††thanks: These two authors contributed equally to this work.††thanks: These
two authors contributed equally to this work.

# Higher order topological state induced by $d$-wave competing orders in high-
Tc superconductor based heterostructure

Xiaoming Wang1 Yu-Xuan Li2 Tao Zhou1 Corresponding author<EMAIL_ADDRESS>1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum
Materials, School of Physics and Telecommunication Engineering, Guangdong-Hong
Kong Joint Laboratory of Quantum Matter, and Frontier Research Institute for
Physics, South China Normal University, Guangzhou 510006, China
2Centre for Quantum Physics, Key Laboratory of Advanced Optoelectronic Quantum
Architecture and Measurement (MOE), School of Physics, Beijing Institute of
Technology, Beijing 100081, China

###### Abstract

We introduce a two-dimensional Chern insulator in proximity to a $d$-wave
pseudogap state of the high-Tc superconducting material as an effective
platform to realize the higher order topological system. The proximity-induced
$d$-density-wave (DDW) order in the Chern insulator layer serves as an
effective mass. The edge states will be fully gapped by this DDW order. In the
real space, the sign of the DDW order parameter changes at the system corners
due to the $d$-wave factor, leading to the gapless corner states, indicating
that this system may be in a higher order topological state. The higher order
topology in this coupled system is confirmed based on the calculation of the
edge polarization and the quadrupole moment. In the superconducting state
where the superconducting order and the DDW order coexist, the Majorana corner
states emerge.

††preprint: APS/123-QED

##  I Introduction

Higher order topology has gained much attention in recent years [13, 12, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29]. For a $d$-dimensional n-th order topological material, the
system is fully gapped at its $d-1$ dimensional boundaries, while it hosts
gapless states at its d-n dimensional boundaries $(n\geq 2)$. In a two-
dimensional higher order topological material, the gapless modes emerge at the
corners between different fully gapped one-dimensional edges. In the past
several years, the higher order topology has been proposed to exist in the
three-dimensional Bismuth material [3, 19], the twisted angle graphene [16],
the graphdiyne [17], and some heterostructure systems and artificial systems
[20, 21, 22, 23, 24, 25, 26, 27, 28, 29].

A number of proposals for the realization of higher order topological
superconductors demand unconventional superconducting pairing term, for
instance, the $d$-wave pairing term [20, 21, 22, 23, 24, 25]. For a
heterostructure with an additional superconducting pairing term being added to
a first order topological system, an energy gap may be induced by the
superconducting pairing term. Then, the original gapless system edges may be
fully gapped. For a $d$-wave superconductor, the sign of the pairing term
changes at the system corners. As a result, the effective mass may change and
the band inversion occurs at these corners, leading to the possible Majorana
corner states. Now it is well accepted that the cuprate high-Tc
superconducting material has the $d$-wave pairing symmetry. Therefore, the
heterostructure with the high-Tc superconducting material may provide a useful
platform to realize the higher order topological superconductor or the
Majorana corner states.

The high-Tc superconductivity in the family of the cuprate materials has been
studied for more than three decades. One of the most important issues is the
pseudogap behaviour in the underdoped region of the phase diagram [30]. So far
the origin of the pseudogap is still an open question. A possible explanation
is that the pseudogap may be due to a certain competing order [4, 5, 33, 34].
In the topological superconducting system with the high-Tc superconductor
platform, the interplay between the competing order and the topological
behaviour is also of interest [35, 36]. The additional energy gap induced by
the competing order may change the topological behaviour, so that the
topological behaviour may be used to detect the competing orders [35]. And
recently it was indicated that a charge density wave order may be used to
modulate the gap size and the Majorana zero modes [36].

Experimentally, it was indicated that the pseudogap in the high-Tc
superconducting material has the similar momentum dependence with the
superconducting gap. Therefore, generally a $d$-wave type factor may exist in
the possible competing pseudogap order [4, 5, 33, 37, 38, 39]. The existence
of the $d$-wave factor may lead to the mass change and the band inversion at
the system corners. Therefore, it is naturally to ask an interesting question
“can the $d$-wave type pseudogap in high-Tc superconducting materials be used
to realize the higher order topology?”. If the answer is positive, more
effective proposals may be brought forward to realize the higher order
topological states. On the other hand, this may be used to detect and resolve
different competing orders of the high-Tc superconductors, providing more
insight for the physical origin of the pseudogap behaviour in the high-Tc
superconductors.

In this paper, we consider a heterostructure of a two-dimensional Chern
insulator in proximity of a high-Tc superconducting material with the
$d$-density wave (DDW) order [40]. Our numerical results indicate that the
gapless edge states of a Chern insulator are gapped out by the DDW order. The
gapless states emerge at the system corners. These results propose that this
coupled system is indeed a higher order topological system. The higher order
topology is confirmed further through calculating the edge polarization with
the Wilson loop method and the quadrupole moment.

The rest of the paper is organized as follows. In Sec. II, we introduce the
model and present the relevant formalism. In Sec. III, we report numerical
calculations and discuss the obtained results. Finally, we give a brief
summary in Sec. IV.

##  II Model and formalism

We start with a Hamiltonian in a two-dimensional square lattice considering
the coupling of a Chern insulator and an additional DDW order term [4, 5],
with the whole Hamiltonian being expressed as,

$H=H_{C}+H_{D}.$ (1)

$H_{C}$ is a two-band model describing the Chern insulator [41, 42],

$H_{C}=\sum_{{\bf k}\sigma}\sigma\varepsilon_{\bf k}c^{\dagger}_{{\bf
k}\sigma}c_{{\bf k}\sigma}+\sum_{\bf k}(\lambda_{\bf k}c^{\dagger}_{{\bf
k}\uparrow}c_{{\bf k}\downarrow}+h.c.),$ (2)

with $\sigma$ takes $+$ and $-$ for the spin-up and spin-down quasiparticles,
respectively. $\varepsilon_{\bf k}=-2t(\cos k_{x}+\cos k_{y})-m$ and
$\lambda_{\bf k}=2\lambda_{0}(\sin k_{x}+i\sin k_{y})$, represent the spin
polarized hopping and the spin-orbital coupling term, respectively. $m$ is the
effective Zeeman field strength. The above Chern insulator model may be
realized experimentally in the Hg1-xMnxTe/Cd1-xMnxTe quantum well [41] or the
magnetic topological insulators (Bi,Sb)2Te3 [2] and MnBi8Te13 [3].

$H_{D}$ is a proximity induced DDW term, with

$\displaystyle H_{D}=\sum_{\bf k}(\Delta_{\bf k}c^{\dagger}_{{\bf
k}\sigma}c_{{\bf k}+{\bf Q}\sigma}+h.c.),$ (3)

Following Refs. [4, 5], we consider the $d$-wave form factor with $\Delta_{\bf
k}=2i\Delta_{0}(\cos k_{x}-\cos k_{y})$ and the wavevector ${\bf Q}$ is taken
as $(\pi,\pi)$.

To study the edge states of the system, we would like to consider a cylinder
geometry with considering the periodic boundary condition along the $y$
direction and the open boundary condition along the $x$-direction. Then we
define a partial Fourier transformation along the $x$-direction with
$c^{\dagger}_{{\bf
k}\sigma}=\frac{1}{{\sqrt{N_{x}}}}\sum_{x}c^{\dagger}_{{k_{y}}\sigma}(x)e^{ik_{x}x}$.
The Hamiltonian is rewritten as,

$\displaystyle H_{C}=$ $\displaystyle-t\sum\limits_{k_{y},x,\sigma}[\sigma
c^{\dagger}_{k_{y}\sigma}(x)c_{k_{y}\sigma}(x+1)+h.c.]$ (4)
$\displaystyle-\lambda_{0}\sum\limits_{k_{y},x}[ic^{\dagger}_{k_{y}\uparrow}(x)c_{k_{y}\downarrow}(x+1)+h.c.]$
$\displaystyle+\lambda_{0}\sum\limits_{k_{y},x}[ic^{\dagger}_{k_{y}\uparrow}(x)c_{k_{y}\downarrow}(x-1)+h.c.]$
$\displaystyle-\sum\limits_{k_{y},x,\sigma}\sigma(m+2t\cos
k_{y})c^{\dagger}_{k_{y}\sigma}(x)c_{k_{y}\sigma}(x)$
$\displaystyle+2\lambda_{0}\sum\limits_{k_{y},x}[i\sin
k_{y}c^{\dagger}_{k_{y}\uparrow}(x)c_{k_{y}\downarrow}(x)+h.c.],$

and

$\displaystyle H_{D}=$
$\displaystyle\sum\limits_{k_{y}x\sigma}[(-1)^{R_{x}}i\Delta_{0}c^{\dagger}_{{k_{y}\sigma}}(x)c_{{k_{y}+\pi\sigma}}(x+1)+h.c.]$
$\displaystyle-2\sum\limits_{k_{y}x\sigma}[(-1)^{R_{x}}i\Delta_{0}\cos
k_{y}c^{\dagger}_{{k_{y}\sigma}}(x)c_{{k_{y}+\pi\sigma}}(x)+h.c.].$

The above Hamiltonian can be rewritten as a $4N_{x}\times 4N_{x}$ matrix
$\hat{M}({k_{y}})$ with
$H=\sum_{k_{y}}\Psi^{\dagger}(k_{y})\hat{M}(k_{y})\Psi(k_{y})$. $N_{x}$ is the
number of the sites along the $x$ direction. The vector
$\Psi^{\dagger}(k_{y})$ is expressed as,

$\Psi^{\dagger}(k_{y})=(C_{1},C_{2},\cdots,C_{N_{x}}),$ (6)

with
$C_{i}=(c^{\dagger}_{i\uparrow}(k_{y}),c^{\dagger}_{i\downarrow}(k_{y}),c^{\dagger}_{i\uparrow}(k_{y}+\pi),c^{\dagger}_{i\downarrow}(k_{y}+\pi))$.

The higher order topology of the system can be studied through calculating the
edge polarization [13, 12, 3, 4, 5]. We define a Wilson loop matrix with the
elements being written as,

$\displaystyle[W(x,k_{y})]_{ij}=$ $\displaystyle\langle
u^{i}(x,k_{y})|P^{1}(x,k_{y}+\Delta k_{y})$ (7) $\displaystyle
P^{2}(x,k_{y}+2\Delta k_{y})$ $\displaystyle\dots P^{N-2}(x,k_{y}+(N-2)\Delta
k_{y})$ $\displaystyle|u^{j}(x,k_{y})\rangle.$

$P^{l}(x,k_{y}+l\Delta k_{y})$ ($l=1,2,3,\dots,N-2$) is the projection
operator with $P^{l}(x,k_{y}+l\Delta
k_{y})=\sum_{n}\left|u^{n}({x,k_{y}+l\Delta k_{y}})\right\rangle\left\langle
u^{n}(x,k_{y}+l\Delta k_{y})\right|$. $\left|u^{n}(x,k_{y}+l\Delta
k_{y})\right\rangle$ are the eigenvectors of the matrix $\hat{M}({k_{y}})$,
with only the occupied bands being considered. Diagonalizing the Wilson loop
matrix, we have,

$\displaystyle
W(x,k_{y})\left|\nu^{j}_{k_{y}}\right\rangle=e^{i2\pi\nu^{j}_{x}}\left|\nu^{j}_{k_{y}}\right\rangle,$
(8)

where $j$ is corresponding to the occupied band index. $\nu^{j}_{x}$ is
defined as a Wannier spectrum. The tangential polarization as a function of
lattice site $R_{x}$ is given as,

$\displaystyle p_{y}(R_{x})=\sum^{occu}\rho^{j}(R_{x})\nu^{j}_{x},$ (9)

where
$\rho^{j}(R_{x})=\frac{1}{N_{x}}\sum_{k_{y},\sigma,n}\left|[u^{j}(R_{x},k_{y})]^{n,\sigma}[\nu_{k_{y}}^{j}]^{n}\right|^{2}$.
The edge polarization at the $x$ direction is given as

$p^{edge,x}_{y}=\sum_{R_{x}=1}^{R_{x}=N_{x}/2}p_{y}(R_{x}).$ (10)

For a higher order topological system, the gapless states are expected to
emerge at the system corners. In this case one needs to consider the open
boundary condition along both two directions. The Hamiltonian should be
expressed in the real space through a full Fourier transformation, with

$\displaystyle H=$ $\displaystyle-t\sum\limits_{{\bf
i},{\alpha},\sigma}[\sigma c^{\dagger}_{{\bf i}\sigma}c_{{\bf
i}+\hat{\alpha}\sigma}+h.c.]-\sum\limits_{{\bf i},\sigma}\sigma
mc^{\dagger}_{{\bf i}\sigma}c_{{\bf i}\sigma}$ (11)
$\displaystyle-\lambda_{0}\sum\limits_{\bf i}[ic^{\dagger}_{{\bf
i}\uparrow}c_{{\bf i}+\hat{x}\downarrow}-ic^{\dagger}_{{\bf i}\uparrow}c_{{\bf
i}-\hat{x}\downarrow}+h.c.]$ $\displaystyle+\lambda_{0}\sum\limits_{\bf
i}[c^{\dagger}_{{\bf i}\uparrow}c_{{\bf
i}+\hat{y}\downarrow}-c^{\dagger}_{{\bf i}\uparrow}c_{{\bf
i}-\hat{y}\downarrow}+h.c.]$ $\displaystyle-\sum\limits_{{\bf
i}\sigma}[(-1)^{R_{x}+R_{y}}i\Delta_{0}(c^{\dagger}_{{\bf i}\sigma}c_{{\bf
i}+\hat{x}\sigma}-c^{\dagger}_{{\bf i}\sigma}c_{{\bf i}+\hat{y}\sigma})$
$\displaystyle+h.c.].$

The above real space Hamiltonian can be written as a $2N\times 2N$ matrix with
$H=\Psi^{\dagger}\hat{M}\Psi$ ($N=N_{x}N_{y}$ is the number of the sites).
Then following Refs. [9, 10, 11, 12], we define the quadrupole moment, with

$\displaystyle
q_{xy}=\frac{1}{2\pi}\operatorname{Im}\log\left[\operatorname{det}\left(U^{\dagger}\hat{Q}U\right)\sqrt{\operatorname{det}\left(\hat{Q}^{\dagger}\right)}\right],$
(12)

where $\hat{Q}\equiv\exp\left[i2\pi\hat{q}_{xy}\right]$. $\hat{q}_{xy}$ is the
position operator and can be written as a diagonalized matrix with the
elements being $R_{x}R_{y}/N$. The matrix $U$ is constructed by column-wise
packing all of occupied eigenstates.

The Green’s function can be defined through diagonalizing the above
Hamiltonian matrix $\hat{M}$. The matrix elements of the Green’s function is
expressed as,

$\displaystyle G(\omega)_{ij}=\sum_{n}\dfrac{u_{in}u^{\dagger}_{jn}}{\omega-
E_{n}+i\Gamma},$ (13)

where $u_{in}$ and $E_{n}$ are the eigenvectors and the eigenvalues of the
Hamiltonian matrix.

In the real space, the local density of states at the site $i$ is calculated
from the real space Green’s function, with

$\displaystyle\rho_{i}(\omega)=-\frac{1}{\pi}\sum^{2}_{p=1}\mathrm{Im}G(\omega)_{l+p,l+p},$
(14)

where $l=2(i-1)$.

In the following presented results, the parameters are set as $t=1$,
$\lambda_{0}=1$, $m=1$, and $\Delta_{0}=0.4$. We have checked numerically that
our main conclusions are not sensitive with these parameters.

##  III results and discussion

Figure 1: (a) The energy bands considering the open boundary condition along
the $x$-direction with $\Delta_{0}=0$. (b) The energy bands in the presence of
the DDW modulation. (c) The zero energy LDOS in the presence of the DDW order
in a $20\times 20$ lattice with open boundaries along both the $x$ and the $y$
directions. (d) The corresponding eigenvalues of the Hamiltonian in the
$20\times 20$ lattice.

We first study the possible edge states with considering the open boundary
condition along the $x$ direction and periodic boundary condition along the
$y$ direction. The energy bands obtained by diagonalizing the $4N_{x}\times
4N_{x}$ Hamiltonian matrix $\hat{M}(k_{y})$ with $\Delta_{0}=0$ and
$\Delta_{0}=0.4$ are presented in Figs. 1(a) and 1(b), respectively. Without
the DDW term ($\Delta_{0}=0$), the matrix $\hat{M}(k_{y})$ can be block
diagonalized as two $2N_{x}\times 2N_{x}$ matrices, expressed as $H(k_{y})$
and $H(k_{y}+\pi)$, respectively. With the cylinder geometry, the energy bands
are gapless and the chiral edge states exist, as is seen in Fig. 1(a). When
the DDW term is added to the system with $\Delta_{0}=0.4$, as is seen in Figs.
1(b), it is clearly that the chiral edge states are fully gapped. This
indicates that in the presence of the DDW term, the whole system is not a
first order topological system.

We now consider the open boundary condition along both $x$ and $y$ directions
and diagonalizing the $2N\times 2N$ Matrix $\hat{M}$. The zero energy LDOS
spectrum is displayed in Fig. 1(c). As is seen, the zero energy corner states
indeed exist. We present the the eigenvalues of the Matrix in Fig. 1(d). Four
zero energy eigenvalues protected by an energy gap are seen clearly. The
numerical results indicate that this system may indeed be a higher order
topological system.

An intuitive picture for the higher order topology can be provided based on
the edge theory [20]. When the DDW order is zero, the linear gapless states
emerge at the system edge as the effective Zeeman field $0<\mid m\mid<2$ [42].
As $m=0$, the system is a two-dimensional Weyl semimetal and the linear
gapless states also exist. Therefore, without the DDW term, the system edges
are gapless as $\mid m\mid<2$. In the presence of the DDW term, as is shown in
Fig. 1, with the cylinder geometry, the system edges are fully gapped and the
DDW term acts as an effective mass term. Similar to the $d$-wave
superconducting order discussed in Ref. [20], here due to the existence of the
$d$-wave factor in the DDW order, the sign of the DDW term changes at the
system corners, leading to the gapless corner states, as is shown in Fig.
1(c).

Figure 2: (a) The Wannier spectral $\nu_{x}$ as a function of the effective
Zeeman field $m$. (b) The edge polarization
($p^{edge,x}_{y}$,$p^{edge,y}_{x}$) and quadrupole moments $q_{xy}$ as a
function of $m$.

Now let us verify the high order topology of this system through calculating
the edge polarization [13, 12, 3, 4, 5] and the quadrupole moment [9, 10, 11,
12]. With a cylinder geometry, the Wannier spectrum is obtained through
diagonalizing the Wilson loop matrix [Eq.(8)]. The Wannier spectrum $\nu_{x}$
as a function of the effective Zeeman field $m$ is plotted in Fig. 2(a). The
edge polarization calculated from Eq.(10) and the quadrupole moment from
Eq.(12) are plotted in Fig. 2(b). As is seen, as $\mid m\mid<2$, the Wannier
spectra is half quantized to 0.5, leading to the half quantized edge
polarization at this parameter regime. The numerical results of the quadrupole
moment also obtain the same conclusion. These results confirm that the system
is indeed a higher order topological one [13, 12, 3, 4, 5, 9, 10, 11, 12].

It was pointed out that there are some difficulties for the method of
calculating the quadrupole moment from Eq.(12), namely, the improper choice of
the origin point and using odd number lattice size may lead to quadruple
moments being at zero or not quantized, even when the system is a second order
topological one [45]. Here the origin point is not arbitrarily chosen. It is
set at the site $(R_{x},R_{y})=(1,1)$. The system size is chosen as an even
number along both the $x$ and $y$ directions. The difficulties proposed in
Ref. [45] is partly avoided. Moreover, it was proposed that the quadruple
moments can be calculated through the corner charge and the edge polarizations
[13, 12]. We have checked numerically that with this method, the obtained
quadrupole moment is the same with that obtained from Eq.(12) [40].

In a high-Tc superconducting material, the superconducting order and the
possible pseudogap order may coexist in the underdoped region [30]. It is
useful to study the robustness of the corner states in presence of the
superconducting order. Considering an additional $d$-wave superconducting
pairing term with $H_{dSC}=\Delta_{d}\sum_{\bf k}[(\cos k_{x}-\cos
k_{y})c^{\dagger}_{{\bf k}\uparrow}c^{\dagger}_{-{\bf k}\downarrow}+h.c.]$.
For the cylinder geometry with the open boundary along the $x$-direction, the
whole Hamiltonian includes both the electron part and the hole part. It can be
written as the $8N_{x}\times 8N_{x}$ matrix. In the real space with $N$
lattice sites, the Hamiltonian can be written as a $4N\times 4N$ matrix. In
presence of the $d$-wave superconducting order, we plotted the energy bands of
the Hamiltonian with the cylinder geometry without and with the DDW order in
Figs. 3(a) and 3(b), respectively. The intensity plot of the zero energy LDOS
with the coexistence of the superconducting order and the DDW order in the
real space is presented in Fig. 3(c). The corresponding low energy eigenvalues
are plotted in Fig. 3(d). Here the energy spectra are qualitatively the same
when the superconducting order exists, namely, in the pure superconducting
state ($\Delta_{d}\neq 0$ and $\Delta_{0}=0$), the system has gapless edge
states with a cylinder geometry. In the presence of the DDW term, an energy
gap is opened by the DDW order. In the real space with the open boundaries,
the gapless corner states emerge. These results indicate that the system is
still a second order topological system when the superconducting order and the
DDW order coexist. And in this case, the gapless corner states are the
Majorana zero modes.

Figure 3: Similar to Fig. 1 but in presence of an additional $d$-wave pairing
order with $\Delta_{d}=0.5$

Due to the particle-hole symmetry, there are eight zero energy eigenvalues for
the real space Hamiltonian matrix, as is seen in Fig. 3(d), indicating that
there are two zero energy modes at each corner. This is consistent with the
Chern number of the system when the DDW order is absent. For the Chern
insulator system, the Chern number is 1 as $0<m<2$ [42]. Then with the DDW
order, each corner has one gapless corner mode. With the $d$-wave
superconducting order, the system becomes a topological superconductor [46,
47, 10, 49]. Due to the particle-hole symmetry, the Chern number becomes 2. As
a result, with the DDW order, each corner contains two gapless Majorana zero
modes. Note that these two zero modes are unstable. They are not protected by
a certain symmetry. We have numerically checked that the zero modes disappear
and the energy bands become fully gapped with an additional chemical potential
term. Previously, it was discussed and verified that without the symmetry
protection, the Majorana zero modes with the equal spin at the same site may
exist, while they are indeed unstable. They may interfere with each other and
annihilate as the finite energy quasiparticles [50, 51, 52, 53].

On the other hand, previously, the superconductor/Chern-insulator coupled
systems have been studied intensively [46, 47, 10, 49]. The Chern number
depends on the chemical potential and the magnitude of the superconducting
order parameter. A stable topological superconducting phase with the Chern
number being 1 can indeed be realized with proper parameters. We have checked
numerically that this stable topological superconductor would also become a
higher order one in the presence of an additional DDW term. In addition, in
this case, the open system has only one stable Majorana zero mode at each
corner.

Finally, we would discuss the significance and outlook of our present work.
Firstly, we here proposed and demonstrated that the higher order topology can
indeed be realized through considering the $d$-wave form pseudogap order with
the high-Tc superconductor platform. This is of interest and worth further
studies. Secondly, our present proposal can be extended to other topological
systems or other platforms. For instance, we have checked that the higher
order topological superconducting state can be realized for a topological-
superconductor/DDW coupled system. On the other hand, the DDW state is often
called as the staggered flux state [4], which can be easily realized and
controlled in the cold atom system [11]. Also, the pairing state, the spin-
orbital coupling, and various topological states have been successfully
realized in the cold atom system [7, 8, 9, 58, 59, 60, 61]. Thus the cold atom
system may provide an alternative platform to realize our proposal
experimentally [40]. At last, the competing orders in high-Tc cuprate
materials may be rather complex. Also, various density wave orders are
verified to exist in many unconventional superconductors. The relation between
the superconductivity and the density-wave order has been an important issue
and may relate to the mechanism of the unconventional superconductivity.
Moreover, the interplay between the density-wave orders and the nontrivial
topology is of interest and importance, attracting broad interest recently
[62, 63, 64, 65, 66, 67, 68]. We expect that our present studies can be
extended to other possible competing density-wave orders and other families of
unconventional superconductors, providing more insight about the relationship
between the density-wave orders, the unconventional superconductivity, and the
nontrivial topology.

##  IV Summary

In summary, we have proposed that the higher order topology can be realized in
the pseudogap regime of a high-Tc superconducting material coupled with a
Chern insulator. With a $d$-wave form pseudogap order ($d$-density-wave
order), the gapless edge states in the Chern insulator will be fully gapped.
The $d$-density-wave order term acts as an effective mass of the system. The
$d$-wave factor changes sign at the system corner, leading to the higher order
topological state. The higher order topology is confirmed through calculating
the edge polarization and the quadrupole moment. In the underdoped high-Tc
superconducting material, the $d$-wave superconducting order may coexist with
a $d$-wave density wave order. In this case, the gapless corner states still
exist, while each corner contains two zero modes, due to the particle-hole
symmetry.

###### Acknowledgements.

This work was supported by the NSFC (Grant No. 12074130), the Natural Science
Foundation of Guangdong Province (Grant No. 2021A1515012340), and Science and
Technology Program of Guangzhou (Grant No. 202102080434).

## References

  * Benalcazar  _et al._ [2017a] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Quantized electric multipole insulators, Science 357, 61 (2017a).
  * Benalcazar  _et al._ [2017b] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators, Phys. Rev. B 96, 245115 (2017b).
  * Schindler  _et al._ [2018] F. Schindler, A. M. Cook, M. G. Vergniory, Z. Wang, S. S. P. Parkin, B. A. Bernevig, and T. Neupert, Higher-order topological insulators, Sci. Adv. 4, eaat0346 (2018).
  * Franca  _et al._ [2018] S. Franca, J. van den Brink, and I. C. Fulga, An anomalous higher-order topological insulator, Phys. Rev. B 98, 201114 (2018).
  * Liu  _et al._ [2021] Z.-R. Liu, L.-H. Hu, C.-Z. Chen, B. Zhou, and D.-H. Xu, Topological excitonic corner states and nodal phase in bilayer quantum spin hall insulators, Phys. Rev. B 103, L201115 (2021).
  * Volpez  _et al._ [2019] Y. Volpez, D. Loss, and J. Klinovaja, Second-order topological superconductivity in $\pi$-junction Rashba layers, Phys. Rev. Lett. 122, 126402 (2019).
  * Pan  _et al._ [2019] X.-H. Pan, K.-J. Yang, L. Chen, G. Xu, C.-X. Liu, and X. Liu, Lattice-symmetry-assisted second-order topological superconductors and Majorana patterns, Phys. Rev. Lett. 123, 156801 (2019).
  * Wu  _et al._ [2020] Y.-J. Wu, J. Hou, Y.-M. Li, X.-W. Luo, X. Shi, and C. Zhang, In-plane Zeeman-field-induced Majorana corner and hinge modes in an $s$-wave superconductor heterostructure, Phys. Rev. Lett. 124, 227001 (2020).
  * Kang  _et al._ [2019] B. Kang, K. Shiozaki, and G. Y. Cho, Many-body order parameters for multipoles in solids, Phys. Rev. B 100, 245134 (2019).
  * Roy [2019] B. Roy, Antiunitary symmetry protected higher-order topological phases, Phys. Rev. Research 1, 032048 (2019).
  * Wheeler  _et al._ [2019] W. A. Wheeler, L. K. Wagner, and T. L. Hughes, Many-body electric multipole operators in extended systems, Phys. Rev. B 100, 245135 (2019).
  * Li  _et al._ [2020] C.-A. Li, B. Fu, Z.-A. Hu, J. Li, and S.-Q. Shen, Topological phase transitions in disordered electric quadrupole insulators, Phys. Rev. Lett. 125, 166801 (2020).
  * Song  _et al._ [2017] Z. Song, Z. Fang, and C. Fang, $(d-2)$-dimensional edge states of rotation symmetry protected topological states, Phys. Rev. Lett. 119, 246402 (2017).
  * Langbehn  _et al._ [2017] J. Langbehn, Y. Peng, L. Trifunovic, F. von Oppen, and P. W. Brouwer, Reflection-symmetric second-order topological insulators and superconductors, Phys. Rev. Lett. 119, 246401 (2017).
  * Călugăru  _et al._ [2019] D. Călugăru, V. Juričić, and B. Roy, Higher-order topological phases: A general principle of construction, Phys. Rev. B 99, 041301 (2019).
  * Park  _et al._ [2019] M. J. Park, Y. Kim, G. Y. Cho, and S. Lee, Higher-order topological insulator in twisted bilayer graphene, Phys. Rev. Lett. 123, 216803 (2019).
  * Sheng  _et al._ [2019] X.-L. Sheng, C. Chen, H. Liu, Z. Chen, Z.-M. Yu, Y. Zhao, and S. A. Yang, Two-dimensional second-order topological insulator in graphdiyne, Phys. Rev. Lett. 123, 256402 (2019).
  * Xu  _et al._ [2019] Y. Xu, Z. Song, Z. Wang, H. Weng, and X. Dai, Higher-order topology of the axion insulator $\mathrm{EuIn}_{2}\mathrm{As}_{2}$, Phys. Rev. Lett. 122, 256402 (2019).
  * Noguchi  _et al._ [2021] R. Noguchi, M. Kobayashi, Z. Jiang, K. Kuroda, T. Takahashi, Z. Xu, D. Lee, M. Hirayama, M. Ochi, T. Shirasawa, P. Zhang, C. Lin, C. Bareille, S. Sakuragi, H. Tanaka, S. Kunisada, K. Kurokawa, K. Yaji, A. Harasawa, V. Kandyba, A. Giampietri, A. Barinov, T. K. Kim, C. Cacho, M. Hashimoto, D. Lu, S. Shin, R. Arita, K. Lai, T. Sasagawa, and T. Kondo, Evidence for a higher-order topological insulator in a three-dimensional material built from van der waals stacking of bismuth-halide chains, Nat. Mater. 20, 473 (2021).
  * Yan  _et al._ [2018] Z. Yan, F. Song, and Z. Wang, Majorana corner modes in a high-temperature platform, Phys. Rev. Lett. 121, 096803 (2018).
  * Wang  _et al._ [2018] Q. Wang, C. C. Liu, Y. M. Lu, and F. Zhang, High-Temperature Majorana corner states, Phys. Rev. Lett. 121, 186801 (2018).
  * Liu  _et al._ [2018] T. Liu, J. J. He, and F. Nori, Majorana corner states in a two-dimensional magnetic topological insulator on a high-temperature superconductor, Phys. Rev. B 98, 245413 (2018).
  * Zhang  _et al._ [2019] R. X. Zhang, W. S. Cole, X. Wu, and S. Das Sarma, Higher-order topology and nodal topological superconductivity in Fe(Se,Te) heterostructures, Phys. Rev. Lett. 123, 167001 (2019).
  * Zhu [2019] X. Zhu, Second-order topological superconductors with mixed pairing, Phys. Rev. Lett. 122, 236401 (2019).
  * Li and Zhou [2021] Y.-X. Li and T. Zhou, Rotational symmetry breaking and partial majorana corner states in a heterostructure based on high-${T}_{c}$ superconductors, Phys. Rev. B 103, 024517 (2021).
  * Xue  _et al._ [2018] H. Xue, Y. Yang, F. Gao, Y. Chong, and B. Zhang, Acoustic higher-order topological insulator on a kagome lattice, Nat. Mater. 18, 108 (2018).
  * Serra-Garcia  _et al._ [2018] M. Serra-Garcia, V. Peri, R. Susstrunk, O. R. Bilal, T. Larsen, L. G. Villanueva, and S. D. Huber, Observation of a phononic quadrupole topological insulator, Nature 555, 342 (2018).
  * Ni  _et al._ [2018] X. Ni, M. Weiner, A. Alù, and A. B. Khanikaev, Observation of higher-order topological acoustic states protected by generalized chiral symmetry, Nat. Mater. 18, 113 (2018).
  * Peterson  _et al._ [2018] C. W. Peterson, W. A. Benalcazar, T. L. Hughes, and G. Bahl, A quantized microwave quadrupole insulator with topologically protected corner states, Nature 555, 346 (2018).
  * Mueller [2017] E. J. Mueller, Review of pseudogaps in strongly interacting fermi gases, Rep. Prog. Phys. 80, 104401 (2017).
  * Nayak [2000] C. Nayak, Density-wave states of nonzero angular momentum, Phys. Rev. B 62, 4880 (2000).
  * Chakravarty  _et al._ [2001] S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Hidden order in the cuprates, Phys. Rev. B 63, 094503 (2001).
  * Li  _et al._ [2006] J.-X. Li, C.-Q. Wu, and D.-H. Lee, Checkerboard charge density wave and pseudogap of high-${T}_{c}$ cuprate, Phys. Rev. B 74, 184515 (2006).
  * Fradkin  _et al._ [2015] E. Fradkin, S. A. Kivelson, and J. M. Tranquada, Colloquium: Theory of intertwined orders in high temperature superconductors, Rev. Mod. Phys. 87, 457 (2015).
  * Zhou  _et al._ [2019] T. Zhou, Y. Gao, and Z. D. Wang, Detecting competing orders through the edge states in the heterostructures with high-${T}_{c}$ superconductors, Phys. Rev. B 99, 104517 (2019).
  * Li  _et al._ [2022] M. Li, G. Li, L. Cao, X. Zhou, X. Wang, C. Jin, C.-K. Chiu, S. J. Pennycook, Z. Wang, and H.-J. Gao, Ordered and tunable majorana-zero-mode lattice in naturally strained lifeas, Nature 606, 890 (2022).
  * Fujita  _et al._ [2014] K. Fujita, M. H. Hamidian, S. D. Edkins, C. K. Kim, Y. Kohsaka, M. Azuma, M. Takano, H. Takagi, H. Eisaki, S. ichi Uchida, A. Allais, M. J. Lawler, E.-A. Kim, S. Sachdev, and J. C. S. Davis, Direct phase-sensitive identification of a d-form factor density wave in underdoped cuprates, Proc. Natl. Acad. Sci. U.S.A. 111, E3026 (2014).
  * Comin  _et al._ [2015] R. Comin, R. Sutarto, F. He, E. H. da Silva Neto, L. Chauviere, A. Frano, R. Liang, W. N. Hardy, D. A. Bonn, Y. Yoshida, H. Eisaki, A. J. Achkar, D. G. Hawthorn, B. Keimer, G. A. Sawatzky, and A. Damascelli, Symmetry of charge order in cuprates, Nat. Mater. 14, 796 (2015).
  * Forgan  _et al._ [2015] E. M. Forgan, E. Blackburn, A. T. Holmes, A. K. R. Briffa, J. Chang, L. Bouchenoire, S. D. Brown, R. Liang, D. Bonn, W. N. Hardy, N. B. Christensen, M. V. Zimmermann, M. Hucker, and S. M. Hayden, The microscopic structure of charge density waves in underdoped YBa2Cu3O6.54 revealed by x-ray diffraction, Nat. Commun. 6, 10064 (2015).
  * [40] See supplemental material for higher order topological state induced by $d$-wave competing orders in high-$\mathrm{T}_{c}$ superconductor based heterostructure.
  * Qi  _et al._ [2006] X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, General theorem relating the bulk topological number to edge states in two-dimensional insulators, Phys. Rev. B 74, 045125 (2006).
  * Qi  _et al._ [2008] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Topological field theory of time-reversal invariant insulators, Phys. Rev. B 78, 195424 (2008).
  * Chang  _et al._ [2013] C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Experimental observation of the quantum anomalous hall effect in a magnetic topological insulator, Science 340, 167 (2013).
  * Hu  _et al._ [2020] C. Hu, L. Ding, K. N. Gordon, B. Ghosh, H.-J. Tien, H. Li, A. G. Linn, S.-W. Lien, C.-Y. Huang, S. Mackey, J. Liu, P. V. S. Reddy, B. Singh, A. Agarwal, A. Bansil, M. Song, D. Li, S.-Y. Xu, H. Lin, H. Cao, T.-R. Chang, D. Dessau, and N. Ni, Realization of an intrinsic ferromagnetic topological state in $\mathrm{MnBi}_{8}\mathrm{Te}_{13}$, Sci. Adv. 6, eaba4275 (2020).
  * Ono  _et al._ [2019] S. Ono, L. Trifunovic, and H. Watanabe, Difficulties in operator-based formulation of the bulk quadrupole moment, Phys. Rev. B 100, 245133 (2019).
  * Qi  _et al._ [2010] X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Chiral topological superconductor from the quantum hall state, Phys. Rev. B 82, 184516 (2010).
  * Wang  _et al._ [2015] J. Wang, Q. Zhou, B. Lian, and S.-C. Zhang, Chiral topological superconductor and half-integer conductance plateau from quantum anomalous hall plateau transition, Phys. Rev. B 92, 064520 (2015).
  * Liu  _et al._ [2014] X.-J. Liu, K. T. Law, and T. K. Ng, Realization of 2d spin-orbit interaction and exotic topological orders in cold atoms, Phys. Rev. Lett. 112, 086401 (2014).
  * Viyuela  _et al._ [2018] O. Viyuela, L. Fu, and M. A. Martin-Delgado, Chiral topological superconductors enhanced by long-range interactions, Phys. Rev. Lett. 120, 017001 (2018).
  * Wang and Zhou [2022] X. Wang and T. Zhou, Fragile topology in nodal-line semimetal superconductors, New J. Phys. 24, 083013 (2022).
  * Ono  _et al._ [2020] S. Ono, H. C. Po, and H. Watanabe, Refined symmetry indicators for topological superconductors in all space groups, Sci. Adv. 6, eaaz8367 (2020).
  * Hwang  _et al._ [2019] Y. Hwang, J. Ahn, and B.-J. Yang, Fragile topology protected by inversion symmetry: Diagnosis, bulk-boundary correspondence, and wilson loop, Phys. Rev. B 100, 205126 (2019).
  * Po  _et al._ [2018] H. C. Po, H. Watanabe, and A. Vishwanath, Fragile topology and wannier obstructions, Phys. Rev. Lett. 121, 126402 (2018).
  * Aidelsburger  _et al._ [2011] M. Aidelsburger, M. Atala, S. Nascimbène, S. Trotzky, Y.-A. Chen, and I. Bloch, Experimental realization of strong effective magnetic fields in an optical lattice, Phys. Rev. Lett. 107, 255301 (2011).
  * Liu  _et al._ [2009] X.-J. Liu, M. F. Borunda, X. Liu, and J. Sinova, Effect of induced spin-orbit coupling for atoms via laser fields, Phys. Rev. Lett. 102, 046402 (2009).
  * Lin  _et al._ [2011] Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, Spin–orbit-coupled bose–einstein condensates, Nature 471, 83 (2011).
  * Chin  _et al._ [2010] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010).
  * Lohse  _et al._ [2015] M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, A thouless quantum pump with ultracold bosonic atoms in an optical superlattice, Nat. Phys. 12, 350 (2015).
  * Nakajima  _et al._ [2016] S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, Topological thouless pumping of ultracold fermions, Nat. Phys. 12, 296 (2016).
  * Song  _et al._ [2019] B. Song, C. He, S. Niu, L. Zhang, Z. Ren, X.-J. Liu, and G.-B. Jo, Observation of nodal-line semimetal with ultracold fermions in an optical lattice, Nat. Phys. 15, 911 (2019).
  * Ghatak  _et al._ [2020] A. Ghatak, M. Brandenbourger, J. van Wezel, and C. Coulais, Observation of non-hermitian topology and its bulk–edge correspondence in an active mechanical metamaterial, Proc. Natl. Acad. Sci. U.S.A. 117, 29561 (2020).
  * Lin and Bradlyn [2021] K.-S. Lin and B. Bradlyn, Simulating higher-order topological insulators in density wave insulators, Phys. Rev. B 103, 245107 (2021).
  * Hu  _et al._ [2017] L.-H. Hu, C.-C. Chen, C.-X. Liu, F.-C. Zhang, and Y. Zhou, Topological charge-density and spin-density waves in InAs/GaSb quantum wells under an in-plane magnetic field, Phys. Rev. B 96, 075130 (2017).
  * Gooth  _et al._ [2019] J. Gooth, B. Bradlyn, S. Honnali, C. Schindler, N. Kumar, J. Noky, Y. Qi, C. Shekhar, Y. Sun, Z. Wang, B. A. Bernevig, and C. Felser, Axionic charge-density wave in the weyl semimetal (TaSe4)2I, Nature 575, 315 (2019).
  * Wieder  _et al._ [2020] B. J. Wieder, K.-S. Lin, and B. Bradlyn, Axionic band topology in inversion-symmetric weyl-charge-density waves, Phys. Rev. Research 2, 042010 (2020).
  * Shi  _et al._ [2021] W. Shi, B. J. Wieder, H. L. Meyerheim, Y. Sun, Y. Zhang, Y. Li, L. Shen, Y. Qi, L. Yang, J. Jena, P. Werner, K. Koepernik, S. Parkin, Y. Chen, C. Felser, B. A. Bernevig, and Z. Wang, A charge-density-wave topological semimetal, Nat. Phys. 17, 381 (2021).
  * Qin  _et al._ [2020] F. Qin, S. Li, Z. Du, C. Wang, W. Zhang, D. Yu, H.-Z. Lu, and X. Xie, Theory for the charge-density-wave mechanism of 3d quantum hall effect, Phys. Rev. Lett. 125, 206601 (2020).
  * Tang  _et al._ [2019] F. Tang, Y. Ren, P. Wang, R. Zhong, J. Schneeloch, S. A. Yang, K. Yang, P. A. Lee, G. Gu, Z. Qiao, and L. Zhang, Three-dimensional quantum hall effect and metal–insulator transition in ZrTe5, Nature 569, 537 (2019).

Supplemental material for higher order topological state induced by $d$-wave
competing orders in high-Tc superconductor based heterostructure

##  S-1 Experimental realization of the higher order topological state with
the high-Tc superconductor platform

In the main text, we proposed that the higher order topological state may be
realized through adding an additional DDW term to the Hamiltonian of the Chern
insulator. Now let us discuss in more detail the experimental realization of
this proposal. Let us first focus on the heterostructure system. The schematic
illustration is presented in Fig. S1, namely, a Chern insulator is grown on an
underdoped high-Tc cuprate superconductor. Experimentally, the Chern insulator
can be realized in the Hg1-xMnxTe/Cd1-xMnxTe quantum well [1] or the intrinsic
magnetic topological insulators (Bi,Sb)2Te3 [2] and MnBi8Te13 [3]. The DDW
order may exist in an underdoped cuprate superconductor [4, 5, 6]. We expect
that in this heterostructure, the DDW term can be proximity induced to the
Chern insulator layer. Then this setup is promising for our present proposal.
In the main text, we have shown that the existence of the zero modes can be
explored through the LDOS [Fig. 1(c) in the main text]. Experimentally, the
LDOS spectra can be measured through the scanning tunneling microscope (STM)
experiments, as illustrated in Fig. S1.

Figure S1: Illustration of the heterostructure. The blue layer represents an
underdoped High-Tc superconductor. The brown layer represents a monolayer
Chern insulator. The red dots indicate the corner modes.

##  S-2 Experimental realization with the cold atom platform

As we have mentioned in the main text, the cold atom system is expected to be
an alternative platform to realize our proposal. The spin-orbital coupling has
been proposed theoretically and realized experimentally in the cold atom
system [7, 8]. The $s$-wave superfluid pairing state is also realized and
controlled through the $s$-wave Feshbach resonance [9]. Moreover, it was
proposed that both the quantum anomalous Hall effect and the chiral
topological superfluid phase with the Chern number $C=1$ can be realized in
the cold atom system within the experimentally accessible parameter regimes
[10].

In the cold atom system, the hopping term can be driven and controlled by the
Raman field [11]. Considering a spatial dependent field $V(x)$ with
$V(x)=V_{0}\cos[{\bf Q_{x}}\cdot R_{x}]$, a periodic modulation pattern along
the $x$-direction will be generated by such a field. Using two spatial
dependent Raman fields along the $x$-direction and $y$-direction, the
effective hopping term with
$M_{x}=iM_{0}(-1)^{R_{x}+R_{y}}c^{\dagger}(R_{x},R_{y})c(R_{x}+1,R_{y})+h.c.$,
and
$M_{y}=-iM_{0}(-1)^{R_{x}+R_{y}}c^{\dagger}(R_{x},R_{y})c(R_{x},R_{y}+1)+h.c.$
can be created artificially. The additional factor $(-1)^{R_{x}+R_{y}}$ is due
to the periodic modulation of the field $V({\bf r})$. Note that $M_{x}+M_{y}$
is just the DDW term presented in Eq.(11) of the main text. As illustrated in
Fig. S2, when adding this additional DDW term to the chiral topological state
proposed in Ref. [10], the system will become a second order topological one.
Moreover, here the $C=1$ topological superfluid state can be realized [10],
therefore, we expect that with the cold atom platform, the stable Majorana
zero modes may exist at the system corners.

Figure S2: Illustration of the realization of the second order topological
state in the cold atom system. The experimental setup for the chiral
topological state is proposed in Ref. [10].

.

##  S-3 Corner charge

Figure S3: The site-dependent charge density considering the open boundary
condition, with the system Hamiltonian being described by Eq.(11) of the main
text. The parameters are set as $t=1$, $\lambda_{0}=1$, $m=1$, and
$\Delta_{0}=0.4$.

For a topologically nontrivial material, it can host fractional charge at the
system corner due to the symmetry constrain. The site-dependent charge density
is expressed as (using the electronic charge $e$ as the unit),

$\rho\left(R_{x},R_{y}\right)\equiv\sum_{n=1}^{N_{\text{occ
}}}\left|u^{n}\left(R_{x},R_{y}\right)\right|^{2},$ (S1)

where $u^{n}\left(R_{x},R_{y}\right)$ is the n-th component of the eigenstate
$\left|u^{n}\right\rangle$ for the Hamiltonian [Eq.(11) in the main text],
with only occupied energy bands being considered. The numerical results for
the charge density is presented in Fig. S3. As is shown, the charge density at
the system bulk background is 1.0. The corner charge is calculated from the
charge density relative to the background charge density. Here the fractional
corner charge with $Q^{\text{corner}}=0.5$ is obtained [12, 13]. The
quadrupole moment, the edge polarization and the corner charge should satisfy
the following relation [12, 13],

$\displaystyle q_{xy}=p^{edge,y}_{x}+p^{edge,x}_{y}-Q^{\text{corner}}.$ (S2)

In the main text, we have obtained that $p^{edge,y}_{x}=p^{edge,x}_{y}=0.5$.
Then we have $q_{xy}=0.5$, consistent with the result calculated by Eq.(12) of
the main text.

## References

  * Qi  _et al._ [2006] X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, General theorem relating the bulk topological number to edge states in two-dimensional insulators, Phys. Rev. B 74, 045125 (2006).
  * Chang  _et al._ [2013] C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji, Y. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-C. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-K. Xue, Experimental observation of the quantum anomalous hall effect in a magnetic topological insulator, Science 340, 167 (2013).
  * Hu  _et al._ [2020] C. Hu, L. Ding, K. N. Gordon, B. Ghosh, H.-J. Tien, H. Li, A. G. Linn, S.-W. Lien, C.-Y. Huang, S. Mackey, J. Liu, P. V. S. Reddy, B. Singh, A. Agarwal, A. Bansil, M. Song, D. Li, S.-Y. Xu, H. Lin, H. Cao, T.-R. Chang, D. Dessau, and N. Ni, Realization of an intrinsic ferromagnetic topological state in $\mathrm{MnBi}_{8}\mathrm{Te}_{13}$, Sci. Adv. 6, eaba4275 (2020).
  * Nayak [2000] C. Nayak, Density-wave states of nonzero angular momentum, Phys. Rev. B 62, 4880 (2000).
  * Chakravarty  _et al._ [2001] S. Chakravarty, R. B. Laughlin, D. K. Morr, and C. Nayak, Hidden order in the cuprates, Phys. Rev. B 63, 094503 (2001).
  * Forgan  _et al._ [2015] E. M. Forgan, E. Blackburn, A. T. Holmes, A. K. R. Briffa, J. Chang, L. Bouchenoire, S. D. Brown, R. Liang, D. Bonn, W. N. Hardy, N. B. Christensen, M. V. Zimmermann, M. Hucker, and S. M. Hayden, The microscopic structure of charge density waves in underdoped YBa2cu3o6.54 revealed by x-ray diffraction, Nat. Commun. 6, 10064 (2015).
  * Liu  _et al._ [2009] X.-J. Liu, M. F. Borunda, X. Liu, and J. Sinova, Effect of induced spin-orbit coupling for atoms via laser fields, Phys. Rev. Lett. 102, 046402 (2009).
  * Lin  _et al._ [2011] Y.-J. Lin, K. Jiménez-García, and I. B. Spielman, Spin–orbit-coupled bose–einstein condensates, Nature 471, 83 (2011).
  * Chin  _et al._ [2010] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Feshbach resonances in ultracold gases, Rev. Mod. Phys. 82, 1225 (2010).
  * Liu  _et al._ [2014] X.-J. Liu, K. T. Law, and T. K. Ng, Realization of 2d spin-orbit interaction and exotic topological orders in cold atoms, Phys. Rev. Lett. 112, 086401 (2014).
  * Aidelsburger  _et al._ [2011] M. Aidelsburger, M. Atala, S. Nascimbène, S. Trotzky, Y.-A. Chen, and I. Bloch, Experimental realization of strong effective magnetic fields in an optical lattice, Phys. Rev. Lett. 107, 255301 (2011).
  * Benalcazar  _et al._ [2017a] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Electric multipole moments, topological multipole moment pumping, and chiral hinge states in crystalline insulators, Phys. Rev. B 96, 245115 (2017a).
  * Benalcazar  _et al._ [2017b] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes, Quantized electric multipole insulators, Science 357, 61 (2017b).


# Black holes and gravitational waves in three-dimensional f(R) gravity

Hongsheng Zhang${}^{1,2~{}}$111Electronic address<EMAIL_ADDRESS>Dao-
Jun Liu ${}^{1,2~{}}$ 222Electronic address<EMAIL_ADDRESS>Xin-Zhou Li1
333Electronic address<EMAIL_ADDRESS>1Center for Astrophysics, Shanghai
Normal University, 100 Guilin Road, Shanghai 200234, China
2State Key Laboratory of Theoretical Physics, Institute of Theoretical
Physics, Chinese Academy of Sciences, Beijing, 100190, China

###### Abstract

In the three-dimensional pure Einstein gravity, the geometries of the vacuum
space-times are always trivial, and gravitational waves (gravitons) are
strictly forbidden. For the first time, we find a vacuum circularly symmetric
black hole with nontrivial geometries in $f(R)$ gravity theory, in which a
true singularity appears. In this frame with nontrivial geometry, a
perturbative gravitational wave does exist. Beyond the perturbative level, we
make a constructive proof of the existence of a gravitational wave in $f(R)$
gravity, where the Birkhoff-like theorem becomes invalid. We find two classes
of exact solutions of circularly symmetric pure gravitational wave radiation
and absorption.

gravitational wave black hole thermodynamics

###### pacs:

04.20.-q, 04.70.-s, 04.30.-w

††preprint: arXiv:

##  I Introduction

Pure gravity in three-dimensional space-time has been extensively studied over
many aspects, for a review see carlip . But its status is still unsettled. It
is well-known that the three-dimensional pure gravity is trivial, in the sense
that the manifold must be a maximally symmetric space locally, and thus
gravitational wave is forbidden. That is to say, the three-dimensional gravity
becomes a topological theory: the differences among different configurations
in this theory only arose when we consider the boundary effects. A very simple
and intuitionistic explanation of this point is shown as follows. The Weyl
tensor always vanishes in three-dimensional space-time, and thus the Riemann
tensor can be expressed in Ricci tensor. From Einstein equation the Ricci
tensor is proportional to the metric in pure gravity. Therefore, the Riemann
tensor is completely determined by the metric (without the derivative terms of
the metric), which defines a maximally symmetric space-time. There are only a
few non-propagating configurations, just like the solutions of time
independent Schrödinger equation in quantum mechanics, can exist in this frame
deser . Clearly, a graviton does not exist since there is no space-time ripple
that can emerge and propagate, which implies that the quantum theory of three-
dimensional gravity may be of absence.

The arguments of non-existence of quantum theory in three-dimensional gravity
are impaired by the equivalence of three-dimensional gravity and Chern-Simons
gauge theory. The Chern-Simons quantum gauge theory does exist, and
furthermore, is renormalizable. So one expects that a well-defined quantum
theory of the three-dimensional gravity also exists. It is worth noting that
the equivalence between the three-dimensional gravity and Chern-Simons theory
is constrained in local classical and perturbative sense. They have several
important differences when one observes their global non-perturbative
properties. For example, in the gravity aspect, it is natural to sum over all
possible topologies of the space-time manifold. But in the aspect of Chern-
Simons gauge formulation, it is difficult to find the necessity to do so. As
there are important differences between three-dimensional gravity and Chern-
Simons formulation, one may simply give up the efforts to find a quantum
theory for three-dimensional gravity. However, the discovery of BTZ black hole
forces us to reconsider this problem btz . The BTZ black holes have horizon,
mass, entropy, and angular momentum, though its local geometry is just anti-de
Sitter. Especially, these physical quantities obey the thermodynamic laws,
like a usual four-dimensional black hole. A significant discovery in 1970’s is
that gravity is inherently related to thermodynamics. Several deep and
remarkable approaches have been made in gravi-thermodynamics, for example the
black radiation Hawking . Recently we find that the spherically symmetric
metrics can be derived directly from thermodynamic considerations zhang .
Generally the macroscopic thermodynamic properties have microscopic
statistical origins. The existence an entropy proportional to the perimeter of
the horizon hints that there are rich freedoms in the BTZ space-time. Thus it
is natural to explore whether we can find propagating modes by introducing
some new freedoms into the theory.

The propagation freedoms in three dimensional (modified) gravity have been
investigated. Some interesting points of this problem can be found in carlip ;
witten . Generally, modified gravity permits extra freedoms than general
relativity, for example the dilaton field in scalar-tensor gravity. The
additional freedoms in modified gravity may lead non-trivial geometries in
3-dimensions. A topological massive gravity is introduced in deser2 , in which
a spin-2 graviton with propagation degree appears. The Hilbert-Einstein action
together with a special combination of $R^{2}$-terms is equivalent to a Pauli-
Fierz action at the linearized level bht . This new massive gravity has been
studied in bt2 . Exact black hole solutions with non-trivial geometries have
been found in this massive gravity frame in three dimensions ott . A parity-
preserving model of three-dimensional gravity with propagating torsion freedom
in asymptotic AdS space in studied in Blago . In this paper, we make a
different approach towards this problem from the view of a higher derivative
gravity theory. We present an exact solution in a three-dimensional higher
derivative gravity as a heuristic example to display a possible way to
introduce graviton.

In the next section we show a black hole solution in $R^{d+1}$ gravity. In
section III we present the gravitational wave solution in $R^{d+1}$ gravity,
and we conclude this paper in section IV.

##  II The black hole solution

It is wellknown that three-dimensional Einstein gravity does not permit a
graviton. The theory is non-renormalizable even if we consider an off-shell
graviton, since the gravitational constant has a dimension of length. The
studies of the renormalizability of gravity leads to higher derivative theory
stelle , for some early thoughts on this issue, see dewitt . $f(R)$ gravity
naturally encodes the essential properties of higher derivative theories,
meanwhile it is relatively tractable compared to some more complicated
considerations, for a review see soti . Once the higher derivative terms are
introduced, they will dominate the behaviour of the space-time dynamics at
high energy region, such as the regions around the big bang singularity or a
black hole singularity. We shall show that a true singularity is possible in
three-dimensional pure $f(R)$ gravity, while there is no true singularity in
three-dimensional Einstein gravity. Accompanying to the appearance of the true
singularity, the local geometry becomes non-trivial, and thus an on-shell
graviton (gravitational wave) is also possible.

We start from an action in $f(R)$ gravity,

$\displaystyle S=\frac{1}{16\pi G}\left(\int_{\cal M}d^{3}x\sqrt{-{\rm
det}(g)}~{}R^{d+1}+\int_{\cal\partial M}d^{2}x\sqrt{-{\rm
det}(h)}~{}2B_{o}\right),$ (1)

where $G$ is the three-dimensional constant with a dimension of length, ${\cal
M}$ is the manifold in consideration and ${\cal\partial M}$ denotes its
boundary, $g$ labels the metric on ${\cal M}$ and $h$ labels its induced
metric on ${\cal\partial M}$, $R$ is the Ricci scalar for $g$, and $d$ is a
constant. With a proper boundary term $B_{o}$ the corresponding field equation
reads,

$\displaystyle(1+d)R^{d}R_{ab}-\frac{1}{2}R^{d+1}g_{ab}-(1+d)\nabla_{a}\nabla_{b}R^{d}+g_{ab}(1+d)\square
R^{d}=0.$ (2)

A general circularly symmetric metric with 2 Killing fields reads,

$ds^{2}=-A(r)dt^{2}+\frac{1}{B(r)}dr^{2}+r^{2}d\theta^{2},$ (3)

Then, the $\theta$-$\theta$ component of (2) and its trace read

$\left[\frac{\partial^{2}}{\partial
r^{2}}+\left(\frac{A^{\prime}}{2A}+\frac{B^{\prime}}{2B}\right)\left(\frac{\partial}{\partial
r}-\frac{1}{r}\right)-\frac{R}{2(1+d)B}\right]R^{d}=0$ (4)

and

$\left[\frac{\partial^{2}}{\partial
r^{2}}+\left(\frac{1}{r}+\frac{A^{\prime}}{2A}+\frac{B^{\prime}}{2B}\right)\frac{\partial}{\partial
r}-\frac{(1-2d)R}{4(1+d)B}\right]R^{d}=0,$ (5)

respectively, where the Ricci scalar $R$ is given by

$R=-B\left[\left(\frac{A^{\prime}}{A}\right)^{\prime}+\left(\frac{A^{\prime}}{A}+\frac{B^{\prime}}{B}\right)\left(\frac{1}{r}+\frac{A^{\prime}}{2A}\right)\right].$
(6)

(4) and (5) are closed for the two functions $A(r)$ and $B(r)$. Obviously,
$A=C_{1}$, $B=C_{2}$ ($C_{1}$ and $C_{2}$ are two constants) is a solution
with

$R_{ab}=0,$ (7)

and

$R=0,$ (8)

for the field equation (2). This is a trivial Minkowskian one.

Besides this Minkowskian solution, we find that there exist non-trivial
solutions for this three-dimensional theory. Although there is no general
method to solve such non-linear equations as the field equation (2) at fourth-
order, we find a proper ansatz under which the analytical forms of $A$ and $B$
can be obtained. Now we introduce the following ansatz,

$R=-\frac{kL}{r^{p}},$ (9)

where $L>0$, and $k=-1,~{}0,+1$. With this additional ansatz, we find the
following solution,

$A=Br^{\frac{2pd(1+pd)}{1-pd}},$ (10)
$B=k\frac{L(1-dp)r^{2-p}}{2(d+1)\left(d^{2}(2d+1)p^{3}-p+2\right)}+Cr^{\frac{2p^{2}d^{2}}{pd-1}},$
(11)

and

$p=\frac{1-2d}{(1+2d)d},$ (12)

where $C$ is an integration constant. One can check that (9) are satisfied for
$A$ and $B$ in (10) and (11). The situations of different $k$ are something
like the 3 different spatial geometries at $r=$ constant in the case of
topological Schwarzschild solution. However, we emphasize that the local
spatial geometries at $r=$ constant are the same for different $k$ in the
above solution, because a one-dimensional object can not be intrinsically
curved.

From the ansatz, one explicitly sees that the geometries are non-trivial when
$k=\pm 1$. A true singularity appears at $r=0$ for $p>0$, and at $r\to\infty$
when $p<0$. The Ricci scalar (9) vanishes when $k=0$, but it is still a curved
space, which can be seen from the Kretschmann scalar,

$R_{abcd}R^{abcd}=\frac{C^{2}(1-2d)^{2}\left(1+12d^{2}\right)}{2d^{2}(1+2d)^{2}}r^{-\frac{1+12d^{2}}{d+2d^{2}}}.$
(13)

$p=0$ (corresponding to $d=1/2$) is a special point, at which the solution
degenerates to a Minkowskian one for all 3 cases of $k$. This means that
$R^{3/2}$-gravity permits a Minkowskian state. For a general $k$, (13) becomes

$\displaystyle
R_{abcd}R^{abcd}=\frac{C^{2}(1-2d)^{2}\left(1+12d^{2}\right)}{2d^{2}(1+2d)^{2}}r^{-\frac{1+12d^{2}}{d+2d^{2}}}$
$\displaystyle+$
$\displaystyle\frac{(1-2d)^{2}}{2(1-6d)d(d+1)}CkLr^{\frac{8}{2d+1}-\frac{3}{2d}-3}$
(14) $\displaystyle+$
$\displaystyle\frac{24d^{4}+40d^{3}+10d^{2}-8d+1}{36d^{4}+60d^{3}+13d^{2}-10d+1}L^{2}k^{2}r^{\frac{8}{2d+1}-\frac{2}{d}}.$

It is clear that the the Kretschmann scalar is divergent when $d\to 0$, which
implies that this solution is a nonperturbative one compared to the trivial
Minkowskian solution. This property also can be understood by our ansatz. $R$
is not a continuous function when $d=0$ ($p\to\infty$): It is a step (the step
goes to infinity) function at $r=1$. Hence the metric is not regular either at
the point $d=0$ in the parameter space.

The true singularity is enclosed by an apparent horizon $l(r)=0$, which
satisfies,

$h^{ab}\partial_{a}l\partial_{b}l=0,$ (15)

where $h$ is the induced metric,

$h=-A(r)dt^{2}+\frac{1}{B(r)}dr^{2}.$ (16)

The apparent horizon dwells at,

$r_{a}=\left(-\frac{C(6d-1)(d+1)}{d(d+1/2)kL}\right)^{\frac{2d}{6d-1}}.$ (17)

A physical horizon requires,

$C(6d-1)(d+1)d(d+1/2)k<0.$ (18)

For the case with $k=0$, a horizon cannot appear and generally the singularity
becomes a naked one.

##  III Gravitational wave

Since the geometries of the three-dimensional spacetime can be non-trivial,
naturally the space-time ripples become possible. And thus we can discuss the
perturbative gravitational wave in the background of the black holes. Beyond
perturbative level, an exact solution says the final words of the existence of
gravitational wave. We find that the following metric,

$ds^{2}=-\overline{A}(r,u)du^{2}\pm{2}{\overline{B}(r)}dudr+r^{2}d\theta^{2},$
(19)

solves (2), where

$\overline{A}(r,u)=\frac{1}{2r}\left(\frac{d(1+2d)kLr^{3}}{6d^{2}+5d-1}+2r^{\frac{1}{2d}}{\cal
G}(u)\right),$ (20)

and

$\overline{B}(r)=r^{\frac{1/2-d}{d(1+2d)}}.$ (21)

Here ${\cal G}(u)$ is an arbitrary $C^{4}$-function of $u$. The source of the
above metric reads,

$T=\mp\frac{|k|d}{4\pi(1+2d)}r^{-\frac{4d}{1+2d}}\frac{d{\cal G}}{du}du\otimes
du,$ (22)

which is independent of $L$. It is a null source that satisfies the condition

$g^{ab}T_{ab}=0.$ (23)

However, it is not a three-dimensional electromagnetic field, since three-
dimensional electromagnetic field has a non-zero trace. The cases $k=\pm 1$
describe dynamical space-times with sources (22). Surely, gravitational waves
propagate in these space-times.

(19) is closely related to the black hole solution in the last section. $\cal
G$=constant is a degenerated case, which is just (3) in a different
coordinates. One can demonstrate this point by the following coordinates
transformation,

$u=t\mp\int
dr\left[\frac{kL(1-dp)r^{2-p}}{2(d+1)\left(d^{2}(2d+1)p^{3}-p+2\right)}+Cr^{\frac{2p^{2}d^{2}}{pd-1}}\right]r^{\frac{pd(1+pd)}{1-pd}},$
(24)

where we set $\cal G$=$C$. We note that if $\cal G$ is not a constant, we
cannot obtain a metric like (3) through the above transformation. Drawing an
analogy with four-dimensional Vaidya metric vai , we can regard the case with
“+” and $\frac{d{\cal G}}{du}>0$ in (19) as an absorbing star, and the case
with “$-$” and $\frac{d{\cal G}}{du}<0$ in (19) as a shining star in three-
dimensional space-time, notwithstanding, the matter particles that they are
absorbing or shining are not quanta of electromagnetic fields. We emphasize
that three-dimensional Einstein gravity also permits non-trivial geometries in
a sourced space-time, such as a charged BTZ cbtz . So in principle, it also
permits gravitational waves with sources QNM3 .

The case $k=0$ is of special interests, which describes a circularly symmetric
pure gravitational wave. In this case the situation with “+” and $\frac{d{\cal
G}}{du}>0$ still describes an absorbing star, while “$-$” and $\frac{d{\cal
G}}{du}<0$ for a shining star. However, there is nothing but gravity energy to
absorbing or shining in this case: it is a solution for absorption or
radiation of gravitational wave. It realizes two impossibilities in the
corresponding Einstein gravity: The first one is a circularly symmetric
gravitational wave, and the second one is gravitational wave in three-
dimensional vacuum. In this case the Ricci scalar $R$ vanishes. Gravity is the
source of gravity, which is like the case of vacuum Schwarzschild solution in
4-dimensions in some degree. The difference is that in the latter case Weyl
tensor curves the space, while in the former case the Ricci tensor curves the
space, for the Weyl tensor always vanishes in 3-dimensions. The gravitational
waves curve the space-time, which can be demonstrated by the contraction of
Ricci tensor,

$R_{ab}R^{ab}=\pm\frac{(2d-1)r^{-\frac{1+12d^{2}}{d+2d^{2}}}\left(\left(2d-1-12d^{2}+24d^{3}\right){\cal
G}^{2}+4d(1+2d)r^{\frac{4d}{1+2d}}\frac{d{\cal
G}}{du}\right)}{8d^{2}(1+2d)^{2}}.$ (25)

One sees that a Birkhoff-like theorem becomes invalid. The essential point is
that a vanishing stress energy doe not imply a vanishing Ricci tensor in
$f(R)$ gravity. From the above equation (25), it is easy to see that the
metric (19) comes back to a Minkowskian one when ${\cal G}(u)=0$, that is to
say $\overline{A}=0$. So this dynamical space-time can be treated as a
superposition of a gravitational wave $\overline{A}du^{2}$ to a Minkowskian
background ${\overline{B}(r)}dudr+r^{2}d\theta^{2}$ in the double null
coordinates. Since $\cal G$ is an arbitrary function of $u$, this metric can
describe any circularly symmetric gravitational wave in three-dimensional
space-time. It can be tiny, as a fluctuation on the Minkowskian background.
And it can also be large, as a nonperturbative gravitational wave. In any case
it is an exact solution. So, in this work, we find a kind of proper “object”
to quantize. After quantization, they correspond to the on-shell gravitons.
This may offer a useful clue to the gravity quantization in three-dimensional
space-time.

##  IV Conclusion

In three-dimensional pure Einstein gravity, the local geometry is always
trivial, therefore, gravitational wave can not propagate in such a space-time.
This yields some confusions of the corresponding quantum theory. It seems that
the microscopic states in a BTZ black hole do not exist, since there is no
wave in such a configuration. This also yields some confusions in the three-
dimensional black hole thermodynamics. The quantum theory itself promise us
some clues about this problem. A field, even in vacuum state, still shows some
physical effects of itself, for example the Casimir effect. Classical fields
with a zero stress energy has no effects on the gravitational field equation.
However, when quantum effects are considered, they may yield higher derivative
terms, like $f(R)$ theory, in the gravitational field equationbir . Once the
higher derivative terms are introduced, they will make significant and
profound effects in three-dimensional space-time.

First, as we have seen in this paper, non-trivial geometry and gravitational
wave emerges naturally in higher derivative theories, which is radically
different from those in the standard Einstein gravity. This maybe helpful to
solve the problem of the entropy of three-dimensional black hole. A theory
with tiny non-zero $d$ may have enough microstates to yield the black hole
entropy. It is not surprised that the quantum backreaction (resulting a higher
derivative gravity) to illuminate this problem, since fundamentally the
microscopic theory of black hole entropy is quantum theory. Second, to include
the higher derivative terms is helpful to control the divergence in graviton
scattering process, remarkably increasing the renormalizability of the theory.
Thus it is worthy of studying the detailed properties of three-dimensional
$f(R)$ gravity. We find a circularly symmetric black hole solution with non-
trivial geometries in $f(R)$ gravity. The spherically symmetric metrics can be
derived directly from thermodynamic considerations zhang , so we expect the
black hole solution obtained here may also have deep relations with
thermodynamics and can be obtained in a similar manner. Besides, a dynamical
solution is also obtained. When $k=\pm 1$ the solution describes an absorbing
or shining star, depending on the sign of the stress energy. When $k=0$, it
describes a dynamical space-time without source, which becomes a pure
gravitational wave space-time. It is a circular wave, where the Birkhoff-like
theorem is invalid. This solution can be treated as a wave propagating on a
Minkowskian background. Because of the arbitrary function ${\cal G}$ in this
solution, in principle it can describe any wave with circularly symmetry.
Therefore, the solution is a concrete example of the gravitational waves in
three-dimensional space-time. After quantization, it may shed light on the
microscopic origin of three-dimensional black hole, and further more, on the
quantum theory of three-dimensional gravity.

Acknowledgments. H. Z. thanks S. Carlip for helpful discussions. This work is
supported in part by the Program for Professor of Special Appointment (Eastern
Scholar) at Shanghai Institutions of Higher Learning, National Education
Foundation of China under grant No. 200931271104, and National Natural Science
Foundation of China under Grant No. 11075106 and 11275128, and Shanghai
Commission of Science and technology under Grant No. 12ZR1421700.

## References

  * (1) S. Carlip, Class. Quant. Grav. 22, R85 (2005) [gr-qc/0503022].
  * (2) S. Deser, R. Jackiw, and G. ’t Hooft, Annals Phys. 152, 220 (1984).
  * (3) M. Bañados, C. Teitelboim, and J. Zanelli, Phys. Rev. Lett. 69, 1849(1992).
  * (4) S. Hawking, Nature 248, 30 (1974).
  * (5) H. Zhang, S. A. Hayward, X.-H. Zhai, X.-Z. Li, Phys. Rev. D89, 064052 (2014) [arXiv:1304.3647 [physics.gen-ph]]; H. Zhang, Y. Hu and X. Z. Li, Phys. Rev. D 90, 024062 (2014) [arXiv:1406.0577 [gr-qc]]; H. Zhang and X. Z. Li, Phys. Lett. B 737 (2014) 395 [arXiv:1406.1553 [gr-qc]].
  * (6) E. Witten, arXiv:0706.3359 [hep-th].
  * (7) S. Deser, R. Jackiw and S. Templeton, Phys. Rev. Lett. 48, 975 (1982).
  * (8) E. A. Bergshoeff, O. Hohm and P. K. Townsend, Phys. Rev. Lett. 102, 201301 (2009) [arXiv:0901.1766 [hep-th]].
  * (9) E. Bergshoeff, O. Hohm, W. Merbis, A. J. Routh and P. K. Townsend, arXiv:1404.2867 [hep-th]; E. A. Bergshoeff, O. Hohm and P. K. Townsend, Phys. Rev. D 79, 124042 (2009) [arXiv:0905.1259 [hep-th]]; Y. Liu and Y. -W. Sun, Phys. Rev. D 79, 126001 (2009) [arXiv:0904.0403 [hep-th]].
  * (10) J. Oliva, D. Tempo and R. Troncoso, JHEP 0907, 011 (2009) [arXiv:0905.1545 [hep-th]].
  * (11) M. Blagojevic and B. Cvetkovic, Phys. Rev. D 85, 104003 (2012) [arXiv:1201.4277 [gr-qc]].
  * (12) K. S. Stelle, Phys. Rev. D16, 953 (1977).
  * (13) R. Utiyama and B. De Witt, J. Math. Phys. 3, 608 (1962); S. Deser, in Gauge Theories and Modern Field Theory, edited by R. Arnowit and P. Nath, MIT Press (1975).
  * (14) T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451 (2010) [arXiv:0805.1726 [gr-qc]].
  * (15) P. C. Vaidya, Proc. Indian Acad. Sci. A33, 264 (1951); R. W. Lindquist, R. A. Schwartz, and C. W. Misner, Phys. Rev. 137, B1364 (1965).
  * (16) M. Bañados, M. Henneaux, C. Teitelboim, and J. Zanelli, Phys. Rev. D48, 1506 (1993)[gr-qc/9302012].
  * (17) V. Cardoso and J. P. S. Lemos, Phys. Rev. D 63, 124015 (2001) [gr-qc/0101052].
  * (18) N. Birrell and P. Davies, Quantum Fields in Curved Space, Cambrige University Press (1984).