text:
\section{Introduction}

The conventional classifications on the phases of matter are rooted in the Landau paradigm of symmetry breaking theory \cite{McGreevy:2022oyu}. 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 \cite{wen}.
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) \cite{Landsteiner:2015lsa, Landsteiner:2015pdh} and nodal line semimetals (NLSM) \cite{Liu:2018bye, Liu:2020ymx} have been explicitly constructed from the holographic duality. Both holographic WSM and NLSM are shown to possess nontrivial topological invariants \cite{Liu:2018djq}. Remarkably, holographic WSM exhibits interesting effects inherited from the boundary states \cite{Ammon:2016mwa}. 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 \cite{Landsteiner:2019kxb} for a review on the developments.\footnote{Other interesting developments can be found in e.g. \cite{Landsteiner:2016stv, Copetti:2016ewq, Grignani:2016wyz, Ammon:2018wzb, Baggioli:2018afg, Liu:2018spp, Ji:2019pxx, Song:2019asj, Juricic:2020sgg, Baggioli:2020cld, Kiczek:2020qsw, BitaghsirFadafan:2020lkh, Zhao:2021pih, Grandi:2021bsp, Rodgers:2021azg, Ji:2021aan, Zhao:2021qfo, Grandi:2021jkj}. } 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 \cite{Fidkowski:2003nf, Festuccia:2005pi, Grinberg:2020fdj}. In the case that the thermal states are described by the black holes with simple Kasner singularities, it has been shown recently in \cite{Liu:2021hap} 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.\footnote{Other studies on the geometric aspects of  black hole singularities can be found in e.g. \cite{Frenkel:2020ysx,  Hartnoll:2020rwq, Hartnoll:2020fhc, Liu:2022rsy,
Cai:2020wrp, Wang:2020nkd, An:2021plu, Grandi:2021ajl, Mansoori:2021wxf, Dias:2021afz, Sword:2021pfm, Cai:2021obq, Henneaux:2022ijt, Caceres:2022smh, Bhattacharya:2021nqj, An:2022lvo,
Auzzi:2022bfd, Mirjalali:2022wrg, Hartnoll:2022snh, Caceres:2022hei, Hartnoll:2022rdv, Sword:2022oyg}. } 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 \cite{Liu:2018bye, Liu:2018djq}. 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 \cite{Grinberg:2020fdj}. 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. \ref{sec2}, 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. \ref{sec3}, we will  review the holographic NLSM and then also study its interior geometry and the proper time of the timelike geodesics. Sec. \ref{sec4} is devoted to the conclusions and open questions. The details of calculations are in the appendices.

\section{Inside holographic Weyl semimetal}
\label{sec2}

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 \cite{Landsteiner:2015lsa, Landsteiner:2015pdh} is
\begin{align}
\label{eq:actionwsm}
\begin{split}
  S=& \int d^5x \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}
\end{align}
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 $D_a\Phi=\partial_a\Phi - iq A_a\Phi$ 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
\begin{align}
\label{eq:ansatz}
\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}
\end{align}
The equations of motion for the fields can be found in the appendix \ref{app:hwsm}.
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
\begin{equation}
\lim_{r\to\infty} A_z=b\,,~~~\lim_{r\to\infty} r\phi=M\,,
\end{equation}
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. \ref{fig:ahe}. 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.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.57\textwidth]{fig-ahe.pdf}
~~
\end{center}
\vspace{-0.3cm}
\caption{\small
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.}
\label{fig:ahe}
\end{figure}

\subsection{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. \ref{fig:phiosc}, 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 \cite{Hartnoll:2020fhc}, since there is no inner horizon any more for holographic WSM. Similar oscillation behavior has been found previously in neutral helical black holes \cite{Liu:2022rsy}.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.46\textwidth]{fig-phi-in-002.pdf}
~~
\includegraphics[width=0.46\textwidth]{fig-phi-in-fixM.pdf}
\end{center}
\vspace{-0.3cm}
\caption{\small  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 $\phi$.
}
\label{fig:phiosc}
\end{figure}

\subsection{Behaviors of Kasner exponents}
\label{subsec:wsmkas}
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
\begin{align}
\label{eq:asns}
    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)\,,
\end{align}
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 \eqref{eq:scasym1},\eqref{eq:scasym2},\eqref{eq:scasym3} 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 \eqref{eq:nbwsm}. Moreover, as we shall see later, the axial gauge field $A_z$ is determined by the ansatz \eqref{eq:asns}.

Near the singularity the equations of motion (\ref{eq:eom}) can be simplified under the assumption that the ignored terms are subleading which will be numerically checked afterward,
\begin{align}\label{eq:eomns}
\begin{split}
     u''+\frac{h'}{2h}u'-\left( f''+ \frac{f'h'}{2h}  \right)\frac{u}{f}&=0\,,\\
     \frac{f''}{f}+\frac{u''}{2u}-\frac{f'^2}{4f^2} + \frac{f'u'}{fu}+\frac{1}{2} \phi'^2 &=0\,,\\
    \frac{1}{2}\phi'^2 - \frac{u'}{2u}\left( \frac{f'}{f} + \frac{h'}{2h} \right)  - \frac{f'h'}{2fh} - \frac{f'^2}{4f^2}&=0\,,\\
    A_z''+\left( \frac{f'}{f} - \frac{h'}{2h}+ \frac{u'}{u} \right)A_z'&=0\,,\\
    \phi'' + \left( \frac{f'}{f}+\frac{h'}{2h}+ \frac{u'}{u} \right)\phi'&=0\,.
\end{split}
\end{align}
Substituting \eqref{eq:asns} into \eqref{eq:eomns}, we obtain
\begin{align}
\label{eq:relations}
n_h=2\,(1-n_u-n_f)\,, ~~~~~
n_{\phi}=\pm \sqrt{(2n_f+n_u)(1-n_u)- \frac{3n_f^2}{2}} \,.
\end{align}
We can also solve the fourth equation in  \eqref{eq:eomns} to obtain at leading order $A_z$
\begin{align}
\label{eq:nhazwsm}
    A_z \simeq A_{zs0} +  A_{zs1} (r-r_s)^{n_h} \,.
\end{align
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 \eqref{eq:asns} and \eqref{eq:nhazwsm}.

Note that in the above equations \eqref{eq:eomns}, we have assumed that the terms ignored are
subleading. More explicitly, we have assumed
\begin{align}
\label{eq:paregime}
    n_u<2\,,~~~n_f+n_u<1\,, ~~~

    2 n_f + n_u>0\,
\end{align}
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 \eqref{eq:asns} and \eqref{eq:nhazwsm}.

There are two radical conserved charge associated to the scaling symmetries of the system,
\begin{eqnarray}
\label{eq:cc1}
    Q_1&=&\sqrt{h}(u'f-uf')\,,\\
\label{eq:cc2}
    Q_2&=& u'\sqrt{h}f- \frac{h'}{\sqrt{h}}uf -A_z A_z' \frac{uf}{\sqrt{h}}\,.
\end{eqnarray}
We have used them to check the accuracy of the numerics. Moreover,
evaluate them at the horizon and at the singularity we obtain
\begin{eqnarray}\label{eq:cc1s}
    4\pi T f_1 \sqrt{h_1} =Ts &=& u_0 f_0 \sqrt{h_0} (n_f-n_u)\\

&=&   \frac{u_0f_0}{\sqrt{h_0}} \left( n_h A_{zs0}A_{zs1} - h_0 (2n_f+3n_u-2) \,\right)
\end{eqnarray}
where $s$ is the density of entropy.
From \eqref{eq:cc1s}, we have $n_f>n_u$ in addition to the constraints \eqref{eq:paregime}.
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 (\ref{eq:ansatz}, \ref{eq:asns}) and performing the coordinate transformation
\begin{equation}\label{eq:coortr}
\tau=-\frac{2}{\sqrt{n_0}(n_u-2)}(r-r_s)^{(2-n_u)/2}\,,\end{equation}
we obtain the Kasner form for the fields
\begin{align}
\label{eq:kasnerform}
\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}
\end{align}
where
\begin{align}
    \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{2 n_{\phi}}{2-n_u} \,.
    \end{split}
\end{align}
Note that $A_z$ is a constant at the leading order.
Using the relations \eqref{eq:relations}, the above Kasner exponents can be expressed in terms of $n_u$ and $n_f$,
\begin{align}
\label{eq:ptnurel}
\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}
\end{align}
Note that the sign of $p_\phi$ in \eqref{eq:ptnurel} can only be determined from numerics.
They satisfy the following Kasner relations
\begin{align}
\label{eq:kasrel}
    p_t+2p_x+p_z=1\,,~~~~~
    p_t^2 + 2p_x^2+ p_z^2 +p_{\phi}^2=1\,.
\end{align}
It indicates that only two of the four Kasner exponents are independent.

In Fig. \ref{fig:wsmkas}, 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.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.444\textwidth]{fig-pt.pdf}~~
\includegraphics[width=0.444\textwidth]{fig-pp.pdf}\\
\includegraphics[width=0.444\textwidth]{fig-px.pdf}~~
\includegraphics[width=0.444\textwidth]{fig-pz.pdf}
\end{center}
\vspace{-0.3cm}
\caption{\small  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.}
\label{fig:wsmkas}
\end{figure}

Note that in \cite{Liu:2018bye}, 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.

\subsection{Proper time of timelike geodesics}

One of interesting connection between the interior geometry and the boundary observable is given in  \cite{Grinberg:2020fdj} 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
\begin{align}
    \frac{E^2}{g_{tt}}+ g_{rr} {\dot r}^2 = -1\,,
\end{align}
from which we obtain
\begin{align}
    \frac{d\tau}{dr} = \frac{1}{\sqrt{E^2-u}}\,.
\end{align}
The proper time from the horizon to the singularity of a particle with $E=0$ (i.e. the longest time) is
\begin{align}
    \tau_s= \int_{r_s}^{r_h} \frac{dr}{\sqrt{-u}} \,.
\end{align}
The plots of $\tau_s$ as a function of $M/b$ for different $T/b$ are shown in Fig. \ref{fig:taus}.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.47\textwidth]{fig-taus.pdf}
\end{center}
\vspace{-0.3cm}
\caption{\small  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).}
\label{fig:taus}
\end{figure}

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 $\text{Im}(m)<0$ \cite{Grinberg:2020fdj}. 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 \cite{Baggioli:2018afg}. 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.

\section{Inside holographic nodal line semimetal}
\label{sec3}

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. \ref{fig:wsmkas}. 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. \ref{fig:taus}. 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 \cite{Liu:2018bye, Liu:2020ymx}.

The action for the holographic NLSM \cite{Liu:2020ymx} is
\begin{align}
\label{eq:actionnlsm}
\begin{split}
     S&= \int d^5x \,\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( i B_{ab} H^*_{cde} - i B^*_{ab} H_{cde} \big) -V_2(B_{ab}) - \lambda |\Phi|^2 B^*_{ab}B^{ab} \bigg]\,,
\end{split}
\end{align}
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.
$D_a = \nabla_a - i q_1 A_a$ 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
\begin{align}
H_{abc} = \partial_a B_{bc} + \partial_b B_{ca} + \partial_c B_{ab} - i q_2 A_a B_{bc} - i q_2 A_b B_{ca} - i q_2 A_c B_{ab}\,,
\end{align}
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 \cite{Liu:2020ymx}.
The potential terms are chosen as
\begin{align}
V_1 = m_1^2 |\Phi|^2 + \frac{\lambda_1}{2} |\Phi|^4\,,\qquad V_2 = m_2^2 B_{ab}^* B^{ab}\,,
\end{align}
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 \eqref{eq:actionnlsm} 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
\begin{align}
\label{eq:nlansatz}
    \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}
\end{align}
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 \ref{app:nlsm}. 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,
\begin{equation}
\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\,,
\end{equation}
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 \cite{Liu:2020ymx} 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.

\subsection{Kasner exponents}

Close to the singularity $r\to r_s$, similar to the holographic WSM case we again take the ansatz
\begin{align}
\label{eq:asns2}
    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)\,,
\end{align}
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
\begin{align}
\label{eq:nhnlsm}
\begin{split}
    \frac{u''}{u} - \frac{f''}{f} + \frac{h'}{2h} \left( \frac{u'}{u} - \frac{f'}{f}  \right)&=0\,,\\
     \frac{u''}{2u}+\frac{f''}{f}-\frac{f'^2}{4f^2} + \frac{f'u'}{fu}+\frac{1}{2} \phi'^2 &=0\,,\\
    \frac{f'^2}{4f^2} +\frac{f'h'}{2fh} + \frac{u'}{2u}\left( \frac{f'}{f} + \frac{h'}{2h} \right)-\frac{1}{2}\phi'^2 &=0\,,\\
    \phi'' + \big( \frac{f'}{f}+\frac{h'}{2h}+ \frac{u'}{u} \big)\phi'&=0\,,\\
    \mathcal{B}_{tz}'- \frac{\eta \sqrt{h}}{2f}( \lambda \phi^2) \mathcal{B}_{xy} &=0\,,\\
    \mathcal{B}_{xy}'- \frac{\eta f}{2\sqrt{h} u}( \lambda \phi^2) \mathcal{B}_{tz} &=0\,.
\end{split}
\end{align}
The first four equations in \eqref{eq:nhnlsm} are the same as the ones in holographic WSM. Similarly, we obtain
\begin{align}
n_h=2\,(1-n_u-n_f)\,, ~~~~~
n_{\phi}=\pm \sqrt{(2n_f+n_u)(1-n_u)- \frac{3n_f^2}{2}} \,.
\end{align}
From the last two equations in \eqref{eq:nhnlsm} we have the following leading order solutions for the two form fields near the singularity
\begin{equation}
\label{eq:2formnh}
\mathcal{B}_{xy}\sim \mathcal{B}_{xy0}+\dots\,,~~~~
\mathcal{B}_{tz}\sim \mathcal{B}_{tz0}+\dots\,,~~~~
\end{equation}
where the dots are subleading terms of form $(r-r_s)^{2-n_u-2 n_f}(\log (r-r_s))^2$ 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 \eqref{eq:nhnlsm} we have assumed that the ignored terms are subleading. More explicitly, we have assumed
\begin{equation}
\label{eq:hnls-paregime}
    n_u<2\,,~~~2n_u+n_h<2\,, ~~~
    n_u+2n_f<2\,.
\end{equation}
Note that the last two inequalities of above are consistent with the assumptions used in obtaining  \eqref{eq:2formnh}.
We have checked numerically that the inequalities \eqref{eq:hnls-paregime} are satisfied for the parameters we have considered.

Similar to the discussion in section \ref{subsec:wsmkas}, we can make
 a coordinate transformation \eqref{eq:coortr} to write the metric \eqref{eq:asns2} into the Kasner form as \eqref{eq:kasnerform} with the parameters \eqref{eq:ptnurel} and the Kasner relations \eqref{eq:kasrel}. Here the leading order of the two form fields are constant close to the singularity.

The two conserved charges of the scaling symmetries are
\begin{eqnarray}
    Q_1&=& \frac{8}{\eta} \mathcal{B}_{tz} \mathcal{B}_{xy} + \frac{u}{\sqrt{h}} (f'h-fh')\,,\\
    Q_2&=& \frac{f}{\sqrt{h}} (u'h-uh')\,.
\end{eqnarray}
Evaluate them at the horizon and at the singularity we obtain
\begin{equation}
\frac{8}{\eta}\mathcal{B}_{xy0}\mathcal{B}_{tz0}=u_0f_0\sqrt{h_0}(n_f-n_h)
\end{equation}
and
\begin{align}
   4\pi T f_1 \sqrt{h_1} =Ts= u_0 f_0 \sqrt{h_0} (2-2 n_f-3 n_u)
\end{align}
where $s$ is the density of entropy. We have checked the above relations numerically.

In Fig. \ref{fig:nlsmkas}, 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 \eqref{eq:kasrel} 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.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.444\textwidth]{hnls-pt.pdf}~~
\includegraphics[width=0.444\textwidth]{hnls-pp.pdf}\\
\includegraphics[width=0.444\textwidth]{hnls-px.pdf}~~
\includegraphics[width=0.444\textwidth]{hnls-pz.pdf}
\end{center}
\vspace{-0.3cm}
\caption{\small  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.}
\label{fig:nlsmkas}
\end{figure}

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. \ref{fig:topNLSMkas}. Nevertheless, at low enough temperature we see that the Kasner exponents are nearly constant.
\vspace{-0.1cm}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.244\textwidth]{fig-hnlsm-pt-0.pdf}
\includegraphics[width=0.244\textwidth]{fig-hnlsm-px-0.pdf}
\includegraphics[width=0.244\textwidth]{fig-hnlsm-pz-0.pdf}
\includegraphics[width=0.244\textwidth]{fig-hnlsm-tau-0.pdf}
\end{center}
\vspace{-0.4cm}
\caption{\small  Plots of Kasner exponents and $\tau_s$ for holographic NLSM as a function of $T/b$ when $M/b=0$. }
\label{fig:topNLSMkas}
\end{figure}

\subsection{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. \ref{fig:nlsmtaus}, 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 $M/b$.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.47\textwidth]{hnls-taus.pdf}
\end{center}
\vspace{-0.4cm}
\caption{\small  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).}
\label{fig:nlsmtaus}
\end{figure}

\section{Conclusion and discussion}
\label{sec4}

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. \ref{fig:wsmkas} and Fig. \ref{fig:nlsmkas}.

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. \cite{Juricic:2020sgg, Ji:2021aan}, to check if they are universal feature of topological semimetals.

\vspace{.8cm}
\subsection*{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.

\vspace{.6 cm}

\section{Introduction}

The conventional classifications on the phases of matter are rooted in the Landau paradigm of symmetry breaking theory \cite{McGreevy:2022oyu}. 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 \cite{wen}.
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) \cite{Landsteiner:2015lsa, Landsteiner:2015pdh} and nodal line semimetals (NLSM) \cite{Liu:2018bye, Liu:2020ymx} have been explicitly constructed from the holographic duality. Both holographic WSM and NLSM are shown to possess nontrivial topological invariants \cite{Liu:2018djq}. Remarkably, holographic WSM exhibits interesting effects inherited from the boundary states \cite{Ammon:2016mwa}. 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 \cite{Landsteiner:2019kxb} for a review on the developments.\footnote{Other interesting developments can be found in e.g. \cite{Landsteiner:2016stv, Copetti:2016ewq, Grignani:2016wyz, Ammon:2018wzb, Baggioli:2018afg, Liu:2018spp, Ji:2019pxx, Song:2019asj, Juricic:2020sgg, Baggioli:2020cld, Kiczek:2020qsw, BitaghsirFadafan:2020lkh, Zhao:2021pih, Grandi:2021bsp, Rodgers:2021azg, Ji:2021aan, Zhao:2021qfo, Grandi:2021jkj}. } 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 \cite{Fidkowski:2003nf, Festuccia:2005pi, Grinberg:2020fdj}. In the case that the thermal states are described by the black holes with simple Kasner singularities, it has been shown recently in \cite{Liu:2021hap} 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.\footnote{Other studies on the geometric aspects of  black hole singularities can be found in e.g. \cite{Frenkel:2020ysx,  Hartnoll:2020rwq, Hartnoll:2020fhc, Liu:2022rsy,
Cai:2020wrp, Wang:2020nkd, An:2021plu, Grandi:2021ajl, Mansoori:2021wxf, Dias:2021afz, Sword:2021pfm, Cai:2021obq, Henneaux:2022ijt, Caceres:2022smh, Bhattacharya:2021nqj, An:2022lvo,
Auzzi:2022bfd, Mirjalali:2022wrg, Hartnoll:2022snh, Caceres:2022hei, Hartnoll:2022rdv, Sword:2022oyg}. } 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 \cite{Liu:2018bye, Liu:2018djq}. 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 \cite{Grinberg:2020fdj}. 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. \ref{sec2}, 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. \ref{sec3}, we will  review the holographic NLSM and then also study its interior geometry and the proper time of the timelike geodesics. Sec. \ref{sec4} is devoted to the conclusions and open questions. The details of calculations are in the appendices.

\section{Inside holographic Weyl semimetal}
\label{sec2}

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 \cite{Landsteiner:2015lsa, Landsteiner:2015pdh} is
\begin{align}
\label{eq:actionwsm}
\begin{split}
  S=& \int d^5x \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}
\end{align}
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 $D_a\Phi=\partial_a\Phi - iq A_a\Phi$ 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
\begin{align}
\label{eq:ansatz}
\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}
\end{align}
The equations of motion for the fields can be found in the appendix \ref{app:hwsm}.
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
\begin{equation}
\lim_{r\to\infty} A_z=b\,,~~~\lim_{r\to\infty} r\phi=M\,,
\end{equation}
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. \ref{fig:ahe}. 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.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.57\textwidth]{fig-ahe.pdf}
~~
\end{center}
\vspace{-0.3cm}
\caption{\small
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.}
\label{fig:ahe}
\end{figure}

\subsection{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. \ref{fig:phiosc}, 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 \cite{Hartnoll:2020fhc}, since there is no inner horizon any more for holographic WSM. Similar oscillation behavior has been found previously in neutral helical black holes \cite{Liu:2022rsy}.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.46\textwidth]{fig-phi-in-002.pdf}
~~
\includegraphics[width=0.46\textwidth]{fig-phi-in-fixM.pdf}
\end{center}
\vspace{-0.3cm}
\caption{\small  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 $\phi$.
}
\label{fig:phiosc}
\end{figure}

\subsection{Behaviors of Kasner exponents}
\label{subsec:wsmkas}
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
\begin{align}
\label{eq:asns}
    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)\,,
\end{align}
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 \eqref{eq:scasym1},\eqref{eq:scasym2},\eqref{eq:scasym3} 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 \eqref{eq:nbwsm}. Moreover, as we shall see later, the axial gauge field $A_z$ is determined by the ansatz \eqref{eq:asns}.

Near the singularity the equations of motion (\ref{eq:eom}) can be simplified under the assumption that the ignored terms are subleading which will be numerically checked afterward,
\begin{align}\label{eq:eomns}
\begin{split}
     u''+\frac{h'}{2h}u'-\left( f''+ \frac{f'h'}{2h}  \right)\frac{u}{f}&=0\,,\\
     \frac{f''}{f}+\frac{u''}{2u}-\frac{f'^2}{4f^2} + \frac{f'u'}{fu}+\frac{1}{2} \phi'^2 &=0\,,\\
    \frac{1}{2}\phi'^2 - \frac{u'}{2u}\left( \frac{f'}{f} + \frac{h'}{2h} \right)  - \frac{f'h'}{2fh} - \frac{f'^2}{4f^2}&=0\,,\\
    A_z''+\left( \frac{f'}{f} - \frac{h'}{2h}+ \frac{u'}{u} \right)A_z'&=0\,,\\
    \phi'' + \left( \frac{f'}{f}+\frac{h'}{2h}+ \frac{u'}{u} \right)\phi'&=0\,.
\end{split}
\end{align}
Substituting \eqref{eq:asns} into \eqref{eq:eomns}, we obtain
\begin{align}
\label{eq:relations}
n_h=2\,(1-n_u-n_f)\,, ~~~~~
n_{\phi}=\pm \sqrt{(2n_f+n_u)(1-n_u)- \frac{3n_f^2}{2}} \,.
\end{align}
We can also solve the fourth equation in  \eqref{eq:eomns} to obtain at leading order $A_z$
\begin{align}
\label{eq:nhazwsm}
    A_z \simeq A_{zs0} +  A_{zs1} (r-r_s)^{n_h} \,.
\end{align
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 \eqref{eq:asns} and \eqref{eq:nhazwsm}.

Note that in the above equations \eqref{eq:eomns}, we have assumed that the terms ignored are
subleading. More explicitly, we have assumed
\begin{align}
\label{eq:paregime}
    n_u<2\,,~~~n_f+n_u<1\,, ~~~

    2 n_f + n_u>0\,
\end{align}
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 \eqref{eq:asns} and \eqref{eq:nhazwsm}.

There are two radical conserved charge associated to the scaling symmetries of the system,
\begin{eqnarray}
\label{eq:cc1}
    Q_1&=&\sqrt{h}(u'f-uf')\,,\\
\label{eq:cc2}
    Q_2&=& u'\sqrt{h}f- \frac{h'}{\sqrt{h}}uf -A_z A_z' \frac{uf}{\sqrt{h}}\,.
\end{eqnarray}
We have used them to check the accuracy of the numerics. Moreover,
evaluate them at the horizon and at the singularity we obtain
\begin{eqnarray}\label{eq:cc1s}
    4\pi T f_1 \sqrt{h_1} =Ts &=& u_0 f_0 \sqrt{h_0} (n_f-n_u)\\

&=&   \frac{u_0f_0}{\sqrt{h_0}} \left( n_h A_{zs0}A_{zs1} - h_0 (2n_f+3n_u-2) \,\right)
\end{eqnarray}
where $s$ is the density of entropy.
From \eqref{eq:cc1s}, we have $n_f>n_u$ in addition to the constraints \eqref{eq:paregime}.
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 (\ref{eq:ansatz}, \ref{eq:asns}) and performing the coordinate transformation
\begin{equation}\label{eq:coortr}
\tau=-\frac{2}{\sqrt{n_0}(n_u-2)}(r-r_s)^{(2-n_u)/2}\,,\end{equation}
we obtain the Kasner form for the fields
\begin{align}
\label{eq:kasnerform}
\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}
\end{align}
where
\begin{align}
    \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{2 n_{\phi}}{2-n_u} \,.
    \end{split}
\end{align}
Note that $A_z$ is a constant at the leading order.
Using the relations \eqref{eq:relations}, the above Kasner exponents can be expressed in terms of $n_u$ and $n_f$,
\begin{align}
\label{eq:ptnurel}
\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}
\end{align}
Note that the sign of $p_\phi$ in \eqref{eq:ptnurel} can only be determined from numerics.
They satisfy the following Kasner relations
\begin{align}
\label{eq:kasrel}
    p_t+2p_x+p_z=1\,,~~~~~
    p_t^2 + 2p_x^2+ p_z^2 +p_{\phi}^2=1\,.
\end{align}
It indicates that only two of the four Kasner exponents are independent.

In Fig. \ref{fig:wsmkas}, 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.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.444\textwidth]{fig-pt.pdf}~~
\includegraphics[width=0.444\textwidth]{fig-pp.pdf}\\
\includegraphics[width=0.444\textwidth]{fig-px.pdf}~~
\includegraphics[width=0.444\textwidth]{fig-pz.pdf}
\end{center}
\vspace{-0.3cm}
\caption{\small  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.}
\label{fig:wsmkas}
\end{figure}

Note that in \cite{Liu:2018bye}, 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.

\subsection{Proper time of timelike geodesics}

One of interesting connection between the interior geometry and the boundary observable is given in  \cite{Grinberg:2020fdj} 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
\begin{align}
    \frac{E^2}{g_{tt}}+ g_{rr} {\dot r}^2 = -1\,,
\end{align}
from which we obtain
\begin{align}
    \frac{d\tau}{dr} = \frac{1}{\sqrt{E^2-u}}\,.
\end{align}
The proper time from the horizon to the singularity of a particle with $E=0$ (i.e. the longest time) is
\begin{align}
    \tau_s= \int_{r_s}^{r_h} \frac{dr}{\sqrt{-u}} \,.
\end{align}
The plots of $\tau_s$ as a function of $M/b$ for different $T/b$ are shown in Fig. \ref{fig:taus}.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.47\textwidth]{fig-taus.pdf}
\end{center}
\vspace{-0.3cm}
\caption{\small  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).}
\label{fig:taus}
\end{figure}

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 $\text{Im}(m)<0$ \cite{Grinberg:2020fdj}. 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 \cite{Baggioli:2018afg}. 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.

\section{Inside holographic nodal line semimetal}
\label{sec3}

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. \ref{fig:wsmkas}. 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. \ref{fig:taus}. 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 \cite{Liu:2018bye, Liu:2020ymx}.

The action for the holographic NLSM \cite{Liu:2020ymx} is
\begin{align}
\label{eq:actionnlsm}
\begin{split}
     S&= \int d^5x \,\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( i B_{ab} H^*_{cde} - i B^*_{ab} H_{cde} \big) -V_2(B_{ab}) - \lambda |\Phi|^2 B^*_{ab}B^{ab} \bigg]\,,
\end{split}
\end{align}
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.
$D_a = \nabla_a - i q_1 A_a$ 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
\begin{align}
H_{abc} = \partial_a B_{bc} + \partial_b B_{ca} + \partial_c B_{ab} - i q_2 A_a B_{bc} - i q_2 A_b B_{ca} - i q_2 A_c B_{ab}\,,
\end{align}
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 \cite{Liu:2020ymx}.
The potential terms are chosen as
\begin{align}
V_1 = m_1^2 |\Phi|^2 + \frac{\lambda_1}{2} |\Phi|^4\,,\qquad V_2 = m_2^2 B_{ab}^* B^{ab}\,,
\end{align}
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 \eqref{eq:actionnlsm} 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
\begin{align}
\label{eq:nlansatz}
    \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}
\end{align}
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 \ref{app:nlsm}. 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,
\begin{equation}
\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\,,
\end{equation}
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 \cite{Liu:2020ymx} 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.

\subsection{Kasner exponents}

Close to the singularity $r\to r_s$, similar to the holographic WSM case we again take the ansatz
\begin{align}
\label{eq:asns2}
    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)\,,
\end{align}
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
\begin{align}
\label{eq:nhnlsm}
\begin{split}
    \frac{u''}{u} - \frac{f''}{f} + \frac{h'}{2h} \left( \frac{u'}{u} - \frac{f'}{f}  \right)&=0\,,\\
     \frac{u''}{2u}+\frac{f''}{f}-\frac{f'^2}{4f^2} + \frac{f'u'}{fu}+\frac{1}{2} \phi'^2 &=0\,,\\
    \frac{f'^2}{4f^2} +\frac{f'h'}{2fh} + \frac{u'}{2u}\left( \frac{f'}{f} + \frac{h'}{2h} \right)-\frac{1}{2}\phi'^2 &=0\,,\\
    \phi'' + \big( \frac{f'}{f}+\frac{h'}{2h}+ \frac{u'}{u} \big)\phi'&=0\,,\\
    \mathcal{B}_{tz}'- \frac{\eta \sqrt{h}}{2f}( \lambda \phi^2) \mathcal{B}_{xy} &=0\,,\\
    \mathcal{B}_{xy}'- \frac{\eta f}{2\sqrt{h} u}( \lambda \phi^2) \mathcal{B}_{tz} &=0\,.
\end{split}
\end{align}
The first four equations in \eqref{eq:nhnlsm} are the same as the ones in holographic WSM. Similarly, we obtain
\begin{align}
n_h=2\,(1-n_u-n_f)\,, ~~~~~
n_{\phi}=\pm \sqrt{(2n_f+n_u)(1-n_u)- \frac{3n_f^2}{2}} \,.
\end{align}
From the last two equations in \eqref{eq:nhnlsm} we have the following leading order solutions for the two form fields near the singularity
\begin{equation}
\label{eq:2formnh}
\mathcal{B}_{xy}\sim \mathcal{B}_{xy0}+\dots\,,~~~~
\mathcal{B}_{tz}\sim \mathcal{B}_{tz0}+\dots\,,~~~~
\end{equation}
where the dots are subleading terms of form $(r-r_s)^{2-n_u-2 n_f}(\log (r-r_s))^2$ 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 \eqref{eq:nhnlsm} we have assumed that the ignored terms are subleading. More explicitly, we have assumed
\begin{equation}
\label{eq:hnls-paregime}
    n_u<2\,,~~~2n_u+n_h<2\,, ~~~
    n_u+2n_f<2\,.
\end{equation}
Note that the last two inequalities of above are consistent with the assumptions used in obtaining  \eqref{eq:2formnh}.
We have checked numerically that the inequalities \eqref{eq:hnls-paregime} are satisfied for the parameters we have considered.

Similar to the discussion in section \ref{subsec:wsmkas}, we can make
 a coordinate transformation \eqref{eq:coortr} to write the metric \eqref{eq:asns2} into the Kasner form as \eqref{eq:kasnerform} with the parameters \eqref{eq:ptnurel} and the Kasner relations \eqref{eq:kasrel}. Here the leading order of the two form fields are constant close to the singularity.

The two conserved charges of the scaling symmetries are
\begin{eqnarray}
    Q_1&=& \frac{8}{\eta} \mathcal{B}_{tz} \mathcal{B}_{xy} + \frac{u}{\sqrt{h}} (f'h-fh')\,,\\
    Q_2&=& \frac{f}{\sqrt{h}} (u'h-uh')\,.
\end{eqnarray}
Evaluate them at the horizon and at the singularity we obtain
\begin{equation}
\frac{8}{\eta}\mathcal{B}_{xy0}\mathcal{B}_{tz0}=u_0f_0\sqrt{h_0}(n_f-n_h)
\end{equation}
and
\begin{align}
   4\pi T f_1 \sqrt{h_1} =Ts= u_0 f_0 \sqrt{h_0} (2-2 n_f-3 n_u)
\end{align}
where $s$ is the density of entropy. We have checked the above relations numerically.

In Fig. \ref{fig:nlsmkas}, 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 \eqref{eq:kasrel} 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.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.444\textwidth]{hnls-pt.pdf}~~
\includegraphics[width=0.444\textwidth]{hnls-pp.pdf}\\
\includegraphics[width=0.444\textwidth]{hnls-px.pdf}~~
\includegraphics[width=0.444\textwidth]{hnls-pz.pdf}
\end{center}
\vspace{-0.3cm}
\caption{\small  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.}
\label{fig:nlsmkas}
\end{figure}

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. \ref{fig:topNLSMkas}. Nevertheless, at low enough temperature we see that the Kasner exponents are nearly constant.
\vspace{-0.1cm}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.244\textwidth]{fig-hnlsm-pt-0.pdf}
\includegraphics[width=0.244\textwidth]{fig-hnlsm-px-0.pdf}
\includegraphics[width=0.244\textwidth]{fig-hnlsm-pz-0.pdf}
\includegraphics[width=0.244\textwidth]{fig-hnlsm-tau-0.pdf}
\end{center}
\vspace{-0.4cm}
\caption{\small  Plots of Kasner exponents and $\tau_s$ for holographic NLSM as a function of $T/b$ when $M/b=0$. }
\label{fig:topNLSMkas}
\end{figure}

\subsection{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. \ref{fig:nlsmtaus}, 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 $M/b$.

\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.47\textwidth]{hnls-taus.pdf}
\end{center}
\vspace{-0.4cm}
\caption{\small  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).}
\label{fig:nlsmtaus}
\end{figure}

\section{Conclusion and discussion}
\label{sec4}

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. \ref{fig:wsmkas} and Fig. \ref{fig:nlsmkas}.

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. \cite{Juricic:2020sgg, Ji:2021aan}, to check if they are universal feature of topological semimetals.

\vspace{.8cm}
\subsection*{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.

\vspace{.6 cm}

redpajama_set_name: RedPajamaArXiv


text:
Philo of Alexandria: On the Life of Abraham
Introduction, Translation, and Commentary
Philo of Alexandria Commentary Series, Volume: 6
Authors: Ellen Birnbaum and John M. Dillon
On the Life of Abraham displays Philo's philosophical, exegetical, and literary genius at its best. Philo begins by introducing the biblical figures Enos, Enoch, Noah, Abraham, Isaac, and Jacob as unwritten laws. Then, interweaving literal, ethical, and allegorical interpretations, Philo presents the life and achievements of Abraham, founder of the Jewish nation, in the form of a Greco-Roman bios, or biography. Ellen Birnbaum and John Dillon explain why and how this work is important within the context of Philo's own oeuvre, early Jewish and Christian exegesis, and ancient philosophy. They also offer a new English translation and detailed analyses, in which they elucidate the meaning of Philo's thought, including his perplexing notion that Israel's ancestors were laws in themselves.
Not Yet Published
Login via Institution
Prices from (excl. VAT):
€189.00$227.00
Please choose a format
Hardback: €189.00$227.00
E-Book: €189.00$227.00
View PDF Flyer
Ellen Birnbaum (Ph.D., 1992, Columbia University) is author of The Place of Judaism in Philo's Thought: Israel, Jews, and Proselytes (Scholars Press, 1996) and several other studies of Philo and the ancient Alexandrian Jewish community.
/br> John Dillon is Regius Professor of Greek (Emeritus) at Trinity College Dublin. His chief publications are in the area of the Platonic tradition, but he has also published extensively on Philo, and in the field of Patristics.
Students and scholars of Philo, biblical literature, ancient Judaism, classics, ancient philosophy, and early Christianity.
Copyright Year:
Biblical Interpretations, Biblical Studies, Ancient Judaism, Biblical Studies, Greek & Latin Literature, Classical Studies, History & Culture, Jewish Studies, Ancient Philosophy, Philosophy

redpajama_set_name: RedPajamaCommonCrawl


My name is Brittany Perry and I am a licensed IN Realtor and owner of Divine Investment Partners in Indianapolis, IN. I have taught Financial Literacy at Community Action of Greater Indianapolis where I facilitated the Pre-Purchase Counseling program for three years.
I am a native of Indianapolis, IN and call it home with my husband and 2 daughters. We enjoy the people, seasons, culture and activities offered to us as residents of this growing city.
In my practice of real estate I am well known for my awesome customer service, consistent communication and ensuring a positive experience for all parties in a transaction. My approach to a transaction is "it's about the client experience and relationship." I know what it means to communicate and I know what it takes to successfully close a transaction making the buying or selling experience positive and seamless!
Updated: 23rd April, 2019 3:06 AM.


A study last year showed that energy drinks can change heart rhythms. A team of researchers at the University of Bonn in Germany scanned the hearts of 17 people an hour after they'd consumed an energy drink. They found the left ventricle of the heart, which pumps blood around the body, was contracting harder after the drink. Frane's public health agency has said young people who drink energy drinks are at risk of sleep disorders, daytime drowsiness and developing addiction to other substances.
Hence the warning: 'Because of their composition, these beverages have a stimulating effect which, when associated with certain other behaviours (alcohol consumption, sports, etc), can give rise to serious cardiac accidents in consumers with common genetic predispositions which frequently go undiagnosed. It is therefore recommended that avoiding the consumption of so-called energy drinks in association with alcohol or during physical exercise should be encouraged.
In 2008, 21-year-old Chioe Leach in East Yorkshire, collapsed and died after drinking Red Bull and a cocktail containing vodka and caffeine at a student party. A post-motem examination found that Chloe probably had an undiagnosed heart condition, and the caffeine she consumed may have triggered faulty electrical activity in her heart. The same year, 14-year-old Naomi Haynes was hospitalised after her energy drink habit reached 50 cans a week. Doctors warned her that her consumption of the drinks was putting her at a risk of a heart attack. Recently, 19-year-old Joshua Merrick died after downing a high-caffeine drink.
Meanwhile, in the U.S, the Food and Drug Administration is said to be investigating a number of deaths and hospitalisations linked to energy drinks. Two mothers are suing the manufacturer of Monster over the deaths of their children from heart attacks. The drinks have also been linked to emotional problems and suicide. Aside from the worries over caffeine energy drinks are extremely high in sugar: one 250ml can of Red Bull contains five-and – a half cubes, for example. Given the growing body of evidence that implicates sugar in a host of health problems, this is fuelling calls for the drinks to be more strongly regulated. With many children drinking several energy drinks a day, there are also concerns over addiction.
Caffeine on its own is not addictive, but it does create dependency, says Peter Rogers, professor of biological psychology at Bristol University and a leading expert on caffeine. According to him: ''Caffeine creates a state of dependency where if one stops taking it, one feels tired, fatigued, headachy and studies show this happens in children as well. Louise Van de Valde believes her son, Jordan, developed an unhealthy relationship with Red Bull at the age of 14.
Now, you've got an impressive array of wines this festive period ,do you know how to store them?
Experts advise you keep unopened wine in a cool, dark place, and store bottles on their sides to keep the corks moist. Dried corks shrink easily and allow air to seep into the wine. White wines may also be stored in the refrigerator, though it isn't necessary. You can simply chill the wine in the fridge two hours before serving.
If you have left-over wine after it's been opened, replace the cork and place the bottle in the fridge. It should keep for three to five days. If you prefer to drink red wine at a warmer temperature, remove it from the fridge and let it sit at room temperature until it warms to your liking. Left-over wine can be used within three to five days after it has been opened.
How to serve wine: White wine is generally best served between 45o F and 55oF, so keep it in the fridge. If it's too cool for your taste, let it sit on a counter top for about 15 minutes before drinking. Red wine is generally best served between 32oF and 35oF.
If the cork breaks in half while you're opening the bottle, gently screw the corkscrew into it and pull it out. If the cork shatters, push it into the bottle, then pour the whine through a cheesecloth or coffee filter into glasses or another container to strain out the pieces. Are you familiar with the vino lingo? ''Sweet: ''Dry''', ''oakey''-what does it all means? Aroma vs ….Bouquet: Aroma is the smell of a young wine. Bouquet refers to the aroma a wine takes on after it has aged in the bottle. Body; the weight of wine in the mouth. Wines are usually described as being either light, medium or full bodies.
Dry: A descriptive terms for wine with no perceptible sugar content.
Finish: The final after-taste of a wine, very good wines tend to have relatively long finishes.
Sweet: A description term for wine with some amount of sugar.
Noise: A wine term synonymous with aroma (e.g. you might say,"''The nose of this wine reminds me of cherries''). Oakey, a descriptive term for a wine that has a pronounced oak flavour, generally as a result of aging the wine in new, small oak barrels. Chardonnay is often oakey.
Tanning: A group of beneficial compounds in wine that come mainly from the grape's skins and seed. Tanning gives wine structure and, because it acts as a natural preservative allow wine to age.
How to Serve Your Wine: If you had unlimited space and an unlimited budget, you could buy a different glass for practically every type of wine. But for the rest of us, here's a simple guide to glassware.
Red Wine: Choose a glass with a broader bowl. This allows you to enjoy more of the bouquet of red wines.
White Wine: Narrower glasses work well for lighter white wines, where the aroma is more delicate.
One Glass: Don't worry if your budget or space permits only one type of wine glass.


Member support is a cornerstone of ASCL's work. Richard Tanton explains the different services and functions provided.
According to our research, one of the key reasons that leaders join ASCL is to tap into the individual support and advice offered through the Member Support Directorate. Alongside its reputation for being able to help in challenging situations, people fi nd it reassuring and encouraging knowing that their professional association has the necessary expertise and knowledge to provide the support that they need in a range of situations.
The directorate is staffed by experienced colleagues with many years of senior school and college leadership. Leading and managing the work falls to Deputy Director Lesley Cooper, Hotline Leader David Snashall and myself as director.
The Hotline is live from 9am to 5pm every day of the year (yes, that does include Christmas Day, Boxing Day and New Year's Day and we do receive calls) and it is usually the first port of call for any member. When it began, its primary function was as a port of call when a member considered that their employment was potentially at risk, and that remains a significant aspect to the service. But members should phone the Hotline if they have other concerns around the issues that they face in an ever-changing and ever-challenging landscape.
In the last 18 months, Hotline has had its busiest ever period. From 1 September to 31 December 2014, we received 2,656 calls to our Hotline. On average we take about 30-40 calls a day on issues such as Ofsted inspections, curriculum developments, pensions, a possible restructuring, a funding concern, how to implement the new recommendations on pay and conditions and any number of other issues. Most questions can be answered on the spot or, if more technical, later that day. In 5 to 6 per cent of cases, the member's call is referred on to one of ASCL's specialists who use their expertise to provide the necessary detailed answer.
Over the last academic year, directorate staff have referred more than 1,100 calls to ASCL regional officers, whose experience of leading schools, together with their knowledge of the regional landscape, are huge assets in analysing a member's specific situation and in considering the options available at any particular time. A referral is made in total confidence.
Casework can be astonishingly complex and a typical individual case handled by a regional officer can last anything from three to 48 months. Regional officers and field officers will nearly always seek to meet with a member, at either their workplace or other venue, to explore the individual situation and any relevant paperwork. From there, depending on the nature of the case, there may be dialogue with employers, heads of HR, the Advisory, Conciliation and Arbitration Service (ACAS) or CEOs of academy chains, as well as discussion with ASCL's legal team (see below) or with me, if appropriate.
There is no typical day for a regional or field officer. They may travel up to 12,000 miles a year, handling up to 35 cases at a time, and are quite frequently working in situations that are in the public domain, and therefore very media-sensitive. The officers possess a depth of skill and knowledge to represent an individual ASCL member in an individual set of circumstances while also looking to scope as many possible options for that person.
The directorate has close working relationships with other areas of ASCL. We meet and discuss custom and practice with Leora Cruddas, ASCL Director of Policy, and have close working ties with our specialists. Whether it is through individual casework, meetings at Council or looking to establish a network of connections with the leading academy chains and, more recently, multi-academy trusts, that cross-fertilisation of practice and knowledge invariably improves the service to our members.
The recent work on embedding a sense of regional identity for ASCL is an example of that cross-fertilisation in action. Regional pilots in the North East and East of England for 2014-15 have been led by the regional officers, working with coordinators Alison Edwards and Joe Pajak. The regional programmes showed that a bespoke series of events, based on regional input and need, are an excellent way to establish a sense of ASCL regionalisation. T Three more regions will be included for 2015 in the North West, South West and East Midlands, to be joined by the other regions in 2016, and this work has meant working closely with both ASCL Professional Development and ASCL Communications. The regional conferences in the summer term, organised by the Member Support Directorate, will epitomise not only these close working links but also regional identity. Dates for the conferences will be published shortly and speakers include ASCL's President Peter Kent, as well as regional school commissioners and regional Ofsted directors.
Many of you may have phoned the Hotline on occasion and have met your regional or field officer at an ASCL event; many will also have had contact with ASCL's Legal Specialist Richard Bird at one of his training courses or have read one of Richard's articles in Leader. It's less likely that you will have met Tim Glover, ASCL's Senior Solicitor, who heads a team of employment law solicitors, which includes Sarah Linden and Aman Patel. Their expertise is a significant feature of member support and it is a reflection of the increased casework and importance of the legal team's work that we will be employing a fourth solicitor from March 2015.
As a member, you may need to seek advice from our solicitors at some point, for example, to consider a potential employment tribunal claim or to ask their opinion about a discussion you are having with an employer. Our solicitors work closely with regional and field officers to ensure that members receive comprehensive, bespoke advice on a confidential basis. Meanwhile, our bespoke training programmes ensure that all member support staff receive regular and appropriate employment law updates. It means that their sound working knowledge is further enhanced by the experience, knowledge and pragmatism of our solicitors.
Thanks in large part to the work of regional and field officers in negotiating settlements we manage to keep the number of tribunal claims issued to a minimum. In 2013-14, the legal department advised on 239 settlement agreements negotiated by regional and field officers (excluding those agreements negotiated by the solicitors on their own cases). Where claims to a tribunal are made, proceedings are often protracted and the average length of a claim is 430 days. We have had great success in representing our members before tribunal claims and particularly so in the last two years, which is a tribute to both regional and legal colleagues.
Over the last 18 months we have been increasingly asked to advise on safeguarding matters and this has been reflected in a higher number of referrals to the National College for Teaching and Leadership (NCTL) and the Disclosure and Barring Service (DBS). Where this is unfortunately the case, our solicitors will represent the individual member and will seek external legal advice if and when appropriate to that specific case. I hope this whistle-stop guide to our work in member support has given you an insight into the service we offer to our members and the quality of the team available to you as an ASCL member.
There is no typical day for a regional or field officer. They may travel up to 12,000 miles a year, handling up to 35 cases at a time, and are quite frequently working in situations that are in the public domain, and therefore very media-sensitive.


Media history is millions, even billions, of years old. That is the premise of Jussi Parikka's pioneering and provocative book, A Geology of Media, which argues that to adequately understand contemporary media culture we must set out from material realities that precede media themselves—Earth's history, geological formations, minerals, and energy.
Jussi Parikka's A Geology of Media really expands what media theory can do. The materiality of media is no longer restricted to questions of economies or technics but extends all the way to its molecular composition. It connects the fast calculations of digital time to the deepest of temporalities, that of the earth itself. An essential contribution to a media theory for the Anthropocene.
Media history is millions, even billions, of years old. That is the premise of this pioneering and provocative book, which argues that to adequately understand contemporary media culture we must set out from material realities that precede media themselves—Earth's history, geological formations, minerals, and energy. And to do so, writes Jussi Parikka, is to confront the profound environmental and social implications of this ubiquitous, but hardly ephemeral, realm of modern life.
Exploring the resource depletion and material resourcing required for us to use our devices to live networked lives, Parikka grounds his analysis in Siegfried Zielinski's widely discussed notion of deep time—but takes it back millennia. Not only are rare earth minerals and many other materials needed to make our digital media machines work, he observes, but used and obsolete media technologies return to the earth as residue of digital culture, contributing to growing layers of toxic waste for future archaeologists to ponder. Parikka shows that these materials must be considered alongside the often dangerous and exploitative labor processes that refine them into the devices underlying our seemingly virtual or immaterial practices.
A Geology of Media demonstrates that the environment does not just surround our media cultural world—it runs through it, enables it, and hosts it in an era of unprecedented climate change. While looking backward to Earth's distant past, it also looks forward to a more expansive media theory—and, implicitly, media activism.
A Geology of Media does not complete or close down an area of research, but rather opens one up. This book is vital to any continuing consideration of media today.
A powerful, illuminating, passionate book rewriting the history and future of media from a much needed materialist perspective.
Parikka prods us to think big, to get past our primordial inhibitions, to look beyond mass media consumerism.
Parikka's book offers refreshing insights into the materiality of digital technologies - that have radically changed cinema too - and can make us place past and theoretical debates into the present.
Parikka points readers toward a more expansive media theory in ways that no other researcher has.
A satisfying and challenging book.
Parikka's invaluable book will prompt a myriad of important conversations within his discipline over the nature of media and technology.
A Geology of Media offers a greatly expanded definition of media materialism, productively redefines the scope of media archaeology, and nuances the discourse of ecocritical media theory with its emphasis on the importance of the nonorganic world.
With A Geology of New Media, Parikka not only expands and vitalizes the fields of media theory and media history, he also forces the humanities at large to rethink its methods and objectives.
A Geology of Media is an excellent book, which mixes cultural theory and history with geological science and contemporary art.
Radical in its far-reaching and interdisciplinary approach, and welcome for being so, the scope of A Geology of Media reflects its topical intricacy whilst reshaping the arenas of discourse in which interrogations of an evolving, non-discrete, complex of media cultures can take place.
A provocative book that succeeds in proposing a potentially vast field of study.
A Geology of Media provides rich theoretic interventions and examples that expand on the increasing scholarship on the Anthropocene, materiality, and waste.
A welcome contribution to this relevant area of study.
Jussi Parikka deeply has examined the rawest matter of media.
In A Geology of Media, Jussia Parikka offers a refreshingly raw materiality approach to media studies.
What is virtual, or photographic, fracking?
Silicon Valley's rhetoric of magical innovation relies on a hidden abode of rare earth mining and hydro-cooled server farms. A review of Jussi Parikka's A GEOLOGY OF MEDIA.
Humans are a doubly young species — we haven't been around for long, and we don't live for long either. Review of Jussi Parikka's book.
Review of Jussi Parikka's A GEOLOGY OF MEDIA.
In A Geology of Media, Jussi Parikka continues his innovative conflation of media production with the changed concepts of nature.
Parikka understands the human as an after-effect of the material and formal construction of its contexts, expanding the horizon again, however, to understand the human as an after-effect of geological time.


text:
Irish Water invests in new Water Recycling Jet-Vac Unit to ensure delivery of first class service for Fingal customers
3 August 2018 Go back to News
Recycling Jet-Vac Unit
The new state of the art Water Recycling Jet-Vac Unit will replace 20 year old traditional jetting units
Irish Water has invested in a new Water Recycling Jet-Vac unit to replace ageing units in Fingal. The new Recycler is part of the Irish Water's ongoing strategy to improve the service delivered to customers.
The upgrade will see a new Recycler unit coming into service in Fingal to replace the existing units currently operating some of which are more than 20 years old. This will ensure that staff working in Fingal will be able to deliver a more effective and efficient service to our customers.
The new, modern unit will enable Fingal County Council staff working on behalf of Irish Water to carry out their work in greater safety, while also being able to set up the Recycler on site in a shorter time frame when dealing with blockages and general maintenance in the sewer network. The new high tech truck has an enhanced CCTV system to identity where blockages are in the sewers which will enable faster cleaning of the sewers, remove blockages, and complete high pressure cleaning. Similar Recycling Jet-Vac units are in use in other water utility companies in the UK and Europe and will enable operators to carry out this essential sewer maintenance and emergency repair work safely and efficiently.
Bringing this new vehicle into service will also mean that there will be fewer breakdowns, improved working conditions for staff and a better service on site when emergencies arise.
Martin Carroll of Irish Water, commented: "We are delighted to be in a position to hand over this new Recycler unit to Fingal County Council. This investment in a modern, efficient truck will enable personnel to carry out their work in a much safer and more efficient manner, ultimately improving the service we offer to our customers.
"Having a safe, properly managed modern unit will enable more effective inspection, servicing and maintenance of Irish Water's wastewater network. The new Recycler will enable us to efficiently resolve hundreds of wastewater blockages across Fingal.
"The ageing Jet Vac units that are being replaced had to be filled with clean drinking water from local hydrants. This drinking water is much too valuable for cleaning sewers. A key priority for Irish Water when investing in the new Recycler unit is that future sewer cleaning will be carried out using recycled water. This is a sustainable solution for cleaning and maintaining our wastewater network.
"This investment is in line with Irish Water's commitment to developing a systematic approach to the operation and maintenance of the water and wastewater networks in order to deliver a quality service to all our customers."
Speaking about the new recycler, Gilbert Power, Director of Services Fingal County Council said "The new recycler is part of Irish Water and Fingal's ongoing strategy to improve the service delivered to customers. This will bring new levels of efficiency to the sewer cleaning that has not been seen before in Ireland."
The new Wiedemann/UTS truck recently came into service. Extensive consultation has taken place with the local authority staff and management to ensure that this vehicle is fit for purpose and will meet all operational requirements.
Irish Water, in partnership with local authorities, plans to invest over €10m in upgrading its fleet of vans and Jet Vacuum trucks nationally over the coming 18 months. This will be essential in order to deliver on the company's commitment to deliver quality, fit-for-purpose water services for all our customers. Nationally, 40% of the fleet of water service vehicles is more than six years old and 20% is more than 10 years old.
Martin Carroll Irish Water, Dave O'Rourke Fingal County Council, Willie Garbutt Ervia, Michael King Fingal County Council and David Breen Ervia
Repairs to burst water main close to Vartry water treatment plant continue
Further progress made on the Howth Water Supply Scheme, safeguarding water supply for homes and businesses in the area
Irish Water and Fingal County Council are working to restore water supply to customers in parts of Blanchardstown following a burst water main


Urzila Carlson has been a part of the local comedy scene for a while now but I believe she hit the comedic jackpot last year with her now famous catchphrase "don't go falling in love now!". Since then she has become an undeniable local favorite.
Don't let her South African accent throw you; she now calls Aotearoa home and we have been more than happy to return the favor, proudly embracing her as our own. The Long Flight To Freedom, put simply, is an autobiographical recount of how Urzila came to be in New Zealand.
Throughout the hour, she shares candidly a collection of comical stories that span from birth to childhood and finally to present time. Dreaded family holidays at the farm, a memorable family team building exercise involving a resident mouse, tips on dealing with noisy neighbors and her genius plan on disciplining her soon-to-be-born "Jesus baby" – Urzila bares all. Her casual delivery and laidback style made it feel more like a conversation among friends which suited the narrative perfectly.
There is no doubt that Urzila was born to do comedy. From start to finish she had the audience hooked on her every word. Her ability to garner laughs from the crowd so effortlessly coupled with her skill in delivering material so naturally that it feels completely unscripted is a real gift and mark of a brilliant comedian.
This is an entertaining hour of getting-to-know-Urzila that will bring endless laughs. Everybody needs to see this extremely talented lady live – here's hoping for an encore, and in a bigger venue!
The Whimsical Banana rates The Long Flight To Freedom: 5/5 bananas!
Urzila Carlson has already sold out her entire season but you can still see her in action as co-host at Le Comique this Sunday May 12th – for more details and tickets to this event, click here.