text:
\section{Introduction}

Averaging operations has been studied a lot in matrix theory ad operators. Many notions of means of positive definite matrices have been posed and studied extensively.
The metric geometric mean of two positive definite matrices $A$ and $B$ is defined as
\begin{equation*}
A \# B := A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2},
\end{equation*}
which was first introduced by Pusz and Woronowicz \cite {PW75} in 1975 and further studied by Kubo and Ando~\cite{KA79} in the 1970s. Since then, it has attracted a great deal of attention in the past several decades. Though the definition looks awkward, it is indeed a natural generalization of the classical geometric mean $\sqrt {ab}$ of two positive numbers $a, b$ \cite{Bh}. Later Kubo and Ando~\cite{KA79} established the metric geometric mean of positive definite operators, then many properties and applications including the operator inequality have been developed.

Based on the metric geometric mean, the spectral geometric mean of two positive definite $A, B$ was introduced by Fiedler and Pt\'ak \cite {FP97}
\begin{equation*}\label{eqn:sm}
A \natural B := (A^{-1}\# B)^{1/2}A(A^{-1}\# B)^{1/2}.
\end{equation*}
They named it as spectral geometric mean because the square of $A \natural B$ is similar to $AB$, which means that the eigenvalues of their spectral mean are the positive square roots of the corresponding eigenvalues of $AB$ \cite [Theorem 3.2 and Remark 3.4] {FP97}.
As the spectral geometric mean is based on the metric geometric mean form the view of the formulations, it possesses some important properties of the metric geometric mean and has been studied ~\cite{KL15, LL04, LMW21, Li12}. However, unlike the metric geometric mean, not many results have been obtained on the spectral geometric mean. Thus, this paper aims to obtain new results on the spectral geometric mean and its extension, namely, the weighted spectral geometric mean.

The structure of this article is organized as follows. We recall the fundamental properties of the weighted spectral geometric mean, and also provide a new proof of Lie-Trotter-Kato formula for the weighted spectral geometric mean in Section~\ref{Sec:Sp-Geo}. In Section~\ref{Sec:Solution} we study at the weighted spectral geometric mean from the point of view as the unique solution of the nonlinear equations. In Section~\ref{Sec:Geodesic} we prove that the weighted spectral geometric mean is a geodesic with respect to the semi-metric $d(A, B) = 2 \Vert \log (A^{-1} \# B) \Vert$, and give the counterexample of the convexity. In Section~\ref{Sec:Tolerance} we construct the equivalent conditions that $A^{-1}$ and $B$ have two kinds of tolerance relations. We also prove in Section~\ref{Sec:Pinch} that one positive $n$-tuple can be obtained by at most $n$ spectral pinches consecutively from another positive $n$-tuple if the log-majorization relation exists.

\section{Spectral geometric mean}~\label{Sec:Sp-Geo}

Let $B(\mathcal{H})$ be the Banach space of all bounded linear operators on a Hilbert space $\mathcal{H}$ with inner product $\langle \cdot, \cdot \rangle$, and let $S(\mathcal{H}) \subset B(\mathcal{H})$ be the closed subspace of all self-adjoint linear operators. We say that $A \in S(\mathcal{H})$ is positive semi-definite (positive definite) if $\langle x, Ax \rangle \geq (>) 0$ for all (nonzero, respectively) vector $x \in \mathcal{H}$. We denote as $\mathbb{P} \subset S(\mathcal{H})$ the open convex cone of all positive definite operators. The group $\textrm{GL}$ of all invertible operators transitively acts on $\mathbb{P}$ via congruence transformation. On the finite-dimensional setting $\mathcal{H} = \mathbb{C}^{m}$ we write as $\mathbb{P}_{m}$ the open convex cone of all $m \times m$ positive definite matrices.

The weighted spectral geometric mean of $A, B \in \mathbb{P}$ is a curve defined by
\begin{displaymath}
t \in \mathbb{R} \mapsto A \natural_{t} B := (A^{-1} \# B)^{t} A (A^{-1} \# B)^{t}.
\end{displaymath}
Note that $A \natural_{0} B = A$ and $A \natural_{1} B = B$. We simply write as $A \natural B = A \natural_{1/2} B$.

We list fundamental properties of the weighted spectral geometric mean.
\begin{lemma} \label{L:SG} \cite{HL07}
Let $A, B\in \mathbb{P}$ and $s, t, u \in \mathbb{R}$.
\begin{itemize}
\item[(1)] $A \natural_{t} B = A^{1-t} B^{t}$ if $A$ and $B$ commute.
\item[(2)] $(a A) \natural_{t} (b B) = a^{1-t} b^{t} (A \natural_{t} B)$ for any $a, b > 0$.
\item[(3)] $U^{*} (A \natural_{t} B) U = (U^{*} A U) \natural_{t} (U^{*} B U)$ for any unitary operator $U$.
\item[(4)] $A \natural_{t} B = B \natural_{1-t} A$.
\item[(5)] $(A \natural_{s} B) \natural_{t} (A \natural_{u} B) = A \natural_{(1-t)s + t u} B$.
\item[(6)] $(A \natural_{t} B)^{-1} = A^{-1} \natural_{t} B^{-1}$.
\end{itemize}
\end{lemma}

\begin{remark}
By \cite[Proposition 2.3]{HK} the boundedness of weighted spectral geometric mean for Loewner order has been shown:
\begin{equation} \label{E:bound}
2^{1 + t} (A + B^{-1})^{-t} - A^{-1} \leq A \natural_{t} B \leq [ 2^{1 + t} (A^{-1} + B)^{-t} - A ]^{-1}
\end{equation}
for any $A, B \in \mathbb{P}_{m}$ and $t \in (0,1)$. Using \eqref{E:bound} we provide another proof of Lie-Trotter-Kato formula for the weighted spectral geometric mean:
\begin{equation} \label{E:Lie-Trotter-Kato}
\lim_{s \to 0} \, \left( A^{s} \natural_{t} B^{s} \right)^{1/s} = \exp ((1-t) \log A + t \log B).
\end{equation}

Without loss of generality, assume that $A, B \geq \alpha I$ for some $0 < \alpha < 1$. For any $s > 0$ by \eqref{E:bound}
\begin{displaymath}
2^{1 + t} (A^{s} + B^{-s})^{-t} - A^{-s} \leq A^{s} \natural_{t} B^{s} \leq [ 2^{1 + t} (A^{-s} + B^{s})^{-t} - A^{s} ]^{-1}.
\end{displaymath}
Set $f(s) := 2^{1 + t} (A^{s} + B^{-s})^{-t} - A^{-s}$ on the interval
\begin{displaymath}
I: 0 < s < \min \left\{ t, \frac{1}{-1 + \log_{2} (\alpha^{t-1} + \alpha^{-t-1})} \right\}.
\end{displaymath}
Since $\alpha^{t-1} + \alpha^{-t-1} \geq 2 \alpha^{-1} > 2$ by the arithmetic-geometric mean inequality, we have $\log_{2} (\alpha^{t-1} + \alpha^{-t-1}) > 1$. Since $\displaystyle s < \frac{1}{-1 + \log_{2} (\alpha^{t-1} + \alpha^{-t-1})}$ and $B^{-t} \leq \alpha^{-t} I$, we have $2^{1 + \frac{1}{s}} > \alpha^{t-1} + \alpha^{-t-1}$, and
\begin{displaymath}
A^{t-1} + A^{-1/2} B^{-t} A^{-1/2} \leq A^{t-1} + \alpha^{-t} A^{-1} \leq \left( \alpha^{t-1} + \alpha^{-t-1} \right) I < 2^{1 + \frac{1}{s}} I.
\end{displaymath}
Taking congruence transformation by $A^{1/2}$ yields $A^{t} + B^{-t} < 2^{1 + \frac{1}{s}} A$.

It has been shown in \cite{BS} that the map $\displaystyle F(p) = \left( \frac{A^{p} + B^{p}}{2} \right)^{1/p}$ is operator monotone for $p \in (-\infty, -1] \cup [1, \infty)$. So for $\displaystyle \frac{t}{s} > 1$
\begin{displaymath}
\frac{A + B^{-1}}{2} \leq \left( \frac{A^{\frac{t}{s}} + B^{- \frac{t}{s}}}{2} \right)^{\frac{s}{t}}.
\end{displaymath}
Substituting $A, B$ into $A^{s}, B^{s}$, applying the monotonicity of the $t$-power map, and using the preceding argument we obtain
\begin{displaymath}
\left( \frac{A^{s} + B^{-s}}{2} \right)^{t} \leq \left( \frac{A^{t} + B^{-t}}{2} \right)^{s} < 2 A^{s}.
\end{displaymath}
Thus,
\begin{displaymath}
2^{1 + t} (A^{s} + B^{-s})^{-t} - A^{-s} = 2 \left[ \left( \frac{A^{s} + B^{-s}}{2} \right)^{-t} - \frac{1}{2} A^{-s} \right] > 0,
\end{displaymath}
that is, $f(s)$ is positive definite for any $s \in I$. Since
\begin{displaymath}
f'(s) = 2^{1+t} (-t) (A^{s} + B^{-s})^{-t-1} (A^{s} \log A - B^{-s} \log B) + A^{-s} \log A
\end{displaymath}
we can see by matrix calculus that
\begin{displaymath}
\lim_{s \to 0} \ln f(s)^{1/s} = \lim_{s \to 0} \frac{\ln f(s)}{s} = f'(0) = -t (\log A - \log B) + \log A = (1-t) \log A + t \log B.
\end{displaymath}
Hence, $\displaystyle \lim_{s \to 0} f(s)^{1/s} = \exp ((1-t) \log A + t \log B)$. Similarly, we can obtain
\begin{displaymath}
\lim_{s \to 0} [ 2^{1 + t} (A^{-s} + B^{s})^{-t} - A^{s} ]^{-1/s} = \exp((1-t) \log A + t \log B).
\end{displaymath}
Thus, we conclude \eqref{E:Lie-Trotter-Kato}.
\end{remark}

\section{Unique solution of nonlinear equations} \label{Sec:Solution}

From the Riccati Lemma: the metric geometric mean $A \# B$ is the unique positive definite solution $X \in \mathbb{P}$ of the Riccati equation $X A^{-1} X = B$, we obtain the following proposition.
\begin{proposition} \label{P:sp-mean}
Let $A, B \in \mathbb{P}$ and $t \in \mathbb{R}$. Then $A \natural_{t} B$ is a unique positive definite solution $X \in \mathbb{P}$ of the equation
\begin{displaymath}
A^{-1} \# X = (A^{-1} \# B)^{t}.
\end{displaymath}
\end{proposition}

\begin{theorem}
For non-zero $t \in \mathbb{R}$ let $G_{t}: \mathbb{P} \times \mathbb{P} \to \mathbb{P}$ be a map satisfying $G_{t}(A, A) = A$ for all $A \in \mathbb{P}$ and
\begin{equation} \label{E:condition}
G_{t}(A, B) = I \ \Longrightarrow \ B = A^{1 - \frac{1}{t}} \ \textrm{for any} \ A, B \in \mathbb{P}.
\end{equation}
Then $A \natural_{t} B$ for $A, B \in \mathbb{P}$ is a unique solution $X \in \mathbb{P}$ of the equation
\begin{displaymath}
G_{t}(A \# X^{-1}, B \# X^{-1}) = I.
\end{displaymath}
\end{theorem}

\begin{proof}
Set $U = A \# X^{-1}$ and $V = B \# X^{-1}$ for $X \in \mathbb{P}$. Then $U, V \in \mathbb{P}$ and $G_{t}(U, V) = I$, so by \eqref{E:condition}
\begin{displaymath}
V = U^{1 - \frac{1}{t}}.
\end{displaymath}
Since $X = U^{-1} A U^{-1} = V^{-1} B V^{-1}$ by the Riccati Lemma, we have
\begin{displaymath}
A = U V^{-1} B V^{-1} U = U^{\frac{1}{t}} B U^{\frac{1}{t}}.
\end{displaymath}
Again by the Riccati Lemma $U^{\frac{1}{t}} = A \# B^{-1}$, and hence, by self-duality of the metric geometric mean: $(A \# B)^{-1} = A^{-1} \# B^{-1}$
\begin{displaymath}
X = U^{-1} A U^{-1} = (A \# B^{-1})^{-t} A (A \# B^{-1})^{-t} = (A^{-1} \# B)^{t} A (A^{-1} \# B)^{t} = A \natural_{t} B.
\end{displaymath}
This completes the proof.
\end{proof}

\begin{remark}
A two-variable mean on a set $X$ is a binary operation $G: X \times X \to X$ satisfying the idempotency: $G(x, x) = x$ for all $x \in X$.  There are numerous examples of two-variable mean on the open cone $\mathbb{P}$ satisfying the idempotency and \eqref{E:condition}, such as the metric geometric mean and spectral geometric mean.
\end{remark}

The following show that a unique solution for the system of equations is given by a pair of metric geometric mean and spectral geometric mean.
\begin{theorem}
For any $A, B \in \mathbb{P}$ and $t \in \mathbb{R}$
\begin{displaymath}
\left\{
  \begin{array}{ll}
    A = X^{-t} Y X^{-t} & \hbox{$(1)$} \\
    B = X^{1-t} Y X^{1-t} & \hbox{$(2)$}
  \end{array}
\right.
\end{displaymath}
has a unique solution $(X, Y) = (A^{-1} \# B, A \natural_{t} B)$.
\end{theorem}

\begin{proof}
By (1) $Y = X^{t} A X^{t}$, and then (2) reduces to $B = X A X$. By the Riccati Lemma $X = A^{-1} \# B$, so
\begin{displaymath}
Y = (A^{-1} \# B)^{t} A (A^{-1} \# B)^{t} = A \natural_{t} B.
\end{displaymath}
\end{proof}

\section{Geodesic property}~\label{Sec:Geodesic}

A curve $\gamma: I \to X$ from an interval $I$ to the metric space $(X, d)$ is called a \emph{geodesic} if there exists a constant $c \geq 0$ such that for any $t \in I$ there is a neighborhood $J$ containing $t$ such that
\begin{displaymath}
d(\gamma(t_{1}), \gamma(t_{2})) = c |t_{1} - t_{2}|
\end{displaymath}
for any $t_{1}, t_{2} \in J$. This is a generalized notion of the geodesic in Riemannian manifold. If the above equality with $c = 1$ holds for any $t_{1}, t_{2} \in I$ the geodesic is called a minimizing geodesic or shortest path. The arc length of a minimizing geodesic between two points defines a distance function for any Riemannian manifold, which makes it into a metric space.

We consider the map $d: \mathbb{P} \times \mathbb{P} \to [0, \infty)$ given by
\begin{displaymath}
d(A, B) = 2 \Vert \log (A^{-1} \# B) \Vert,
\end{displaymath}
where $\Vert \cdot \Vert$ denotes the operator norm. From \cite[Lemma 2]{Kim21} it is a semi-metric: it satisfies the axioms of metric except the triangle inequality. Furthermore, it is invariant under the homogeneity, inversion, and unitary congruence transformation.

\begin{theorem}
The weighted geometric mean is a geodesic for the semi-metric $d$, that is,
for any $s, t \in \mathbb{R}$
\begin{displaymath}
d(A \natural_{s} B, A \natural_{t} B) = |s-t| d(A, B).
\end{displaymath}
\end{theorem}

\begin{proof}
When $s = 1$
\begin{displaymath}
\begin{split}
d(A \natural_{1} B, A \natural_{t} B) & = d(B, B \natural_{1-t} A) = 2 \Vert \log (B^{-1} \# (B \natural_{1-t} A)) \Vert \\
& = 2 \Vert \log (B^{-1} \# A)^{1-t} \Vert = |1-t| d(A, B).
\end{split}
\end{displaymath}
The first equality follows from Lemma \ref{L:SG} (4), and the third equality follows from Proposition \ref{P:sp-mean}.

Let $s \neq 1$. Then one can write $\displaystyle A \natural_{t} B = (A \natural_{s} B) \natural_{\frac{t-s}{1-s}} B$ by Lemma \ref{L:SG} (5). Then
\begin{displaymath}
\begin{split}
d(A \natural_{s} B, A \natural_{t} B) & = d \left( A \natural_{s} B, (A \natural_{s} B) \natural_{\frac{t-s}{1-s}} B \right) \\
& = 2 \left\Vert \log \left[ (A \natural_{s} B)^{-1} \# B \right]^{\frac{t-s}{1-s}} \right\Vert \\
& = 2 \left| \frac{t-s}{1-s} \right| \cdot \left\Vert \log \left[ (A \natural_{s} B)^{-1} \# B \right] \right\Vert \\
& = \left| \frac{t-s}{1-s} \right| d(A \natural_{s} B, B) \\
& = \left| \frac{t-s}{1-s} \right| \cdot |1-s| d(A, B) = |t-s| d(A, B).
\end{split}
\end{displaymath}
The second equality follows from Proposition \ref{P:sp-mean}, and the fifth equality follows from the preceding argument.
\end{proof}

\begin{remark}
The weighted metric geometric mean satisfies the convexity for the Thompson metric:
\begin{displaymath}
d_{T}(A \#_{t} B, A \#_{t} C) \leq t d_{T}(B, C)
\end{displaymath}
for $A, B, C \in \mathbb{P}$ and $t \in [0,1]$, where $d_{T}(A, B) = \Vert \log A^{-1/2} B A^{-1/2} \Vert$ denotes the Thompson metric. It gives a generalized convexity
\begin{displaymath}
d_{T}(A \#_{s} B, C \#_{t} D) \leq (1-s) d_{T}(A, C) + s d_{T}(B, D) + |s-t| d_{T}(C, D)
\end{displaymath}
for $s, t \in [0,1]$, and plays a very important role to extension of two-variable geometric mean to multi-variable geometric means as Ando-Li-Mathias mean \cite{ALM}, Bini-Meini-Poloni mean \cite{BMP}, and Cartan mean \cite{Ho, LP14}.

Unfortunately, the weighted spectral geometric mean does not satisfy the convexity for the semi-metric $d$. In other words,
\begin{displaymath}
d(A \natural_{t} B, A \natural_{t} C) \leq t d(B, C)
\end{displaymath}
does not always hold for $A, B, C \in \mathbb{P}$ and $t \in [0,1]$. For a counterexample, let
\begin{displaymath}
A =
\left(
  \begin{array}{cc}
    12.9638 & 8.0820 \\
    8.0820 & 10.9249 \\
  \end{array}
\right), \
B =
\left(
  \begin{array}{cc}
    11.3531 & 9.1847 \\
    9.1847 & 11.9930 \\
  \end{array}
\right), \
C =
\left(
  \begin{array}{cc}
    21.8929 & -10.7568 \\
    -10.7568 & 39.9958 \\
  \end{array}
\right).
\end{displaymath}
Using MATLAB we can get $d(A \natural_{1/3} B, A \natural_{1/3} C) \approx 0.9328$ and $\frac{1}{3} d(B, C) \approx 0.9266$.
\end{remark}

For the uniqueness of a minimal geodesic for the Thompson metric on $\mathbb{P}_{2}$ we have from \cite[Lemma 2.4]{CGL}
\begin{equation} \label{E:linear-sum}
A \#_{t} B = L_{1-t} (\lambda) A + L_{t} (\lambda) B
\end{equation}
for any $A, B \in \mathbb{P}_{2}$ and $t \in [0,1]$, where $\lambda$ is an eigenvalue of $A B^{-1}$ and
\begin{displaymath}
L_{t}(\lambda) =
\left\{
  \begin{array}{ll}
    \displaystyle \frac{\lambda^{t} - \lambda^{-t}}{\lambda - \lambda^{-1}}, & \hbox{$\lambda \neq 1$;} \\
    t, & \hbox{$\lambda = 1$.}
  \end{array}
\right.
\end{displaymath}
The formula \eqref{E:linear-sum} is a generalization of the consequence in \cite[Proposition 4.1.12]{Bh}
\begin{displaymath}
A \# B = \frac{A + B}{\sqrt{\det (A + B)}}
\end{displaymath}
for any $A, B \in \mathbb{P}_{2}$ with determinant $1$. It is a natural question whether or not the spectral geometric mean on $\mathbb{P}$ can be written as a linear sum. It is obvious for commuting $A, B \in \mathbb{P}_{2}$ that the spectral geometric mean has a linear form:
\begin{displaymath}
A \natural_{t} B = A^{1-t} B^{t} = A \#_{t} B = L_{1-t} (\lambda) A + L_{t} (\lambda) B.
\end{displaymath}
On the other hand, it does not hold in general for non-commuting positive definite operators $A$ and $B$.

\begin{theorem}
The linear sum for two-variable spectral geometric mean $A \natural_{t} B$ of non-commuting $A, B \in \mathbb{P}$ for $t \in (0,1)$ is not available.
\end{theorem}

\begin{proof}
For $A, B \in \mathbb{P}$ which $A B \neq B A$, assume that there exist $x, y > 0$ such that $A \natural_{t} B = x A + y B$ for $t \in (0,1)$. Then by Proposition \ref{P:sp-mean}
\begin{displaymath}
\begin{split}
(A^{-1} \# B)^{t} & = A^{-1} \# (x A + y B) \\
& = A^{-1/2} (x A^{2} + y A^{1/2} B A^{1/2})^{1/2} A^{-1/2}.
\end{split}
\end{displaymath}
Since the map $A \mapsto A^{1/2}$ is operator concave on $\mathbb{P}$,
\begin{displaymath}
(A^{-1} \# B)^{t} \geq A^{-1/2} \left[ x A + y (A^{1/2} B A^{1/2})^{1/2} \right] A^{-1/2} = x I + y A^{-1} \# B.
\end{displaymath}
It says that all positive eigenvalues $\lambda$ of a positive definite operator $A^{-1} \# B$ must satisfy
\begin{displaymath}
\lambda^{t} \geq x + y \lambda.
\end{displaymath}
Set $f(\lambda) := \lambda^{t} - x - y \lambda$. Then $f''(\lambda) = t(t-1) \lambda^{t-2} < 0$, and $f(\lambda) \to -x$ as $\lambda \to 0^{+}$. Since $\lambda$ is arbitrarily positive, it does not hold that $f(\lambda) \geq 0$.
\end{proof}

\section{Tolerance relation}~\label{Sec:Tolerance}

A tolerance relation on a set, first recognized by Poincar\'{e} \cite{Poin}, is a reflexive and symmetric relation. One can see that it is like a congruence, except the assumption of transitivity. Tolerance relations provide a convenient tool for studying indiscernibility and indistinguishability phenomena. We study in this section the effect of tolerance relation on spectral geometric means.

For $A, B \in \mathbb{P}_{m}$ we define $A \sim B$ if there exist $a, b > 0$ such that
\begin{center}
$\sqrt{a b} = \det(A^{-1} B)^{\frac{1}{m}}$ \ and \ $\sigma(A^{-1} B) = \{ a, b \}$.
\end{center}
Note that $\sim$ is a tolerance relation on $\mathbb{P}_{m}$. Furthermore, it is invariant under Riemannian isometries such as the inversion and congruence transformation.

\begin{remark} \label{R:tolerance}
If $A$ and $B$ are linearly independent and $A \sim B$, then $m$ is even, and $A \sim B$ if and only if $A^{-1/2} B A^{-1/2}$ is similar to
\begin{displaymath}
D_{m} (a, b) := \textrm{diag}(\underset{m/2}{\underbrace{a, \dots, a}}, \underset{m/2}{\underbrace{b, \dots, b}})
\end{displaymath}
for some $a, b > 0$.
\end{remark}

From the known result, we obtain
\begin{lemma} \cite{Lim21} \label{L:supplement}
For $A, B \in \mathbb{P}_{m}$ with $\det A = \det B = 1$, $A \sim B$ if and only if
\begin{displaymath}
A \# B = \frac{1}{\sqrt[m]{\det(A + B)}} (A + B).
\end{displaymath}
\end{lemma}
We provide several equivalent conditions that $A^{-1}$ and $B$ have the tolerance relation.

\begin{theorem} \label{T:tolerance}
Let $A, B \in \mathbb{P}_{m}$ with $\det A = \det B = 1$, and $t \in \mathbb{R}$. Then the following are equivalent.
\begin{itemize}
\item[(1)] $A^{-1} \sim B$;
\item[(2)] $\displaystyle A^{-1} \# B = \frac{1}{\sqrt[m]{\det(I + A B)}} (A^{-1} + B)$;
\item[(3)] $\displaystyle A \natural_{t} B = \frac{1}{\sqrt[m]{\det(I + A B)^{2t}}} (A^{-1} + B)^{t} A (A^{-1} + B)^{t}$.
\end{itemize}
\end{theorem}

\begin{proof}
From Lemma \ref{L:supplement}, it is obvious that (1) and (2) are equivalent.
\begin{itemize}
\item[(2)] $\Rightarrow (3)$: Applying (2) to definition of the weighted spectral geometric mean, we can easily obtain the formula (3).

\item[(3)] $\Rightarrow (2)$: If $t = 0$ it is trivial. Assume that (3) holds for any non-zero real number $t$. Set $X := (A^{-1} \# B)^{t}$. Then by definition of the weighted spectral geometric mean
\begin{displaymath}
X A X = \frac{1}{\det(I + A B)^{2t/m}} (A^{-1} + B)^{t} A (A^{-1} + B)^{t}.
\end{displaymath}
By the Riccati Lemma and using the definition of metric geometric mean,
\begin{displaymath}
\begin{split}
X & = A^{-1} \# \left[ \frac{1}{\det(I + A B)^{2t/m}} (A^{-1} + B)^{t} A (A^{-1} + B)^{t} \right] \\
& = \frac{1}{\det(I + A B)^{t/m}} (A^{-1} + B)^{t}.
\end{split}
\end{displaymath}
Taking the $1/t$ power on both sides, we obtain (2).
\end{itemize}
\end{proof}

\begin{corollary} \label{C:sp-mean}
Let $A, B \in \mathbb{P}_{m}$ with $\det A = \alpha$ and $\det B = \beta$, and $t \in (0,1)$. Then $A^{-1} \sim B$ implies
\begin{displaymath}
A \natural_{t} B = \frac{(\alpha \beta)^{3t/m}}{\det((\alpha \beta)^{1/m} I + A B)^{2t/m}} (A^{-1} + (\alpha \beta)^{-1/m} B)^{t} A (A^{-1} + (\alpha \beta)^{-1/m} B)^{t}.
\end{displaymath}
\end{corollary}

\begin{proof}
Assume that $A^{-1} \sim B$ for $A, B \in \mathbb{P}_{m}$. That is, there exist $a, b > 0$ such that
\begin{center}
$\sqrt{a b} = \det(A B)^{\frac{1}{m}}$ \ and \ $\sigma(A B) = \{ a, b \}$.
\end{center}
Let $A_{1} = \alpha^{-1/m} A$ and $B_{1} = \beta^{-1/m} B$. Then
\begin{displaymath}
\begin{split}
\sigma(A_{1} B_{1}) & = (\alpha \beta)^{-1/m} \sigma(A B) = \left\{ (\alpha \beta)^{-1/m} a, (\alpha \beta)^{-1/m} b \right\}, \\
\det(A_{1} B_{1})^{1/m} & = (\alpha \beta)^{-1/m} \det (A B)^{1/m} = (\alpha \beta)^{-1/m} \sqrt{a b}.
\end{split}
\end{displaymath}
So $A_{1}^{-1} \sim B_{1}$. Since $A_{1}, B_{1} \in \mathbb{P}_{m}$ have determinant $1$, Theorem \ref{T:tolerance} yields
\begin{displaymath}
A_{1} \natural_{t} B_{1} = \frac{1}{\sqrt[m]{\det(I + A_{1} B_{1})^{2t}}} (A_{1}^{-1} + B_{1})^{t} A_{1} (A_{1}^{-1} + B_{1})^{t}.
\end{displaymath}
By a simple calculation together with the joint homogeneity of weighted spectral geometric mean in Lemma \ref{L:SG} (2), we can obtain the desired formula.
\end{proof}

\begin{remark}
Since any $A, B \in \mathbb{P}_{2}$ have tolerance relation $A^{-1} \sim B$,
\begin{displaymath}
A \natural_{t} B = \frac{(\alpha \beta)^{t/2}}{\det((\alpha \beta)^{1/2} I + A B)^{t}} (\sqrt{\alpha \beta} A^{-1} + B)^{t} A (\sqrt{\alpha \beta} A^{-1} + B)^{t}.
\end{displaymath}
Since $\det((\alpha \beta)^{1/2} I + A B) = \sqrt{\alpha \beta} (2 \sqrt{\alpha \beta} + \tr (A B))$, we have
\begin{displaymath}
A \natural_{t} B = \left[ \frac{1}{2 \sqrt{\alpha \beta} + \tr (A B)} \right]^{t} (\sqrt{\alpha \beta} A^{-1} + B)^{t} A (\sqrt{\alpha \beta} A^{-1} + B)^{t},
\end{displaymath}
which is already provided in \cite[Corollary 2]{Kim21}. So Corollary \ref{C:sp-mean} is a generalization of \cite[Corollary 2]{Kim21} under tolerance relation.
\end{remark}

There is another invariant tolerance relation on $\mathbb{P}_{m}$. For $A, B\in \mathbb{P}_{m}$, we define $A\sigma B$ if there exist $a, b>0$ such that $\sigma (A^{-1}B) = \{ a, b\}$. From the recent result, we know that it can be characterized by the linear sum for the two-variable weighted metric geometric mean $A\#_t B$. Note that for $t\in \mathbb{R}$ and $a, b>0$, define
\[L_{a, b}(t) = \frac{ab^t - ba^t}{a-b}.
\]

\begin{lemma}\cite{Lim21} \label{L:linear}
For $A, B \in \mathbb{P}_m$ and $0<t<1$, $A\sigma B$ if and only if $t\mapsto A\#_t B$ lies in the positive linear span of $A$ and $B$, that is,
\[A\#_t B = L_{a, b}(t) A+ L_{a^{-1}, b^{-1}}(1-t) B,\]
where $\sigma(A^{-1}B) = \{ a, b\}$.
\end{lemma}

Motivated by Lemma~\ref{L:linear}, we provide several equivalent conditions that $A^{-1}$ and $B$ have the tolerance relation.

\begin{theorem}
Let $A, B \in \mathbb{P}_{m}$ and $0<t<1$. Then the following are equivalent.
\begin{itemize}
\item[(1)] $A^{-1} \sigma B$;
\item[(2)] $\displaystyle A^{-1} \# B = \frac{\sqrt{a}-\sqrt{b}}{a-b} (\sqrt{ab}A^{-1} + B)$;
\item[(3)] $\displaystyle A \natural_{t} B = \frac{\sqrt{a}-\sqrt{b}}{a-b} (\sqrt{ab}A^{-1} + B)^{t} A (\sqrt{ab}A^{-1} + B)^{t}$.
\end{itemize}
\end{theorem}

\begin{proof}
By Lemma \ref{L:linear}, it is obvious that (1) and (2) are equivalent for the case $t=1/2$.
\begin{itemize}
\item[(2)] $\Rightarrow (3)$: Applying (2) to definition of the weighted spectral geometric mean, we can easily obtain the formula (3).

\item[(3)] $\Rightarrow (2)$: If $t = 0$ it is trivial. Assume that (3) holds for any non-zero real number $t$. Set $X := (A^{-1} \# B)^{t}$. Then by definition of the weighted spectral geometric mean
\begin{displaymath}
X A X = \left(\frac{\sqrt{a}-\sqrt{b}}{a-b}\right)^{2t} (\sqrt{ab}A^{-1} + B)^t A (\sqrt{ab}A^{-1} + B)^t.
\end{displaymath}
By the Riccati Lemma and using the definition of metric geometric mean,
\begin{displaymath}
\begin{split}
X & = A^{-1} \# \left[ \left(\frac{\sqrt{a}-\sqrt{b}}{a-b}\right)^{2t}(\sqrt{ab}A^{-1} + B)^t A (\sqrt{ab}A^{-1} + B)^t \right] \\
& = \left(\frac{\sqrt{a}-\sqrt{b}}{a-b}\right)^{t} (\sqrt{ab}A^{-1} + B)^{t}.
\end{split}
\end{displaymath}
Taking the $1/t$ power on both sides, we obtain (2).
\end{itemize}
\end{proof}

\section{Spectral pinch}~\label{Sec:Pinch}

Let $\alpha, \beta \in \mathbb R^m$. We say that $\beta$ is a \emph{pinch} of $\alpha$ (see \cite{MOA11}, page 17), if
\begin{displaymath}
\beta = t \alpha +(1-t) Q \alpha
\end{displaymath}
for some $t \in [0,1]$, where $Q$ is the $m \times m$ permutation matrix that interchanges two coordinates. This means that $\beta$ is obtained by $\alpha$ under the weighted arithmetic mean $t I + (1-t)Q$. It is well known that if $\beta$ is majorized by $\alpha$, denoted by $\beta \prec \alpha$, then $\beta$ can be obtained by applying at most $m$ pinches consecutively, starting from $\alpha$. The converse is clearly true.

Later Dinh, Ahsani and Tam~\cite{DAT16} provided the definition of geometric pinch.
Let $\alpha = (\alpha_{1}, \dots, \alpha_{m}), \beta = (\beta_{1}, \dots, \beta_{m}) \in \mathbb R_+^m$, where $\mathbb R_+^m$ denotes the set of all positive $m$-tuples.
We say that $\beta$ is a \emph {geometric pinch} of $\alpha$ if
\[
{\mbox{diag}\,} (\beta_1, \dots, \beta_m) = (Q^T {\mbox{diag}\,} (\alpha_1, \dots, \alpha_m) Q) \#_t {\mbox{diag}\,} (\alpha_1, \dots, \alpha_m)
\]
for some $t\in [0,1]$ and some transposition matrix $Q$.
They also proved the following theorem related to the log-majorization.

\begin{theorem} \cite{DAT16} \label{T:geo-pinch}
Let $\alpha, \beta \in \mathbb R_+^m$.
If $\beta$ is log-majorized by $\alpha$, denoted by $\beta \prec_{\log} \alpha$, then $\beta$ can be obtained by applying at most $m$ geometric pinches consecutively, starting from $\alpha$.
\end{theorem}

Naturally we define and consider the spectral pinch.
We say that $\beta$ is a \emph {spectral pinch} of $\alpha$ for $\alpha, \beta \in \mathbb R_+^m$ if
\[
{\mbox{diag}\,} (\beta_1, \dots, \beta_m) = (Q^T {\mbox{diag}\,} (\alpha_1, \dots, \alpha_m) Q)\natural_t {\mbox{diag}\,} (\alpha_1, \dots, \alpha_m)
\]
for some $t \in [0,1]$ and some transposition matrix $Q$.

\begin{theorem} \label{T:sp-pinch}
Let $\alpha, \beta \in \mathbb R_+^m$.
If $\beta \prec_{\log} \alpha$, then $\beta$ can be obtained by applying at most $m$ spectral pinches consecutively, starting from $\alpha$.
\end{theorem}

\begin{proof}
	If $\beta \prec_{\log} \alpha$, that is, $\log \beta \prec \log \alpha$, then we know $\log \beta$ can be obtained by applying at most $n$ pinches consecutively starting from $\log \alpha$.
	Let $\log \hat{\alpha}$ be a pinch of $\log \alpha$. Without loss of generality, we may assume that the pinch occurs on the first two coordinates.
	So $(\hat{\alpha_1}, \hat{\alpha_2})\prec_{\log} (\alpha_1, \alpha_2)$ and thus $\hat{\alpha_1} = \alpha_1^t\alpha_2^{1-t}$ and $\hat{\alpha_2} = \alpha_2^t\alpha_1^{1-t}$ for some $t\in [0,1]$. Let $P$ denote the matrix corresponding to the transposition switching the first two coordinates. Then by Lemma \ref{L:SG} (1)
\begin{displaymath}
\begin{split}
& (P^T {\mbox{diag}\,}(\alpha_1, \alpha_2, \alpha_3, \dots, \alpha_m) P) \natural_t {\mbox{diag}\,}(\alpha_1, \alpha_2, \alpha_3, \dots, \alpha_m) \\
& = {\mbox{diag}\,}(\alpha_2, \alpha_1, \alpha_3, \dots, \alpha_m) \natural_t {\mbox{diag}\,}(\alpha_1, \alpha_2, \alpha_3, \dots, \alpha_m) \\
& = {\mbox{diag}\,}(\alpha_1^t \alpha_2^{1-t}, \alpha_2^t \alpha_1^{1-t}, \alpha_3, \dots, \alpha_m).
\end{split}
\end{displaymath}
Then repeat the process to conclude that there exist $t_1, \dots, t_k \in [0,1]$ and transposition matrices $P_1, \dots, P_k$ such that
\[
{\mbox{diag}\,} \alpha^{(i+1)} := (P_i^T ({\mbox{diag}\,} \alpha^{(i)}) P_i) \natural_{t_i} {\mbox{diag}\,} \alpha^{(i)}, \quad i=1, \dots, k,
\]
where $\alpha^{(1)} := \alpha$ and $\alpha^{(k+1)} := \beta$.
\end{proof}

\begin{remark}
Let $A, B \in \mathbb{P}_{m}$. Note that $A \prec_{\log} B$ if $\lambda(A) \prec_{\log} \lambda(B)$, where $\lambda(A)$ denotes the $m$-tuple of positive eigenvalues of $A$. We can generalize the definition of geometric and spectral pinches such as $B$ is a geometric pinch (spectral pinch) of $A$ if
\begin{center}
$A = Q^{T} B Q \#_{t} B$ \ ($A = Q^{T} B Q \natural_{t} B$ respectively)
\end{center}
for some $t \in [0,1]$ and some transposition matrix $Q$. If $A, B$ are diagonal matrices then they are the same as the original definitions so Theorem \ref{T:geo-pinch} and Theorem \ref{T:sp-pinch} hold for $A \prec_{\log} B$. Since $B$ and $Q^{T} B Q$ do not commute in general, geometric pinch and spectral pinch are different. It would be an interesting problem to find relationships between the log-majorization and geometric (spectral) pinch of $A$ and $B$.
\end{remark}

\vspace{1cm}

\textbf{Acknowledgement}

The work of L. Gan was supported by the AMS-Simons Travel Grant 2022-2024.
The work of S. Kim was supported by the National Research Foundation of Korea grant funded by the Korea government (MSIT) (No. NRF-2022R1A2C4001306).

\bibliographystyle{abbrv}


text:
\section{Introduction}
\label{intro}

In the end of seventies Mansfield and Boyd \cite{mansfield78} introduced the concept of twist-like solutions or twistons, as a $180^0$ twist of several $CH_2$ groups in a crystalline polyethylene molecule. The twiston emerges as consequence of the large torsional flexibility of the crystalline chain, and it appears in the plane orthogonal to the chain direction. Since the seminal work of Mansfield and Boyd, solutions like twistons are part of different branches in the scientific literature, covering subjects as organic chains \cite{su79}, or even as the description of deformations in nanoribbons \cite{savin17} and also as consequence of anharmonic interactions in nanotubes \cite{savin}.

Most part of the applications of twistons are based on well-known soliton solutions or in numerical calculations. In the search for a different analytical approach of the twistons proposed by \cite{mansfield78}, Bazeia and collaborators \cite{bazeia99,bazeia00} introduced a methodology based on classical field theory, which remarkably describes the expected behavior of a twist-like model. The model described in \cite{bazeia99,bazeia00} also presents an infinite number of degenerated states, due to the form of its potential.

An attempt to break such a degeneracy was proposed by \cite{dutra10}, where the authors added a perturbation function to the interaction potential. Despite the fact that such a perturbation function broke the degenerated states, it yielded to equations of motion which were non-longer analytical. Therefore, only numerical twistons were found in this last approach. In this study we introduce a new route to treat the degeneracy issue, keeping the twist-like solutions analytical. Such a methodology is grounded in the so-called extension method, and the effective twiston model is build naturally.

The discussions tailored in this letter are divided as follows: Firstly we review some generalities about the analytical treatment of twistons. After that, we present the general aspects about the extension method. There we explain carefully how the method works and how it can be implemented in the context of twiston-like models. Then, we construct a new twiston model by applying the extension method, and we derive a new family of potentials which share the same analytical solutions. Finally we present our final remarks and future perspectives.

\section{Generalities}
\label{gen}
We begin this generalities by reviewing the connection between molecular dynamics and classical field theory previously investigated by \cite{bazeia99,bazeia00,dutra10}. Let us firstly assume that we are dealing with a rigid molecular group, which can be described through cilindrical coordinates. Therefore, the kinetic energy of the chain is such that
\begin{equation}
T=\frac{1}{2}\,m\,r_0^{\,2}\,\sum_{n}\,\left[\dot{\phi}_n^{\,2}+\frac{c^{\,2}}{r_0^{\,2}}\,\dot{\chi}_n^{\,2}\right]\,,
\end{equation}
here $r_0$ is the equilibrium position of the radial coordinate, $c$ is the longitudinal distance between adjacent molecular groups, and we are neglecting the radial degree of freedom.

Thus, a Lagrangian formalism for such a chain can be written as
\begin{equation}
L=T-U(\phi_{n},\chi_{n})\,,
\end{equation}
where $U$ is the intermolecular potential. In the continuum limit the coordinates $\phi_n$ and $\chi_n$ are going to behave like real fields $\phi(x,t)$ and $\chi(x,t)$. Therefore, the  twistons' dynamics can be modeled through a classical relativistic Lagrangian composed by these two scalar fields, whose explicit form is
\begin{equation}
\label{gen_01}
{\cal L}=\frac{1}{2}\,\partial_{\,\mu}\phi\,\partial^{\,\mu}\phi+\frac{1}{2}\,\partial_{\,\mu}\chi\,\partial^{\,\mu}\chi-V(\phi,\chi)\,,
\end{equation}
as discussed by \cite{mansfield78, zhang94, bazeia99, bazeia00, dutra10}. Here $\mu=0,1$ but we are going to work with statical fields, which do not change the physical properties of the model, since time dependent fields can be derived taking a Lorentz boost. In order to find analytical topological defects, as the agents of the twist movement, we assume that the potential is positive defined and written in terms of a superpotential such as
\begin{equation}
\label{gen_02}
V(\phi,\chi)=\frac{1}{2}\,W_{\,\phi}^{\,2}+\frac{1}{2}\,W_{\,\chi}^{\,2}\,,
\end{equation}
where, $W_{\,\phi}=\partial\,W/\partial\,\phi$, and $W_{\,\chi}=\partial\,W/\partial\,\chi$. If we deal with statical one-dimensional fields, the total energy related with these defects is given by
\begin{eqnarray}
\label{gen_03}
E &=&-\int_{-\infty}^{+\infty}\,dx\,{\cal L} \\ \nonumber
&=&
\int_{-\infty}^{+\infty}\,dx\,\left[\frac{1}{2}\,\left(\frac{d\,\phi}{d\,x}\right)^{\,2}+\frac{1}{2}\,\left(\frac{d\,\chi}{d\,x}\right)^{\,2}+V(\phi,\chi)\right]\,.
\end{eqnarray}
Let us make use of the so-called BPS method (Bogomol'niy-Prasad-Sommerfield) \cite{bps,bps2}, to rewrite the total amount of energy as
\begin{equation}
\label{gen_04}
E=\frac{1}{2}\,\int_{-\infty}^{+\infty}\,dx\,\bigg[\left(\frac{d\,\phi}{d\,x}\mp W_{\,\phi}\right)+ \left(\frac{d\,\chi}{d\,x}\mp W_{\,\chi}\right)
\pm 2\,\left(\frac{d\,\phi}{d\,x}\,W_{\,\phi}+\frac{d\,\chi}{d\,x}\,W_{\,\chi}\right)\bigg]\,.
\end{equation}
Therefore, imposing the first-order differential equations
\begin{equation}
\label{gen_05}
\frac{d\,\phi}{d\,x}=W_{\,\phi}(\phi,\chi)\,; \qquad \frac{d\,\chi}{d\,x}=W_{\,\chi}(\phi,\chi)\,,
\end{equation}
as constraints which minimize the energy, we yield to the well-known BPS energy
\begin{equation}
\label{gen_06}
E_{BPS}=\left|\Delta\,W\right|\,; \qquad \Delta\,W= W_{+\infty}-W_{-\infty}\,,
\end{equation}
where
\begin{equation}
W_{+\infty}=W\left[\phi(x(+\infty)),\chi(x(+\infty))\right]\,; \qquad
W_{-\infty}=W\left[\phi(x(-\infty)),\chi(x(-\infty))\right]\,.
\end{equation}
Here we point that the solutions from the coupled differential equations presented in $(\ref{gen_05})$ also satisfy the equations of motion coming from $(\ref{gen_01})$.

One possible model introduced by Bazeia and collaborators \cite{bazeia99,bazeia00}, and also studied in \cite{dutra10} consists in the potential
\begin{equation}
\label{gen_07}
V=\frac{1}{2}\,\left[\lambda\,\phi\,(\phi^{\,2}-\pi^{\,2})+\mu\,\phi\,\chi^{\,2}\right]^{\,2}+\frac{1}{2}\,\left(\mu\,\phi^{\,2}\,\chi\right)^{\,2}\,,
\end{equation}
where $\lambda$ and $\mu$ are real constants. An interesting point to be noted about this model is that the potential $V$ presents a line of zeros when $\phi=0$, as one can see in Fig. \ref{FIG1}. The first-order differential equations for such a potential are
\begin{equation}
\label{gen_08}
\frac{d\,\phi}{d\,x}=\lambda\,\phi\,(\phi^{\,2}-\pi^{\,2})+\mu\,\phi\,\chi^{\,2}\,; \qquad \frac{d\,\chi}{d\,x}=\mu\,\phi^{\,2}\,\chi\,,
\end{equation}
yielding to the analytical twistons
\begin{equation}
\label{gen_09}
\phi(x)=\pm\,\pi\,\sqrt{\frac{1}{2}\,\left(1-\tanh(\mu\,\pi^{\,2}\,x)\right)}\,; \qquad \chi(x)=\pm\,\pi\,\sqrt{\frac{\lambda}{\mu}-1}\,\sqrt{\frac{1}{2}\,\left(1+\tanh(\mu\,\pi^{\,2}\,x)\right)}\,,
\end{equation}
which were calculated in \cite{bazeia99,bazeia00}. In this last expression we realize that field $\phi$ is responsible for the torsion motion of the polymer chain, while field $\chi$ is equivalent to its longitudinal motion. We also see that the previous solutions are valid only if $\lambda/\mu\,>\,1$, as pointed by \cite{bazeia99,bazeia00}.

\begin{figure}
 \includegraphics[width=0.45\columnwidth]{v1} \hspace{0.5 cm} \includegraphics[width=0.45\columnwidth]{v2}
 \caption{Potential $V$ for $\lambda=7.30\,\pi^{\,-2}$, $\mu=0.67$. The graphics show a line of zeros for $\phi=0$, besides two other vacua in $\phi=\pm\pi$ and $\chi=0$.}
 \label{FIG1}
\end{figure}

The superpotential used to described this analytical model is
\begin{equation}
\label{gen_10}
W(\phi,\chi)=\frac{\lambda}{2}\,\phi^{\,2}\,\left(\frac{\phi^{\,2}}{2}-\pi^{\,2}\right)+\frac{\mu}{2}\,\phi^{\,2}\,\chi^{\,2}\,,
\end{equation}
which is constructed from the first-order differential equations $(\ref{gen_05})$. Then, we substitute the superpotential together with the analytical solutions $(\ref{gen_09})$ into Eq. $(\ref{gen_06})$, yielding to the BPS energy
\begin{equation}
\label{gen_11}
E_{BPS}=\frac{\lambda}{4}\,\pi^{\,4}\,.
\end{equation}
This result was derived in \cite{bazeia00}, where the authors worked with $|\lambda|\,\pi^{\,2}=7.30\,\mbox{kcal/mol}$. Such a value was based on the experimental data for the energy contribution of the localized twisted region \cite{mansfield78}. This experimental constraint yields to $E=17.99\,\mbox{kcal/mol}$ as the energy for the topological twistons. As mentioned in \cite{bazeia00}, this energy agrees quite well with other methods to describe twistons as one can see in \cite{mansfield78, skinner84, zhang94}.

There is one important point to be noted about the line of zeros presented in Fig. \ref{FIG1}, which was discussed in details by de Souza Dutra {\it et al.} in \cite{dutra10}. As was shown before, field $\phi$ is responsible to twist the molecule chain by an integer multiple of $\pi$, while field $\chi$ produces a longitudinal unit shift for the chain, see $(\ref{gen_09})$. Such a longitudinal shift should be of order of the longitudinal distance between adjacent molecular groups. However, once the line of zeros presents degenerated energy for any value of $\chi_v$, this means that it would be possible to break the chain without any extra energy cost than that necessary to make an unit shift \cite{dutra10}.

In reference \cite{dutra10}, the authors proposed a route to break this degenerated energy configuration, by adding a perturbation function to the original potential. The procedure worked nicely, however the solutions for the fields should be numerically derived.  In order to keep an analytical description for the twistons, we are going to apply the so-called extension method \cite{bls} to build another two-field models which are going to have well defined vacua to field $\chi$. This methodology rescues the physical meaning of the analytical twistons derived in \cite{bazeia00,bazeia99}, despites the degeneracy issue.

\section{The extension method}
\label{extmet}

In this section we present some generalities about the extension method firstly introduced in \cite{bls}. In order to apply such a method, let us verify that the first-order differential equations from \eqref{gen_05} can be rearranged as
\begin{equation} \label{sec2_eq1}
\phi_{\,\chi}=\frac{\phi^{\,\prime}}{\chi^{\,\prime}}=\frac{W_{\,\phi}(\phi,\chi)}{W_{\,\chi}(\phi,\chi)}\,,
\end{equation}
whose integration lead us to analytical orbits between $\phi$ and $\chi$, or in other words, we find $\phi=f(\chi)$. Such a mapping procedure is approached in a different context by the so-called deformation method \cite{blm}.

In the deformation method, we work with two standard classical field Lagrangians, whose forms are
\begin{equation} \label{sec2_eq2}
{\cal L}=\frac{1}{2}\,\partial_{\,\mu}\,\phi\,\partial^{\,\mu}\,\phi-V(\phi)\,; \qquad {\cal L}_d =\frac{1}{2}\,\partial_{\,\mu}\,\chi\,\partial^{\,\mu}\,\chi-\widetilde{V}(\chi)\,,
\end{equation}
where the real fields $\phi$ and $\chi$ are statical, one dimensional, and are solutions of  the equations of motion
\begin{equation} \label{sec2_eq3}
-\phi^{\,\prime\,\prime}+V_{\,\phi}=0\,; \qquad -\chi^{\,\prime\,\prime}+\widetilde{V}_{\,\chi}=0\,.
\end{equation}
Moreover, the BPS defects should also satisfy the first-order differential equations
\begin{equation} \label{sec2_eq4}
\phi^{\,\prime}=W_{\,\phi}(\phi)\,; \qquad \chi^{\,\prime}=W_{\,\chi}(\chi)\,,
\end{equation}
where  $V$ and $\widetilde{V}$ are given by
\begin{equation} \label{sec2_eq5}
V=\frac{W_{\,\phi}^{\,2}}{2}\,; \qquad \widetilde{V}=\frac{W_{\,\chi}^{\,2}}{2}\,.
\end{equation}
The connection between these two models is mediated by a function $f$, which is named deformation function. Therefore, a connection of type $\phi=f(\chi)$ yields to the constraints
\begin{equation} \label{sec2_eq6}
W_{\phi}(\phi)=W_{\,\chi}(\chi)\,f_{\,\chi}\,; \qquad \phi_{\,\chi}=f_{\,\chi}=\frac{W_{\,\phi}}{W_{\,\chi}}\bigg|_{\phi\rightarrow\chi}\,.
\end{equation}

The reader can see that the previous relation is very close to the one presented in Eq.  \eqref{sec2_eq1}. The basis for the extension method is that we can rearrange Eq. \eqref{sec2_eq6} to write it as an effective differential equation of a two-field model. The procedure works as follows: let us rewrite  \eqref{sec2_eq6} as
\begin{equation} \label{sec2_eq7}
\phi_{\,\chi}=\frac{W_{\,\phi}}{W_{\,\chi}}=\frac{a_1\,W_{\,\phi}(\chi)+a_2\,W_{\,\phi}(\phi,\chi)+a_3\,W_{\,\phi}(\phi)+c_1\,g(\chi)+c_2\,g(\phi,\chi)+c_3\,g(\phi)}{b_1\,W_{\,\chi}(\chi)+b_2\,W_{\,\chi}(\phi,\chi)+b_3\,W_{\,\chi}(\phi)}\,,
\end{equation}
where $W_{\phi}(\chi)$, $W_{\phi}(\phi,\chi)$, and $W_{\phi}(\phi)$ are equivalent since they are constructed with the deformation function $f(\chi)$ and with its inverse $\chi=f^{\,-1}(\phi)$. The analogous procedure is repeated for the different forms of $W_{\chi}$, as well as for the extra function $g$.

This last function is used to connect fields $\phi$ and $\chi$ in the effective two scalar field model. Moreover, the complete equivalence of the previous equation with  \eqref{sec2_eq6} imposes the constraints $a_1+a_2+a_3=1$, $b_1+b_2+b_3=1$, and $c_1+c_2+c_3=0$.

Consequently, once Eq. \eqref{sec2_eq7} has the same structure of Eq. \eqref{sec2_eq1}, we can identify the following terms
\begin{equation} \label{sec2_eq8}
W_{\,\phi}=a_1\,W_{\,\phi}(\chi)+a_2\,W_{\,\phi}(\phi,\chi)+a_3\,W_{\,\phi}(\phi)+c_1\,g(\chi)+c_2\,g(\phi,\chi)+c_3\,g(\phi)\,;
\end{equation}
\begin{equation} \label{sec2_eq9}
W_{\,\chi}=b_1\,W_{\,\chi}(\chi)+b_2\,W_{\,\chi}(\phi,\chi)+b_3\,W_{\,\chi}(\phi)\,,
\end{equation}
where $W_{\,\phi\,\chi}$ and $W_{\,\chi\,\phi}$ must commutate, which means that
\begin{equation}  \label{sec2_eq10}
W_{\,\chi\,\phi}=W_{\,\phi\,\chi}\,,
\end{equation}
yielding to a constraint for the function $g$, given by
\begin{equation}  \label{sec2_eq11}
b_2\,W_{\,\chi\,\phi}(\phi,\chi)+b_3\,W_{\,\chi\,\phi}(\phi)=a_1\,W_{\,\phi\,\chi}(\chi)+a_2\,W_{\,\phi\,\chi}(\phi,\chi)+c_1\,g_{\,\chi}(\chi)+c_2\,g_{\,\chi}(\phi,\chi)\,.
\end{equation}
Then, in order to have an unique form of $g$, we must choose either $c_1$ or $c_2$ equals to zero. Another interesting feature about this extension method is that the analytical solutions of the deformed one field systems are going to automatically satisfy the equations of motion of the effective two-field model. Such a methodology was successfully applied to the construction of new two and three scalar fields cosmological models, as one can see in \cite{ms, smvf}.

\section{Extended Twistons}
\label{extwist}

Let us consider two one-field models which obey the first-order differential equations
\begin{equation} \label{twist01}
\phi^{\,\prime}=W_{\phi}=\mu\,  \phi\,  \left(\phi ^2-\pi ^2\right)\,; \qquad \chi^{\,\prime}=\mu \, \chi\,  \left(\pi ^2-\frac{\chi ^2}{\gamma^{\,2}}\right)\,; \qquad \gamma=\sqrt{\frac{\lambda}{\mu}-1}\,,
\end{equation}
which are satisfied by the analytical defects
\begin{equation} \label{twist02}
\phi(x)=\pi \, \sqrt{\frac{1}{2} \left(1-\tanh \left(\pi ^2\, \mu\,  x\right)\right)}\,;\qquad \chi(x)=\pi \, \gamma \sqrt{\frac{1}{2}\, \left(\tanh \left(\pi ^2\, \mu \, x\right)+1\right)}\,.
\end{equation}

From the previous equations, we realize that the mapping between fields $\phi$ and $\chi$ is mediated by
\begin{equation} \label{twist03}
\phi=f(\chi)=\sqrt{\pi^{\,2}-\frac{\chi^{\,2}}{\gamma^{\,2}}}\,; \qquad \chi=f^{\,-1}(\phi)= \gamma\,\sqrt{\pi^{\,2}-\phi^{\,2}}\,.
\end{equation}
Then, we are able to use these ingredients to write two different and equivalent forms of $W_{\,\phi}$ and $W_{\,\chi}$, respectively, yielding to
\begin{eqnarray} \label{twist04}
&&
W_{\,\phi}(\phi ,\chi)=\lambda\,\phi\,\left(\phi^{\,2}-\pi^{\,2}\right)+\mu\,\chi^{\,2}\,\phi\,; \qquad W_{\,\phi}(\phi)=\mu\,\left(\phi^{\,2}-\pi^{\,2}\right)\,\phi\,; \\ \nonumber
&&
W_{\,\chi}(\chi)=\mu\,\chi\,\left(\pi^{\,2}-\frac{\chi^{\,2}}{\gamma^{\,2}}\right) \,; \qquad W_{\,\chi}(\phi ,\chi)=\mu\,\chi\,\phi^{\,2}\,.
\end{eqnarray}
Here  $W_{\phi}(\chi)$ and $W_{\chi}(\phi)$ were neglected once they presented terms with rational powers of $\phi$ and $\chi$. The procedure of neglecting these forms is equivalent to choose $a_1=0$ and $b_3=0$. Therefore, choosing $c_1=0$, the constraint \eqref{sec2_eq11} results in
\begin{equation} \label{twist05}
c_2\,g_\chi=2\,\mu\,\left(b_2-a_2\right)\,\chi\,\phi\,,
\end{equation}
whose a direct integration lead us to
\begin{equation} \label{twist06}
c_2\,g(\phi,\chi)=\mu\,\left(b_2-a_2\right)\,\chi^{\,2}\,\phi\,; \qquad c_2\,g(\phi) =\mu\gamma^{\,2}\,\left(b_2-a_2\right)\,\left(\pi^{\,2}-\phi^{\,2}\right)\,\phi\,,
\end{equation}
where the last was obtained using $\chi=f^{\,-1}(\phi)$.

\begin{figure}[ht!]
 \includegraphics[width=0.45\columnwidth]{v3} \hspace{0.5 cm} \includegraphics[width=0.45\columnwidth]{v3c}
 \caption{In the left panel we show potential $V$ for $\lambda=7.30\,\pi^{\,-2}$, $\mu=0.67$ and $b_2=0$. In the right panel we present a contour of $V$, were the solid and the dotted red curves are the analytical orbits connecting fields $\phi$ and $\chi$.}
 \label{FIG2}
\end{figure}

\begin{figure}[h!]
 \includegraphics[width=0.45\columnwidth]{v4} \hspace{0.5 cm} \includegraphics[width=0.45\columnwidth]{v4c}
 \caption{In the left panel we show potential $V$ for $\lambda=7.30\,\pi^{\,-2}$, $\mu=0.67$ and $b_2=2$. In the right panel we present a contour of $V$, were the solid and the dotted red curves are the analytical orbits connecting fields $\phi$ and $\chi$.}
 \label{FIG3}
\end{figure}

By substituting all these ingredients in \eqref{sec2_eq8} and \eqref{sec2_eq9} we find that
\begin{equation} \label{twist07}
W_{\,\phi}=\left(b_2\,\gamma^{\,2}+1\right)\,\mu\,\left(\phi^{\,2}-\pi^{\,2}\right)\,\phi+b_2\,\mu\,\chi^{\,2}\,\phi\,,
\end{equation}
and
\begin{equation} \label{twist08}
W_{\,\chi}=(1-b_2)\,\mu\,\left(\pi^{\,2}-\frac{\chi^{\,2}}{\gamma^{\,2}}\right)\,\chi+b_2\,\mu\,\chi\,\phi^{\,2}\,.
\end{equation}
The integration of $W_{\phi}$ and of $W_{\chi}$ in respect to $\phi$ and $\chi$, respectively results in the effective superpotential
\begin{equation} \label{twist09}
W(\phi,\chi)=\frac{1}{2}\, (1-b_2)\, \mu\, \left(\pi^{\,2}-\frac{\chi ^2}{2\, \gamma^{\,2}}\right)\,\chi ^2+\frac{1}{2} \,\left(b_2\,\gamma^{\,2}+1\right)\,\mu\, \left(\frac{\phi ^2}{2}-\pi ^2\right)\, \phi ^2 +\frac{1}{2}\,b_2\, \mu\,  \chi ^2 \,\phi ^2\,.
\end{equation}
Moreover, from Eq. \eqref{gen_02} we can derive an analytical expression for potential $V$ given by
\begin{eqnarray} \label{twist10}
V(\phi,\chi) &=& \frac{\mu^{\,2}}{2}\,\phi^{\,2}\,\left[\left(b_2\,\gamma^{\,2}+1\right)\,\left(\phi^{\,2}-\pi^{\,2}\right)+b_2\,\chi^{\,2}\right]^{\,2} \\ \nonumber
&+&
\frac{\mu^{\,2}}{2}\,\chi^{\,2}\,\left[(1-b_2)\,\left(\pi^{\,2}-\frac{\chi^{\,2}}{\gamma^{\,2}}\right)+b_2\,\phi^{\,2}\right]^{\,2}\,.
\end{eqnarray}

In order to find analytical models consistent with the longitudinal movement of the molecular chain (which here is normalized to $\chi_v=\pm1$), we are going to constraint $\lambda$ and $\mu$ as
\begin{equation}
\mu=\frac{\pi^{\,2}}{1+\pi^{\,2}}\,\lambda\,; \qquad \gamma= \frac{1}{\pi}\,.
\end{equation}
Consequently, choosing $\lambda=7.30\,\pi^{\,-2}$ we find that $\mu\approx 0.67$, yielding to a family of physical potentials which can describe the twiston behavior naturally. The features of such potentials are unveiled in the graphics of Figs. \ref{FIG2} and \ref{FIG3}. There, we show how the $b_2$ parameter changes the shape of potential $V$. In Fig. \ref{FIG2} we built a graphic for $b_2=0$, which unveils the presence of nine vacua, whose four are connected by analytical orbits. The solid red orbit consist in the set of analytical solutions from Eq. $(\ref{twist02})$, while the dotted red orbit is simply given by
\begin{equation} \label{twist11}
\phi(x)=-\pi \, \sqrt{\frac{1}{2} \left(1-\tanh \left(\pi ^2\, \mu\,  x\right)\right)}\,;\qquad \chi(x)=-\pi \, \gamma \sqrt{\frac{1}{2}\, \left(\tanh \left(\pi ^2\, \mu \, x\right)+1\right)}\,.
\end{equation}
Due the symmetry of such a potential, the previous orbit also satisfies its equations of motion.  Besides, in Fig. \ref{FIG3} we plotted $V$ for $b_2=0.5$, unveiling a new set of vacua of our potential. Again, four of these vacua are connected by orbits $(\ref{twist02})$ and $(\ref{twist11})$. Moreover, we realized that the case $b_2=1$ is equivalent to the potential analyzed by Bazeia {\it et al.} in \cite{bazeia99,bazeia00}. Another interesting fact is that this new models present
\begin{equation} \label{twist12}
E_{BPS}=\frac{\lambda}{4}\,\pi^{\,4}=17.99\,,
\end{equation}
as its BPS energy, which is the same one derived in \cite{bazeia99,bazeia00}.

\section{Final remarks}
\label{remarks}

In this letter we applied the extension method to find analytical twistons for the crystalline polyethylene. For the first time, an analytical twiston model was constructed by a non-trivial combination of two one field systems. The analytical solutions used in this methodology can satisfy an entire family of two scalar field models, whose potentials do not present a line of zeros. Therefore, these new models agree with a physically consistent  longitudinal shift for the molecular chain, which should be proportional to the distance between adjacent molecular groups. Such a shift costs finite energy whose value is exactly the same computed in \cite{bazeia99,bazeia00}. We believe that the procedure presented in this letter can be applied to other interesting scenarios where the a two scalar fields model describes a molecular dynamics like polyaniline \cite{mge91}, leucoemeraldine \cite{mgme}  or fullerene \cite{bazeia02}. There are other models describing crystalline polyethylene based on sine-Gordon potentials, as the one investigated in \cite{hammad}, which could be revisited with this method. Moreover, we also intend to apply this procedure in the context of biological fluids, following the lines of \cite{sh12}, and in a three-field description of twistons as pointed in \cite{bazeia00,bazeia99}.

\acknowledgements

The authors would like to thank CNPq, and CAPES (Brazilian agencies) for support.

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