k_{+}&-\xi_{-}({\bm{k}})\end{array}\right),$ (73) $\displaystyle{\cal
H}_{-}^{-}({\bm{k}})=\left(\begin{array}[]{cccccc}\xi_{+}({\bm{k}})&\alpha
k_{+}&0&0&-d_{{\rm out}-}({\bm{k}})&-\psi_{\rm out}({\bm{k}})\\\ \alpha
k_{-}&\xi_{-}({\bm{k}})&\sqrt{2}t_{\perp}&0&\psi_{\rm out}({\bm{k}})&d_{{\rm
out}+}({\bm{k}})\\\ 0&\sqrt{2}t_{\perp}&\xi_{-}({\bm{k}})&d_{{\rm
in}+}({\bm{k}})&0&0\\\ 0&0&d_{{\rm
in}+}^{\ast}({\bm{k}})&-\xi_{-}({\bm{k}})&0&-\sqrt{2}t_{\perp}\\\ -d_{{\rm
out}-}^{\ast}({\bm{k}})&\psi_{\rm
out}({\bm{k}})^{\ast}&0&0&-\xi_{+}({\bm{k}})&\alpha k_{-}\\\ -\psi_{\rm
out}({\bm{k}})^{\ast}&d_{{\rm
out}+}^{\ast}({\bm{k}})&0&-\sqrt{2}t_{\perp}&\alpha
k_{+}&-\xi_{-}({\bm{k}})\end{array}\right),$ (80)

where $\xi_{\pm}({\bm{k}})=\xi({\bm{k}})\mp\mu_{\rm{B}}H_{z}$, $k_{\pm}=\sin
k_{y}\pm i\sin k_{x}$, and $d_{{\rm out/in}\pm}({\bm{k}})=d_{{\rm
out/in}}^{\,x}({\bm{k}})\pm id_{{\rm out/in}}^{\,y}({\bm{k}})$. Although we do
not know their counterpart in existing systems, we have shown that both
$S+p$-wave and $D+p$-wave PDW states are topological mirror superconductors
with the mirror Chern number, $\nu_{\rm M}=1$. Instead of discussing the
details of the calculation, we show an intuitive understanding of the
$S+p$-wave PDW state. In this case, the gap does not close when we
adiabatically change the parameter as $t_{\perp}\rightarrow 0$. In the limit,
the layers are isolated. The two outer layers are noncentrosymmetric
$S+p$-wave superconductors which are mostly trivial as we discussed for
bilayers. On the other hand, the sector Hamiltonian of the inner layer is
equivalent to the spinless chiral $p$-wave superconductor, known as a
topological superconductor hosting Majorana fermions Read and Green (2000);
Ivanov (2001). Indeed, this part gives nontrivial Chern numbers $\nu_{\pm}=\pm
1$ leading to $\nu_{\rm M}=1$. Although such an intuitive understanding has
not been obtained for the $D+p$-wave state, we numerically obtained the same
topological invariant, $\nu_{\rm M}=1$.

Finally, we discuss the quad-layer PDW state Yoshida  _et al._ (2017). By
symmetry, the Rashba ASOC, spin-singlet gap function, and spin-triplet one
follow the layer dependence,

$\displaystyle{}[\alpha_{1},\psi_{1}(\bm{k}),\bm{d}_{1}(\bm{k})]$
$\displaystyle=$ $\displaystyle[\alpha,\psi(\bm{k}),\bm{d}(\bm{k})],$ (81a)
$\displaystyle{}[\alpha_{2},\psi_{2}(\bm{k}),\bm{d}_{2}(\bm{k})]$
$\displaystyle=$
$\displaystyle[\alpha^{\prime},\psi^{\prime}(\bm{k}),\bm{d}^{\prime}(\bm{k})],$
(81b) $\displaystyle{}[\alpha_{3},\psi_{3}(\bm{k}),\bm{d}_{3}(\bm{k})]$
$\displaystyle=$
$\displaystyle[-\alpha^{\prime},-\psi^{\prime}(\bm{k}),\bm{d}^{\prime}(\bm{k})],$
(81c) $\displaystyle{}[\alpha_{4},\psi_{4}(\bm{k}),\bm{d}_{4}(\bm{k})]$
$\displaystyle=$ $\displaystyle[-\alpha,\psi(\bm{k}),\bm{d}(\bm{k})].$ (81d)

Setting $\psi(\bm{k}):=\Delta_{d}(\cos k_{x}-\cos k_{y})$ and
$\bm{d}(\bm{k}):=a\left(\sin k_{y},\sin k_{x},0\right)^{T}$ with real numbers
$\Delta_{d}$ and $a$ in accordance with the $B$-representation of the $C_{4}$
point group, we numerically obtain the phase diagram in Fig. 16.

Figure 16:  (a) and (b): Chern numbers of mirror sectors $\nu_{+}$ and
$\nu_{-}$ against the magnetic field $\mu_{B}H$ and the chemical potential
$\mu$, respectively Yoshida  _et al._ (2017). The Chern numbers are shown by
numbers enclosed with solid lines. (c): Chern numbers as functions of the
magnetic field. (d) The mirror Chern number and the total Chern number as
functions of the magnetic field. These data are obtained for $t=1.0$,
$t_{\perp}=0.1$, $\alpha=0.3$, $\alpha^{\prime}=0.2$,
$\Delta_{d}=\Delta^{\prime}_{d}=0.05$, $a=-0.01$, and $a^{\prime}=-0.0067$. In
panels (c) and (d), the chemical potential is fixed to $\mu=-0.1$.

In Fig. 16, we plot the Chern numbers $\nu_{\pm}$ against the Zeeman field
$\mu_{B}H$ and the chemical potential $\mu$. This figure is obtained by
employing the Fukui-Hatsugai-Suzuki method Fukui  _et al._ (2005). In Figs.
16(a) and (b), we can see that the mirror Chern number and the total Chern
number take $(\nu_{\mathrm{M}},\nu_{\mathrm{tot}})=(8,0)$ when the magnetic
field is weak, predicting eight pairs of helical Majorana edge modes. The
presence of these modes has been confirmed by the computation of energy
spectrum Yoshida  _et al._ (2017) (see Fig. 17 in the next section).

###  IV.4 Summary and outlook of this section

In this section, we have shown the field-induced transition from the even-
parity superconducting state to the odd-parity state, which arises from the
unique electronic structure in locally noncentrosymmetric multilayer systems.
Based on this finding, we propose a way to realize odd-parity TSC by spin-
singlet Cooper pairing, which are ubiquitous in materials.

The field-induced odd-parity superconducting state, named PDW state, was
identified as a topological crystalline superconductor protected by mirror
reflection symmetry. The topological invariant, mirror Chern number, shows
non-monotonic dependence on the number of layers (see Table. 1). For the
$d$-wave superconductors, the representative mirror Chern numbers are
$\nu_{\rm M}=4$, $1$, and $8$ for bilayers, trilayers, and quad-layers,
respectively. All of these topological phases have intriguing properties. The
bilayer PDW state is an analog of topological $d$-wave superconductivity
discussed in Secs. II and III. The trilayer PDW state may host stable Majorana
fermion on boundaries because $\nu_{\rm M}\in 2\mathbb{Z}+1$. As we show in
the next section, the quad-layer PDW state is a testbed for the reduction of
TSC by interactions, because $\nu_{\rm M}\in 8\mathbb{Z}$. These results shed
light on the strongly correlated electron systems with multilayer structures
as a fascinating platform of the TSC. For example, cuprate superconductors
naturally form the multilayer structure Gotlieb  _et al._ (2018), while recent
developments of the heavy-fermion superlattice enable artificial control of
the multilayer structure Shimozawa  _et al._ (2016); Naritsuka  _et al._
(2021).

Very recently, the field-induced odd-parity superconducting state has been
reported in the bulk CeRh2As2 having the locally noncentrosymmetric bilayer
structure Khim  _et al._ (2021). Motivated by this discovery, interest on the
locally noncentrosymmetric superconductivity and the PDW state is increasing
Skurativska  _et al._ (2021); Möckli and Ramires (2021a); Schertenleib  _et
al._ (2021); Möckli and Ramires (2021b). Thus, attention is naturally paid to
the possible TSC in CeRh2As2. Classification of topological phases is
different from the case in this section, because not the mirror symmetry but
the nonsymmorphic glide symmetry is preserved in CeRh2As2. To clafiry TSC in
CeRh2As2, topological analysis combined with a first-principles band
calculation has been carried out. It shows that the PDW state in CeRh2As2 is a
topological crystalline superconductor protected by the nonsymmorphic glide
symmetry and specified by the glide $\mathbb{Z}_{2}$ invariant Nogaki  _et
al._ (2021).

##  V  Reduction of topological classification by interactions

So far, we have supposed that electron correlations are negligible. However,
extensive theoretical studies have elucidated that the electron correlations
may induce novel topological phenomena which have not been observed for free
fermions. In particular, Fidkowski and Kitaev Fidkowski and Kitaev (2010) have
pointed out that electron correlations may change the classification results
which play an essential role for the material searching Schnyder  _et al._
(2008); Kitaev (2009); Ryu  _et al._ (2010). Namely, they have analyzed the
one-dimensional topological superconductors which follow the
$\mathbb{Z}$-classification at the non-interacting level and have found that
one can gap out the Majorana edge modes without breaking the relevant symmetry
when the number of the gapless Majorana modes is multiple of eight. This fact
indicates that the electron correlations change the topological classification
from $\mathbb{Z}$ to $\mathbb{Z}_{8}$ Fidkowski and Kitaev (2010). In this
review, we refer to this type of phenomena as reduction of the topological
classification because the number of possible topological phases is reduced by
electron correlations. Further extensive theoretical studies have elucidated
that the reduction occurs at any dimensions Turner  _et al._ (2011); Fidkowski
and Kitaev (2011); Yao and Ryu (2013); Ryu and Zhang (2012); Qi (2013); Gu and
Wen (2014); Kapustin  _et al._ (2015); Lu and Vishwanath (2012); Levin and
Stern (2012); Hsieh  _et al._ (2014); Isobe and Fu (2015); Yoshida and
Furusaki (2015); Fidkowski  _et al._ (2013); Wang  _et al._ (2014); Metlitski
_et al._ (2014); Wang and Senthil (2014); You and Xu (2014); Morimoto  _et
al._ (2015), enhancing its significance.

In spite of the above theoretical progress, there are few experimental studies
addressing this issue because of the absence of candidate platforms.
Therefore, it would make variable advance to theoretically propose a possible
testbed. In this section, we point out the possibility that a
$\mathrm{CeCoIn_{5}}/\mathrm{YbCoIn_{5}}$ superlattice system is a feasible
platform of the reduction of the topological classification from
$\mathbb{Z}\times\mathbb{Z}$ to $\mathbb{Z}\times\mathbb{Z}_{8}$ Yoshida  _et
al._ (2017).

In the absence of correlations, the heavy fermions confined in the layer of
$\mathrm{CeCoIn_{5}}$ can show the topological crystalline superconductivity
following $\mathbb{Z}\times\mathbb{Z}$, which we have already discussed in
Sec. IV. For quad-layer (bilayer) of $\mathrm{CeCoIn_{5}}$, the eight (four)
pairs of helical Majorana modes emerge. In this section, we see that electron
correlations can gap out the helical Majorana modes without breaking the
symmetry when the number of the helical modes are multiple of eight. Thus,
quad-layer $\mathrm{CeCoIn_{5}}$ can be a testbed of the reduction of the
topological classification from $\mathbb{Z}\times\mathbb{Z}$ to
$\mathbb{Z}\times\mathbb{Z}_{8}$.

###  V.1  $\mathrm{CeCoIn_{5}}/\mathrm{YbCoIn_{5}}$ superlattice

The $\mathrm{CeCoIn_{5}}/\mathrm{YbCoIn_{5}}$ superlattice system is a typical
example of experimentally realizable 2D heavy-fermion superconductors Mizukami
_et al._ (2011); Goh  _et al._ (2012); Shimozawa  _et al._ (2014, 2016);
Naritsuka  _et al._ (2021). In this system, the thickness of
$\mathrm{CeCoIn_{5}}$-layers can be tuned at the atomic level. Proximity
effects between $\mathrm{CeCoIn_{5}}$ and $\mathrm{YbCoIn_{5}}$ are suppressed
due to large mismatch of the Fermi velocity She and Balatsky (2012); Yoshida
_et al._ (2012); Yamanaka  _et al._ (2015). Thus, 2D heavy fermions emerge in
$\mathrm{Ce}$-layers which show superconductivity around $1\mathrm{K}$. In the
following, we discuss the 2D superconducting phase in the subsystem composed
of $\mathrm{CeCoIn_{5}}$.

The CeCoIn5 multilayers may be the platform of the model Eq. (50), because of
the following facts. (i) Mirror reflection symmetry is locally broken owing to
the presence of $\mathrm{YbCoIn}_{5}$-layers Goh  _et al._ (2012); Shimozawa
_et al._ (2014, 2016); Maruyama  _et al._ (2012). (ii) Bulk CeCoIn5 is a
$d_{x^{2}-y^{2}}$-wave superconductor Matsuda  _et al._ (2006b). (iii) The
system is close to the Pauli limit Tayama  _et al._ (2002). (iv) The CeCoIn5
superlattice is affected by the strong spin-orbit coupling Goh  _et al._
(2012); Maruyama  _et al._ (2012); Shimozawa  _et al._ (2016, 2014). Thus, we
may expect that the field-induced odd-parity superconducting state, namely,
the PDW state is realizable in the CeCoIn5 superlattice. The PDW phase of the
quad-layer CeCoIn5 can be described by the BdG Hamiltonian Eq. (IV.3) with Eq.
(81).

Topological phase diagram of the model has been shown in Fig. 16. For the
numerical calculation, we set parameters for quad-layer CeCoIn5 as follows.
(i) Intralayer hopping: taking into account mass renormalization, we assume
the intralayer hopping approximately $3\mathrm{meV}$. (ii) Pairing potentials:
we adopt the pairing potential, $\Delta_{d}=0.05t$, in accordance with the
experimental observation Mizukami  _et al._ (2011) showing the transition
temperature approximately $T_{\rm c}\simeq 1\mathrm{K}$. (iii) Rashba spin-
orbit coupling: we set $\alpha=0.3t$. First-principles calculations indicate
typical spin-splitting in the heavy fermions as $1000\mathrm{K}$ Samokhin  _et
al._ (2004). Here, we have taken into account the renormalization factor. (iv)
Interlayer hopping: we set the interlayer hopping as $t_{\perp}=0.1t$. This is
consistent with the angular dependence of the upper critical field, which
indicates the quasi-2D electronic structure and the interlayer hopping weaker
than the Rashba spin-orbit coupling Goh  _et al._ (2012); Shimozawa  _et al._
(2016).

Figures 16(a) and 16(b) reveal the mirror Chern number and the total Chern
number $(\nu_{\mathrm{M}},\nu_{\mathrm{tot}})=(8,0)$ in the weak magnetic
field region, predicting the eight pairs of helical Majorana edge modes
localized at each edge. In order to verify this bulk-edge correspondence, we
calculate the energy spectrum $E(k_{y})$ by imposing the open (periodic)
boundary condition for the $x$\- ($y$-) direction. In Fig. 17, we observe
eight Majorana edge modes in the $\lambda=+1$ mirror sector which is
consistent with the Chern number $\nu_{+}=8$ in the bulk. For the $\lambda=-1$
mirror sector, the Chern number $\nu_{-}=-8$ predicts the eight Majorana edge
modes propagating in the opposite direction, which has also been verified by
numerics.

Figure 17:  Energy spectrum of the mirror sector Hamiltonian $\mathcal{H}_{+}$
under the open (periodic) boundary condition for the $x$\- ($y$-) direction.
Here, we plot the data around $k_{y}=\pm\pi/2$. The edge modes localized
around the edge $x=1$ ($x=L$) are shown with red (blue) symbols. The red lines
are for the guide of the eyes. The data are obtained for the following
parameter set in order to suppress the finite size effects: $t=1.0$,
$t_{\perp}=0.1$, $\alpha=\alpha^{\prime}=0.3$,
$\Delta_{d}=\Delta^{\prime}_{d}=0.4$, $\mu_{\rm B}H=\mu=0$, and $L=300$. At
this parameter set, the topology of the bulk is characterized by
$(\nu_{\mathrm{M}},\nu_{\mathrm{tot}})=(8,0)$.

With the above data, we expect that eight pairs of helical Majorana modes
emerge in the quad-layer $\mathrm{CeCoIn_{5}}$ of CeCoIn5/YbCoIn5
superlattices. We stress that the number of the helical Majorana modes can be
tuned in experiments; e.g., for the bilayer $\mathrm{CeCoIn_{5}}$, four pairs
of helical Majorana modes may emerge (see Sec. IV).

###  V.2  Gappability of the helical Majorana modes

Here, we show that the helical Majorana modes can be gapped out without
breaking the mirror reflection symmetry when the number of helical edge modes
is multiple of eight.

The effective Hamiltonian for the helical edge modes is written as

$\displaystyle
H_{\mathrm{edge}}=\sum_{\lambda\alpha}\int\\!dx\;\left[\mathrm{sgn}(\lambda)\eta_{\lambda,\alpha}(x)(-iv\partial_{x})\eta_{\lambda,\alpha}(x)\right],$
(82)

where the summation is taken over $\lambda=\pm$ and $\alpha=1,\cdots,8$, and
$\mathrm{sgn}(\lambda)$ takes $1$ ($-1$) for $\lambda=+$ $(-)$. We note that
the single pair of helical Majorana modes cannot be gapped out without
breaking the symmetry Fidkowski and Kitaev (2010); Yao and Ryu (2013); Ryu and
Zhang (2012); Qi (2013); Hsieh  _et al._ (2014); You and Xu (2014); Morimoto
_et al._ (2015). Before analyzing gappability of the multiple helical Majorana
modes, we discuss the symmetry beyond the quadratic Hamiltonian. The particle-
hole symmetry changes to the symmetry of fermion number parity
$P_{f}=(-1)^{N_{f}}$ with the fermion number operator $N_{f}$ when we go
beyond the quadratic Hamiltonian Fidkowski and Kitaev (2010). Therefore, the
relevant symmetry to the helical gapless edge modes is the symmetry of mirror
reflection and fermion number parity.

Now, by employing bosonization approach, let us see that one can gap out the
eight pairs of helical Majorana modes without breaking the relevant symmetry.
To simplify the analysis, we rewrite the two pairs of helical Majorana
fermions $\eta_{\lambda,\alpha}(x)$ ($\alpha=1,\cdots,8$) with a pair of
helical complex fermions
$f^{\dagger}_{\lambda,\alpha}:=\eta_{\lambda,2\alpha-1}(x)-i\eta_{\lambda,2\alpha}(x)$
($\alpha=1,\cdots,4$),

$\displaystyle\mathcal{L}_{\mathrm{edge}}=\int\\!dx\;\left[K_{IJ}\partial_{\tau}\phi_{I}\partial_{x}\phi_{J}-V_{IJ}\partial_{x}\phi_{I}\partial_{x}\phi_{J}\right].$
(83)

Here, $\phi_{I}(x)$ ($I=1,\cdots,N$) denotes the bosonic field defined as
$f_{I}(x):=\kappa_{I}e^{-i\phi_{I}(x)}/\sqrt{2\pi\alpha}$ with the fermion
operator $\bm{f}:=(f_{+,1},f_{-,1},f_{+,2},f_{-,2},\cdots,f_{+,N},f_{-,N})$.
The matrix $K:=\sigma^{z}\otimes\mbox{1}\mbox{l}_{N\times N}$ describes the
chirality (i.e., whether the modes propagate to left or right).
$V=v\mbox{1}\mbox{l}_{2N\times 2N}$ denotes the velocity. With transformation
laws of fermions
$R\eta_{\lambda,\alpha}(x)R^{-1}=-\lambda\eta_{\lambda,\alpha}(x)$ and
$P_{f}\eta_{\lambda,\alpha}(x)P_{f}^{-1}=-\eta_{\lambda,\alpha}(x)$, we obtain

$\displaystyle R{\bm{\phi}}R^{-1}$ $\displaystyle=$
$\displaystyle\bm{\phi}+\pi(1,0,1,0,\cdots,1,0),$ (84a) $\displaystyle
P_{f}{\bm{\phi}}P^{-1}_{f}$ $\displaystyle=$
$\displaystyle\bm{\phi}+\pi(1,1,\cdots,1,1).$ (84b)

Here, the mirror reflection operator is defined as
$R=e^{-i\frac{\pi}{2}N_{f}}e^{-i\pi S^{z}}P_{z}$ Pre  with $P_{z}$ denoting
the exchange of the layers and the $z$-component of the total spin operator
$S^{z}$. We note that introducing a set of cosine terms
$\cos(\bm{l}^{T}_{i}\cdot\bm{\phi})$ with $n$-independent integral vectors
$\bm{l}$’s gaps out $n$-helical edge modes where $\bm{l}$’s satisfy the
Haldane’s criteria $\bm{l}^{T}_{i}K\bm{l}_{j}=0$ for all pair of
$i,j=1,\cdots,n$ Haldane (1995).

Examining the presence/absence of the symmetry-protected gapless edge modes
elucidates whether the system is topological/trivial. In terms of the
bosonization approach, it can be found by analyzing whether there exist cosine
terms gapping out all of the edge modes without symmetry breaking. We note
that there are two ways to break symmetry: (i) introducing a cosine term which
is not invariant under the transformation; (ii) spontaneous symmetry breaking.

As a first step, we discuss the case of $N=1$ where we can see a pair of
helical Majorana modes which are symmetry-protected. Firstly, we note that
vectors $\bm{l}$’s satisfying the Haldane’s criteria are written as

$\displaystyle\bm{l}=l(1,-1),\ l(1,1)$ $\displaystyle\quad l\in\mathbb{Z}.$
(85)

If $l$ is odd, the cosine term $\cos(\bm{l}^{T}\cdot\bm{\phi})$ is not
invariant under applying $R$. If $l$ is even, the cosine term
$\cos(\bm{l}^{T}\cdot\bm{\phi})$ preserves the symmetry. We note, however,
that reflection symmetry is spontaneously broken because applying the operator
$R$ shifts
$\langle\bm{l}^{T}\cdot\phi\rangle/l\to\langle\bm{l}^{T}\cdot\phi\rangle/l+\pi$
(mod $2\pi$). Thus, there is no cosine term gapping out the edge mode without
symmetry breaking. Therefore, the single helical Majorana mode is symmetry-
protected.

In a similar way, we can discuss the case of $N=2$. In this case, two linearly
independent vectors $\bm{l}$’s describing cosine terms are required in order
to gap out all of the edge modes. Namely, at least a pair of helical Majorana
modes survives even in the presence of cosine terms preserving the symmetry
Yoshida and Kawakami (2016); Zhang  _et al._ (2016b).

In the case of $N=4$, we can see that the eight pairs of helical Majorana
modes are no longer symmetry-protected; all of these modes can be gapped out
without symmetry breaking. Specifically, introducing the cosine terms
described by the following four independent vectors $\bm{l}$’s gaps out all of
edge modes:

$\displaystyle{}\bm{l}^{T}_{1}=(1,0|1,0|0,-1|0,-1),$
$\displaystyle\bm{l}^{T}_{2}=(0,1|0,1|-1,0|-1,0),$
$\displaystyle{}\bm{l}^{T}_{3}=(1,1|-1,-1|0,0|0,0),$
$\displaystyle\bm{l}^{T}_{4}=(0,0|0,0|1,1|-1,-1).$

Introducing these cosine terms do not break the symmetry because the following
relations hold for arbitrary $i$ ($i=1,\cdots,4$):

$\displaystyle R\bm{l}^{T}_{i}\cdot\bm{\phi}R^{-1}$ $\displaystyle=$
$\displaystyle\bm{l}^{T}_{i}\cdot\bm{\phi}\quad(\mathrm{mod}\,\,2\pi),$ (87a)
$\displaystyle P_{f}\bm{l}^{T}_{i}\cdot\bm{\phi}P^{-1}_{f}$ $\displaystyle=$
$\displaystyle\bm{l}^{T}_{i}\cdot\bm{\phi}\quad(\mathrm{mod}\,\,2\pi).$ (87b)

These cosine potentials describe two-body interactions which can be induced by
intralayer antiferromagnetic interactions. Therefore, we can conclude that for
$N=4$, the system is topologically trivial because of the electron
correlations. This means that the topological classification is reduced from
$\mathbb{Z}\times\mathbb{Z}$ to $\mathbb{Z}\times\mathbb{Z}_{8}$ Yoshida  _et
al._ (2017).

We note that the problem for $N=3$ is reduced to the one for $N=-1$ because we
know that the system is topologically trivial for $N=4$. In a similar way as
the $N=1$ case, we can see that the Majorana modes are symmetry-protected for
$N=-1$.

Therefore, we end up with the conclusion that the system characterized by
$(\nu_{\mathrm{M}},\nu_{\mathrm{tot}})=(8,0)$ is topologically trivial in the
presence of electron correlations, indicating that the
$\mathrm{CeCoIn_{5}}/\mathrm{YbCoIn_{5}}$ superlattice system may serve as a
testbed of the reduction of topological classification.

###  V.3  Summary of this section

In this section, we have pointed out that the superlattice system of
$\mathrm{CeCoIn_{5}}/\mathrm{YbCoIn_{5}}$ can be a testbed of the reduction of
topological classification,
$\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}\times\mathbb{Z}_{8}$. We have shown
that the system may host eight pairs of helical edge modes protected by the
mirror reflection symmetry. In addition, we have shown that these eight pairs
of helical Majorana modes can be gapped out by interactions without symmetry
breaking. For experimental observation of the reduction, the STM measurement
is a promising possibility. So far, the STM measurement has been carried out
to detect the Majorana modes localized around the edge of the one-dimensional
topological superconductors. In addition, recently, it became possible to
apply the STM measurement to the heterostructures of $\mathrm{CeCoIn_{5}}$
Haze  _et al._ (2018). Thus, we consider that the STM measurement may
experimentally support the reduction of topological classification by
observing the following behaviors: the system hosts Majorana modes for the
bilayer or trilayer of $\mathrm{CeCoIn_{5}}$ while it does not for the quad-
layer systems.

We note that after the proposal for the
$\mathrm{CeCoIn_{5}}/\mathrm{YbCoIn_{5}}$ superlattice Yoshida  _et al._
(2017) another platform of the reduction was also proposed; a one-dimensional
system of cold atoms showing $\mathbb{Z}\to\mathbb{Z}_{4}$ Yoshida  _et al._
(2018). For this system, quantitative calculations are available by the
density-matrix renormalization group (DMRG) analysis. For the 2D testbed,
further numerical analysis, which takes into account the details of electronic
structure and electron correlations, are awaited. This issue is left as a
future work.

##  VI Summary and Outlook

In this review, we have discussed the TSC based on the non-chiral $d$-wave
superconductors. Our proposals rely on the globally or locally
noncentrosymmetric crystal structure, where the spin-orbit coupling may affect
superconducting properties. For noncentrosymmetric $d$-wave superconductors,
the Zeeman term in combination with the spin-orbit coupling induces
effectively chiral gap function and makes the bulk topological invariant
nontrivial. The mechanism of TSC was explained in analogy with the quantum
anomalous Hall insulator in the Haldane model. In locally noncentrosymmetric
superconductors of either $s$-wave or $d$-wave symmetry, the paramagnetic
pair-breaking effect causes the even-odd parity transition. The resulting
topological crystalline superconductivity was clarified.

Different from most proposals for topological $s$-wave superconductors, the
topological $d$-wave superconductivity discussed here is realizable without
fine-tuning of parameters, such as spin-orbit coupling, Zeeman term, and
chemical potential. The platform is the 2D heterostructures of strongly
correlated electron systems, and the controllability of recently developed
$d$-wave superconductor heterostructures may enable various topological
superconducting phases. For example, the quad-layer $d$-wave superconductor
specified by the mirror Chern number $8$ may be a testbed for the reduction
topological classification, which has not been demonstrated so far.

Previous studies mainly focused on $s$-wave and spin-triplet superconductors.
However, both of them have disadvantages. Although the $s$-wave
superconductors are ubiquitous, fine-tuning of parameters is usually required
for realizing TSC. The spin-triplet superconductors are possibly topological
as it is. However, candidates are limited in nature. In addition to these
directions, the topological $d$-wave superconductivity clarified here may open
a new way for searching TSC.

Progress in the topological $d$-wave superconductivity is further on-going
recently. For instance, chiral $d$-wave superconductivity in twisted bilayer
cuprate superconductors has been proposed Can  _et al._ (2021a, b), and
experimental efforts are being conducted for realization Zhao  _et al._
(2021). Furthermore, a newly discovered superconductor CeRh2As2 Khim  _et al._
(2021) may be a bulk candidate of locally noncentrosymmetric topological
superconductors Nogaki  _et al._ (2021). Identification of TSC and Majorana
fermions and clarification of the unique properties are awaited for future
studies.

##  Acknowledgements

The authors are grateful to Tomohiro Yoshida, Daisuke Maruyama, Tatsuya
Watanabe, Manfred Sigrist, and Norio Kawakami for collaboration in the
original works. The authors would like to thank Swee Kuan Goh, Yuji Matsuda,
Takasada Shibauchi, Masaaki Shimozawa for fruitful discussions on the
experiments. This work was partly supported by JSPS KAKENHI Grant No.
JP21102506, JP23102709, JP24740230, JP25103711, JP15H05855, JP15H05884,
JP16H00991, JP18H01140, JP18H05842, JP18H04225, JP18H05227, JP18H01178,
JP19H01838, JP20H0515, JP20H04627, JP21K13880, and JP21K13850 and by SPIRITS
2020 of Kyoto University. K.T. was supported by the U.S. Department of Energy
(DOE), Office of Science, Basic Energy Sciences (BES), under Contract No.
AC02-05CH11231 within the Ultrafast Materials Science Program (KC2203). The
numerical calculations were performed on the supercomputer at the Institute
for Solid State Physics in the University of Tokyo.

##  Appendix A Derivation of energy dispersion

We show the outline of the derivation of Eq. (12). It is convenient to work
with the Nambu spinor
${\Psi^{\prime}_{\bm{k}}}^{\dagger}=(c^{\dagger}_{\bm{k}\uparrow},c_{\bm{k}\downarrow}^{\dagger},c_{-\bm{k}\downarrow},-c_{-\bm{k}\uparrow})$
rather than $\Psi^{\dagger}_{\bm{k}}$. Then, we obtain the BdG Hamiltonian
corresponding to ${\Psi^{\prime}_{\bm{k}}}^{\dagger}$,

$\displaystyle\mathcal{H}^{\prime}_{\mathrm{BdG}}(\bm{k})$
$\displaystyle=(\xi(\bm{k})+\bm{g}(\bm{k})\cdot\bm{\sigma})\tau_{z}-\bm{h}\cdot\bm{\sigma}$
$\displaystyle\quad+\\{(\psi(\bm{k})+\bm{d}(\bm{k})\cdot\bm{\sigma})\tau_{+}+\mathrm{H.c.}\\},$
(88)

with $\bm{h}=\mu_{B}\bm{H}$ and $\alpha\bm{g}(\bm{k})\to\bm{g}(\bm{k})$ for
simplicity. The symbol $\tau_{\mu}$ ($\mu=0,1,2,3$) represents the Pauli
matrices in the Nambu space, with $\tau_{\pm}\equiv(\tau_{x}\pm i\tau_{y})/2$.
The essential procedure of the derivation is a somewhat technical unitary
transformation $U_{s}=\exp(-i\pi\hat{g}(\bm{k})\cdot\bm{\sigma}\tau_{z}/4)$.
The transformed BdG Hamiltonian
$U_{s}\mathcal{H}^{\prime}_{\mathrm{BdG}}U_{s}^{\dagger}$ has the same form as
Eq. (88) with

$\displaystyle\bm{g}(\bm{k})\to\bm{g}_{s}(\bm{k})=\bm{g}(\bm{k})-\hat{g}(\bm{k})\times\bm{h},$
(89)
$\displaystyle\bm{h}\to\bm{h}_{s}(\bm{k})=(\hat{g}(\bm{k})\cdot\bm{h})\,\hat{g}(\bm{k}),$
(90)
$\displaystyle\psi(\bm{k})\to\psi_{s}(\bm{k})=-i\bm{d}(\bm{k})\cdot\hat{g}(\bm{k}),$
(91)
$\displaystyle\bm{d}(\bm{k})\to\bm{d}_{s}(\bm{k})=-i\psi(\bm{k})\hat{g}(\bm{k})+\hat{g}(\bm{k})\times(\bm{d}(\bm{k})\times\hat{g}(\bm{k})).$
(92)

When we consider the case $\bm{h}\perp\bm{g}(\bm{k})$ (such as for Rashba
systems under $\bm{h}\parallel\hat{z}$), $\bm{h}_{s}=0$ and
$U_{s}\mathcal{H}^{\prime}_{\mathrm{BdG}}U_{s}^{\dagger}$ is formally
equivalent with noncentrosymmetric superconductors in the absence of Zeeman
field. Thus, Eqs. (12) are obtained by using the formula (5) (Eq. (5) is valid
for complex-valued $\psi(\bm{k})$ and $\bm{d}(\bm{k})$ as well). In the
presence of a parallel Zeeman field, it causes Pauli pair-breaking effect by
$\pm\bm{h}_{s}(\bm{k})\cdot\hat{g}_{s}(\bm{k})$, leading to the full formula
Eq. (12). For more details, see Ref. Daido and Yanase, 2016.

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