from
optical contrast and through Raman spectroscopy. The flake was then
transferred onto gold contact probes pre-patterned on hBN substrates.
Subsequently, a flake of NbSe2 exfoliated inside a glove box (with oxygen and
moisture levels maintained at less than one ppm) was transferred on top of the
MoS2 flake. The thickness of the NbSe2 flake was estimated from its optical
contrast to be $\sim 15$ nm. Before extraction from inside the glove box, the
heterostack of SL-MoS2/NbSe2 was covered by a hBN flake of thickness $\sim 30$
nm to protect it from environmental degradation. Subsequently, the stack was
annealed at $200^{\circ}$ C to increase the coupling between the layers.

##  III Results and Discussion

For the initial characterization of superconducting properties of the
heterostructures, electrical transport measurements were done using a DC
current source and a nano voltmeter in a four-probe configuration (Fig. 1(a)).
The temperature dependence of resistance $R$ shows a metallic behavior at high
temperatures followed by a transition to a superconducting state (Fig. 1(c))
at $T^{0}_{C}=6$ K. From the $R$ versus $T$ plot and also from non-linear
current-voltage relations, we estimate $T_{BKT}$ to be $6.1\pm 0.1$ K.(see
Supplementary Information S1 for details, also Ref. [18]).

We investigated the fluctuation statistics of the system around the $T_{BKT}$
using a 4-probe resistance fluctuation spectroscopy technique [35, 20, 21] –
the details are discussed in Supplementary Information S2. Briefly, the device
was current biased, the voltage developed across it pre-amplified and detected
by a dual-phase lock-in-amplifier (LIA). The demodulated output of the LIA was
digitized at a sampling rate of $1024$ points/s using a $16$ bit analog-to-
digital conversion card and transferred in the computer memory for further
processing. The biasing current was always maintained at a value much smaller
than the critical current of the superconductor. The acquired time series of
resistance fluctuations were decimated and filtered digitally to eliminate
aliasing and related digital artifacts. The power spectral density of the
resistance fluctuation $S_{R}\left(f\right)$ was then calculated over a
frequency range $4$ mHz–$4$ Hz.

Fig. 2(a) is a plot of the time traces of resistance fluctuations for our
device measured at a few representative temperatures, $T$. The fluctuations
increase in amplitude with $T$ approaching $T_{BKT}$ from above. The
corresponding $S_{R}\left(f\right)$ were found to have a frequency dependence
of the form $S_{R}\left(f\right)\propto 1/{f^{\alpha}}$ (Fig. 2(b)). One can
see that the power spectral density $S_{R}\left(f\right)$ increases by several
orders of magnitude with decreasing temperature, reflecting the observed
increased resistance fluctuation in Fig. 2(a). Additionally, the value of the
exponent $\alpha$ for $f<0.5$ Hz (the method of evaluation the exponent is
discussed in Supplementary Information S3) increases monotonically from $\sim
1$ at higher temperature to $\sim 2.4$ near $T_{BKT}$ (Fig. 3(a)). There can
be two possible reasons for this increase in $\alpha$ – (i) transition of the
system across different vortex phases, e.g., from an ordered to a disordered
regime [22] or (ii) fluctuation in the domain parameter of different phases
across the transition range [23]. Discriminating between these two scenarios
requires further analysis and is beyond the scope of the current letter.

The relative variance of resistance fluctuations (we refer to this as the
noise) was evaluated by integrating $S_{R}\left(f\right)$ over the bandwidth
of measurement [35, 20, 21]:

$\mathcal{R}=\frac{\left<\delta
R^{2}\right>}{\left<R^{2}\right>}=\frac{1}{R^{2}}\int S_{R}(f)df.$ (1)

Fig. 3(b) shows the plots of relative variance and the normalized resistance
against $T/T_{critical}$ ($T/T_{BKT}$) for the heterostructure region. We
observe that $\mathcal{R}$ in the normal state is $\sim 10^{-10}$. This value
is almost five orders of magnitude lower than that reported for a typical
semiconducting TMD [24] attesting to the high quality of our heterostructure.

With decreasing $T$, $\mathcal{R}$ increases by nearly six orders of magnitude
over a very narrow temperature window near $T_{BKT}$. As we discuss later,
this divergence in noise can be well explained in terms of a percolation
network model of superconducting fluctuations [14]. Moreover according to the
percolation model in the transition regime the system should follow the
relation, $S_{R}\left(f\right)/R^{2}\leavevmode\nobreak\ \propto R^{-l_{rs}}$
where $l_{rs}$ is the percolation exponent which takes up the value $\sim 0.9$
in the classical picture [36]. In our case, the exponent $l_{rs}$ comes out to
be $0.89\pm 0.03$ (see Supplementary Information S4), establishing the
percolative nature of the system. Notably, when compared with that of the
pristine NbSe2, the noise in the heterostructure is almost an order higher
around the respective transition temperatures (c.f. Fig. 3(b)).

Before we proceed further, the effect of thermal fluctuations on the measured
noise needs to be considered. $dR/dT$ diverges close to the critical
temperature for a superconductor. Consequently, any minor fluctuation in
temperature can give rise to large resistance fluctuations near $T_{BKT}$. To
eliminate this trivial effect as the the origin of the large resistance
fluctuations seen in our device, we evaluated the relative contribution of
temperature fluctuations to the noise using the relation
$\left[\left(dR/{dT}\right)\times\left(\delta T/R\right)\right]^{2}$. Here
$\delta T$ is the temperature fluctuation in the measurement system, which has
been measured to be $<5$ mK in our case. The evaluated value of this relative
variance at $T_{BKT}$ is $\sim 10^{-7}$ (see Supplementary Information S5).
This value is at least two orders of magnitude smaller than the measured
$\mathcal{R}$ near $T_{BKT}$, thus ruling out any significant contributions of
temperature fluctuations in the measured noise.

In Fig. 4 we present a phenomenological explanation of the effect of
percolation dynamics on the resistance fluctuations in a 2D superconductor in
terms of a percolation network model of superconducting fluctuations [14]. The
squares below the plot show the microscopic status of the system schematically
in terms of a superconducting-normal network in different $T$-regimes.

Region-I is a purely superconducting phase. On approaching $T_{BKT}$ from
below, small patches of dissipative (metallic) domains begin nucleating in the
superconducting background (region-II). With increasing temperature,
fluctuations in the superconducting order parameter result in the formation of
a dynamic network of interconnected superconducting and normal (dissipative)
regions [36]. This effect is especially severe in the case of 2D
superconductors. The enhanced noise in this $T$-regime has two major
components – (i) resistance fluctuations in dissipative regions; (ii)
fluctuations in the number/size of the superconducting clusters [26, 27].
Beyond $T=T_{BKT}$, the system crosses into region III, where the proportion
of superconducting and non-superconducting domains become almost equal. At
this temperature (which we denote as $T_{max}$), the resistance of the device
is nearly half of the normal state resistance, and the resistance fluctuation
is at its maximum. The other boundary of region III comes at $T_{BCS}$, which
is at $\sim 6.5$ K for the system. For $T>T_{BCS}$, the fraction of the
superconducting phase decreases sharply with increasing $T$ till the entire
system becomes dissipative. Consequent to this decrease in electronic phase
segregation of the system, the variance of resistance fluctuations decreases
as the system approaches the metallic phase.

We turn now to the nature of the correlations between the fluctuating entities
at different $T$-ranges. In 2D superconductors undergoing BKT transition, the
XY model predicts the fluctuations to be non-Gaussian around $T_{BKT}$ [28].
These non-Gaussian resistance fluctuations have emerged as a unique signature
of BKT physics and have successfully been used to discriminate between 2D and
3D superconductors [17, 14]. We quantify the non-Gaussianity of the resistance
fluctuations through their ‘Second spectrum,’ which is the four-point
correlation function of $\delta R$, calculated over a frequency octave
($f_{l},f_{h}$). Being extremely sensitive to the presence of non-Gaussian
components (NGC), this parameter is a highly effective tool to probe
correlation in a system [29, 30]. To estimate the second spectrum, repeated
measurements of the PSD, $S_{R}\left(f\right)$ is done over a selected
frequency range ($f_{l},f_{h}$). The power spectrum of this series over a
frequency octave gives the second spectrum [29, 30]:

$\begin{split}S^{f_{1}}_{R}\left(f_{2}\right)=\int_{0}^{\infty}\left<\delta
R^{2}\left(t\right)\delta R^{2}\left(t+\tau\right)\right>\cos{\left(2\pi
f_{2}\tau\right)}d\tau,\\\
\sigma^{\left(2\right)}={\int_{0}^{{f_{l}}-{f_{h}}}S^{f_{1}}_{R}\left(f_{2}\right)df_{2}}\Big{/}{\left[\int_{f_{l}}^{f_{h}}S_{R}\left(f\right)df\right]}^{2}\end{split}$
(2)

Here $f_{1}$ is the center frequency of the chosen octave and $f_{2}$ is the
spectral frequency. $\sigma^{\left(2\right)}$ is the normalized second
spectrum; it equals 3 for Gaussian fluctuations.

Fig. 5 shows the plot of measured $\sigma^{\left(2\right)}$ as a function of
$T/{T_{BKT}}$ for the heterostructure region. As can be observed, while
decreasing the temperature, $\sigma^{\left(2\right)}$ increases from a
baseline value of $\sim 3$ at higher $T$ to $\sim 30$ near $T_{BKT}$ before
decaying again to the Gaussian base value for $T<T_{BKT}$. This enhancement of
$\sigma^{\left(2\right)}$ in the narrow window of $T$ in Region-III
establishes clearly the appearance of non-Gaussian resistance fluctuations
near $T=T_{BKT}$. In contrast, for the pristine NbSe2 region
$\sigma^{\left(2\right)}$ remains at the baseline value of $\sim 3$ (black
solid circle in Fig. 5) throughout the temperature range around the transition
indicating a Gaussian distribution of fluctuations as expected for a 3D
superconductor.

Non-Gaussian fluctuations in superconductors can have different origins – (1)
long-range correlation among the vortices as has been observed in previous
studies [17], (2) the dominance of percolation kinetics around superconducting
transition seen in inhomogeneous superconductors [14], and (3) dynamic current
redistribution which appears as a consequence of substantial transport
inhomogeneity and large local resistivity fluctuations that translate to the
necessary condition of $\delta R/R>>1$ [30]. The third cause can be
immediately ruled out by noting that in our system$\delta R/R<<1$. To
discriminate between the remaining two scenarios, note that a comparison of
the $T$ dependence of $\mathcal{R}$ and $\sigma^{\left(2\right)}$ (Fig. 5)
reveals that significant fluctuations in the resistance extends beyond the
onset of normal state (i.e. $R/R_{N}=1$) at $T_{BCS}$ of $\sim 6.5$ K. In the
low-temperature limit, it extends to $T<T_{BKT}$. On the other hand, the
deviation from the Gaussian value in $\sigma^{\left(2\right)}$ stays confined
in region III within $T_{BKT}<T<T_{BCS}$. This deviation indicates that the
increase of $\sigma^{\left(2\right)}$ that marks the existence of non-Gaussian
fluctuations in the fluctuation is a consequence of an independent process
which is unlikely to be the percolation kinetics that dominates the spectrum
of resistance fluctuations. This strongly suggests that the first scenario of
correlated vortices is at play in inducing the non-Gaussian fluctuations in
the system, as has been reported earlier for clean, homogeneous
superconductors [17]. Further theoretical and experimental studies are
essential to establish unequivocally if this is the case.

##  IV Conclusion

In summary, we have studied the carrier dynamics of SL-MoS2/NbSe2
heterostructures near the superconducting transition by probing the low-
frequency conductance fluctuations of the system. The first spectrum
(resistance noise) shows signatures of the percolative nature of the
superconducting transition. We provide a phenomenological explanation of the
different phase-space regions around the transition temperature in terms of a
percolative microstructure picture and correlate the resulting fluctuations
with it. Furthermore, we establish the presence of strong correlations in the
system around $T_{BKT}$ arising most probably from the interacting vortices
and thus established that the superconducting transition in the system is of
the universal BKT type.

Acknowledgments: The authors acknowledge device fabrication facilities in
NNFC, CeNSE, IISc. A.B. acknowledges funding from SERB (No. HRR/2015/000017)
and DST (No. DST/SJF/PSA01/2016-17)

Figure 1: (a) A schematic of the device structure. (b) False colour
Differential Interference Contrast image of the device with different colors
defining different layers of the heterostructure – SL-MoS2(green) and pristine
NbSe2 (orange) and overlap region (light-red). (c) Temperature dependence of
the four-probe resistance of the heterostructure.

Figure 2: (a) Time series of resistance fluctuations of the heterostructure
region at a few representative temperatures. (b) Plots of
$S_{R}\left(f\right)$ as function of frequency at the same values of $T$ as in
(a).

Figure 3: (a) Plot of exponent $\alpha$ versus $T/T_{BKT}$ for
heterostructure. (b) Plots of the relative variance of resistance fluctuations
$\mathcal{R}$ for heterostructure (solid green circles) and for the pristine
3D NbSe2 region (open orange triangles) as function of $T/{T_{critical}}$.
Here $T/T_{critical}$ is $T/T_{BKT}$ for the heterostructure and $T/T^{0}_{c}$
for the pristine NbSe2. On the right-axis are plotted the normalized
resistance $R/R_{N}$ for the heterostructure (red line).

Figure 4: (a) Plots of the variance of noise $\left<\delta R^{2}\right>$
(left-axis, yellow filled circles) and of the normalized resistance (right-
axis, black solid line) as a function of $T/{T_{BKT}}$ for the heterostructure
region. The color gradient indicates the transition from superconducting
(blue) to normal state (red). (b) Schematics representing the microscopic
status of the system in each electronic phase (for details see text). The
color bar gives the value of $R/R_{N}$ – blue represents the zero resistance
i.e. superconducting state and red represents the normal state.

Figure 5: Plots of the variance of resistance fluctuations $\left<\delta
R^{2}\right>$ (filled yellow circles, left axis) and the normalized second
spectrum $\sigma^{\left(2\right)}$ for heterostructure (filled blue circles,
right axis) and for pristine 3D NbSe2 (filled black circles, right axis) as a
function of $T/T_{critical}$. Here $T/T_{critical}$ is $T/T_{BKT}$ for the
heterostructure and $T/T^{0}_{c}$ for the pristine NbSe2. The color gradient
has the same connotation as in Fig. 4. (for details see text).

## Supplementary Information

##  S1 Evaluation of BKT transition temperature

Figure S1: (a) Zero magnetic field current-voltage characteristics of the
heterostructure with varying temperature ranging from $4.5$ K to $10.0$ K. The
red dashed line represents the range of current within which linear fit was
done for each curve. (b) Plot of $\gamma$ as a function of temperature, $T$ to
evaluate $T_{BKT}$ for heterostructure. (c) Plot of $(dlnR/dT)^{-{2/3}}$ as a
function of temperature, $T$ for the heterostructure where the intersect of
the black dashed line with the x axis gives $T_{BKT}$.

For the initial characterization of superconducting properties of the
heterostructure, electrical transport measurements were done using a DC
current source and a nano voltmeter in a four-probe configuration. As reported
in our previous work, the superconductivity in system is of 2D nature. We thus
expect the observed superconducting transition to be of
Berezinskii–Kosterlitz–Thouless (BKT) type. For a 2D-superconductor there
exists a characteristic temperature, $T_{BKT}$ below which a finite electric
current can unbind the vortex-antivortex pairs system, giving rise to a
dissipation which is reflected in the current-voltage characteristic as a non-
linear behavior of the form, $V\propto I^{\gamma\left(T\right)}$ [31, 32].
$\gamma$ is a temperature dependent exponent that takes the value $3$ at
$T=T_{BKT}$ and eventually goes to $1$ in the normal ohmic state. Fig. S1(a)
shows the zero field DC non-linear current-voltage characteristics. The value
of the exponent $\gamma$, evaluated through linear fitting of each curve
within the marked region, are shown in Fig. S1(b). From this plot, we evaluate
$T_{BKT}$ to be $6.13$ K. We also evaluated $T_{BCS}$ (defined as the
temperature where onset of transition occurs or in other words where the IV
becomes linear i.e. $\gamma=1$) to be $6.5$ K.

The BKT transition temperature can also be obtained from resistance vs
temperature plot as near $T_{BKT}$ the resistance takes the form
$R=R_{0}\mathrm{exp}[-b_{R}/(T-T_{BKT})^{1/2}]$ (where $b_{R}$ gives the
vortex-antivortex interaction strength) [31, 33, 34]. To evaluate $T_{BKT}$ we
reduced the formula to a form,
$\left(dlnR/dT\right)^{-2/3}=\left(2/b_{R}\right)^{2/3}\left(T-T_{BKT}\right)$.
As shown in Fig. S1(c), the intercept of the plot of
$\left(dlnR/dT\right)^{-2/3}$ vs $T$ gives $T_{BKT}$ as to be $6.14$ K.

##  S2 Details of noise measurement technique

Figure S2: Schematic diagram of the noise measurement setup. The sample is
probed in four-probe geometry. PA represents the pre-amplifier and LIA
represents the dual phase lock-in amplifier.

We investigated the fluctuation statistics of the system around
superconducting transition through analysis of the zero field temperature
dependent resistance fluctuations acquired using an ac technique that allows
us to measure the fluctuations from system and the background simultaneously
[35]. Fig. S2 is schematic of the measurement setup. The sample was current-
biased using the sine wave output of a lock-in amplifier (SR830). A resistor,
$R_{series}$ in series with the sample controls the current through it. The
value of the excitation current was always maintained to be lower than the
critical current of the superconductor. The voltage developed across the
sample was detected using the dual-phase lock-in-amplifier coupled with a
preamplifier (SR$552$). The excitation frequency of the current was kept at
the eye of the noise figure of the preamplifier to minimize the contribution
of amplifier noise in measured the background noise. The time constant were
set to be $30$ ms with a filter roll off of $24$ dB/octave - this subsequently
determines the upper cutoff frequency of the power spectral density (PSD). The
output of the LIA was digitized at a sampling rate of 1024 points/s using a
$16$ bit analog-to-digital conversion card and transferred in the computer
memory for further processing. The in-phase channel (X-channel) picks up the
excess noise from the sample as well as the background whereas the quadrature
channel (Y-channel) picks up only the fluctuations from background. At every
temperature, the time series of the resistance fluctuations was acquired for a
duration of $30$ minutes ($\sim 1.8X10^{6}$ data points). These were
subsequently decimated with a factor of 128 and digitally filtered to
eliminate aliasing and related digital artifacts. These filtered time series
were then used to calculate the power spectral density (PSD) over the specific
frequency range. The PSD of the sample noise were finally obtained by
subtracting the PSD of X-channel fluctuation from that of the Y-channel.

##  S3 Evaluation of the exponent $\alpha$ from $S_{R}\left(f\right)$

Figure S3: Plot of $S_{R}\left(f\right)$ as a function frequency, $f$ for two
different temperatures around the transition, $8$ K (violet) and $6.2$ K
(brown). The dashed lines show the linear fits to the plots. The boxes with
each plot show value of the slope for the respective fitting.

As mentioned in the main manuscript, the power spectral density,
$S_{R}\left(f\right)$ has a frequency dependency $S_{R}\left(f\right)\propto
1/{f^{\alpha}}$. To evaluate the exponent $\alpha$ we plotted
$S_{R}\left(f\right)$ as function of frequency, $f$ in log-log scale, as shown
in Fig. S3. The slope of these plots gives the value of $\alpha$. As can be
seen here the slope i.e. $\alpha$ is $\sim 1$ at $8$ K in which the system is
in normal state whereas at $6.2$ K which is near to $T_{BKT}$ the value
becomes $\sim 2.4$.

##  S4 Classical picture of percolation

Figure S4: Plot of $S_{R}/R^{2}$ at $0.0625$ Hz as function of $R$(olive solid
circle). Both the axis are in log scale. The red line is the linear fit to the
data that yields a slope of $\sim 0.89$

For a system having percolative nature it follows that the spectral density of
relative resistance fluctuation at a certain frequency,
$S_{R}\left(f\right)/R^{2}$ grows as power law to the decreasing resistance
towards superconductivity with a form given by [36]

$\frac{S_{R}\left(f\right)}{R^{2}}\leavevmode\nobreak\ \propto R^{-l_{rs}}$
(S1)

where $l_{rs}$ is the percolation exponent, which takes the value $\sim 0.9$
in the classical picture. As can be seen in Fig. S4 the percolation exponent
for the heterostructure comes out to be $0.89\pm 0.03$ which matches quite
well with the classical percolation picture.

##  S5 Contribution of temperature fluctuations to the measured noise

Figure S5: (a) Plot of $T-T_{setpoint}$ as a function time measured at $12$ K.
(b) Plot of the measured relative variance ${\left<\delta
R^{2}\right>}/{\left<R\right>^{2}}$ for heterostructure (olive solid circles)
and that estimated from thermal fluctuation (blue solid circles) as function
of $T/{T_{BKT}}$. The lines are guide to the eye. The right-axis shows a plot
of the resistance for the heterostructure region (solid red line).

As mentioned in the main manuscript, we have evaluated the relative
contribution of the temperature fluctuation of the measurement system to the
measured noise. The temperature stability of a system depends mainly on the
PID value of the temperature controller used in the experiment. We have fixed
this PID value in such a way that we were able to have a temperature
fluctuation, $\delta T<5$ mK at all temperatures at which noise measurements
were done.

In Fig S5(a), we show a plot of $T-T_{setpoint}$ versus time over a period of
30 minutes. This is the typical time for a single noise run. Here
$T_{setpoint}$ is the target temperature value (in this case, 12 K), and
$T(t)$ is the instantaneous value of temperature. From Fig. S5(a), the maximum
fluctuation is about 3 mK indicating that taking $5$ mK as $\delta T$ in our
calculation is a safe choice. Fig. S5(b) shows the plots of the measured
relative variance (olive solid circles) and that estimated from temperature
fluctuations. One can see that near $T_{BKT}$, the value of relative variance
of resistance fluctuations estimated from the temperature fluctuations is
almost two orders of magnitude smaller than the measured relative variance of
resistance fluctuations, $\mathcal{R}$ of the heterojunction. This establishes
that temperature fluctuations play a negligibly small role in the measured
noise.

##  S6 Noise data from another device

Figure S6:  Plots of the relative variance of resistance fluctuations,
$\left<\delta R^{2}\right>/\left<R^{2}\right>$ for the heterostructure (solid
olive circles) and for the pristine 3D NbSe2 region (solid orange circles) as
function of $T/{T_{critical}}$ for device D2. Here $T/T_{critical}$ is
$T/T_{BKT}$ for the heterostructure and $T/T^{0}_{c}$ for the pristine NbSe2.
On the right-axis are plotted the normalized resistance $R/R_{N}$ for the
heterostructure (red line).

Fig. S6 shows the relative variance of resistance fluctuations of another
device D2 having identical structure to the device D1 whose data were
presented in the main text. As is evident from the plot, the data from D2 is
very similar to that from D1 – it shows percolative transition near $T_{BKT}$.
Similar to the data for device D1, for D2 also we observe an order of
magnitude higher value of the relative variance for heterostructure region in
comparison to pristine NbSe2 section of the device near $T/T_{BKT}$.

Figure S7: Plots of the variance of noise $\left<\delta R^{2}\right>$ (left-
axis, olive solid circles) and of the normalized Resistance (right-axis, red
solid line) as a function of $T/{T_{BKT}}$ for the heterostructure region of
device D2.

Fig. S7 shows the variance of the resistance fluctuations (olive solid circle,
left axis) along with the normalized resistance, $R/R_{N}$ (red line, right
axis) of the heterostructure region measured for device D2 as a function of
$T/T_{BKT}$. As can be seen, the increased fluctuation extends beyond
$T_{BCS}$ into the normal state just as for D1 in the main text.

Figure S8: Plots of the variance of resistance fluctuations $\left<\delta
R^{2}\right>$ (solid olive circles, left axis) and the normalized second
spectrum $\sigma^{\left(2\right)}$ for heterostructure (solid red circles,
right axis) and for pristine 3D NbSe2 (solid blue circles, right axis) as a
function of $T/T_{critical}$ for device D2. Here $T/T_{critical}$ is
$T/T_{BKT}$ for the heterostructure and $T/T^{0}_{c}$ for the pristine NbSe2

The deviation of the normalized second spectrum, $\sigma^{\left(2\right)}$
from the baseline value of $3$ in Fig. S8 are constricted within the region
bounded by $T_{BKT}$ and $T_{BCS}$ suggesting that, like device D1 in main
text, the non-Gaussian nature for device D2 also occurs due to the long range
correlations between the vortex-antivortex pair around the transition.
Moreover, as expected we observed $\sigma^{\left(2\right)}$ to be $\sim 3$ for
pristine NbSe2 region of device D2 throughout the temperature range indicating
the Gaussian nature of the fluctuations. This similarity in the evaluated
results for the two different devices of similar heterostructure thus proves
that the main observed phenomenons are inherent to the system and not device
specific.

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