Now we want to proceed as in Remark 3.43. We start by assuming $A$ to be the
finite collection of Lemma 3.44, and then successively add matching fields to
$A$.

For each point in the variety $\mathcal{N}_{2}$, there is an index $1\leq
i*\leq n$ with $z_{1,i^{*}n}\neq 0$, since $P_{f}^{1}=Z_{1}e_{n}$ and
$P_{s}^{2}=0$ don’t vanish simultaneously by the recursive formula (4). So
this is especially true for points in the stratum $\mathcal{B}_{n}$. The tupel
${x=(z_{1,nj},z_{2,i^{*}1},...,z_{2,i^{*}n})}$, $j\neq i^{*}$, is of (Type 8)
and we claim that the corresponding vector field $\partial_{x}^{3}$ satisfies
$\partial_{x}^{3}(z_{1,nj})=(z_{1,ni^{*}})^{n}\det(Z_{3})\neq 0$ and
$\partial_{x}^{3}(z_{3,ij})=0$ for all $1\leq i\leq j\leq n$. By construction,
there holds $\partial_{x}^{3}(P_{i}^{3})=0$ for $1\leq i\leq n$. The recursive
formula (4) implies

$P_{i}^{3}=z_{1,ni}+\sum_{l=1}^{n}z_{3,il}P_{n+l}^{2}.$

Hence

$\displaystyle
0=\partial_{x}^{3}(P_{i}^{3})=\partial_{x}^{3}(z_{1,ni})+\sum_{l=1}^{n}z_{3,il}\partial_{x}^{3}(P_{n+l}^{2}),\quad
1\leq i\leq n.$

Suppose that $\partial_{x}^{3}(P_{n+l}^{2})=0$, for all $1\leq l\leq n$. Then
we obtain a contradiction

$0=\partial_{x}^{3}(P_{j}^{3})=\partial_{x}^{3}(z_{1,nj})\neq 0.$

Therefore, the flow of $\partial_{x}^{3}$ through points of $\mathcal{B}_{n}$
leaves $\mathcal{N}_{2}$. We add $\partial_{x}^{3}$ (actually its $n$ push-
forwards with respect to the biholomorphisms from Lemma 3.15) to the
collection $A$ and by Lemma 3.44, we get

$\mathcal{F}^{4}_{y}\cap\mathcal{B}_{n}\subset
C_{A}(\mathcal{F}^{4}_{y}\cap\mathcal{N}_{1})\subset
C_{A}(\mathcal{F}^{4}_{y}\cap\mathcal{U}_{4}).$

Now, let’s prove the claim. The vector field $\partial_{x}^{3}$ is constant in
the directions $\tfrac{\partial}{\partial z_{3,ij}}$, by definition. So we
only need to check $\partial_{x}^{3}(z_{1,jn})\neq 0$. By definition, there
holds $\partial_{x}^{3}(z_{1,jn})=\det(A)$ for
$A:=\begin{pmatrix}\frac{\partial}{\partial{{z}_{2,i^{*}1}}}P_{f}^{3}&\cdots&\frac{\partial}{\partial{{z}_{2,i^{*}n}}}P_{f}^{3}\end{pmatrix}$.
Note that

$\frac{\partial}{\partial{{z}_{2,ij}}}P_{f}^{3}=\begin{pmatrix}I_{n}&Z_{3}\end{pmatrix}\begin{pmatrix}0&0\\\
\tilde{E}_{ij}&0\end{pmatrix}\begin{pmatrix}Z_{1}e_{n}\\\
e_{n}\end{pmatrix}=Z_{3}\tilde{E}_{ij}Z_{1}e_{n}.$

Therefore, by definition of $F_{i^{*}}$ (see paragraph before Corollary 3.7),

$A=\begin{pmatrix}Z_{3}\tilde{E}_{i^{*}1}Z_{1}e_{n}&\cdots&Z_{3}\tilde{E}_{i^{*}n}Z_{1}e_{n}\end{pmatrix}=Z_{3}F_{i^{*}}(Z_{1}e_{n})=Z_{3}\begin{pmatrix}z_{1,i^{*}n}&&&&\\\
&\ddots&&&\\\ z_{1,1n}&\cdots&z_{1,i^{*}n}&\cdots&z_{1,nn}\\\ &&&\ddots&\\\
&&&&z_{1,i^{*}n}\end{pmatrix}$

and hence

$\partial_{x}^{3}(z_{1,jn})=\det(A)=(z_{1,i^{*}n})^{n}\det(Z_{3})\neq 0.$

In a next step, we assume $1\leq k\leq n-1$ and we divide the stratum
$\mathcal{B}_{k}$ into two more strata. Define

$\mathcal{C}_{k}:=\\{\vec{Z}_{4}\in\mathcal{B}_{k}:Z_{3}e_{i}\neq 0\Rightarrow
z_{1,ni}=0,\forall 1\leq i\leq n\\}$

and consider a point $\vec{Z}_{4}\in\mathcal{B}_{k}\setminus\mathcal{C}_{k}$.
By definition, there is an index $1\leq j^{*}\leq n$ such that
$Z_{3}e_{j^{*}}\neq 0$ and $z_{1,j^{*}n}\neq 0$. Furthermore, again by
definition, we assume that the matrix $Z_{3}$ has rank $k$. By the
complementary bases theorem, we can choose complementary indices
$\\{i_{1},...,i_{n-k}\\}\dot{\cup}\\{j_{1},...,j_{k}\\}=\\{1,...,n\\}$ such
that $j^{*}\in\\{j_{1},...,j_{k}\\}$ and such that the vectors
$Z_{3}e_{j_{1}},...,Z_{3}e_{j_{k}}$ form a basis of the image
$\mathrm{Im}(Z_{3})$. Choose $i\in\\{i_{1},...,i_{n-k}\\}$, then the tupel
$x=(z_{1,i_{1}n},...,z_{1,i_{n-k}n},z_{2,j^{*}j_{1}},...,z_{2,j^{*}j_{k}},z_{3,ii})$
is of (Type 7). Since
$\frac{\partial}{\partial{{z}_{3,ii}}}P_{f}^{3}=\tilde{E}_{ii}P_{s}^{2}=0$ on
$\mathcal{N}_{2}$, the vector field $\partial_{x}^{3}$ is given by
$\pm\det(B)\frac{\partial}{\partial{{z}_{3,ii}}}$ on $\mathcal{N}_{2}$, where

$\displaystyle B$
$\displaystyle=\begin{pmatrix}\frac{\partial}{\partial{{z}_{1,i_{1}n}}}P_{f}^{3}&\cdots&\frac{\partial}{\partial{{z}_{1,i_{n-k}n}}}P_{f}^{3}&\frac{\partial}{\partial{{z}_{2,j^{*}j_{1}}}}P_{f}^{3}&\cdots&\frac{\partial}{\partial{{z}_{2,j^{*}j_{k}}}}P_{f}^{3}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}(I_{n}+Z_{3}Z_{2})e_{i_{1}}&\cdots&(I_{n}+Z_{3}Z_{2})e_{i_{n-k}}&Z_{3}\tilde{E}_{j^{*}j_{1}}Z_{1}e_{n}&\cdots&Z_{3}\tilde{E}_{j^{*}j_{k}}Z_{1}e_{n}\end{pmatrix}.$

We’re going to show that $B$ is regular. At first, observe that by some basic
linear algebra arguments, we get

$\mathrm{span}\\{\tilde{E}_{j^{*}j_{1}}Z_{1}e_{n},...,\tilde{E}_{j^{*}j_{k}}Z_{1}e_{n}\\}=\mathrm{span}\\{z_{1,j^{*}n}e_{j_{1}},...,z_{1,j^{*}n}e_{j_{k}}\\},$

if $j^{*}\in\\{j_{1},...,j_{k}\\}$ and $z_{1,j^{*}n}\neq 0$. By multi-
linearity of the determinant operator, this implies

$\det(B)=\pm(z_{1,j^{*}n})^{k}\det\begin{pmatrix}(I_{n}+Z_{3}Z_{2})e_{i_{1}}&\cdots&(I_{n}+Z_{3}Z_{2})e_{i_{n-k}}&Z_{3}e_{j_{1}}&\cdots&Z_{3}e_{j_{k}}\end{pmatrix}$

and by the complementary bases theorem, applied to the matrix

$\begin{pmatrix}I_{n}&Z_{3}\\\ 0&I_{n}\end{pmatrix}\begin{pmatrix}I_{n}&0\\\
Z_{2}&I_{n}\end{pmatrix}=\begin{pmatrix}I_{n}+Z_{3}Z_{2}&Z_{3}\\\
Z_{2}&I_{n}\end{pmatrix},$

this is non-vanishing, i.e. $\det(B)\neq 0$ in the given point
$\vec{Z}_{4}\in\mathcal{B}_{k}\setminus\mathcal{C}_{k}$. Next consider the
$(k+1)\times(k+1)$ submatrix of $Z_{3}$

$Z:=\begin{pmatrix}z_{3,ii}&z_{3,ij_{1}}&\cdots&z_{3,ij_{k}}\\\
z_{3,j_{1}i}&z_{3,j_{1}j_{1}}&\cdots&z_{3,j_{1}j_{k}}\\\
\vdots&\vdots&\ddots&\vdots\\\
z_{3,j_{k}i}&z_{3,j_{k}j_{1}}&\cdots&z_{3,j_{k}j_{k}}\end{pmatrix},$

for $i\in\\{i_{1},...,i_{n-k}\\}$. Its determinant $\det(Z)$, written as a
function in $z_{3,ii}$, is given by

$\det(Z)=\det((Z_{3})_{i_{1},...,i_{n-k}})z_{3,ii}+\alpha,$

where $\alpha\in\mathbb{C}$ is constant in $z_{3,ii}$. There holds $\det(Z)=0$
on $\mathcal{B}_{k}$, since the rank of $Z_{3}$ is $k$ on this stratum. Apply
the vector field $\partial_{x}^{3}$ to the equation $\det(Z)=0$. Using the
same notation as in Example 3.47, this gives us

$\partial_{x}^{3}(\det(Z))=\det(B)\frac{\partial}{\partial{{z}_{3,ii}}}\det(Z)=\det(B)\det((Z_{3})_{i_{1},...,i_{n-k}})\neq
0.$

Hence the flow of $\partial_{x}^{3}$ through the given point
$\vec{Z}_{4}\in\mathcal{B}_{k}\setminus\mathcal{C}_{k}$ leaves the set
$\mathcal{A}_{k}$ and intersects the stratum $\mathcal{B}_{k+1}$. Hence there
exists a finite collection $A\subset\mathcal{Q}_{4}$ of complete fiber-
preserving holomorphic vector fields such that

$\mathcal{F}^{4}_{y}\cap\mathcal{B}_{k}\setminus\mathcal{C}_{k}\subset
C_{A}(\mathcal{F}^{4}_{y}\cap\mathcal{B}_{k+1})$

for each generic fiber $\mathcal{F}^{4}_{y}$.

Now, consider a point $\vec{Z}_{4}\in\mathcal{C}_{k}$. Choose again a
complementary set of indices
$\\{i_{1},...,i_{n-k}\\}\dot{\cup}\\{j_{1},...,j_{k}\\}=\\{1,...,n\\}$ such
that the vectors $Z_{3}e_{j_{1}},...,Z_{3}e_{j_{k}}$ form a basis of the image
$\mathrm{Im}(Z_{3})$. By definition of $\mathcal{C}_{k}$, there holds
$z_{1,j_{1}n}=...=z_{1,j_{k}n}=0$. Recall that by definition of
$\mathcal{N}_{2}$, there is an index $1\leq i^{*}\leq n$ with
$z_{1,i^{*}n}\neq 0$. In particular, there holds $Z_{3}e_{i^{*}}=0$, again by
definition of $\mathcal{C}_{k}$. The vector fields $\gamma^{2}_{jj,i^{*}}$,
$j\in\\{j_{1},...,j_{k}\\}$, from Lemma 3.17 are given by

$\gamma^{2}_{jj,i^{*}}=(z_{1,i^{*}n})^{2}\frac{\partial}{\partial{{z}_{2,jj}}},\quad
j\in\\{j_{1},...,j_{k}\\},$

in the given point $\vec{Z}_{4}\in\mathcal{C}_{k}$. Consider the tupel
${x=(z_{1,n1},...,z_{1,nn},z_{3,i^{*}i^{*}})}$ of (Type 5). Its corresponding
vector field $\partial_{x}^{3}$ is of the form
$\pm\det(I_{n}+Z_{3}Z_{2})\frac{\partial}{\partial{{z}_{3,i^{*}i^{*}}}}$ in
$\vec{Z}_{4}$. Apply a suitable composition to the equation
$z_{3,i^{*}i^{*}}=0$, namely

$\displaystyle\gamma^{2}_{j_{k}j_{k},i^{*}}\circ\cdots\circ\gamma^{2}_{j_{1}j_{1},i^{*}}\circ\partial_{x}^{3}(z_{3,i^{*}i^{*}})$
$\displaystyle=\pm\gamma^{2}_{j_{k}j_{k},i^{*}}\circ\cdots\circ\gamma^{2}_{j_{1}j_{1},i^{*}}(\det(I_{n}+Z_{3}Z_{2}))$
$\displaystyle=\pm(z_{1,i^{*}n})^{2k}\frac{\partial}{\partial{{z}_{2,j_{k}j_{k}}}}\circ\cdots\circ\frac{\partial}{\partial{{z}_{2,j_{1}j_{1}}}}(\det(I_{n}+Z_{3}Z_{2})).$

Note that
$\frac{\partial}{\partial{{z}_{2,jj}}}(I_{n}+Z_{3}Z_{2})=Z_{3}\tilde{E}_{jj}$
which means that the $j$-th column $(I_{n}+Z_{3}Z_{2})e_{j}$ is the only
column in $I_{n}+Z_{3}Z_{2}$ depending on the variable $z_{2,jj}$. In
particular, the product formula for determinants (12) and a suitable
rearrangement of the columns yields

$\frac{\partial}{\partial{{z}_{2,j_{k}j_{k}}}}\circ\cdots\circ\frac{\partial}{\partial{{z}_{2,j_{1}j_{1}}}}(\det(I_{n}+Z_{3}Z_{2}))=\pm\det((I_{n}+Z_{3}Z_{2})e_{i_{1}},...,(I_{n}+Z_{3}Z_{2})e_{i_{n-k}},Z_{3}e_{j_{1}},...,Z_{3}e_{j_{k}}).$

Another application of the complementary bases theorem implies

$\gamma^{2}_{j_{k}j_{k},i^{*}}\circ\cdots\circ\gamma^{2}_{j_{1}j_{1},i^{*}}\circ\partial_{x}^{3}(z_{3,i^{*}i^{*}})\neq
0.$

By Lemma 3.34, we have found a suitable finite collection
$A\subset\mathcal{Q}_{4}$ such that

$\mathcal{F}^{4}_{y}\cap\mathcal{C}_{k}\subset
C_{A}(\mathcal{F}^{4}_{y}\cap\mathcal{B}_{k}\setminus\mathcal{C}_{k})$

for all generic fibers $\mathcal{F}^{4}_{y}$.

In summary, we have found a finite collection $A\subset\mathcal{Q}_{4}$
satisfying

$\mathcal{F}^{4}_{y}\cap\mathcal{B}_{k}\subset
C_{A}(\mathcal{F}^{4}_{y}\cap\mathcal{B}_{k+1}),$

for each $1\leq k\leq n-1$ and each generic fiber $\mathcal{F}^{4}_{y}$.

In a last step, consider a point $\vec{Z}_{4}\in\mathcal{B}_{0}$. Then the
field $\partial_{x}^{3}$, corresponding to the tupel
$x=(z_{1,1n},...,z_{1,nn},z_{3,nn})$ of (Type 7), is given by

$\partial_{x}^{3}=\pm\det(I_{n}+Z_{3}Z_{2})\frac{\partial}{\partial{{z}_{3,nn}}}=\pm\frac{\partial}{\partial{{z}_{3,nn}}}.$

Therefore, there is a finite $A\subset\mathcal{Q}_{4}$ with
$\mathcal{F}^{4}_{y}\cap\mathcal{B}_{0}\subset
C_{A}(\mathcal{F}^{4}_{y}\cap\mathcal{B}_{1})$ for each generic fiber.

It remains to put everything together and apply the properties of the closure
operator $C_{A}$. More precisely, we have found a finite
$A\subset\mathcal{Q}_{4}$ such that

$C_{A}(\mathcal{F}^{4}_{y}\cap\mathcal{B}_{k})\subset
C_{A}(\mathcal{F}^{4}_{y}\cap\mathcal{B}_{k+1})$

for each $0\leq k\leq n-1$ and each generic fiber. And since we already know

$\mathcal{F}^{4}_{y}\cap\mathcal{B}_{n}\subset
C_{A}(\mathcal{F}^{4}_{y}\cap\mathcal{U}_{4})$

for each generic fiber, the proof is complete. ∎

####  3.7.4 The Spanning theorem for $K\geq 5$

The strategy for $K\geq 5$ is very similar to that for $K=4$. Let’s write the
function $Q_{f}^{K-2}Q_{s}^{K-2}$ in terms of the notation $\tilde{P}^{K}$
(see Lemma 3.15). Recall

$\tilde{P}^{K}=\begin{cases}P_{f}^{K}&\text{if }K=2k+1\\\ P_{s}^{K}&\text{if
}K=2k\end{cases}$

and the recursive formula (5)

$\tilde{P}^{K}=\tilde{P}^{K-2}+Z_{K}\tilde{P}^{K-1}.$

Then we get,

$Q_{f}^{K-2}Q_{s}^{K-2}=\tilde{P}^{K-2}_{1}\cdots\tilde{P}^{K-2}_{n}\tilde{P}^{K-3}_{1}\cdots\tilde{P}^{K-3}_{n}$

for all $K\geq 5$.

######  Lemma 3.48.

Let $K\geq 5$ and define

$\mathcal{N}_{1}:=\\{\vec{Z}_{K}\in\mathcal{W}_{K}:\tilde{P}^{K-2}_{1}\cdots\tilde{P}^{K-2}_{n}\tilde{P}^{K-3}_{1}\cdots\tilde{P}^{K-3}_{n}=0,\tilde{P}^{K-2}\neq
0\\}.$

Then there is a finite collection $A\subset\mathcal{Q}_{K}$ such that

$\mathcal{F}^{K}_{y}\cap\mathcal{N}_{1}\subset
C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{U}_{K})$

for each generic fibers $\mathcal{F}^{K}_{y}$.

###### Proof.

In a first step, consider a point $\vec{Z}_{K}$ in the set

$X_{1}=\\{\vec{Z}_{K}\in\mathcal{N}_{1}:\tilde{P}^{K-3}_{1}\cdots\tilde{P}^{K-3}_{n}\neq
0\\}.$

By definition of this set, we find indices $1\leq i,i^{*}\leq n$ with
$\tilde{P}^{K-2}_{i}(\vec{Z}_{K})=0$ and
$\tilde{P}^{K-2}_{i^{*}}(\vec{Z}_{K})\neq 0$. Pick the vector field
$\partial_{x_{i}}^{K-1}$ corresponding to the tupel
$x_{i}=(z_{K-2,ii},z_{K-1,i^{*}1},...,z_{K-1,i^{*}n})$ of (Type 2). It is of
the form

$\pm(\tilde{P}^{K-2}_{i^{*}})^{n}\frac{\partial}{\partial{{z}_{K-2,ii}}}+\sum_{l=1}^{n}\alpha_{l}\frac{\partial}{\partial{{z}_{K-1,i^{*}l}}}$

for some suitable holomorphic functions $\alpha_{1},...,\alpha_{n}$. By the
recursive formula (5), we get

$\tilde{P}^{K-2}_{i}=\tilde{P}^{K-4}_{i}+e_{i}^{T}Z_{K-2}\tilde{P}^{K-3}$

and therefore

$\partial_{x_{i}}^{K-1}(\tilde{P}^{K-2}_{i})=\pm(\tilde{P}^{K-2}_{i^{*}})^{n}\tilde{P}^{K-3}_{i}\neq
0.$

This applies for every $1\leq i\leq n$ with $\tilde{P}^{K-2}_{i}=0$. Hence
there is a finite collection $A\subset\mathcal{Q}_{K}$ such that

$\mathcal{F}^{K}_{y}\cap X_{1}\subset
C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{U}_{K})$

for each generic fiber $\mathcal{F}^{K}_{y}$.

In a second step, consider a point $\vec{Z}_{K}=(\vec{Z}_{K-1},Z_{K})$ in the
set

$X_{2}=\\{\vec{Z}_{K}\in\mathcal{N}_{1}:\tilde{P}^{K-3}_{1}\cdots\tilde{P}^{K-3}_{n}=0,\tilde{P}^{K-3}\neq
0\\}.$

Observe that the projection of $\vec{Z}_{K}$ to the first component
$\vec{Z}_{K-1}$ is contained in a generic $(K-1)$-fiber
$\mathcal{F}^{K-1}_{\tilde{y}}$ and in $\mathcal{W}_{K-1}$. By the induction
hypothesis, there is a finite collection $B\subset\mathcal{Q}_{K-1}$ such that

$\vec{Z}_{K-1}\in
C_{B}(\mathcal{F}^{K-1}_{\tilde{y}}\cap\\{\vec{Z}_{K-1}:\tilde{P}^{K-3}_{1}\cdots\tilde{P}^{K-3}_{n}\neq
0\\}).$

Put the corresponding pullbacks into the collection $A$. This gives us a
finite collection $A\subset\mathcal{Q}_{K}$ such that

$\mathcal{F}^{K}_{y}\cap X_{2}\subset C_{A}(\mathcal{F}^{K}_{y}\cap
X_{1})\subset C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{U}_{K})$

for each generic fiber $\mathcal{F}^{K}_{y}$.

In a third step, consider a point $\vec{Z}_{K}=(\vec{Z}_{K-1},Z_{K})$ in the
set

$X_{3}=\\{\vec{Z}_{K}\in\mathcal{N}_{1}:\tilde{P}^{K-3}=0\\}.$

Define the subset

$\tilde{X}_{3}=\\{(\vec{Z}_{K-1},Z_{K})\in
X_{3}:\vec{Z}_{K-1}\in\mathcal{W}_{K-1}\\}.$

By Lemma 3.45, we can apply the induction hypothesis, to obtain a finite
collection $A\subset\mathcal{Q}_{K}$ such that

$\mathcal{F}^{K}_{y}\cap\tilde{X}_{3}\subset C_{A}(\mathcal{F}^{K}_{y}\cap
X_{2})$

for each generic fiber $\mathcal{F}^{K}_{y}$. Observe that
$\mathcal{W}_{2k+1}=\mathcal{W}_{2k}\times\mathbb{C}^{\frac{n(n+1)}{2}}$,
hence the two sets $X_{3}$ and $\tilde{X}_{3}$ coincide for $K=2k+1$. Now, let
us assume $K=2k$ and consider a point $\vec{Z}_{K}\in
X_{3}\setminus\tilde{X}_{3}$. This implies
$Z_{1}e_{n}=Z_{3}e_{n}=...=Z_{2k-3}e_{n}=0$. Pick the vector field
$\partial_{x}^{K-1}$ corresponding to the tupel
$x=(z_{K-3,1n},...,z_{K-3,nn},z_{K-1,nn})$ of (Type 5). It satisfies

$\partial_{x}^{K-1}(z_{K-3,nn})=\det(e_{1}+Z_{K-1}Z_{K-2}e_{1},...,e_{n-1}+Z_{K-1}Z_{K-2}e_{n-1},e_{n})$

in $\vec{Z}_{K}$. Also consider the fields $\partial_{x_{i}}^{K-1}$ and
$\partial_{y_{i}}^{K-1}$, $i=1,...,n-1$, corresponding to the tupels
$x_{i}=(z_{K-2,ii},z_{K-1,1n},...,z_{K-1,nn})$ and
$y_{i}=(z_{K-1,ii},z_{K-1,1n},...,z_{K-1,nn})$ of (Type 2) and (Type 3),
respectively. They are of the form
$\partial_{x_{i}}^{K-1}=\frac{\partial}{\partial{{z}_{K-2,ii}}}$ and
$\partial_{y_{i}}^{K-1}=\frac{\partial}{\partial{{z}_{K-1,ii}}}$ on
$X_{3}\setminus\tilde{X}_{3}$. With the very same reasoning as in the third
step of Lemma 3.44, we obtain a finite collection $A\subset\mathcal{Q}_{K}$
such that

$\mathcal{F}^{K}_{y}\cap X_{3}\setminus\tilde{X}_{3}\subset
C_{A}(\mathcal{F}^{K}_{y}\cap X_{2})$

for each generic fiber $\mathcal{F}^{K}_{y}$. In summary, we obtain a finite
collection $A\subset\mathcal{Q}_{K}$ such that

$\mathcal{F}^{K}_{y}\cap X_{3}\subset C_{A}(\mathcal{F}^{K}_{y}\cap
X_{2})\subset C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{U}_{K})$

for each generic fiber $\mathcal{F}^{K}_{y}$ and for each $K\geq 5$.

In a last step, observe that $\mathcal{N}_{1}=X_{1}\cup X_{2}\cup X_{3}$. We
put all the involved vector fields above together and obtain a finite
collection $A\subset\mathcal{Q}_{K}$ satisfying

$\mathcal{F}^{K}_{y}\cap\mathcal{N}_{1}\subset
C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{U}_{K})$

for each generic fiber $\mathcal{F}^{K}_{y}$. This finishes the proof. ∎

######  Lemma 3.49.

Let $K\geq 5$ and define

$\mathcal{N}_{2}:=\\{\vec{Z}_{K}\in\mathcal{W}_{K}:\tilde{P}^{K-2}=0\\}.$

Then there is a finite collection $A\subset\mathcal{Q}_{K}$ such that

$\mathcal{F}^{K}_{y}\cap\mathcal{N}_{2}\subset
C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{U}_{K})$

for each generic fibers $\mathcal{F}^{K}_{y}$.

###### Proof.

The proof of this lemma follows more or less the strategy of Lemma 3.45. We
stratify $\mathcal{N}_{2}$ in the following way. For $0\leq k\leq n$, define

$\mathcal{A}_{k}=\\{\vec{Z}_{K}\in\mathcal{N}_{2}:\mathrm{rank}(Z_{K-1})\leq
k\\}$

and as a convention let $\mathcal{A}_{-1}:=\emptyset$. Then each stratum

$\mathcal{B}_{k}:=\mathcal{A}_{k}\setminus\mathcal{A}_{k-1},\quad 0\leq k\leq
n,$

consists of those points $\vec{Z}_{K}\in\mathcal{N}_{2}$ with
$\mathrm{rank}(Z_{K-1})=k$.

In a first step, we prove the following claim: for
$\vec{Z}_{K}=(\vec{Z}_{K-2},Z_{K-1},Z_{K})\in\mathcal{N}_{2}$, the projection
to the first component $\vec{Z}_{K-2}$ is contained in a generic $(K-2)$-fiber
and in $\mathcal{W}_{K-2}$. The first part of the claim follows directly from
the fact that $\tilde{P}^{K-3}$ and $\tilde{P}^{K-2}$ don’t vanish
simultaneously, by the recursive formula (4). For the second part of the
claim, recall that $\vec{Z}_{K}\in\mathcal{W}_{K}$ by definition of
$\mathcal{N}_{2}$. Assume that $\vec{Z}_{K-2}\not\in\mathcal{W}_{K-2}$ and
observe

$\lceil\tfrac{K-3}{2}\rceil=\begin{cases}k-1&\text{if }K=2k+1\\\ k-1&\text{if
}K=2k.\end{cases}$

Then there holds $Z_{1}e_{n}=Z_{3}e_{n}=...=Z_{2k-3}e_{n}=0$ and by Lemma 3.9,
we conclude $\Phi_{2k-2}=e_{2n}$. In the case $K=2k$, this contradicts
$0=\tilde{P}^{K-2}=P_{s}^{2k-2}=e_{n}$. And in the case $K=2k+1$, this leads
to

$0=\tilde{P}^{K-2}=P_{f}^{2k-1}=Z_{2k-1}e_{n},$

which contradicts $\vec{Z}_{K}\in\mathcal{W}_{K}$. This proves the claim. We
are now able to apply the induction hypothesis (the  (Spanning theorem). for
$K-2$) to $\vec{Z}_{K-2}$ and assume without loss of generality that
$\tilde{P}^{K-4}_{1}\cdots\tilde{P}^{K-4}_{n}\neq 0$.

For the second step, observe that for each point in the variety
$\mathcal{N}_{2}$, there is an index $1\leq j\leq n$ with
$\tilde{P}^{K-3}_{j}\neq 0$, since $\tilde{P}^{K-3}$ and $\tilde{P}^{K-2}$
don’t vanish simultaneously. So this is especially true for points
$\vec{Z}_{K}$ in the stratum $\mathcal{B}_{n}$. Consider the tupel
$x=(z_{K-3,ii},z_{K-2,j1},...,z_{K-2,jn})$, $i\neq j$, of (Type 2). The
corresponding vector field $\partial_{x}^{K-1}$ is of the form

$\alpha\frac{\partial}{\partial{{z}_{K-3,ii}}}+\sum_{r=1}^{n}\beta_{r}\frac{\partial}{\partial{{z}_{K-2,jr}}}$

for some suitable holomorphic functions $\alpha,\beta_{1},...,\beta_{n}$. By
construction there holds ${\partial_{x}^{K-1}(\tilde{P}^{K-1})=0}$.
Furthermore, there holds
$\tilde{P}^{K-1}=\tilde{P}^{K-3}+Z_{K-1}\tilde{P}^{K-2}$ by the recursive
formula (5). Hence

$0=\partial_{x}^{K-1}(\tilde{P}^{K-1})=\partial_{x}^{K-1}(\tilde{P}^{K-3})+Z_{K-1}\partial_{x}^{K-1}(\tilde{P}^{K-2}),$

which implies that if $\tilde{P}^{K-2}$ is in the kernel of
$\partial_{x}^{K-1}$, then $\tilde{P}^{K-3}$ is in that kernel too.

Observe that $\tilde{P}^{K-4}$ and $\tilde{P}^{K-5}$ are in the kernel of
$\partial_{x}^{K-1}$, since they don’t depend on the matrices $Z_{K-3}$ and
$Z_{K-2}$. Therefore

$\partial_{x}^{K-1}(\tilde{P}^{K-3})=\partial_{x}^{K-1}(Z_{K-3})\tilde{P}^{K-4}=\alpha\tilde{E}_{ii}\tilde{P}^{K-4}=\alpha\tilde{P}^{K-4}_{i},$

and by the previous paragraph, we may assume $\tilde{P}^{K-4}_{i}\neq 0$. It
remains to compute $\alpha$. First, let’s compute the derivatives

$\frac{\partial}{\partial{{z}_{K-2,jr}}}\tilde{P}^{K-1}=Z_{K-1}\frac{\partial}{\partial{{z}_{K-2,jr}}}\tilde{P}^{K-2}=Z_{K-1}\tilde{E}_{jr}\tilde{P}^{K-3},\quad
1\leq r\leq n.$

This gives us

$\displaystyle\alpha$
$\displaystyle=\det(Z_{K-1}\tilde{E}_{j1}\tilde{P}^{K-3},...,Z_{K-1}\tilde{E}_{jn}\tilde{P}^{K-3})$
$\displaystyle=\det(Z_{K-1})\det(\tilde{E}_{j1}\tilde{P}^{K-3},...,\tilde{E}_{jn}\tilde{P}^{K-3})$
$\displaystyle=\det(Z_{K-1})\det\begin{pmatrix}\tilde{P}^{K-3}_{j}&&&&\\\
&\ddots&&&\\\
\tilde{P}^{K-3}_{1}&\cdots&\tilde{P}^{K-3}_{j}&\cdots&\tilde{P}^{K-3}_{n}\\\
&&&\ddots&\\\ &&&&\tilde{P}^{K-3}_{j}\end{pmatrix}$
$\displaystyle=(\tilde{P}^{K-3}_{j})^{n}\det(Z_{K-1})\neq 0.$

Hence $\partial_{x}^{K-1}(\tilde{P}^{K-3})\neq 0$ and therefore also
$\partial_{x}^{K-1}(\tilde{P}^{K-2})\neq 0$. We conclude that there is a
finite $A\subset\mathcal{Q}_{K}$ such that

$\mathcal{F}^{K}_{y}\cap\mathcal{B}_{n}\subset
C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{N}_{1})\subset
C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{U}_{K})$

for each generic fiber $\mathcal{F}^{K}_{y}$.

In a third step, let’s assume $1\leq k\leq n-1$ and we divide the stratum
$\mathcal{B}_{k}$ into two more strata. Define

$\mathcal{C}_{k}:=\\{\vec{Z}_{K}\in\mathcal{B}_{k}:Z_{K-1}e_{i}\neq
0\Rightarrow\tilde{P}^{K-3}_{i}=0,\forall 1\leq i\leq n\\}$

and consider a point $\vec{Z}_{K}\in\mathcal{B}_{k}\setminus\mathcal{C}_{k}$.
By definition, there is an index $1\leq j^{*}\leq n$ such that
$Z_{K-1}e_{j^{*}}\neq 0$ and $\tilde{P}^{K-3}_{j^{*}}\neq 0$. Moreover, also
by definition of the stratum $\mathcal{B}_{k}$, we assume that the matrix
$Z_{K-1}$ has rank $k$. By the complementary bases theorem, we can choose
complementary indices
$\\{i_{1},...,i_{n-k}\\}\dot{\cup}\\{j_{1},...,j_{k}\\}=\\{1,...,n\\}$ such
that $j^{*}\in\\{j_{1},...,j_{k}\\}$ and such that the vectors
$Z_{K-1}e_{j_{1}},...,Z_{K-1}e_{j_{k}}$ form a basis of the image
$\mathrm{Im}(Z_{K-1})$. Choose $i,i^{*}\in\\{i_{1},...,i_{n-k}\\}$, then the
tupel
$x=(z_{K-3,i^{*}i_{1}},...,z_{K-3,i^{*}i_{n-k}},z_{K-2,j^{*}j_{1}},...,z_{K-2,j^{*}j_{k}},z_{K-1,ii})$
is of (Type 7). Since
$\frac{\partial}{\partial{{z}_{K-1,ii}}}\tilde{P}^{K-1}=\tilde{E}_{ii}\tilde{P}^{K-2}=0$
on $\mathcal{N}_{2}$, the vector field $\partial_{x}^{K-1}$ is given
$\pm\det(B)\frac{\partial}{\partial{{z}_{K-1,ii}}}$ on $\mathcal{N}_{2}$,
where

$\displaystyle B$
$\displaystyle=\begin{pmatrix}\frac{\partial}{\partial{{z}_{K-3,i^{*}i_{1}}}}\tilde{P}^{K-1}&\cdots&\frac{\partial}{\partial{{z}_{K-3,i^{*}i_{n-k}}}}\tilde{P}^{K-1}&\frac{\partial}{\partial{{z}_{K-2,j^{*}j_{1}}}}\tilde{P}^{K-1}&\cdots&\frac{\partial}{\partial{{z}_{K-2,j^{*}j_{k}}}}\tilde{P}^{K-1}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}(I_{n}+Z_{K-1}Z_{K-2})e_{i_{1}}&\cdots&(I_{n}+Z_{K-1}Z_{K-2})e_{i_{n-k}}&Z_{K-1}\tilde{E}_{j^{*}j_{1}}\tilde{P}^{K-3}&\cdots&Z_{K-1}\tilde{E}_{j^{*}j_{k}}\tilde{P}^{K-3}\end{pmatrix}.$

A similar argument as in Lemma 3.45 implies that $B$ is regular in
$\vec{Z}_{K}$. Moreover, we apply the same strategy from the mentioned lemma,
to conclude that the flow of $\partial_{x}^{K-1}$ through $\vec{Z}_{K}$ leaves
the set $\mathcal{A}_{k}$ and intersects the stratum $\mathcal{B}_{k+1}$.
Hence there exists a finite collection $A\subset\mathcal{Q}_{K}$ such that

$\mathcal{F}^{K}_{y}\cap\mathcal{B}_{k}\setminus\mathcal{C}_{k}\subset
C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{B}_{k+1})$

for each generic fiber $\mathcal{F}^{K}_{y}$.

Next, consider a point $\vec{Z}_{K}\in\mathcal{C}_{k}$. Choose again a
complementary set of indices
$\\{i_{1},...,i_{n-k}\\}\dot{\cup}\\{j_{1},...,j_{k}\\}=\\{1,...,n\\}$ such
that the vectors $Z_{K-1}e_{j_{1}},...,Z_{K-1}e_{j_{k}}$ form a basis of the
image $\mathrm{Im}(Z_{K-1})$. By definition of $\mathcal{C}_{k}$, there holds
$\tilde{P}^{K-3}_{j_{1}}=...=\tilde{P}^{K-3}_{j_{k}}=0$. Recall that there is
an index $1\leq i^{*}\leq n$ with $\tilde{P}^{K-3}_{i^{*}}\neq 0$, since
$\tilde{P}^{K-3}$ and $\tilde{P}^{K-2}$ don’t vanish simultaneously. The
vector fields $\gamma^{K-2}_{jj,i^{*}},j\in\\{j_{1},...,j_{k}\\}$, from Lemma
3.17, are given by

$\gamma^{K-2}_{jj,i^{*}}=(\tilde{P}^{K-3}_{i^{*}})^{2}\frac{\partial}{\partial{{z}_{K-2,jj}}},\quad
j\in\\{j_{1},...,j_{k}\\},$

in $\vec{Z}_{K}$. Now consider the tupel
$x=(z_{K-3,n1},...,z_{K-3,nn},z_{K-1,i^{*}i^{*}})$ of (Type 5). Its
corresponding vector field $\partial_{x}^{K-1}$ is of the form
$\pm(\tilde{P}^{K-4}_{n})^{n}\det(I_{n}+Z_{K-1}Z_{K-2})\frac{\partial}{\partial{{z}_{K-1,i^{*}i^{*}}}}$
in $\vec{Z}_{K}$. As in Lemma 3.45, we have

$\gamma^{K-2}_{j_{k}j_{k},i^{*}}\circ\cdots\circ\gamma^{K-2}_{j_{1}j_{1},i^{*}}\circ\partial_{x}^{K-1}(z_{K-1,i^{*}i^{*}})\neq
0.$

This implies the existence of a finite collection $A\subset\mathcal{Q}_{K}$
such that

$\mathcal{F}^{K}_{y}\cap\mathcal{C}_{k}\subset
C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{B}_{k}\setminus\mathcal{C}_{k})$

for each generic fiber $\mathcal{F}^{K}_{y}$. In summary, this leads to

$\mathcal{F}^{K}_{y}\cap\mathcal{B}_{k}\subset
C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{B}_{k+1})$

for each generic fiber $\mathcal{F}^{K}_{y}$ and $1\leq k\leq n-1$.

It remains to consider a point $\vec{Z}_{K}$ in the stratum $\mathcal{B}_{0}$,
that is, we assume $Z_{K-1}=0$. Choose again the vector field
$\partial_{x}^{K-1}$ corresponding to the tupel
$x=(z_{K-3,n1},...,z_{K-3,nn},z_{K-1,i^{*}i^{*}})$ of (Type 5). Then

$\partial_{x}^{K-1}(z_{K-1,i^{*}i^{*}})=\pm(\tilde{P}^{K-4}_{n})^{n}\underbrace{\det(I_{n}+Z_{K-1}Z_{K-2})}_{=1}\neq
0,$

and therefore there is a finite collection $A\subset\mathcal{Q}_{K}$ such that

$\mathcal{F}^{K}_{y}\cap\mathcal{B}_{0}\subset
C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{B}_{1})$

for each generic fiber $\mathcal{F}^{K}_{y}$.

For the final step, we only need to put everything together. We have

$C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{B}_{0})\subset
C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{B}_{1})\subset\cdots\subset
C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{B}_{n})\subset
C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{U}_{K})$

for each generic fiber $\mathcal{F}^{K}_{y}$, and since
$\mathcal{N}_{2}=\bigcup_{k=0}^{n}\mathcal{B}_{k}$, we obtain

$\mathcal{F}^{K}_{y}\cap\mathcal{N}_{2}\subset
C_{A}(\mathcal{F}^{K}_{y}\cap\mathcal{U}_{K})$

for each generic fiber $\mathcal{F}^{K}_{y}$. This finishes the proof. ∎

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# The possible coexistence of superconductivity and topological electronic
states in 1T-RhSeTe

Tengdong Zhang1 Rui Fan3 Yan Gao1 Yanling Wu1 Xiaodan Xu1 Dao-Xin Yao2
<EMAIL_ADDRESS>Jun Li1<EMAIL_ADDRESS>1Key Laboratory for
Microstructural Material Physics of Hebei Province, School of Science, Yanshan
University, Qinhuangdao 066004, China.
2State Key Laboratory of Optoelectronic Materials and Technologies, Guangdong
Provincial Key Laboratory of Magnetoelectric Physics and Devices, School of
Physics, Sun Yat-Sen University, Guangzhou 510275, Peoples Republic of China.
3School of Physics, Beijing Institute of Technology, Beijing 100081, China

###### Abstract

Transition metal dichalcogenides (TMDs), exhibit a range of crystal structures
and topological quantum states. The 1$T$ phase, in particular, shows promise
for superconductivity driven by electron-phonon coupling, strain, pressure,
and chemical doping. In this theoretical investigation, we explore 1$T$-RhSeTe
as a novel type of TMD superconductor with topological electronic states. The
optimal doping structure and atomic arrangement of 1$T$-RhSeTe are
constructed. Phonon calculations validate the integrity of the constructed
doping structure. The analysis of the electron-phonon coupling (EPC) using the
Electron-phonon Wannier (EPW) method has confirmed the existence of a robust
electron-phonon interaction in 1$T$-RhSeTe, resulting in total EPC constant
$\lambda$ = 2.02, the logarithmic average frequency $\omega_{\text{log}}$ =
3.15 meV and $T_{c}$ = 4.61 K, consistent with experimental measurements and
indicative of its classification as a BCS superconductor. The band structure
analysis revealed the presence of Dirac-like band crossing points. The
topological non-trivial electronic structures of the 1$T$-RhSeTe are confirmed
via the evolution of Wannier charge centers (WCCs). Collectively, these
distinctive properties underscore 1$T$-RhSeTe as a possible candidate for a
topological superconductor, warranting further investigation into its
potential implications and applications.

††preprint: APS/123-QED

##  I INTRODUCTION

Figure 1: Top and side views of the crystal structure of 1$T$-RhSeTe and
RhTe2: (a) type-I doping 1$T$-RhSeTe, where Se replaces Te atoms along lines;
(b) type-II doping 1$T$-RhSeTe, where Se replaces Te atoms within plane; (c)
1$T$-RhTe2.

Transition metal dichalcogenides (TMDs) are compounds represented by the
chemical formula MX2, where M denotes transition metals such as Mo, W, and Ni,
and X represents chalcogen elements such as S, Se, and Te. Featuring a layered
structure, TMDs display a remarkable degree of diversity and exhibit a variety
of topological quantum states, including Weyl semimetallic, topological
insulator, and quantum spin Hall effect [1, 2, 3, 4]. TMDs can adopt various
crystal structures, with the most prevalent being hexagonal (2-$H$),
rhombohedral (3-$R$), and trigonal (1-$T$) phases. The 2-$H$ phase typically
manifests as a semiconductor and is the most thermodynamically stable, while
the 1-$T$ phase exhibits metallic behavior and possesses distinctive
topological properties [5, 6, 7].

The superconducting behaviors of 1-$T$ phase TMDs are intrinsic or induced by
chemical doping or applied strain [8, 9, 10]. For example, both 1$T$-TiSe2 and
1$T$-PdTe2 exhibit semimetallic characteristics, and the topological
superconductivity can be driven by electron-phonon coupling. Experimental
observations have revealed that 1$T$-TiSe2 exhibits a critical temperature
($T_{c}$) of 2 K under a pressure of 3 GPa, while the $T_{c}$ of PdTe2 is 1.7
K [11, 12]. In the 1-$T$ phases of PtTe2, PdSe2, NiTe2, and PdTe2, the Dirac
cone is tilted, and forms a type-II Dirac topological point [13, 14, 15, 16].
Furthermore, the Dirac point in NiTe2 is situated at the Fermi energy level of
0.80 meV, and the position of the Fermi energy can be adjusted through strain,
pressure, and chemical doping [10]. 1$T$-phase TMDs hold significant promise
for inducing superconductivity with topological quantum states.

In the broadest sense, a superconductor is regarded as topological if any of
the topological numbers is nonzero [17, 18]. Topological superconductors have
emerged as a fascinating area of research in quantum computing, due to their
unique physical properties and potential applications [19, 20, 21, 22, 23].
They offer an ideal platform for studying fundamental quantum physics
problems, such as non-abelian anyons and quantum entanglement, and propose
possible solutions for building future quantum computers that can achieve
fault-tolerant quantum computation [22, 24, 23]. Topological superconductors
are closely associated with Majorana fermions, which are exotic particles that
are their own antiparticles [25, 23].

A novel Dirac semimetal, 1$T$-RhSeTe, has been identified as a superconductor
with $T_{c}$ = 4.72 K [26]. In 1$T$-RhSeTe, selenium (Se) replaces tellurium
(Te) atoms in the 1$T$ phase of RhTe2, a TMD previously reported to lack
superconductivity and topological electronic states [27, 28]. This
substitution induces superconductivity and topological electronic states,
which is a possible candidate for topological superconductors.

This study employs density-functional theory (DFT) within first-principles
calculations to examine the electronic bands, phonon, superconductivity, and
topological characteristics of 1$T$-RhSeTe. Analysis of the electron-phonon
coupling and Eliashberg spectral function yields the electron-phonon coupling
strength ($\lambda$ = 2.02 ) and $T_{c}$ = 4.61 K.

Furthermore, a Dirac cone crosses near the Fermi energy in the absence of SOC.
However, the presence of the SOC effect result in the Dirac cone becoming
gapped, thereby manifesting a Dirac-type surface state that is protected by
time-reversal symmetry. The calculated $\mathbb{Z}_{2}$ topological index is
$\mathbb{Z}_{2}$ = 1, indicating that 1$T$-RhSeTe is in a strongly topological
electronic state.

##  II Computational Details

Table 1: Lattice constants for type-I, type-II 1$T$-RhSeTe, compare with experiment and 1$T$-RhTe2 Structure | Lattices(Å) | Energy | Ref.
---|---|---|---
Type-I | a, b = 3.794, c = 5.363 | -32.036 eV | calculate
Type-II | a, b = 3.818, c = 5.291 | -32.012 eV | calculate
Exper. | a, b = 3.798, c = 5.389 |  | Ref.[26]
RhTe2 | a, b = 3.938, c = 5.513 |  | Ref.[27]

We use the projector-augmented wave method implemented in the Vienna Ab initio
Simulation Package (VASP) to perform density-functional theory (DFT)
calculations for the structure optimization [29]. The generalized gradient
approximation (GGA) and the Perdew Burke-Ernzerhof (PBE) function are used to
treat the electron exchange-correlation potential[30, 31]. In the computation,
the plane wave cut-off energy is set at 700 eV, a Monkhorst-Pack k lattice
with a spacing of 2$\pi\times$ 0.03 Å-1 is employed, and the structure is
optimized using optB88-vdW van der Waals (vdW) interaction corrections [32]
until the force on each atom less than 0.015 eV$\times$ Å-1.

Quantum Espresso (QE) is used for DFT calculations including self-consistent,
band structure, and phonon calculations [33]. The PBE generalization of GGA is
the basis for the exchange-correlation generalization. The calculations used
kinetic energy cutoff values of 70 Ry for the wave function and 700 Ry for the
potential function, a 6 $\times$ 6 $\times$ 6 k-grid, and a 6 $\times$ 6
$\times$ 6 q-point grid. The self-consistent convergence threshold is set to 1
$\times$ 10-10 Ry. The d-orbitals of the Rh atoms, the p-orbitals of the Te
atoms, and the p-orbitals of the Se atoms are selected as the projected
orbits, and the 6 $\times$ 6 $\times$ 6 k-mesh grid is set up to construct the
tight-binding Hamiltonian for the established structure by the software
package wannier90 based on maximally localized wannier functions (MLWFs) [34,
35].

The Electron-phonon Wannier (EPW) code is employed to obtain the electron-
phonon coupling and BCS superconductivity. The k-mesh and q-mesh grids are
uniformly increased to 20 $\times$ 20 $\times$ 20 for more accurate results.
The EPW code solves the Eliashberg equation and related superconductivity
equations to calculate electron-phonon coupling strength and $T_{c}$. The
electron-phonon coupling constant $\lambda$, which can be calculated by

$\lambda=2\int\frac{\alpha^{2}\left(\omega\right)}{\omega}d\omega,$ (1)

where $\alpha^{2}F(\Omega)$ is the Eliashberg spectral function, the phonon
density of states weighted by an electron-phonon coupling matrix element at
each k-point and frequency. The superconducting $T_{c}$ is determined with the
McMillan equation modified by Allen-Dynes formula [36, 37]

$T_{c}=\frac{f_{1}f_{2}\omega_{log}}{1.2}\mathrm{exp}\left[\frac{-1.04\left(1+\lambda\right)}{\lambda\left(1-0.62\mu^{*}\right)-\mu^{*}}\right],$
(2)

where $\mu^{*}$ is the effective screened Coulomb repulsion constant typically
ranging from 0.1 to 0.16 eV, and the logarithmically averaged frequency
$\omega_{log}$ can be defined as

$\omega_{log}=\mathrm{exp}\left[\frac{2}{\lambda}\int\frac{d\omega}{\omega}\alpha^{2}F(\omega)\ln{\omega}\right].$
(3)

We use WannierTools[38] to get WCCs [39] to reflect the topological properties
of time-reversal invariant systems. $\mathbb{Z}_{2}$ topological invariant
[40] is given by the number of times $L$ any line crosses a WCCs line when
going from $k_{i}$ = 0 to $k_{i}$ = $\pi$.

$\mathbb{Z}_{2}=L\mod 2$ (4)

The topological properties of three-dimensional materials can be classified
using the $\mathbb{Z}_{2}$ topological index, classified by a set of indices
$(\nu;\,\nu_{x}\,\nu_{y}\,\nu_{z})$ defined through the 2D invariants on the
time-reversal invariant planes in the Brillouin zone

$\nu=\mathbb{Z}_{2}(k_{i}=0)+\mathbb{Z}_{2}(k_{i}=0.5)\mod 2$ (5)
$\nu_{i}=\mathbb{Z}_{2}(k_{i}=0.5)$ (6)

where $k_{i}$ is in reduced coordinates. A system is referred to as a strong
topological electronic state if $\nu$ = 1.

##  III results and discussion

Figure 2: The phonon spectrum of the 1$T$-RhSeTe, phonon density of states
(PHDOS), and Eliashberg spectral function $\alpha^{2}F(\omega)$. Figure 3:
Band structure and density of states of the 1$T$-RhSeTe (a) without SOC and
(b)including SOC obtained from DFT calculations. There is a Dirac point
between $D$ and $\Gamma$.

The 1$T$-RhSeTe sample obtained in the experiment was prepared by doping Te
with Se, with an exact stoichiometry of Rh1.01Se1.05Te0.94. Two potential
doping structures, designated as type-I and type-II, are illustrated in Fig.
1(a) and (b) respectively. In the Type-I doping of 1$T$-RhSeTe, Te atoms are
substituted by Se atoms in a linear configuration, resulting in a zigzag
arrangement of Te and Se atoms. This arrangement disrupts the inherent triple
rotational symmetry of 1$T$-RhTe2 and leads to a slightly shorter Se-Rh bond
compared to the Te-Rh bond, thereby reducing the symmetry to P21/m (No. 11).
In contrast, Type-II doping 1$T$-RhSeTe involves the replacement of Te atoms
by Se atoms within a plane, where Te and Se atoms are distributed in planes,
disrupting only the pre-existing -3 symmetry and reducing the symmetry to P3m1
(No. 156). The symmetry of Type-I doped 1$T$-RhSeTe, which is P21/m (No. 11),
is lower than that of Type-II doping 1$T$-RhSeTe [P3m1 (No. 156)]. Both of
these symmetries are lower than the P-3m1 (No. 164) symmetry of 1$T$-RhTe2.

The lattice constants of the two structures following structural relaxation
are presented in Tab 1. Type-II doping of 1$T$-RhSeTe occupies a larger
portion of the lattice within the plane, and the two different lengths of Se-
Rh and Te-Rh bonds are forced to twist onto the same plane, resulting in a
higher energy state compared to Type-I of doping 1$T$-RhSeTe. Significantly,
the lattice constants of Type-I doping 1$T$-RhSeTe are more closely aligned
with experimental measurements, suggesting that the experimental measured
1T-RhSeTe is more likely to conform to Type-I doping. Consequently, we only
consider the Type-I doping structure of 1$T$-RhSeTe in the subsequent
analysis.

Phonon dispersion represents a fundamental approach to assessing the the
dynamic stability of a system. The calculated phonon dispersions of
1$T$-RhSeTe are presented in Fig. 2. The absence of imaginary phonon modes in
1T-RhSeTe is indicative of its dynamic stability. The phonon dispersion can be
classified into two distinct regions based on the vibration frequency. In the
high-frequency region ($\omega$ $\geq$ 110 cm -1), the majority of the phonons
are composed of the vibration modes of Se and Te atoms. The Se atoms, which
have the lowest mass, contribute the most to the higher-frequency phonons. The
Te atoms engage in a seesaw-like resonance around the neighboring Rh atoms
with Se atoms, thereby contributing to an additional portion of high-frequency
phonons. In the low-frequency region, the phonons are formed by a combination
of the vibrational modes of Rh and Te atoms, as these atoms are heavier and
more localized in frequency space than Se atoms.

The calculated Eliashberg spectral and integrated electron-phonon coupling
strength are plotted in the right panel of Fig. 2. The distribution of the
Eliashberg spectral function,$\alpha^{2}F(\Omega)$, in the high-frequency
region is almost the same as in the low-frequency region. Notably, the
vibration frequencies of the heavier elements, including Rh, Se, and Te,
predominantly fall within the low-frequency range of 0-200 cm-1. This behavior
mirrors findings from electron-phonon coupling calculations in heavier element
superconductors, such as LaBi3 and LaCuSb2, where electron-phonon interactions
are concentrated in the lower vibration frequency range[41, 42].

The electron-phonon coupling in the low-frequency region peaks at around
$\omega$ $\sim$ 25 cm -1, with obvious in-plane vibration modes around the M-K
line. As indicated by Eq. (1), these in-plane vibration modes can result in a
more pronounced electron-phonon coupling effect, thereby enhancing
superconductivity. Around $\omega$ $\sim$ 180 cm -1 in the high-frequency
region, there is another electron-phonon coupling peak, which is mainly
dominated by the seesaw-like vibration modes of Se and Te atoms.

Utilizing MLWFs, we derive the isotropic Eliashberg spectral function
$\alpha^{2}F(\omega)$ for 1$T$-RhSeTe. The integration of
$\alpha^{2}F(\omega)$ yields the total EPC constant $\lambda$ = 2.02 and the
logarithmic average frequency $\omega_{\text{log}}$ = 3.15 meV, indicating a
strong electron-phonon coupling in the material. $T_{c}$ is estimated using
the modified McMillan equation by Allen and Dynes. We select $\mu^{*}$ = 0.16
eV considering the heavy elements Rh, Se, and Te. Consequently, we obtain
$T_{c}$ = 4.61 K, which is remarkably close to the experimental value of 4.72
K [26]. This calculation lends credence to the system’s classification as a
BCS superconductor, bolstering the validity of our methods.

The energy band structure of 1$T$-RhSeTe is determined using QE, with and
without SOC, as illustrated in Fig. 3. In the absence of SOC, the four energy
bands closest to the Fermi level are distinctly separated from the remaining
bands, exhibiting clear differentiation. It is noteworthy that the two energy
bands in closest proximity to the Fermi level exhibit a band crossing along
the D-$\Gamma$ direction. This crossing point is located very close to the
Fermi level (-0.01 eV) and exhibits a linear dispersion, resulting in the
formation of Dirac fermions that are analogous to the Dirac points observed in
graphene. The Fermi velocity at this point is $v_{f}$ = 1.71 $\times$ 105
ms-1, which is much lower than that of graphene ($\sim$ $\times$ 106 ms-1).
This reduction in Fermi velocity can be attributed to the fact that these two
bands are primarily derived from the p orbitals of Te. Given that the atomic
mass of Te is significantly greater than that of C, the effective Fermi
velocity is consequently much smaller than that observed in graphene.

Figure 4: The evolution of WCCs in the six planes of in Brillouin Zone. The
$\mathbb{Z}_{2}$ invariants are (a) $k_{1}$ = 0.0, $k_{2}-k_{3}$ plane:
$\mathbb{Z}_{2}$ = 1 (b) $k_{1}$ = 0.5, $k_{2}-k_{3}$ plane: $\mathbb{Z}_{2}$
= 0 (c) $k_{2}$ = 0.0, $k_{1}-k_{3}$ plane: $\mathbb{Z}_{2}$ = 1 (d) $k_{2}$ =
0.5, $k_{1}-k_{3}$ plane: $\mathbb{Z}_{2}$ = 0 (e) $k_{3}$ = 0.0,
$k_{1}-k_{2}$ plane: $\mathbb{Z}_{2}$ = 0 (f) $k_{3}$ = 0.5, $k_{1}-k_{2}$
plane: $\mathbb{Z}_{2}$ = 1.

A peak in the density of states (DOS) is observed approximately 0.1 eV below
the Fermi level, predominantly contributed by the Te atoms. This van Hove
singularity in the electron band, associated with the Te atoms, is beneficial
for the formation of superconductivity. Additionally, it is noteworthy that
this van Hove singularity originates from a flat band along the $\Gamma$-M
direction. In conjunction with the aforementioned analysis of the low-
frequency phonons contributed by Te atoms, it is proposed that this in-plane
$\Gamma$-M flat band is pivotal for the emergence of superconductivity.
Building upon the findings of previous on cuprate superconductors, we propose
that electron doping or applying pressure could significantly enhance the
superconductivity of 1$T$-RhSeTe, potentially approaching the McMillan limit.

In consideration of the SOC effect, the calculated band structure and DOS for
1$T$-RhSeTe are presented in Fig. 3(b). In comparison to the band structures
without the SOC effect, the band structures undergo significant changes due to
the pronounced SOC resulting from the large atomic mass. Each energy band
splits into two bands when including SOC, demonstrating that the 1$T$-RhSeTe
system adheres to time-reversal symmetry. The energy and momentum positions of
the Dirac cone in the $\Gamma$-D line are roughly 151 meV (another suspected
Dirac cone at 115 MeV) above the Fermi energy level. The results of the
calculation indicated the presence of topological non-trivial states in
1$T$-RhSeTe.

FIG.4 illustrates the evolution of WCCs in the six planes of the Brillouin
Zone. Three planes exhibit a topologically nontrivial state, leading to the
determination of the topological $\mathbb{Z}_{2}$ index for 1$T$-RhSeTe
represented as (1 ; 0 0 1). The first $\mathbb{Z}_{2}$ index is the strong
index, while the last three $\mathbb{Z}_{2}$ numbers are the weak indices.
This topological $\mathbb{Z}_{2}$ index indicates that 1$T$-RhSeTe has a
strongly topological electronic state.

##  IV Conclusions

In conclusion, theoretical calculations have been conducted on 1$T$-RhSeTe,
which has been identified as a type of superconductor exhibiting topological
electronic states. A doping structure and atomic arrangement for 1$T$-RhSeTe
are constructed that optimizes the material’s properties. The integrity of the
constructed doping structure is validated through phonon calculations. The
analysis of the electron-phonon coupling using the EPW method has confirmed
the existence of a robust electron-phonon interaction in 1$T$-RhSeTe. This has
led to the determination of total EPC constant $\lambda$ = 2.02, the
logarithmic average frequency $\omega_{\text{log}}$ = 3.15 meV and $T_{c}$ =
4.61 K, consistent with experimental measurements and indicative of its
classification as a BCS superconductor. The band structure analysis revealed
the presence of Dirac-like band crossing points, van Hove singularities, and a
flat band. The confirmation of the topological non-trivial electronic
structures of 1$T$-RhSeTe is achieved through the evolution of WCCs. The
coexistence of superconductivity and topological electronic states in
1$T$-RhSeTe indicates that it may be a promising candidate for topological
superconductivity.

##  V ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China
(Grant No. 12204400), Science Research Project of Hebei Education Department
(Grant No. QN2022169), Natural Science Foundation of Hebei Province (Grant No.
A2022203010), Innovation Capability Improvement Project of Hebei province
(Grant No. 22567605H). T. Z. and R. F. contributed equally to this work.

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on studies of carbon-chain molecules that are presented in this article. K.T.
appreciates Professor Masao Saito (National Astronomical Observatory of Japan)
for giving his advice and continuous encouragement. K.T. is supported by JSPS
KAKENHI grant No.JP20K14523. P. G acknowledges the support from the Chalmers
Initiative of Cosmic Origins Postdoctoral Fellowship. We would like to thank
Professor Fumitaka Nakamura (National Astronomical Observatory of Japan) and
Professor Kazuhito Dobashi (Tokyo Gakugei University) for providing original
data of mapping observations of carbon-chain species toward TMC-1 obtained by
the Nobeyama 45m radio telescope. We would also like to thank Dr. Emmanuel E.
Etim for his comments and suggestions. J.C.T. acknowledges support from ERC
Advanced Grant MSTAR.

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