Ling-Long Gao${}^{\,a,b}$111Email<EMAIL_ADDRESS>Yan
Liu${}^{\,a,b}$222Email<EMAIL_ADDRESS>and Hong-Da
Lyu${}^{\,a,b}$333Email<EMAIL_ADDRESS>
aCenter for Gravitational Physics, Department of Space Science
and International Research Institute of Multidisciplinary Science,
Beihang University, Beijing 100191, China

bPeng Huanwu Collaborative Center for Research and Education,
Beihang University, Beijing 100191, China

# Black hole interiors
in holographic topological semimetals

###### Contents

  1. 1 Introduction
  2. 2 Inside holographic Weyl semimetal
    1. 2.1 Inner structures
    2. 2.2 Behaviors of Kasner exponents
    3. 2.3 Proper time of timelike geodesics
  3. 3 Inside holographic nodal line semimetal
    1. 3.1 Kasner exponents
    2. 3.2 Proper time of timelike geodesics
  4. 4 Conclusion and discussion
  5. A Equations in holographic WSM
  6. B Equations in holographic NLSM

##  1 Introduction

The conventional classifications on the phases of matter are rooted in the
Landau paradigm of symmetry breaking theory [1]. Over the past thirty years,
new states of matter have been found which are beyond the concept of Landau
paradigm. One example is the topological states of matter, including the
quantum Hall states, topological insulators, topological semimetals and so on
[2]. Different from the conventional Landau paradigm, there is no symmetry
breaking during the topological phase transition and it attracts lots of
research attention.

In recent years, the strongly interacting topological Weyl semimetals (WSM)
[3, 4] and nodal line semimetals (NLSM) [5, 6] have been explicitly
constructed from the holographic duality. Both holographic WSM and NLSM are
shown to possess nontrivial topological invariants [7]. Remarkably,
holographic WSM exhibits interesting effects inherited from the boundary
states [8]. These features suggest that the physical properties associated to
topology from the weakly coupled field theories persist in the strongly
coupled topological systems from the holography. Moreover, the systems could
go through a topological phase transition to a topologically trivial semimetal
phase, see [9] for a review on the developments.444Other interesting
developments can be found in e.g. [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27].  In the holographic WSM, during the topological
phase transition the anomalous Hall conductivity could be served as an order
parameter, while in the holographic NLSM it is not clear about the order
parameters. Whether possible universal “order parameter” exist for the
topological phase transitions? What is the topological nature in the
topological phase from holography? These are elusive problems we aim to
explore from the holographic duality.

In holography, the thermal states are dual to black hole geometries in the
bulk. The black hole interior is expected to encode important information of
the dual field theory [28, 29, 30]. In the case that the thermal states are
described by the black holes with simple Kasner singularities, it has been
shown recently in [31] that the order of the thermal phase transition in the
dual field theory is connected to the behavior of the Kasner exponents of the
black hole singularity.555Other studies on the geometric aspects of black hole
singularities can be found in e.g. [32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53].  For the topological phase
transitions in holographic topological semimetals at finite temperature, the
systems experience a smooth crossover from a topological phase, a critical
phase to a trivial phase. Although the phase crossover is different from
thermal phase transitions, it is still interesting to explore the interior
geometries in holographic topological semimetals, in order to uncover possible
universal behavior during the topological phase transitions.

It turns out that there exist both universal and special behaviors of the
singularities in holographic topological semimetals. The universal behavior is
similar to the topological nature of topological phase and might give hints to
the problems we raised for the topological semimetals, while the special
features can be understood from the fact that the holographic WSM and the
holographic NLSM share similarities and also differences in the constructions
as emphasized in [5, 7]. More precisely, in both cases, two matters fields are
added which play same role from the point of view of the boundary field
theory, while they play different roles in the bulk geometry. In the boundary
field theory, one of the two matter fields is to deform the Dirac point into
two Weyl nodes or a nodal line, while the other matter field is to gap the
system. In the bulk, in the topological phase of holographic WSM the IR
geometry of Schwarzschild black hole is not deformed by the matter fields,
while the backreaction of the matter fields on the gravitational geometry is
quite strong in IR in the topological phase of holographic NLSM. We will see
that these two different situations lead to different properties of the black
hole singularities in the topological phases.

It is known that the information of the interior geometry can be probed from
the geodesics which correspond to certain correlators in the dual field
theory. For example, the proper time from the horizon to the singularity can
be extracted from the thermal one point function of certain heavy operator
[30]. We will compute this quantity in the bulk and study its behavior in the
topological phases and trivial phases.

This paper is organized as follows. In Sec. 2, we will first review the
holographic WSM and then study its interior geometry as well as the proper
time of the timelike geodesics. In Sec. 3, we will review the holographic NLSM
and then also study its interior geometry and the proper time of the timelike
geodesics. Sec. 4 is devoted to the conclusions and open questions. The
details of calculations are in the appendices.

##  2 Inside holographic Weyl semimetal

In this section we first briefly review the holographic WSM which describes a
topological phase transition from topological WSM phase to a trivial semimetal
phase. Then we study the interior geometry of the black hole solutions and
discuss the possible universal behavior of the black hole singularities as
well as the interior geometry. We also comment on the possible observable as
the role of “order parameter” during the topological phase transition.

The action of the holographic WSM [3, 4] is

$\displaystyle\begin{split}S=&\int
d^{5}x\sqrt{-g}\,\bigg{[}\frac{1}{2\kappa^{2}}\big{(}R+\frac{12}{L^{2}}\big{)}-\frac{1}{4}\mathcal{F}^{2}-\frac{1}{4}F^{2}+\frac{\alpha}{3}\epsilon^{abcde}A_{a}\big{(}F_{bc}F_{de}+3\mathcal{F}_{bc}\mathcal{F}_{de}\big{)}\\\
&~{}~{}~{}~{}~{}-(D_{a}\Phi)^{*}(D^{a}\Phi)-V(\Phi)\bigg{]}\,,\end{split}$
(2.1)

where two gauge fields are dual to vector and axial currents respectively. A
special Chern-Simons structure is introduced to match the Wald identity for
these currents. An axially charged scalar field $\Phi$ is also introduced in
the model with the source interpreted as the mass term. Note that
$D_{a}\Phi=\partial_{a}\Phi-iqA_{a}\Phi$ where $A_{a}$ is the axial $U(1)$
gauge potential, and $V(\Phi)=m^{2}|\Phi|^{2}+\frac{\lambda}{2}|\Phi|^{4}$. We
set $2\kappa^{2}=L=1$.

We focus on the finite temperature and use the following ansatz

$\displaystyle\begin{split}ds^{2}&=-udt^{2}+\frac{dr^{2}}{u}+f(dx^{2}+dy^{2})+hdz^{2}\,,\\\
A&=A_{z}dz\,,\quad\Phi=\phi\,.\end{split}$ (2.2)

The equations of motion for the fields can be found in the appendix A. In the
following we consider $m^{2}=-3,q=1,\lambda=1/10.$ Generalization to the other
values of the parameters is straightforward.

We use the following boundary conditions for the matter fields

$\lim_{r\to\infty}A_{z}=b\,,~{}~{}~{}\lim_{r\to\infty}r\phi=M\,,$ (2.3)

where $b$ is the time reversal symmetry breaking parameter which play the role
of splitting a Dirac point into two Weyl points, and $M$ is the mass parameter
which gaps the Dirac point. The competing between these two effects leads to
interesting topological phase transitions. The system is completely determined
by the dimensionless parameters $T/b,M/b$.

In the weakly coupled WSM, the quantum topological phase transition could be
manifest from both the band structure and equivalently the behavior of the
anomalous Hall conductivity. In the strongly coupled model from holography,
the anomalous Hall conductivity behaves similarly to the weakly coupled case,
indicating that there is a topological phase transition, as shown in Fig. 1.
The lines in red, blue and purple are for $T/b=0.05,0.02,0.01$ respectively.
The transition becomes sharp at zero temperature and the dashed gray line is
the critical value of the transition at zero temperature.

Figure 1: Plot of anomalous Hall conductivity as a function of $M/b$ at the
temperatures $T/b=0.05$ (red), $0.02$ (blue), $0.01$ (purple). The gray dashed
line is the critical value of $M/b$ of the quantum phase transition at zero
temperature.

###  2.1 Inner structures

The phase transitions could be parameterized by the anomalous Hall
conductivity which is completely determined by the horizon value of the axial
gauge field $A_{z}$. Given the possible connection between the physics inside
and outside the horizon, it is interesting to study the black hole inner
structures during the topological phase transitions.

From the black hole solutions we have obtained, we could integrate the system
further to the singularity since the geometry is smooth at the horizon. We
find that at low temperature, the matter field $\phi$ oscillates inside the
horizon only in the topological phase (i.e. $M/b<0.744$). The typical behavior
is shown in Fig. 2, where the oscillation regime of the scalar field $\phi$
(which has been rescaled according to $\phi/\phi_{h}$) as a function of
$r/r_{h}$ at fixed $T/b$ (left) or $M/b$ (right) are plotted respectively. We
find that when we fix the temperature $T/b$, the times of oscilation become
less when we increase $M/b$ from $0$ to $(M/b)_{c}$. Furthermore, when we fix
$M/b<(M/b)_{c}$, the lower temperature, the more times that $\phi$ oscillates.
Note that the other fields do not show any oscillation from the horizon to the
singularity.

Different from the holographic superconductor cases, the oscillation here is
not related to the collapse of Einstein-Rosen bridge [34], since there is no
inner horizon any more for holographic WSM. Similar oscillation behavior has
been found previously in neutral helical black holes [35].

Figure 2: The plots of $\phi/\phi_{h}$ along radial direction in the
oscillation region at fixed $T/b=0.02$ (left) while $M/b=0.1$ (purple), $0.4$
(blue), $0.6$ (orange), $0.74$ (red), as well as at fixed $M/b=0.1$ (right)
while $T/b=0.05$ (red), $0.02$ (blue), $0.01$ (purple). Here $\phi_{h}$ is the
horizon value of $\phi$.

###  2.2 Behaviors of Kasner exponents

The interior solution can be further integrated to the singularity. Near the
singularity $r_{s}$, we assume that at the leading order the fields behave as

$\displaystyle u\sim-u_{0}(r-r_{s})^{n_{u}}\,,~{}~{}~{}f\sim
f_{0}(r-r_{s})^{n_{f}}\,,~{}~{}~{}h\sim
h_{0}(r-r_{s})^{n_{h}}\,,~{}~{}~{}\phi\sim n_{\phi}\ln(r-r_{s})\,,$ (2.4)

where $u_{0},f_{0},h_{0}$ and $n_{u},n_{f},n_{h},n_{\phi}$ are all constants.
Here $u_{0},f_{0},h_{0}$ depend on the scaling symmetry in (A.3),(A.4),(A.5)
while $n_{u},n_{f},n_{h},n_{\phi}$ are not. Also note that here $r_{s}$ is not
necessarily to be zero since there is a shift symmetry of the system $r\to
r+\alpha$ along the radial direction which was used to set the boundary
behavior (A.8). Moreover, as we shall see later, the axial gauge field $A_{z}$
is determined by the ansatz (2.4).

Near the singularity the equations of motion (A.6) can be simplified under the
assumption that the ignored terms are subleading which will be numerically
checked afterward,

$\displaystyle\begin{split}u^{\prime\prime}+\frac{h^{\prime}}{2h}u^{\prime}-\left(f^{\prime\prime}+\frac{f^{\prime}h^{\prime}}{2h}\right)\frac{u}{f}&=0\,,\\\
\frac{f^{\prime\prime}}{f}+\frac{u^{\prime\prime}}{2u}-\frac{f^{\prime
2}}{4f^{2}}+\frac{f^{\prime}u^{\prime}}{fu}+\frac{1}{2}\phi^{\prime
2}&=0\,,\\\ \frac{1}{2}\phi^{\prime
2}-\frac{u^{\prime}}{2u}\left(\frac{f^{\prime}}{f}+\frac{h^{\prime}}{2h}\right)-\frac{f^{\prime}h^{\prime}}{2fh}-\frac{f^{\prime
2}}{4f^{2}}&=0\,,\\\
A_{z}^{\prime\prime}+\left(\frac{f^{\prime}}{f}-\frac{h^{\prime}}{2h}+\frac{u^{\prime}}{u}\right)A_{z}^{\prime}&=0\,,\\\
\phi^{\prime\prime}+\left(\frac{f^{\prime}}{f}+\frac{h^{\prime}}{2h}+\frac{u^{\prime}}{u}\right)\phi^{\prime}&=0\,.\end{split}$
(2.5)

Substituting (2.4) into (2.5), we obtain

$\displaystyle
n_{h}=2\,(1-n_{u}-n_{f})\,,~{}~{}~{}~{}~{}n_{\phi}=\pm\sqrt{(2n_{f}+n_{u})(1-n_{u})-\frac{3n_{f}^{2}}{2}}\,.$
(2.6)

We can also solve the fourth equation in (2.5) to obtain at leading order
$A_{z}$

$\displaystyle A_{z}\simeq A_{zs0}+A_{zs1}(r-r_{s})^{n_{h}}\,.$ (2.7)

Note that the leading term $A_{zs0}$ can be rescaled to be $1$, while
$A_{zs1}$ could be determined from the radial conserved quantities as will be
discussed later. Thus there are only two independent parameters in (2.4) and
(2.7).

Note that in the above equations (2.5), we have assumed that the terms ignored
are subleading. More explicitly, we have assumed

$\displaystyle n_{u}<2\,,~{}~{}~{}n_{f}+n_{u}<1\,,~{}~{}~{}2n_{f}+n_{u}>0\,.$
(2.8)

Numerically we have checked that all the above relations are satisfied for the
parameters we have considered, which indicates that the singularities are
stable and of form (2.4) and (2.7).

There are two radical conserved charge associated to the scaling symmetries of
the system,

$\displaystyle Q_{1}$ $\displaystyle=$
$\displaystyle\sqrt{h}(u^{\prime}f-uf^{\prime})\,,$ (2.9) $\displaystyle
Q_{2}$ $\displaystyle=$ $\displaystyle
u^{\prime}\sqrt{h}f-\frac{h^{\prime}}{\sqrt{h}}uf-
A_{z}A_{z}^{\prime}\frac{uf}{\sqrt{h}}\,.$ (2.10)

We have used them to check the accuracy of the numerics. Moreover, evaluate
them at the horizon and at the singularity we obtain

$\displaystyle 4\pi Tf_{1}\sqrt{h_{1}}=Ts$ $\displaystyle=$ $\displaystyle
u_{0}f_{0}\sqrt{h_{0}}(n_{f}-n_{u})$ (2.11) $\displaystyle=$
$\displaystyle\frac{u_{0}f_{0}}{\sqrt{h_{0}}}\left(n_{h}A_{zs0}A_{zs1}-h_{0}(2n_{f}+3n_{u}-2)\,\right)$
(2.12)

where $s$ is the density of entropy. From (2.11), we have $n_{f}>n_{u}$ in
addition to the constraints (2.8). Moreover, the above two conserved
quantities give the relations $n_{h}A_{zs0}A_{zs1}=h_{0}(3n_{f}+2n_{u}-2)$
which turns out to be zero in the topological phase at low temperature where
$A_{zs1}=3n_{f}+2n_{u}-2=0$.

Starting from (2.2, 2.4) and performing the coordinate transformation

$\tau=-\frac{2}{\sqrt{n_{0}}(n_{u}-2)}(r-r_{s})^{(2-n_{u})/2}\,,$ (2.13)

we obtain the Kasner form for the fields

$\displaystyle\begin{split}ds^{2}&=-d\tau^{2}+c_{t}\tau^{2p_{t}}dt^{2}+c_{x}\tau^{2p_{x}}(dx^{2}+dy^{2})+c_{z}\tau^{2p_{z}}dz^{2}\,,\\\
\phi&=p_{\phi}\log\tau+c_{\phi}\,,\end{split}$ (2.14)

where

$\displaystyle\begin{split}p_{t}&=\frac{n_{u}}{2-n_{u}}\,,~{}~{}~{}~{}p_{x}=\frac{n_{f}}{2-n_{u}}\,,~{}~{}~{}~{}p_{z}=\frac{n_{h}}{2-n_{u}}\,,~{}~{}~{}~{}p_{\phi}=\frac{2n_{\phi}}{2-n_{u}}\,.\end{split}$
(2.15)

Note that $A_{z}$ is a constant at the leading order. Using the relations
(2.6), the above Kasner exponents can be expressed in terms of $n_{u}$ and
$n_{f}$,

$\displaystyle\begin{split}&p_{t}=\frac{n_{u}}{2-n_{u}}\,,~{}~{}p_{x}=\frac{n_{f}}{2-n_{u}}\,,~{}~{}p_{z}=\frac{2(1-n_{u}-n_{f})}{2-n_{u}}\,,~{}~{}p_{\phi}=\pm\frac{\sqrt{4(2n_{f}+n_{u})(1-n_{u})-6n_{f}^{2}}}{2-n_{u}}\,.\end{split}$
(2.16)

Note that the sign of $p_{\phi}$ in (2.16) can only be determined from
numerics. They satisfy the following Kasner relations

$\displaystyle
p_{t}+2p_{x}+p_{z}=1\,,~{}~{}~{}~{}~{}p_{t}^{2}+2p_{x}^{2}+p_{z}^{2}+p_{\phi}^{2}=1\,.$
(2.17)

It indicates that only two of the four Kasner exponents are independent.

In Fig. 3, we show the Kasner exponents as a function of $M/b$ at different
temperatures $T/b=0.05$ (red), $0.02$ (blue), $0.01$ (purple). We find that at
low temperature, the Kasner exponents in the Weyl semimetal phase take the
same value of the Schwarzschild black hole (e.g. within the difference of
order less than $10^{-9}$ between $M/b=0.5$ and $M/b=0$ at $T/b=0.01$). This
reminds us the topological feature in terms of the black hole singularity. It
is related to the fact that the matter fields do not backreact relevantly to
the Schwarzschild solution in the topological phase, i.e. the probe limit of
system in terms of matter fields in the Schwarzschild black hole background
works well. We have also checked that inside the black holes, in the
topological phase the matter fields obtained from the backreacted case match
well with the solutions obtained from the probe limit. In the quantum critical
regime, the Kasner exponents oscillate. While in the trivial phase, the Kasner
exponent does not have any oscillate behavior.

Figure 3: Plots of Kasner exponents as a function of $M/b$. For all cases we
have $T/b=0.05$ (red), $0.02$ (blue), $0.01$ (purple). The dashed gray
vertical lines are the Kasner exponents of five dimension Schwarzschild black
hole.

Note that in [5], a paradigm for constructing the topological phase was
proposed and the holographic Weyl semimetal belongs to the first type, where
the matter fields are irrelevant in the IR of the Schwarzschild black hole. It
seems likely that in any topological phase of this kind, the singularities are
of Kasner form taking values of Schwarzschild black hole.

###  2.3 Proper time of timelike geodesics

One of interesting connection between the interior geometry and the boundary
observable is given in [30] that the proper time of radial timelike geodesic
can be encoded in the thermal one point functions of heavy operators. It is
thus interesting to study the proper time of radial timelike geodesics to see
if it has specific behavior during the topological phase transitions.

We consider radial timelike geodesic for which
$g_{tt}{\dot{t}}^{2}+g_{rr}{\dot{r}}^{2}=-1$, where the dot denotes the
derivative with respect to the proper time $\tau$. Along the geodesic there is
a conserved charge $E=-g_{tt}\dot{t}$ which can be interpreted as energy. Then
the equation of motion of the geodesic becomes

$\displaystyle\frac{E^{2}}{g_{tt}}+g_{rr}{\dot{r}}^{2}=-1\,,$ (2.18)

from which we obtain

$\displaystyle\frac{d\tau}{dr}=\frac{1}{\sqrt{E^{2}-u}}\,.$ (2.19)

The proper time from the horizon to the singularity of a particle with $E=0$
(i.e. the longest time) is

$\displaystyle\tau_{s}=\int_{r_{s}}^{r_{h}}\frac{dr}{\sqrt{-u}}\,.$ (2.20)

The plots of $\tau_{s}$ as a function of $M/b$ for different $T/b$ are shown
in Fig. 4.

Figure 4: Plots of the proper time $\tau_{s}$ from the horizon to the
singularity as a function of $M/b$ at different temperatures $T/b=0.05$ (red),
$0.02$ (blue), $0.01$ (purple).

The proper time from the horizon to the singularity in the topological phase
is equal to the case of Schwarzschild black hole $\tau_{s}=\pi/4$ (e.g. within
the difference of order less than $10^{-4}$ between $M/b=0.5$ and $M/b=0$ at
$T/b=0.01$). This is expected from the fact that in the topological phase at
low temperature the interior of the black holes match well with the
Schwarzschild black hole. In the topologically trivial phase $\tau_{s}$ is
monotonically decreasing. Moreover, $\tau_{s}$ shows a jump behavior and takes
a maximum value in the critical regime. Note that $\tau_{s}$ is encoded in the
thermal one point function of heavy operators in the form of
$\langle\mathcal{O}\rangle\propto e^{-im\tau_{s}}$ where the complexified mass
$m$ has $\text{Im}(m)<0$ [30]. One might use this thermal one point function
as the “order” parameter for the topological phase transition. The behavior of
the proper time also reminds us the behavior of the dimensionless information
screening length in [14]. One obvious difference is that the information
screening length is determined by the quantities at the horizon, while
$\tau_{s}$ is determined by the geometry from the horizon to the singularity.

##  3 Inside holographic nodal line semimetal

In the previous section, we have seen that the interior of the black hole
geometries for the holographic WSM exhibit interesting behavior. In the
topological WSM phase, the Kasner exponents of the dual geometries take the
same value of the Schwarzschild black hole at low temperature, as shown in
Fig. 3. Moreover, the dual operator which encodes the proper time from the
horizon to the singularity could be served as an “order parameter” during the
topological phase transition, as shown in Fig. 4. To check if these behaviors
are universal for any topological phase transitions, in this section we study
the other topological phase transition model from holography, i.e. the
holographic NLSM model which describes a phase transition from the topological
NLSM phase to a trivial semimetal phase [5, 6].

The action for the holographic NLSM [6] is

$\displaystyle\begin{split}S&=\int
d^{5}x\,\sqrt{-g}\,\bigg{[}\frac{1}{2\kappa^{2}}\big{(}R+\frac{12}{L^{2}}\big{)}-\frac{1}{4}\mathcal{F}^{2}-\frac{1}{4}F^{2}+\frac{\alpha}{3}\epsilon^{abcde}A_{a}\big{(}F_{bc}F_{de}+3\mathcal{F}_{bc}\mathcal{F}_{de}\big{)}\\\
&-(D_{a}\Phi)^{*}(D^{a}\Phi)-V_{1}(\Phi)-\frac{1}{6\eta}\epsilon^{abcde}\big{(}iB_{ab}H^{*}_{cde}-iB^{*}_{ab}H_{cde}\big{)}-V_{2}(B_{ab})-\lambda|\Phi|^{2}B^{*}_{ab}B^{ab}\bigg{]}\,,\end{split}$
(3.1)

where $\mathcal{F}_{ab}=\partial_{a}V_{b}-\partial_{b}V_{a}$ is the vector
gauge field strength. $F_{ab}=\partial_{a}A_{b}-\partial_{b}A_{a}$ is the
axial gauge field strength. $D_{a}=\nabla_{a}-iq_{1}A_{a}$ is the covariant
derivative and $q_{1}$ is the axial charge of scalar field. $\alpha$ is the
Chern-Simons coupling. $B_{ab}$ is an antisymmetric complex two form field
with the field strength

$\displaystyle
H_{abc}=\partial_{a}B_{bc}+\partial_{b}B_{ca}+\partial_{c}B_{ab}-iq_{2}A_{a}B_{bc}-iq_{2}A_{b}B_{ca}-iq_{2}A_{c}B_{ab}\,,$
(3.2)

where $q_{2}$ is the axial charge of the two form field. $\eta$ is the Chern-
Simons coupling strength of the two form field. The introduction of the Chern-
Simons terms while not canonical kinetic term for the two form field follows
from the self-duality condition of the two form operator in the weakly coupled
theory [6]. The potential terms are chosen as

$\displaystyle
V_{1}=m_{1}^{2}|\Phi|^{2}+\frac{\lambda_{1}}{2}|\Phi|^{4}\,,\qquad
V_{2}=m_{2}^{2}B_{ab}^{*}B^{ab}\,,$ (3.3)

where $m^{2}_{1}$ and $m_{2}^{2}$ are the mass parameters of the scalar field
and the two form field. The $\lambda$ term in the action (3.1) denotes the
interaction between the scalar field and the two form field. We set
$2\kappa^{2}=L=1$.

Similar to the holographic WSM, we focus on the finite temperature solution
and take the ansatz

$\displaystyle\begin{split}ds^{2}&=-udt^{2}+\frac{dr^{2}}{u}+f(dx^{2}+dy^{2})+hdz^{2}\,,\\\
\Phi&=\phi\,,\\\ B_{xy}&=-B_{yx}=\mathcal{B}_{xy}\,,\\\
B_{tz}&=-B_{zt}=i\mathcal{B}_{tz}\,.\end{split}$ (3.4)

Plugging the above ansatz into the equations of motion, we could obtain the
dynamical equations of the fields, which can be found in the appendix B. In
the following we choose $m_{1}^{2}=-3,m_{2}^{2}=1,\eta=2$ and
$q_{1}=q_{2}=1,\lambda=1,\lambda_{1}=0.1$ for simplicity.

With the following boundary conditions,

$\lim_{r\to\infty}r\phi=M\,,~{}~{}~{}\lim_{r\to\infty}\frac{\mathcal{B}_{xy}}{r}=\lim_{r\to\infty}\frac{\mathcal{B}_{tz}}{r}=b\,,$
(3.5)

we can integrate the system from the boundary to the horizon. Different from
the holographic WSM, in holographic NLSM there is no sharp “order parameter”
like anomalous Hall conductivity. Nevertheless, it was found in [6] that at
zero temperature, the dual fermionic spectral function shows multiple Fermi
surfaces with the topology of nodal lines when $M/b<(M/b)_{c}$ while it is
gapped when $M/b>(M/b)_{c}$. This indicates that the system undergoes a
topological phase transitions from topological NLSM to topologically trivial
semimetal phase.

With the regularity condition near the horizon, the system can be further
integrated to the singularity. In the following we will discuss the interior
geometries and singularities of the system.

###  3.1 Kasner exponents

Close to the singularity $r\to r_{s}$, similar to the holographic WSM case we
again take the ansatz

$\displaystyle u\sim-u_{0}(r-r_{s})^{n_{u}}\,,~{}~{}~{}f\sim
f_{0}(r-r_{s})^{n_{f}}\,,~{}~{}~{}h\sim
h_{0}(r-r_{s})^{n_{h}}\,,~{}~{}~{}\phi\sim n_{\phi}\ln(r-r_{s})\,,$ (3.6)

where $u_{0},f_{0},h_{0}$ and $n_{u},n_{f},n_{h},n_{\phi}$ are all constants.
The other two matter fields $\mathcal{B}_{tz}$ and $\mathcal{B}_{xy}$ will be
determined by the above ansatz.

The equations of motion can be simplified close to the singularity under the
assumption that the ignored terms are subleading

$\displaystyle\begin{split}\frac{u^{\prime\prime}}{u}-\frac{f^{\prime\prime}}{f}+\frac{h^{\prime}}{2h}\left(\frac{u^{\prime}}{u}-\frac{f^{\prime}}{f}\right)&=0\,,\\\
\frac{u^{\prime\prime}}{2u}+\frac{f^{\prime\prime}}{f}-\frac{f^{\prime
2}}{4f^{2}}+\frac{f^{\prime}u^{\prime}}{fu}+\frac{1}{2}\phi^{\prime
2}&=0\,,\\\ \frac{f^{\prime
2}}{4f^{2}}+\frac{f^{\prime}h^{\prime}}{2fh}+\frac{u^{\prime}}{2u}\left(\frac{f^{\prime}}{f}+\frac{h^{\prime}}{2h}\right)-\frac{1}{2}\phi^{\prime
2}&=0\,,\\\
\phi^{\prime\prime}+\big{(}\frac{f^{\prime}}{f}+\frac{h^{\prime}}{2h}+\frac{u^{\prime}}{u}\big{)}\phi^{\prime}&=0\,,\\\
\mathcal{B}_{tz}^{\prime}-\frac{\eta\sqrt{h}}{2f}(\lambda\phi^{2})\mathcal{B}_{xy}&=0\,,\\\
\mathcal{B}_{xy}^{\prime}-\frac{\eta
f}{2\sqrt{h}u}(\lambda\phi^{2})\mathcal{B}_{tz}&=0\,.\end{split}$ (3.7)

The first four equations in (3.7) are the same as the ones in holographic WSM.
Similarly, we obtain

$\displaystyle
n_{h}=2\,(1-n_{u}-n_{f})\,,~{}~{}~{}~{}~{}n_{\phi}=\pm\sqrt{(2n_{f}+n_{u})(1-n_{u})-\frac{3n_{f}^{2}}{2}}\,.$
(3.8)

From the last two equations in (3.7) we have the following leading order
solutions for the two form fields near the singularity

$\mathcal{B}_{xy}\sim\mathcal{B}_{xy0}+\dots\,,~{}~{}~{}~{}\mathcal{B}_{tz}\sim\mathcal{B}_{tz0}+\dots\,,~{}~{}~{}~{}$
(3.9)

where the dots are subleading terms of form
$(r-r_{s})^{2-n_{u}-2n_{f}}(\log(r-r_{s}))^{2}$ and
$(r-r_{s})^{2n_{f}}(\log(r-r_{s}))^{2}$ respectively. Here we have assumed
$2-n_{u}-2n_{f}>0$ and $n_{f}>0$, otherwise the leading solution of the two
form field might be divergent. Similar to the holographic WSM, these constants
of the two form field depend on the scaling symmetry of the system.

Note that in (3.7) we have assumed that the ignored terms are subleading. More
explicitly, we have assumed

$n_{u}<2\,,~{}~{}~{}2n_{u}+n_{h}<2\,,~{}~{}~{}n_{u}+2n_{f}<2\,.$ (3.10)

Note that the last two inequalities of above are consistent with the
assumptions used in obtaining (3.9). We have checked numerically that the
inequalities (3.10) are satisfied for the parameters we have considered.

Similar to the discussion in section 2.2, we can make a coordinate
transformation (2.13) to write the metric (3.6) into the Kasner form as (2.14)
with the parameters (2.16) and the Kasner relations (2.17). Here the leading
order of the two form fields are constant close to the singularity.

The two conserved charges of the scaling symmetries are

$\displaystyle Q_{1}$ $\displaystyle=$
$\displaystyle\frac{8}{\eta}\mathcal{B}_{tz}\mathcal{B}_{xy}+\frac{u}{\sqrt{h}}(f^{\prime}h-fh^{\prime})\,,$
(3.11) $\displaystyle Q_{2}$ $\displaystyle=$
$\displaystyle\frac{f}{\sqrt{h}}(u^{\prime}h-uh^{\prime})\,.$ (3.12)

Evaluate them at the horizon and at the singularity we obtain

$\frac{8}{\eta}\mathcal{B}_{xy0}\mathcal{B}_{tz0}=u_{0}f_{0}\sqrt{h_{0}}(n_{f}-n_{h})$
(3.13)

and

$\displaystyle 4\pi
Tf_{1}\sqrt{h_{1}}=Ts=u_{0}f_{0}\sqrt{h_{0}}(2-2n_{f}-3n_{u})$ (3.14)

where $s$ is the density of entropy. We have checked the above relations
numerically.

In Fig. 5, we show the Kasner exponents for the holographic NLSM as functions
of $M/b$ at different temperature $T/b=0.05$ (red), $0.02$ (blue), $0.01$
(purple). We find that at low temperature, the Kasner exponents
$p_{t},p_{x},p_{z}$ of the metric fields in the NLSM semimetal phase are
almost constant in the topological phase (e.g. within the difference of order
less than $1\%$ between $M/b=0.5$ and $M/b=0$ at $T/b=0.01$), which is quite
similar to the holographic WSM, while $p_{\phi}$ changes a lot in the
topological phase. Note that this is consistent with the Kasner relations
(2.17) since $p_{\phi}$ is small. It is expected that at extremely low
temperature, the properties of the Kasner exponents in the holographic NLSM
might be the same as those in the holographic WSM, i.e. all the Kasner
exponents are constant. Due to numerical difficulty we have not explored such
a low temperature regime.

Figure 5: Plots of Kasner exponents for holographic NLSM as a function of
$M/b$. For all cases we have $T/b=0.05$ (red), $0.02$ (blue), $0.01$ (purple).
The horizontal dashed gray lines represent the Kasner exponents for $M/b=0$ at
$T/b=0.01$. The vertical dashed gray lines represent the quantum critical
point at zero temperature.

Different from the holographic WSM where the geometry is the same as
Schwarzschild black hole with a constant nonzero $A_{z}$ when $M/b=0$. Here
when $M/b=0$, in the holographic NLSM, due to the fact that the matter fields
backreact to the IR geometry and the Kasner exponents are no longer the
constant exponents of Schwarzschild black hole and instead they depend on
$T/b$, as shown in the first three pictures in Fig. 6. Nevertheless, at low
enough temperature we see that the Kasner exponents are nearly constant.

Figure 6: Plots of Kasner exponents and $\tau_{s}$ for holographic NLSM as a
function of $T/b$ when $M/b=0$.

###  3.2 Proper time of timelike geodesics

Similar to the holographic WSM, we can also discuss the proper time from the
horizon to the singularity in holographic NLSM. In Fig. 7, we show the proper
time $\tau_{s}$ as a function of $M/b$ at different temperatures. Again we see
that at low temperature, the proper time is almost a constant in the
topological phase (e.g. within the difference of order less than
$5\text{\textperthousand}$ between $M/b=0.5$ and $M/b=0$ at $T/b=0.01$), which
shows a topological behavior under the changes of the systems. Similar to the
holographic WSM, we could take the operator which encodes the information of
$\tau_{s}$ as the order parameter for the topological phase transition in
holographic NLSM. In the trivial phase, the proper time $\tau_{s}$ is
monotonically decreasing when we increase $M/b$.

Figure 7: Plots of the proper time $\tau_{s}$ from the horizon to the
singularity as a function of $M/b$ at different temperatures $T/b=0.05$ (red),
$0.02$ (blue), $0.01$ (purple).

##  4 Conclusion and discussion

We have studied the interior geometries of black holes in two different
holographic topological semimetals. We find that the singularities of the
geometries are of simple Kasner form, together with a constant one form gauge
potential or constant two form fields. In the topological WSM phase, all the
Kasner exponents are constant taking values of Schwarzschild black hole at low
temperature. In the topological NLSM phase, the Kasner exponents of the metric
fields are also almost constant (the difference is of order less than $1\%$ at
$T/b=0.01$), while the Kasner exponent of the scalar field is small and
changes a bit in the topological phase. Moreover, we find the proper times
from the horizon to the singularity are nearly constant in both holographic
WSM and holographic NLSM. These features seem to be of topological in the
sense that they stay as constant during the changes of physical parameters of
the systems. The proper time in the trivial phases of the two holographic
semimetal decreases when we increase $M/b$.

In addition to the above universal behavior, specific behaviors inside the
horizon are also found. In the topological phase of holographic WSM, we find
the oscillations of the matter field $\phi$ inside the horizon at low
temperature. In other phases we have not found any oscillations of fields. The
Kasner exponents oscillate in the critical regime of holographic WSM. There is
no oscillation of background fields in holographic NLSM. In the trivial phases
of the two holographic semimetals, the Kasner exponents behave differently,
where the details can be found in Fig. 3 and Fig. 5.

It would be interesting to connect the topological features of Kasner
exponents and the proper times in the topological phases of the two
holographic semimetals to the topological invariants. It is known that they
can be extracted from the correlators of heavy operators. It is very
interesting to determine the precise observables associated to these
quantities to understand the role played by topology. This would shed light on
the universal theories describing the topological semimetals. Meanwhile, it is
also interesting to check the behavior of these physical quantities in the
topological phases of other holographic topological semimetals, e.g. [18, 25],
to check if they are universal feature of topological semimetals.

### Acknowledgments

We are grateful to Matteo Baggioli, Karl Landsteiner, Ya-Wen Sun, Xin-Meng Wu,
Jun-Kun Zhao for useful discussions. This work is supported by the National
Natural Science Foundation of China grant No.11875083.

##  Appendix A Equations in holographic WSM

In this appendix we list the useful equations for calculating the geometries
in holographic WSM in section 2.

The equations of motion for the action (2.1) are

$\displaystyle\begin{split}R_{ab}-\frac{1}{2}g_{ab}(R+12)-T_{ab}&=0\,,\\\
\nabla_{b}F^{ba}+\alpha\epsilon^{abcde}(F_{bc}F_{de}+\mathcal{F}_{bc}\mathcal{F}_{de})-iq\left(\Phi^{*}(D_{a}\Phi)-\Phi(D_{a}\Phi)^{*}\right)&=0\,,\\\
\nabla_{b}\mathcal{F}^{ba}+2\alpha\epsilon^{abcde}F_{bc}\mathcal{F}_{de}&=0\,,\\\
D_{a}D^{a}\Phi-m^{2}\Phi-\lambda\Phi^{*}\Phi^{2}&=0\,,\end{split}$ (A.1)

where

$\displaystyle T_{ab}=$
$\displaystyle\frac{1}{2}(\mathcal{F}_{ac}\mathcal{F}_{b}^{~{}c}-\frac{1}{4}g_{ab}\mathcal{F}^{2})+\frac{1}{2}(F_{ac}F_{b}^{~{}c}-\frac{1}{4}g_{ab}F^{2})+\frac{1}{2}((D_{a}\Phi)^{*}D_{b}\Phi+(D_{b}\Phi)^{*}D_{a}\Phi)$
$\displaystyle-\frac{1}{2}g_{ab}((D_{c}\Phi)^{*}D^{c}\Phi+V(\Phi))\,$ (A.2)

and $D_{a}\Phi=\partial_{a}\Phi-iqA_{a}\Phi$.

There are three different scaling symmetries of the system

$\displaystyle(x,y)\rightarrow a(x,y)\,,f\rightarrow a^{-2}f\,;$ (A.3)
$\displaystyle z\rightarrow az\,,h\rightarrow a^{-2}h\,,A_{z}\rightarrow
a^{-1}A_{z}\,;$ (A.4) $\displaystyle r\rightarrow ar\,,(t,x,y,z)\rightarrow
a^{-1}(t,x,y,z)\,,(u,f,h)\rightarrow a^{2}(u,f,h)\,,A_{z}\rightarrow
aA_{z}\,.$ (A.5)

For the ansatz (2.2), we have equations

$\displaystyle\begin{split}u^{\prime\prime}+\frac{h^{\prime}}{2h}u^{\prime}-\Big{(}f^{\prime\prime}+\frac{f^{\prime}h^{\prime}}{2h}\Big{)}\frac{u}{f}&=0\,,\\\
\frac{f^{\prime\prime}}{f}+\frac{u^{\prime\prime}}{2u}-\frac{f^{\prime
2}}{4f^{2}}+\frac{f^{\prime}u^{\prime}}{fu}-\frac{6}{u}+\frac{\phi^{2}}{2u}\Big{(}m^{2}+\frac{\lambda}{2}\phi^{2}-\frac{q^{2}A_{z}^{2}}{h}\Big{)}-\frac{A_{z}^{\prime
2}}{4h}+\frac{1}{2}\phi^{\prime 2}&=0\,,\\\ \frac{1}{2}\phi^{\prime
2}+\frac{6}{u}-\frac{u^{\prime}}{2u}\Big{(}\frac{f^{\prime}}{f}+\frac{h^{\prime}}{2h}\Big{)}-\frac{f^{\prime}h^{\prime}}{2fh}-\frac{f^{\prime
2}}{4f^{2}}+\frac{A_{z}^{\prime
2}}{4h}-\frac{\phi^{2}}{2u}\Big{(}m^{2}+\frac{\lambda}{2}\phi^{2}-\frac{q^{2}A_{z}^{2}}{h}\Big{)}&=0\,,\\\
A_{z}^{\prime\prime}+\Big{(}\frac{f^{\prime}}{f}-\frac{h^{\prime}}{2h}+\frac{u^{\prime}}{u}\Big{)}A_{z}^{\prime}-\frac{2q^{2}\phi^{2}}{u}A_{z}&=0\,,\\\
\phi^{\prime\prime}+\Big{(}\frac{f^{\prime}}{f}+\frac{h^{\prime}}{2h}+\frac{u^{\prime}}{u}\Big{)}\phi^{\prime}-\frac{1}{u}\,\Big{(}\frac{q^{2}A_{z}^{2}}{h}+m^{2}+\lambda\phi^{2}\Big{)}\phi&=0\,.\end{split}$
(A.6)

Near the horizon $r=r_{h}$, the fields can be expanded as follows,

$\displaystyle\begin{split}u&=4\pi T(r-r_{h})+\cdots\,,\\\
f&=f_{1}-f_{1}A_{z2}\frac{2m^{2}\phi_{1}^{2}+\lambda\phi_{1}^{4}-24}{6A_{z1}q^{2}\phi_{1}^{2}}(r-r_{h})+\cdots\,,\\\
h&=h_{1}-\left(A_{z1}A_{z2}+h_{1}A_{z2}\frac{2m^{2}\phi_{1}^{2}+\lambda\phi_{1}^{4}-24}{6A_{z1}q^{2}\phi_{1}^{2}}\right)(r-r_{h})+\cdots\,,\\\
A_{z}&=A_{z1}+A_{z2}(r-r_{h})+\cdots\,,\\\
\phi&=\phi_{1}+A_{z2}\frac{A_{z1}^{2}q^{2}+h_{1}(m^{2}+\lambda\phi_{1}^{2})}{2A_{z1}h_{1}q^{2}\phi_{1}^{2}}(r-r_{h})+\cdots\,,\end{split}$
(A.7)

where $T=\frac{\phi_{1}^{2}q^{2}A_{z1}}{2\pi A_{z2}}$. Note that there is a
shift symmetry $r\to r+\alpha$ along the radial direction which can be used to
fix $r_{h}$ to be any value and we choose $r_{h}=1$. There are five free
parameters $T,f_{1},h_{1},A_{z1},\phi_{1}$ and we can use the scaling
symmetries (A.3, A.4) to fix $f_{1}=1,h_{1}=1$ respectively. Then we can shoot
three parameters $T,A_{z1},\phi_{1}$ to obtain the parameters $T,M,b$ of
boundary field theory, i.e. the two dimensionless parameters $T/b,M/b$
according the scaling symmetry in (A.5) (we work in unit $b=1$).

When $r\to\infty$, the UV expansions are

$\displaystyle\begin{split}u&=r^{2}-\frac{M^{2}}{3}+\frac{M^{4}(2+3\lambda)}{18}\frac{\ln
r}{r^{2}}-\frac{M_{b}}{r^{2}}+\cdots\,,\\\
f&=r^{2}-\frac{M^{2}}{3}+\frac{M^{4}(2+3\lambda)}{18}\frac{\ln
r}{r^{2}}+\frac{f_{3}}{r^{2}}+\cdots\,,\\\
h&=r^{2}-\frac{M^{2}}{3}+\frac{M^{4}(2+3\lambda)+9b^{2}M^{2}q^{2}}{18}\frac{\ln
r}{r^{2}}+\frac{h_{3}}{r^{2}}+\cdots\,,\\\ A_{z}&=b-bM^{2}q^{2}\frac{\ln
r}{r^{2}}+\frac{\eta}{r^{2}}+\cdots\,,\\\
\phi&=\frac{M}{r}-\frac{(3b^{2}Mq^{2}+2M^{3}+3\lambda M^{3}))}{6}\frac{\ln
r}{r^{3}}+\frac{O}{r^{3}}+\cdots\,,\end{split}$ (A.8)

where $h_{3}=\frac{1}{72}M(-72O+9b^{2}Mq^{2}+M^{3}(14+9\lambda))-2f_{3}$.

Note that in order to match the expansion (A.8) we should use the shift
symmetry of the system $r\to r+\alpha$ which could change the location of the
horizon/singularity.

##  Appendix B Equations in holographic NLSM

In this appendix, we list the calculations for the geometries in holographic
NLSM in section 3.

The equations of motion for the action (3.1) are

$\displaystyle\begin{split}R_{ab}-\frac{1}{2}g_{ab}(R+12)-T_{ab}&=0\,,\\\
\nabla_{b}\mathcal{F}^{ba}+2\alpha\epsilon^{abcde}F_{bc}\mathcal{F}_{de}&=0\,,\\\
\nabla_{b}F^{ba}+\alpha\epsilon^{abcde}(F_{bc}F_{de}+\mathcal{F}_{bc}\mathcal{F}_{de})-iq_{1}\left(\Phi^{*}D^{a}\Phi-(D^{a}\Phi)^{*}\Phi\right)+\frac{q_{2}}{\eta}\epsilon^{abcde}B_{bc}B^{*}_{de}&=0\,,\\\
D_{a}D^{a}\Phi-\partial_{\Phi^{*}}V_{1}-\lambda\Phi B_{ab}^{*}B^{ab}&=0\,,\\\
\frac{i}{3\eta}\epsilon_{abcde}H^{cde}-m_{2}^{2}B_{ab}-\lambda\Phi^{*}\Phi
B_{ab}&=0\,,\end{split}$ (B.1)

where

$\displaystyle T_{ab}$
$\displaystyle=\frac{1}{2}(\mathcal{F}_{ac}\mathcal{F}_{b}^{~{}c}-\frac{1}{4}g_{ab}\mathcal{F}^{2})+\frac{1}{2}(F_{ac}F_{b}^{~{}c}-\frac{1}{4}g_{ab}F^{2})+\frac{1}{2}\big{(}(D_{a}\Phi)^{*}D_{b}\Phi+(D_{b}\Phi)^{*}D_{a}\Phi\big{)}$
$\displaystyle+(m_{2}^{2}+\lambda|\Phi|^{2})(B^{*}_{ac}B_{b}^{~{}c}+B^{*}_{bc}B_{a}^{~{}c})-\frac{1}{2}g_{ab}\big{(}(D_{c}\Phi)^{*}D^{c}\Phi+V_{1}+V_{2}+\lambda|\Phi|^{2}B^{*}_{cd}B^{cd}\big{)}\,.$
(B.2)

With the ansatz (3.4), the equations are

$\displaystyle\begin{split}\frac{u^{\prime\prime}}{u}-\frac{f^{\prime\prime}}{f}+\frac{h^{\prime}}{2h}\left(\frac{u^{\prime}}{u}-\frac{f^{\prime}}{f}\right)-\frac{4}{u}(m_{2}^{2}+\lambda\phi^{2})\left(\frac{\mathcal{B}_{tz}^{2}}{uh}+\frac{\mathcal{B}_{xy}^{2}}{f^{2}}\right)&=0\,,\\\
\frac{u^{\prime\prime}}{2u}+\frac{f^{\prime\prime}}{f}-\frac{f^{\prime
2}}{4f^{2}}+\frac{f^{\prime}u^{\prime}}{fu}-\frac{6}{u}+\frac{1}{u}(m_{2}^{2}+\lambda\phi^{2})\left(\frac{\mathcal{B}_{tz}^{2}}{uh}+\frac{\mathcal{B}_{xy}^{2}}{f^{2}}\right)~{}~{}~{}~{}&\,\\\
+\frac{\phi^{2}}{2u}\left(m_{1}^{2}+\frac{\lambda_{1}\phi^{2}}{2}\right)+\frac{\phi^{\prime
2}}{2}&=0\,,\\\ \frac{f^{\prime
2}}{4f^{2}}+\frac{f^{\prime}h^{\prime}}{2fh}+\frac{u^{\prime}}{2u}\left(\frac{f^{\prime}}{f}+\frac{h^{\prime}}{2h}\right)-\frac{6}{u}+\frac{1}{u}(m_{2}^{2}+\lambda\phi^{2})\left(-\frac{\mathcal{B}_{tz}^{2}}{uh}+\frac{\mathcal{B}_{xy}^{2}}{f^{2}}\right)~{}~{}~{}~{}&\,\\\
+\frac{\phi^{2}}{2u}\left(m_{1}^{2}+\frac{\lambda_{1}\phi^{2}}{2}\right)-\frac{1}{2}\phi^{\prime
2}&=0\,,\\\
\mathcal{B}_{tz}^{\prime}-\frac{\eta\sqrt{h}}{2f}(m_{2}^{2}+\lambda\phi^{2})\mathcal{B}_{xy}&=0\,,\\\
\mathcal{B}_{xy}^{\prime}-\frac{\eta
f}{2\sqrt{h}u}(m_{2}^{2}+\lambda\phi^{2})\mathcal{B}_{tz}&=0\,,\\\
\phi^{\prime\prime}+\phi^{\prime}\left(\frac{u^{\prime}}{u}+\frac{f^{\prime}}{f}+\frac{h^{\prime}}{2h}\right)-\left(m_{1}^{2}+\lambda_{1}\phi^{2}-\frac{2\lambda\mathcal{B}_{tz}^{2}}{uh}+\frac{2\lambda\mathcal{B}_{xy}^{2}}{f^{2}}\right)\frac{\phi}{u}&=0\,.\end{split}$
(B.3)

There are three different scaling symmetries of the system

$\displaystyle(x,y)\rightarrow a(x,y)\,,f\rightarrow
a^{-2}f\,,\mathcal{B}_{xy}\rightarrow a^{-2}\mathcal{B}_{xy}\,;$ (B.4)
$\displaystyle z\rightarrow az\,,h\rightarrow
a^{-2}h\,,\mathcal{B}_{tz}\rightarrow a^{-1}\mathcal{B}_{tz}\,;$ (B.5)
$\displaystyle r\rightarrow ar\,,(t,x,y,z)\rightarrow
a^{-1}(t,x,y,z)\,,(u,f,h,\mathcal{B}_{xy},\mathcal{B}_{tz})\rightarrow
a^{2}(u,f,h,\mathcal{B}_{xy},\mathcal{B}_{tz})\,.$ (B.6)

Near the horizon $r\to r_{h}$, the fields can be expanded as follows,

$\displaystyle\begin{split}u&=4\pi T(r-r_{h})+\cdots\,,\\\
f&=f_{1}-\frac{4\mathcal{B}_{xy2}\left(8\mathcal{B}_{xy1}^{2}(m_{2}^{2}+\lambda\phi_{1}^{2})+f_{1}^{2}(2m_{1}^{2}\phi_{1}^{2}+\lambda_{1}\phi_{1}^{4}-24)\right)}{3\mathcal{B}_{xy1}f_{1}\eta^{2}(m_{2}^{2}+\lambda\phi_{1}^{2})^{2}}(r-r_{h})+\cdots\,,\\\
h&=h_{1}-\frac{4h_{1}\mathcal{B}_{xy2}\left(4\mathcal{B}_{xy1}^{2}(m_{2}^{2}+\lambda\phi_{1}^{2})-f_{1}^{2}(2m_{1}^{2}\phi_{1}^{2}+\lambda_{1}\phi_{1}^{4}-24)\right)}{3\mathcal{B}_{xy1}f_{1}\eta^{2}(m_{2}^{2}+\lambda\phi_{1}^{2})^{2}}(r-r_{h})+\cdots\,,\\\
\mathcal{B}_{xy}&=\mathcal{B}_{xy1}+\mathcal{B}_{xy2}(r-r_{h})+\cdots\,,\\\
\mathcal{B}_{tz}&=\frac{\eta\sqrt{h_{1}}\mathcal{B}_{xy1}(m_{2}^{2}+\lambda\phi_{1}^{2})}{2f_{1}}(r-r_{h})+\cdots\,,\\\
\phi&=\phi_{1}+\frac{4\mathcal{B}_{xy2}\phi_{1}\left(2\lambda\mathcal{B}_{xy1}^{2}+f_{1}^{2}(m_{1}^{2}+\lambda_{1}\phi_{1}^{2})\right)}{\mathcal{B}_{xy1}f_{1}^{2}\eta^{2}(m_{2}^{2}+\lambda\phi_{1}^{2})^{2}}(r-r_{h})+\cdots\,,\end{split}$
(B.7)

where
$T=\frac{\mathcal{B}_{xy1}\eta^{2}(m_{2}^{2}+\lambda\phi_{1}^{2})^{2}}{16\pi\mathcal{B}_{xy2}}$.
The strategy of the numerics the same as the holographic WSM. We first use the
shift symmetry $r\to r+\alpha$ to fix $r_{h}=1$. Then we also have five free
parameters $T,f_{1},h_{1},\mathcal{B}_{xy1},\phi_{1}$ and we can use the
scaling symmetries (B.4, B.5) to fix $f_{1}=1,h_{1}=1$ respectively. After
that we have only three near horizon parameters
$T,\mathcal{B}_{xy1},\phi_{1}$, from which we obtain $T,M,b$ in the dual field
theory, which are equivalently two dimensionless parameters $T/b,M/b$
according the scaling symmetry (B.6).

Near the boundary $r\to\infty$, we have

$\displaystyle\begin{split}u&=r^{2}-2b^{2}-\frac{M^{2}}{3}+\frac{8b^{4}+M^{4}(2+3\lambda_{1})}{18}\frac{\ln
r}{r^{2}}-\frac{M_{b}}{r^{2}}+\cdots\,,\\\
f&=r^{2}-\frac{M^{2}}{3}+\frac{8b^{4}+M^{4}(2+3\lambda_{1})}{18}\frac{\ln
r}{r^{2}}+\frac{f_{3}}{r^{2}}+\cdots\,,\\\
h&=r^{2}-2b^{2}-\frac{M^{2}}{3}+\frac{8b^{4}+M^{4}(2+3\lambda_{1})}{18}\frac{\ln
r}{r^{2}}+\frac{h_{3}}{r^{2}}+\cdots\,,\\\
\mathcal{B}_{xy}&=br+\frac{2b^{3}\ln r}{r}+\frac{b_{2}}{r}+\cdots\,,\\\
\mathcal{B}_{tz}&=br-\frac{2b^{3}\ln
r}{r}-\frac{b\left(b^{2}+M^{2}(1+\lambda)\right)+b_{2}}{r}+\cdots\,,\\\
\phi&=\frac{M}{r}-\frac{M^{3}(2+3\lambda_{1})}{6}\frac{\ln
r}{r^{3}}+\frac{O}{r^{3}}+\cdots\,,\end{split}$ (B.8)

where
$b_{2}=\frac{1}{48b}\left(-56b^{4}+72(2f_{3}+h_{3})-8b^{2}M^{2}(2+3\lambda)-M^{4}(14+9\lambda_{1})+72MO\right)$.

Note that to match the expansion (B.8) we should use the shift symmetry $r\to
r+\alpha$ which could change the location of the horizon/singularity .

## References

  * [1] J. McGreevy, Generalized Symmetries in Condensed Matter, [arXiv:2204.03045].
  * [2] X. -G. Wen, Zoo of quantum-topological phases of matter, Rev. Mod. Phys. 89, 41004 (2017) [arXiv:1610.03911].
  * [3] K. Landsteiner and Y. Liu, The holographic Weyl semi-metal, Phys. Lett. B 753 (2016), 453-457 [arXiv:1505.04772].
  * [4] K. Landsteiner, Y. Liu and Y. W. Sun, Quantum phase transition between a topological and a trivial semimetal from holography, Phys. Rev. Lett. 116 (2016) no.8, 081602 [arXiv:1511.05505].
  * [5] Y. Liu and Y. W. Sun, Topological nodal line semimetals in holography, JHEP 12 (2018), 072 [arXiv:1801.09357].
  * [6] Y. Liu and X. M. Wu, An improved holographic nodal line semimetal, JHEP 05 (2021), 141 [arXiv:2012.12602].
  * [7] Y. Liu and Y. W. Sun, Topological invariants for holographic semimetals, JHEP 10 (2018), 189 [arXiv:1809.00513].
  * [8] M. Ammon, M. Heinrich, A. Jiménez-Alba and S. Moeckel, Surface States in Holographic Weyl Semimetals, Phys. Rev. Lett. 118 (2017) no.20, 201601 [arXiv:1612.00836].
  * [9] K. Landsteiner, Y. Liu and Y. W. Sun, Holographic topological semimetals, Sci. China Phys. Mech. Astron. 63 (2020) no.5, 250001 [arXiv:1911.07978].
  * [10] K. Landsteiner, Y. Liu and Y. W. Sun, Odd viscosity in the quantum critical region of a holographic Weyl semimetal, Phys. Rev. Lett. 117 (2016) no.8, 081604 [arXiv:1604.01346].
  * [11] C. Copetti, J. Fernández-Pendás and K. Landsteiner, Axial Hall effect and universality of holographic Weyl semi-metals, JHEP 02 (2017), 138 [arXiv:1611.08125].
  * [12] G. Grignani, A. Marini, F. Pena-Benitez and S. Speziali, AC conductivity for a holographic Weyl Semimetal, JHEP 03, 125 (2017) [arXiv:1612.00486].
  * [13] M. Ammon, M. Baggioli, A. Jiménez-Alba and S. Moeckel, A smeared quantum phase transition in disordered holography, JHEP 04, 068 (2018) [arXiv:1802.08650].
  * [14] M. Baggioli, B. Padhi, P. W. Phillips and C. Setty, Conjecture on the Butterfly Velocity across a Quantum Phase Transition, JHEP 07, 049 (2018) [arXiv:1805.01470].
  * [15] Y. Liu and J. Zhao, Weyl semimetal/insulator transition from holography, JHEP 12, 124 (2018) [arXiv:1809.08601].
  * [16] X. Ji, Y. Liu and X. M. Wu, Chiral vortical conductivity across a topological phase transition from holography, Phys. Rev. D 100, no.12, 126013 (2019) [arXiv:1904.08058].
  * [17] G. Song, J. Rong and S. J. Sin, Stability of topology in interacting Weyl semi-metal and topological dipole in holography, JHEP 10, 109 (2019) [arXiv:1904.09349].
  * [18] V. Juričić, I. Salazar Landea and R. Soto-Garrido, Phase transitions in a holographic multi-Weyl semimetal, JHEP 07, 052 (2020) [arXiv:2005.10387].
  * [19] M. Baggioli and D. Giataganas, Detecting Topological Quantum Phase Transitions via the c-Function, [arXiv:2007.07273].
  * [20] B. Kiczek, M. Rogatko and K. I. Wysokiński, Anomalous Hall conductivity of the holographic $\mathbb{Z}_{2}$ Dirac semimetals, [arXiv:2010.13095].
  * [21] K. Bitaghsir Fadafan, A. O’Bannon, R. Rodgers and M. Russell, A Weyl semimetal from AdS/CFT with flavour, JHEP 04 (2021), 162 [arXiv:2012.11434].
  * [22] J. Zhao, Momentum relaxation in a holographic Weyl semimetal, Phys. Rev. D 104, no.6, 066003 (2021).
  * [23] N. Grandi, V. Juričić, I. Salazar Landea and R. Soto-Garrido, Towards holographic flat bands, JHEP 05 (2021), 123 [arXiv:2103.01690].
  * [24] R. Rodgers, E. Mauri, U. Gürsoy and H. T. C. Stoof, Thermodynamics and transport of holographic nodal line semimetals, JHEP 11 (2021), 191 [arXiv:2109.07187].
  * [25] X. Ji, Y. Liu, Y. W. Sun and Y. L. Zhang, A Weyl-Z2 semimetal from holography, JHEP 12 (2021), 066 [arXiv:2109.05993].
  * [26] J. Zhao, Momentum relaxation of holographic Weyl semimetal from massive gravity, Eur. Phys. J. C 82 (2022) no.4, 300 [arXiv:2111.14068].
  * [27] N. Grandi, V. Juričić, I. S. Landea and R. Soto-Garrido, Engineering holographic flat fermionic bands, Phys. Rev. D 105 (2022) no.8, L081902 [arXiv:2112.12198].
  * [28] L. Fidkowski, V. Hubeny, M. Kleban and S. Shenker, The Black hole singularity in AdS / CFT, JHEP 02, 014 (2004) [arXiv:hep-th/0306170].
  * [29] G. Festuccia and H. Liu, Excursions beyond the horizon: Black hole singularities in Yang-Mills theories. I., JHEP 04 (2006), 044 [arXiv:hep-th/0506202].
  * [30] M. Grinberg and J. Maldacena, Proper time to the black hole singularity from thermal one-point functions, JHEP 03 (2021), 131 [arXiv:2011.01004].
  * [31] Y. Liu, H. D. Lyu and A. Raju, Black hole singularities across phase transitions, JHEP 10 (2021), 140 [arXiv:2108.04554].
  * [32] A. Frenkel, S. A. Hartnoll, J. Kruthoff and Z. D. Shi, Holographic flows from CFT to the Kasner universe, JHEP 08 (2020), 003 [arXiv:2004.01192].
  * [33] S. A. Hartnoll, G. T. Horowitz, J. Kruthoff and J. E. Santos, Gravitational duals to the grand canonical ensemble abhor Cauchy horizons, JHEP 10 (2020), 102 [arXiv:2006.10056].
  * [34] S. A. Hartnoll, G. T. Horowitz, J. Kruthoff and J. E. Santos, Diving into a holographic superconductor, SciPost Phys. 10 (2021) no.1, 009 [arXiv:2008.12786].
  * [35] Y. Liu and H. D. Lyu, Interior of helical black holes, JHEP 09 (2022), 071 [arXiv:2205.14803].
  * [36] R. G. Cai, L. Li and R. Q. Yang, No Inner-Horizon Theorem for Black Holes with Charged Scalar Hair, [arXiv:2009.05520].
  * [37] Y. Q. Wang, Y. Song, Q. Xiang, S. W. Wei, T. Zhu and Y. X. Liu, Holographic flows with scalar self-interaction toward the Kasner universe, [arXiv:2009.06277].
  * [38] Y. S. An, L. Li and F. G. Yang, No Cauchy horizon theorem for nonlinear electrodynamics black holes with charged scalar hairs, Phys. Rev. D 104 (2021) no.2, 024040 [arXiv:2106.01069].
  * [39] N. Grandi and I. Salazar Landea, Diving inside a hairy black hole, JHEP 05 (2021), 152 [arXiv:2102.02707].
  * [40] S. A. H. Mansoori, L. Li, M. Rafiee and M. Baggioli, What’s inside a hairy black hole in massive gravity? [arXiv:2108.01471].
  * [41] O. J. C. Dias, G. T. Horowitz and J. E. Santos, Inside an asymptotically flat hairy black hole, JHEP 12, 179 (2021) [arXiv:2110.06225].
  * [42] L. Sword and D. Vegh, Kasner geometries inside holographic superconductors, [arXiv:2112.14177].
  * [43] R. G. Cai, C. Ge, L. Li and R. Q. Yang, Inside anisotropic black hole with vector hair, JHEP 02 (2022), 139 [arXiv:2112.04206].
  * [44] M. Henneaux, The final Kasner regime inside black holes with scalar or vector hair, [arXiv:2202.04155].
  * [45] E. Caceres, A. Kundu, A. K. Patra and S. Shashi, Trans-IR Flows to Black Hole Singularities, [arXiv:2201.06579].
  * [46] A. Bhattacharya, A. Bhattacharyya, P. Nandy and A. K. Patra, Bath deformations, islands, and holographic complexity, Phys. Rev. D 105 (2022) no.6, 066019 [arXiv:2112.06967].
  * [47] Y. S. An, L. Li, F. G. Yang and R. Q. Yang, Interior Structure and Complexity Growth Rate of Holographic Superconductor from M-Theory, [arXiv:2205.02442].
  * [48] R. Auzzi, S. Bolognesi, E. Rabinovici, F. I. S. Massolo and G. Tallarita, On the time dependence of holographic complexity for charged AdS black holes with scalar hair, [arXiv:2205.03365].
  * [49] M. Mirjalali, S. A. Hosseini Mansoori, L. Shahkarami and M. Rafiee, Probing inside a charged hairy black hole in massive gravity, JHEP 09 (2022), 222 [arXiv:2206.02128].
  * [50] S. A. Hartnoll, Wheeler-DeWitt states of the AdS-Schwarzschild interior, [arXiv:2208.04348].
  * [51] E. Caceres and S. Shashi, Anisotropic Flows into Black Holes, [arXiv:2209.06818].
  * [52] S. A. Hartnoll and N. Neogi, AdS Black Holes with a Bouncing Interior, [arXiv:2209.12999].
  * [53] L. Sword and D. Vegh, What lies beyond the horizon of a holographic p-wave superconductor, JHEP 12 (2022), 045 [arXiv:2210.01046].


# Extremely low-energy collective modes in a quasi-one-dimensional system

Z. X. Wei  State Key Laboratory of Electronic Thin Films and Integrated
Devices, University of Electronic Science and Technology of China, Chengdu
610054, China  S. Zhuang  Beijing National Laboratory for Condensed Matter
Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190,
China  Y. L. Su  State Key Laboratory of Electronic Thin Films and Integrated
Devices, University of Electronic Science and Technology of China, Chengdu
610054, China  L. Cheng  State Key Laboratory of Electronic Thin Films and
Integrated Devices, University of Electronic Science and Technology of China,
Chengdu 610054, China  H. D. Zhou  Department of Physics and Astronomy,
University of Tennessee, Knoxville, Tennessee 37996, USA  Z. Jiang  School of
Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA  H. Weng
Beijing National Laboratory for Condensed Matter Physics, Institute of
Physics, Chinese Academy of Sciences, Beijing 100190, China  J. Qi
<EMAIL_ADDRESS>State Key Laboratory of Electronic Thin Films and Integrated
Devices, University of Electronic Science and Technology of China, Chengdu
610054, China

###### Abstract

We have investigated the quasiparticle dynamics and collective excitations in
the quasi-one-dimensional material ZrTe5 using ultrafast optical pump-probe
spectroscopy. Our time-domain results reveal two coherent oscillations having
extremely low energies of $\hbar\omega_{1}\sim$0.33 meV (0.08 THz) and
$\hbar\omega_{2}\sim$1.9 meV (0.45 THz), which are softened as the temperature
approaches two different critical temperatures ($\sim$54 K and $\sim$135 K).
We attribute these two collective excitations to the amplitude mode of charge
density wave instabilities in ZrTe5 with tremendously small nesting wave
vectors. Furthermore, scattering with the $\hbar\omega_{2}$ mode may result in
a peculiar quasiparticle decay process with a timescale of $\sim$1-2 ps below
the transition temperature $T^{*}$ ($\sim$135 K). Our findings provide pivotal
information for studying the fluctuating order parameters and their associated
quasiparticle dynamics in various low-dimensional topological systems and
other materials.

In a many-body system, interaction between the quasiparticles, and/or
interaction between the quasparticles and other quantized modes may bring
about various broken symmetry ground states G book 1994 , e.g. superconducting
state, density wave state, and magnetically ordered state, accompanied
simultaneously by abundant collective excitations, typical examples of which
are phonons, density waves (charge or spin) and magnons obeying the Bose-
Einstein statistics. Delicate balance among different interactions and phases
might cause the fluctuating characteristics in some order parameters or
collective excitations with a finite correlation time $\tau_{F}$ G book 1994 ;
Kivelson_RMP_2003 .

ZrTe5 is a quasi-one-dimensional material with highly anisotropic crystal
lattice and electronic structures Weng_PRX_2014 . Their subtle correlations
Weng_PRX_2014 ; J sciadv 2019 ; C PRX 2020  make this material enormously
interesting. On the one hand, ZrTe5 is associated with the topological phases
of matter, i.e. the topological insulator (TI) G PRL 2016 ; R PRX 2016 ;
Weng_PRX_2014 ; Y PRB 2018 ; H PRB 2017 ; R PRX 2016  or the Dirac semimetal
RY PRB 2015 ; Y NC 2016 ; Wang 2018 , which have been experimentally
investigated by different equilibrium or quasi-equilibrium techniques, e.g.
electronic transport Niu PRB2017 ; Terry PRB 1999 , infrared optical
spectroscopy RY PRB 2015 ; ZG PNAS 2017 ; Jiang PRL2020 ; Jiang PRB 2017 ; B
PRL 2018 , scanning tunneling microscope Li PRL 2016 ; R PRX 2016 , and angle-
resolved photoemission spectroscopy (ARPES) R PRX 2016 ; H PRB 2017 . On the
other hand, many unusual electronic responses such as the chiral magnetic
effect Li_NPhy_2016 , anomalous Hall effect Liang NP 2018 , and 3D quantum
Hall effect Tang Nature 2019  have been reported.

It is believed that those exotic phenomena are connected to the peculiar
fermionic and bosonic excitations in ZrTe5 with the anomalous phase transition
around temperature $T^{*}$, where the resistivity presents a peak FJ PRB 1981
. In particular, the static charge density wave (CDW) stabilized by the
external magnetic field was employed to interpret the 3D quantum Hall effect
at temperatures below 20 K Tang Nature 2019 ; Qin_PRL_2020 , while the
collective Dirac polarons were proposed to explain the sharp resistivity peak
around $T^{*}$ Fu_PRL_2020 . However, the fluctuating nature of any unique
collective excitation(s) in ZrTe5 has never been observed or elucidated,
although the related dynamics could be critical to understand the emergent
phenomena in this material previously observed via the quasi-equilibrium
probing techniques.

Figure 1:  (a) Typical $\Delta R(t)/R$ data as a function of temperature at a
pump fluence of $\sim$0.5 $\mu$J/cm2. Inset: $\Delta R(t)/R$ in short
timescale. The red line is the pump-probe cross correlation profile. (b) and
(c) Typical fitted curves (red solid lines) for $\Delta R(t)/R$ data using Eq.
(1) at 5 K and 200 K, respectively. The blue, green and purple lines are for
components with decay times $\tau_{1}$, $\tau_{2}$, and $\tau_{s}$,
respectively.

By contrast, the ultrafast optical pump-probe technique is a powerful tool
that not only can reveal the quasiparticle dynamics associated with the
topological properties Gedik PRL2011 ; Wang PRL 2016  but also is capable of
unveiling the collective phenomena in a broad correlation time- or length
scale Yusupov_PRL_2008 ; Torchinsky_NM_2013 ; Liu PRL 2020 . Here, using
ultrafast optical pump-probe spectroscopy our measurements for the first time
reveal two extremely low-energy collective amplitude mode oscillations
($\sim$0.33 meV and 1.9 meV) at low temperatures in the high-quality ZrTe5
single crystals note1 , which have a transition temperature $T^{*}$ of
$\sim$135 K. We demonstrate these two modes arising from the fluctuating CDW
triggered by the acoustic phonons. We also observe a novel quasiparticle
dynamics associated with this type of bosonic mode emerging below $T^{*}$ with
a ps-timescale.

All measurements were performed using an ultrafast laser with a repetition
rate of 80 MHz, a pulse width of $\sim$55 fs (FWHM), and a center wavelength
tunable between 780 nm and 820 nm. Detailed information on experimental setup,
sample preparation and additional data can also be found in the Supplemental
Material Suppl  and Refs. Wang PRL 2016 ; Qi PRL 2013 ; Liu PRL 2020 ; Jiang
PRB 2017 ; Jiang PRL2020 ; W J 2016 . Figure 1 (a) shows the typical measured
signals at low and high temperatures. Upon photoexcitation, all the $\Delta
R(t)/R$ signals clearly exhibit an instantaneous rise, succeeded by lateral
relaxation processes. Besides, rich damped ultrafast oscillations are
superimposed on the non-oscillating decay background. These coherent
oscillations can verify the high quality of our samples, as will be discussed
in detail later.

We first focus on the non-oscillatory signals. A close inspection of the
rising curve (see the inset to Fig. 1(a)) reveals that it actually comprises
two parts in the time-domain: (1) one exactly overlaps with the pump-probe
cross correlation profile; (2) the other one shows relatively slow rising
before the $\Delta R/R$ signal reaches the maximum value. We identify that the
relative slow-rising can be well described by a sub-ps exponential decay
process having amplitude with a different sign from that of the subsequent
relaxation dynamics O PRL 2013 ; Qi APL 2010  (See also Supplemental Material
for further details Suppl ). However, the succeeding relaxation process does
not exactly follow a single exponential decay at all investigated
temperatures. Rather, an extra decay process can be unambiguously observed
below a critical temperature, which is nearly the same as the transition
temperature $T^{*}\sim$135 K. According to these observations, we can fit our
data using the following formula D PRL 2002 ; Wang PRL 2016 ,

$\Delta
R(t)/R=(A_{1}e^{-\frac{t}{\tau_{1}}}+A_{2}e^{-\frac{t}{\tau_{2}}}+A_{s}e^{-\frac{t}{\tau_{s}}}+C)\otimes
G(t),$ (1)

where $A_{j}$ and $\tau_{j}$ ($j=1,2,s$) are the amplitudes and decay times,
respectively. $A_{1}>0$ and $A_{2},A_{s}\leq 0$. Here, $\tau_{1}$ and
$\tau_{s}$ represent the sub-ps and extra decay processes, respectively.
$\tau_{2}$ is the subsequent decay after $\tau_{1}$ or $\tau_{s}$. $C$ is a
constant, and $G(t)$ is an Gaussian function standing for the pump-probe cross
correlation. As seen in Figs. 1(b) and (c), the fitted curves are in excellent
agreement with the experimental data at low and high temperatures.
Specifically, we found that $\tau_{1}$ is less than $\sim$0.3 ps, $\tau_{2}$
is $\sim$5-18 ps, and $\tau_{s}$ is $\sim$1-2 ps. The decay process
characterized by $\tau_{s}$ emerges below $\sim 135$ K ($T^{*}$), or $A_{s}=0$
for $T\geq T^{*}$. The extracted $\tau_{j}$ ($j=1,2,s$) as a function of
temperature are shown in Fig. 2.

In general, the relaxation processes in conventional nonmagnetic materials
disclosed by $\Delta R(t)/R$ can include the e-e thermalization, the e-ph
scattering, the e-h recombination, and the thermal diffusion processes D Rev
Mod Phys ; J Book 2013 ; R 2002 . The e-e thermalization usually has a
timescale less than $\sim$100 femtoseconds. If this process is faster than the
excitation pulse and out of the time resolution limit, it is often concealed
in the initial rising signal, which then follows the pump-probe cross
correlation profile. This is well consistent with our experiments. On the
other hand, the thermal diffusion process and the e-h recombination across a
direct gap with radiative decay have timescales larger than $\sim$100 ps, and
clearly do not contribute to the three processes characterized by $\tau_{j}$
($j=1,2,s$) shown in Figs. 1 and 2.

Figure 2:  (a) $\tau_{1}$ as a function of temperature. Red line is a fit via
TTM. (b) Schematic of the hot carrier excited by $h\nu$ photon cooling via
e-ph scattering. (c) $\tau_{2}$ as a function of temperature. Red line is a
fit based on the phonon-assisted e-h recombination process. Inset:
$1/\tau_{2}$ as a function of pump fluence at 5 K. (d) Schematic of the
phonon-assisted e-h recombination. (e) $\tau_{s}$ as a function of
temperature. Red line is a fit via TTM. (f) Schematic of the hot carrier
cooling excited by $h\nu$ photon involved with unknown bosonic excitations
emerging below $T^{*}$.

Since $\tau_{1}$ and $\tau_{2}$ decay processes exist at all investigated
temperatures, and experience a continuous change across $T^{*}$, they should
represent a generic dynamical responses of ZrTe5, independent of any physics
underlying the transition around $T^{*}$. In fact, the sub-ps process
characterized by $\tau_{1}$ can be attributed to the photoexcited hot-carrier
cooling via e-ph scattering (see Fig. 2(b)), which in turn leads to the
optical phonon emission. Its $T$-dependence is quite similar with those of
e-ph thermalization times observed in metals, semiconductors and topological
materials Liang APL 2014 ; M PRB 2005 ; R PRB 1995 ; Qi APL 2010 ; J Book 2013
; M Nano Lett 2012 . Such $T$-dependent behavior can be described by the two-
temperature model (TTM) Allen PRL 1987 ; R PRB 1995 . Fig. 2(a) demonstrates
that the fitted results using TTM agree very well with the experimental data.
The fluence-dependent $\tau_{1}$ further confirms the effectiveness of TTM
(see Supplemental Material Suppl ). Employing
$\tau^{-1}=3\hbar\lambda\langle\omega^{2}\rangle(\pi k_{B}T_{e})^{-1}$ Allen
PRL 1987 , we can further obtain the dimensionless e-ph coupling constant
$\lambda$ to be $\sim$0.17 via $\tau_{1}(T)$. During the calculations, we
obtained $\lambda\langle\omega^{2}\rangle\simeq 1.8$ $\times 10^{25}$
Hz${}^{2}\simeq$78.2 meV2, where the second moment of the phonon spectrum
$\langle\omega^{2}\rangle$ is $\sim$6.2$\times 10^{4}$ K2 extracted via the
Debye model.

Figure 3:  (a) Extracted oscillations for three typical temperatures. (b) The
temperature-dependent Fourier transform spectra for the extracted
oscillations. (c) The two low energy modes, $\omega_{1}$ and $\omega_{2}$, as
a function of temperature in the frequency domain. The red dashed curves are
guide lines. (d) At 5 K, wavelength dependence of the two low energy modes,
$\omega_{1}$ and $\omega_{2}$, indicated by the red dashed lines.

The $\tau_{2}$ decay process, with a timescale of $\sim$3-18 ps, illustrates a
totally different $T$-dependent behavior as that of the $\tau_{1}$ process
(see Fig. 2(c)). After the e-ph thermalization, the nonequilibrium electrons
(holes) can still accumulate in the conduction (valence) bands. We might thus
consider the phonon-assisted e-h recombination She PRB 2013 ; Dai PRB 2015 ,
where the electron and hole recombine with the assistance of e-ph scattering
between the electron and hole pockets, as illustrated in Fig. 2(d). In fact,
the conduction and valence band extrema for ZrTe5 are in close proximity
within the Brillouin zone Weng_PRX_2014 ; Monserrat B Physical Review Research
2019 , in favor of such interband e-h scatterings. Therefore, we assign the
$\tau_{2}$ decay to this type of recombination process. This inference is not
only consistent with the fluence ($F$) dependence of $\tau_{2}^{-1}$ (inset of
Fig. 2(c)) D PRL 2010 ; Liu PRL 2020 , but also is strongly justified by the
excellent agreement between the experimental $\tau_{2}$ and the theoretical
fitted results based on such recombination Suppl , as shown in the Fig. 2(c).

The peculiar $\tau_{s}$ process emerges below $T^{*}$, implying that the
$\tau_{s}$ decay should be associated with the physics behind the transition.
At first glance, $T$-dependent behavior of $\tau_{s}$ well resembles that of
$\tau_{1}(T)$. So we suspect that this $\tau_{s}$ process should
microscopically resemble the e-ph scattering but involves unknown bosonic
excitations emerging below $T^{*}$. Numerically, we still can use
$\tau^{-1}=3\hbar\lambda\langle\omega^{2}\rangle(\pi k_{B}T_{e})^{-1}$ to fit
$\tau_{s}$ based on the TTM Suppl . However, the fitted parameter
$\lambda\langle\omega^{2}\rangle$ ($\sim$5.47 meV2) only gives an upper limit
because the TTM now evolves into a three temperature model Wang PRB 2018 . At
this stage, although the type of boson associated with $\tau_{s}$ is unknown
(to be identified later), we can tell its energy is not very high and probably
close to scale of the acoustic phonons based on its timescale (1-2 ps) Gedik
PRL2011 .

One may argue whether the $\tau_{s}$ process is associated with the Fermi
level shifting from the valence band to the conduction band as the temperature
decreases across $T^{*}$ Zhang NC 2017 . However, previous ultrafast optical
studies on graphene I PRB 2020  demonstrated that when the Fermi level was
continuously tuned across the Dirac point, no emergence or disappearance of
any quasiparticle decay channel was observed except for change of the e-ph
relaxation time. Therefore, this effect should be irrelevant here.

Figure 4:  $T$-dependence of the (a) $\omega_{1}$ and (b) $\omega_{2}$ modes.
The red lines are fit described in the main text. (c) and (d) are
$T$-dependent damping rates of the amplitudon for $\omega_{1}$ and
$\omega_{2}$ modes, respectively. The red lines are fits using a power law
($\propto T^{\alpha}$).

Now let us focus on the “rich” oscillations observed in $\Delta R(t)/R$. Fig.
3(a) shows the typical time domain signals at temperatures below and above
$T^{*}$. The temperature evolution of the oscillatory signals can be revealed
by the Fourier transform data in the frequency domain (see Fig. 3(b)), where
five modes with distinct frequencies are clearly observed.

In general, the oscillations with THz frequency in pump-probe spectroscopy
arise from the coherent optical phonon modes near the $\Gamma$ point due to
the coherent Raman scattering or displacive excitation H PRB 1992 . The exact
modes can be revealed by a direct comparison with the Raman spectroscopy data
Taguchi I Solid state communications 1983 ; Landa G Solid state communications
1984  (see also the Supplemental Material Suppl ). We thus obtain that three
modes with center frequencies near $\sim$1.2 THz, $\sim$3.7 THz, and $\sim$4.6
THz are attributed to the $A_{g}$ phonon modes. These modes are
characteristics of a high quality ZrTe5 single crystal, first-time observed
simultaneously in the time-resolved measurements. Temperature-dependence of
all these modes can be well described by the anharmonic phonon model Suppl .

Unexpectedly, as shown in Figs. 3(b)-(d), we also clearly observed two low
energy modes with frequencies of $\omega_{1}/2\pi\sim$0.08 THz and
$\omega_{2}/2\pi\sim$0.45 THz, respectively. These two modes cannot be
attributed to the optical phonons since their frequencies are well under the
low frequency limit of the Raman-active optical phonons, i.e. $\sim$1 THz,
obtained by our theoretical calculations Suppl . In specific, the $\omega_{1}$
mode, emerging below $\sim$60 K, exhibits a clear softening as the temperature
increases. The $\omega_{2}$ mode is non-observable above $\sim T^{*}$. One
might first suspect they originate from the low energy acoustic phonons.
However, the estimated phonon frequency and the corresponding probe-wavelength
dependent shift significantly deviate from our observations (see Fig. 3(d) and
Supplemental Material Suppl ). Therefore, we can exclude such a possibility J
PRB 2006 ; D PRB 2005 ; R PRL 2017 .

Since in ZrTe5 the long-range CDW order stabilized using the strong external
magnetic field has been proposed to explain the quantum Hall effect Tang
Nature 2019 ; Qin_PRL_2020 , these two modes, thus, might be attributed to the
CDW instabilities triggered by the low-energy acoustic phonon mode G book 1994
. The dynamical CDW may give rise two collective excitations: (1) the
amplitude mode (amplitudon) exhibits a gap when the wave vector approaches
zero, and (2) the phase mode (phason) is gapless. Because the phason has
similar dispersion relation as that of acoustic phonon, it is expected to bear
the same wavelength dependence R PRL 2017  and thus cannot contribute to these
two low energy modes. We thus finally arrive at the most reasonable candidate
for these two modes, i.e. the amplitudon of CDW.

Particularly, we notice that $T$ dependence of these two modes is quite
similar with the temperature evolution of the fluctuating CDW G book 1994 ; R
PRL 2017 ; Yusupov_PRL_2008 , as seen in Figs. 4(a) and (b). $\omega_{1}$ and
$\omega_{2}$ as a function $T$ can well be fitted by a mean-field-like $T$
dependence Yusupov_PRL_2008 ; G book 1994 :
$\omega_{A}=\omega_{A}(0)(1-T/T_{C})^{\beta}$, where $T_{C}$ is defined as a
transition temperature. Satisfactory fits to the data in Figs. 4(a)-(b) give
note2 : $\beta_{\omega_{1}}\simeq$0.25 and $\beta_{\omega_{2}}\simeq$0.1
corresponding to $T_{C}^{\omega_{1}}\simeq 54$ K and $T_{C}^{\omega_{2}}\simeq
135$ K, respectively. $\beta_{\omega_{1}}$ having a value of $\sim$1/4
indicates that $\omega_{1}$ mode has a pure one-dimensional origin note2 . In
fact, such character is consistent with our theoretical calculations Suppl .
Specifically, the obtained phonon linewidth in ZrTe5 has a quasi-one-
dimensional character, i.e. the corresponding values of three acoustic modes
along the $\Gamma-Z$ direction (inter-plane) are in general larger than those
along others. Since the e-ph interactions dominate the order parameter
Qin_PRL_2020 , it is not surprising that a fluctuating CDW can emerge along
the inter-plane direction ($b$-axis). By contrast, the $\omega_{2}$ mode
cannot belong to a simple one-dimensional case. The intricate $\omega_{2}$
mode might have higher dimensional characteristics, and needs to be unveiled
by further investigations.

The $T$-dependent damping rate $\Gamma_{1}$ ($\Gamma_{2}$) for the
$\omega_{1}$ ($\omega_{2}$) mode can be extracted by fitting the oscillatory
signals via a damped sine function Suppl , as shown in Figs. 4(c) and (d). In
order to compare with the theoretical calculations Takada_PRB_1985 ;
Richard_SSC_1993 , we used a power law ($\propto T^{\alpha}$) to fit the data,
and obtained that $\Gamma_{1}$ and $\Gamma_{2}$ scale well as $T$ and
$T^{0.6}$, respectively. Such $T$-scalings clearly deviate from the
theoretical intrinsic damping Takada_PRB_1985 ; Richard_SSC_1993 ;
Torchinsky_NM_2013 , i.e. $T^{2}$ away from $T_{C}$ and $T^{5}$ close to
$T_{C}$, respectively. Indeed, decay of the amplitudon includes both the
intrinsic damping and correlation time of the fluctuating CDW. Therefore,
$\Gamma_{1}^{-1}$ and $\Gamma_{2}^{-1}$ here provide a lower bound of the
correlation time $(\tau_{F})_{1}$ and $(\tau_{F})_{2}$.

As is known, the acoustic phonon may soften significantly at $q=2k_{f}$ upon
the phase transition of CDW G book 1994 . In the $q=$0 limit,
$\omega_{A}\simeq\lambda^{1/2}\omega_{2k_{f}}$, where $\omega_{2k_{f}}$ is the
frequency of the softening phonon above the phase transition connected to the
frequency of amplitudon ($\omega_{A}$) via the e-ph coupling constant
$\lambda$. Therefore, using a linear dispersion approximation with sound
velocity of $v_{s}\sim 10^{3}$ m/s Wang_JAP_2018 , $k_{f}$ can be estimated to
have values of 0.053 Å-1 and 0.34 Å-1 for $\omega_{1}$ and $\omega_{2}$ modes,
respectively. Although we do not have accurate knowledge of anisotropic
$v_{s}$ associated with $\omega_{1}$ and $\omega_{2}$ modes, based on these
two estimated values we can conclude that their corresponding modulation wave
vectors should be very small. Such uncommonly small $k_{f}$ values indicate
that our observed density waves have tremendously long lattice modulation
periods in the real space. These results are directly due to the extremely low
energies of $\omega_{1}$ and $\omega_{2}$ modes. Interestingly, these derived
values are consistent with the work demonstrating the 3D quantum Hall effect
Tang Nature 2019 .

Since $\omega_{1}$ and $\omega_{2}$ modes now can be attributed to the
fluctuating CDW that obeys the Bose-Einstein statistics, they can naturally be
assigned to the bosons contributing to the $\tau_{s}$ decay process emerging
below $T^{*}$. However, the $\omega_{2}$ mode is the mostly possible candidate
due to two reasons: (1) appearance temperature of this mode coincides with
that of the $\tau_{s}$ process; (2) No clear anomalies at $T_{C}^{\omega_{1}}$
were found in the $T$-dependent $\tau_{s}$ and $A_{s}$ Suppl ; (3) energy of
the $\omega_{1}$ mode is nearly smaller by an order than that of the
$\omega_{2}$ mode so that the timescale of energy exchange between the
nonequilibrium quasiparticles and $\omega_{1}$ mode can mix up with the long
thermal diffusion process, where large number of low-energy acoustic phonons
will overwhelm the CDW modes.

In summary, using the ultrafast optical spectroscopy we present a detailed
investigation on the quasi-one-dimensional system ZrTe5. Our results reveal
that below $T^{*}$, there emerges a new quasiparticle decay process with a
timescale of $\sim$1-2 ps, in addition to the two relaxation dynamics
persisting at all temperatures with timescales of $\sim$0.15-0.3 ps and
$\sim$3-18 ps, respectively. We address that this new decay process is most
likely due to appearance of some novel collective excitations in ZrTe5, while
the other two can be attributed to the electron-phonon (e-ph) scattering and
the phonon-assisted electron-hole (e-h) recombination processes. Surprisingly,
our experimental data for the first time unambiguously reveal two coherent
oscillations with extremely small energies occurring below two distinct
temperatures $\sim$54 K and $\sim$135 K, respectively. We argue that they
arise from the amplitude mode of fluctuating CDW triggered by the acoustic
phonons, and can well explain the new observed quasiparticle decay below
$T^{*}$. Whether these findings are connected to the other emergent phenomena
in this system deserves further explorations.

We acknowledge the valuable discussion from Hrvoje Petek. This work was
supported by the National Natural Science Foundation of China (Grants Nos.
11974070, 11734006, 11925408, 11921004), the Frontier Science Project of
Dongguan (2019622101004), the National Key R&D Program of China (Grant Nos.
2016YFA0300600 and 2018YFA0305700), the Strategic Priority Research Program of
Chinese Academy of Sciences (Grant No. XDB33000000), the K. C. Wong Education
Foundation (Grant No. GJTD-2018-01), the Beijing Natural Science Foundation
(Grant No. Z180008), the Beijing Municipal Science and Technology Commission
(Grant No. Z191100007219013), and the CAS Interdisciplinary Innovation Team.

## References

  * (1) G. Grüner, Density waves in solids (Perseus, Cambridge, MA, 1994).
  * (2) S. A. Kivelson, I. P. Bindloss, E. Fradkin, V. Oganesyan, J. M. Tranquada, et al., Rev. Mod. Phys. 75, 1201 (2003).
  * (3) H. Weng, X. Dai, and Z. Fang, Phys. Rev. X 4, 011002 (2014).
  * (4) J. Mutch, W. C. Chen, P. Went, T. Qian, I. Z. Wilson, et al., Sci. Adv. 5, eaav9771 (2019).
  * (5) C. Vaswani, L. L. Wang, D. H. Mudiyanselage, Q. Li, P. M. Lozano, et al., Phys. Rev. X 10, 021013 (2020).
  * (6) R. Wu, J. Z. Ma, S. M. Nie, L. X. Zhao, X. Huang, et al., Phys. Rev. X 6, 021017 (2016).
  * (7) G. Manzoni, L. Gragnaniello, G. Autès, T. Kuhn, A. Sterzi, et al., Phys. Rev. Lett. 117, 237601 (2016).
  * (8) H. Xiong, J. A. Sobota, S. L. Yang, H. Soifer, A. Gauthier, et al., Phys. Rev. B 95, 195119 (2017).
  * (9) Y. Y. Lv, B. Bin Zhang, X. Li, K. W. Zhang, X. B. Li, et al., Phys. Rev. B 97, 115137 (2018).
  * (10) R. Y. Chen, S. J. Zhang, J. A. Schneeloch, C. Zhang, Q. Li, et al., Phys. Rev. B 92, 075107 (2015).
  * (11) Y. Liu, X. Yuan, C. Zhang, Z. Jin, A. Narayan, et al., Nat. Commun. 7, 12516 (2016).
  * (12) J. Wang, J. Niu, B. Yan, X. Li, R. Bi, et al., Proc. Natl. Acad. Sci. U. S. A. 115, 9145 (2018).
  * (13) T. M. Tritt, N. D. Lowhorn, R. T. Littleton, A. Pope, C. R. Feger, and J. W. Kolis, Phys. Rev. B 60, 7816 (1999).
  * (14) J. Niu, J. Wang, Z. He, C. Zhang, X. Li, et al., Phys. Rev. B 95, 035420 (2017).
  * (15) Y. Jiang, Z. L. Dun, H. D. Zhou, Z. Lu, K. W. Chen, et al., Phys. Rev. B 96, 041101 (2017).
  * (16) Z. G. Chen, R. Y. Chen, R. D. Zhong, J. Schneeloch, C. Zhang, et al., Proc. Natl. Acad. Sci. U. S. A. 114, 816 (2017).
  * (17) B. Xu, L. X. Zhao, P. Marsik, E. Sheveleva, F. Lyzwa, et al., Phys. Rev. Lett. 121, 187401 (2018).
  * (18) Y. Jiang, J. Wang, T. Zhao, Z. L. Dun, Q. Huang, et al., Phys. Rev. Lett. 125, 046403 (2020).
  * (19) X. B. Li, W. K. Huang, Y. Y. Lv, K. W. Zhang, C. L. Yang, et al., Phys. Rev. Lett. 116, 176803 (2016).
  * (20) Q. Li, D. E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosić, et al., Nat. Phys. 12, 550 (2016).
  * (21) T. Liang, J. Lin, Q. Gibson, S. Kushwaha, M. Liu, et al., Nat. Phys. 14, 451 (2018).
  * (22) F. Tang, Y. Ren, P. Wang, R. Zhong, J. Schneeloch, et al., Nature 569, 537 (2019).
  * (23) F. J. DiSalvo, R. M. Fleming, and J. V. Waszczak, Phys. Rev. B 24, 2935 (1981).
  * (24) F. Qin, S. Li, Z. Z. Du, C. M. Wang, W. Zhang, et al., Phys. Rev. Lett. 125, 206601 (2020).
  * (25) B. Fu, H. W. Wang, and S. Q. Shen, Phys. Rev. Lett. 125, 256601 (2020).
  * (26) D. Hsieh, F. Mahmood, J. W. McIver, D. R. Gardner, Y. S. Lee, and N. Gedik, Phys. Rev. Lett. 107, 077401 (2011).
  * (27) M. C. Wang, S. Qiao, Z. Jiang, S. N. Luo, and J. Qi, Phys. Rev. Lett. 116, 036601 (2016).
  * (28) R. V. Yusupov, T. Mertelj, J. H. Chu, I. R. Fisher, and D. Mihailovic, Phys. Rev. Lett. 101, 246402 (2008).
  * (29) D. H. Torchinsky, F. Mahmood, A. T. Bollinger, I. Božović, and N. Gedik, Nat. Mater. 12, 387 (2013).
  * (30) Y. P. Liu, Y. J. Zhang, J. J. Dong, H. Lee, Z. X. Wei, et al., Phys. Rev. Lett. 124, 057404 (2020).
  * (31) The unique properties of ZrTe5 do not manifest them in previous time-resolved optical experiments Zhang PRB 2019 ; Li PRB 2020 , primarily due to the critical deficiencies in these measurements such as the usage of unreliable samples, evidenced by the absence of various coherent Raman modes characterizing the quality of ZrTe5.
  * (32) X. Zhang, H. Y. Song, X. C. Nie, S. B. Liu, Y. Wang, et al., Phys. Rev. B 99, 125141 (2019).
  * (33) N. Li, W. Liang, and S. N. Luo, Phys. Rev. B 101, 014304 (2020).
  * (34) See Supplemental Material for additional data, more experimental and theoretical details, which includes Refs. Weng_PRX_2014 ; Jiang PRB 2017 ; Li_NPhy_2016 ; 1 ; 4 ; 5 ; 6 ; 7 ; 9 ; Landa G Solid state communications 1984 ; 10 ; 11 ; 12 ; Qi PRL 2013 ; Balkanski M PRB 1983 ; Menendez J PRB 1984 ; Tang H PRB 1991 ; 17 ; 18 ; 19 ; 20 ; 22 ; 23 ; 24 ; 25 ; 26 ; 27 ; 28 .
  * (35) J. Qi, T. Durakiewicz, S. A. Trugman, J. X. Zhu, P. S. Riseborough, et al., Phys. Rev. Lett. 111, 057402 (2013).
  * (36) T. Dekorsy, G. C. Cho, and H. Kurz, in Light Scattering in Solids VIII, Ed. by M. Cardona and G. Güntherodt, (Springer, Berlin, 2000)
  * (37) P. B. Allen, Phys. Rev. B 6, 2577 (1972).
  * (38) R. H. Groeneveld, R. Sprik, and A. Lagendijk, Phys. Rev. B 51, 11433 (1995).
  * (39) A. A. Lopez, Phys. Rev. 175, 823 (1968).
  * (40) A. Zwick, G. Landa, R. Carles, M. A. Renucci, and A. Kjekshus, Solid State Commun 44, 89 (1982).
  * (41) R. Merlin, Solid State Commun. 102, 207 (1997).
  * (42) G. A. Garrett, T. F. Albrecht, J. F. Whitaker, and R. Merlin, Phys. Rev. Lett. 77, 3661 (1996).
  * (43) I. Gdor, T. Ghosh, O. Lioubashevski, and S. Ruhman, J. Phys. Chem. Lett. 8, 1920 (2017).
  * (44) G. Batignani, C. Ferrante, G. Fumero, and T. Scopigno, J. Phys. Chem. Lett. 10, 7789 (2019).
  * (45) Z. Guo, H. Gu, M. Fang, B. Song, W. Wang, X. Chen, C. Zhang, H. Jiang, L. Wang, and S. Liu, ACS Materials Lett. 3, 525 (2021).
  * (46) V. L. Ginzburg, Sov. Phys. Usp. 5, 649 (1963).
  * (47) Y. Toda, F. Kawanokami, T. Kurosawa, M. Oda, I. Madan, T. Mertelj, V. V. Kabanov, and D. Mihailovic, Phys. Rev. B 90, 094513(2014).
  * (48) A. Kogar, A. Zong, P. E. Dolgirev, X. Z. Shen, et al., Nat. Phys. 16, 159 (2019).
  * (49) S. Furuseth, L. Brattas, and A. Kjekshus. J. Acta Chemica Scandinavica 27, 2367 (1973).
  * (50) P. Giannozzi, S. Baroni, et al., J. Phys. Condens. Matter 21, 395502 (2009).
  * (51) P. Giannozzi, O. Andreussi, et al., J. Phys. Condens. Matter 29, 465901 (2017).
  * (52) S. Baroni, S. d. Gironcoli, A. D. Corso, and P. Giannozzi, Rev. Mod. Phys. 73, 515 (2001).
  * (53) D. R. Hamann. Phys. Rev. B 88, 085117 (2013).
  * (54) F. Giustino, M. L. Cohen, and S. G. Louie. Phys. Rev. B 76, 165108 (2007).
  * (55) S. Poncé, E.R. Margine, C. Verdi, and F. Giustino. Comput. Phys. Commun. 209, 116 (2016).
  * (56) G. Landa, A. Zwick, R. Carles, M. A. Renucci, and A. Kjekshus, Solid State Commun. 49, 1095 (1984).
  * (57) M. Balkanski, R. F. Wallis, and E. Haro, Phys. Rev. B 28, 1928 (1983).
  * (58) J. Menéndez and M. Cardona, Phys. Rev. B 29, 2051 (1984).
  * (59) H. Tang and I. P. Herman, Phys. Rev. B 43, 2299 (1991).
  * (60) W. Yu, Y. Jiang, J. Yang, Z. L. Dun, H. D. Zhou, et al., Sci. Rep. 6, 353357 (2016).
  * (61) J. Qi, X. Chen, W. Yu, P. Cadden-Zimansky, D. Smirnov, et al., Appl. Phys. Lett. 97, 182102 (2010).
  * (62) J. P. Hinton, J. D. Koralek, G. Yu, E. M. Motoyama, Y. M. Lu, et al., Phys. Rev. Lett. 110, 217002 (2013).
  * (63) D. J. Hilton and C. L. Tang, Phys. Rev. Lett. 89, 146601 (2002).
  * (64) J. Shah, Ultrafast spectroscopy of semiconductors and semiconductor nanostructures. (Springer Science Business Media, 2013).
  * (65) D. N. Basov, R. D. Averitt, D. Van Der Marel, M. Dressel, and K. Haule, Rev. Mod. Phys. 83, 471 (2011).
  * (66) R. D. Averitt and a J. Taylor, J. Phys. Condens. Matter 14, R1357 (2002).
  * (67) R. H. M. Groeneveld, R. Sprik, and A. Lagendijk, Phys. Rev. B 51, 11433 (1995).
  * (68) M. Hase, K. Ishioka, J. Demsar, K. Ushida, and M. Kitajima, Phys. Rev. B 71, 184301 (2005).
  * (69) M. Hajlaoui, E. Papalazarou, J. Mauchain, G. Lantz, N. Moisan, et al., Nano Lett. 12, 3532 (2012).
  * (70) L. Cheng, C. La-O-Vorakiat, C. S. Tang, S. K. Nair, B. Xia, et al., Appl. Phys. Lett. 104, 211906 (2014).
  * (71) P. B. Allen, Phys. Rev. Lett. 59, 1460 (1987).
  * (72) Y. M. Sheu, Y. J. Chien, C. Uher, S. Fahy, and D. A. Reis, Phys. Rev. B 87, 075429 (2013).
  * (73) Y. M. Dai, J. Bowlan, H. Li, H. Miao, S. F. Wu, et al., Phys. Rev. B 92, 161104 (2015).
  * (74) B. Monserrat and A. Narayan, Phys. Rev. Res. 1, 033181 (2019).
  * (75) D. H. Torchinsky, G. F. Chen, J. L. Luo, N. L. Wang, and N. Gedik, Phys. Rev. Lett. 105, 027005 (2010).
  * (76) M. C. Wang, H. S. Yu, J. Xiong, Y. F. Yang, S. N. Luo, et al., Phys. Rev. B 97, 155157 (2018).
  * (77) Y. Zhang, C. Wang, L. Yu, G. Liu, A. Liang, et al., Nat. Commun. 8, 15512 (2017).
  * (78) I. Katayama, K. I. Inoue, Y. Arashida, Y. Wu, H. Yang, et al., Phys. Rev. B 101, 245408 (2020).
  * (79) H. J. Zeiger, J. Vidal, T. K. Cheng, E. P. Ippen, G. Dresselhaus, and M. S. Dresselhaus, Phys. Rev. B 45, 768 (1992).
  * (80) I. Taguchi, A. Grisel, and F. Levy, Solid State Commun. 46, 299 (1983).
  * (81) D. Lim, V. K. Thorsmølle, R. D. Averitt, Q. X. Jia, K. H. Ahn, et al., Phys. Rev. B 71, 134403 (2005).
  * (82) J. K. Miller, J. Qi, Y. Xu, Y. J. Cho, X. Liu, et al., Phys. Rev. B 74, 113313 (2006).
  * (83) R. Y. Chen, S. J. Zhang, M. Y. Zhang, T. Dong, and N. L. Wang, Phys. Rev. Lett. 118, 107402 (2017).
  * (84) $\beta$ will arrive at 1/4 under the condition of $\omega_{A}\gg$$v_{f}q$ in one-dimensional case G book 1994 .
  * (85) S. Takada, K. Y. M. Wong, and T. Holstein, Phys. Rev. B 32, 4639 (1985).
  * (86) J. Richard and J. Chen, Solid State Commun. 86, 485 (1993).
  * (87) C. Wang, H. Wang, Y. B. Chen, S. H. Yao, and J. Zhou, J. Appl. Phys. 123, 175104 (2018).