Yan Liua111Email<EMAIL_ADDRESS>and Ya-Wen Sunb,c222Email:
<EMAIL_ADDRESS>aCenter for Gravitational Physics, Department of Space
Science, Beihang University, Beijing 100191, China
bSchool of physics $\&$ CAS Center for Excellence in Topological Quantum
Computation, University of Chinese Academy of Sciences, Beijing 100049, China
cKavli Institute for Theoretical Sciences, University of Chinese Academy of
Sciences, Beijing 100049, China

###### Contents

  1. 1 Introduction
  2. 2 A short review of relativistic hydrodynamics
  3. 3 Topological hydrodynamic modes
    1. 3.1 A simple introduction of topological states of matter
    2. 3.2 Effective Hamiltonian
    3. 3.3 Gapping the system
    4. 3.4 Topologically nontrivial nodes
      1. 3.4.1 The single 4D hydrodynamic system
      2. 3.4.2 Two 2D hydrodynamic systems
      3. 3.4.3 More possibilities with topological hydrodynamic modes
  4. 4 Possible origin for non-conservation terms of $T^{\mu\nu}$
    1. 4.1 Origin I: Effective external tensor matter field
    2. 4.2 Origin II: Gravitational field, non-inertial reference frame
      1. 4.2.1 Symmetry of the single 4D system
      2. 4.2.2 non-inertial reference frame
  5. 5 Topological invariant
  6. 6 Transport properties
    1. 6.1 Single 4D heat transport
    2. 6.2 2D+2D heat transport
  7. 7 $\mathcal{O}(k^{2})$ effects
  8. 8 Ward identities and holographic realization
    1. 8.1 Ward identities
    2. 8.2 Holographic realization
      1. 8.2.1 Prescription for holographic Ward identities
      2. 8.2.2 Ward identities in holographic non-inertial frame systems
  9. 9 Conclusions and discussions
  10. A Hydrodynamics with momentum dissipation

##  1 Introduction

Topological states of quantum matter have been studied extensively in
condensed matter physics during the last decade [1, 2]. Well known topological
states include anomalous Hall effect, topological insulators, topological
superconductors, which are gapped, and various kinds of topological
semimetals, e.g. Weyl/nodal line/Dirac semimetals, which are gapless
topological states of matter.

The earliest topological states of matter were all found in quantum electronic
systems, however, soon it was realized that the topological property is not
fundamentally quantum but is associated with the wave property of the system.
In 2008, topological states in classical/bosonic systems have been predicted
in several systems in [3, 4, 5]. Moreover, classical topological states of
matter were realized experimentally in gyromagnetic photonic crystals at
microwave frequencies [5]. Ever since then, the study in classical topological
states of matter has been grown rapidly, see e.g. [6] and references therein.
It has been found that many classical systems have nontrivial topological
states too, e.g. topological optical/sound systems (see e.g. [7, 8, 9] and
references therein), which have also been confirmed experimentally.

In a recent paper [10], we have shown that semimetal-like topologically
nontrivial modes could be found in relativistic hydrodynamics by weakly
breaking the conservation of energy momentum. Hydrodynamics describes the
classical behavior of a system of continuum away from its local equilibrium in
the long wavelength and low frequency limit. The dynamics of a system in the
hydrodynamic limit is governed by conservation equations of conserved
currents, including the energy momentum tensor and the conserved charges of
internal symmetries. In the simplest case with the only conserved quantity
being the energy momentum tensor, we found that a special form of non-
conservation terms of the energy momentum tensor in the conservation equation
could deform the spectrum of hydrodynamics to a shape similar to that of
topological semimetals.

We provided an explicit example with this kind of behavior in a 4D
hydrodynamic system in [10] and in that case, the nontrivial topology of the
system requires the protection of a special spacetime symmetry. In this paper,
we generalize the findings to more systems with similar behavior and provide
more detailed calculations on various aspects. Besides the single 4D system
found in [10], we find several interacting hydrodynamic systems with two
energy momentum sectors exchanging energy and momentum with each other as well
as with the environment. Besides the fact that the crossing nodes in these
systems will not become gapped under small perturbations, we will also show
evidence of the nontrivial topology from the calculation of topological
invariants. We also calculate the second order effects in detail in this paper
for the single 4D system and show explicitly that the dissipative second order
part contributes to an imaginary part in the dispersion relation of the
hydrodynamic modes. This imaginary part shows a jump at the topologically
nontrivial crossing node, seeming to provide another piece of evidence that
the states near the nodes are not adiabatically connected to each other and
the crossing nodes are topologically nontrivial.

The non-conservation terms of the energy momentum tensor could come from an
appropriately chosen external matter field or a gravitational field. In the
latter case, we could start from a flat spacetime and perform a specific
reference frame transformation. The new spacetime could be viewed as a non-
inertial reference frame. This suggests that topologically trivial
hydrodynamic modes could become topologically nontrivial viewed by an observer
accelerating in a specific way.

As hydrodynamics has been studied extensively for strongly coupled systems in
holography with many important results [11, 12], e.g. the prediction of a KSS
bound as the lower bound for the shear viscosity over entropy ratio [13],
which has been confirmed in experiments, we also propose a holographic
realization for the single 4D system that have topologically nontrivial
hydrodynamic modes. We start from ordinary AdS/CFT correspondence and perform
a specific reference frame transformation to get a non-inertial reference
frame version of AdS/CFT correspondence. We develop a new method to calculate
holographic Ward identities for the energy momentum tensor without calculating
out all components of the Green functions. Using this method we show that the
holographic Ward identities from our new holographic set-up match to the Ward
identities of the hydrodynamic systems with specific non-conservation terms of
the energy momentum tensor. This provides the evidence that we have found a
strongly coupled holographic system which has topologically nontrivial
hydrodynamic modes.

In the rest of the paper, we will first review some basics of relativistic
hydrodynamics in section 2. In section 3, we introduce the notion of effective
Hamiltonians and show several systems which have topologically nontrivial
hydrodynamic modes by adding non-conservation terms of the energy momentum
tensor. In section 4 we show the possible origins for the non-conservation
terms of the energy momentum tensor, emphasizing the role of non-inertial
reference frames. The symmetry of the single 4D system is derived and the
symmetry needed for the protection of the topological nodes is also obtained
in this section with the help of the external metric field. We calculate the
nontrivial topological invariants in section 5 and the effects of these extra
terms for transport properties in section 6. In section 7, we consider second
order effects to these systems. In section 8 we introduce the holographic
realization for the single 4D system and match the Ward identities from
holography to the hydrodynamic ones. Section 9 devotes to conclusions and open
questions.

##  2 A short review of relativistic hydrodynamics

Hydrodynamics is the universal low energy theory for systems at long distance
and late time. It could describe a variety of physical systems ranging from
matter at large scales in the universe, the quark-gluon plasma [14], to Weyl
semimetals [15, 16] and graphenes [17] in the laboratory. At small momentum
and frequency, perturbations of a hydrodynamic system away from the
equilibrium would produce sound and transverse modes [18]. These modes are
gapless whose poles are at $\omega={\bf k}=0$, which reflects the fact that
energy momentum is conserved. In this paper we focus on the simplest
hydrodynamic systems with no internal charges whose only conserved quantity is
the energy momentum tensor.

The conservation equation for the energy momentum tensor in $d+1$ dimensions
is

$\displaystyle\partial_{\mu}T^{\mu\nu}$ $\displaystyle=$ $\displaystyle 0\,.$
(2.1)

Up to the first order in derivative, the constitutive equation for the energy
momentum tensor in the Landau frame is

$\displaystyle T^{\mu\nu}$ $\displaystyle=$ $\displaystyle\epsilon
u^{\mu}u^{\nu}+P\Delta^{\mu\nu}-\eta\Delta^{\mu\alpha}\Delta^{\nu\beta}\big{(}\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha}-\frac{2}{d}\eta_{\alpha\beta}\partial_{\sigma}u^{\sigma}\big{)}-\zeta\Delta^{\mu\nu}\partial_{\alpha}u^{\alpha}+\mathcal{O}(\partial^{2})\,,$

where $\Delta_{\mu\nu}=\eta_{\mu\nu}+u_{\mu}u_{\nu}$, $\epsilon$, $P$ are the
energy densities and pressure and $\eta$, $\zeta$ are the shear and bulk
viscosities. For conformally invariant systems, the bulk viscosity $\zeta=0.$
For $d=1$ conformal fluid, the first order transport coefficients $\eta$ and
$\zeta$ are both zero.

At equilibrium the system has an energy density $T^{00}=\epsilon$ and pressure
$T^{ii}=P$. With small perturbations slightly away from equilibrium, the
system would respond to the perturbations and develop hydrodynamic modes. The
equations for perturbations of the energy momentum tensor are

$\displaystyle\partial_{\mu}\delta T^{\mu\nu}=0\,,$ (2.2)

where in four dimensions

$\displaystyle\begin{split}\delta T^{00}&=\delta\epsilon\,,\\\ \delta
T^{0i}&=\delta T^{i0}=(\epsilon+p)\delta u^{i}=\delta\pi^{i}\,,\\\ \delta
T^{ij}&=g^{ij}v_{s}^{2}\delta\epsilon-\frac{\eta}{\epsilon+P}(\partial^{i}\delta\pi^{j}+\partial^{j}\delta\pi^{i})+\frac{\frac{2}{3}\eta-\zeta}{\epsilon+P}g^{ij}\partial_{\alpha}\delta\pi^{\alpha}\,\end{split}$
(2.3)

with $v_{s}=\sqrt{\frac{\partial P}{\partial\epsilon}}$. We keep the momenta
in all directions nonzero for later convenience, i.e. ${\bf
k}=(k_{x},k_{y},k_{z})$ in four dimensions. After solving the conservation
equation and diagonalizing the solutions, there will be four eigenmodes of the
system. Two of them are the sound modes propagating in the direction of ${\bf
k}=(k_{x},k_{y},k_{z})$ with the dispersion relation $\omega=\pm
v_{s}k-i\frac{\Gamma_{s}}{2}k^{2}$ , where
$\Gamma_{s}=(\frac{4}{3}\eta+\zeta)/(\epsilon+P)$. The other two are
transverse modes with $\omega=-i\frac{\eta}{\epsilon+P}k^{2}$. These modes
also lead to poles in the Green functions of various components of the energy
momentum tensor at low frequency.

To the first order in $k$, dissipative terms disappear and the spectrums of
the four modes are real, which cross each other at $\omega={\bf k}=0$ (see the
left plot in Fig. 1).333A recent study on the stability issue for the first
order hydrodyanmics can be found in [19]. This spectrum looks similar to the
spectrum of Dirac semimetals (the middle plot in Fig. 1), except that we have
two flat bands here. In the next section we will show that a gap could also
open in these hydrodynamic modes after introducing non-conservation terms for
the energy momentum tensor and with more non-conservation terms, the
hydrodynamic system could develop gapless topologically nontrivial modes
similar to the Weyl semimetal states (the right plot in Fig. 1).

Figure 1: The spectrums in relativistic hydrodynamics for $\omega,k\ll T$
(left), Dirac semimetal (middle) and Weyl semimetal (right).

##  3 Topological hydrodynamic modes

The spectrum of the relativistic hydrodynamic modes crosses at $\omega={\bf
k}=0$ as a consequence of energy momentum conservation. To deform the spectrum
of the hydrodynamic modes and find nontrivial topological structure in the
hydrodynamic modes, we introduce non-conservation terms for the energy
momentum tensor in (2.1) and make sure that the non-conservation terms are
small enough to stay within the hydrodynamic limit. The non-conservation of
energy and momentum could come from a certain external system which couples to
the hydrodynamic system that we study. We assume that the constitutive
equations for perturbations of the hydrodynamic system being considered do not
change or the change does not affect the final spectrum and the change will be
discussed in section 4, while the conservation equations would change due to
interchange of energy momentum with external systems. In this section, we
consider dissipativeless non-conservation terms that deform the spectrum of
the hydrodynamic modes. Before that we will first introduce some basic
knowledge about topological states of matter and calculations of topological
invariants.

###  3.1 A simple introduction of topological states of matter

In this subsection, we introduce very briefly some basic knowledge about
topological states of matter, especially gapless ones. Topological states of
matter have attracted a lot of interest among condensed matter physicists
during the last years. Topological states of matter are states that have
topologically nontrivial band structure in the momentum space. Here we use the
terminology “bands” as in electronic systems and in fact “bands” could refer
to any structure of the spectrum from the eigenvalues of a Hamiltonian.
Various kinds of gapped and gapless topological states of matter have been
found. Gapped topological states of matter include topological insulators,
anomalous Hall effect, topological superconductors, etc., and gapless
topological states of matter include Weyl semimetals, nodal line semimetals,
etc..

In gapped topological states of matter, the bands are separated by a nonzero
energy interval and because of the nontrivial topology of the band structure
the system cannot be adiabatically deformed to a trivial vacuum state without
closing the gap. Gapless topological states of matter have connected
conduction and valence bands in the spectrum, and they cannot be adiabatically
deformed to a trivial gapless state, which is a state whose gap could open due
to arbitrarily small perturbations of the system. Gapless topological states
could not be gapped by an infinitesimal perturbation of the system.

Fig. 2 gives an illustration of band crossings that behave differently in
their topological properties. The left plot in Fig. 2 shows an accidental band
crossing which would be gapped under an arbitrarily small perturbation and the
right plot in Fig. 2 shows a topologically nontrivial band crossing which is
still gapless under small perturbations.

In the figure each curve with the same color denotes the same set of
eigenstates with an eigenvalue depending on $k$ as a continuous function
$E_{1,2}(k)$. The same set of eigenstates and their eigenvalues would deform
continuously under parameter change of the system. The two curves in each
figure indicate two sets of eigenstates with two different eigenvalues
$E_{1,2}(k)$. At certain values of $k=k_{0}$, the two eigenvalues coincide
with $E_{1}(k_{0})=E_{2}(k_{0})$ and we have band crossings at these $k_{0}$.
Then at each of these band crossing points, there are two degenerate
eigenstates. Then in the left figure of Fig. 2, under a small perturbation
that moves the upper band up while the lower band down, the system would
immediately develop a gap, while in the right figure, under a small
perturbation that modifies the curves slightly, the band crossings could not
vanish no matter how the two curves move. The right figure represents a
topologically nontrivial gapless state while the left one gives a trivial
gapless state.

Figure 2: An accidental band crossing (left) and a topologically nontrivial
band crossing (right).

For topologically nontrivial gapless states, the behavior in the previous
paragraph could be explained by the fact that band crossing nodes are
protected by a nontrivial topological structure and each possesses a
nontrivial topological invariant. For example, a Weyl node is protected by a
nontrivial topological monopole charge, which could be calculated from the
surrounding first Chern number in the three dimensional momentum space. Thus
to know if a gapless state is topologically nontrivial or not, there are two
ways which should be consistent with each other: one is to see if arbitrarily
small perturbations could gap the nodes and the other is to calculate the
topological invariant and see if it is different from the value obtained for a
trivial vacuum state. More details about calculation of topological invariants
for gapless systems in various dimensions would be found in Section 5.

###  3.2 Effective Hamiltonian

Before calculating the hydrodynamic modes in different situations, we first
review our definition of the effective Hamiltonian in [10], which is a
physical quantity parallel to the Hamiltonian matrix of topological electronic
systems [20]. In the case of electronic systems, Dirac fermions are four
component fields, thus the Hamiltonian from $i\partial_{t}\Psi=H\Psi$ is a
matrix whose eigenvalues give the spectrum of the electronic system. In
hydrodynamics, the modes that we consider are perturbations of the energy and
momentum densities, thus there are also multiple components of fields in the
equation. We could in principle define a similar matrix of effective
Hamiltonian whose eigenvalues give the spectrum of the hydrodynamic modes.

We start from the conservation equation of energy momentum tensor which
determines the dynamics of hydrodynamic modes. Substituting the constitutive
equations for the perturbations (2.3) into $\partial_{\mu}\delta
T^{\mu\nu}=0,$ we could rewrite the equations for the perturbations of energy
and momentum densities into

$\displaystyle i\partial_{t}\Psi=H\Psi$ (3.1)

where we have defined

$\displaystyle\Psi=\begin{pmatrix}\delta\epsilon\\\ \delta\pi^{x}\\\
\delta\pi^{y}\\\
\delta\pi^{z}\end{pmatrix}\,,~{}~{}~{}~{}~{}~{}H=\begin{pmatrix}0&~{}~{}k_{x}&~{}~{}k_{y}&~{}~{}k_{z}\\\
k_{x}v_{s}^{2}&~{}~{}0&~{}~{}0&~{}~{}0\\\
k_{y}v_{s}^{2}&~{}~{}0&~{}~{}0&~{}~{}0\\\
k_{z}v_{s}^{2}&~{}~{}0&~{}~{}0&~{}~{}0\end{pmatrix}\,$ (3.2)

at leading order in $k$, i.e. omitting dissipative terms at
$\mathcal{O}(k^{2})$.

In analogy to the electronic systems, in this way we have defined an effective
Hamiltonian matrix $H$ whose eigenvalues give the spectrum of hydrodynamic
modes.444Note that this effective Hamiltonian is different from the
hydrodynamic Hamiltonian, e.g. in section 2.4 of [18] in the sense that this
effective Hamiltonian matrix $\Psi^{T}(-k)H\Psi(k)$ gives the “kinetic” part
of the Hamiltonian while the latter only counts in the potential part
associated with the source terms. The four eigenvalues of the matrix
Hamiltonian above give the sound modes $\omega=\pm
v_{s}\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$ and double copies of transverse
modes $\omega=0$. The form (3.2) is the “free” Hamiltonian matrix for a
conserved energy momentum tensor.

###  3.3 Gapping the system

With exact energy momentum conservation the hydrodynamic modes are “band
crossed” at $\omega={\bf k}=0$. However, when the energy momentum tensor is
not conserved, the modes might become gapped. To introduce nontrivial
topological structure into the spectrum of hydrodynamic modes, the first step
is to generate gaps in the modes as in electronic systems. In this section we
will directly modify the effective Hamiltonian to find terms that we need and
later in section 4 we will show how these terms could arise from external
fields.

We modify the conservation equation for $T^{\mu\nu}$ to

$\displaystyle\begin{split}\partial_{\mu}\delta T^{\mu t}&=m_{1}\delta
T^{tx}\,,\\\ \partial_{\mu}\delta T^{\mu x}&=-m_{2}v_{s}^{2}\delta
T^{tt}\,,\\\ \partial_{\mu}\delta T^{\mu y}&=0\,,\\\ \partial_{\mu}\delta
T^{\mu z}&=0\,,\end{split}$ (3.3)

where $m_{1,2}$ are parameters leading to a gap in the spectrum of the
hydrodynamic modes. To stay in the hydrodynamic limit, $m_{1,2}$ has to be
small compared to $T$. Here we have introduced non-conservation of energy and
momentum in the $x$ direction and this non-conservation has to be in a
specific form as indicated above. The amount of non-conserved
energy/$x$-direction momentum density is proportional to the $x$-direction
momentum/energy density and the sign of $m_{1,2}$ has to be the same. This
sign could be either positive or negative indicating that the modes could
either gain or lose energy. For simplicity, in the following we will choose
$m_{1}=m_{2}=m$.

At leading order in $k$ the effective Hamiltonian becomes

$\displaystyle H=\begin{pmatrix}0&~{}~{}k_{x}-im&~{}~{}k_{y}&~{}~{}k_{z}\\\
(k_{x}+im)v_{s}^{2}&~{}~{}0&~{}~{}0&~{}~{}0\\\
k_{y}v_{s}^{2}&~{}~{}0&~{}~{}0&~{}~{}0\\\
k_{z}v_{s}^{2}&~{}~{}0&~{}~{}0&~{}~{}0\end{pmatrix}\,$ (3.4)

and $k^{2}$ effects will be studied in section 7. The four eigenvalues of the
matrix $H$ are now

$\displaystyle\omega=\pm v_{s}\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}+m^{2}}$
(3.5)

and double sectors of $\omega=0$ . This shows that $m$ terms indeed gap the
sound modes while do not change the transverse modes. Fig. 3 shows the “band
structure” of the system at nonzero $m$. We can see that sound modes only
exist above/below a certain frequency while transverse modes are not affected.

Figure 3: The spectrum of the modified hydrodynamics with dynamical equation
(3.3). Here $m_{1}=m_{2}=m$ and it plays the role to gap the spectrum of sound
modes. Here and in all the following pictures $\omega,k,m\ll T$.

Effects in the spectrum here would be the same if the momentum $m$ term is in
the $y$ or $z$ direction instead of the $x$ direction, or if $m$ terms in all
$x$, $y$ and $z$ directions exist. This is important later when we need to
check carefully if the topological structure of the modes would not be
destroyed by these $m$ terms in all directions or only in certain directions.
Here we also emphasize that gapping the system is a first step to realize
topologically nontrivial gapless states as we need to show that these mass
terms which could gap the standard hydrodynamic system will not gap the
topologically nontrivial gapless state that we introduce later confirming that
they are indeed topologically nontrivial. However we are not saying that the
gapped state in this section is topologically nontrivial.

###  3.4 Topologically nontrivial nodes

After gapping the system, we could continue to search for possible
modifications to the effective Hamiltonian that change the topologically
trivial crossing nodes of the hydrodynamic modes into nontrivial ones with
more complicated “band structures”. This procedure is similar to introducing
time reversal symmetry breaking terms in a Dirac system to obtain a
topologically nontrivial Weyl semimetal. Here we need to find a topologically
nontrivial spectrum by introducing modifications to the effective Hamiltonian.
The possible modifications here are not arbitrary as most deformations of the
effective Hamiltonian would give non-meaningful results. We have tried a lot
of possible deformations to finally find the ones that could realize the
topologically nontrivial spectrum here.

There are more than one ways to realize this and we list several possibilities
in this section, including the single 4D hydrodynamic system found in [10],
and interacting hydrodynamic systems with two sectors who further exchange
energy and momentum with each other.555Hydrodynamical systems with multiple
sectors of energy momentum tensors and currents have also been studied in e.g.
[16]. Further possibilities and possible ways to get topologically nontrivial
gapped hydrodynamic modes will be left for future work.

####  3.4.1 The single 4D hydrodynamic system

The first possibility is to take a 3+1D hydrodynamic system of (3.3) and
further break the translational symmetry in the $y$ and $z$ directions, as
found in [10]. The momentum in the two directions is non-conserved in a
specific form so that the resulting spectrum of the hydrodynamic modes would
develop crossing nodes.

More precisely, the conservation equation is now

$\displaystyle\begin{split}\partial_{\mu}\delta T^{\mu t}&=m_{1}\delta
T^{tx}\,,\\\ \partial_{\mu}\delta T^{\mu x}&=-m_{2}v_{s}^{2}\delta
T^{tt}\,,\\\ \partial_{\mu}\delta T^{\mu y}&=b_{1}v_{s}\delta T^{tz}\,,\\\
\partial_{\mu}\delta T^{\mu z}&=-b_{2}v_{s}\delta T^{ty}\,,\end{split}$ (3.6)

where $m_{1,2}$ terms are mass terms while $b_{1,2}$ terms change the momentum
position of the crossing nodes in the spectrum. Here we have used different
values of mass and $b$ terms in equations for generality and in the following
we will take $m_{1}=m_{2}=m$ and $b_{1}=b_{2}=b$ for simplicity.

Note that in the attempt to modify the effective Hamiltonian to get
topologically nontrivial hydrodynamic modes, we have to keep the modified
effective Hamiltonian in a Hermitian form or similar to a Hermitian matrix at
order $\mathcal{O}(k)$. The eigenvalues of $H$ would then be real and there
would be no dissipative effects at order $\mathcal{O}(k)$.

After substituting the constitutive equation (2.3) into the conservation
equation above, the equation (3.6) can be written as

$\displaystyle i\partial_{t}\Psi=H\Psi$ (3.7)

where

$\displaystyle\Psi=\begin{pmatrix}\delta\epsilon\\\ \delta\pi^{x}\\\
\delta\pi^{y}\\\
\delta\pi^{z}\end{pmatrix}\,,~{}~{}~{}~{}~{}~{}H=\begin{pmatrix}0&~{}~{}k_{x}+im&~{}~{}k_{y}&~{}~{}k_{z}\\\
(k_{x}-im)v_{s}^{2}&~{}~{}0&~{}~{}0&~{}~{}0\\\
k_{y}v_{s}^{2}&~{}~{}0&~{}~{}0&~{}~{}ibv_{s}\\\
k_{z}v_{s}^{2}&~{}~{}0&~{}~{}-ibv_{s}&~{}~{}0\end{pmatrix}\,.$ (3.8)

Redefine $\delta\epsilon\to\frac{1}{v_{s}}\delta\epsilon$, it is easy to see
that $H$ is similar to a Hermitian matrix where $v_{s}$ becomes an overall
scaling factor, thus this effective $H$ has real eigenvalues. In the following
for simplicity we will ignore the factor of $v_{s}$ and when necessary it can
be taken back by an inverse redefinition of
$\delta\epsilon\to\frac{1}{v_{s}}\delta\epsilon$.

The spectrum of the hydrodynamic modes are now

$\displaystyle\omega=\pm\frac{1}{\sqrt{2}}\sqrt{b^{2}+k^{2}+m^{2}\pm\sqrt{(k_{x}^{2}+m^{2}-b^{2})^{2}+(k_{y}^{2}+k_{z}^{2})^{2}+2(k_{y}^{2}+k_{z}^{2})(k_{x}^{2}+m^{2}+b^{2})}}\,,$

where $k=\sqrt{k_{x}^{2}+k_{y}^{2}+k_{z}^{2}}$.

Figure 4 shows this spectrum as functions of $k_{x}$ for $k_{y}=k_{z}=0$ in
three different situations: $m<b$, $m=b$ and $m>b$ as well as for
$k_{y}>0,k_{z}=0$ at $m<b$. The effect of $b$ terms is to lift and lower the
two transverse flat bands to symmetric positions of opposite sides of the $k$
axis. The effect of $m$ terms is still to gap the two sound modes. Now the
sound and transverse modes are mixed with each other. In this way, the modes
have band crossings at nonzero values of $k$ for $m<b$.

Figure 4: The spectrum of the modified hydrodynamics with dynamical equation
(3.6). From left to right: in the first three plots we have $m<b$, $m=b$ and
$m>b$ respectively and $k_{y}=k_{z}=0$. The fourth plot is for
$k_{y}>0,k_{z}=0$ and $m<b$.

From figure 4, we could see that for $m<b$ now we have four band crossing
nodes at $k_{y}=k_{z}=0$ while $k_{x}\neq 0$, though $\omega\neq 0$ for these
modes. Note that in condensed matter terminology, this kind of system is
called gapless as there are band crossings in the spectrum. However, this is
different from the usual standard “hydrodynamic” definition of gapless, which
means that $\omega(k=0)=0$. Thus here we do not follow the condensed matter
terminology to call this kind of system gapless. Instead, we call them band
crossings from here on. Similar to the Weyl semimetal case, these four nodes
are still points in the expanded space of $\omega$, $k_{x}$, $k_{y}$ and
$k_{z}$ as can be seen from the fourth plot in figure 4. For $m=b>0$, the
system becomes critical with 2 nodes and for $m>b$ the band crossings in the
system disappear again. This behavior is qualitatively similar to the
topological phase transition of a Weyl semimetal [21]. For $m<b$, now there
could still be band crossings though small perturbations of these $m$ terms
could previously gap the system and the gap is still there at $\omega=0$. This
property is a defining feature of a topologically nontrivial gapless state,
which we call the topologically nontrivial band crossing state here to avoid
the conflict with terminology in hydrodynamics. We will show later that this
is in fact a symmetry protected topological band crossing state.

Note that different from the Weyl semimetal case, here the band crossings are
not at $\omega=0$. Similar to what happens in condensed matter physics, the
nodes do not need to be at $\omega=0$ for them to be topologically protected
nodes and the important ingredient is that the nodes that we study need to be
real band crossings instead of accidental band touchings. Topologically
nontrivial band crossing states are those states that possess a spectrum with
no gap at any $\omega$ and at the same time the spectrum would not develop a
gap under small perturbations. In this sense, we need two bands to cross so
there would be band crossings instead of all bands being fully separated in
the spectrum (again at any $\omega$) and for the band crossings to be
topologically nontrivial we need the band crossings to be stable under small
perturbations. Also another reason that the nodes do not need to be at
$\omega=0$ is that the absolute value of energy is not important and we
usually study effective excitations near each of the band crossing points so
the energy at the band crossings could each be set to zero effectively.

We also need to emphasize here that the definition of a topologically band
crossing state is a state whose band crossings would not disappear under small
perturbations. This is because as long as a band crossing could disappear
under small perturbations, this would mean that the band crossed state is
topologically equivalent to the trivial vacuum. Thus for a topologically
nontrivial band crossing state, by being topologically protected, we refer to
the fact that the state remains band crossed under small perturbations of the
system which usually could gap a system. This indicates that the band crossing
points should be singular points in the momentum space. Being topologically
inequivalent to the trivial vacuum, this topologically nontrivial band
crossing state should possess a nontrivial topological invariant which takes a
different value compared to the trivial vacuum, e.g. the two nodes of a Weyl
semimetal possess nontrivial chiral charges which are different from the
trivial value of zero thus the two nodes cannot disappear under small
perturbations of the system. The topology of states depends on the Hamiltonian
or equivalently the wave functions and is an intrinsic property of the
Hamiltonian.

The $m$ terms in (3.8) do not eliminate the four band crossing nodes in the
$m<b$ case, however, if we have $m$ terms in the $y$ or $z$ directions, the
band crossing will disappear no matter how small the $y$ or $z$ mass
parameters are as shown in figure 5. In this situation, there are no crossing
nodes anymore. According to the definition above, the nodes are not
topologically protected as they could be destroyed by $m$ terms in the $y$ or
$z$ directions. These in fact belong to symmetry protected topologically band
crossing states, which could only endure perturbations that respect the
symmetry. Here the nodes should be topologically nontrivial under the
protection of the symmetry that forbids the $m$ terms in the $y$ and $z$
directions as will be further confirmed in section 5, i.e. the symmetry that
would be broken by $m$ terms in the $y$ and $z$ directions. As a rough
estimate, the symmetry that we need here should be related to the
translational symmetry in the $y$ and $z$ directions. This symmetry is a
restriction to the system while not the solutions. The explicit form of this
symmetry will be shown in section 4.2.1 from a more accurate calculation. In
this sense, the system (3.6) experiences a symmetry protected topological
phase transition which happens at the critical point $m=b$.

Another thing that we need to emphasize here is that the fact that $m$ terms
in the $y$ and $z$ directions would eliminate the band crossings in the system
has nothing to do with the fact that the system has no band crossing at
nonzero $k_{y}$, $k_{z}$ and also $k_{x}$ away from the node values. The
notion of band crossing only requires a band crossing in the whole momentum
space so though the system has no band crossing at nonzero $k_{y}$, $k_{z}$
and also $k_{x}$ away from the node values, the system is still a band crossed
state because of the band crossing at $k_{y}=k_{z}=0$. This latter fact is
used to show that the band crossings in this case are nodal points in the
momentum space while not nodal lines or surfaces and this property is the same
as Weyl semimetals.

Figure 5: The spectrum of the hydrodynamic equation (3.6) modified by an extra
mass term in the $y$ or $z$ direction or both these two directions. This plot
is for $k_{y}=k_{z}=0$ and $m<b$. The four nodes disappear.

Several remarks are in order:

  1. 1.

For the hydrodynamical modes $\omega(k=0)$ is not zero anymore due to the non-
conservation of energy, i.e. energy is continuously pumped into or out of the
system;

  2. 2.

In the case $m<b$, the crossing nodes have nonzero $\omega$ and $k$. This
indicates that two degenerate copies of modes would suddenly arise at a
special value of $\omega$ and $k$, leading to a sudden rise in the amplitude
of sound. Besides this possible observational effect, the effect of these
energy momentum non-conservation terms in transport coefficients will be shown
in section 6.1.

  3. 3.

These crossing nodes at $m<b$ are dissipative when order $k^{2}$ terms are
taken into account as we will see in section 7. This is different from the
$\omega=k=0$ nodes which are real poles in hydrodynamics with unbroken
translational symmetries.

There is one extra point that we need to emphasize, the new $m$ and $b$ terms
above are not dissipative so they only change the shape of the spectrum while
do not introduce any imaginary parts in the dispersion relation. Readers who
are interested in the difference between the two systems could find more
details in A.

####  3.4.2 Two 2D hydrodynamic systems

The second choice is to start from two separately conserved hydrodynamic
systems. After introducing weak interchange of energy and momentum between the
two systems, there will be more possibilities for the existence of topological
band crossing modes.

We start from the simplest case, where we have two 1+1D hydrodynamic systems
exchanging energy and momentum with each other weakly. In this case, we have
two variables to solve ($T^{00}$ and $T^{0x}$) in each sector and four in
total. The conservation equation under consideration has the form of

$\displaystyle\begin{split}\partial_{\mu}\delta T_{L}^{\mu t}&=m_{1}\delta
T_{L}^{tx}+b_{1}\delta T_{R}^{tt}\,,\\\ \partial_{\mu}\delta T_{L}^{\mu
x}&=-m_{1}v_{sL}^{2}\delta T_{L}^{tt}+b_{1}\delta T_{R}^{tx}\,,\\\
\partial_{\mu}\delta T_{R}^{\mu t}&=m_{2}\delta T_{R}^{tx}-b_{2}\delta
T_{L}^{tt}\,,\\\ \partial_{\mu}\delta T_{R}^{\mu x}&=-m_{2}v_{sR}^{2}\delta
T_{R}^{tt}-b_{2}\delta T_{L}^{tx}\,,\end{split}$ (3.9)

in which there are two different sectors of matter and they can exchange both
energy and momentum as well as gain or lose energy and momentum from or to the
environment. Furthermore, we assume that the parameters $m_{1,2},b_{1,2}\ll T$
to be in the hydrodynamic limit. The $m$ terms are the same as in the single
4D case, except that now there is only one spatial direction. Both $m$ and $b$
terms involve exchange of energy and momentum within the two sectors as well
as with the environment. For simplicity we shall focus on the case
$m_{1}=m_{2}=m,b_{1}=b_{2}=b,v_{sL}=v_{sR}=v_{s},$ and $v_{s}$ could be set to
be $1$ as shown in section 3.4.1.

In two dimensions, the first order corrections to the ideal fluid components
of energy momentum tensors are zero. The constitutive equations for the
perturbations of energy momentum tensors are

$\displaystyle\begin{split}\delta T^{00}&=\delta\epsilon\,,\\\ \delta
T^{x0}&=\delta T^{0x}=(\epsilon+p)\delta u^{x}=\delta\pi\,,\\\ \delta
T^{xx}&=\frac{\partial
P}{\partial\epsilon}\delta\epsilon=v_{s}^{2}\delta\epsilon\,,\end{split}$
(3.10)

where we have omitted the subscript $L$ and $R$ for simplicity.

In momentum space, the equation (3.9) can be written as

$\displaystyle i\partial_{t}\Psi=H\Psi$ (3.11)

where

$\displaystyle\Psi=\begin{pmatrix}\delta\epsilon_{L}\\\ \delta\pi_{L}\\\
\delta\epsilon_{R}\\\
\delta\pi_{R}\end{pmatrix}\,,~{}~{}~{}~{}~{}~{}H=\begin{pmatrix}0&~{}~{}k_{x}+im&~{}~{}ib&~{}~{}0\\\
(k_{x}-im)v_{s}^{2}&~{}~{}0&~{}~{}0&~{}~{}ib\\\
-ib&~{}~{}0&~{}~{}0&~{}~{}k_{x}+im\\\
0&~{}~{}-ib&~{}~{}(k_{x}-im)v_{s}^{2}&~{}~{}0\end{pmatrix}\,.$ (3.12)

The eigenvalues of $H$ give

$\displaystyle\omega=\pm b\pm\sqrt{m^{2}+k_{x}^{2}}v_{s}\,.$ (3.13)

Without $m$ and $b$ terms there are two sets of two sound modes and these
modes will mix together when $m$ and $b$ become nonzero. $m$ is to gap the
hydrodynamical modes and $b$ is to separate the crossing nodes of the
hydrodynamic modes, thus generating non-trivial topological structure in the
spectrum. The structure of the spectrum here looks very similar to the band
structure of the Weyl semimetal. From figure 6 we can see that for $m<b$, the
system is in the topological phase with two crossing nodes. For $m=b$, the
system is in the critical point where the two nodes merge into one. As effects
of $m$ and $b$ terms cannot cancel each other even when $m=b$, in this case
the spectrum is not the same as when $T^{\mu\nu}$ is conserved. For $m>b$, the
system is in the trivial phase where no band crossing exists. More evidence of
the nontrivial topological structure from the calculation of Berry phase will
be shown in section 5.

Figure 6: The spectrum of the modified hydrodynamics with dynamical equation
(3.9). From left to right, $m<b$, $m=b$ and $m>b$ respectively.

Note that in this case there is only one spatial direction, so these nodes in
the topological phase will not disappear due to new mass terms. Also as we
mentioned above, there are no dissipative terms in 2D, so that the nodes in
this case are non-dissipative at order $\mathcal{O}(k^{2})$, which is
different from the single 4D case. The same as the single 4D case, there
should be a sudden rise in amplitude of sounds at a certain momentum for
$m<b$, however, different from the 4D case, now the frequency is zero.

####  3.4.3 More possibilities with topological hydrodynamic modes

We can generalize the 2D+2D system above to higher spatial dimensions to get
more possible topological hydrodynamic modes. To generalize the $b$ terms,
there are several choices. We may generalize the $b$ terms to all spatial
dimensions or only to some of them. In the following we show the
generalization to 3D+3D and 4D+4D systems with $b$ interaction terms in some
or all of the dimensions. The spectrum is much more complicated in these cases
and we will mainly list the qualitative features of these systems in this
subsection.

I. 3D+3D/4D+4D with $b$ interaction terms not in all directions

For 3D+3D systems, to have $b$ interaction terms not in all directions, the
only possible effective Hamiltonian matrix is

$\displaystyle
H_{3D+3D,I}=\begin{pmatrix}0&~{}~{}k_{x}+im&~{}~{}k_{y}&~{}~{}ib&~{}~{}0&~{}~{}0\\\
k_{x}-im&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}ib&~{}~{}0\\\
k_{y}&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0\\\
-ib&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}k_{x}+im&~{}~{}k_{y}\\\
0&~{}~{}-ib&~{}~{}0&~{}~{}k_{x}-im&~{}~{}0&~{}~{}0\\\
0&~{}~{}0&~{}~{}0&~{}~{}k_{y}&~{}~{}0&~{}~{}0\\\ \end{pmatrix}\,.$ (3.14)

For 4D+4D there are two possibilities for the $b$ terms

$\displaystyle
H_{4D+4D,I1}=\begin{pmatrix}0&~{}~{}k_{x}+im&~{}~{}k_{y}&~{}~{}k_{z}&~{}~{}ib&~{}~{}0&~{}~{}0&~{}~{}0\\\
k_{x}-im&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}ib&~{}~{}0&~{}~{}0\\\
k_{y}&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0\\\
k_{z}&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0\\\
-ib&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}k_{x}+im&~{}~{}k_{y}&~{}~{}k_{z}\\\
0&~{}~{}-ib&~{}~{}0&~{}~{}0&~{}~{}k_{x}-im&~{}~{}0&~{}~{}0&~{}~{}0\\\
0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}k_{y}&~{}~{}0&~{}~{}0&~{}~{}0\\\
0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}k_{z}&~{}~{}0&~{}~{}0&~{}~{}0\\\
\end{pmatrix}\,,$ (3.15)

and

$\displaystyle\begin{split}H_{4D+4D,I2}=\begin{pmatrix}0&~{}~{}k_{x}+im&~{}~{}k_{y}+im&~{}~{}k_{z}&~{}~{}ib&~{}~{}0&~{}~{}0&~{}~{}0\\\
k_{x}-im&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}ib&~{}~{}0&~{}~{}0\\\
k_{y}-im&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}ib&~{}~{}0\\\
k_{z}&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0\\\
-ib&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}k_{x}+im&~{}~{}k_{y}+im&~{}~{}k_{z}\\\
0&~{}~{}-ib&~{}~{}0&~{}~{}0&~{}~{}k_{x}-im&~{}~{}0&~{}~{}0&~{}~{}0\\\
0&~{}~{}0&~{}~{}-ib&~{}~{}0&~{}~{}k_{y}-im&~{}~{}0&~{}~{}0&~{}~{}0\\\
0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}k_{z}&~{}~{}0&~{}~{}0&~{}~{}0\\\
\end{pmatrix}\,.\end{split}$ (3.16)

Here $\Psi=\begin{pmatrix}\Psi_{1}\\\ \Psi_{2}\end{pmatrix}\,,$ with
$\Psi_{1,2}$ denoting the 3 or 4 modes for the first and the second systems
separately and we have ignored $v_{s}$ factors. For these systems, there are
no simple analytical expressions for the spectrum.

Numerically we could see that the spectrum looks similar to that in 2D+2D with
slight differences. The spectrums of $H_{3D+3D,I}$ and $H_{4D+4D,I}$ are
almost the same with the latter having two more flat bands corresponding to
two transverse modes (see Figs. 7 and 9). In the following we list the
qualitative behavior for the spectrum in all three cases above.

$H_{3D+3D,I}$: As shown in figure 7, we could see that in the 3D+3D case at
$m<b$ the crossing nodes form two circles pinned together at two opposite
points in $\omega$, $k_{x}$, $k_{y}$ space. With an extra $m$ term in the $y$
direction, the two pinned nodes will disappear but the two circles of crossing
nodes remain. Figure 8 is an illustration in the $\omega$, $k_{x}$, $k_{y}$
space for the crossing nodes only, which can be viewed as an extended space
version of the left two and right four crossing points in figure 7. In the
left figure the two circles of crossing nodes are pinned by two opposite
points without $m$ term in the $y$ direction, while in the right plot the two
pinned points disappear with $m$ term in the $y$ direction and now two circles
do not intersect with each other. This indicates that the two pinned nodes are
topologically nontrivial protected by the symmetry that forbids the $m$ term
in the $y$ direction while the two circles of crossing nodes are topologically
nontrivial without the need of symmetry protection. In section 5 we will
confirm the topological structure of these nodes from the calculation of the
Berry phase. In this case, the nodes at nonzero frequency are supposed to be
dissipative with higher order terms considered.

Figure 7: The spectrum of the modified hydrodynamics of (3.14) $H_{3D+3D,I}$
with $m<b$ and $k_{y}=0$ (left) and $k_{y}>0$ (right). The spectrum of
$H_{3D+3D,I}$ with an extra $m$ term in the $y$ direction at $k_{y}=0$ is the
same as the right plot.

Figure 8: The illustration plot in the $\omega$, $k_{x}$, $k_{y}$ space for
the crossing nodes of $3D+3D$ hydrodynamical systems (3.14) (left) and the
crossing nodes of its generalization with an extra $m$ term in the $y$
direction (right).

$H_{4D+4D,I1}$: As now there are three spatial dimensions, there are two extra
flat bands corresponding to two transverse modes and the spectrum looks more
complicated. With all spatial dimensions taken into account, the circles in
the $H_{3D+3D,I}$ case now become spheres in the $\omega,k_{x},k_{y},k_{z}$
space. As there is a rotational symmetry in the $y$-$z$ plane, we can use
$k_{1}=\sqrt{k_{y}^{2}+k_{z}^{2}}$ in the consideration of the spectrum. In
the $\omega$, $k_{x}$, $k_{1}$ space, the behavior of the spectrum is the
similar for $H_{4D+4D,I}$ to the $H_{3D+3D,I}$ case as shown in figure 9.
Compared to the $H_{3D+3D,I}$ case, there are four additional crossing nodes
in the $\omega,k_{x}$ space due to two extra flat bands in $H_{4D+4D,I}$ case,
i.e. four additional circles of crossing nodes in the $\omega,k_{x},k_{1}$
space. With an extra $m$ term in the $y$ direction, the pinned circles will
become two non-intersecting circles, indicating the two spheric nodes in the
$\omega,k_{x},k_{y},k_{z}$ space are topologically nontrivial without the need
of symmetry protection.

Figure 9: The spectrum of the modified hydrodynamics of (3.15) $H_{4D+4D,I1}$
with $m<b$ and $k_{1}=\sqrt{k_{y}^{2}+k_{z}^{2}}=0$ (left) and $k_{1}>0$
(right). The spectrum of $H_{4D+4D,I1}$ with an extra $m$ term in the $y$
direction at $k_{1}=0$ is the same as the right plot.

$H_{4D+4D,I2}$: This system is much more complicated and we only describe the
qualitative behavior briefly. As could be seen from figure 10, in this case
the nodes form two large and non-intersecting circles and one small circle in
the middle in the $\omega,k_{x},k_{y}$ space at $k_{z}=0$. For nonzero
$k_{z}$, the one small circle becomes two non-intersecting circles at opposite
$\omega$, meaning that in the $\omega,k_{x},k_{z}$ space at $k_{y}=0$, the
nodes are similar to the $3D+3D$ case above, where the two small circles are
pinned at two opposite points of $k_{x}$ at $k_{z}=0$ as in figure 10. In the
$k_{x},k_{y},k_{z}$ space, the nodes become two large spheres and two small
spheres. This behavior will not change as long as there is no mass term in the
$z$ direction, i.e. $m$ could appear in either $k_{x}$ or $k_{y}$ or both
$k_{x}$ and $k_{y}$ directions. With a mass term in the $z$ direction, the
same as in the $3D+3D$ case with the $m$ term in the $y$ direction, in the
$\omega,k_{x},k_{z}$ space, the two small circles in the middle which are
pinned at two points at opposite values of $k_{x}$ would become two separate
circles.

Figure 10: The spectrum of the modified hydrodynamics (3.16) of $H_{4D+4D,I2}$
with $m<b$. We set $k_{z}=0$ and increase $k_{y}$ from the left to right for
the first three plots. In the right plot $k_{y}=0$ while a nonzero $k_{z}$.

II. 3D+3D/4D+4D with $b$ interaction terms in maximal dimensions

The second choice is to generalize the $b$ terms to all the spatial
directions. For the 3D+3D case, there are two choices for the $m$ terms: to
have them in both spatial directions or only one spatial direction.

First we have $m$ terms in both $x$ and $y$ directions. The effective
Hamiltonian is

$\displaystyle
H_{3D+3D,II}=\begin{pmatrix}0&~{}~{}k_{x}+im&~{}~{}k_{y}+im&~{}~{}ib&~{}~{}0&~{}~{}0\\\
k_{x}-im&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}ib&~{}~{}0\\\
k_{y}-im&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}ib\\\
-ib&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}k_{x}+im&~{}~{}k_{y}+im\\\
0&~{}~{}-ib&~{}~{}0&~{}~{}k_{x}-im&~{}~{}0&~{}~{}0\\\
0&~{}~{}0&~{}~{}-ib&~{}~{}k_{y}-im&~{}~{}0&~{}~{}0\\\ \end{pmatrix}\,.$ (3.17)

We could also have the $m$ term only in one direction, the $k_{x}$ direction,
and the qualitative behavior for the spectrum does not change at all. In
figure 11 we show the spectrum for the $H_{3D+3D,II}$ case in the
$\omega,k_{x}$ space at $k_{y}=0$ and $k_{y}\neq 0$. We could see that in the
$\omega,k_{x},k_{y}$ space there are three circles of band crossing nodes with
two large ones and one small one. All these band crossing nodes will not
disappear due to the $m$ term in any direction. When $m$ terms in both
directions disappear, the middle circle becomes a point.

Figure 11: The spectrum of the modified hydrodynamics with (3.17)
$H_{3D+3D,II}$ at $m<b$ and increasing from $k_{y}=0$ to larger values from
left to right.

We continue to the 4D+4D case. From the 3D+3D case we have seen that with
maximal $b$ terms there will still be crossing band nodes even when $m$ terms
in all spatial directions are turned on. Thus here for simplicity we have $m$
terms in all spatial directions and for cases with only one or two $m$ terms
the behavior would be qualitatively the same.

The effective Hamiltonian is

$\displaystyle
H_{4D+4D,II}=\begin{pmatrix}0&~{}~{}k_{x}+im&~{}~{}k_{y}+im&~{}~{}k_{z}+im&~{}~{}ib&~{}~{}0&~{}~{}0&~{}~{}0\\\
k_{x}-im&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}ib&~{}~{}0&~{}~{}0\\\
k_{y}-im&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}ib&~{}~{}0\\\
k_{z}-im&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}ib\\\
-ib&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}0&~{}~{}k_{x}+im&~{}~{}k_{y}+im&~{}~{}k_{z}+im\\\
0&~{}~{}-ib&~{}~{}0&~{}~{}0&~{}~{}k_{x}-im&~{}~{}0&~{}~{}0&~{}~{}0\\\
0&~{}~{}0&~{}~{}-ib&~{}~{}0&~{}~{}k_{y}-im&~{}~{}0&~{}~{}0&~{}~{}0\\\
0&~{}~{}0&~{}~{}0&~{}~{}-ib&~{}~{}k_{z}-im&~{}~{}0&~{}~{}0&~{}~{}0\\\
\end{pmatrix}\,.$

Now there is an SO(3) symmetry in the $x,y,z$ directions. Again there is no
analytic result for the spectrum, and numerically we could see from figure 12
that for $m<b$, in the $\omega,k_{x},k_{y}$ space at $k_{z}=0$, the crossing
nodes form two large circles at opposite values of $\omega$ and one small
circle at $\omega=0$. In the $k_{x},k_{y},k_{z}$ space, the nodes become three
spheres, which are codimension $1$ surfaces in the momentum space.

Figure 12: The spectrum of the modified hydrodynamics with $H_{4D+4D,II}$ at
$m<b,k_{z}=0$ and increasing from $k_{y}=0$ to larger values from left plot to
right plot.

Though we have not written out explicitly, in all the cases above when $m$ and
$b$ values change, there will be phase transitions between fully trivial
spectrum (for large enough $m/b$) and spectrum with crossing nodes which are
topologically nontrivial (protected by special spacetime symmetries in certain
directions in some cases). Also as may have been noticed, all topologically
nontrivial nodes (without symmetry protection) are codimension one manifolds
while topologically nontrivial nodes that require the protection of spacetime
symmetries in $n$ dimensions are codimension $n+1$ manifolds in the $\omega$,
${\bf k}$ space.

##  4 Possible origin for non-conservation terms of $T^{\mu\nu}$

We have added a specific set of non-conservation terms in the effective
Hamiltonian to get topologically nontrivial hydrodynamic modes. In this
section, we consider possible origin for these $m$ and $b$ terms. The non-
conservation of $T^{\mu\nu}$ could come from external fields that couple to
the system we are studying, i.e. the system constantly loses or gains energy
or momentum to the external fields.

In the case of momentum dissipation terms that have been studied extensively
[25], the external fields could be scalar fields [26, 27] or vector fields
[28] that break translational symmetry. However, for the $m$ and $b$ terms in
section 3.4.1, it has been explicitly checked that scalar or vector external
fields cannot give this form of energy momentum non-conservation. The simplest
way to realize such extra terms is to introduce an external rank two symmetric
tensor field $f_{\mu\nu}$. This external tensor field could either be an
effective external tensor matter field that couples to the energy momentum
tensor of the system in a specific way or could be viewed as an external
gravitational field. In the latter case, the non-conservation of $T^{\mu\nu}$
could even be the effect of observing in a non-inertial reference frame. In
the following we show more details implementing the two possibilities. In this
section, we will mainly focus on the 4D case of section 3.4.1. More general
cases will be studied in future work.

###  4.1 Origin I: Effective external tensor matter field

We consider an external symmetric matter field $f_{\mu\nu}$ to be the source
that breaks energy momentum conservation and it couples to the operator
$O^{\mu\nu}$ in the Lagrangian. This contributes an extra
$f_{\mu\nu}O^{\mu\nu}$ term in the Lagrangian of the system effectively. With
this extra term the energy momentum of the system will not be conserved as it
can be transferred to the external system whose energy momentum will not be
counted into the system that we study.

We could calculate the non-conservation equation for $T^{\mu\nu}$ using the
method introduced in section 5 of [29]. The Lagrangian of the system is now

$\displaystyle\mathcal{L}=\mathcal{L}_{0}+f_{\mu\nu}O^{\mu\nu}\,.$ (4.1)

We assume that $\mathcal{O}(f_{\mu\nu})\sim\mathcal{O}(k)$ so that the
conservation of energy momentum tensor is weakly broken. The whole action with
restored metric $S=\int d^{4}x\sqrt{-g}\mathcal{L}$ is diffeomorphism
invariant and under infinitesimal coordinate transformations
$x_{\mu}^{\prime}=x_{\mu}+\epsilon_{\mu}$, the change in the action is

$\displaystyle\delta S=\int
d^{4}x\bigg{(}\frac{\delta(\sqrt{-g}\mathcal{L})}{\delta g_{\mu\nu}}\delta
g_{\mu\nu}+\sqrt{-g}\frac{\delta(f_{\alpha\beta}O^{\alpha\beta})}{\delta
f_{\mu\nu}}\delta f_{\mu\nu}\bigg{)}=0\,,$ (4.2)

where

$\displaystyle\delta
g_{\mu\nu}=\nabla_{\mu}\epsilon_{\nu}+\nabla_{\nu}\epsilon_{\mu}\,,~{}~{}~{}~{}~{}~{}\delta
f_{\mu\nu}=\epsilon^{\rho}\partial_{\rho}f_{\mu\nu}+f_{\rho\nu}\partial_{\mu}\epsilon^{\rho}+f_{\rho\mu}\partial_{\nu}\epsilon^{\rho}\,.$
(4.3)

Note that we have used the fact that $\mathcal{L}_{0}$ does not depend on
$f_{\mu\nu}$. Here only $g_{\mu\nu}$ and $f_{\mu\nu}$ are external fields
while $T_{\mu\nu}$ defined as

$\displaystyle
T^{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L})}{\delta
g_{\mu\nu}}\,.$ (4.4)

could be composed of internal fields whose variations vanish due to their
equations of motion. We have also assumed that $\epsilon^{\mu}$ vanishes at
the boundary so all surface terms are ignored.

Substituting (4.3) into (4.2) and omit terms at order of $\mathcal{O}(k^{2})$
or higher, we get the non-conservation equation for $T^{\mu\nu}$

$\displaystyle\partial_{\mu}T^{\mu\nu}=O^{\rho\mu}(\partial_{\nu}f_{\rho\mu}-2\partial_{\mu}f_{\rho\nu})\,.$
(4.5)

To get the non-conservation terms that we need, this $O^{\mu\nu}$ in the
formula above should be $T^{\mu\nu}$ and the nonzero components of
$f_{\mu\nu}$ are

$\displaystyle
f_{tt}=f_{xx}=mx\,,~{}~{}f_{tx}=f_{xt}=\frac{1}{2}mt(v_{s}^{2}+1)\,,~{}~{}$
(4.6) $\displaystyle
f_{ty}=f_{yt}=-\frac{1}{2}bv_{s}z\,,~{}~{}f_{tz}=f_{zt}=\frac{1}{2}bv_{s}y\,.$
(4.7)

Note that here when $O^{\mu\nu}=T^{\mu\nu}$, it seems that we need some kind
of “fine-tuning” to make sure that $T^{\mu\nu}$ defined from (4.4) is the same
as the operator $T^{\mu\nu}$ in $f_{\mu\nu}T^{\mu\nu}$ which itself also
determines the forms of the operator $T^{\mu\nu}$ from (4.4). In fact here as
$\mathcal{O}(\partial f_{\mu\nu})\sim\mathcal{O}(k)$, the term
$f_{\mu\nu}T^{\mu\nu}$ in the Lagrangian only contributes a higher order term
in the definition of $T^{\mu\nu}$ thus does not contribute to the leading
order equation of the non-conservation equation for $T^{\mu\nu}$. Therefore we
could directly take
$O^{\mu\nu}=T^{\mu\nu}_{(0)}=\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{0})}{\delta
g_{\mu\nu}}$, which is more natural, and $T^{\mu\nu}_{(0)}=T^{\mu\nu}$ at
order $\mathcal{O}(k^{0})$. At the fundamental level, the only possibility for
$f_{\mu\nu}$ to have this kind of coupling is the metric field of the next
subsection, however, we could not rule out the possibility of an effective
coupling in this form in an effective theory, e.g. in elastic theories, though
we do not have any concrete examples at hand.

###  4.2 Origin II: Gravitational field, non-inertial reference frame

This is a more natural and at the same time very interesting possibility.666We
thank Karl Landsteiner for pointing this out. As we need an external symmetric
tensor field which couples to the energy momentum tensor in the form of
$f_{\mu\nu}T^{\mu\nu}$, it is natural to consider if this external field could
be a metric field. With this possibility the spacetime would be nontrivial and
the energy momentum tensor would not be conserved in the form of
$\partial_{\mu}T^{\mu\nu}=0$ but as $\nabla_{\mu}T^{\mu\nu}=0$. In this way,
when we have a nontrivial metric field, $\partial_{\mu}T^{\mu\nu}=0$ will not
hold anymore.

We start from the following covariant form of energy momentum conservation
equation

$\displaystyle\nabla_{\mu}\delta T^{\mu\nu}=0\,.$ (4.8)

We assume that the new spacetime metric is
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$, thus we get

$\displaystyle\partial_{\mu}\delta
T^{\mu\nu}=-\frac{1}{2}\partial_{\alpha}h\delta
T^{\alpha\nu}-\frac{1}{2}\eta^{\nu\beta}(2\partial_{\mu}h_{\alpha\beta}-\partial_{\beta}h_{\mu\alpha})\delta
T^{\mu\alpha}\,.$ (4.9)

Again, we have assumed that $\mathcal{O}(\partial
h_{\mu\nu})\sim\mathcal{O}(k)$ and only kept leading order in $k$ terms. To
get the exact $m$ and $b$ terms in the effective Hamiltonian of the single 4D
system, we could choose the components of $h_{\mu\nu}$ to be

$\displaystyle
h_{tt}=h_{xx}=mx\,,~{}~{}h_{tx}=h_{xt}=\frac{1}{2}mt(v_{s}^{2}+1)\,,~{}~{}$
(4.10) $\displaystyle
h_{ty}=h_{yt}=-\frac{1}{2}bv_{s}z\,,~{}~{}h_{tz}=h_{zt}=\frac{1}{2}bv_{s}y$
(4.11)

while all the other components are zero. With this choice, the conservation
equation (4.8) reduces to (3.6).777We assume that we are working in a large
but finite volume of spacetime and $m$, $b$ are so small that $mx,by,bz,mt\ll
1$ in the finite volume. In fact this requires $m\ll k$ so we assume that
$\mathcal{O}(k^{2})<\mathcal{O}(m)<\mathcal{O}(k)$. In this way we still keep
all terms at order $m,~{}b$ and order $k$ so all the calculations are not
affected. We ignore the boundary effects as the volume is large.

####  4.2.1 Symmetry of the single 4D system

In this subsection, we obtain the symmetry of this single 4D symmetry from the
Killing vector of the new metric and then introduce the symmetry that we need
to protect the topology of the crossing nodes, i.e. the symmetry that would be
broken by $m$ terms in the $y$ and $z$ directions.

In the system of (3.6), original Poincare symmetry is broken and new spacetime
symmetries of the system could be found from the new metric
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$. From the covariant conservation
equation $\nabla_{\mu}T^{\mu\nu}=0$, the isometry of $g_{\mu\nu}$ could keep
the equation (3.6) unchanged. The Killing vectors generating the isometry of
the new spacetime metric $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ are

$\displaystyle
K_{\mu}=\sum_{i=0}^{3}a_{i}\chi_{i}+\sum_{i=1}^{6}c_{i}\theta_{i}$ (4.12)

where $a_{0},...,a_{3},c_{1},...,c_{6}$ are constants with

$\displaystyle\begin{split}\chi_{0}&=\left(1-\frac{mx}{2},~{}-\frac{mt}{2},~{}0,~{}0\right)\,,~{}~{}~{}~{}\chi_{1}=\left(\frac{mtv_{s}^{2}}{2},~{}1+\frac{mx}{2},~{}0,~{}0\right)\,,~{}~{}\\\
\chi_{2}&=\left(-\frac{bzv_{s}}{4},~{}0,~{}1,~{}-\frac{btv_{s}}{4}\right)\,,~{}~{}~{}~{}\chi_{3}=\left(\frac{byv_{s}}{4},~{}0,~{}\frac{btv_{s}}{4},~{}1\right)\,,~{}~{}\end{split}$
(4.13)

and

$\displaystyle\begin{split}\theta_{1}&=\left(-\frac{m(x^{2}+v_{s}^{2}t^{2})}{4}+x,~{}-t-\frac{mtx}{2},~{}0,~{}0\right)\,,~{}~{}\\\
\theta_{2}&=\left(\big{(}1-\frac{mx}{2}\big{)}y,~{}-\frac{mty}{2},~{}\big{(}-1+\frac{mx}{2}\big{)}t,~{}\frac{bt^{2}v_{s}}{4}\right)\,,~{}~{}\\\
\theta_{3}&=\left(\big{(}1-\frac{mx}{2}\big{)}z,~{}-\frac{mtz}{2},~{}-\frac{bt^{2}v_{s}}{4},~{}\big{(}-1+\frac{mx}{2}\big{)}t\right)\,,~{}~{}\\\
\theta_{4}&=\left(\frac{mtyv_{s}^{2}}{2}+\frac{bxzv_{s}}{4},~{}-\frac{btzv_{s}}{4}+y\big{(}1+\frac{mx}{2}\big{)},~{}-\frac{m(x^{2}+v_{s}^{2}t^{2})}{4}-x,~{}\frac{btxv_{s}}{4}\right)\,,~{}~{}\\\
\theta_{5}&=\left(\frac{mtzv_{s}^{2}}{2}-\frac{bxyv_{s}}{4},~{}\frac{btyv_{s}}{4}+z\big{(}1+\frac{mx}{2}\big{)},~{}-\frac{btxv_{s}}{4},~{}-\frac{m(x^{2}+v_{s}^{2}t^{2})}{4}-x\right)\,,~{}~{}\\\
\theta_{6}&=\left(-\frac{bv_{s}(y^{2}+z^{2})}{4},~{}0,~{}z,~{}-y\right)\,.~{}~{}\end{split}$
(4.14)

These Killing vectors generate the isometry of the metric $g_{\mu\nu}$, i.e.
with a coordinate transformation $x^{\mu}\to x^{\mu}+sK_{\mu}$ where $s$ is an
infinitesimal constant, the metric $g_{\mu\nu}$ would not change. These
coordinate transformations can be viewed as the spacetime symmetries of the
hydrodynamic system with dynamical equation (3.6). We have ten independent
coordinate transformations which keep the system invariant. $\chi_{0}$ is a
combination of translation in the $t$ direction and a boost in the $t$-$x$
directions. $\chi_{1}$ is a combination of translation in the $x$ direction as
well as a boost in the $t$-$x$ directions.

Among these ten Killing vectors, the ones that forbid $m$ terms in the $y$ and
$z$ directions could be chosen to be $\chi_{2}$ and $\chi_{3}$ which are
combinations of translations in $y$ and $z$ directions and boosts in the
$t$-$y$ and $t$-$z$ directions. Thus, in the single 4D case, the symmetries
that protect the topologically nontrivial band crossing modes could be the two
spacetime symmetries generated by $\chi_{2}$ and $\chi_{3}$.

We have solved the infinitesimal spacetime symmetry of the system and besides
this infinitesimal coordinate transformation, there are other possible large
symmetries of the system that could forbid the $m$ terms in the two
directions, e.g. reflection symmetries in the two directions. Also in symmetry
protected topological states of matter, the symmetry that is needed to protect
the topological structure does not need to be the whole symmetry that forbids
the $m$ term. It could be that only a subgroup of the symmetry is at work to
prevent the band crossings from being destroyed. Thus, the symmetry that is
required for the protection of the topological structure could be a special
subgroup of the whole symmetry. Here the symmetry that we need for the
protection of the topological structure in the single 4D case is in fact the
reflection symmetry $M$ in both $y$ and $z$ directions, i.e.
$M:y\to-y,z\to-z$. This is not an infinitesimal coordinate transformation of
the spacetime, but we could see that the $M$ symmetry forbids both $m$ terms
in the $y$ and $z$ directions as these terms violate the reflection symmetry
in the $y$ or $z$ directions. This could be seen from the fact that
$T^{yt}\to-T^{yt}$ under the symmetry transformation and the mass term in the
$y$ or $z$ directions would violate the reflection symmetry in these two
directions. We will see in the next section how this protecting symmetry could
be used to calculate the topological invariants.

Note that not all systems that were studied in section 3 require a protecting
symmetry and some of the systems are topologically band crossing states
without requiring any symmetry.

####  4.2.2 non-inertial reference frame

This graviton field $h_{\mu\nu}$ could come from sources of massive matter and
more intriguingly it could also come from a coordinate transformation from the
flat Minkowski metric, where $x_{\mu}^{\prime}=x_{\mu}+\xi_{\mu}$ with

$\displaystyle\xi_{\mu}=\bigg{(}\frac{mxt}{2},~{}~{}\frac{mx^{2}}{4}+\frac{mt^{2}}{4}v_{s}^{2},~{}~{}-\frac{b}{4}v_{s}zt,~{}~{}\frac{b}{4}v_{s}yt\bigg{)}\,.$
(4.15)

This is an intriguing result as usually a nontrivial gravitational field could
not be transformed to a flat spacetime globally but only locally. It could be
checked that this new metric field has all components of the Riemann tensor
vanishing at leading order, thus could be transformed to the flat spacetime.
Though equivalent to a flat spacetime, $h_{\mu\nu}$ could still be viewed as a
non-trivial gravitational field according to the equivalence principle.

This result suggests that in a specific non-inertial frame, we could observe
hydrodynamic modes that are topologically protected even when they are
topologically trivial in the original inertial frame. A third possible
circumstance to have this nonzero $h_{\mu\nu}$ could be in analog gravity
systems, where certain materials could give rise to effective hydrodynamic
equations as if there exists a nontrivial gravitational field.

Note that in this case, with nonzero components of $h_{\mu\nu}$ the
constitutive equations for $T_{\mu\nu}$ could also be written into a covariant
form thus leading to extra terms compared to the original constitutive
equations. However, it can be explicitly checked that these extra terms do not
change the spectrum at leading $k\sim m,b$ order up to a rescaling of
parameters $m$, $b$ and $v_{s}$. More details could be found in the appendix
of [10].

##  5 Topological invariant

The band crossing behavior in the hydrodynamic modes in this paper is very
similar to that of the Weyl semimetal or other types of gapless semimetals,
raising the question if these modes are indeed topologically protected modes.
In section 3 we have shown explicitly that these band crossing nodes would not
disappear by small perturbations thus have already shown that they should be
topologically protected. Note that in several systems we studied in section 3,
the perturbations need to obey certain required symmetry and in other systems
we do not require any protecting symmetry. In this section we further confirm
that the band crossing hydrodynamical modes are indeed topologically
nontrivial modes or symmetry protected topological modes by the existence of
nontrivial topological invariants for different cases. Before the detailed
calculation of topological invariants in our system, we will first introduce
some basic ingredients in the calculation of topological invariants for
topological band crossing states of matter.
Topological invariants for band crossing topological states

A topological invariant is a quantity that does not change unless there is a
topological phase transition from a topologically nontrivial phase to a
trivial one and the quantity has to have different values for the
topologically nontrivial and trivial phases. This quantity could be a number
or a property which cannot be identified with a number. For topological band
crossing modes, the crossing node is a singular point in the momentum space
which cannot disappear due to small perturbations, thus could be associated
with a nontrivial topological invariant which is different from the value
obtained for a trivial vacuum state. For accidental band crossing modes, the
crossing node could be disappear due to an arbitrarily small perturbation,
thus should possess trivial topological invariants.

For a Weyl semimetal, the topological charge is the chirality charge that
could be calculated from the integral of the Berry curvature on an
infinitesimal sphere enclosing the Weyl point in the three dimensional
momentum space. The topological invariant for nontrivial Weyl nodes are either
$1$ or $-1$ in contrast to the trivial value of $0$. For a nodal line
semimetal, as the nodal line spans in one momentum direction, there are only
two spatial directions left and the topological invariant is the Berry phase
accumulated along the circle that links the nodal line [30, 31, 32]. A
nontrivial nodal line semimetal would have a Berry phase of $\pi$ in contrast
to $0$ for an accidental one.

Note for band crossing states, the calculation of the topological invariant
depends crucially on the dimensionality of the system. For band crossing
states, the nodes are momentum space singular points for Green functions and
any physical calculation has to avoid passing through this point. Thus for a
three spatial dimensional Weyl semimetal, the topological charge is an
integration of the Berry curvature on a two dimensional sphere surrounding the
node (left figure of Fig. 13), while for a three spatial dimensional nodal
line semimetal, as the nodes form a circle, the topological charge is an
integration of the Berry phase along a one dimensional manifold: a circle
linking the circle of the nodes (middle figure of Fig. 13).

For some systems, the topological charge has to be calculated on an even lower
dimensional manifold, i.e. on zero dimensional manifolds. This could happen
for a nodal line semimetal in two spatial dimensions. More commonly, this
could come from symmetry protected topological state, where the calculation of
the topological invariant has to be on a high symmetric point thus reducing
the dimension of the system to effectively zero dimension. An example is the
mirror symmetry protected topological nodal line semimetal (right figure of
Fig. 13), where the nodes have to be protected by a mirror symmetry, i.e.
small perturbations that violate the mirror symmetry could destroy the band
crossings of the system while perturbations that do not violate the mirror
symmetry would not. In symmetry protected topological states, topological
invariants have to be calculated at the high symmetric points in the momentum
space, which would give a nontrivial result, and the topological invariants
calculated directly (not at the high symmetric points in the momentum space)
would give a trivial result. The requirement of the high symmetric points
would usually reduce the dimensionality of the system to lower dimensions.

Fig. 13 shows the three typical cases where the calculation of the topological
charge is on a two/one/zero dimensional manifold respectively. The left plot
in Fig. 13 is for a typical Weyl node, where the calculation of the
topological invariant is an integration of the Berry curvature on a two
dimensional sphere (magenta sphere). The middle plot in Fig. 13 is for a
typical nodal line semimetal, where the calculation of the topological
invariant is an integration of the Berry phase on a one dimensional circle
(magenta circle). The right plot in Fig. 13 is for a mirror symmetry protected
topological nodal line semimetal [30], where the calculation of the
topological invariant has to be on the high symmetric plane, which reduces the
dimensionality for the calculation to zero.

Figure 13: Three typical cases where the calculation of the topological charge
is on a two/one/zero dimensional manifold respectively. In the left figure,
the crossing nodes are two points. In the right two figures blue curves
indicate the crossing nodes and the spectrum at other momenta have not been
plotted.

Here for all the cases in this paper, the manifold enclosing the node is zero
dimensional effectively, either because the system is symmetry protected so
the calculation of the topological invariant needs to be at a high symmetric
point or because the nodes form a co-dimension one surface. Compared to the
sphere in three spatial dimensions and the circle in two spatial dimensions
surrounding the singular nodes, in the cases here we only have two points
surrounding the singular nodes, as illustrated in the right figure of Fig. 13.

In this zero dimensional case, the calculation of the topological invariant is
very different. We first briefly review the calculation of the topological
invariant of the mirror symmetry protected topological nodal line semimetal in
[30] as an example here and then perform our calculation following this
calculation later in this section. We will also show that this calculation is
in fact equivalent to the requirement of the existence of an undetermined
Berry phase.

The mirror symmetry protected topological nodal line semimetal is illustrated
in the right figure of Fig. 13. To calculate the topological invariant, we
have to perform the calculation at the high symmetric points of the system,
which is the plane invariant under the reflection, shown in shade in the
figure. The nodal line also resides on this plane and we pick two points on
the two sides of the nodal loop, denoted by $p_{1}$ and $p_{2}$ separately,
e.g. we could denote the outer point in the right figure of Fig. 13 as $p_{1}$
and the inner one as $p_{2}$. As in the right figure of Fig. 13 only the nodes
are plotted, at each point $p_{i}$, there should be a gapped spectrum, i.e.
one band above the mirror plane and one band below the mirror plane (occupied
band), which has not been plotted out in Fig. 13 and is shown in 14.

Figure 14: $p_{1}$ and $p_{2}$ denote the outer and inner points of the nodal
loop.

Then as explained in Sec.II A of [30], the topological invariant is defined to
be $\xi_{0}=N_{1}-N_{2}$ (equation (6) in [30]), where $N_{i}$ refers to the
number of occupied bands that have mirror symmetry eigenvalue of $+1$ at the
point $p_{i}$. Here as the mirror symmetry commutes with the Hamiltonian at
the high symmetric point, the energy eigenstates carry their Mirror symmetry
eigenvalues. Consider Fig. 14 as the $k_{y}=0$ spectrum of a nodal line
semimetal and $p_{i}$ points are the two inner and outer points of the left
node, as shown in Fig. 14. The two curves with different colors have the
mirror symmetry eigenvalue 1 and -1 separately as each curve is for a set of
eigenstates with the same eigenvalue $E_{i}(k)$ which also has the same mirror
symmetry eigenvalue. Thus if at $p_{1}$, the number of occupied $+1$ mirror
eigenvalue band is 1 and at the other point $p_{2}$, the number of occupied
$+1$ mirror eigenvalue band would be zero as could seen from Fig. 14. Thus the
topological invariants could be calculated from the defining formula to be
$\pm 1$ for the two nodes. Thus the system is a mirror symmetry protected
topologically nontrivial state. Note that in general, the symmetry does not
need to be the full symmetry of the system, and one subgroup of the symmetry
that is enough for the calculation of the topological invariant.

In fact we could see from the calculation above that this calculation of the
topological invariant relies on the fact that the two occupied states (states
with energy lower than the crossing nodes) at the two sides of the node belong
to two different sets of eigenstates so that they have distinct mirror
symmetry eigenvalues. The existence of a nontrivial topological invariant
defined this way is also equivalent to the fact that the band crossings cannot
be destroyed by small perturbations. This is because the two occupied states
at the two sides of the node behave differently under small deformations of
the system. When the system is perturbed by a small perturbation, each set of
eigenstates will get a small deformation and they will still cross at some
point as long as the perturbation is small enough. However, if the two lowest
energy states at the two sides of the node belong to the same origin of
eigenstates (as the left figure of Fig. 2), when each band gets a small
perturbation, it is possible that the band crossing would disappear, e.g. if
the upper band goes up while the lower band goes down.

Then the question becomes how do we distinguish whether they belong to the
same origin of eigenstates or to two different sets of eigenstates. In the
calculation above, different types of eigenstates are characterized by
different eigenvalues of the mirror reflection symmetry. In fact, a more
direct way is to see if the eigenstates at the two sides the nodes are
orthogonal or equal to each other up to a relative phase. If they are
orthogonal to each other, they surely belong to two different sets of
eigenstates.

Figure 15: $|n_{1}\rangle$ and $|n2\rangle$ are the left and right limiting
states near one node.

At the singular node, the effective Hamiltonian would have two degenerate
orthogonal eigenstates with the same eigenvalue. At the left and right limits
of the singular mode in the occupied band (the band with energy lower than the
crossing nodes), the two states $|n_{1}\rangle$ and $|n_{2}\rangle$ should be
extremely close to one of these two degenerate eigenstates as shown in Fig.
15. If the left and right limit states of the same upper or lower band are two
different eigenstates, i.e. if they are orthogonal to each other, then the
left and right limits cannot be connected together by performing a small
perturbation as the band crossing point is a singular point which cannot
disappear from a small perturbation. This means that two bands cannot be
separated with a small perturbation.

Thus we could also use this fact to define a topological invariant: if the
left and right limit states of the node in the same occupied band are
orthogonal to each other ($|n_{1}\rangle$ and $|n_{2}\rangle$ states in Fig.
15), the crossing node is a topologically nontrivial one and for trivial
crossing nodes, the two states should be the same to each other up to a
relative phase. The latter case gives a topological invariant the same as the
trivial vacuum state indicating a topologically trivial state. As mentioned
above, a topological invariant could be a number or could be a
quantity/property that is not a number as long as the property distinguishes
different topological classes. Here it provides an example of a topological
invariant which is not a number.

Thus a node would possess a nontrivial topological invariant when an
undetermined Berry phase exists, i.e. the two left and right limit states are
orthogonal, which makes the Berry phase an undetermined one. We name this
method the undetermined Berry phase method, however, it in fact refers to the
fact of orthogonality of adjacent states on the two sides of the nodes. This
method is more useful for systems without a protecting symmetry while the
calculation has to be on zero dimensional manifolds. In the following, we will
use the first method (a modified version of the formula $\xi_{0}=N_{1}-N_{2}$)
to calculate the topological invariant of the single 4D case and show that it
is equivalent to the second method, i.e. the undetermined Berry phase method.
Then we mainly use the second method to calculate the topological invariants
for the more complicated systems. We emphasize again that these two methods
are equivalent and they both rely on the fact that the two lower energy states
on the two sides of the singular node belong to two different kinds of
eigenstates which behave differently under deformations of the system. More
details could also be found below for each case.

Before we proceed there is another point that we need to emphasize: the
topological structure of the systems that we discussed above is a consequence
of the spectrum while not directly related to the modification of the
effective Hamiltonians or the modified Ward identities due to the non-
conservation terms of the energy momentum tensor. This is similar to what
happens to Weyl semimetal: the deformed Hamiltonian with a time reversal
symmetry breaking terms produces the spectrum of a topological Weyl semimetal
but the topological structure is not directly related to the deformed
Hamiltonian.

Finally we need to emphasize that, though the calculation of topological
invariants was first done for quantum electronic systems, in principle
topological invariants are also properties of the wave phenomenology while not
essentially associated to the quantum nature. In classical wave systems,
topological invariants could also be defined in the same way as for quantum
systems, e.g. Berry phases are thought to be quantum quantities as they are
usually associated with quantum mechanical interference, however, it can in
principle occur wherever phase interference phenomena exist and are governed
by Hermitian eigenvalue problems, as in classical topological systems [4].

The single 4D system

For the single 4D system in section 3.4.1, there are four nodes for $m<b$ when
the $y$ and $z$ directions have no $m$ terms. The nodes disappear with $m$
terms in the $y$ and $z$ directions, thus the nodes should be topologically
nontrivial ones protected by the special spacetime symmetry in the $y$ and $z$
directions shown in section 4.2. Here we first review the calculation of the
topological invariant for this system as in [10].

The calculation of the topological invariant for a symmetry protected state is
different from the calculation for an ordinary topological state. For systems
protected by a certain symmetry, we could calculate the topological invariant
at a high symmetric point in the momentum space. The same to the case of
mirror symmetry protected nodal line semimetals discussed in [30], at the high
symmetric point the protecting symmetry $M$ commutes with the effective
Hamiltonian and the eigenstates would each have an eigenvalue of the
reflection symmetry $M$ in our system: $M:y\to-y,z\to-z$. We have to find the
eigenvalues of the states under this symmetry.

The high symmetric point here should be $k_{y}=k_{z}=0$, which is invariant
under the reflection symmetry transformation $M$. Note that the high symmetric
point is the one whose wave function is invariant under the protecting
symmetry and here the high symmetric point being $k_{y}=k_{z}=0$ is not
related to the fact that the system has no band crossing at nonzero $k_{y}$
and $k_{z}$. Also the existence of a high symmetric point does not mean that
the symmetry of the system requires $k_{y}=k_{z}=0$: the symmetry is a
restriction to the system while not the solutions and we only need
$k_{y}=k_{z}=0$ for the calculation of the topological invariant for a
symmetry protected topological state.

The solutions are symmetric for $\omega\to-\omega$ and we could focus on the
lower two nodes in figure 16. In the figure, the different two eigenstates of
the effective Hamiltonian are denoted in different colors. In the language of
the first method of the topological invariant calculation, the two eigenstates
have two different eigenvalues under the symmetry $M$. Now we calculate the
topological invariant using this method in detail.

The formula for the topological invariant should be $\xi=N_{1}-N_{2}$, where
in our case here, $N_{i}$ is the number of occupied bands at point $p_{i}$
with eigenvalue of the reflection symmetry $M$ to be $1$. In figure 16,
$|n_{i}>$ is the occupied state at point $p_{i}$ and we have
$|n_{1}\rangle=\frac{1}{\sqrt{2}}\big{(}0,0,-i,1\big{)}$ and
$|n_{2}\rangle=\frac{1}{\sqrt{\frac{1}{v_{s}^{2}}-1}}\Big{(}\frac{\sqrt{m^{2}+k_{x}^{2}}}{v_{s}(m+ik_{x})},i,0,0\Big{)}$.
As already shown in section 4.2.1, the protecting symmetry required here is
the reflection symmetry in both $y$ and $z$ directions. Under this symmetry,
the state $|n_{1}\rangle$ has eigenvalue $-1$ and the state $|n_{2}\rangle$
has eigenvalue $1$. Thus from the formula $\xi=N_{1}-N_{2}$, the topological
invariant at each node is $\xi=1$ or $\xi=-1$, which is a nontrivial value in
contrast to the trivial value of $0$, confirming that the nodes are protected
by a nontrivial topological charge.

Now we calculate the topological invariant using the second method, i.e. the
fact of $|n_{i}\rangle$ being orthogonal to each other. For the left node at
$k_{x}=k_{1}$, the green solutions at the left limit $k_{x}\to k_{1-}$ and the
right limit $k_{x}\to k_{1+}$ are denoted as $|n_{1}\rangle$ and
$|n_{2}\rangle$ separately. If the node is an accidental crossing which is not
topologically protected, the two bands should easily be separated by a gap and
the two solutions $|n_{1}\rangle$ and $|n_{2}\rangle$ should be equal to each
other or at most differ by a relative phase. If the node is indeed a
topologically protected one, the two solutions should be different indicating
that there exists a singular point in between the two states at $k_{x}=k_{1}$.
In all the systems that we study here we could still use a Berry phase between
the two states $e^{-i\alpha}=\frac{\langle n_{1}|n_{2}\rangle}{|\langle
n_{1}|n_{2}\rangle|}$ to denote the topological invariant here where when the
Berry phase is an undetermined one, i.e. the two states are orthogonal to each
other, the system is topologically nontrivial as the two states can not be
connected without passing a singularity.

Figure 16: The left and right limit of the eigenstates $|n_{1}\rangle$ and
$|n_{2}\rangle$ at the crossing point of the modified hydrodynamics with
dynamical equation (3.6). Note that in this figure, different from some of the
previous illustration figures, each color denotes a separate band, which,
however, may come from different types of eigenstates of the system as we use
the fact wether the value of the energy is larger or lower than the crossing
node to distinguish between different bands while we use the continuous
functions of eigenvalues to distinguish between different sets of eigenstates
of the system.

In this single 4D case,
$|n_{1}\rangle=\frac{1}{\sqrt{2}}\big{(}0,0,-i,1\big{)}$ and
$|n_{2}\rangle=\frac{1}{\sqrt{\frac{1}{v_{s}^{2}}-1}}\Big{(}\frac{\sqrt{m^{2}+k_{x}^{2}}}{v_{s}(m+ik_{x})},i,0,0\Big{)}$.
We could easily see that $\langle n_{1}|n_{2}\rangle=0$, which means that the
Berry phase is undetermined and the two states cannot connect together
smoothly without passing through a singular point in between. Thus the two
bands cannot be separated easily by a gap without going through a topological
phase transition. Similar behavior of an undetermined Berry phase has also
happened for the holographic nodal line semimetals [32, 33].

This result confirms that the four nodes in this case are topologically
nontrivial protected by the special spacetime symmetry in the $y$ and $z$
directions. At the same time, the Berry phase accumulated through the whole
circle around this node would be trivial indicating that it is indeed
topologically trivial without the special spacetime symmetries in the $y$ and
$z$ directions.

The 2D+2D system

This is a topological band crossing state without the protection of a
symmetry. For the 2D+2D case, there is only one spatial direction, i.e. the
$x$ direction. As shown in figure 17 different bands are plotted in different
colors. There are two nodes when $m<b$. Again, we denote the two solutions of
the same blue band on the two sides of the singular node as $|n_{1}\rangle$
and $|n_{2}\rangle$. Similar to the 4D case, we have $\langle
n_{1}|n_{2}\rangle=0$ from numerical calculations, which means that the Berry
phase is undetermined. This proves that the two nodes are topologically
nontrivial as the two bands cannot separate without passing through a
topological phase transition, and in this case it does not require the
existence of any symmetry.

Figure 17: The left and right limits of eigenstates $|n_{1}\rangle$ and
$|n_{2}\rangle$ at the crossing point of the modified hydrodynamics with
dynamical equation (3.9).

The 3D+3D/4D+4D case I

The structure of the spectrum in this case has been described in section
3.4.3. Here we calculate the Berry phase numerically and confirm the
topological structure in that section. In the 3D+3D case, as shown in the left
plot of figure 18, for $m<b$ there are two nodes in the $\omega,k_{x}$ plane
at $k_{y}=0$ which look qualitatively the same as in the 2D+2D case. In this
3D+3D case, the two nodes in fact form two circles with the extra $k_{y}$
direction. Again, we use $|n_{1}\rangle$ and $|n_{2}\rangle$ to denote the two
states from the same band at the left and right limits of the node
$k_{x}=k_{1}$. At the high symmetric point $k_{y}=0$ we have checked
numerically that $\langle n_{1}|n_{2}\rangle=0$, which again gives
undetermined Berry phase, meaning that the two nodes in this case are
topologically nontrivial protected by the special spacetime symmetry that
forbids the $m$ term in the $y$ direction. This symmetry should be similar to
the one of the single 4D case and the explicit form of this symmetry will be
left for future work.

Figure 18: The left and right limit of the eigenstates $|n_{1}\rangle$ and
$|n_{2}\rangle$ at the crossing point of the modified hydrodynamics with
dynamical equation (3.14). The left plot is for the case of a single $m$ term
while the right plot is for the case with $m$ terms in both $x$ and $y$
directions.

Besides these two nodes, we have shown that with an $m$ term in the $y$
direction, the two nodes disappear and form four nodes as shown in the right
plot of figure 18, and now we have two separated circles with the extra
spatial dimension. We have checked explicitly by numerical calculations of the
Berry phase that these new four circles are still topologically nontrivial,
which do not require any protection of symmetry for the 3D+3D case. We
summarize this behavior in figure 8. where we can see that two circles of
nodes are pinned together at two nodes. The two nodes are topologically
nontrivial under the protection of a special spacetime symmetry in the $y$
direction and the two circles are topologically nontrivial without the need of
any symmetry.

For the 4D+4D case, as we have two more spatial directions, depending on the
existence of one or two extra mass terms, the topological structure would be
more complicated. However, qualitatively the behavior is the same as the 3D+3D
case above and we skip the details here.

The 3D+3D/4D+4D systems with maximal $b$ terms

In fact the 2D+2D case above also belongs to this category, i.e. it has
maximal $b$ terms in its spatial direction: the $k_{x}$ direction. These three
cases are qualitatively the same which are topologically nontrivial without
protection of any symmetry and have crossing nodes forming codimension $1$
surfaces in the momentum space. Thus in all three cases, the results for the
Berry phase are also similar. As the nodes form a codimension $1$ surface,
there is only one extra independent direction left: the two sides of the
surface. To check whether these nodes are topologically nontrivial we only
need to check if the two states on the two sides of the surface could be the
same or not.

It has been checked numerically that the at the two sides of the crossing
nodes, the eigenvectors of the effective Hamiltonians are orthogonal to each
other in the limit close to the nodes, meaning that the Berry phase is
undetermined and the two states cannot be the same by small deformations. Thus
the codimension $1$ nodes in these cases are topologically nontrivial and it
does not require the protection of any symmetry.

In summary, in all cases of this section we have undetermined Berry phase,
which makes the nature of all these cases qualitatively the same. The nodes
are either topologically nontrivial (when the nodes form codimension $1$
surfaces) or topologically nontrivial under protection of certain spacetime
symmetries of extra spatial dimensions (when the nodes form surfaces with
codimension larger than $1$).

##  6 Transport properties

In this section we show how the extra energy momentum non-conervation terms
affect the transport properties of the systems besides changing the spectrum
of the hydrodynamic modes. Without other conserved currents, the most
interesting transport coefficient in these systems is the heat transport. In
the following we provide details for the calculation of the heat transport for
the single 4D case in [10]. and generalize to the 2D+2D case, while other
cases would have qualitatively similar properties.

###  6.1 Single 4D heat transport

Heat transport can be computed from the linear response theory. We will follow
the calculations in [18, 23] for the momentum dissipative hydrodynamics system
to compute the heat transport for our new hydrodynamical system.

We first consider the single 4D system studied in [10] and in section 3.4.1.
We focus on the case with $m_{1}=m_{2}=m$ and $b_{1}=b_{2}=b$. Perturbing the
system of (3.6) by the fluctuations $(\delta T/T,\delta u^{x},\delta
u^{y},\delta u^{z})$ which couple to
$(\delta\epsilon,\delta\pi_{x},\delta\pi_{y},\delta\pi_{z})$ in the
conservation equation (3.6), and performing Laplace and Fourier
transformations in time and spatial directions respectively we obtain

$\displaystyle\begin{split}(\omega\delta\epsilon-i\delta\epsilon^{(0)})-(k_{x}\delta\pi_{x}+k_{y}\delta\pi_{y}+k_{z}\delta\pi_{z})-im\delta\pi_{x}&=0\,,\\\
(\omega\delta\pi_{x}-i\delta\pi_{x}^{(0)})-v_{s}^{2}k_{x}\delta\epsilon-\frac{i\eta}{\epsilon+P}\vec{k}^{2}\delta\pi_{x}+\frac{i\left(\frac{1}{3}\eta+\zeta\right)}{\epsilon+P}k_{x}k_{i}\delta\pi_{i}+imv_{s}^{2}\delta\epsilon&=0\,,\\\
(\omega\delta\pi_{y}-i\delta\pi_{y}^{(0)})-v_{s}^{2}k_{y}\delta\epsilon-\frac{i\eta}{\epsilon+P}\vec{k}^{2}\delta\pi_{y}+\frac{i\left(\frac{1}{3}\eta+\zeta\right)}{\epsilon+P}k_{y}k_{i}\delta\pi_{i}-ibv_{s}\delta\pi_{z}&=0\,,\\\
(\omega\delta\pi_{z}-i\delta\pi_{z}^{(0)})-v_{s}^{2}k_{z}\delta\epsilon-\frac{i\eta}{\epsilon+P}\vec{k}^{2}\delta\pi_{z}+\frac{i\left(\frac{1}{3}\eta+\zeta\right)}{\epsilon+P}k_{z}k_{i}\delta\pi_{i}+ibv_{s}\delta\pi_{y}&=0\,,\end{split}$
(6.1)

where
$k_{i}\delta\pi_{i}=k_{x}\delta\pi_{x}+k_{y}\delta\pi_{y}+k_{z}\delta\pi_{z}$
and $\epsilon^{(0)},\delta\pi_{i}^{(0)}$ are the perturbations at initial
time.

Solving these equations and following [18, 23] we obtain the Green function at
$k_{y}=k_{z}=0$

$\displaystyle\begin{split}G_{\pi_{x}\pi_{x}}(\omega,k_{x})&=-(\epsilon+P)\frac{(k_{x}^{2}+m^{2})v_{s}^{2}+i\frac{\eta}{\epsilon+P}\omega
k_{x}^{2}}{(k_{x}^{2}+m^{2})v_{s}^{2}+i\frac{\eta}{\epsilon+P}\omega
k_{x}^{2}-\omega^{2}}\,,\\\
G_{\pi_{y}\pi_{y}}(\omega,k_{x})&=G_{\pi_{z}\pi_{z}}(\omega,k_{x})=-(\epsilon+P)\frac{b^{2}v_{s}^{2}+k_{x}^{2}\frac{\eta}{\epsilon+P}(k_{x}^{2}\frac{\eta}{\epsilon+P}+i\omega)}{b^{2}v_{s}^{2}+(i\omega+\frac{\eta}{\epsilon+P}k_{x}^{2})^{2}}\,,\\\
G_{\pi_{y}\pi_{z}}(\omega,k_{x})&=-G_{\pi_{z}\pi_{y}}(\omega,k_{x})=\frac{i\omega(\epsilon+P)dv_{s}}{b^{2}v_{s}^{2}+(i\omega+\frac{\eta}{\epsilon+P}k_{x}^{2})^{2}}\,.\end{split}$
(6.2)

From the Kubo formula

$\displaystyle\kappa(\omega,k_{x})=\frac{i}{\omega
T}\big{(}G^{R}_{\pi\pi}(\omega,k_{x})-G^{R}_{\pi\pi}(0,k_{x})\big{)}$ (6.3)

we have

$\displaystyle\begin{split}\kappa_{xx}(\omega,k_{x})&=-\frac{i\omega(\epsilon+P)}{T\left((k_{x}^{2}+m^{2})v_{s}^{2}+i\frac{\eta}{\epsilon+P}\omega
k_{x}^{2}-\omega^{2}\right)}\,,\\\
\kappa_{yy}(\omega,k_{x})&=\kappa_{zz}(\omega,k_{x})=-\frac{k_{x}^{2}\eta+i\omega(\epsilon+P)}{T\left(b^{2}v_{s}^{2}+(i\omega+\frac{\eta}{\epsilon+P}k_{x}^{2})^{2}\right)}\,,\\\
\kappa_{yz}(\omega,k_{x})&=-\kappa_{zy}(\omega,k_{x})=\frac{(\epsilon+P)bv_{s}}{T\left(b^{2}v_{s}^{2}+(i\omega+\frac{\eta}{\epsilon+P}k_{x}^{2})^{2}\right)}\,.\end{split}$
(6.4)

From the above formulas, it is easy to see that when $m=b=0$, all components
of the DC heat transport diverge. For generic $m$ and $b$, we have vanishing
DC heat transport $\kappa_{xx}(0,0),\kappa_{yy}(0,0)$ and $\kappa_{zz}(0,0)$
while $\kappa_{yz}(0,0)=-\kappa_{zy}(0,0)=\frac{\epsilon+P}{Tbv_{s}}.$ These
$m$ and $b$ terms eliminate the unphysical divergence of DC heat transports
and lead to interesting vanishing DC heat transport behavior.

###  6.2 2D+2D heat transport

We continue to the calculations of 2D+2D heat transports using the linear
response theory. Perturbing the system of (3.9) by the fluctuations $(\delta
T/T,\delta u^{x})_{L,R}$ which couple to $(\delta\epsilon,\delta\pi)_{L,R}$ in
the conservation equation (3.9), and performing Laplace and Fourier
transformations in time and spatial directions respectively we obtain

$\displaystyle\begin{split}\omega\delta\epsilon_{L}-i\delta\epsilon_{L}^{(0)}&=(k_{x}+im_{1})\delta\pi_{L}+ib_{1}\delta\epsilon_{R}\,,\\\
\omega\delta\pi_{L}-i\delta\pi_{L}^{(0)}&=(k_{x}-im_{1})v_{sL}^{2}\delta\epsilon_{L}+ib_{1}\delta\pi_{R}\,,\\\
\omega\delta\epsilon_{R}-i\delta\epsilon_{R}^{(0)}&=(k_{x}+im_{2})\delta\pi_{R}-ib_{2}\delta\epsilon_{L}\,,\\\
\omega\delta\pi_{R}-i\delta\pi_{R}^{(0)}&=(k_{x}-im_{2})v_{sR}^{2}\delta\epsilon_{R}-ib_{2}\delta\pi_{L}\,.\end{split}$
(6.5)

We consider the simplest case with the assumption that these two 2D systems
$L$\- and $R$\- sectors have the same constitutive equations and
$m_{1}=m_{2}=m,b_{1}=b_{2}=b$ and $v_{sL}=v_{sR}=v_{s}$. With equations (6.5)
one can obtain the Green function following the linear response theory [18,
23]

$\displaystyle\begin{split}G_{\pi_{L}\pi_{L}}(\omega,k_{x})&=G_{\pi_{R}\pi_{R}}(\omega,k_{x})=\frac{(\epsilon+P)(\omega^{2}(b^{2}+(m^{2}+k_{x}^{2})v_{s}^{2})-(b^{2}-(m^{2}+k_{x}^{2})v_{s}^{2})^{2})}{\left((k_{x}^{2}+m^{2})v_{s}^{2}-\omega^{2}-b^{2}\right)^{2}-4b^{2}\omega^{2}}\,,\\\
G_{\pi_{L}\pi_{R}}(\omega,k_{x})&=-G_{\pi_{R}\pi_{L}}(\omega,k_{x})=\frac{-ib\omega(\epsilon+P)(b^{2}-(m^{2}+k_{x}^{2})v_{s}^{2}-\omega^{2})}{\left((k^{2}+m^{2})v_{s}^{2}-\omega^{2}-b^{2}\right)^{2}-4b^{2}\omega^{2}}\,.\end{split}$
(6.6)

From Kubo formula (6.3) we obtain

$\displaystyle\kappa_{LL}(\omega,k_{x})=\kappa_{RR}(\omega,k_{x})=-\frac{i\omega(\epsilon+P)\left(b^{2}+(k_{x}^{2}+m^{2})v_{s}^{2}-\omega^{2}\right)}{T\left(b^{4}+\left((k_{x}^{2}+m^{2})v_{s}^{2}-\omega^{2}\right)^{2}-2b^{2}\left((k_{x}^{2}+m^{2})v_{s}^{2}+\omega^{2}\right)\right)}\,,$
$\displaystyle\kappa_{LR}(\omega,k_{x})=-\kappa_{RL}(\omega,k_{x})=\frac{(\epsilon+P)\left(b^{2}-(k_{x}^{2}+m^{2})v_{s}^{2}-\omega^{2}\right)}{T\left(b^{4}+\left((k_{x}^{2}+m^{2})v_{s}^{2}-\omega^{2}\right)^{2}-2b^{2}\left((k_{x}^{2}+m^{2})v_{s}^{2}+\omega^{2}\right)\right)}\,.$

When $m=b=0$, we have divergent DC heat transports
$\kappa_{LL}(0,0),\kappa_{RR}(0,0)$. For general $m$ and $b$, we have
vanishing DC heat transport. The poles of heat transports are exactly the same
as the spectrum of the effective Hamiltonian (3.12) in section 3.4.2. This is
due to the fact that there are no dissipation terms at the first derivative
term in the constitutive equation. It is easy to check that when close to the
poles (3.13), the heat transports diverge.

##  7 $\mathcal{O}(k^{2})$ effects

Up to now we have focused on the first order in $k$ effect, where energy
momentum non-conservation terms change the spectrum of the modes, while there
are no dissipative terms at leading order. In this section, we consider
$\mathcal{O}(k^{2})$ effects, which in original hydrodynamics provide
dissipative effects and lead to imaginary parts in the poles.

As an example, we consider the single 4D case studied in section 3.4.1. We
take into account all the second order in $k$ terms in the constitutive
equations. The effective Hamiltonian up to $\mathcal{O}(k^{2})$ becomes

$\displaystyle H=\begin{pmatrix}0&~{}~{}k_{x}+im&~{}~{}k_{y}&~{}~{}k_{z}\\\
(k_{x}-im)v_{s}^{2}&~{}~{}\frac{-i\eta
k^{2}-i(\frac{\eta}{3}+\zeta)k_{x}^{2}}{\epsilon+P}&~{}~{}-i\frac{(\frac{\eta}{3}+\zeta)k_{x}k_{y}}{\epsilon+P}&~{}~{}-i\frac{(\frac{\eta}{3}+\zeta)k_{x}k_{z}}{\epsilon+P}\\\
k_{y}v_{s}^{2}&~{}~{}-i\frac{(\frac{\eta}{3}+\zeta)k_{x}k_{y}}{\epsilon+P}&~{}~{}\frac{-i\eta
k^{2}-i(\frac{\eta}{3}+\zeta)k_{y}^{2}}{\epsilon+P}&~{}~{}-i\frac{(\frac{\eta}{3}+\zeta)k_{y}k_{z}}{\epsilon+P}+ibv_{s}\\\
k_{z}v_{s}^{2}&~{}~{}-i\frac{(\frac{\eta}{3}+\zeta)k_{x}k_{z}}{\epsilon+P}&~{}~{}-i\frac{(\frac{\eta}{3}+\zeta)k_{y}k_{z}}{\epsilon+P}-ibv_{s}&~{}~{}\frac{-i\eta
k^{2}-i(\frac{\eta}{3}+\zeta)k_{z}^{2}}{\epsilon+P}\end{pmatrix}\,.$ (7.1)

The $k^{2}$ terms are dissipative as we can see that they make the effective
Hamiltonian matrix non-Hermitian. Here we still keep terms at $m\sim b$ order
while not $m^{2}\sim b^{2}$ order assuming that $m\sim k^{2}$ in this part.

From the eigenvalues of (7.1) we find that the real part has not changed while
imaginary parts appear. At the four nodes, the imaginary parts are not zero
indicating that the four nodes are dissipative in comparison to nondissipative
nodes at $\omega=0$ in the usual hydrodynamics.

As we could see from figure 19, the imaginary part for each of the band has a
jump at the crossing nodes at $k_{y}=0$ in the $k_{x}$ axis, i.e. the
imaginary parts of the same band are different at the left and right limits of
the singular node. This behavior is similar to the behavior of the eigenstates
when calculating the Berry phase and this provides another piece of evidence
of the existence of a symmetry protected topological singular node.

Figure 19: The real and imaginary parts for the lower two bands of the
modified hydrodynamics with (7.1) in the left plot of figure 4. The same color
in the two plots corresponds to the same band.

For the 2D+2D case, there are no dissipative terms and for other cases the
results are similar which we skip here. Note that the Berry phase results
would not be affected by $k^{2}$ terms as these at most change the zero result
of $|\langle n_{1}|n_{2}\rangle|$ to a small number proportional to $k^{2}$
and this suggests that the two states are still far from being equivalent for
which $|\langle n_{1}|n_{2}\rangle|=1$.

##  8 Ward identities and holographic realization

The physics of hydrodynamics has been studied extensively in holography for
strongly coupled systems [11, 12]. Many interesting results have been
obtained, e.g. the famous KSS bound for the shear viscosity over entropy
density ratio [13]. In this section we focus on the holographic realization of
the topological hydrodynamic modes above. As a first step, we need to have a
holographic system with the same non-conservation equations for the energy
momentum tensor. In holography, to break translational symmetry, we could
either introduce external fields into the system or start from massive gravity
[34] which breaks diffeomorphism itself. In this paper, we choose to view
$h_{\mu\nu}$ as the gravitational field that comes from a reference frame
transformation from the original inertial reference frame. We will start from
an ordinary holographic system and perform a reference frame transformation to
obtain a holographic system whose ward identities for the energy momentum
tensor have the same form as those from the non-conservation equations in
previous sections.

Here we focus on the single 4D case with only one energy momentum tensor. We
will first derive the Ward identities for the energy momentum tensor for the
hydrodynamic system with non-conservation terms from section 3.4.1. Then we
show how in holography we could get these general Ward identities without
calculating out all the Green function components.

###  8.1 Ward identities

For conserved energy momentum tensor, the Ward identities are [35]

$\displaystyle k_{\mu}(G^{\mu\nu\lambda\rho}-\eta^{\nu\lambda}\langle
T^{\mu\rho}\rangle-\eta^{\nu\rho}\langle
T^{\mu\lambda}\rangle-\eta^{\lambda\rho}\langle
T^{\mu\nu}\rangle+\eta^{\mu\nu}\langle T^{\lambda\rho}\rangle)=0\,,$ (8.1)

where $G^{\mu\nu\lambda\rho}$ is the Green function for the energy momentum
tensor. This Ward identity could easily be checked holographically by
calculating out all components of Green functions for $T^{\mu\nu}$.

Now we calculate the Ward identities for the single 4D system with non-
conservation terms of $T^{\mu\nu}$. In the case that these non-conservation
terms come from a gravitational field, we could start from the covariant
conservation equation $\nabla_{\mu}T^{\mu\nu}=0$ and differentiate it with
respect to $g_{\lambda\rho}$, we obtain the Ward identities in the momentum
space

$\displaystyle\begin{split}k_{\mu}\bigg{[}-G^{\mu\nu,\lambda\rho}(k)+g^{\lambda\rho}\langle
T^{\mu\nu}\rangle+g^{\nu\lambda}\langle T^{\mu\rho}\rangle+g^{\nu\rho}\langle
T^{\mu\lambda}\rangle-g^{\mu\nu}\langle T^{\lambda\rho}\rangle\bigg{]}&\\\
-i\bigg{[}\big{(}g^{\mu\rho}\Gamma^{\lambda}_{~{}\alpha\mu}+g^{\mu\lambda}\Gamma^{\rho}_{~{}\alpha\mu}\big{)}\langle
T^{\alpha\nu}\rangle+\big{(}g^{\nu\rho}\Gamma^{\lambda}_{~{}\alpha\mu}+g^{\nu\lambda}\Gamma^{\rho}_{~{}\alpha\mu}\big{)}\langle
T^{\alpha\mu}\rangle\bigg{]}&\\\
-i\Big{[}\Gamma^{\mu}_{~{}\mu\alpha}G^{\alpha\nu,\lambda\rho}(k)+\Gamma^{\nu}_{~{}\mu\alpha}G^{\mu\alpha,\lambda\rho}(k)\Big{]}=0\,,~{}~{}~{}&\end{split}$
(8.2)

where $\Gamma^{\nu}_{~{}\mu\alpha}$ is the Christoffel tensor for the metric
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$. To the first order in $h_{\mu\nu}$ we
could rewrite the Ward identities above as

$\displaystyle
k_{\mu}G^{\mu\nu,\lambda\rho}(k)+i\Big{[}\Gamma^{(1)\mu}_{~{}~{}~{}~{}\mu\alpha}G^{\alpha\nu,\lambda\rho}(k)+\Gamma^{(1)\nu}_{~{}~{}~{}~{}\mu\alpha}G^{\mu\alpha,\lambda\rho}(k)\Big{]}+\text{contact
terms}=0\,,$ (8.3)

where the explicit form of the contact terms is omitted. With nonzero
$h_{\mu\nu}$ several components of $\Gamma^{(1)\nu}_{~{}~{}~{}~{}\mu\alpha}$
would be nonzero and contribute extra terms to the Ward identities compared to
the ones in hydrodynamics systems with conserved $T^{\mu\nu}$.

Note that we could as well choose to treat $h_{\mu\nu}$ as an external
effective matter field as in section 4.1. We start from the non-conservation
equation (4.5) and perform the same procedure as above. The final result is
exactly the same as (8.3).

###  8.2 Holographic realization

In this section, we show that the Ward identities obtained in the previous
subsection could be reproduced from a holographic system whose boundary theory
lives in a non-inertial frame,888Previous studies on holography for field
theories on curved spacetime could be found in e.g. [36]. i.e. we start from
the usual AdS Schwartzchild black hole and perform coordinate transformations
so that the boundary metric becomes $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$. In
this way, we get a non-inertial reference frame version of AdS/CFT
correspondence and obtain a holographic realization of the non-conservation
equations for $T^{\mu\nu}$ introduced in the single 4D hydrodynamic system.
More evidence, e.g. direct calculations of hydrodynamic modes and Green
functions as well as holographic realizations for other systems in section 3
will be provided in future work.

In the rest of this section we will first show how to derive Ward identities
for an ordinary holographic system without calculating out all components of
the Green functions. Then we use the same procedure to get the Ward identities
for the holographic non-inertial reference frame system and show that they are
exactly the same as (8.3).

####  8.2.1 Prescription for holographic Ward identities

To calculate Green functions for the energy momentum tensor in holography, we
switch on perturbations of the metric field and solve all the fields from
their equations of motion. Then we substitute the solutions into the action to
get the on-shell action and differentiate the on-shell action with the source
terms to get corresponding Green functions. For holographic systems with
simple gravity backgrounds, this calculation is simple, however, with
$h_{\mu\nu}$ fields turned on and $k_{x}$, $k_{y}$, $k_{z}$ all nonzero, the
on-shell action and the solutions would be much more complicated. Here we show
that we do not need the details of the on-shell action and could still get the
Ward identities for the energy momentum tensor.

For the purpose of calculating the Green functions for $T^{\mu\nu}$ we have
ten independent fields of $\delta g_{\mu\nu}$, which are the
$tt,tx,ty,tz,xx,xy,xz,yy,yz,zz$ components. Here for simplicity we could
denote these ten fields as $\phi_{i}$, $i\in\\{1...,10\\}$. $\delta g_{r\mu}$
could also be nonzero but will finally be eliminated from the on-shell action
using its equation of motion.

The action of $\delta g_{\mu\nu}(\vec{k})$ could be written as

$\displaystyle
S\supset\int\frac{drd^{4}k}{(2\pi)^{4}}\bigg{(}W_{1}^{ij}\phi_{i}^{\prime\prime}(-\vec{k})\phi_{j}(\vec{k})+W_{2}^{ij}\phi_{i}^{\prime}(-\vec{k})\phi_{j}^{\prime}(\vec{k})+W_{3}^{ij}\phi_{i}^{\prime}(-\vec{k})\phi_{j}(\vec{k})+W_{4}^{ij}\phi_{i}(-\vec{k})\phi_{j}(\vec{k})\bigg{)},$

where $\phi_{i}(\vec{k})$ are functions of $r$ only and here
$\vec{k}=(\omega,k_{x},k_{y},k_{z})$ and $W_{1,2,3,4}^{ij}$ are functions of
$r$ whose explicit form depends on the system. Equations of motion could be
derived from this action and after substituting the solutions into this
action, the on-shell action that is relevant to the Green functions becomes

$\displaystyle S_{\rm on-
shell}\supset\int\frac{d^{4}k}{(2\pi)^{4}}W_{2}^{ij}\phi_{i}^{\prime}(-\vec{k})\phi_{j}(\vec{k})\Big{|}_{r_{h}}^{r_{b}}+\cdots,$
(8.4)

where $\cdots$ are terms that are only related to the contact terms and
$\phi_{i}(\vec{k})$ in the above formula have to be substituted by their on-
shell solutions. $r_{h}$ and $r_{b}$ denote the horizon and the boundary
separately. For AdS $W_{1,2,3,4}^{ij}$ could be calculated for all components,
however, we show in the following that we are still able to derive the Ward
identities without the explicit form of these terms, which manifest the
physical origin of the Ward identities more clearly.

Due to the diffeomorphism invariance of the gravity system, the action has to
be invariant under coordinate transformations. Under coordinate
transformations generated by $\epsilon_{\mu}(t,x,y,z)$, fields transform as

$\displaystyle\delta
g_{\mu\nu}=\nabla_{\mu}\epsilon_{\nu}+\nabla_{\nu}\epsilon_{\mu}\,,$ (8.5)

where $\nabla$ is performed with respect to the background AdS metric. In this
way each component $\delta g_{\mu\nu}$ cannot be invariant under general
coordinate transformations, while some combinations of several components of
$\delta g_{\mu\nu}$ could be a gauge invariant and the action must be written
into sums of gauge invariant combinations due to its diffeomorphism
invariance.

We choose the nonzero components of $\epsilon_{\mu}$ to be
$\epsilon_{t,x,y,z}(\vec{k})$ with dependence on momenta in all directions and
$r$ dependence does not affect the calculations. From (8.5) six independent
gauge invariants $Z_{i},i=1,...,6$ under these general gauge transformations
could be found in total and all possible sums of these gauge invariants are
also gauge invariants. The six independent gauge invariants are

$\displaystyle\begin{split}Z_{1}&=\frac{\delta
g_{xx}}{2k_{x}^{2}}+\frac{\delta g_{tx}}{\omega k_{x}}+\frac{\delta
g_{tt}}{2\omega^{2}}\,,~{}~{}~{}Z_{2}=\frac{\delta
g_{yy}}{2k_{y}^{2}}+\frac{\delta g_{ty}}{\omega k_{y}}+\frac{\delta
g_{tt}}{2\omega^{2}}\,,\\\ Z_{3}&=\frac{\delta
g_{zz}}{2k_{z}^{2}}+\frac{\delta g_{tz}}{\omega k_{z}}+\frac{\delta
g_{tt}}{2\omega^{2}}\,,~{}~{}~{}Z_{4}=\frac{\delta
g_{xx}}{2k_{x}^{2}}-\frac{\delta g_{xy}}{k_{x}k_{y}}+\frac{\delta
g_{yy}}{2k_{y}^{2}}\,,\\\ Z_{5}&=\frac{\delta g_{xx}}{2k_{x}^{2}}-\frac{\delta
g_{xz}}{k_{x}k_{z}}+\frac{\delta
g_{zz}}{2k_{z}^{2}}\,,~{}~{}~{}Z_{6}=\frac{\delta
g_{yy}}{2k_{y}^{2}}-\frac{\delta g_{yz}}{k_{y}k_{x}}+\frac{\delta
g_{zz}}{2k_{z}^{2}}\,.\end{split}$ (8.6)

The most general form of the on-shell action could be written as

$\displaystyle
S\supset\int\frac{d^{4}k}{(2\pi)^{4}}G_{ij}(r)Z_{i}^{\prime}(-\vec{k})Z_{j}(\vec{k})\Big{|}_{r_{h}}^{r_{b}}\,,$
(8.7)

where $G_{ij}$ is a real and symmetric function matrix with $G_{ij}=G_{ji}$,
$i,j=1,\dots,6$. The Green functions could be calculated from the on-shell
action above as functions of the variables $G_{ij}$, for example,

$\displaystyle G_{tt,tt}=\frac{\delta^{2}S}{\delta(\delta
g_{tt}^{b})\delta(\delta
g_{tt}^{b})}=\frac{1}{2\omega^{4}}(G_{11}+2G_{12}+2G_{13}+G_{22}+2G_{23}+G_{33})\,,$
(8.8)

and

$\displaystyle G_{tx,tx}=\frac{1}{4}\frac{\delta^{2}S}{\delta(\delta
g_{tx}^{b})\delta(\delta g_{tx}^{b})}=\frac{1}{2k_{x}^{2}\omega^{2}}G_{11}\,,$
(8.9)

where $\delta g^{b}_{\mu\nu}$ denotes the boundary value of these fields.

Green functions have 55 independent components while $G_{ij}$ have 21
independent variables. Expressing all Green function components using $G_{ij}$
and eliminating $G_{ij}$ from these expressions, we are left with 34
identities. In this system, the Ward identity we see from the field theory
derivation has 40 identities, 6 of which are independent that could be derived
from other 34 identities. Thus the number of total independent Ward identities
are the same from the two sides. We have checked through tedious calculations
that after eliminating $G_{ij}$ the remaining 34 Ward identities are exactly
the Ward identities (8.3).

Note that the appearance of any $k_{\mu}$ in the expression of the action
indicates that a derivative is taken in that direction. From the combinations,
it seems that the total derivatives including the $r$ derivatives may exceed
two which should be the highest order of derivatives in Einstein gravity. This
is because the on-shell action is obtained from solving the constraint
equations of $\delta g_{r\mu}$ and substituting the expressions of $\delta
g_{r\mu}$ into the action, which brings higher orders into the action.

####  8.2.2 Ward identities in holographic non-inertial frame systems

Now let us come to the system with energy momentum non-conservation as in
section 3.4.1. To obtain the deformed metric in the bulk after a reference
frame transformation to the non-inertial frame, we first choose the reference
frame transformation $\xi_{\mu}$ to be the same as in the field theory part.
The AdS background now becomes
$g_{\mu\nu}^{\text{bulk}}=g^{AdS}_{\mu\nu}+h_{\mu\nu}^{\text{bulk}}$, where
$h_{\mu\nu}^{\text{bulk}}=\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu}$ with
$\nabla_{\mu}$ performed with respect to the bulk background metric.

At leading order in $m$ and $b$, only nonzero components of
$h_{\mu\nu}^{\text{bulk}}$ at the boundary are

$\displaystyle\begin{split}h_{tt}^{\text{bulk}}=mxg_{tt}^{\text{bulk}},~{}h_{tx}^{\text{bulk}}=\frac{1}{2}m(1+v_{s}^{2})g_{tt}^{\text{bulk}},~{}h_{xx}^{\text{bulk}}=mxg_{xx}^{\text{bulk}},\\\
h_{ty}^{\text{bulk}}=-\frac{1}{2}bv_{s}zg_{tt}^{\text{bulk}},h_{tz}^{\text{bulk}}=\frac{1}{2}bv_{s}yg_{tt}^{\text{bulk}}.\end{split}$
(8.10)

Note that in the bulk the expressions for these fields would be more
complicated, but we only need the boundary expressions in deriving the Green
functions.

With the new metric, the form of the on-shell action would be different from
the AdS one, nevertheless, it can still be written as sums of gauge invariant
terms. However, now the combinations (8.6) are not gauge invariant under
coordinate transformations anymore. We need to derive the new gauge invariant
combinations in this new background.

We choose $\epsilon_{\mu}$ to be $\epsilon_{\mu}(t,x,y,z)$, where
$\mu=t,x,y,z$. As the background metric now depends on $t,x,y,z$, we keep the
$t,x,y,z$ dependence of $\epsilon_{\mu}$. We could write out the
transformations of all components of $\delta g_{\mu\nu}$ under this general
coordinate transformation and look for all independent combinations of the
components which remain unchanged under this coordinate transformation. After
we find out the gauge invariant combinations we can transform to the $k$ space
as the background metric only has contributions at $k=0$ though it has
nontrivial dependence on the coordinates. Up to leading order in $m$ and $b$,
the new gauge invariant combinations could be found to be

$\displaystyle\begin{split}Z_{1}&=\frac{\delta
g_{xx}}{2k_{x}^{2}}+\frac{\delta g_{tx}}{\omega k_{x}}+\frac{\delta
g_{tt}}{2\omega^{2}}-\frac{im\delta g_{tt}}{4k_{x}\omega^{2}}+\frac{im\delta
g_{tx}}{2k_{x}^{2}\omega}-\frac{imv_{s}^{2}\delta
g_{xx}}{4k_{x}\omega^{2}}\,,\\\ Z_{2}&=\frac{\delta
g_{yy}}{2k_{y}^{2}}+\frac{\delta g_{ty}}{\omega k_{y}}+\frac{\delta
g_{tt}}{2\omega^{2}}-\frac{imv_{s}^{2}\delta
g_{xx}}{4k_{x}\omega^{2}}-\frac{ibv_{s}\delta g_{zz}}{4k_{y}k_{z}\omega}\,,\\\
Z_{3}&=\frac{\delta g_{zz}}{2k_{z}^{2}}+\frac{\delta g_{tz}}{\omega
k_{z}}+\frac{\delta g_{tt}}{2\omega^{2}}-\frac{imv_{s}^{2}\delta
g_{xx}}{4k_{x}\omega^{2}}+\frac{ibv_{s}\delta g_{yy}}{4k_{y}k_{z}\omega}\,,\\\
Z_{4}&=\frac{\delta g_{xx}}{2k_{x}^{2}}-\frac{\delta
g_{xy}}{k_{x}k_{y}}+\frac{\delta g_{yy}}{2k_{y}^{2}}-\frac{im\delta
g_{xx}}{4k_{x}^{3}}\,,\\\ Z_{5}&=\frac{\delta g_{xx}}{2k_{x}^{2}}-\frac{\delta
g_{xz}}{k_{x}k_{z}}+\frac{\delta g_{zz}}{2k_{z}^{2}}\,,\\\ Z_{6}&=\frac{\delta
g_{yy}}{2k_{y}^{2}}-\frac{\delta g_{yz}}{k_{y}k_{x}}+\frac{\delta
g_{zz}}{2k_{z}^{2}}-\frac{im\delta g_{xx}}{4k_{x}^{3}}\,.\end{split}$ (8.11)

The on-shell action at the boundary is now

$\displaystyle
S\supset\int\frac{d^{4}k}{(2\pi)^{4}}G_{ij}(r)Z_{i}^{\prime}(-\vec{k})Z_{j}(\vec{k})\Big{|}_{r_{b}}+...\,,$
(8.12)

where $\cdots$ terms are horizon contributions and contact terms related
parts. $G_{ij}$ is still a real and symmetric function matrix, while the
values should be different from the AdS ones. Now the Green functions could
again be obtained from the on-shell action. As an example we write out two of
the components here

$\displaystyle\begin{split}G_{tt,tt}=\frac{\delta^{2}S}{\delta(\delta
g_{tt}^{b})\delta(\delta
g_{tt}^{b})}=&\frac{1}{2\omega^{4}}(G_{11}+2G_{12}+2G_{13}+G_{22}+2G_{23}+G_{33})\\\
&~{}~{}-\frac{im}{k_{x}\omega^{4}}\bigg{(}G_{12}+G_{13}+G_{22}+2G_{23}+G_{33}\bigg{)}\,,\end{split}$
(8.13)

and

$\displaystyle\begin{split}G_{tx,tx}&=\frac{1}{4}\frac{\delta^{2}S}{\delta(\delta
g_{tx}^{b})\delta(\delta g_{tx}^{b})}\\\
&=\frac{1}{2k_{x}^{2}\omega^{2}}G_{11}-\frac{im}{2}\bigg{(}-\frac{G_{11}v_{s}^{2}}{k_{x}\omega^{4}}-\frac{G_{12}v_{s}^{2}}{k_{x}\omega^{4}}-\frac{G_{13}v_{s}^{2}}{k_{x}\omega^{4}}+\frac{G_{11}}{k_{x}^{3}\omega^{2}}-\frac{G_{14}}{k_{x}^{3}\omega^{2}}-\frac{G_{15}}{k_{x}^{3}\omega^{2}}\bigg{)}\,.\end{split}$
(8.14)

There are extra $m$ and $b$ terms in the expressions for the Green functions
now. Again we could eliminate all $G_{ij}$ and get the resulting Ward
identities. Here we take a simpler method: by substituting the resulting Green
functions which are functions of $G_{ij}$ into the field theory Ward
identities, we could see that all the $G_{ij}$ cancel out in the Ward
identities leaving only components of the Green functions and the field theory
Ward identities indeed hold in holography ignoring all contact term
contributions. This confirms that the holographic set-up indeed gives the
holographic system of the hydrodynamic system with topological modes.

Note that the Ward identities in the new non-inertial reference frame come
from covariance of the energy momentum conservation. Here in holography, the
on-shell action would still transform back to the AdS one after a backward
reference frame transformation. The difference in the Ward identities reflects
non-inertial effects.

More details about Green functions and the hydrodynamic modes from Green
functions would be reported in a future work. We will also consider other
possibilities of holographic realizations, e.g. viewing $h_{\mu\nu}$ as
external fields while not gravitons from reference frame transformations or
from massive gravities. Holographic systems for other dimensions and for
systems with gapped hydrodynamic topological modes will be looked at, too.

##  9 Conclusions and discussions

We have studied topological hydrodynamic modes in relativistic hydrodynamics
by introducing non-conservation terms for the energy momentum tensor. There
are several systems where we could get topologically nontrivial hydrodynamic
modes. Depending on different forms of the $T^{\mu\nu}$ non-conservation
terms, the resulting crossing nodes in the spectrum of hydrodynamic modes
could either be topologically nontrivial under the protection of special
spacetime symmetries in certain dimensions or directly topologically
nontrivial. We confirm their nontrivial topology from the calculation of
nontrivial topological invariants for all these systems. We also studied the
dissipative $O(k^{2})$ effects and show that different from the original
$\omega={\bf k}=0$ crossing nodes for relativistic hydrodynamics, the crossing
nodes in these systems are mostly dissipative. The discontinuous imaginary
parts for the hydrodynamic modes provides another piece of evidence of the
nontrivial topology of the crossing nodes.

The non-conservation terms for $T^{\mu\nu}$ could come from an effective
external rank two symmetric matter field or from a gravitational field. In the
latter case, the system could be viewed as to be in a non-inertial reference
frame indicating that topologically nontrivial modes could arise from trivial
modes by transforming to a specific non-inertial reference frame. We propose a
holographic realization of one of these system, starting from ordinary AdS/CFT
correspondence and perform a transformation to a non-inertial frame. We derive
the Ward identities for the non-conserved equation of $T^{\mu\nu}$ and match
them to the holographic realization of this system.

There are many open questions and possible generalizations at this moment
which we hopefully will address in the future. The following is an incomplete
list of them.

  * •

We have focused on band crossing hydrodynamic modes. Is it possible to also
find topologically nontrivial modes without band crossings in a classical
relativistic hydrodynamic system?

  * •

What would be possible experimental observable effects besides the transport
coefficients discussed in this paper and the sudden increase of amplitude at
the crossing node? Would the graphene system be a possible arena for the test
of the theoretical predictions in this paper?

  * •

Is it possible to have similar topological modes in non-relativistic
hydrodynamics999Similar topological modes with deformed effective Hamiltonians
in a non-relativistic system has been studied in [37] for Lamb waves/
acoustic-gravity waves in earth physics. We thank Karl Landsteiner for
bringing this reference to us. in non-inertial frame, which might be easier
for laboratory tests?

  * •

We have shown that topologically trivial hydrodynamic modes could turn into
nontrivial ones in a non-inertial reference frame. Would this behavior also
hold for other topological states of matter, e.g. topological electronic
systems? If yes, could this property be tested in a laboratory resting in a
non-inertial reference frame?

  * •

Is it possible to find a similar $h_{\mu\nu}$ field from analog gravity
systems, i.e. from certain materials?

  * •

We have focused on hydrodynamic systems without any other conserved charges
except the energy momentum tensor. Would the structure be more complicated
with more conserved charges?

  * •

Is it possible to find other holographic realizations of the single 4D system,
e.g. from external matter field, or massive gravity [38]?

  * •

What are possible holographic set-ups for the 2D+2D systems? Possibly a bi-
metric gravitational theory [39] in the bulk could provide us a holographic
example.

  * •

All the effective Hamiltonians studied in this paper are similar to Hermitian
matrices. In our search for effective Hamiltonians that could give rise to
interesting spectrum that we need, we focused on Hermitian matrices as most
non-Hermitian matrices would give complex eigenvalues. This raises the
interesting question if non-Hermitian matrices with PT symmetry could be found
which also give similar while real spectrum and topological states here. Could
we also have similar holographic non-Hermitian physics as in [40]?

  * •

Though we have shown that the holographic set-up indeed reproduces the Ward
identities for the system with $m$ and $b$ terms, what is the topological
structure in the bulk? Is this related to some new global properties of
spacetime?

  * •

In holography, hydrodynamic modes on the boundary corresponds to gravitational
field in the bulk. What would the work in this paper imply for the physics of
gravitational waves in asymptotically flat spacetime? Is it possible to find
possible topological modes in gravitational waves?

  * •

It would be interesting to study the effect field theory description of
hydrodynamics with nontrivial topological modes along the lines developed in
[41, 42, 43].

### Acknowledgments

We would like to thank Matteo Baggioli, Rong-Gen Cai, Hyun-Sik Jeong, Shu Lin,
Wen-Bin Pan, Koenraad Schalm and especially Karl Landsteiner for useful
discussions. This work is supported by the National Key R&D Program of China
(Grant No. 2018FYA0305800). The work of Y.L. was also supported by the
National Natural Science Foundation of China grant No.11875083. The work of
Y.W.S. has also been partly supported by starting grants from University of
Chinese Academy of Sciences and Chinese Academy of Sciences, and by the Key
Research Program of Chinese Academy of Sciences (Grant No. XDPB08-1), the
Strategic Priority Research Program of Chinese Academy of Sciences, Grant No.
XDB28000000, and by the National Natural Science Foundation of China Grant No.
12035016..

##  Appendix A Hydrodynamics with momentum dissipation

In this appendix we show the effects of momentum dissipation terms in the
dispersions of hydrodynamic modes for comparison. In contrast to the single 4D
topological system in the main text, momentum dissipation terms that have been
studied extensively in hydrodynamics and holography [22, 23, 24, 25],
contribute to imaginary parts of the spectrum, thus indicating that they are
dissipative terms, different from the terms for topological modes above. Note
that we added this appendix only for readers not to mix our system with the
momentum dissipation systems which have been studied a lot in holography. In
the system in this appendix, momentum dissipation could be caused by external
fields as well as broken diffeomorphism invariance and in all cases that have
been studied $\Gamma$ is real while not imaginary. Even if the spectrum in
that case could be real, that has no relation to topology, at least not any
that we could directly see. Our system has a nontrivial topological structure
not because their spectrum is real, i.e. we are not trying to find a real
spectrum in hydrodynamics and claim that this is topologically nontrivial
because it has real spectrum. Real spectrum is a very basic and necessary
requirement for us to analyze the topological structure of the system but it
does not mean that as long as we have real spectrum, we would have a
nontrivial topological structure. Whether the system has a nontrivial
topological structure depends on the structure of the effective Hamiltonian or
equivalently the structure of the spectrum.

For relativistic hydrodynamics with weak momentum dissipation, the
conservation equations change to [22, 23, 25]

$\displaystyle\partial_{\mu}\delta T^{\mu t}$ $\displaystyle=$ $\displaystyle
0\,,$ $\displaystyle\partial_{\mu}\delta T^{\mu i}$ $\displaystyle=$
$\displaystyle\Gamma\delta T^{ti}\,,$

where $\Gamma\ll 1/T.$ The momentum is almost conserved and weakly broken.

Up to the first order in derivatives, the four dimensional conservation
equation can be written as

$\displaystyle i\partial_{t}\Psi=H\Psi$ (A.1)

where

$\displaystyle\Psi=\begin{pmatrix}\delta\epsilon_{L}\\\ \delta\pi_{L}\\\
\delta\epsilon_{R}\\\
\delta\pi_{R}\end{pmatrix}\,,~{}~{}~{}~{}~{}~{}H=\begin{pmatrix}0&~{}~{}k_{x}&~{}~{}k_{y}&~{}~{}k_{z}\\\
v_{s}^{2}k_{x}&~{}~{}-i\Gamma&~{}~{}0&~{}~{}0\\\
v_{s}^{2}k_{y}&~{}~{}0&~{}~{}-i\Gamma&~{}~{}0\\\
v_{s}^{2}k_{z}&~{}~{}0&~{}~{}0&~{}~{}-i\Gamma\end{pmatrix}\,.$ (A.2)

It is easy to see that $H$ is not Hermitian and the spectrum is

$\displaystyle\omega=\bigg{(}-i\Gamma,~{}-i\Gamma,~{}\frac{1}{2}\big{(}-i\Gamma-\sqrt{4v_{s}^{2}k^{2}-\Gamma^{2}}\big{)},~{}\frac{1}{2}\big{(}-i\Gamma+\sqrt{4v_{s}^{2}k^{2}-\Gamma^{2}}\big{)}\bigg{)}\,.$
(A.3)

These terms contribute to imaginary parts of the spectrum, indicating that
they are dissipative terms, different from the terms for topological modes
studied in this paper.

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