# Yamabe solitons on conformal Sasaki-like almost contact B-metric manifolds

Mancho Manev  University of Plovdiv Paisii Hilendarski, Faculty of Mathematics
and Informatics, Department of Algebra and Geometry, 24 Tzar Asen St., Plovdiv
4000, Bulgaria & Medical University – Plovdiv, Faculty of Pharmacy, Department
of Medical Physics and Biophysics, 15A Vasil Aprilov Blvd, Plovdiv 4002,
Bulgaria<EMAIL_ADDRESS>
###### Abstract.

A Yamabe soliton is defined on arbitrary almost contact B-metric manifold,
which is obtained by a contact conformal transformation of the Reeb vector
field, its dual contact 1-form, the B-metric, and its associated B-metric. The
cases when the given manifold is cosymplectic or Sasaki-like are studied. In
this way, manifolds from one of the main classes of the studied manifolds are
obtained. The same class contains the conformally equivalent manifolds of
cosymplectic manifolds by the usual conformal transformation of the B-metric.
An explicit 5-dimensional example of a Lie group is given, which is
characterized in relation to the obtained results.

###### Key words and phrases:

Yamabe soliton, almost contact B-metric manifold, almost contact complex
Riemannian manifold, Sasaki-like manifold

###### 2010 Mathematics Subject Classification:

Primary 53C25, 53D15, 53C50; Secondary 53C44, 53D35, 70G45

## Introduction

Richard Hamilton introduced the notion of the Yamabe flow in 1988 (see [8]) as
an apparatus for constructing metrics with constant scalar curvature. The
significance of this problem from the point of view of mathematical physics is
that the Yamabe flow corresponds to the case of fast diffusion of the porous
medium equation [4].

A self-similar solution of the Yamabe flow, defined on a Riemannian manifold
or a pseudo-Riemannian manifold, is called a Yamabe soliton and is determined
by ([1])

$\frac{1}{2}\mathcal{L}_{v}g=(\tau-\sigma)g,$

where $\mathcal{L}_{v}g$ stands for the Lie derivative of the metric $g$ along
the vector field $v$, $\tau$ denotes the scalar curvature of $g$ and $\sigma$
is a constant. Many authors have studied the Yamabe soliton on different types
of manifolds since the introduction of this concept (see e.g. [2, 3, 5, 7,
14]).

In the present paper we begin the study of the mentioned Yamabe solitons on
almost contact B-metric manifolds. The geometry of these manifolds is
significantly influenced by the presence of two B-metrics, which are
interconnected by the almost contact structure.

It is a well-known fact that the Yamabe flow preserves the conformal class of
the metric. So this gives us a reason to study Yamabe solitons and conformal
transformations together. Contact conformal transformations were studied in
[11], which transform not only the metric but also the Reeb vector field and
its associated contact 1-form through the pair of B-metrics. According to this
work, the class of almost contact B-metric manifolds, which is closed under
the action of contact conformal transformations, is the direct sum of the four
main classes among the eleven basic classes of these manifolds known from the
classification of Ganchev-Mihova-Gribachev in [6]. Main classes are called
those whose manifolds are characterized by the fact that the covariant
derivative of the structure tensors with respect to the Levi-Civita connection
of some of the B-metrics is expressed only by the pair of B-metrics and the
corresponding traces.

We study Yamabe solitons on two of the simplest types of manifolds among those
studied, namely cosymplectic and Sasaki-like. The former have parallel
structure tensors with respect to the Levi-Civita connections of the
B-metrics. The latter are those whose complex cone is a Kähler manifold with a
pair of Norden metrics, i.e. again with parallel structure tensors with
respect to the Levi-Civita connections of the metrics. Note that the class of
Sasaki-like manifolds does not contain cosymplectic manifolds, although they
are in each of the eleven basic classes of the classification used.

We find that the manifolds thus constructed in both cases belong to one of the
main classes, the only one which contains the conformally equivalent manifolds
of the cosymplectic ones by the usual conformal transformations.

The present paper is organized as follows. Section 1 is devoted to the basic
concepts of almost contact B-metric manifolds, contact conformal
transformations of the structure tensors on them and the introduction of the
notion of a Yamabe soliton on a transformed almost contact B-metric manifold.
In Section 2 and Section 3, we study the constructed manifolds when the
initial manifold is cosymplectic and Sasaki-like, respectively. In Section 4,
we provide an explicit example of a Lie group as a 5-dimensional manifold
equipped with the structures studied.

##  1\. Almost contact B-metric manifolds, contact conformal transformations
and Yamabe solitons

Let $(M,\varphi,\xi,\eta,g)$ be an almost contact manifold with B-metric or an
_almost contact B-metric manifold_ , i.e. $M$ is a $(2n+1)$-dimensional
differentiable manifold with an almost contact structure $(\varphi,\xi,\eta)$
consisting of an endomorphism $\varphi$ of the tangent bundle, a vector field
$\xi$, its dual 1-form $\eta$ as well as $M$ is equipped with a pseudo-
Riemannian metric $g$ of signature $(n+1,n)$, such that the following
algebraic relations are satisfied [6]

(1)
$\displaystyle\varphi\xi=0,\quad\varphi^{2}=-\iota+\eta\otimes\xi,\quad\eta\circ\varphi=0,\quad\eta(\xi)=1,$
(2) $\displaystyle g(X,Y)=-g(\varphi X,\varphi Y)+\eta(X)\eta(Y)$

for arbitrary $X$, $Y$ of the algebra $\mathfrak{X}(M)$ on the smooth vector
fields on $M$, where $\iota$ stands for the identity transformation on
$\mathfrak{X}(M)$.

The manifolds $(M,\varphi,\xi,\eta,g)$ are also known as _almost contact
complex Riemannian manifolds_ (see e.g. [9]).

Further, $X$, $Y$, $Z$ will stand for arbitrary elements of $\mathfrak{X}(M)$
or vectors in the tangent space $T_{p}M$ of $M$ at an arbitrary point $p$ in
$M$.

The associated B-metric $\tilde{g}$ of $g$ on $M$ is defined by

$\tilde{g}(X,Y)=g(X,\varphi Y)\allowbreak+\eta(X)\eta(Y)$

and it is also of signature $(n+1,n)$ as its counterpart $g$.

The fundamental tensor $F$ of type (0,3) on
$(M,\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
is defined by

$F(X,Y,Z)=g\bigl{(}\left(\nabla_{X}\varphi\right)Y,Z\bigr{)},$

where $\nabla$ is the Levi-Civita connection of $g$. The following properties
of $F$ are consequences of (1) and (2):

$\begin{split}F(X,Y,Z)&=F(X,Z,Y)\\\\[4.0pt] &=F(X,\varphi Y,\varphi
Z)+\eta(Y)F(X,\xi,Z)+\eta(Z)F(X,Y,\xi)\end{split}$

and relations of $F$ with $\nabla\xi$ and $\nabla\eta$ are:

(3) $\left(\nabla_{X}\eta\right)Y=g\left(\nabla_{X}\xi,Y\right)=F(X,\varphi
Y,\xi).$

The following 1-forms, known as Lee forms of the manifold, are associated with
$F$:

$\theta=g^{ij}F(E_{i},E_{j},\cdot),\qquad\theta^{*}=g^{ij}F(E_{i},\varphi
E_{j},\cdot),\qquad\omega=F(\xi,\xi,\cdot),$

where $g^{ij}$ are the components of the inverse matrix of $g$ with respect to
a basis $\left\\{E_{i};\xi\right\\}$ $(i=1,2,\dots,2n)$ of $T_{p}M$. The
following general identities for the Lee forms of
$(M,\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
are known from [10]

(4) $\theta^{*}\circ\varphi=-\theta\circ\varphi^{2},\qquad\omega(\xi)=0.$

A classification of the almost contact B-metric manifolds in terms of $F$ is
given in [6]. This classification includes eleven basic classes
$\mathcal{F}_{1}$, $\mathcal{F}_{2}$, $\dots$, $\mathcal{F}_{11}$. Their
intersection is the special class $\mathcal{F}_{0}$ defined by condition for
the vanishing of $F$. Hence $\mathcal{F}_{0}$ is the class of cosymplectic
B-metric manifolds, where the structures $\varphi$, $\xi$, $\eta$, $g$,
$\tilde{g}$ are $\nabla$-parallel.

Using the pair of B-metrics $g$ and $\tilde{g}$ as well as $\eta\otimes\eta$,
in [10], the author and K. Gribachev introduced the so-called contact
conformal transformation of the B-metric $g$ into a new B-metric $\bar{g}$ for
$(M,\varphi,\xi,\eta,g)$. Later, in [11], this transformation is generalized
as a contact conformal transformation that gives an almost contact B-metric
structure $(\varphi,\bar{\xi},\bar{\eta},\bar{g})$ as follows

(5)
$\begin{array}[]{l}\bar{\xi}=e^{-w}\xi,\qquad\bar{\eta}=e^{w}\eta,\\\\[4.0pt]
\bar{g}=e^{2u}\cos{2v}\,g+e^{2u}\sin{2v}\,\tilde{g}+\left(e^{2w}-e^{2u}\cos{2v}-e^{2u}\sin{2v}\right)\eta\otimes\eta,\end{array}$

where $u,v,w$ are differentiable functions on $M$.

The corresponding tensors $\bar{F}$ and $F$ are related by [11], see also [13,
(22)],

(6) $\displaystyle 2\bar{F}$ $\displaystyle(X,Y,Z)=2e^{2u}\cos{2v}\,F(X,Y,Z)$
$\displaystyle+e^{2u}\sin{2v}\left[F(\varphi Y,Z,X)-F(Y,\varphi
Z,X)+F(X,\varphi Y,\xi)\eta(Z)\right]$
$\displaystyle+e^{2u}\sin{2v}\left[F(\varphi Z,Y,X)-F(Z,\varphi
Y,X)+F(X,\varphi Z,\xi)\eta(Y)\right]$
$\displaystyle+(e^{2w}-e^{2u}\cos{2v})\left[F(X,Y,\xi)+F(\varphi Y,\varphi
X,\xi)\right]\eta(Z)$
$\displaystyle+(e^{2w}-e^{2u}\cos{2v})\left[F(X,Z,\xi)+F(\varphi Z,\varphi
X,\xi)\right]\eta(Y)$
$\displaystyle+(e^{2w}-e^{2u}\cos{2v})\left[F(Y,Z,\xi)+F(\varphi Z,\varphi
Y,\xi)\right]\eta(X)$
$\displaystyle+(e^{2w}-e^{2u}\cos{2v})\left[F(Z,Y,\xi)+F(\varphi Y,\varphi
Z,\xi)\right]\eta(X)$
$\displaystyle-2e^{2u}\left[\cos{2v}\,\alpha(Z)+\sin{2v}\,\beta(Z)\right]g(\varphi
X,\varphi Y)$
$\displaystyle-2e^{2u}\left[\cos{2v}\,\alpha(Y)+\sin{2v}\,\beta(Y)\right]g(\varphi
X,\varphi Z)$
$\displaystyle-2e^{2u}\left[\cos{2v}\,\beta(Z)-\sin{2v}\,\alpha(Z)\right]g(X,\varphi
Y)$
$\displaystyle-2e^{2u}\left[\cos{2v}\,\beta(Y)-\sin{2v}\,\alpha(Y)\right]g(X,\varphi
Z)$ $\displaystyle+2e^{2w}\eta(X)\left[\eta(Y)\mathrm{d}w(\varphi
Z)+\eta(Z)\mathrm{d}w(\varphi Y)\right],$

where we use the following notations for brevity

(7)
$\alpha=\mathrm{d}u\circ\varphi+\mathrm{d}v,\qquad\beta=\mathrm{d}u-\mathrm{d}v\circ\varphi.$

In the general case, the relations between the Lee forms of the manifolds
$(M,\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
and $(M,\varphi,\bar{\xi},\bar{\eta},\bar{g})$ are the following (see [11])

(8)
$\bar{\theta}=\theta+2n\,\alpha,\qquad\bar{\theta}^{*}=\theta^{*}+2n\,\beta,\qquad\bar{\omega}=\omega+\mathrm{d}w\circ\varphi.$

######  Definition 1.1.

We say that the B-metric $\bar{g}$ generates a Yamabe soliton with potential
the Reeb vector field $\bar{\xi}$ and soliton constant $\bar{\sigma}$ on a
conformal almost contact B-metric manifold
$(M,\varphi,\bar{\xi},\bar{\eta},\bar{g})$, if the following condition is
satisfied

(9)
$\frac{1}{2}\mathcal{L}_{\bar{\xi}}\bar{g}=(\bar{\tau}-\bar{\sigma})\bar{g},$

where $\bar{\tau}$ is the scalar curvature of $\bar{g}$.

It is well known the following expression of the Lie derivative in terms of
the covariant derivative with respect to the Levi-Civita connection
$\bar{\nabla}$ of $\bar{g}$

(10)
$\left(\mathcal{L}_{\bar{\xi}}\bar{g}\right)(X,Y)=\bar{g}\left(\bar{\nabla}_{X}\bar{\xi},Y\right)+\bar{g}\left(X,\bar{\nabla}_{Y}\bar{\xi}\right).$

##  2\. The case when the given manifold is cosymplectic

In this section, we consider
$(M,\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
to be a cosymplectic manifold with almost contact B-metric structure, i.e. an
$\mathcal{F}_{0}$-manifold, defined by covariant constant structure tensor
$\varphi$ (and consequently the same condition for $\xi$, $\eta$, $\tilde{g}$
is valid) with respect to $\nabla$ of $g$. Therefore $F=0$ and (6) takes the
form

(11) $\displaystyle\bar{F}(X,Y,Z)=$
$\displaystyle-e^{2u}\left[\cos{2v}\,\alpha(Z)+\sin{2v}\,\beta(Z)\right]g(\varphi
X,\varphi Y)$
$\displaystyle-e^{2u}\left[\cos{2v}\,\alpha(Y)+\sin{2v}\,\beta(Y)\right]g(\varphi
X,\varphi Z)$
$\displaystyle-e^{2u}\left[\cos{2v}\,\beta(Z)-\sin{2v}\,\alpha(Z)\right]g(X,\varphi
Y)$
$\displaystyle-e^{2u}\left[\cos{2v}\,\beta(Y)-\sin{2v}\,\alpha(Y)\right]g(X,\varphi
Z)$ $\displaystyle+e^{2w}\eta(X)\left[\eta(Y)\mathrm{d}w(\varphi
Z)+\eta(Z)\mathrm{d}w(\varphi Y)\right].$

From the latter equality, bearing in mind the general identity (3) for the
manifolds under consideration, we obtain

$\displaystyle\bar{g}\left(\bar{\nabla}_{X}\bar{\xi},Y\right)=$
$\displaystyle-e^{2u-w}\left[\cos{2v}\,\beta(\xi)-\sin{2v}\,\alpha(\xi)\right]g(\varphi
X,\varphi Y)$
$\displaystyle+e^{2u-w}\left[\cos{2v}\,\alpha(\xi)+\sin{2v}\,\beta(\xi)\right]g(X,\varphi
Y)$ $\displaystyle+e^{w}\eta(X)\mathrm{d}w(\varphi^{2}Y).$

Combining the last expression with (7) and (10) gives the following

(12) $\displaystyle\left(\mathcal{L}_{\bar{\xi}}\bar{g}\right)(X,Y)=$
$\displaystyle-2e^{2u-w}\left[\cos{2v}\,\mathrm{d}{u}(\xi)-\sin{2v}\,\mathrm{d}{v}(\xi)\right]g(\varphi
X,\varphi Y)$
$\displaystyle+2e^{2u-w}\left[\cos{2v}\,\mathrm{d}{v}(\xi)+\sin{2v}\,\mathrm{d}{u}(\xi)\right]g(X,\varphi
Y)$
$\displaystyle+e^{w}\left[\eta(X)\mathrm{d}w(\varphi^{2}Y)+\eta(Y)\mathrm{d}w(\varphi^{2}X)\right].$

######  Theorem 2.1.

An almost contact B-metric manifold that is cosymplectic can be transformed by
a contact conformal transformation of type (5) so that the transformed
B-metric is a Yamabe soliton with potential the transformed Reeb vector field
and a soliton constant $\bar{\sigma}$ if and only if the functions $(u,v,w)$
of the used transformation satisfy the conditions

(13)
$\mathrm{d}u(\xi)=0,\qquad\mathrm{d}v(\xi)=0,\qquad\mathrm{d}w=\mathrm{d}w(\xi)\eta.$

Moreover, the obtained Yamabe soliton has a constant scalar curvature with
value $\bar{\tau}=\bar{\sigma}$ and the obtained almost contact B-metric
manifold belongs to the subclass of the main class $\mathcal{F}_{1}$
determined by the conditions:

(14)
$\bar{\theta}=2n\left\\{\mathrm{d}u\circ\varphi-\mathrm{d}v\circ\varphi^{2}\right\\},\qquad\bar{\theta}^{*}=-2n\left\\{\mathrm{d}u\circ\varphi^{2}+\mathrm{d}v\circ\varphi\right\\}.$

###### Proof.

Is we assume that $\bar{g}$ generates a Yamabe soliton with potential
$\bar{\xi}$ and a soliton constant $\bar{\sigma}$ on
$(M,\varphi,\bar{\xi},\bar{\eta},\bar{g})$, then due to (9) and (12) we have
the following

(15) $\begin{split}&\
-e^{2u-w}\left\\{\left[\cos{2v}\,\mathrm{d}{u}(\xi)-\sin{2v}\,\mathrm{d}v(\xi)\right]g(\varphi
X,\varphi Y)\right.\\\\[4.0pt] &\phantom{\hskip
48.0pt}\left.-\left[\sin{2v}\,\mathrm{d}{u}(\xi)+\cos{2v}\,\mathrm{d}v(\xi)\right]g(X,\varphi
Y)\right\\}\\\\[4.0pt]
&+\frac{1}{2}e^{w}\left\\{\mathrm{d}w(\varphi^{2}X)\eta(Y)+\mathrm{d}w(\varphi^{2}Y)\eta(X)\right\\}\\\\[4.0pt]
&=(\bar{\tau}-\bar{\sigma})\left\\{-e^{2u}\left[\cos{2v}\,g(\varphi X,\varphi
Y)-\sin{2v}\,g(X,\varphi Y)\right]\right.\\\\[4.0pt]
&\phantom{=(\bar{\tau}-\bar{\sigma})\left\\{\right.}\left.+e^{2w}\eta(X)\eta(Y)\right\\}.\end{split}$

The substitution $\xi$ for $Y$ in the latter equality gives the following

(16)
$\begin{split}\mathrm{d}w(\varphi^{2}X)=2e^{w}(\bar{\tau}-\bar{\sigma})\eta(X),\end{split}$

which is valid if and only if

(17) $\bar{\tau}=\bar{\sigma}.$

Therefore, (16) implies the condition $\mathrm{d}w\circ\varphi=0$. In this
case, because of (15), we obtain $\mathrm{d}u(\xi)=\mathrm{d}v(\xi)=0$, which
completes (13).

Let us remark that $\mathcal{L}_{\bar{\xi}}\bar{g}$ vanishes because of (17)
and (9), i.e. $\bar{\xi}$ is a Killing vector field in the considered case.

After that, we apply the following relations to (11), which are equivalent to
the last formula in (5):

(18) $\begin{array}[]{l}\bar{g}(\varphi X,\varphi Y)=e^{2u}\cos{2v}\,g(\varphi
X,\varphi Y)-e^{2u}\sin{2v}\,g(X,\varphi Y),\\\\[4.0pt] \bar{g}(X,\varphi
Y)=e^{2u}\cos{2v}\,g(X,\varphi Y)+e^{2u}\sin{2v}\,g(\varphi X,\varphi
Y).\end{array}$

Then we use (13) and the formula $\bar{\eta}=e^{w}\eta$ from (5) to obtain the
following expression

(19) $\displaystyle\bar{F}(X,Y,Z)=$ $\displaystyle\ \bar{g}(\varphi X,\varphi
Y)\,\alpha(\varphi^{2}Z)+\bar{g}(X,\varphi Y)\,\beta(\varphi^{2}Z)$
$\displaystyle+\bar{g}(\varphi X,\varphi
Z)\,\alpha(\varphi^{2}Y)+\bar{g}(X,\varphi Z)\,\beta(\varphi^{2}Y).$

Now, we substitute the Lee forms of $\bar{F}$ from (8) in (19) and get

$\displaystyle\bar{F}(X,Y,Z)=$ $\displaystyle\
\frac{1}{2n}\Bigl{\\{}\bar{g}(\varphi X,\varphi
Y)\,\bar{\theta}(\varphi^{2}Z)+\bar{g}(X,\varphi
Y)\,\bar{\theta}^{*}(\varphi^{2}Z)$ $\displaystyle\hskip
20.0pt+\bar{g}(\varphi X,\varphi
Z)\,\bar{\theta}(\varphi^{2}Y)+\bar{g}(X,\varphi
Z)\,\bar{\theta}^{*}(\varphi^{2}Y)\Bigr{\\}}.$

The last expression of $\bar{F}$ means that the obtained manifold belongs to
the main class $\mathcal{F}_{1}$, according to the classification in [6], and
the corresponding Lee forms have the following properties:

$\bar{\theta}=-\bar{\theta}\circ\varphi^{2},\qquad\bar{\theta}^{*}=-\bar{\theta}^{*}\circ\varphi^{2},\qquad\bar{\omega}=0.$

Finally, taking into account (8), (13) and the vanishing of $\theta$,
$\theta^{*}$ and $\omega$ for any cosymplectic B-metric manifold, we find the
expressions of the Lee forms of the transformed manifold as in (14). ∎

As a result from (13), we obtain that the case in the present section is
possible when the functions $(u,v,w)$ of the contact conformal transformation
in (5) satisfy the conditions:

  * •

$u$ and $v$ are constants on the vertical distribution
$\mathcal{V}=\operatorname{span}\xi=\ker\varphi$;

  * •

$w$ is a constant on the horizontal (contact) distribution
$\mathcal{H}=\ker\eta=\operatorname{im}\varphi$.

##  3\. The case when the given manifold is Sasaki-like

In the present section, we suppose that the given almost contact B-metric
manifold
$(M,\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
is Sasaki-like, i.e. the complex cone $M\times\mathbb{R}^{-}$ is a Kähler
manifold with Norden metric also known as a holomorphic complex Riemannian
manifold [9].

If
$(M,\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
is Sasaki-like, then the following condition is met:

$\nabla_{X}\xi=-\varphi X.$

The Sasaki-like condition in terms of $F$ given in [9] is the following

(20) $F(X,Y,Z)=g(\varphi X,\varphi Y)\eta(Z)+g(\varphi X,\varphi Z)\eta(Y).$

Then, by virtue of (3), (6), (10) for
$(M,\varphi,\bar{\xi},\bar{\eta},\bar{g})$ and (20), we get

(21) $\begin{split}\left(\mathcal{L}_{\bar{\xi}}\bar{g}\right)(X,Y)=&\
2e^{2u-w}\left\\{\langle\sin{2v}\,\mathrm{d}{u}(\xi)-\cos{2v}\,\left[1-\mathrm{d}v(\xi)\right]\rangle
g(X,\varphi Y)\right.\\\\[4.0pt]
&\left.-\langle\cos{2v}\,\mathrm{d}{u}(\xi)+\sin{2v}\left[1-\mathrm{d}v(\xi)\right]\rangle
g(\varphi X,\varphi Y)\right\\}\\\\[4.0pt]
&+e^{w}\left\\{\mathrm{d}w(\varphi^{2}X)\eta(Y)+\mathrm{d}w(\varphi^{2}Y)\eta(X)\right\\}.\end{split}$

######  Theorem 3.1.

An almost contact B-metric manifold that is Sasaki-like can be transformed by
a contact conformal transformation of type (5) so that the transformed
B-metric is a Yamabe soliton with potential the transformed Reeb vector field
and a soliton constant $\bar{\sigma}$ if and only if the functions $(u,v,w)$
of the used transformation satisfy the conditions

(22)
$\mathrm{d}u(\xi)=0,\qquad\mathrm{d}v(\xi)=1,\qquad\mathrm{d}w=\mathrm{d}w(\xi)\eta.$

Moreover, the obtained Yamabe soliton has a constant scalar curvature with a
value $\bar{\tau}=\bar{\sigma}$ and the transformed almost contact B-metric
manifold belongs to a subclass of the main class $\mathcal{F}_{1}$ determined
by:

(23)
$\bar{\theta}=2n\left\\{\mathrm{d}u\circ\varphi-\mathrm{d}v\circ\varphi^{2}\right\\},\qquad\bar{\theta}^{*}=2n\left\\{\mathrm{d}u-\mathrm{d}v\circ\varphi\right\\}.$

###### Proof.

The expression in (21) and the assumption that $\bar{g}$ generates a Yamabe
soliton with potential $\bar{\xi}$ and a soliton constant $\bar{\sigma}$ on
$(M,\varphi,\bar{\xi},\bar{\eta},\bar{g})$ imply the following

(24) $\begin{split}&\
e^{2u-w}\left\\{\langle\sin{2v}\,\mathrm{d}{u}(\xi)-\cos{2v}\,\left[1-\mathrm{d}v(\xi)\right]\rangle
g(X,\varphi Y)\right.\\\\[4.0pt] &\hskip
35.0pt\left.-\langle\cos{2v}\,\mathrm{d}{u}(\xi)+\sin{2v}\left[1-\mathrm{d}v(\xi)\right]\rangle
g(\varphi X,\varphi Y)\right\\}\\\\[4.0pt]
&+\frac{1}{2}e^{w}\left\\{\mathrm{d}w(\varphi^{2}X)\eta(Y)+\mathrm{d}w(\varphi^{2}Y)\eta(X)\right\\}\\\\[4.0pt]
&=(\bar{\tau}-\bar{\sigma})\left\\{e^{2u}\left[\sin{2v}\,g(X,\varphi
Y)-\cos{2v}\,g(\varphi X,\varphi Y)\right]\right.\\\\[4.0pt]
&\phantom{=(\bar{\tau}-\bar{\sigma})\left\\{\right.}\left.+e^{2w}\eta(X)\eta(Y)\right\\}.\end{split}$

An obvious consequence for $Y=\xi$ is the following

$\mathrm{d}w(\varphi^{2}X)=2e^{w}(\bar{\tau}-\bar{\sigma})\eta(X),$

which is satisfied if and only it the following conditions are fulfilled

(25) $\displaystyle\bar{\tau}=\bar{\sigma},$ (26)
$\displaystyle\mathrm{d}w=\mathrm{d}w(\xi)\eta.$

Due to (25) and (9), we have the vanishing of
$\mathcal{L}_{\bar{\xi}}\bar{g}$, which means that $\bar{\xi}$ is a Killing
vector field in this case as well.

Applying (25) and (26) in (24), we obtain

$\begin{split}&\
e^{2u-w}\left\\{\langle\sin{2v}\,\mathrm{d}{u}(\xi)-\cos{2v}\,\left[1-\mathrm{d}v(\xi)\right]\rangle
g(X,\varphi Y)\right.\\\\[4.0pt] &\hskip
35.0pt\left.-\langle\cos{2v}\,\mathrm{d}{u}(\xi)+\sin{2v}\left[1-\mathrm{d}v(\xi)\right]\rangle
g(\varphi X,\varphi Y)\right\\}=0.\end{split}$

The latter equality is valid for arbitrary vector fields if and only if the
following conditions are satisfied

$\mathrm{d}u(\xi)=0,\qquad\mathrm{d}v(\xi)=1.$

Substitute (20) and (22) into (6) to get

(27) $\displaystyle\bar{F}(X,Y,Z)=$ $\displaystyle\
\frac{e^{2u}}{2n}\Bigl{\\{}\left[\cos{2v}\\{\alpha(Z)-\eta(Z)\\}+\sin{2v}\,\beta(Z)\right]g(\varphi
X,\varphi Y)$ $\displaystyle\hskip
24.0pt+\left[\cos{2v}\,\beta(Z)-\sin{2v}\\{\alpha(Z)-\eta(Z)\\}\right]g(X,\varphi
Y)$ $\displaystyle\hskip
24.0pt+\left[\cos{2v}\\{\alpha(Y)-\eta(Y)\\}+\sin{2v}\,\beta(Y)\right]g(\varphi
X,\varphi Z)$ $\displaystyle\hskip
24.0pt+\left[\cos{2v}\,\beta(Y)-\sin{2v}\\{\alpha(Y)-\eta(Y)\\}\right]g(X,\varphi
Z)\Bigr{\\}}.$

Using our assumption that
$(M,\allowbreak{}\varphi,\allowbreak{}\xi,\allowbreak{}\eta,\allowbreak{}{}g)$
is Sasaki-like, we have [9]

$\theta=-2n\,\eta,\qquad\theta^{*}=\omega=0.$

Then, by notations (7) and conditions (22), formulae (8) yield

(28)
$\bar{\theta}=2n\left[\alpha-\eta\right],\qquad\bar{\theta}^{*}=2n\,\beta,\qquad\bar{\omega}=0.$

Equalities (7) and (22) imply the relation $\beta=-\alpha\circ\varphi,$ which
together with the first identity in (4) helps to obtain the following
expressions

(29)
$\alpha=-\frac{\bar{\theta}\circ\varphi^{2}}{2n}+\eta,\qquad\beta=-\frac{\bar{\theta}\circ\varphi}{2n}.$

Substitute (29) into (27) and get the formula

$\displaystyle\bar{F}(X,Y,Z)=$ $\displaystyle\
\frac{e^{2u}}{2n}\Bigl{\\{}\left[\cos{2v}\,\bar{\theta}(\varphi^{2}Z)+\sin{2v}\,\bar{\theta}(\varphi
Z)\right]g(\varphi X,\varphi Y)$ $\displaystyle\hskip
24.0pt+\left[\cos{2v}\,\bar{\theta}(\varphi
Z)-\sin{2v}\,\bar{\theta}(\varphi^{2}Z)\right]g(X,\varphi Y)$
$\displaystyle\hskip
24.0pt+\left[\cos{2v}\,\bar{\theta}(\varphi^{2}Y)+\sin{2v}\,\bar{\theta}(\varphi
Y)\right]g(\varphi X,\varphi Z)$ $\displaystyle\hskip
24.0pt+\left[\cos{2v}\,\bar{\theta}(\varphi
Y)-\sin{2v}\,\bar{\theta}(\varphi^{2}Y)\right]g(X,\varphi Z)\Bigr{\\}}.$

Then, we apply (18) in the last equality and obtain the following expression

$\displaystyle\bar{F}(X,Y,Z)=$ $\displaystyle\
\frac{1}{2n}\Bigl{\\{}\bar{\theta}(\varphi^{2}Z)\bar{g}(\varphi X,\varphi
Y)+\bar{\theta}(\varphi Z)\bar{g}(X,\varphi Y)$ $\displaystyle\hskip
24.0pt+\bar{\theta}(\varphi^{2}Y)\bar{g}(\varphi X,\varphi
Z)+\bar{\theta}(\varphi Y)\bar{g}(X,\varphi Z)\Bigr{\\}}.$

The obtained form of $\bar{F}$ coincides with the definition
$(M,\varphi,\bar{\xi},\bar{\eta},\bar{g})$ to belong to the basic class
$\mathcal{F}_{1}$, according to the classification of Ganchev-Mihova-Gribachev
in [6].

Finally, the expression of the Lee forms of the transformed manifold given in
(23) follows from (28) and (7). ∎

##  4\. Example

We recall a known example of a Sasaki-like manifold, given in [9] as Example
2. A Lie group $G$ of dimension $5$ is considered to have a basis of left-
invariant vector fields $\\{E_{0},\dots,E_{4}\\}$ defined by the following
commutators for $\lambda,\mu\in\mathbb{R}$:

$\begin{array}[]{ll}[E_{0},E_{1}]=\lambda E_{2}+E_{3}+\mu
E_{4},&[E_{0},E_{2}]=-\lambda E_{1}-\mu E_{3}+E_{4},\\\\[4.0pt]
[E_{0},E_{3}]=-E_{1}-\mu E_{2}+\lambda E_{4},&[E_{0},E_{4}]=\mu
E_{1}-E_{2}-\lambda E_{3}.\end{array}$

An invariant almost contact B-metric structure is then defined on $G$ by

$\begin{array}[]{rl}&g(E_{0},E_{0})=g(E_{1},E_{1})=g(E_{2},E_{2})=1\\\\[4.0pt]
&g(E_{3},E_{3})=g(E_{4},E_{4})=-1,\\\\[4.0pt] &g(E_{i},E_{j})=0,\quad
i,j\in\\{0,1,2,3,4\\},\;i\neq j,\\\\[4.0pt] &\xi=E_{0},\quad\varphi
E_{1}=E_{3},\quad\varphi E_{2}=E_{4}.\end{array}$

It is verified that the constructed manifold $(G,\varphi,\xi,\eta,g)$ is an
almost contact B-metric manifold that is Sasaki-like.

In [12], the components $R_{ijkl}=R(e_{i},e_{j},e_{k},e_{l})$ of its curvature
tensor are calculated and the type of the corresponding Ricci tensor $\rho$ is
found. Namely, it is $\rho=4\,\eta\otimes\eta$ and the scalar curvature is
$\tau=4$.

Using the non-zero ones of $R_{ijkl}$ determined by the following equalities:

$\begin{array}[]{l}R_{0110}=R_{0220}=-R_{0330}=-R_{0440}=1,\\\\[4.0pt]
R_{1234}=R_{1432}=R_{2341}=R_{3412}=1,\qquad R_{1331}=R_{2442}=1\end{array}$

and the properties $R_{ijkl}=-R_{jikl}=-R_{ijlk}$, we compute that the
associated quantity $\tau^{*}$ of $\tau$ is zero, i.e.

$\tau^{*}=g^{ij}\rho(E_{i},\varphi E_{j})=0.$

Now, we define the following functions on
$\mathbb{R}^{5}=\left\\{(x^{0},x^{1},x^{2},x^{3},x^{4})\right\\}$:

(30)
$\begin{split}u&=\frac{1}{2}\ln\left\\{\left[(x^{1})^{2}+(x^{3})^{2}\right]\left[(x^{2})^{2}+(x^{4})^{2}\right]\right\\},\\\\[4.0pt]
v&=\arctan\frac{x^{1}x^{4}+x^{2}x^{3}}{x^{3}x^{4}-x^{1}x^{2}}+x^{0},\\\\[4.0pt]
w&=x^{0}.\end{split}$

The non-zero ones between their partial derivatives are the following:

$\begin{array}[]{ll}\dfrac{\partial u}{\partial x^{1}}=-\dfrac{\partial
v}{\partial x^{3}}=\dfrac{x^{1}}{(x^{1})^{2}+(x^{3})^{2}},&\dfrac{\partial
u}{\partial x^{2}}=-\dfrac{\partial v}{\partial
x^{4}}=\dfrac{x^{2}}{(x^{2})^{2}+(x^{4})^{2}},\\\\[9.0pt] \dfrac{\partial
u}{\partial x^{3}}=\dfrac{\partial v}{\partial
x^{1}}=\dfrac{x^{3}}{(x^{1})^{2}+(x^{3})^{2}},&\dfrac{\partial u}{\partial
x^{4}}=\dfrac{\partial v}{\partial
x^{2}}=\dfrac{x^{4}}{(x^{2})^{2}+(x^{4})^{2}},\\\\[9.0pt] \dfrac{\partial
v}{\partial x^{0}}=\dfrac{\partial w}{\partial x^{0}}=1.\end{array}$

Then, for an arbitrary vector field $X=X^{i}\frac{\partial}{\partial x^{i}}$,
$i\in\\{0,1,2,3,4\\}$, we have

$\varphi X=-X^{3}\frac{\partial}{\partial x^{1}}-X^{4}\frac{\partial}{\partial
x^{2}}+X^{1}\frac{\partial}{\partial x^{3}}+X^{2}\frac{\partial}{\partial
x^{4}},\quad X^{0}=\eta(X),\quad\xi=\frac{\partial}{\partial x^{0}}.$

We verify immediately that the functions defined by (30) satisfy the
properties

$\mathrm{d}u=-\mathrm{d}v\circ\varphi,\qquad\mathrm{d}u(\xi)=0,\qquad\mathrm{d}v(\xi)=1,\qquad\mathrm{d}w=\eta.$

Let us consider a contact conformal transformation defined by (5), where the
functions $(u,v,w)$ are determined as in (30).

Then, the transformed manifold $(G,\varphi,\bar{\xi},\bar{\eta},\bar{g})$ is
an $\mathcal{F}_{1}$-manifold with a Yamabe soliton with potential $\bar{\xi}$
and a constant scalar curvature $\bar{\tau}=\bar{\sigma}$, according ot
Theorem 3.1. Moreover, taking into account (23), we obtain the corresponding
Lee forms as follows

$\bar{\theta}=4n\,\mathrm{d}u\circ\varphi,\qquad\bar{\theta}^{*}=4n\,\mathrm{d}u,\qquad\bar{\omega}=0.$

According to Proposition 8 in [9], the Ricci tensor of an almost contact
B-metric manifold is invariant under a contact homothetic transformation (i.e.
when $u$, $v$, $w$ are constants), and therefore for the corresponding scalar
curvatures we have

$\begin{array}[]{l}\bar{\tau}=e^{-2u}\cos{2v}\,\tau-e^{-2u}\sin{2v}\,\tau^{*}+\left\\{e^{-2w}-e^{-2u}\cos{2v}\right\\}\rho(\xi,\xi),\\\\[4.0pt]
\bar{\tau}^{*}=e^{-2u}\sin{2v}\,\tau+e^{-2u}\cos{2v}\,\tau^{*}-e^{-2u}\sin{2v}\,\rho(\xi,\xi).\end{array}$

Then, bearing in mind the last results, we obtain for our example that

$\bar{\tau}=4\,e^{-2w},\qquad\bar{\tau}^{*}=0.$

Hence, $(G,\varphi,\bar{\xi},\bar{\eta},\bar{g})$ is scalar flat and the
Yamabe soliton constant for $\bar{g}$ is $\bar{\sigma}=4$, and thus the
obtained Yamabe soliton is shrinking.

## Acknowledgements

The author was supported by projects MU21-FMI-008 and FP21-FMI-002 of the
Scientific Research Fund, University of Plovdiv Paisii Hilendarski, Bulgaria.

## Conflict of interest

The author declares that he has no conflict of interest.

## References

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  * [2] H.D. Cao, X. Sun, Y. Zhang. _On the structure of gradient Yamabe solitons_. Math. Res. Lett. 19 (4) (2012), 767–774.
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  * [6] G. Ganchev, V. Mihova, K. Gribachev. _Almost contact manifolds with B-metric_. Math. Balkanica (N.S.) 7 (1993), 261–276.
  * [7] A. Ghosh. _Yamabe soliton and Quasi Yamabe soliton on Kenmotsu manifold_. Math. Slovaca 70 (1) (2020), 151–160.
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  * [9] S. Ivanov, H. Manev, M. Manev. _Sasaki-like almost contact complex Riemannian manifolds_. J. Geom. Phys. 105 (2016), 136–148.
  * [10] M. Manev, K. Gribachev. _Contactly conformal transformations of almost contact manifolds with B-metric_. Serdica Math. J. 19 (1993), 287–299.
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# Ultrafast Relaxation Dynamics of Spin-Density Wave Order
in BaFe2As2 under High Pressures

Ivan Fotev  Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany
Technische Universität Dresden, 01062 Dresden, Germany  Stephan Winnerl
Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  Saicharan
Aswartham  Leibniz IFW Dresden, 01069 Dresden, Germany  Sabine Wurmehl
Leibniz IFW Dresden, 01069 Dresden, Germany  Institute of Solid State and
Materials Physics and Würzburg-Dresden Cluster of Excellence ct.qmat,
Technische Universität Dresden, 01062 Dresden, Germany  Bernd Büchner  Leibniz
IFW Dresden, 01069 Dresden, Germany  Institute of Solid State and Materials
Physics and Würzburg-Dresden Cluster of Excellence ct.qmat, Technische
Universität Dresden, 01062 Dresden, Germany  Harald Schneider  Helmholtz-
Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  Manfred Helm  Helmholtz-
Zentrum Dresden-Rossendorf, 01328 Dresden, Germany  Technische Universität
Dresden, 01062 Dresden, Germany  Alexej Pashkin<EMAIL_ADDRESS>Helmholtz-
Zentrum Dresden-Rossendorf, 01328 Dresden, Germany

###### Abstract

BaFe2As2 is the parent compound for a family of iron-based high-temperature
superconductors as well as a prototypical example of the spin-density wave
(SDW) system. In this study, we perform an optical pump-probe study of this
compound to systematically investigate the SDW order across the pressure-
temperature phase diagram. The suppression of the SDW order by pressure
manifests itself by the increase of relaxation time together with the decrease
of the pump-probe signal and the pump energy necessary for complete
vaporization of the SDW condensate. We have found that the pressure-driven
suppression of the SDW order at low temperature occurs gradually in contrast
to the thermally-induced SDW transition. Our results suggest that the
pressure-driven quantum phase transition in BaFe2As2 (and probably other iron
pnictides) is continuous and it is caused by the gradual worsening of the
Fermi-surface nesting conditions.

##  I Introduction

Iron pnictide high-$T_{c}$ superconductors have become one of the most
intensively studied material systems since their discovery in 2008 [1]. In
contrast to superconducting cuprates, they demonstrate a high sensitivity to
the applied stress, for instance, to the hydrostatic pressure. The magnetic
spin-density wave (SDW) order in a parental compound for these materials can
be suppressed not only by chemical doping, but also by pressure [2]. Moreover,
since parental iron pnictides are metallic even in the absence of doping, they
can undergo pressure-induced SDW-to-superconductor quantum phase transition
(QPT) at low temperatures [3, 4]. Exploring the mechanism of this transition,
the competition and coexistence of the magnetic order and the superconducting
condensate is crucial for the understanding of high-$T_{c}$ superconductivity
in iron pnictides.

For the present study we have chosen BaFe2As2 as a prime example of a well-
known parental iron pnictide compound, in which the electronic state can be
controlled by external pressure. At ambient pressure the onset of the magnetic
SDW order occurs simultaneously with the structural transition from tetragonal
to orthorhombic phase at $T_{\text{SDW}}$=137 K [5]. A chemical doping or an
external pressure application cause a decrease of the transition temperature
$T_{\text{SDW}}$ [6]. The $p$-$T$ phase diagram of BaFe2As2 depicted in Fig. 1
is qualitatively similar to the one of chemically doped compounds: at low
temperatures the system is in the SDW state that is gradually suppressed by
increasing the doping level or pressure until the superconductivity (SC) sets
in [7]. Upon applying high pressures, the structural properties of BaFe2As2
change the same way as in the chemically doped materials [2]. The structural
changes are correlated with the reduction of the Fermi-surface nesting between
the inner hole and electron pockets, which results in the SDW state
suppression above a pressure of $\approx$3 GPa (depending on the experimental
conditions) [2, 8, 3]. This suppression results in the quantum critical
behavior [9] indicating that the pressure- or doping-driven SDW phase
transition should be of the second-order in contrast to the first-order
temperature-induced SDW transition [10].

Figure 1: Schematic pressure-temperature phase diagram of BaFe2As2, mostly
according to [3]. The black dashed line denotes the structural phase
transition that splits out of the SDW transition with doping [6] or pressure
[11]. Broken red and blue lines illustrate $p=\text{const}$ and
$T=\text{const}$ measurement trajectories corresponding to measurements shown
in Figs.2 and 5, respectively.

Measuring the pressure dependence of the SDW or superconducting energy gaps
poses a significant challenge, because it has to be performed on a sample
inside a diamond anvil cell (DAC). This precludes the use of angle-resolved
photoemission spectroscopy (ARPES), which requires a direct access to the
sample’s surface in vacuum. Tunneling spectroscopy under high pressures was
demonstrated for conventional superconductors [12]. However, it has not been
applied to high-$T_{c}$ iron-based superconductors yet. Nevertheless, a recent
tunneling spectroscopy study demonstrates a strong impact of the local surface
strain on the superconducting gap in LiFeAs [13]. Up to now, the SDW and the
superconducting energy gaps in BaFe2As2 under high pressure have been
determined using infrared spectroscopy [14]. That study reveals the
coexistence of the SDW and SC orders at 3.6 GPa since the spectral weight of
the SDW gap excitation is not significantly affected by the emergence of the
superconductivity.

Besides infrared spectroscopy that probes the linear optical response of a
material, nonlinear pump-probe spectroscopy provides another way to estimate
the energy gaps in high-$T_{c}$ superconductors via the analysis of
quasiparticle relaxation dynamics [15]. The main advantage of pump-probe
spectroscopy is its high sensitivity to the evolution of electronic order in
close vicinity of phase transitions. Numerous pump-probe spectroscopy studies
of iron pnictides have demonstrated that the suppression of the SDW or SC
order around the transition temperatures is accompanied by a critical slowing
down of the relaxation dynamics for the given electronic order [16, 17, 18].
Near-infrared pump – mid-infrared probe spectroscopy of parental BaFe2As2 has
shown that the coherent lattice oscillation that modulates the Fe-As-Fe
bonding angle can periodically induce transient SDW ordering, even at
temperatures above $T_{\text{SDW}}$ [19].

Here we report a study of the SDW order in BaFe2As2 across the $p-T$ phase
diagram. Our results demonstrate a critical slowing down of the SDW relaxation
time caused by the suppression of SDW order both by temperature or pressure.
However, in contrast to the abrupt change across the temperature-driven first-
order transition, the increase of the relaxation time caused by pressure at 8
K is gradual, indicating the second-order character of the quantum phase
transition.

##  II Experimental Results

###  II.1 Sample preparation and experimental setup

The studied BaFe2As2 monocrystals were grown by a self-flux high temperature
solution growth technique [20, 5]. The sample, 100$\,\mu$m $\times$ 85$\,\mu$m
in size and a thickness of about 30$\,\mu$m, was initially cleaved from the
BaFe2As2 monocrystal, so that it can fit into the sample chamber of the DAC.
The commercial closed-cycle cryostat system with integrated DAC
(Diacell${}^{\textrm{\textregistered}}$ CCS-DAC from Almax$\cdot$easyLab)
enabled us to perform in situ control of pressure and temperature
independently due to the gas membrane-driven DAC system. A Ti:sapphire
amplifier system operating at the repetition rate of 250 kHz generates 60 fs
long laser pulses utilized in the experiment. The pump and the probe beams
(both with the wavelength of 800 nm) are focused onto the sample inside the
DAC, and the reflected probe signal is collected using a confocal microscopy
scheme. A ruby chip was placed inside the sample chamber, next to the sample,
and used for external pressure calibration [21]. We chose CsI powder as the
pressure-transmitting medium in order to ensure a direct contact between the
sample and the diamond anvil.

###  II.2 Temperature dependence at fixed pressure

Figure 2: (a) Relative differential reflectivity transients and their
monoexponential fits, recorded at $p=1.4$ GPa for several temperatures, and
fixed pump fluence of $75\,\mu$J/cm2. (b) Amplitudes (black) and relaxation
times (blue) from the monoexponential fits. The substantial drop in amplitude
above 100 K, accompanied by the increase in relaxation times, reveals the
phase transition from SDW to metallic state.

Figure 2(a) shows a set of photoinduced differential reflectivity $\Delta
R(t)/R$ measurements, obtained at pump fluence of $F=75\,\mu$J/cm2, and fixed
pressure of 1.4 GPa. Together these measurement points correspond to the
vertical path in the $p$-$T$ diagram, marked by the broken red line in Fig. 1.
At lower temperatures BaFe2As2 is in the SDW phase, and the photoinduced
reflectivity change $\Delta R/R$ reaches its negative maximum quickly after
the photoexcitation at the zero delay time. Then, the signal decays within few
picoseconds to a nearly constant level corresponding to a thermalized hot
state. We assign this monoexponential, fast-decaying part of the signal to the
relaxation of the SDW phase. Its amplitude $|A|$ is proportional to the
concentration of photoexcited quasiparticles: $|A|\propto n_{pe}$ [22, 23,
24]. With increasing temperature $A$ decreases substantially, and near 100 K
the relaxation process slows down. Finally, the pump-probe response at $T=106$
K (red line) does not contain the SDW relaxation signature anymore and
demonstrates only a small signal with opposite sign with a much faster decay.
We assign it to the hot-electron cooling process typical for metals.

Figure 2(b) shows the temperature dependence of the exponential decay
amplitude. Its strong drop around 100 K is related to the SDW phase transition
to the non-magnetic metallic phase. Naturally, this transition temperature
determined for the pressure of 1.4 GPa is lower than $T_{\text{SDW}}=137$ K at
atmospheric pressure. Simultaneously, we observe indications of the critical
slowing down of the relaxation dynamics that was previously reported for
SmFeAsO [18], SrFe2As2 [17] and BaFe2As2 [16, 25]. However, it was not
possible to reliably quantify this effect since close to $T_{\text{SDW}}$,
where the slowing down is expected, the signal amplitude becomes very small
and the fitting uncertainty is too large.

At temperatures well below $T_{\text{SDW}}$ the relaxation time $\tau\approx
1.6$ ps remains nearly constant. This behavior contradicts the bimolecular
recombination model:
$\mathrm{d}n_{pe}/\mathrm{d}t=-2Rn_{T}n_{pe}-Rn_{pe}^{2}$, where $R$ is the
bimolecular recombination coefficient and $n_{T}$ and $n_{pe}$ are the
densities of thermally and photoexcited quasiparticles, respectively. Here the
first term $Rn_{T}n_{pe}$ corresponds to the recombination of a photoexcited
quasiparticle with a thermally populated one, and the second term
$Rn_{pe}^{2}$ represents the recombination of the two photoexcited
quasiparticles [26]. At low temperatures the bimolecular model predicts that
the initial recombination rate $\tau^{-1}$ should increase with $n_{T}$, which
strongly depends on temperature. Similar to our observations, the studies of
SmFeAsO [18] and SrFe2As2 [17] also report $\tau$ that does not increase with
decreasing temperatures. It is interpreted as a hint for an additional
relaxation channel besides the bimolecular recombination at the SDW gap;
namely, an interband scattering of quasiparticles to other, ungapped
electronic bands [17]. Such scattering should be temperature independent and
it should result in an additional, dominating monomolecular recombination term
$-n_{pe}/\tau$ in the rate equation.

###  II.3 Suppression of the SDW relaxation by pressure and temperature

Figure 3: $p-T$ diagram obtained from the amplitudes of the SDW transients
$A$. At the phase transitions to metallic or superconducting states the
amplitude goes to zero (blue color). The individual measurement points are
represented by asterisks.

The data from all temperature-dependent measurements at fixed pressures from
0.5 to 4.4 GPa (vertical scans of the $p$-$T$ diagram) are combined in a $p-T$
diagram, depicted in Fig. 3. The color represents the amplitude of the
normalized monoexponential signal $|A|$: from yellow for 1, to blue for 0.
Therefore, the metallic phase corresponds to the blue region of the diagram
and the density of the SDW condensate is displayed by the brightness of the
yellow color. As expected from Fig. 1, the temperature at which the amplitude
vanishes, i.e. the SDW transition temperature $T_{\text{SDW}}(p)$, decreases
with pressure, and at high pressures the SDW state gets completely suppressed.
From the 4.4 GPa measurements we could not identify SDW $\Delta R(t)/R$
transients anymore, so we assigned zeros to the amplitude of those points.

As shown in Figs. 2(b) and 3, upon increasing temperature the amplitudes
approach smoothly and monotonically zero at the corresponding
$T_{\text{SDW}}(p)$ for each pressure, which was also reported in Ref. 16.
This is in contrast to the other two studies of iron pnictides [17, 18], where
the temperature dependence of $A(T)$ exhibits a small increase relatively
close to $T_{\text{SDW}}$, before starting to drop rapidly: a characteristic
feature of the bottleneck model [27]. However, the pump fluences used in the
present study are much larger than in the other three reports, making the
direct comparison harder.

###  II.4 Relaxation dynamics and its fluence dependence at 8 K

Figure 4: (a) Reflectivity changes $\Delta R(t)/R$ measured at 1.9 GPa and 8 K
for selected pump fluences. The fluence values are given in $\mu$J/cm2. (b)
Fluence dependence of the SDW reflectivity transient amplitudes $|A|$, for
several pressures. $F_{0}$ is the part of the pump fluence $F$ which
penetrates into the sample, after the reflection from its surface. Lines are
the fits with the model of the threshold fluence for condensate vaporization
from [28].

Now let us examine the suppression of the SDW order by pressure in the ground
state, i.e., the path of the quantum phase transition as it is depicted in
Fig. 1. In order to gain more reliable information about the SDW phase we
measured pump-probe response for various pump fluences at each pressure in the
range from 0.8 to 3 GPa. Figure 4(a) presents a set of measurements taken at
different pump fluences at pressure of 1.9 GPa and temperature of 8 K.
Starting from $7\,\mu$J/cm2 the amplitude $A$ grows linearly with pump fluence
$F$, roughly up to $30\,\mu$J/cm2, after which it completely saturates. This
happens because the proportionality between the number of the photoexcited
quasiparticles and the number of the pump photons $n_{pe}\propto F$, which is
valid in the low-fluence regime, breaks down when the pump fluence becomes
high enough for a noticeable depletion of the SDW condensate. Finally, at a
certain threshold vaporization fluence $F_{T}$ the SDW condensate is
completely suppressed, and consequentially no more quasiparticles can be
excited. As a result, $n_{pe}$ and the pump-probe signal $A$ get saturated.

The saturation curves measured at a few chosen pressures are shown in Fig.
4(b). Here $F_{0}=(1-R)F$ is the part of the pump fluence $F$ which penetrates
the sample, after the reflection from its surface. $R=0.2$ is the reflectivity
of BaFe2As2 (at 800 nm) with respect to the diamond that is almost independent
on pressure and temperature [14]. Although the curves look qualitatively
similar, the saturation fluence $F_{T}$ as well as the saturation amplitude
$A_{sat}$ decrease with pressure. We have extracted both parameters from the
fits based on the model introduced in Ref. 28, which takes into account the
lateral Gaussian and exponential depth profiles of the pump and probe beams.
The fitting curves are represented by lines in Fig. 4(b). The obtained values
of $F_{T}(p)$ and $A_{sat}(p)$ for all measured pressures are presented in
Fig. 5(a) with black squares and blue circles, respectively.

Besides the suppression of the pump-probe signal and the decrease of the
saturation fluence, the applied pressure also affects the relaxation time
$\tau$. Figure 5(b) shows its pressure dependence $\tau(p)$, for a fixed $T$=8
K and $F$=30 $\mu$J/cm2. Similarly to the temperature-dependent measurements
[16, 25] the increasing $\tau$ signalizes the vicinity of the phase
transition. As the system approaches the transition pressure, the gap
$\Delta_{\text{SDW}}(p)$ gets smaller, and consequently the SDW recovery time
slows down. It is known that near the transition point the relaxation time is
inversely proportional to the gap $\tau\propto\Delta^{-1}$ [29, 30, 15]. Thus,
all these observations indicate the gradual suppression of the SDW state by
the external pressure leading to the quantum phase transition in BaFe2As2.
Since at 4.4 GPa we do not observe any SDW relaxation signature (see Appendix
A), the pressure of the QPT should be between 3 and 4.4 GPa.

Figure 5: (a) Pressure dependence of the threshold vaporization fluence
$F_{T}$ (black squares) and saturation amplitudes (blue circles). The value of
$F_{T}$ can be considered as the measure of SDW condensation energy. (b)
Pressure dependence of the relaxation time $\tau(p)$ at fixed temperature
$T=8\,K$ and fluence of $30\,\mu$J/cm2.

##  III Discussion

Similar to the case of the superconducting condensate in cuprates [28] it is
useful to compare the energy deposited at the saturation fluence with the
thermodynamical condensation energy that can be obtained from the specific
heat measurements [31]. For the lowest pressure measurement at $p=0.8\,$GPa
the threshold fluence is $F_{T}=29\,\mu$J/cm2 (Fig. 5(a)). In order to
estimate the absorbed energy density we use the optical penetration depth of
BaFe2As2 at $1.55$ eV (800 nm) to be $\lambda_{op}=36\,$nm. This calculation
is based on the reported real parts of dielectric constant
$\varepsilon_{1}\approx 1$ and conductivity $\sigma_{1}=1500\,\Omega^{-1}$cm-1
taken from Ref. 32. Strictly speaking, the mentioned values of
$\varepsilon_{1}$ and $\sigma_{1}$ were measured on the optimally doped
compound, but since the reflectivity $R$ is the same at 1.55 eV for the doped
and parent compounds [33], we assume that this is also holds for
$\varepsilon_{1}$ and $\sigma_{1}$, and consequently, $\lambda_{op}$.
Therefore, the absorbed energy density at the vaporization threshold is
$F_{T}/\lambda_{op}=8\,$J/cm3. On the other hand, the thermodynamic SDW
condensation energy is estimated to be $E_{c}/V=1.6\,$J/cm3, using the
specific heat $C(T)$ data for the parent BaFe2As2 ($p=0\,$GPa) [31]. Thus, the
pump energy necessary for the vaporization of the SDW condensate is about 5
times larger than the condensation energy density of BaFe2As2. A similar
result was reported for La2-xSrxCuO4 cuprate superconductors, and interpreted
as the evidence that a large portion of the pump energy goes to the phonon
subsystem and does not affect the electronic condensate [28]. Therefore, the
saturation amplitude $A_{sat}$ that is related to the full depletion of the
SDW condensate appear to be more reliable parameters for the estimation of the
SDW gap energy.

Figure 6: (a) Temperature dependence of $\sqrt{A(T)}$ (blue squares) from the
measurements at $1.4\,$GPa and $F=75\,\mu$J/cm2, compared with the BaFe2As2
SDW gap $\Delta_{\text{SDW}}(T)$ (gray triangles), extracted from the ARPES
study at $p=0\,$GPa [34]. (b) The evolution of the SDW gap with pressure at
$T=10\,$K estimated as $\sqrt{A_{sat}}$ and $1/\tau$. The highest pressure
point shows the absence of the SDW order at 4.4 GPa. The dashed line
schematically depicts the evolution of $\Delta_{\text{SDW}}(T)$ across the
QPT.

According to the Ginzburg-Landau theory the order parameter scales as
$|\Delta|^{2}\propto n_{cond}\propto A_{sat}$ ($n_{cond}$ is the condensate
density) and $|\Delta|\propto 1/\tau$, when $T\rightarrow T_{\text{SDW}}$.
Figure 6(a) shows the normalized $\sqrt{A(T)}$ dependence at $1.4\,$GPa and
$F=75\,\mu$J/cm2 (blue squares). Based on the results in Fig. 5(a), this
fluence value corresponds to the saturation regime and, thus, we expect that
$|\Delta_{\text{SDW}}|\propto\sqrt{A}$. For comparison, Figure 6(a) also shows
the normalized SDW gap $\Delta_{\text{SDW}}(T)$ (gray triangles), extracted
from the angle-resolved photoemission spectroscopy (ARPES) measurements [34].
Since in contrast to the ARPES study, the depicted pump-probe results were
recorded under high pressure ($p=1.4\,$GPa), the corresponding SDW transition
temperatures are different: for our data $T_{\text{SDW}}\approx$ 100 K and for
the ARPES study $T_{\text{SDW}}=$ 138 K. In order to compare both data sets,
the horizontal axis in Fig. 6(a) is normalized to $T_{\text{SDW}}$. Both match
quite well demonstrating that the relation
$|\Delta_{\text{SDW}}|\propto\sqrt{A_{sat}}$ is fulfilled in broad range of
temperatures below the SDW transition. Some deviation is observed far from the
phase transition ($T<0.3\,T_{\text{SDW}}$) where the Ginzburg-Landau theory is
no longer fully applicable.

Now we apply the same method to evaluate the SDW gap as a function of
pressure. Figure 6(b) shows the pressure dependence of $\Delta_{\text{SDW}}$
at $T$=8 K estimated as $\sqrt{A_{sat}(p)}$ using the saturation amplitude
values from Fig. 5(a). In addition, the evaluated relaxation time $\tau(p)$
for $F=30\,\mu$J/cm2 (Fig. 5(b)) are reliable enough to use them for another
estimation of the SDW gap as $\Delta_{\text{SDW}}\propto 1/\tau(p)$, which are
depicted in Figure 6(b) as dark yellow circles. In general, all data in Fig.
6(b) indicate the trend of a _gradual_ decrease of $\Delta_{\text{SDW}}$ by
about 50% on the pressure increase up to 3 GPa. This corresponds to
$p/p_{\text{SDW}}>0.7$ taking into account that $p_{\text{SDW}}<4.4\,$GPa
according to our results. If we compare Figs. 6(a) and (b), it is apparent
that the phase transition achieved by increasing pressure happens much more
gradually compared to the first-order thermally-driven phase transition.
Possibly, this indicates the second-order character of the pressure-induced
QPT in BaFe2As2 that we schematically depicted by the dashed line in Fig.
6(b). This conclusion agrees with the observation of quantum fluctuations near
the quantum critical point in isovalently doped BaFe2(AsxP1-x)2 [9].
Microscopically the gradual character of the pressure-induced changes can be
understood as the result of a gradual failure of the Fermi-surface nesting
that causes the suppression of the SDW order [2].

Finally, we discuss the possible pressure-induced superconducting phase. Our
data demonstrate that for pressures well above 3 GPa the SDW order is fully
suppressed and, according to the phase diagram (Fig. 1), one expects the onset
of the superconductivity. In order to verify that, we have measured pump-probe
traces $\Delta R(t)/R$ at 8 K and 4.4 GPa (see Appendix A). All pump-probe
taken at various pump fluences show the same relaxation dynamics, which is
much faster than in the SDW state. The absence of fluence-dependent relaxation
dynamics and saturation behavior leads us to the conclusion that the fast
dynamics observed at 4.4 GPa does not come from the superconducting or SDW
state, but rather from the cooling of the hot electrons in the metallic phase.

##  IV Conclusion

We have performed a systematic temperature- and pressure-dependent pump-probe
study in order to investigate the evolution of the SDW order in BaFe2As2.
Measuring the amplitude of the relaxation process has enabled us to construct
the $p-T$ phase diagram that depicts the variation of the SDW order across the
whole range of the applied pressures and temperatures. The comparison of the
saturation fluence with the thermodynamic condensation energy of the SDW state
shows that the absorbed pump energy required for the suppression of the
magnetic order largely exceeds the condensation energy indicating that a large
portion of the absorbed pump energy is transferred to the phonon bath. Using
the parameters of the SDW relaxation process we have estimated the variation
of the SDW gap as a function of temperature and pressure. Our results
demonstrate an apparent difference in the character of the thermally-induced
phase transition and the pressure-driven quantum phase transition. The former
is of the first-order and occurs rather abruptly near $T_{\text{SDW}}$,
whereas the latter occurs gradually and the SDW gap starts to decrease for
pressures well below the critical pressure $p_{\text{SDW}}$. This behavior
suggests a second-order character of the quantum phase transition in BaFe2As2
in agreement with previous studies of quantum fluctuations in this material
[9].

###### Acknowledgements.

We would like to thank Jure Demsar and Ece Uykur for fruitful discussions.
This work was supported by the Deutsche Forschungsgemeinschaft DFG through the
project PA 2113/1-2 and the Priority Programme SPP1458.

##  Appendix A Pump-probe signal at 4.4 GPa

Figure 7: (a) Comparison between the reflectivity transients taken at 3 GPa
(blue), and 4.4 GPa (red), the conditions which should correspond to the SDW
and superconductivity phases, respectively. (b) Fluence dependence of the
$\Delta R(t)/R$ transients at 4.4 GPa. (c) The same data after the offsets
were subtracted, and transients normalized. The lifetimes are around 0.7 ps,
and do not depend on the pump fluence. (d) Maxima of the transients (shown in
(b)) grow linearly up to $F=75\,\mu$J/cm2, and after that sub-linearly, but
without saturating (at least not in the investigated fluence range up to
$226\,\mu$J/cm2). This indicates that the reflectivity transients might come
from the electrons in the metallic phase.

Figure 7(a) shows the change in reflectivity $\Delta R(t)/R$ measured at 8 K
and 4.4 GPa (red line). According to the $p$-$T$ diagram, BaFe2As2 is expected
to be in the superconducting phase at that pressure. For the sake of
comparison, we have plotted also the SDW relaxation transient at 3 GPa (blue
line), which was recorded at the same fluence and temperature. Already the
sign flip and much shorter lifetime of the 4.4 GPa measurement suggest that
the SDW phase has been indeed suppressed, i.e. the phase transition occurred.
The fluence dependence of $\Delta R(t)/R$ transients, recorded at 8 K and 4.4
GPa, are shown in Fig. 7(b). The same data, normalized after subtracting the
offsets, are presented in Fig. 7(c). Fig. 7(d) shows maxima of the
relfectivity transients from Fig. 7(b).

In the fluence range of $F=10–226\,\mu$J/cm2, the initial relaxation time is
practically constant: $\tau\approx 0.7\,$ps, and, although growing linearly up
to $75\,\mu$J/cm2, and sub-linearly at higher fluences, the amplitudes did not
saturate. On the contrary, the characteristic feature of many superconductors
is the decrease of $\tau$ with $F$, which can be explained using the
bimolecular recombination model. Torchinsky et al. [24] found for the
optimally hole-doped Ba0.6K0.4Fe2As2 that at 7 K and the fluences in the range
of 1.4–37 $\mu$J/cm2, the relaxation time decreases by about two orders of
magnitude with increasing fluence. Furthermore, they reported that this
initial, fast relaxation component is always followed by a slow relaxation
process caused by a boson bottleneck, lasting for more than 100 ps [24], which
we also did not observe in our high-pressure experiment. For the optimally
doped Ba0.6K0.4Fe2As2, the saturation fluence of the superconducting state was
estimated to be already at $25\,\mu$J/cm2, at 7 K [26]. The absence of
fluence-dependent relaxation dynamics and saturation behavior (in the fluence
range characteristic for the SDW state) leads us to the conclusion that the
observed relaxation process is not related to the superconducting or SDW
state, but perhaps rather stems from the relaxation of the hot electrons in
the metallic phase.

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22: else if $\mathrm{pos}[i]$ and $!\mathrm{pos}[i+1]$ then

23: if $V[i,i+1]\neq 0$ then$\triangleright$ O(m)

24: $\mathrm{col}_{V}(i+1)\mathrel{+}=\mathrm{col}_{V}(i)$

25: return $(\,R,V\,)\leftarrow(\,PRP,PVP\,)$

### Transposition complexity details

Consider Algorithm 6. The matrices $R,V$ are both $(m\times m)$, where $m$ is
the number of simplices in the underlying filtration. Each transposition
$\mathrm{Tr}(i,i+1)$ requires at most 2 column operations (denoted with
$S_{i}^{j}$ in the algorithm) in both $R$ and $V$, a constant number of
queries to the matrix representation, and exactly 1 row and column exchange.
Column operations take $O(m)$ time. Querying to determine whether an entry at
position $(i,i+1)$ is non-zero depends on the sparse matrix implementation,
but is generally either $O(\log m)$ or $O(m)$. In compressed sparse matrix
representation, column and row exchanges take upwards of $O(z)$ time, where
$z$ is the number of non-zero entries in the matrix, however this can be made
$O(1)$ time by using the data structure described in [2]. This data structure
requires storing an index using at most $O(m\log m)$ bits with each non-zero
entry, as well as several $O(m)$-sized auxiliary arrays to protect the
matrices from row exchanges.