#  Many-body selection rule for quasiparticle pair creations in
centrosymmetric superconductors

Junyeong Ahn<EMAIL_ADDRESS>Department of Physics, Harvard
University, Cambridge, MA 02138, USA  Naoto Nagaosa<EMAIL_ADDRESS>RIKEN
Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
Department of Applied Physics, The University of Tokyo, Bunkyo, Tokyo
113-8656, Japan

###### Abstract

When metal becomes superconducting, new optical excitation channels are
created by particle-hole mixing. These excitation channels contribute
negligibly to optical responses in most superconductors, but they can be
relevant in ultra-strong-coupling superconductors that are close to the Bose-
Einstein condensate regime. Recently, selection rules for these excitations
have been formulated based on single-particle anti-unitary symmetries in the
mean-field theory. While being potentially useful for studying optical
properties of ultra-strong-coupling superconductors, they had fundamental
limitations because significant quantum fluctuations invalidate mean-field
approaches. Here, we use many-body states to formulate an optical selection
rule that does not rely on the mean-field approximation. In this approach, the
physical meaning of the previous selection rules becomes clearer as they are
simply recast as the selection rule for many-body inversion eigenstates, not
involving anti-unitary symmetries. This selection rule applies not only to the
Bogoliubov quasiparticles of Fermi liquids but also to non-Fermi-liquid
quasiparticles and electrically charged bosonic excitations. We also study the
Bogoliubov Fermi surfaces, whose topological stability is closely related to
the selection rule. We provide a many-body formulation of their topological
charges and show that the low-energy optical conductivity of the Bogoliubov
Fermi surfaces depends crucially on their secondary topological charge.
Finally, we discuss the implications of our results to the stability of the
superconducting state.

##  I Introduction

Conventional Bardeen-Cooper-Schrieffer (BCS) superconductors respond to static
electromagnetic fields as if almost all conduction electron pairs contribute
to the supercurrent [1]. This property follows from the optical property of
the BCS model: Cooper pairs are not broken when a spatially uniform electric
field is turning on because their optical spectral weight is zero. The absence
of optical excitations in the BCS model has been attributed to the crystal
momentum conservation imposed by translation symmetries [2, 3].

It has been recently found that an optical selection rule imposed by
additional inversion symmetry is necessary for the absence of pair-breaking
low-energy optical excitations [4, 5]. The selection rule was formulated
within the mean-field Bogoliubov-de Gennes (BdG) formalism, where the relevant
symmetry operation was the single-particle anti-unitary operation defined by
the combination of inversion and particle-hole conjugation [5]. The
establishment of this selection rule provided routes to pair-breaking optical
responses in single-crystal superconductors. Xu, Morimoto, and Moore showed
that Cooper pairs can be optically broken in superconductors without inversion
symmetry [4]. Ahn and Nagaosa showed further that multiband effects can allow
for the optical pair breaking even in the presence of inversion symmetry [5].

However, the applicability of this inversion selection rule in real
superconductors is in question. The optical transition probability of a
Cooper-pair breaking accompanies the factor of $(k_{F}\xi)^{-2}$, where
$k_{F}$ is the Fermi wave vector and $\xi$ is the superconducting coherence
length, which is typically very small. This factor can be non-negligible only
when the Thomas-Fermi screening length $\sim k_{F}^{-1}$ is comparable to the
coherence length, in which case quantum fluctuations give a significant
correction to the mean-field results. Since the selection rule in Refs. [4, 5]
is formulated within the mean-field theory, it is not clear whether it applies
to ultra-strong-coupling superconductors close to the Bose-Einstein condensate
(BEC) regime $(k_{F}\xi<1)$. Addressing this issue is relevant for the optical
study of the BCS-BEC crossover [6, 7] recently observed in superconductors,
including iron-based superconductors [8, 9, 10], twisted bilayer graphene
[11], twisted trilayer graphene [12, 13], and LixZrNCl [14].

Another issue in the mean-field approach is the U(1) gauge symmetry breaking
by Cooper pairing, which leads to the non-conservation of electric charges in
the superconducting state. This may leave some anomalies in physical responses
such as those related to violation of the sum rule [15]. A charge-conserving
approach is desirable to establish an unambiguous result, especially to relate
the optical conductivity to superfluid stiffness through the Ferrel-Glover-
Tinkham (FGT) sum rule [16, 17].

Figure 1:  Optical breaking of a Cooper pair. $q$ is the electric charge of a
quasiparticle. We consider the selection rule imposed on the optical process
where a Cooper pair in a many-body state $|\Omega\rangle$ is excited to
$|\psi\rangle$ with a broken pair of quasiparticles.

In this paper, we promote the results of Refs. [4, 5] to a form that applies
beyond the mean-field theory. We reformulate the inversion selection rule
using many-body states in the charge-conserving theory in Sec. II. Our
selection rule has the advantage of being physically clearer because it does
not involve an anti-unitary operator. This is possible because the particle-
hole symmetry of the BdG Hamiltonian appears due to the redundancy in the
formalism and is not a symmetry of the many-body Hamiltonian. Since there is
no particle-hole symmetry in the many-body approach, we need to consider
inversion symmetry only, which imposes ordinary selection rules as a unitary
symmetry. Namely, we use that, for eigenstates $|\Omega\rangle$ and
$|\psi\rangle$ of the many-body inversion operator $\hat{I}$ with eigenvalues
$\lambda_{\psi}$ and $\lambda_{\Omega}$,

$\displaystyle\braket{\psi}{\hat{\bf J}}{{\Omega}}=0\text{ when
}\lambda_{\psi}\neq-\lambda_{\Omega},$ (1)

where ${\bf\hat{J}}$ is the many-body current operator. Hereafter we put the
hat symbol on operators acting on the many-body Hilbert space. We show below
that the selection rules in Refs. [4, 5] are reproduced when
$|{\Omega}\rangle$ is the ground state and $|\psi\rangle$ is the excited state
with inversion-related electron pairs. This approach applies to non-Fermi
liquid and bosonic quasiparticles also. We connect our result to the previous
mean-field single-particle approach in Sec. III and give some examples of the
consequeces of the selection rule in Sec. IV.

Our selection rule is intimately related to the topological stability of the
Bogoliubov Fermi surfaces [5], which also require the symmetry under the
combination of particle-hole conjugation and inversion in the single-particle
approach. This suggests an interesting optical properties of Bogoliubov Fermi
surfaces. In Sec. V, we reveal low-energy divergent optical responses of
Bogoliubov Fermi surfaces carrying two topological charges. Also, we show how
to express the topological charges of Bogoliubov Fermi surfaces without
involving particle-hole symmetry explicitly, by using a momentum-dependent
many-body ground state. We conclude in Sec. VI by discussing the stability of
unconventional superconductors in light of our selection rule.

Throughout the paper, we assume that the external gauge field is uniform in
space. This neglects the effect of photon momentum ${\bf q}$ that comes as a
small correction of the factor $q^{2}/k_{F}^{2}=(v_{F}/c)^{2}(\hbar
qc/E_{F})^{2}$, where $k_{F}$, $v_{F}$, and $E_{F}=\hbar v_{F}k_{F}$ are Fermi
wave vector, velocity, and energy, $c$ is the velocity of light, and $\hbar
qc$ is the photon energy.

##  II Selection rule in charge-conserving theory

We first introduce the many-body inversion selection rule without mean-field
approximation. Let us recall that the optical conductivity tensor is expressed
by the Kubo formula as

$\displaystyle\sigma^{ab}(\omega)$
$\displaystyle=\frac{\hbar}{V}\sum_{m,n}\frac{\braket{n}{\hat{J}^{a}}{m}\braket{m}{\hat{J}^{b}}{n}}{E_{m}-E_{n}}\frac{-i(\rho_{n}-\rho_{m})}{E_{m}-E_{n}-\hbar\omega}$
$\displaystyle\quad+\frac{ie^{2}}{\omega
V}\left\langle\frac{\partial^{2}\hat{H}}{\partial A^{a}\partial
A^{b}}\right\rangle,$ (2)

where $V$ is the volume of the system, $\hat{H}|n\rangle=E_{n}|n\rangle$ for
the Hamiltonian $\hat{H}$, $\rho_{n}=\braket{n}{\hat{\rho}}{n}$ is the density
matrix element, and $\hat{\bf J}=\partial\hat{H}/\partial{\bf A}$ is the
current operator, and ${\bf A}$ is the external gauge field, and
$\braket{\hat{O}}=\sum_{n}\braket{n}{\hat{O}}{n}\rho_{n}$ in the last line. We
are interested in the resonant transitions between two different energy
levels, so the diamagnetic term in the second line is irrelevant. We study the
constraint of inversion symmetry on the transition amplitude
$\braket{m}{\hat{J}^{a}}{n}$ for specific channels that we now explain. In
this section, we do not assume lattice translation symmetries.

We are interested in the excitation of a pair of quasiparticles created by
$\hat{\gamma}^{\dagger}$ with charge $q$ in the superconducting state, where
the Cooper pair carries charge $2q$. In the simplest case,
$\hat{\gamma}^{\dagger}$ is the electron creation operator and the Cooper pair
is formed by two electrons, but we also consider non-fermionic quasiparticle
excitations. The excited state of our interest has the form

$\displaystyle|\psi_{\alpha}\rangle=\hat{\gamma}^{\dagger}_{\alpha}(\hat{I}\hat{\gamma}^{\dagger}_{\alpha}\hat{I}^{-1})\hat{\cal
C}|{\Omega}\rangle,$ (3)

where $|\Omega\rangle$ is an inversion eigenstate, $\alpha$ is the collective
index containing all orbital, spin, and real-space position or crystal
momentum, $\hat{I}:(t,{\bf x})\rightarrow(t,-{\bf x})$ is an inversion
operator, and the Cooper pair annihilation operator $\hat{\cal C}$ is needed
to conserve the charge in the optical excitation. We suppose that
$[\hat{H},\hat{\gamma}^{\dagger}_{\alpha}]=E_{\gamma,\alpha}\hat{\gamma}^{\dagger}_{\alpha}$,
$[\hat{H},\hat{\cal C}^{\dagger}]=E_{\cal C}\hat{\cal C}^{\dagger}$, and
$[\hat{H},\hat{I}]=0$. Then, if $|\Omega\rangle$ is an energy eigenstate, then
$|\psi_{\alpha}\rangle$ is also an energy eigenstate. Although the excitation
of inversion-asymmetric pairs is also possible in centrosymmetric systems, we
are interested in the excitation of inversion-related pairs [Eq. (3)] because
this channel can be the lowest pair excitation if it is not forbidden by the
selection rule [5]. We consider arbitrary spatial dimensions. It is convenient
to define $\hat{I}$ such that its square may contain an overall U(1) phase
rotation, i.e.,
$\hat{I}^{2}\hat{\gamma}^{\dagger}_{\alpha}\hat{I}^{-2}=\epsilon_{I}\hat{\gamma}^{\dagger}_{\alpha}$,
where $\epsilon_{I}=\pm 1$, although this phase can be removed by re-defining
$\hat{I}$ to contain a phase rotation. For example, the nontrivial factor
$\epsilon_{I}=-1$ can arise due to spin $2\pi$ rotation of fermions when
$\hat{I}$ represent the spatial inversion times a spin $\pi$ rotation in three
dimensions or the twofold rotation around the normal axis in two dimensions.

Equation (1) means that an inversion eigenstate $|\Omega\rangle$ with the
eigenvalue $\lambda_{\Omega}$ can be excited only to those states with
inversion eigenvalue $-\lambda_{\Omega}$ because the current operator has odd
parity under inversion. Let us evaluate the inversion eigenvalue of
$|\psi_{\alpha}\rangle$. We suppose that $\hat{\gamma}^{\dagger}$ satisfies
the commutation relation

$\displaystyle\hat{\gamma}^{\dagger}_{\alpha}\hat{\gamma}^{\dagger}_{\beta}$
$\displaystyle=\eta_{\gamma}\hat{\gamma}^{\dagger}_{\beta}\hat{\gamma}^{\dagger}_{\alpha},$
(4)

where $\eta_{\gamma}=\pm 1$, and transforms under $\hat{I}$ as

$\displaystyle\hat{I}\hat{\gamma}^{\dagger}_{\alpha}\hat{I}^{-1}=\hat{\gamma}^{\dagger}_{\beta}(V_{I})_{\beta\alpha},\quad(V_{I}^{2})_{\beta\alpha}=\epsilon_{I}\delta_{\beta\alpha}.$
(5)

Note that $\hat{\gamma}^{\dagger}_{\alpha}$ may not be a fermionic or bosonic
creation operator, because we do not require canonical commutation relations
$\hat{\gamma}_{\alpha}\hat{\gamma}^{\dagger}_{\beta}+\eta_{\gamma}\hat{\gamma}^{\dagger}_{\beta}\hat{\gamma}_{\alpha}=\delta_{\alpha\beta}$.
It follows from Eqs. (4) and (5) that
$\hat{I}[\hat{\gamma}^{\dagger}_{\alpha}(\hat{I}\hat{\gamma}^{\dagger}_{\alpha}\hat{I}^{-1})]\hat{I}^{-1}=\eta_{\gamma}\epsilon_{I}\hat{\gamma}^{\dagger}_{\alpha}(\hat{I}\hat{\gamma}^{\dagger}_{\alpha}\hat{I}^{-1}).$
The eigenvalues of the Cooper pair and the initial state $|\Omega\rangle$ are

$\displaystyle\hat{I}\hat{\cal C}^{\dagger}\hat{I}^{-1}$
$\displaystyle=\lambda_{\cal C}\hat{\cal C}^{\dagger},$
$\displaystyle\hat{I}|\Omega\rangle$
$\displaystyle=\lambda_{\Omega}|\Omega\rangle,$ (6)

where $\lambda_{\cal C}=\pm 1$ and $\lambda_{\Omega}=\pm 1$. The inversion
eigenvalue of $|\psi_{\alpha}\rangle$ is then
$\eta_{\gamma}\epsilon_{I}\lambda_{\cal C}\lambda_{\Omega}$, so we arrive at
our main result.

$\displaystyle\braket{\psi_{\alpha}}{\hat{\bf J}}{{\Omega}}=0\text{ when
}\lambda_{\cal C}\neq-\eta_{\gamma}\epsilon_{I}.$ (7)

Note that $\lambda_{\Omega}$ does not appear in the final expression.

This selection rule is intimately related to the energy spectrum of the two-
particle state $|\psi_{\alpha}\rangle$. Let us consider the case where
$|\Omega\rangle$ is the superconducting ground state. When
$|\psi_{\alpha}\rangle$ and $|\Omega\rangle$ have the same inversion
eigenvalue, there is band repulsion between them. Therefore, the spectrum of
$|\psi_{\alpha}\rangle$ is gapped in general. More precisely, more than three
parameters (contained in $\alpha$ in our notation) need to be adjusted to
close the gap. For example, in three-dimensional momentum space, the gap can
close at isolated points, generically. The density of states of normal charge
carriers vanishes at zero temperature. On the other hand, when
$|\psi_{\alpha}\rangle$ and $|\Omega\rangle$ have the opposite inversion
eigenvalues, no band repulsion occurs between them. It is possible to close
the gap by tuning only one parameter. Therefore, zero-energy states with a
non-vanishing density of states can appear generically (although not
necessarily 111It is possible to have a gapped spectrum while having a nonzero
oscillator strength for pair productions, $\braket{\psi_{\alpha}}{\hat{\bf
J}}{{\Omega}}\neq 0$, in the clean limit. This is interesting because
Bogoliubov quasiparticles behave like neutral quasiparticles due to perfect
charge screening in superconductors, which makes one expect that Bogoliubov
quasiparticles have zero oscillator strength [18].). In mean-field
superconductors, this means the formation of the so-called Bogoliubov Fermi
surface [19] where $\gamma^{\dagger}_{\alpha}$ is the Bogoliubov quasiparticle
creation operator. As the appearance of such extensive gapless states is
energetically unfavorable in most cases, it is typical to have
$\hat{I}|\psi_{\alpha}\rangle=\lambda_{\Omega}|\psi_{\alpha}\rangle$ such that
$\braket{\psi_{\alpha}}{\hat{\bf J}}{{\Omega}}=0$.

##  III Selection rule in mean-field theory

As the selection rule is based on symmetry, one might expect that mean-field
theory also gives the same result. Here we re-derive the above selection rule
within the mean-field theory of electronic quasiparticles and reproduce the
results in Refs. [4, 5] based on the single-particle approach. Mean-field
theory assumes the nontrivial expectation value of the superconducting order
parameter

$\displaystyle b_{\alpha\beta}$ $\displaystyle=\braket{{\Omega}^{G}_{\rm
MF}}{\hat{c}_{\alpha}\hat{c}_{\beta}}{{\Omega}^{G}_{\rm MF}},$ (8)

where $\hat{c}_{\alpha}$ is the electronic quasiparticle annihilation
operator. A nonzero value of the order parameter means that the mean-field
ground state $|\Omega^{G}_{\rm MF}\rangle$ breaks the charge conservation.
Also, when the order parameter has odd parity, $|\Omega^{G}_{\rm MF}\rangle$
breaks the inversion symmetry of the normal state, while it preserves the
symmetry under the combination of inversion and a phase transformation by
$\pm\pi/2$. Namely, when the order parameter satisfies

$\displaystyle(U_{I})_{\alpha\gamma}b_{\gamma\delta}(U_{I}^{T})_{\delta\beta}=e^{2i\phi}b_{\alpha\beta},$
(9)

where $T$ in the superscript is the matrix transpose, the combination

$\displaystyle\hat{I}_{\phi}=\hat{U}_{\phi}\hat{I}$ (10)

remains a symmetry operator, where

$\displaystyle\hat{U}_{\phi}\hat{c}_{\alpha}\hat{U}_{\phi}^{-1}$
$\displaystyle=e^{i\phi}\hat{c}_{\alpha},$
$\displaystyle\hat{I}\hat{c}_{\alpha}\hat{I}^{-1}$
$\displaystyle=\hat{c}_{\beta}(U_{I})^{*}_{\beta\alpha},$ (11)

and $e^{2i\phi}=\pm 1$. In comparison, note that the superconducting ground
state preserves inversion symmetry and particle number conservation in the
charge-conserving formalism.

The two-particle-excited state we consider in the mean-field theory is thus

$\displaystyle|\psi_{\alpha}^{\rm
MF}\rangle=\hat{\gamma}^{\dagger}_{\alpha}(\hat{I}_{\phi}\hat{\gamma}^{\dagger}_{\alpha}\hat{I}_{\phi}^{-1})|{\Omega}_{\rm
MF}\rangle,$ (12)

where $|{\Omega}_{\rm MF}\rangle$ is an eigenstate of $\hat{I}_{\phi}$,
instead of $\hat{I}$, where $\hat{\gamma}^{\dagger}$ is the Bogoliubov
quasiparticle creation operator. We do not need to include the Cooper pair
annihilation operator $\hat{\cal C}$ here because particle number conservation
is broken here. In fact, the selection rule does not depend on how many
$\hat{\cal C}$ operators we include, since we formulate the selection rule in
terms of $\hat{I}_{\phi}$, under which the Cooper pair is invariant, i.e.,
$\hat{I}_{\phi}\hat{\cal C}\hat{I}_{\phi}^{-1}=\hat{\cal C}$. As we assume
$\hat{I}_{\phi}$ symmetry, a Bogoliubov quasiparticle is transformed to
another Bogoliubov quasiparticle under $\hat{I}_{\phi}$:
$\hat{I}_{\phi}\hat{\gamma}^{\dagger}_{\alpha}\hat{I}_{\phi}^{-1}=\hat{\gamma}^{\dagger}_{\beta}(V_{I_{\phi}})_{\beta\alpha}$.
Using that $\hat{I}_{\phi}{\bf J}\hat{I}_{\phi}^{-1}=-{\bf J}$ and following
the same steps in the previous section, we derive

$\displaystyle\braket{\psi_{\alpha}^{\rm MF}}{\hat{\bf J}}{{\Omega}_{\rm
MF}}=0\text{ when }e^{2i\phi}\neq-\eta_{\gamma}\epsilon_{I}.$ (13)

One can see that this is exactly the selection rule in Eq. (7) by noting that
the inversion eigenvalue of the order parameter, $e^{2i\phi}$, is the same as
that of the Cooper pair, $\lambda_{\cal C}$. This selection was previously
derived for Bogoliubov quasiparticles in multiband BCS superconductors, in
terms of the single-particle symmetry under the combined particle-hole-
conjugation and inversion operation [5].

Let us see how the present approach is related to the previous one in Ref.
[5]. The Bogoliubov quasiparticle annihilation operator in momentum space has
the form

$\displaystyle\hat{\gamma}_{\alpha\bf k}=(u_{\bf
k})_{\alpha\beta}\hat{c}_{\beta\bf k}-(v_{\bf
k})_{\alpha\beta}\hat{c}^{\dagger}_{\beta{\bf-\bf k}},$ (14)

where $\hat{c}_{\beta{\bf\bf k}}$ is the electron annihilation operator, and
we use the Einstein summation convention. $\hat{\gamma}_{\alpha\bf k}$ does
not have a well-defined particle number. Only the number parity is well
defined. Note that we change the notation slightly such that $\alpha$ is the
index for both orbital and spin degrees of freedom but does not include
momentum, while $\alpha$ includes all labels including momentum or position in
Eqs. (3) and (12).

In BCS theory, the Bogoliubov quasiparticle creation operator can be written
as the single-particle particle-hole conjugation ($P$) of the annihilation
operator as

$\displaystyle\hat{\gamma}^{\dagger}_{\alpha-\bf k}$
$\displaystyle=-(v^{*}_{-\bf k})_{\alpha\beta}\hat{c}_{\beta{\bf\bf
k}}+(u^{*}_{-\bf k})_{\alpha\beta}\hat{c}^{\dagger}_{\beta-\bf
k}=(\hat{\gamma}_{\alpha\bf k})^{P},$ (15)

where the single-particle particle-hole conjugation is defined by

$\displaystyle(\hat{\gamma}_{\alpha\bf k})^{P}$
$\displaystyle\equiv\begin{pmatrix}c_{\bf k}&c^{\dagger}_{-\bf
k}\end{pmatrix}U_{P}K\begin{pmatrix}u^{T}_{-\bf k}\\\ -v^{T}_{-\bf
k}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}c_{\bf k}&c^{\dagger}_{-\bf
k}\end{pmatrix}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}K\begin{pmatrix}u^{T}_{-\bf k}\\\ -v^{T}_{-\bf
k}\end{pmatrix},$ (16)

where $K$ is the complex conjugation operator. Therefore, we can write the
inversion pair of $\hat{\gamma}^{\dagger}_{\alpha\bf k}$ as

$\displaystyle\hat{I}_{\phi}\hat{\gamma}^{\dagger}_{\alpha\bf
k}\hat{I}_{\phi}^{-1}$ $\displaystyle=\hat{I}_{\phi}(\hat{\gamma}_{\alpha-\bf
k})^{P}\hat{I}_{\phi}^{-1}=(\hat{\gamma}_{\alpha\bf k})^{I_{\phi}P},$ (17)

where $(\hat{\gamma}_{\alpha\bf k})^{I_{\phi}P}=((\hat{\gamma}_{\alpha\bf
k})^{P})^{I_{\phi}}$, and

$\displaystyle(\hat{\gamma}_{\alpha\bf k})^{I_{\phi}}$
$\displaystyle\equiv\begin{pmatrix}c_{\bf k}&c^{\dagger}_{-\bf
k}\end{pmatrix}U_{I_{\phi}}\begin{pmatrix}u^{T}_{\bf k}\\\ -v^{T}_{\bf
k}\end{pmatrix},$ $\displaystyle U_{I_{\phi}}$
$\displaystyle=\begin{pmatrix}e^{i\phi}U^{*}_{I}&0\\\
0&e^{-i\phi}U_{I}\end{pmatrix}$ (18)

is the single-particle representation of $\hat{I}_{\phi}$ in the Nambu basis
$(c_{\bf k}\;c^{\dagger}_{-\bf k})$. The two-particle state
$|\psi_{\alpha}\rangle$ can thus be viewed equivalently as a state with a
quasiparticle $\hat{\gamma}^{\dagger}_{\alpha\bf k}$ and an anti-quasiparticle
of $(\hat{\gamma}_{\alpha\bf k})^{I_{\phi}P}$.

$\displaystyle|\psi_{\alpha}\rangle=\hat{\gamma}^{\dagger}_{\alpha\bf
k}(\hat{\gamma}_{\alpha\bf k})^{I_{\phi}P}|\Omega_{\rm MF}\rangle.$ (19)

In this picture, the matrix element $\braket{\psi_{\alpha}}{\hat{\bf
J}}{{\Omega}_{\rm MF}}$ describes the transition to the single-particle state
$|\alpha\bf k\rangle$ from the single-particle state
$U_{I_{\phi}}U_{P}K|\alpha\bf k\rangle$.

The particle-number conservation can be recovered from the BCS mean-field
solution as follows [1]. We insert the Cooper pair operator in the definition
of the Bogoliubov quasiparticle [20] so that its particle number is well
defined. $\hat{\gamma}_{\alpha\bf k}=(u_{\bf
k})_{\alpha\beta}\hat{c}_{\beta\bf k}-(v_{\bf
k})_{\alpha\beta}\hat{c}^{\dagger}_{\beta{\bf-\bf k}}\hat{\cal C},$ where
$\hat{\cal C}^{\dagger}=\sum_{\bf k}\hat{c}_{\alpha{\bf k}}^{\dagger}(u_{\bf
k}^{-1}v_{\bf k})_{\alpha\beta}\hat{c}^{\dagger}_{\beta{-\bf k}}$ is the
Cooper pair creation operator. The ground state is projected to the
$N$-particle state. $|{\Omega}^{G}\rangle\propto\hat{P}_{N}e^{\hat{\cal
C}^{\dagger}}|0\rangle\propto(\hat{\cal C}^{\dagger})^{N/2}|0\rangle.$
$|{\Omega}^{G}\rangle$ has a well-defined particle number $N$, and it is an
inversion eigenstate when $\hat{\cal C}^{\dagger}$ has a definite inversion
parity.

##  IV Examples

###  IV.1 Spin-triplet pairing of electrons

Let us first consider the spin-triplet pairing of electrons (or pairing of
spin-polarized fermions) in centrosymmetric systems. We first consider the
spin selection rule. Spin rotation symmetry around the axis of the Cooper pair
spin requires that the two Bogoliubov quasiparticles in
$|\psi_{\alpha}\rangle$ carry the same spin whose direction is parallel to the
Cooper pair spin. Let us suppose that $\hat{z}$ is the Cooper pair spin
direction. The spin-$z$ selection rule imposes
$2s_{z}^{\gamma}+s_{z}^{J}-s_{z}^{\cal C}=2s_{z}^{\gamma}-s_{z}^{\cal C}=0$,
where $s^{O}_{z}$ is the spin of $\hat{O}$ along the $\hat{z}$ direction,
because the current operator does not carry spin. We have $s_{z}^{\gamma}=1/2$
for triplet pairing. In this case, a nontrivial inversion selection rule is
due to the spatial inversion without spin rotation. Since $\eta_{\gamma}=-1$
and $\epsilon_{I}=1$, $\braket{\psi_{\alpha}}{\hat{\bf J}}{{\Omega}}=0$ for
odd-parity pairing $\lambda_{\cal C}=-1$. Note that while spin-triplet pairing
is usually odd-parity pairing, even-parity spin-triplet pairing is not
excluded in multiband systems, because the Cooper pair can form an orbital
singlet.

###  IV.2 Spin-singlet pairing of electrons

When the Cooper pair is a spin singlet of electrons, an optically created
Bogoliubov quasiparticle pair should carry the opposite spin by the spin
conservation. The inversion selection rule is imposed when $\hat{I}$ flip the
spin such that $\hat{I}^{2}=\epsilon_{I}=-1$: $\hat{I}$ is a combination of
spatial inversion and a spin rotation by $\pi$. Then,
$\braket{\psi_{\alpha}}{\hat{\bf J}}{{\Omega}}=0$ for even-parity pairing
$\lambda_{\cal C}=1$.

###  IV.3 Pairing of spin-orbit coupled electrons

If the spin-orbit coupling is significant, it is not useful to think about
spin rotation symmetry. We thus consider spatial inversion without spin
rotation, satisfying $\epsilon_{I}=1$, in spin-orbit coupled electronic
systems. Since $\eta_{\gamma}=-1$ and $\epsilon_{I}=1$,
$\braket{\psi_{\alpha}}{\hat{\bf J}}{{\Omega}}=0$ for odd-parity pairing
$\lambda_{\cal C}=-1$.

When time reversal symmetry is present, it is possible to optically excite a
pair of quasiparticles with zero total momentum even for odd-parity pairing.
This is because the optical excitation of time-reversal-related pairs is not
forbidden: time-reversed partner of a Bogoliubov quasiparticle is different
from the inversion-related partner because time reversal flips spin while
inversion does not. This is in contrast to the systems with spin rotation
symmetry, where the time-reversal-related pair is identical to the pair
related by an effective inversion – defined by the combination of spatial
inversion and a spin $\pi$ rotation.

###  IV.4 Charged bosonic excitations with integer spin

Let us consider the excitation of two charged bosons related by inversion
symmetry. We suppose that the boson carries an integer spin. Since
$\eta_{\gamma}=1$ and $\epsilon_{I}=1$, $\braket{\psi_{\alpha}}{\hat{\bf
J}}{{\Omega}}=0$ when $\lambda_{\cal C}=1$. This can relevant to the optical
excitation of Cooperons (pair density wave excitations) in charge-$4e$
superconductors [21, 22].

###  IV.5 Superconductivity in doped Mott insulators

The Mott insulating phase of the Hubbard model has two different
quasiparticle-like excitations: singly occupied electronic state called holons
and doubly occupied electronic states called doublons, defining the upper and
lower Hubbard bands. Holon and doublon creation operators,
$\hat{\zeta}^{\dagger}_{i\sigma}=\hat{c}^{\dagger}_{i\sigma}\hat{n}_{i\bar{\sigma}}$
and
$\hat{\eta}^{\dagger}_{i\sigma}=\hat{c}_{i\alpha}^{\dagger}(1-\hat{n}_{i\bar{\sigma}})$,
respectively, do not satisfy the full fermionic canonical commutation
relations. However, they satisfy
$\\{\hat{\zeta}^{\dagger}_{i\sigma_{1}},\hat{\zeta}^{\dagger}_{j\sigma_{2}}\\}=\\{\hat{\eta}^{\dagger}_{i\sigma_{1}},\hat{\eta}^{\dagger}_{j\sigma_{2}}\\}=0$.
Also, they transform like electrons under symmetry operations. If these two
properties are inherited to the corresponding excitations in the
superconducting state, such that Eqs. (4) and (5) are satisfied, our selection
rule applies to those excitations. We show in Appendix. A that a model of
superconductivity [23] in a momentum-space analog of the Hubbard model indeed
satisfies these conditions.

##  V Bogoliubov Fermi surfaces

The optical selection rule in superconductors and the stability of the
Bogoliubov Fermi surfaces are closely related as we explain in Sec. II:
Inversion pairs of Bogoliubov quasiparticles can be excited in the class of
superconductors where the Bogoliubov Fermi surfaces are topologically stable
[5]. Therefore, we can expect unique low-energy optical responses from
Bogoliubov Fermi surfaces. In some symmetry classes, Bogoliubov Fermi surfaces
can carry two kinds of topological charges [19, 24]. We show in Sec. V.1 that
doubly charged Bogoliubov Fermi surfaces feature divergent optical responses
in the clean-limit, while singly charged Bogoliubov Fermi surfaces feature
trivial optical responses. We first take the conventional BdG formalism to
define the topological charges and calculate the optical conductivity, where
particle-hole symmetry is crucial for topological charges and the selection
rule. However, as we noted in the introduction and showed in previous sections
that we do not need a particle-hole symmetry in the many-body approach. This
implies that topological charges of Bogoliubov Fermi surfaces can also be
formulated without particle-hole symmetry. We demonstrate this in Sec. V.1.

###  V.1 Optical conductivity

Bogoliubov Fermi surfaces can carry two topological charges in non-interacting
superconductors of ${\bf k}$-local symmetry class D or BDI [24]. Class D is
characterized by the symmetry under the combination of inversion $I$ and
particle-hole conjugation $P$, satisfying $(U_{IP}K)^{2}=1$. Here,
$U_{IP}=U_{I_{\phi}}U_{P}$, and $U_{I_{\phi}}$ and $U_{P}$ are matrix
representations of the symmetry operators for single-particle excitations [See
Eq. (III) and (III).]. There are two topological charges: the Pfaffian
invariant (0D charge) and the Chern number (2D charge). A Bogoliubov Fermi
surface is called doubly charged when it carries both kinds of charges. Class
BDI has additional symmetry under $IT,$ satisfying $(U_{IT}K)^{2}=1$, where
$U_{IT}=U_{I_{\phi}}U_{T}$, and $U_{T}$ is the single-particle representation
of time reversal. Two topological charges in class BDI are the Pfaffian
invariant and the $O(N)$ winding number (1D charge), where $N$ is the number
of bands in the normal state.

Below we consider class BDI in two dimensions and class D and BDI in three
dimensions. We note that we use ${\bf k}$-local symmetry classes [24], which
are distinct from the conventional Altland-Zirnbaurer classes defined by
symmetries local in real space [25, 26].

We focus on doubly charged Bogoliubov Fermi surfaces because singly charged
ones show negligible optical responses. Let us note that the two-band single-
particle effective Hamiltonian of a singly charged Bogoliubov Fermi surface
has the form $H=f({\bf k})\sigma_{z}$. In this effective model, no optical
excitations occur because the single-particle current operator $J^{a}=\partial
H/\partial k^{a}\propto\sigma_{z}$ has zero off-diagonal component. Therefore,
only additional bands beyond the minimal model can contribute to nontrivial
optical responses for singly charged Bogoliubov Fermi surfaces. In contrast,
doubly charged Bogoliubov Fermi surfaces show characteristic nontrivial low-
energy optical responses as we show below.

####  V.1.1 Two dimensions

We consider the simplest model of a doubly charged Bogoliubov Fermi surface in
two dimensions, where the single-particle Bogoliubov-de Gennes Hamiltonian has
the form

$\displaystyle H=\hbar
v(k_{x}\sigma_{x}+k_{y}\sigma_{z})-\mu\tau_{z}+\Delta\tau_{x}.$ (20)

This Hamiltonian has spacetime inversion $IT$ symmetry $U_{IT}H^{*}({\bf
k})(U_{IT})^{-1}=H({\bf k})$ and spatial-inversion-particle-hole-conjugation
$IP$ symmetry $U_{IP}H^{*}({\bf k})(U_{IP})^{-1}=-H({\bf k})$, where
$U_{IT}=1$ and $U_{IP}=\tau_{y}\sigma_{y}$. As the $\sigma$ terms and $\tau$
terms commute with each other, the energy eigenstates are in the form
$|n^{\tau,\sigma}\rangle=|n^{\tau}\rangle\otimes|n^{\sigma}\rangle$, where
$|n^{\tau}\rangle$ and $|n^{\sigma}\rangle$ are two-component states
satisfying
$(-\mu\tau_{z}+\Delta\tau_{x})|n^{\tau}\rangle=s_{\tau}\sqrt{\mu^{2}+\Delta^{2}}|n^{\tau}\rangle$
and
$(k_{x}\sigma_{x}+k_{y}\sigma_{z})|n^{\sigma}\rangle=s_{\sigma}\sqrt{k_{x}^{2}+k_{y}^{2}}|n^{\sigma}\rangle$,
where $s_{\tau}=\pm 1$ and $s_{\sigma}=\pm 1$, such that the energy
eigenvalues are

$\displaystyle
E^{\tau,\sigma}=s_{\tau}\sqrt{\mu^{2}+\Delta^{2}}+s_{\sigma}\hbar
v\sqrt{k_{x}^{2}+k_{y}^{2}}.$ (21)

Band crossing occurs at zero energy when
$k\equiv\sqrt{k_{x}^{2}+k_{y}^{2}}=\sqrt{\mu^{2}+\Delta^{2}}$, which forms a
closed loop [Fig. 2(a)]. We call this the Bogoliubov Fermi surface.

The Bogoliubov Fermi surface is topologically stable because of its nontrivial
0D topological charge protected by $IP$ symmetry. Since $U_{IP}H^{*}({\bf k})$
is an anti-symmetric real matrix, its Pfaffian is well defined and real
valued. The explicit expression of the Pfaffian is ${\rm
Pf}\left[U_{IP}H^{*}({\bf k})\right]=\sqrt{\mu^{2}+\Delta^{2}}-k$ or
$-(\sqrt{\mu^{2}+\Delta^{2}}-k)$. There are two possible choices of the sign
at each ${\bf k}$. However, we require that the Pfaffian is continuous in
${\bf k}$, and this allows only one global choice of the sign. After the
global sign is fixed, the sign of the Pfaffian defines a well-defined 0D
${\mathbb{Z}}_{2}$ topological invariant at each ${\bf k}$. This invariant
changes when there is a band crossing at $E=0$ [19, 24], such that this change
defines the 0D topological charge of the band crossing appearing at zero
energy, which forms the Bogoliubov Fermi surface.

In addition, the Bogoliubov Fermi surface carries an additional 1D topological
charge protected by the simultaneous presence of $IP$ and $IT$ symmetries.
This charge is given by the winding number of the pseudospin ${\bf f}=\hbar
v(k_{x},k_{y})$ for $\sigma$ [24], which is one in our model. See Ref. [24]
for a general definition. This nontrivial winding number adds stability to the
Bogoliubov Fermi surface by protecting it from annihilating itself after being
shrunken to a point.

Let us now calculate the real part of the optical conductivity tensor in the
clean limit, which has the form
$\sigma^{ab}(\omega)=\delta^{ab}(\pi/2\hbar\omega)\int_{\bf
k}\sum_{n,m}f_{nm}|J^{c}_{mn}|^{2}\delta(\omega-\omega_{mn})$. It can be
calculated simply as the current matrix element is separated for the $\tau$
and $\sigma$ parts. $J^{a}_{mn}=\braket{m}{\partial_{\bf A}H}{n}|_{{\bf
A}=0}=\frac{e}{\hbar}\braket{m^{\tau}}{\tau_{z}}{n^{\tau}}\braket{m^{\sigma}}{\partial_{k^{a}}H}{n^{\sigma}}$.
Here, $\tau_{z}$ appears because electrons and holes couple to the gauge field
${\bf A}$ differently through ${\bf k}\rightarrow{\bf k}+(e/\hbar){\bf A}$ and
${\bf k}\rightarrow{\bf k}-(e/\hbar){\bf A}$, respectively. We obtain

$\displaystyle\sigma^{ab}(\omega<\omega_{c})$
$\displaystyle=\delta^{ab}\frac{1}{\hbar\omega}\frac{e^{2}}{4\hbar}\frac{\Delta^{2}}{\omega_{c}}$
(22)

for $\omega_{c}=2\sqrt{\mu^{2}+\Delta^{2}}/\hbar$, while
$\sigma^{ab}(\omega)=\delta^{ab}e^{2}(16\hbar)^{-1}$ is constant for
$\omega>\omega_{c}$. The optical conductivity is plotted in the left panel of
Fig. 2(b).

Figure 2:  Optical conductivity of singly and doubly charged Bogoliubov Fermi
surfaces. (a) Schematic low-energy single-particle band structure. We consider
isotropic Bogoliubov Fermi surfaces centered at ${\bf k}=0$ and show the band
structure along $k_{x}$ passing through ${\bf k}=0$. (b) Optical conductivity.
Blue and red curves corresponds to singly and doubly charged cases. We
consider the form $H=f({\bf k})\sigma_{z}$ for singly charged Bogoliubov Fermi
surfaces: $\sigma^{ab}(\omega)=0$ in this case. We use Eq. (20) and (23) for
doubly charged Bogoliubov Fermi surfaces in 2D and 3D, respectively, with
$\Delta=0.2\hbar\omega_{c}$. $\sigma^{ab}=\sigma^{xx}$ for $b=a$ and
$\sigma^{ab}=0$ for $b\neq a$.

####  V.1.2 Three dimensions

In three dimensions, we begin with a class D Hamiltonian

$\displaystyle H=\hbar
v(k_{x}\sigma_{x}+k_{y}\sigma_{y}+k_{z}\sigma_{z})-\mu\tau_{z}+\Delta_{1}\tau_{x}+\Delta_{2}\tau_{y},$
(23)

which has $IP$ symmetry under $U_{IP}=\tau_{y}\sigma_{y}$. This Hamiltonian
describes a Bogoliubov Fermi surface carrying both the nontrivial Pfaffian
invariant and Chern number 2 (note that we have two copies of identical Weyl
fermions after we continuously deform the system to have
$\mu=\Delta_{1}=\Delta_{2}=0$.). The optical conductivity tensor for
$\omega<\omega_{c}=2\sqrt{\mu^{2}+\Delta^{2}}/\hbar$ is

$\displaystyle\sigma^{ab}(\omega<\omega_{c})$
$\displaystyle=\delta^{ab}\frac{1}{\hbar\omega}\frac{e^{2}}{6\pi\hbar}\frac{\Delta^{2}}{\hbar
v}\left[1+\left(\frac{\omega}{\omega_{c}}\right)^{2}\right],$ (24)

where $\Delta=\sqrt{\Delta_{1}^{2}+\Delta_{2}^{2}}$. When $\omega>\omega_{c}$,
an additional term
$\delta^{ab}e^{2}(6\pi\hbar)^{-1}(\mu/\omega_{c})^{2}(\omega/v)$ appears. See
the right panel of Fig. 2(b).

If we impose $IT$ symmetry with $U_{IT}=1$ additionally, then the Hamiltonian
reduces to the form we presented for two dimensions. It describes a
cylindrical Bogoliubov Fermi surface. The optical conductivity of this has the
same form as the two-dimensional one and features the low-energy $\omega^{-1}$
divergence again.

###  V.2 Alternative definitions of the topological charges

Here we present an alternative expression of the topological invariants of the
non-interacting Bogoliubov Fermi surfaces within the mean-field description.
As we are interested in inversion-symmetric systems, we focus on the effective
Brillouin zone defined by $k_{x}\geq 0$. In the mean-field description, the
degrees of freedom at ${\bf k}$ are decoupled from the degrees of freedom at
the other momenta and are described by the ${\bf k}$-local Hamiltonian
$\hat{H}^{\rm MF}({\bf k})=\hat{c}^{\dagger}_{\alpha\bf k}h_{\alpha\beta\bf
k}\hat{c}_{\beta\bf k}+\frac{1}{2}\hat{c}^{\dagger}_{\alpha\bf
k}\Delta_{\alpha\beta}({\bf k})\hat{c}^{\dagger}_{\beta-\bf
k}+\frac{1}{2}\hat{c}_{\alpha-\bf k}\Delta^{\dagger}_{\alpha\beta}({\bf
k})\hat{c}_{\beta\bf k}+({\bf k}\rightarrow-{\bf k}),$ where the full mean-
field Hamiltonian is $\hat{H}^{\rm MF}=\sum_{{\bf k}:k_{x}>0}\hat{H}^{\rm
MF}({\bf k})+\text{const}.$

We define topological invariants using the ground state $|\Omega^{G}_{\bf
k}\rangle$ of $\hat{H}^{\rm MF}({\bf k})$. The zero-, one-, and two-
dimensional topological invariants are defined by

$\displaystyle\lambda({\bf k})$ $\displaystyle=\braket{\Omega^{G}_{\bf
k}}{\hat{I}_{\phi}}{\Omega^{G}_{\bf k}},$ $\displaystyle w_{1}(C)$
$\displaystyle=\frac{1}{\pi}\oint_{C}d{\bf k}\cdot\braket{\Omega^{G}_{\bf
k}}{\nabla_{\bf k}}{\Omega^{G}_{\bf k}},$ $\displaystyle C_{1}(S)$
$\displaystyle=\frac{1}{2\pi}\oint_{S}d{\bf S}\cdot\nabla_{\bf
k}\times\braket{\Omega^{G}_{\bf k}}{\nabla_{\bf k}}{\Omega^{G}_{\bf k}},$ (25)

where $C$ and $S$ indicate a closed loop and a closed surface, respectively.
$\lambda=\pm 1$ is simply the inversion eigenvalue, $w_{1}=0$ or $1$ (mod $2$)
is the normalized Berry phase (equivalently, the first Stiefel-Whitney number
[27]), and $C_{1}$ is the first Chern number. Here, different from other
quantities, $w_{1}$ for a generic loop $C$ is quantized only when there is
additional time reversal symmetry, in which case $C_{1}=0$.

The topological charges of a Bogoliubov Fermi surface are defined by the
change of topological invariants across the Bogoliubov Fermi surface. As we
noted in the main text, the Bogoliubov Fermi surface appears when
$\lambda({\bf k})$ changes, because the lowest-energy excited state that
preserves inversion symmetry is the two-particle state $|\psi_{\alpha\bf
k}\rangle=\gamma^{\dagger}_{\alpha k}(\hat{I}_{\phi}\gamma^{\dagger}_{\alpha
k}\hat{I}_{\phi}^{-1})|\Omega^{G}_{\bf k}\rangle$. Since $|\psi_{\alpha\bf
k}\rangle$ has the inversion parity opposite to $|\Omega^{G}_{\bf k}\rangle$,
only one parameter needs to be adjusted to close the gap between them, and the
crossing in the momentum space forms the Bogoliubov Fermi surface.

The change of the Berry phase and Chern number in Eq. (V.2) across the
Bogoliubov Fermi surface defines the 1D and 2D topological charges. To see
this, note that Bogoliubov Fermi surfaces can be continuously deformed to the
Fermi surfaces of the normal state by turning off the superconducting pairing.
Since the Berry phase and the Chern number defines a well-defined topological
charge of the Fermi surface (e.g., the Fermi surface of a Weyl point is
characterized by the Chern number jump of the ground state), it is enough to
show that those topological charges remain well defined (i.e., quantized) when
superconductivity comes in. The quantization of the Chern number of a single
state living on a (${\bf k}$-space) closed manifold is well known. As we show
below, the first Stiefel-Whitney number is also quantized by time reversal
symmetry. Therefore, they can serve as a topological charge of the Bogoliubov
Fermi surface.

Let us show that $w_{1}$ is ${\mathbb{Z}}_{2}$-quantized by time reversal
symmetry. Time reversal symmetry of the ground state is represented by
$\hat{T}|\Omega^{G}_{\bf k}\rangle=e^{i\theta_{T}({\bf k})}|\Omega^{G}_{\bf
k}\rangle$, for some $\theta_{T}({\bf k})$. This symmetry gives a constraint
on the Berry connection that $\braket{\Omega^{G}_{\bf k}}{\nabla_{\bf
k}}{\Omega^{G}_{\bf k}}=\frac{1}{2}\nabla_{\bf k}\theta_{T}({\bf k})$, and so
$w_{1}(C)=\frac{1}{2\pi}\oint_{C}\nabla_{\bf k}\theta_{\bf k}$ is an integer-
valued winding number. However, the winding number is defined modulo two
because a phase rotation $|\Omega^{G}_{\bf k}\rangle\rightarrow e^{i\phi_{\bf
k}}|\Omega^{G}_{\bf k}\rangle$ leads to the change $\theta_{T}({\bf
k})\rightarrow\theta_{T}({\bf k})+2\phi({\bf k})$. This proves that
$w_{1}(C)=0$ or $1$ (mod $2$).

##  VI Discussion

One of the implications of our selection rule is that, in the strong pairing
regime, superconductivity is expected to be more robust in centrosymmetric
systems than in non-centrosymmetric systems with similar interaction strength
and Fermi surfaces. This is because nontrivial pair-breaking optical
conductivity in the non-centrosymmetric systems reduces the superfluid
stiffness ${\cal D}^{ab}$, according to the FGT sum rule ${\cal
D}^{ab}=\frac{2}{\pi}\int^{\infty}_{0^{+}}d\omega\left[\sigma^{ab}_{1n}(\omega)-\sigma^{ab}_{1s}(\omega)\right]$,
where $\sigma^{ab}_{1n}$ and $\sigma^{ab}_{1s}$ are the real part of the
conductivity tensor in the normal and superconducting states, respectively.
Berezinskii–Kosterlitz–Thouless (BKT) transition temperature $T_{\rm BKT}$ is
also expected to be reduced because it is proportional to the superfluid
stiffness [28, 29].

Let us discuss how relevant this is in the superconductivity of twisted
layered materials [11, 30, 12, 13, 31, 32, 33, 34, 35, 36]. Robust
superconductivity has been observed in twisted bilayer and trilayer graphene,
having twofold rotation symmetry, but not in other twisted materials lacking
twofold rotation and inversion symmetries. When we assume the ratio
$\Delta_{\rm BCS}/E_{F}\sim 0.3$ of twisted trilayer graphene [12, 13], where
$\Delta_{\rm BCS}=1.76k_{B}T_{c}$, and $T_{c}\sim T_{\rm BKT}$, the relative
enhancement of the superfluid stiffness by the inversion selection rule is a
factor of $(\Delta_{\rm BCS}/E_{F})^{2}\sim 0.1$, which is non-negligible but
still small. Nonetheless, it is worth noting that $(k_{F}\xi)^{2}\sim
1\gg(\Delta_{\rm BCS}/E_{F})^{2}$ in the optimal regime, where $\xi$ is the
superconducting (Ginzburg-Landau) coherence length. This implies that the
superconducting gap is not given by $2\Delta_{\rm BCS}$ but by another
pseudogap of scale $\sim\hbar v_{F}\xi^{-1}$, which is a characteristic of the
BCS-BEC crossover regime [14, 6, 7]. If the creation of inversion-related
quasiparticle pairs depends on the pseudogap rather than $\Delta_{\rm BCS}$,
the inversion selection rule can significantly enhance $T_{\rm BKT}$ in
centrosymmetric twisted materials compared to noncentrosymmetric ones.

The same argument applies to iron-based superconductors [8, 9, 10] and
LixZrNCl [14], which are other few exceptional ultra-strong-coupling
superconductors. They are also in the BCS-BEC crossover regime with a large
pseudogap and $k_{F}\xi\sim 1$. Hence, the effect of the selection rule are
expected to be significant. Since these materials have inversion symmetry, the
suppression of the superconducting optical conductivity by the selection rule
contributes to the robustness of the superconductivity

It is worth noting that, in the strong-coupling regime, another contribution
of the same order $\propto(\Delta/E_{F})^{2}$ becomes relevant to the
superfluid stiffness: the quantum geometric contribution by multiband effects
[37, 38, 39, 40, 41]. While the pair breaking reduces the superfluid
stiffness, the quantum geometry enforces the superfluid stiffness. Interplay
between the two competing effects will be particularly important in the
stability of non-centrosymmetric strong-coupling superconductors.

A large spectral weight in the superconducting state can cause an instability
even in centrosymmetric systems, when the inversion-related pair creation of
Bogoliubov quasiparticles is allowed by multi-orbital effects (i.e., when
$\lambda_{\cal C}=\epsilon_{I}$). In this class of systems, Bogoliubov Fermi
surfaces (namely, the Fermi surface of Bogoliubov quasiparticles) are
topologically protected [19, 24]. Because of this topological protection,
there remains small Fermi surface pockets generically when the superconducting
gap opens. As the pairing strength becomes large enough, the increase of the
optical spectral weight in the superconducting state at low energies cause the
negativity of the superfluid stiffness [42, 43, 44], making it unstable
towards an inversion-broken state.

Furthermore, our mean-field calculations show that, when the Bogoliubov Fermi
surface carries two topological charges, the superfluid stiffness is always
negative in the clean limit because the spectral weight diverges
logarithmically as the temperature approaches zero in the superconducting
state (because $\sigma(\omega)\propto\omega^{-1}$ at low frequencies and the
spectrum is gapless.). This is consistent with other studies of inversion-
breaking instabilities of Bogoliubov Fermi surfaces [45, 46, 47] from
different perspectives. In particular, Ref. [47] showed that there always is
an inversion-breaking instability, analogous to the Cooper instability for
superconductivity. While it was not emphasized, the models used in those works
host doubly charged Bogoliubov Fermi surfaces, which carry both the Pfaffian
invariant and the Chern number.

This instability suggests a new type of topological superconductors, having no
topologically protected gapless surface states. It was recently shown in Ref.
[48] that doubly charged Bogoliubov Fermi surfaces can appear as anomalous
surface states on the twofold-rotation-invariant surfaces of three-dimensional
topological crystalline superconductors. They appear when the superconducting
order parameter preserves time reversal symmetry and changes its sign under
the twofold rotation. Our discussion above implies that the surface state is
unstable towards the breaking of twofold rotational symmetry — an effective
two-dimensional inversion symmetry on the surface. As the spontaneous symmetry
breaking occurs only on the surfaces, the nontrivial bulk topology and the
gapless helical hinge states remain stable. These properties define novel
topological crystalline superconductivity characterized by a spontaneous
symmetry breaking on the surface.

###### Acknowledgements.

J.A. was supported by the Basic Science Research Program through the National
Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant
No. 2020R1A6A3A03037129) and by the Center for Advancement of Topological
Semimetals, an Energy Frontier Research Center funded by the US Department of
Energy Office of Science, Office of Basic Energy Sciences, through the Ames
Laboratory under contract No. DE-AC02-07CH11358. N.N. was supported by JST
CREST Grant Number JPMJCR1874 and JPMJCR16F1, Japan, and JSPS KAKENHI Grant
number 18H03676.

##  Appendix A Philips-Yao-Huang (PYH) model

To test our selection rule for unconventional superconductivity in a non-
Fermi-liquid metal, we consider the mean-field Philips-Yao-Huang (PYH) model
[23]. Here we explicitly demonstrate the selection rule by calculating the
optical conductivity through the exact diagonalization. The mean-field
Hamiltonian is

$\displaystyle\hat{H}^{\rm MF}=\sum_{{\bf k}:k_{x}>0}\hat{H}_{\rm PYH}({\bf
k})+\text{const}.$ (26)

where $\Delta$ is an $s$-wave singlet pairing order parameter, and

$\displaystyle\hat{H}_{\rm PYH}({\bf k})$ $\displaystyle=\xi_{\bf
k}(\hat{n}_{\uparrow\bf k}+\hat{n}_{\downarrow\bf k}+\hat{n}_{\uparrow-\bf
k}+\hat{n}_{\downarrow-\bf k})$ $\displaystyle+U(\hat{n}_{\uparrow\bf
k}\hat{n}_{\downarrow\bf k}+\hat{n}_{\uparrow-\bf k}\hat{n}_{\downarrow-\bf
k})$ $\displaystyle+\Delta(\hat{b}^{\dagger}_{\bf k}+\hat{b}^{\dagger}_{-\bf
k})+\Delta^{*}(\hat{b}_{\bf k}+\hat{b}_{-\bf k}).$ (27)

is a ${\bf k}$-dependent operator that can be represented as a $16\times 16$
matrix acting on the 16 basis states

$\displaystyle\begin{pmatrix}1\\\ \hat{c}_{\uparrow\bf k}^{\dagger}\\\
\hat{c}_{\downarrow\bf k}^{\dagger}\\\ \hat{c}_{\uparrow\bf
k}^{\dagger}\hat{c}_{\downarrow\bf
k}^{\dagger}\end{pmatrix}\otimes\begin{pmatrix}1\\\ \hat{c}_{\uparrow-\bf
k}^{\dagger}\\\ \hat{c}_{\downarrow-\bf k}^{\dagger}\\\ \hat{c}_{\uparrow-\bf
k}^{\dagger}\hat{c}_{\downarrow-\bf k}^{\dagger}\end{pmatrix}|0\rangle.$ (28)

The normal state ($\Delta=0$) of this model corresponds to the Hatsugai-
Kohmoto model [49].

In the superconducting state, the analogs of the Bogoliubov quasiparticle
annihilation operators have the form [23]

$\displaystyle\hat{\gamma}^{l}_{\sigma\bf k}$ $\displaystyle\propto x_{\bf
k}\hat{\zeta}_{\sigma\bf k}-\sigma z_{\bf
k}\hat{\zeta}^{\dagger}_{\bar{\sigma}-\bf k},$
$\displaystyle\hat{\gamma}^{u}_{\sigma\bf k}$ $\displaystyle\propto z_{\bf
k}\hat{\eta}_{\sigma\bf k}-\sigma y_{\bf
k}\hat{\eta}^{\dagger}_{\bar{\sigma}-\bf k},$ (29)

where $\hat{\zeta}^{\dagger}_{\sigma\bf k}=\hat{c}^{\dagger}_{\sigma\bf
k}(1-\hat{n}_{\bar{\sigma}\bf k})$ and $\hat{\eta}^{\dagger}_{\sigma\bf
k}=\hat{c}^{\dagger}_{\sigma\bf k}\hat{n}_{\bar{\sigma}\bf k}$ are the holon
and doublon creation operators for the lower and upper Hubbard bands in the
normal state, respectively. These quasiparticle annihilation operators satisfy
the anticommutation relations

$\displaystyle\\{\hat{\gamma}^{l}_{\sigma_{1}{\bf
k}_{1}},\hat{\gamma}^{l}_{\sigma_{2}{\bf k}_{2}}\\}$
$\displaystyle=\\{\hat{\gamma}^{u}_{\sigma_{1}{\bf
k}_{1}},\hat{\gamma}^{u}_{\sigma_{2}{\bf k}_{2}}\\}=0,$ (30)

while other commutation relations (such as the one between $\hat{\gamma}$ and
$\hat{\gamma}^{\dagger}$) are not fermionic canonical commutation relations.
Here, $i=l,u$ is the index indicating the lower ($l$) or upper ($u$) Hubbard
band, and $\sigma=\uparrow,\downarrow$ is the spin index. Also, they
quasiparticles transforms under inversion as

$\displaystyle\hat{I}\hat{\gamma}^{i\dagger}_{\sigma\bf
k}\hat{I}^{-1}=\hat{\gamma}^{i\dagger}_{\sigma,-\bf k}$ (31)

for $i=l,u$. Accordingly, $\eta_{\gamma}=-\epsilon_{I}=-1$.

The ground state and all the excited states can be simply calculated by
diagonalizing the $16\times 16$ matrix. For concreteness, we consider a one-
dimensional system with $\xi_{\bf k}=-\mu-2t\cos k$ and take $\mu=t=\Delta=1$
and $U=16t$. When $\Delta=0$, the normal state is a hole-doped Mott insulator
that has an extensive ground state degeneracy: spin degeneracy at each k.
After we turn on $\Delta\neq 0$, the unique superconducting ground state is
chosen by the preferred singlet formation.

We find that both of the ground state and the lowest excited two-particle
state of $\hat{H}_{\rm PYH}({\bf k})$ at each ${\bf k}$ have even parity under
inversion, so the current matrix element between them should vanish. This is
consistent with our selection rule that an inversion-related pair of
quasiparticles is not optically excited when $\eta_{\gamma}=-1$ and
$\epsilon_{I}=1$ for even-parity pairing. One can directly see the vanishing
of the relevant matrix element by using the current operator $\hat{\bf
J}=\frac{e}{\hbar}\sum_{\bf k}\frac{\partial\xi_{\bf k}}{\partial{\bf
k}}\left(\hat{n}_{\uparrow\bf k}+\hat{n}_{\downarrow\bf k}\right)$.

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