# Engineering second-order nodal-line semimetals by breaking $\mathcal{PT}$
symmetry and periodic driving

Ming-Jian Gao  Lanzhou Center for Theoretical Physics, Key Laboratory of
Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000,
China  Hong Wu* — Lanzhou Center for Theoretical Physics, Key Laboratory of
Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou 730000,
China  Jun-Hong An* —<EMAIL_ADDRESS>Lanzhou Center for Theoretical
Physics, Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou
University, Lanzhou 730000, China

###### Abstract

Hosting unique drumhead surface states enclosed by nodal lines, topological
nodal-line semimetals exhibit novel transport phenomena. Thus, the exploration
of topological semimetals with different nodal-line structures has attracted
much attention. In this work, we first find a second-order nodal line
semimetal (SONLS), which has the coexisting hinge Fermi arcs and drumhead
surface states, in a $\mathcal{PT}$-symmetry broken system. Then, without
changing the intrinsic parameters, we artificially create exotic hybrid-order
nodal-line semimetals hosted by different quasienergy gaps and fruitful nodal-
line structures including nodal chains, nodal links, crossing line nodes, and
nodal nets by applying a periodic driving on our SONLS. Enriching the
classification of topological semimetals, such Floquet engineered high
tunability of the orders and nodal-line structures of the SONLS sets up a
foundation on exploring its further applications.

##  I Introduction

The rapid development of topological phases of matter [1, 2, 3, 4, 5] not only
enriches the paradigm of condensed matter physics, but also has profound
impacts on many artificial classical systems, such as phononic and photonic
crystals [6, 7, 8], electric-circuit arrays [9], and mechanical system [10].
In the family of topological phases, topological semimetals [5] have attracted
much attention recently. From fundamental-physics perspective, their discovery
successfully generalizes the topological phases from bulk-gapped systems to
bulk-gapless ones. From application perspective, topological semimetals have
exhibited their ability in designing topological devices by using their
chiral-anomaly induced giant magnetoresistance and high carrier mobility [11,
12, 13, 14].

Symmetries play a leading role in classifying topological phases. Various
novel topological semimetals, i.e., Dirac [15, 16, 17, 18, 19, 20, 21, 22, 23,
24], Weyl [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], and nodal-line [36,
37, 38, 39, 40, 41] semimetals, have been found by exploring different kinds
of symmetries. Dirac semimetals present in the systems with spatial inversion
$\mathcal{P}$ and time-reversal $\mathcal{T}$ symmetries. If either of the two
symmetries is broken, then Weyl semimetals may be formed. Nodal-line
semimetals present in the systems with either $\mathcal{P}$ and $\mathcal{T}$
symmetries or mirror symmetry [42]. Recently, a novel semimetal called second-
order nodal-line semimetal (SONLS), which has coexisting hinge Fermi arcs and
drumhead surface states, was proposed in a $\mathcal{PT}$-invariant system
[43, 44, 45]. From the viewpoint of classification of topological phases, an
important question is whether $\mathcal{PT}$ symmetry is an essential
prerequisite for SONLS.

On the other hand, nodal-line semimetals hosting drumhead surface states
enclosed by different nodal-line structures exhibit novel transport features,
which builds the foundation on their measurement and application [46, 47, 48,
49]. Various types of nodal-line structures, e.g., nodal ring [50, 51, 52,
53], nodal link [36], crossing line node [54], nodal chain [46], nodal knot
[55], and nodal net [56], have been discovered in different systems. In
practical applications, one always desires that the transport features caused
by different nodal-line structures can be tuned on demand. This is difficult
to realize in static systems since their properties cannot be changed anymore
once their material samples are fabricated. Coherent control via periodic
driving of external fields dubbed as Floquet engineering has become a
versatile tool in creating novel topological phases [57, 58, 59, 60, 61, 62,
63, 64, 60, 65, 66]. A natural question is that can we realize a free
tunability and conversion of the nodal-line structures and the topological
phases of a SONLS by Floquet engineering.

Figure 1: (a) Schematic diagram of SONLS, with $\gamma$ and $\lambda$ being
the intracell and nearest-neighbor intercell hopping rates, $im/2$ and $iw/2$
being the next-nearest-neighbor intercell ones, and $a/2$ being the interlayer
one. The dashed lines denote the hopping rates with a $\pi$-phase difference
from their solid lines. (b) Energy spectrum of $\mathcal{H}_{0}(\textbf{k})$
(blue lines) and quadrupole moment (green line) and (c) hinge Fermi arc as a
function of $k_{z}$. The red solid (dashed) line is the dispersion relation
along the high-symmetry line $k=0$ ($\pi$). (d) Phase diagram described by
quadrupole moment. We use $\gamma$ = $0.3f$, $a$ = $0.6f$, $\lambda$ = 0, and
the lattice number $N_{x}$ = $N_{y}$ = 40.

Addressing on these questions, we here investigate the SONLS and its Floquet
engineering. We discover a way to realize the SONLS in $\mathcal{PT}$-symmetry
broken systems. It is found that a Dirac semimetal transforms into a SONLS
when a perturbation breaking the $\mathcal{PT}$ symmetry is applied. We also
explore the tunability of the nodal-line structures and the topological phases
of the SONLS by Floquet engineering. Hybrid-order nodal-line semimetal and
fruitful nodal-line structures including nodal chains, nodal links, crossing
line nodes, and nodal nets are created at ease by applying a periodic driving.
The interconversion of different orders of topological semimetal is also
realized by the periodic driving. Enriching the family of topological phases,
our result is helpful to explore their application.

##  II SONLS in $\mathcal{PT}$-symmetry broken system

We first consider a system of spinless fermions moving on a 3D cubic lattice
[see Fig. 1(a)]. Its momentum-space Hamiltonian reads
$\hat{H}_{0}=\sum_{\textbf{k}}$$\hat{C}^{\dagger}_{\textbf{k}}$$\mathcal{H}$(k)$\hat{C}_{\textbf{k}}$
with $\hat{C}^{\dagger}_{\textbf{k}}$ = ($\hat{C}^{\dagger}_{\textbf{k},1}$
$\hat{C}^{\dagger}_{\textbf{k},2}$ $\hat{C}^{\dagger}_{\textbf{k},3}$
$\hat{C}^{\dagger}_{\textbf{k},4}$) and

$\begin{split}\mathcal{H}_{0}(\textbf{k})&=[\gamma+\chi(k_{z})\cos
k_{x}]\Gamma_{5}-\chi(k_{z})\sin k_{x}\Gamma_{3}\\\
&+\,[\gamma+\chi(k_{z})\cos k_{y}]\Gamma_{2}+\chi(k_{z})\sin
k_{y}\Gamma_{1},\end{split}$ (1)

where $\chi(k_{z})=\lambda+a\cos k_{z}$, $\gamma$ is the intarcell hopping
rate, $\lambda$ is the intercell hopping rate, $a$ is interlayer hopping rate
between the nearest-neighbor unit cells, $\Gamma_{i}=\tau_{y}\sigma_{i}$
($i=1,2,3$), and $\Gamma_{5}=\tau_{x}\sigma_{0}$, with $\tau_{i}$ and
$\sigma_{i}$ being Pauli matrices, $\tau_{0}$ and $\sigma_{0}$ being identity
matrices. The system is a 3D generalization of the Benalcazar-Bernevig-Hughes
model [67] by considering the interlayer hopping [35]. It is a Dirac-type
second-order topological semimetal, which is sliced into a family of 2D
$k_{z}$-dependent second-order topological insulators and normal insulators
separated by discrete Dirac points. The system has the chiral symmetry
$\mathcal{S}=\tau_{z}\sigma_{0}$, time-reversal symmetry $\mathcal{T}=K$, with
$K$ being the complex conjugation, and the spatial inversion symmetry
$\mathcal{P}$ = $\tau_{0}\sigma_{y}$. Thus the $\mathcal{PT}$ symmetry of our
system satisfies $\mathcal{(PT)}^{2}=-1$, which is different from the case of
$\mathcal{(PT)}^{2}=1$ in Ref. [43, 44]. The system also possesses the mirror-
rotation symmetries $\mathcal{M}_{x}$ = $\tau_{x}\sigma_{z}$,
$\mathcal{M}_{y}$ = $\tau_{x}\sigma_{x}$, and
$\mathcal{M}_{xy}=[(\tau_{0}+\tau_{z})\sigma_{x}-(\tau_{z}-\tau_{0})\sigma_{z}]/2$.
So the topological phases are described by the Hamiltonian along the high-
symmetry line $k_{x}=k_{y}\equiv k$, which is diagonalized into
$\text{diag}[\mathcal{H}_{0}^{+}(k,k_{z}),\mathcal{H}_{0}^{-}(k,k_{z})]$ with
$\mathcal{H}^{\pm}_{0}(k,k_{z})=\mathbf{h}^{\pm}\cdot{\boldsymbol{\sigma}}$
and $\mathbf{h}^{\pm}=\sqrt{2}[\gamma+\chi(k_{z})\cos k,\pm\chi(k_{z})\sin
k,0]$. A series of Dirac points satisfying $|\gamma|=|\chi(k_{z})|$ when $k=0$
or $\pi$ can be discovered, at which a $k_{z}$-dependent topological phase
transition occurs. This is confirmed by the energy spectrum in Fig. 1(b),
where fourfold degenerate zero-mode states are formed. Their probability
distributions in Fig. 1(c) reveal that they are corner states in the sliced 2D
space and form the hinge Fermi arcs of the 3D second-order topological
semimetal. These corner states are topologically characterized by the
$k_{z}$-dependent quadrupole moment [60, 68, 69]

$P=\Big{[}\frac{\text{Im}\ln\det\mathcal{U}}{2\pi}-\sum_{{\bf n},i;{\bf
m},j}\frac{X_{{\bf n},i;{\bf m},j}}{2L_{x}L_{y}}\Big{]}\text{mod}~{}1.$ (2)

Here the elements of $\mathcal{U}$ read
$\mathcal{U}_{ab}\equiv\langle\psi_{a}|e^{i2\pi
X/(L_{x}L_{y})}|\psi_{b}\rangle$, $|\psi_{\alpha}\rangle$ ($\alpha=a,b$)
satisfying $\hat{H}_{0}|\psi_{a}\rangle=E_{a}|\psi_{a}\rangle$ and
$E_{\alpha}<0$ are the occupied eigen states, and the coordinate $X_{{\bf
n},i;{\bf m},j}=n_{x}n_{y}\delta_{{\bf n}{\bf m}}\delta_{ij}$ with
$i,j=1,\cdots,4$ being the sublattices and $n_{x,y}$ being the numbers of unit
cell. $P=0.5$ signifies the formation of the corner states, see Figs. 1(b) and
1(d).

To generate a SONLS, we add a perturbation on Eq. (1). Different from the
utilization of a $\mathcal{PT}$-invariant perturbation [43, 44], we here
explore the feasibility of generating a SONLS by a $\mathcal{PT}$-symmetry
broken perturbation

$\Delta\mathcal{H}({\bf k})=w\sin k_{x}\tau_{x}\sigma_{x}+m\sin
k_{y}\tau_{x}\sigma_{z},$ (3)

where $w$ and $m$ are the $x$\- and $y$-direction next-nearest-neighbour
intercell hopping rates [see Fig. 1(a)]. Either term in Eq. (3) breaks the
time-reversal symmetry $\mathcal{T}$ and mirror-rotation symmetry
$\mathcal{M}_{xy}$, while preserves the chiral symmetry $\mathcal{S}$, and the
spatial inversion symmetry $\mathcal{P}$, mirror-rotation symmetries
$\mathcal{M}_{x}$ and $\mathcal{M}_{y}$ of $\mathcal{H}_{0}(\textbf{k})$. It
is noted that the two terms in Eq. (3) exhaust all the permitted possibilities
in our system satisfying the above symmetry requirement. We plot the energy
spectrum under the open boundary condition in Fig. 2(a). It is found that each
Dirac point in Fig. 1(b) spreads into a nodal line. The nontrivial quadrupole
moment $P=0.5$ guarantees that the corner nature of the gapped zero-mode state
is preserved. This proves the formation of the 2D sliced second-order
topological phase. It is interesting to find that the regimes within the loops
formed by the nodal lines are 2D sliced Chern insulators. Such first-order
topological phases are characterized by the winding number [70, 3]

$\displaystyle W(k_{y},k_{z})=\frac{1}{4\pi
i}\int_{-\pi}^{\pi}\textrm{Tr}[\mathcal{S}\mathcal{Q}(\textbf{k})\partial_{k_{x}}\mathcal{Q}(\textbf{k})]dk_{x},$
(4)

where $\mathcal{Q}(\textbf{k})=\sum_{l=1,2}[|u_{-l}(\textbf{k})\rangle\langle
u_{-l}(\textbf{k})|-|u_{l}(\textbf{k})\rangle\langle u_{l}(\textbf{k})|]$,
with $|u_{l}(\textbf{k})\rangle$ satisfying
$[\mathcal{H}_{0}(\textbf{k})+\Delta\mathcal{H}(\textbf{k})]|u_{l}(\textbf{k})\rangle=E_{l}(\textbf{k})|u_{l}(\textbf{k})\rangle$.
The chiral symmetry makes $E_{-l}({\bf k})=-E_{l}({\bf k})$. We see from Fig.
2(b) that the regimes with $W=\pm 1$ exactly match with the regimes in the
presence of the nodal lines. This verifies the first-order nature of the
topological insulator phases within the formed nodal-line loops. The
distribution of the nodal lines in the Brillouin zone in Fig. 2(c) indicates
that they resides in the planes of $k_{x}=0$ and $\pi$. The probability
distribution of all the zero-mode states in Fig. 2(d) exhibits the hinge Fermi
arcs in the 2D second-order topological insulator regimes and the drumhead
surface states in the first-order topological insulator regime are separated
by the nodal lines. This confirms the formation of the SONLS. Thus, different
from Ref. [43, 44], we find a way to realize SONLS in a
$\mathcal{PT}$-symmetry broken systems.

Figure 2: (a) Energy spectrum and quadrupole moment (green line) of
$\mathcal{H}_{0}$ \+ $\Delta\mathcal{H}$. (b) Winding number $W(k_{y},k_{z})$
and nodal lines (in red lines) in the $k_{y}$-$k_{z}$ plane. (c) Distribution
of the nodal lines in the Brillouin zone. (d) Coexistence of the hinge Fermi
arcs and the drumhead surface states. We use $m$ = $0.35f$, $w$ = $0$, and
other parameters being the same as Fig. 1. Figure 3: Quasinergy spectrum (a)
and quadrupole moment (b) of the periodically driven system. Winding numbers
$W_{0}$ (c) and $W_{\pi/T}$ (d), with the red lines denoting the projections
of the nodal line on the $k_{y}$-$k_{z}$ plane. Nodal-line distribution of the
zero-mode (e) and $\pi/T$-mode (f) in the Brillouin zone. Hybrid-order nodal-
line semimetal with the coexisting second-order hinge Fermi arcs and the
first-order drumhead surface states in the zero mode (g) and the purely first-
order drumhead surface states in the $\pi/T$ mode (h). We use $\gamma=0.3f$,
$a=0.6f$, $\lambda=0.1f$, $m_{1}=0$, $m_{2}=1.1$, $T_{1}=1.0f^{-1}$, and
$T_{2}=1.5f^{-1}$.

##  III Floquet engineering to SONLS

Being essentially determined by the hopping forms on the lattice, the
topological features and the nodal-line structure of any SONLS in static
systems cannot be changed anymore once their material samples are fabricated.
Without resorting to changing the intrinsic parameters of the materials, we
use the Floquet engineering to conveniently control the topological features
and generate rich nodal-line structures. We consider that the intarcell
hopping rate $m$ in Eq. (3) is periodically driven as

$\begin{split}m(t)=\left\\{\begin{array}[]{ll}m_{1}f,&t\in[nT,nT+T_{1})\\\
m_{2}f,&t\in[nT+T_{1},(n+1)T)\end{array}\right.,\end{split}$ (5)

where $n\in$ $\mathbb{Z}$, $T$ = $T_{1}$ \+ $T_{2}$ is the driving period, and
$f$ is an energy scale to make the driving amplitudes $m_{j}$ ($j=1,2$)
dimensionless. The periodic system does not have a well defined energy
spectrum because its energy is not conserved. However, the Floquet theorem
reveals that $|u_{\alpha}(t)\rangle$ and $\varepsilon_{\alpha}$ in the Floquet
equation
$[\hat{H}(t)-i\partial_{t}]|u_{\alpha}(t)\rangle=\varepsilon_{\alpha}|u_{\alpha}(t)\rangle$
play the same roles as the stationary states and eigen energies in static
systems. They are thus called quasistationary states and quasienergies,
respectively [71, 72]. It is in the quasienergy spectrum that the topological
properties of our periodic system is defined. It can be proven that the
Floquet equation is equivalent to
$\hat{U}_{T}|u_{\alpha}(0)\rangle=e^{-i\varepsilon_{\alpha}T}|u_{\alpha}(0)\rangle$,
where $\hat{U}_{T}=\mathbb{T}e^{-i\int_{0}^{T}\hat{H}(t)dt}$ is the one-period
evolution operator with $\mathbb{T}$ being the time-ordering operator. Thus,
$\hat{U}_{T}$ defines an effective static system
$\hat{H}_{\text{eff}}=iT^{-1}\ln\hat{U}_{T}$ whose energy spectrum exactly
matches with the quasienergy spectrum of our periodic system. Then one can use
the well-developed tool of topological phase transition in static systems to
study periodic systems via $\hat{H}_{\text{eff}}$. Applying the Floquet
theorem on our momentum-space Hamiltonian under the driving protocol (5), we
have

$\mathcal{H}_{\textrm{eff}}(\textbf{k})=iT^{-1}\,\textrm{ln}\,[e^{-i\mathcal{H}_{2}(\textbf{k})T_{2}}e^{-i\mathcal{H}_{1}(\textbf{k})T_{1}}],$
(6)

where $\mathcal{H}_{j}(\textbf{k})$ is the Hamiltonian with $m$ replaced by
$m_{j}$.

Different from the static case, the topological phases of the periodic system
can be carried by the quasienergy gap not only at zero but also at $\pi/T$,
which makes the topological description in static systems insufficient. We can
establish a complete topological description of our periodic system from
$\mathcal{H}_{\text{eff}}({\bf k})$. The first-order topology is described by
the winding number, which requires the chiral symmetries. However,
$\mathcal{H}_{\textrm{eff}}(\textbf{k})$ does not inherit the chiral symmetry
of the static system due to $[\mathcal{H}_{1}({\bf k}),\mathcal{H}_{2}({\bf
k})]\neq 0$. We make two unitary transformations
$G_{l}(\textbf{k})=e^{i(-1)^{l}\mathcal{H}_{l}(\textbf{k})T_{l}/2}$ ($l=1,2$),
which do not change the quasienergy spectrum, to recover the chiral symmetry
and obtain
$\mathcal{\widetilde{H}}_{\textrm{eff},l}(\textbf{k})=iT^{-1}\ln[{G_{l}(\textbf{k}){U}_{T}(\textbf{k})G_{l}^{{\dagger}}(\textbf{k})}]$
[73, 60]. Then two winding numbers $W_{l}$ defined in
$\mathcal{\widetilde{H}}_{\textrm{eff},l}(\textbf{k})$ can be calculated in
the similar manner as the static case. The first-order topologies of
$\mathcal{H}_{\textrm{eff}}(\textbf{k})$ at the quasienergies $\alpha/T$, with
$\alpha=0$ or $\pi$, relate to $W_{l}(\textbf{k})$ as

$W_{\alpha/T}=(W_{1}+e^{i\alpha}W_{2})/2.$ (7)

The number of $\alpha/T$-mode drumhead surface states equals to
$2|W_{\alpha/T}|$. Being same as the static case, the second-order topology is
captured by the quadrupole moment.

Figure 3(a) shows the quasienergy spectrum of $\mathcal{H}_{\textrm{eff}}({\bf
k})$. It can be seen that the second-order corner states are present at the
quasienergy zero, which is witnessed by the nontrivial $P$ in Fig. 3(b). We
also find that large parts of the quasienergy gaps at both of zero and $\pi/T$
are closed. The winding numbers $W_{\alpha/T}$ in Fig. 3(c) and 3(d) reveal
that the regimes with the closed quasienergy gaps exhibit more colorful first-
order topological phases than the static case in Fig. 2(b). In particular, the
large-winding-number phases with $W_{\pi/T}=\pm 2$ and thus the enhanced
numbers of the drumhead surface states, which are absent in the static case,
are present. Combined with the second-order topological phases revealed in
Fig. 3(b), this result demonstrates that a hybrid-order nodal-line semimetal,
with coexisting second-order nodal lines in the zero mode and the first-order
ones in the $\pi/T$ mode, are created by the periodic driving. Figures 3(e)
and 3(f) show the distribution of the zero- and $\pi/T$-mode nodal lines in
the Brillouin zone. We see that the nodal-line structures are dramatically
changed by the periodic driving compared with the static case in Fig. 2(c).
First, the separated nodal loops in the $k_{x}=0$ plane of Fig. 2(c) are
merged. Second, several nodal chains, where the nodal loops in the two planes
of $k_{x}=0$ and $\pi$ are connected by nodal lines, are formed in the zero
mode, and a nodal link, where the three nodal loops in the two planes of
$k_{x}=0$ and $k_{y}=0$ are linked together, are formed in the $\pi/T$ mode.
It indicates that we realize an exotic hybrid-order nodal-line semimetal, with
coexisting second-order nodal chains and first-order nodal links. Such rich
nodal-line structures are difficult to realize in static systems. The hybrid-
order nodal-line topological semimetal is further confirmed by the probability
distributions of the zero- and $\pi/T$-mode states. The second-order hinge
Fermi arcs and the first-order drumhead surface states separated by the nodal
lines in Fig. 3(g) verify the second-order nature of the zero-mode nodal
lines. The purely drumhead surface states in Fig. 3(h) confirms the first-
order nature of the $\pi/T$-mode nodal lines. Therefore, our result reveals
that the periodic driving supplies an efficient tool to adjust and enrich the
nodal-line structure and the topological phases of the SONLS.

Figure 4: Quasinergy spectrum (a) and winding numbers $W_{0}$ (b) and
$W_{\pi/T}$ (c) of the periodically driven system. Nodal-line distribution of
the zero-mode (d) and $\pi/T$-mode (e) in the Brillouin zone. We use
$\gamma=0.5f$, $a=0.6f$, $\lambda=0.3f$, $m_{1}=0$, $m_{2}=1.1$,
$T_{1}=2.8f^{-1}$, and $T_{2}=1.2f^{-1}$.

Furthermore, the periodic driving can also realize the conversion of different
orders of nodal lines and other types of nodal-line structures, both of which
are hard to realize in static systems. The quasienergy spectrum in Fig. 4(a)
shows that the zero-mode corner states disappear and the $\pi/T$-mode gap
persistently closed. Widely changing from $-2$ to $2$, the winding numbers
$W_{\alpha/T}$ in Figs. 4(b) and 4(c) reveals the rich first-order topological
phases in the gap closing regimes both for the zero and $\pi/T$ modes. Thus
the numbers of the drumhead surface states are enhanced. It is interesting to
see that fruitful nodal-line structures are created. The zero mode shows a
nodal link, see Fig. 4(d). The $\pi/T$ mode shows nodal links, nodal chain, a
crossing line node [54], and two nodal nets, see Fig. 4(e). Both of the zero-
and $\pi/T$-mode states are the first-order drumhead surface states. The
results verify that the SONLS in Fig. 2 is converted into the a first-order
nodal-line semimetal with rich topological phases and nodal-line structures
both at the zero and $\pi/T$ modes by the periodic driving.

##  IV Discussion and conclusion

The step like driving protocol is considered just for the convenience of
numerical calculation. Our scheme is generalizable to other driving forms. The
second-order topological semimetal has been predicted in Cd3As2, KMgBi, and
PtO2 [74, 24] and realized in classical acoustic metamaterials [7, 8]. Floquet
engineering has exhibited its ability in creating exotic phases in electronic
material [75, 76], ultracold atoms [77], superconductor qubits [78], and
photonics [79, 80, 81, 82] systems. The progress indicates that our result is
realizable in the recent experimental state of the art.

In summary, we have proposed a scheme to generate SONLS in a
$\mathcal{PT}$-symmetry broken system and explored its diverse variation in
both of the nodal-line structures and the topological orders by the versatile
Floquet engineering. An exotic hybrid-order nodal-line semimetal and abundant
nodal-line structures including nodal chains, nodal links, crossing line
nodes, and nodal nets are created at ease by the periodic driving. This result
enriches the classification of topological semimetals and provides a
convenient way to reduce the practical difficulties in adjusting the nodal
line structures, the Fermi arcs, and drumhead surface states in static
systems. Such an on-demand controllability to the topological semimetal is
helpful in designing novel topological devices simultaneously utilizing the
advantages of two topological orders and various of nodal-line structures of
the semimetal.

##  V Acknowledgments

The work is supported by the National Natural Science Foundation (Grants
No.11875150, No. 11834005, and No. 12047501).

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# Dispersion of plasmons in three-dimensional superconductors

T. Repplinger, M. Gélédan and H. Kurkjian  Laboratoire de Physique Théorique,
Université de Toulouse, CNRS, UPS, 31400, Toulouse, France  S. Klimin and J.
Tempere  TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Anvers,
Belgique

###### Abstract

We study the plasma branch of an homogeneous three-dimensional electron gas in
an $s$-wave superconducting state. We focus on the regime where the plasma
frequency $\omega_{p}$ is comparable to the gap $\Delta$, which is
experimentally realized in cuprates. Although a sum rule guarantees that the
departure of the plasma branch always coincides with the plasma frequency, the
dispersion and lifetime of the plasmons is strongly affected by the presence
of the pair condensate, especially at energies close to the pair-breaking
threshold $2\Delta$. When $\omega_{p}$ is above $1.7\Delta$, the level
repulsion is strong enough to give the plasma branch an anomalous, negative
dispersion with a minimum at finite wavelength. At non-zero temperature and at
$\omega_{p}>2\Delta$, we treat in a non-perturbative way the coupling of
plasmons to the fermionic excitations, and show that a broadened plasma
resonance inside the pair-breaking continuum coexists with an undamped
solution in the band gap. This resonance splitting is associated with the
presence of multiple poles in the analytic continuation of the propagator of
the Cooper pairs.

Introduction: Despite being a very mature experimental platform, supporting
numerous technical applications, superconductors still hold some of the most
fundamental open questions of many-body physics. The impressively high
critical temperature ($T_{c}$) and the unconventional Cooper pairing in
cuprates and iron-based superconductors are the most famous of those
fascinating questions. However, even some properties of conventional Bardeen-
Cooper-Schrieffer (BCS) superconductors are still intensively discussed, such
as the existence of an amplitude collective mode Sacuto2014 ; Measson2019 ;
Shimano2013 , reminiscent of the Higgs mode in high-energy physics.

In fact, even for such usual behavior as plasma oscillations (the collective
modes of the electronic density), superconductors are still not fully
understood. In a pioneering work, Anderson Anderson1958  has shown that the
phononic (Goldstone) branch that exists in a neutral fermionic condensate
acquires a gap corresponding to the plasma frequency $\omega_{p}$ in presence
of long-range Coulomb interaction. This mechanism later became famous due to
its analogy with the phenomenon of mass acquisition in high-energy physics.
The work of Anderson has then been revisited in the context of high-$T_{c}$
superconductivity DasSarma1991 ; Griffin1993 ; Kobelkov1995 ; vanderMarel1995
; Takada1997plasm , and nuclear/neutronic matter Ducoin2011 . While Anderson
focused on the regime of large $\omega_{p}$, the frequency of transverse
plasmons in layered superconductors such as cuprates often lies below the
pair-breaking threshold $2\Delta$ Uchida1992 ; Richard1994 ; Tachiki1997 ,
such that an undamped plasma branch can be expected. The departure of the
plasma branch was shown to always coincide with $\omega_{p}$ Takada1997plasm ,
and as temperature or excitation momenta were varied, a duplication of the
plasma resonance was observed, with a low-energy branch at energies below
$2\Delta$ and a high-energy one above DasSarma1995 ; DasSarma1995manip ;
Yamada1998 .

The existence of such low-energy plasmons was cited as a possible explanation
of the critical temperature increase in cuprates Pentegov2008 . Besides their
importance for the fundamental understanding of superconductivity and long-
range interactions in many-body physics, those modes can also be used in
plasmonics, or to probe and manipulate superconducting materials Cavalleri2013
. Experimental research on Josephson plasmons, which describe transverse
plasmonic excitations is superconducting layers is still very active today
Basov2014 .

At low energy-momentum, plasmons can be described by phenomenological
approaches based on London electrodynamics Buisson1997 , but a microscopic
theory is needed when the eigenfrequency of plasmons approaches the pair-
breaking threshold. In this regime, the theoretical literature is still
hesitant, in particular in the cases where a complex plasma mode describing a
damped resonance is expected. Here, we reveal that remarkable phenomena
affecting the plasma dispersion in presence of superconducting order have been
overlooked. Like Anderson, we consider the reference situation of an isotropic
three-dimensional (3D) $s$-wave superconductor but our study can be readily
extended to layered geometries or anisotropic pairing. Due to the repulsion of
the pair-breaking threshold, the plasma branch acquires a negative curvature
and thus a minimum at non-zero wavenumber when $\omega_{p}$ is between
$1.696\Delta$ and $2\Delta$ at zero temperature. At larger wavenumber, nonzero
temperature or when $\omega_{p}>2\Delta$, the plasma branch enters the pair-
breaking continuum but remains observable with a finite lifetime that we
calculate using recently develop technics higgs ; higgslong ; artlongsk  to
treat the coupling to the fermionic continuum. We find a rich resonance
structure in the density-density response function, with several peaks both
above and below the pair-breaking threshold, which we relate to the existence
of multiple poles in the analytic continuation Nozieres1963 ; higgs ;
artlongsk  of the propagator the density-phase fluctuations. In the
quasiphononic regime $\omega_{p}\ll 2\Delta$ corresponding to the experimental
situation in cuprates DasSarma1995manip ; Tachiki1997 , the plasma
eigenfrequency behaves as $\sqrt{\omega_{p}^{2}+\omega_{q,{\rm n}}^{2}}$ where
$\omega_{q,{\rm n}}$ is the phononic dispersion of the neutral fermion
condensate DasSarma1995 . Finally, we show that when temperature increases
towards $T_{c}$, the plasmons gradually recover their normal (undamped)
dispersion FetterWalecka , except in a region of size $\Delta^{2}/T$ around
the pair-breaking threshold. All these unusal behaviors should have important
consequences on the electromagnetic and transport properties of
superconductors.

Linear response within RPA: We study an homogeneous electron gas evolving in a
cubic volume $V$ with a average density $\rho$, defining the Fermi wavenumber
$\rho=k_{F}^{3}/3\pi^{2}$. Electrons interact through both the long-range
Coulomb potential $V_{C}(r)\propto 1/r$ and a short-range part, responsible
for $s$-wave Cooper pairing, and modelled by a contact potential of coupling
constant $g$:

${V}(\textbf{r}_{1},\textbf{r}_{2})=g\delta(\textbf{r}_{1}-\textbf{r}_{2})+V_{C}(\textbf{r}_{1}-\textbf{r}_{2})$
(1)

We note that a momentum cutoff at the Debye frequency should be used to
regularize the divergence caused by the contact potential, although in the
following discussion this cutoff can safely be send to infinity. In terms of
the electron mass $m$ and wavenumber $q$, the Fourier transform of the Coulomb
potential is $V_{C}(q)=m\omega_{p}^{2}/\rho q^{2}$ (we use $\hbar=k_{B}=1$
throughout the article).

We imagine that the system is driven at fixed frequency $\omega$ and
wavenumber $q$ by an external field (for example an electromagnetic field) and
we study the collective response within linear response theory. The response
function can be computed either using a path integral formalism Randeria1997
with both a pair and density auxiliary fields, or in the Random Phase
Approximation (RPA), neglecting as in Anderson1958  the exchange-scattering
diagrams (the effect of those diagrams is discussed in Ref. Schrieffer1961 ).
In a superconductor, since the density response $\delta\rho$ is coupled to the
fluctuations of the order parameter, in phase $\delta\theta$ and a priori in
modulus $\delta|\Delta|$, the linear response function $\chi$ is a $3\times 3$
matrix:

$\begin{pmatrix}2\textrm{i}\Delta\delta\theta(\textbf{q},\omega)\\\
2\delta|\Delta(\textbf{q},\omega)|\\\
g\delta\rho(\textbf{q},\omega)\end{pmatrix}=\chi(\omega,\textbf{q})\begin{pmatrix}u_{\theta}(\textbf{q})\\\
u_{|\Delta|}(\textbf{q})\\\ u_{\rho}(\textbf{q})\end{pmatrix},$ (2)

where $u_{\theta}$, $u_{|\Delta|}$ and $u_{\rho}$ are respectively the phase,
modulus and density driving fields repdens . The response matrix $\chi$ is
expressed in terms of the bare propagator $\Pi$ as $\chi=-M^{-1}\Pi$ with

$M=\Pi-D\text{ and }D\equiv\begin{pmatrix}V/g&0&0\\\ 0&V/g&0\\\
0&0&V/2V_{C}(\textbf{q})\end{pmatrix}$ (3)

Remark that due to the Coulomb potential $M$ and $\Pi$ do not commute, such
that $\chi$ is not a symmetric matrix. The matrix $\Pi$ was computed for
instance in Refs. Takada1997 ; Griffin2003 ; repdens . We give here its
generic expression

$\Pi_{ij}(z,\textbf{q})=\sum_{\textbf{k}}{\frac{\pi_{ij}^{+}(1-f_{+}-f_{-})}{z^{2}-(\epsilon_{+}+\epsilon_{-})^{2}}-\frac{\pi_{ij}^{-}(f_{+}-f_{-})}{z^{2}-(\epsilon_{+}-\epsilon_{-})^{2}}}$
(4)

in terms of the Fermi-Dirac occupation numbers
$f_{\pm}=1/(1+\text{exp}(\epsilon_{\pm}/T))$, free-fermion
$\xi_{\pm}=\xi_{\textbf{q}/2\pm\textbf{k}}$ and BCS energies
$\epsilon_{\pm}=\epsilon_{\textbf{q}/2\pm\textbf{k}}$ with
$\xi_{\textbf{k}}=k^{2}/2m-\mu$ and
$\epsilon_{\textbf{k}}=\sqrt{\xi_{\textbf{k}}^{2}+\Delta^{2}}$. The
coefficients $\pi_{ij}^{\pm}$ can be deduced from Eq. (36) in repdens . The
first term in Eq. (4) gives rise to the pair-breaking continuum
$\\{\epsilon_{\textbf{q}/2+\textbf{k}}+\epsilon_{\textbf{q}/2-\textbf{k}}\\}_{\textbf{k}}$,
gapped at low $q$ by the pair-breaking threshold $2\Delta$. The second term
exists only at $T\neq 0$ and gives rise to the gapless quasiparticle-quasihole
continuum
$\\{\epsilon_{\textbf{q}/2+\textbf{k}}-\epsilon_{\textbf{q}/2-\textbf{k}}\\}_{\textbf{k}}$

The spectrum of the collective modes is found as the poles of $\chi$, hence as
the zeros of $M$:

$\text{det}M_{\downarrow}(z_{\textbf{q}},\textbf{q})=0$ (5)

The $\downarrow$ sign recalls that when the collective mode is coupled to the
pair-breaking higgs  or quasiparticle-hole continuum artmicro ; artlongsk , it
is complex energy is found only after an analytic continuation of $M$ from
upper to lower half-complex plane.

The present analysis of the plasmon dispersion focuses on the typical weak-
coupling regime of superconductors, with $\Delta$ much smaller than the Fermi
energy $\epsilon_{F}$, and the excitation wavelength comparable to the Cooper
pair size $\xi=k_{F}/2m\Delta$. In this regime, the fluctuation of the modulus
of the order parameter are decoupled from the phase-density fluctuations:

$\text{det}M_{\downarrow}=0\iff
M_{11,\downarrow}M_{33,\downarrow}-M_{13,\downarrow}^{2}=0\text{ or
}M_{22,\downarrow}=0$ (6)

The second condition gives rise the “pair-breaking” or “Higgs” modulus mode
which in the weak-coupling regime is insensitive to Coulomb interactions
Popov1976 ; higgslong . Here, we study the density-phase modes, fulfilling the
first condition.

Anomalous dispersion of long wavelength plasmons: We first study analytically
the plasmon dispersion in the limit $q\ll 1/\xi$, where, by analogy with the
normal case FetterWalecka , one can expect the quadratic law Kobelkov1995 :

$z_{q}=\omega_{0}+\alpha\frac{q^{2}}{2m}+O(q^{4})$ (7)

The expansion in powers of $q$ is more easily performed using the recombined
matrix:

$\widetilde{M}=\begin{pmatrix}M_{11}&zM_{13}+2\Delta M_{11}\\\ zM_{13}+2\Delta
M_{11}&z^{2}M_{33}+4\Delta zM_{13}+4\Delta^{2}M_{11}\end{pmatrix}$ (8)

In particular the origin $\omega_{0}$ of the plasma branch is found simply by
solving $\widetilde{M}_{33}=0$ to lowest order in $q$. We have

$\widetilde{M}_{33}=\frac{\rho
Vq^{2}}{2m}\left(1-\frac{z^{2}}{\omega_{p}^{2}}\right)\\\
+\sum_{\textbf{k}}{\frac{\widetilde{m}_{33}^{+}(1-f_{+}-f_{-})}{z^{2}-(\epsilon_{+}+\epsilon_{-})^{2}}}{-\frac{\widetilde{m}_{33}^{-}(f_{+}-f_{-})}{z^{2}-(\epsilon_{+}-\epsilon_{-})^{2}}}$
(9)

where we have used a sum rule111Explicitely, we have used
$\sum_{\textbf{k}}[\left(1-f_{+}-f_{-}\right)\left(\epsilon_{+}+\epsilon_{-}\right)$
$\left(\epsilon_{+}\epsilon_{-}-\xi_{+}\xi_{-}-\Delta^{2}\right)-\left(f_{+}-f_{-}\right)\left(\epsilon_{+}-\epsilon_{-}\right)\left(\epsilon_{+}\epsilon_{-}+\xi_{+}\xi_{-}+\Delta^{2}\right)]$
$/2\epsilon_{+}\epsilon_{-}=\rho Vq^{2}/2m.$ to simplify the first line, and
we set
$\widetilde{m}_{33}^{\pm}={[\xi_{+}-\xi_{-}]^{2}}\left[\epsilon_{+}\pm\epsilon_{-}\right]\left[\epsilon_{+}\epsilon_{-}\mp\xi_{+}\xi_{-}\mp\Delta^{2}\right]/2\epsilon_{+}\epsilon_{-}$.
The summation on the second line is of order $q^{4}$ and will only affect the
expression of the dispersion parameter $\alpha$. Thus, the origin $\omega_{0}$
of the plasma branch always coincides with the plasma frequency
Takada1997plasm

$\omega_{0}=\omega_{p}$ (10)

This expected results shows that superconductivity does not affect the
departure of the plasma branch. As we now explain, the situation is quite
different for the low-$q$ dispersion and lifetime.

To extract the curvature $\alpha$ of the plasma branch, we expand
$\widetilde{M}_{33}$ to subleading order in $q$, and consider the 2 other
matrix elements

$\displaystyle M_{11}$ $\displaystyle=$
$\displaystyle\sum_{\textbf{k}}\left[\frac{m_{11}^{+}(1-f_{+}-f_{-})}{z^{2}-(\epsilon_{+}+\epsilon_{-})^{2}}-\frac{m_{11}^{-}(f_{+}-f_{-})}{z^{2}-(\epsilon_{+}-\epsilon_{-})^{2}}\right]-\frac{V}{g}$
$\displaystyle\widetilde{M}_{13}$ $\displaystyle=$
$\displaystyle\sum_{\textbf{k}}\left[\frac{\widetilde{m}_{13}^{+}(1-f_{+}-f_{-})}{z^{2}-(\epsilon_{+}+\epsilon_{-})^{2}}-\frac{\widetilde{m}_{13}^{-}(f_{+}-f_{-})}{z^{2}-(\epsilon_{+}-\epsilon_{-})^{2}}\right]$
(11)

with
$m_{11}^{\pm}=(\epsilon_{+}\pm\epsilon_{-})(\epsilon_{+}\epsilon_{-}\pm\xi_{+}\xi_{-}\pm\Delta^{2})/{2\epsilon_{+}\epsilon_{-}}$
and
$\widetilde{m}_{13}^{\pm}=\pm(\epsilon_{+}\pm\epsilon_{-})(2\xi_{+}\xi_{-}+2\Delta^{2}-\epsilon_{+}^{2}-\epsilon_{-}^{2})\Delta/{2\epsilon_{+}\epsilon_{-}}$.
At zero temperature, this yields the fully analytic expression of $\alpha$:

$\alpha=\frac{6\epsilon_{F}}{5\omega_{p}}-\frac{32\epsilon_{F}}{15\omega_{p}}\frac{\Delta^{2}\text{Arcsin}\left({\omega_{p}}/{2\Delta}\right)}{\omega_{p}\sqrt{4\Delta^{2}-\omega_{p}^{2}}}$
(12)

which is shown as a black curve on Fig. 1. This expression remains valid when
$\omega_{p}>2\Delta$ and the plasma branch is embedded in the pair-breaking
continuum. In this case, one should use
$\omega_{p}\to\omega_{p}+\textrm{i}0^{+}$ and $\text{Im}\,\alpha<0$ describes
the nonzero damping rate of plasmons. We note that the repulsion of the pair-
breaking threshold leads to a squareroot divergence of $\text{Re}\,\alpha$ and
$\text{Im}\,\alpha$ when approaching the pair-breaking threshold respectively
from below and above. This opens an interval
$\omega_{p}\in[1.696\Delta,2\Delta[$ where plasmons have an anomalous negative
dispersion at the origin ($\text{Re}\alpha<0$). In the conventional limit
$\omega_{p}\gg 2\Delta$, the second term in (12) becomes negligible, such that
we recover the normal plasmon dispersion $\alpha\to 6\epsilon_{F}/5\omega_{p}$
FetterWalecka . In the opposite « quasiphononic » limit $\omega_{p}\ll
2\Delta$, which corresponds to the experimental situation of Refs.
DasSarma1995manip ; Tachiki1997 , rather than expanding for fixed $z$ as
prescribed by (7), one should expand for $q\to 0$ while keeping $z/v_{F}$
comparable to $q$ DasSarma1995 . This yields222Note that this is consistent
with the behavior of $\alpha$ in the limit $\omega_{p}/\Delta\to 0$.

$z_{\textbf{q}}\underset{\begin{subarray}{c}q\to 0\\\ cq/\omega_{p}\text{
fixed}\end{subarray}}{\longrightarrow}{\sqrt{\omega_{p}^{2}+c^{2}q^{2}}}$ (13)

where $c=v_{F}/\sqrt{3}$ is the speed-of-sound of the weakly-interacting
condensate of neutral fermions.

Figure 1:  Dispersion parameter $\text{Re\,}\alpha$ (multiplied by
$\Delta/\epsilon_{F}$ to have a finite weak-coupling limit), in function of
the plasma frequency at zero (black curve) and high temperature (red curve).
The normal dispersion $6\epsilon_{F}/5\omega_{p}$ is shown by the red dotted
curve. The value where $\alpha$ changes sign at $T=0$ is indicated by the
black dotted line. Inset: the damping parameter $\text{Im\,}\alpha$, which
becomes nonzero inside the pair-breaking continuum $[2\Delta,+\infty[$ (red
area).

At nonzero temperature, $\alpha$ depends on the dimensionless temperature
$\bar{T}=T/\Delta$ and plasma frequency $\bar{\omega}_{p}=\omega_{p}/\Delta$.
We find

$\alpha=\frac{\epsilon_{F}}{\Delta}\left[\frac{6\bar{\omega}_{p}}{5}\left(I_{3}+J_{0}-J_{2}\right)-\frac{8}{3\bar{\omega}_{p}}I_{1}\right]$
(14)

in terms of the dimensionless integrals
$I_{n}=\int_{0}^{+\infty}\textrm{d}\xi\frac{\text{th}({\epsilon/2\bar{T}})}{\epsilon^{n}({\bar{\omega}_{p}^{2}-4\epsilon^{2}})}$
and
$J_{n}=\frac{1}{2\bar{T}\bar{\omega}_{p}^{2}}\int_{0}^{+\infty}\frac{\textrm{d}\xi}{\epsilon^{n}\operatorname{ch}^{2}({\epsilon/2\bar{T}})}$
with $\epsilon=\sqrt{\xi^{2}+1}$. The red curve in Fig. 1 shows $\alpha$ in
the vicinity of the critical temperature $T/T_{c}=0.9989$ ($T/\Delta=10$). We
observe that $\alpha$ tends to its normal limit $6\epsilon_{F}/5\omega_{p}$
uniformly except in a neighborhood of size $\approx\Delta^{2}/T$ around the
pair-breaking threshold $2\Delta$. There, the divergence of the real and
imaginary part is preserved whenever $T<T_{c}$, showing that a regime of
anomalous plasmon dispersion subsists until the transition to the normal
phase. In usual situations, $\omega_{p}$ is fixed in units of the Fermi energy
$\epsilon_{F}$, but the ratio $\omega_{p}/\Delta(T)$ can still be adjusted by
varying the temperature. The negative plasma dispersion will thus eventually
occur when increasing the temperature provided $\omega_{p}<2\Delta(T=0)$. Near
$T_{c}$, we note that the plasma branch may also interact with the phononic
Carlson-Goldman excitations Goldman1973 ; Kobelkov1995 ; Takada1997
describing the motion of the superconducting electrons embedded in a majority
of normal carriers. A convincing description of this phenomenon requires going
beyond the collisionless regime of undamped fermionic quasiparticles
Volkov1997 , which is beyond the scope of this work.

One could be surprised than plasmons remain undamped ($\text{Im}\,\alpha=0$)
for $\omega_{p}<2\Delta$ despite the nonzero temperature, which provides a
decay channel through quasiparticle-quasihole excitations. In fact, to absorb
a plasmon (i.e. to satisfy the resonance condition
$\omega_{p}=\epsilon_{\textbf{q}+\textbf{k}/2}-\epsilon_{\textbf{q}-\textbf{k}/2}$)
quasiparticles need to have a wavenumber $k>2m\omega_{p}/q$. The plasmon
lifetime thus follows an activation law
$\text{Im}z_{q}\propto\textrm{e}^{-2m\omega_{p}^{2}/q^{2}T}$ which is
exponentially suppressed in the limit $\Delta/\epsilon_{F},T/\epsilon_{F}\to
0$ with $\omega_{p},q$ of order $\Delta,1/\xi$. Intrinsic plasmon damping at
$\omega_{p}<2\Delta$ is thus essentially a strong-coupling effect.

We conclude this section by computing the matrix residue $Z_{q}=\lim_{z\to
z_{q}}(z-z_{q})\chi(z,q)$, which quantifies the spectral weight of the plasma
resonance. Writing $\chi=-1-M^{-1}D$ and using
${\textrm{d}}(\text{det}M)/\textrm{d}z\overset{z\to\omega_{p}}{\underset{q\to
0}{\to}}M_{11}M_{22}\textrm{d}\widetilde{M}_{33}/z^{2}\textrm{d}z$ together
with $M_{11}=z^{2}M_{33}/4\Delta^{2}=-zM_{13}/2\Delta$ to leading order in
$q$, we obtain, in the phase-density sector:

$Z_{q}=\Delta\begin{pmatrix}\frac{\omega_{p}}{\rho
g}\frac{4m\Delta}{q^{2}}&1\\\ \frac{\omega_{p}^{2}}{\rho
g}\frac{2m}{q^{2}}&\omega_{p}/2\Delta\end{pmatrix}+O(q^{2})$ (15)

Note that the phase and density excitation channels (respectively first and
second line of $Z_{q}$) dominate respectively in the limits $\omega_{p}\to 0$
and $\omega_{p}\to+\infty$.

Dispersion minimum and resonance splitting at non-vanishing wavenumber:
Outside the limit $q\xi\ll 1$, we study the dispersion of plasmons by
numerically evaluating the $M$ matrix. We first characterize on Fig. 2 the
dispersion minimum of the plasma branch. For $\omega_{p}>1.696$, it is reached
at a nonzero wavenumber $q_{\rm min}$ (blue curve in Fig. 2), such that the
band gap of the plasma branch (black curve) is strictly lower than
$\omega_{p}$. As visible on Fig. 2, this undamped plasma branch at
$\omega_{q}<2\Delta$ persists even when $\omega_{p}>2\Delta$. This is a first
sign of the splitting of the plasma resonance. However, in the normal limit
$\omega_{p}/\Delta\to+\infty$ the undamped branch tends uniformly to $2\Delta$
with a vanishingly small spectral weight.

Figure 2:  The dispersion minimum $\omega_{q_{\rm min}}$ and the wavenumber
$q_{\rm min}$ at which it is reached in function of the plasma frequency. For
$\omega_{p}<1.696$, $q_{\rm min}$ is identically 0 and $\omega_{q_{\rm min}}$
coincides with $\omega_{p}$ (oblique dotted line). In the limit
$\omega_{p}\to+\infty$, $q_{\rm min}$ diverges linearly and $\omega_{q_{\rm
min}}$ tends to $2\Delta$.

Since we expect the plasma branch to eventually enter the pair-breaking
continuum, the characterization of the resonance at finite $q$ requires a
numerical exploration of the analytically continued matrix $M$. At $T=0$, the
pair-breaking threshold $2\Delta$ and the second branching point DasSarma1995

$\omega_{2}=\sqrt{4\Delta^{2}+\epsilon_{F}\frac{q^{2}}{2m}}$ (16)

divide the real axis in three analyticity windows (I, II and III, see the
inset of Fig. 3), each supporting a separate complex root ($\omega_{q}^{\rm
I}$, $z_{q}^{\rm II}$, $z_{q}^{\rm III}$ respectively) of Eq. (5). This
suggests a splitting of the plasma resonance into 3 peaks.

While the quadratic Eq. (7) give the low-$q$ dispersion of $\omega_{q}^{\rm
I}$ and $z_{q}^{\rm III}$ (respectively for $\omega_{p}<2\Delta$ and
$\omega_{p}>2\Delta$), the pole of window II starts from $2\Delta$ and departs
following a non-integer power-law:

$z_{q}^{\rm II}=2\Delta-\frac{\textrm{i}\pm
1}{\sqrt{\Delta}}\sqrt{\frac{8}{3\pi^{2}}\left|1-\frac{4\Delta^{2}}{\omega_{p}^{2}}\right|}\left(\frac{k_{F}q}{2m}\right)^{3/2}+O\left(q^{7/4}\right)$
(17)

where the sign of the real part of $z_{q}^{\rm II}-2\Delta$ is negative if
$\omega_{p}<2\Delta$ and positive otherwise (in either case $z_{q}^{\rm II}$
remains outside the natural interval $[2\Delta,\omega_{2}]$ of window II).
Remarkably, when $\omega_{p}=2\Delta$ the quadratic law reemerges $z_{q}^{\rm
II}=2\Delta-(0.0184+0.9953\textrm{i})\frac{\epsilon_{F}}{\Delta}\frac{q^{2}}{2m}+O(q^{4})$.
Those results are obtained by expanding at low $q$ as prescribed by Eq. (10)
in higgs .

On Fig. 3, we show the dispersion relation of these solutions for
$\omega_{p}=1.9\Delta$. In this case $\omega_{q}^{\rm I}$ supports the main
plasma branch departing in $\omega_{p}$ (while for $\omega_{p}>2\Delta$ the
main branch would be supported by $z_{q}^{\rm III}$), $z_{q}^{\rm II}$ belongs
to an indirect region of the analytic continuation (specifically
$\text{Re}z_{q}^{\rm II}<2\Delta$) and $z_{q}^{\rm III}$ follows rather
closely the angular point $\omega_{2}$. This subtle analytic structure is
reflected in frequency behavior of the density-density response function shown
on Fig. 4. Besides the Dirac peak below $2\Delta$, one (black curve) then two
(grey curves) broadened peaks appear inside the pair-breaking continuum
DasSarma1991  as $q$ is increased. In the time domain, this remarkable
frequency behavior corresponds to an evolution (after e.g. an electromagnetic
quench of the electronic density) where the system oscillated at several
frequencies: an exponentially decaying high-frequency component and a long-
lived low-frequency component.

Figure 3:  The eigenfrequency $\text{Re}z_{q}$ of the plasma branch in
function of the wave vector $q$ (in unit of the inverse pair radius
$\xi=k_{F}/2m\Delta$), with $\omega_{p}=1.9\Delta$. The angular points
$2\Delta$ and $\omega_{2}$ (Eq. (16)) are shown as dotted lines. The analytic
windows are shown in colors: white for $\omega<\omega_{1}$ (window I), blue
for $\omega_{1}<\omega<\omega_{2}$ (window II) and red for $\omega>\omega_{2}$
(window III). The solution of Eq. (6) in each window in shown as a solid line
in the corresponding color. The inset shows their schematic trajectories in
the complex plane after analytic continuation. In window II, the pair-breaking
mode (solution of $M_{22,\downarrow}=0$) is shown as a dashed line. The
dispersion minimum of the undamped solution below $2\Delta$ is shown by the
black dot. Figure 4:  Density-density response of the superconductor in
function of the excitation frequency $\omega$ at fixed plasma frequency
$\omega_{p}=1.9\Delta$ and excitation wave number $q=0.2/\xi$, $0.5\xi$ and
$1.0\xi$ (corresponding to the vertical dotted lines in Fig. 3). The angular
points $2\Delta$ and $\omega_{2}(q)$ are marked by vertical dotted lines.
Besides the Dirac peaks below $2\Delta$, broadened peaks are visible inside
the pair-breaking continuum. Thin dashed lines show the modulus response $\rho
g/\epsilon_{F}M_{22}(\omega+\textrm{i}0^{+})$.

Conclusion: We have described the low-$q$ quadratic dispersion of
superconducting plasmons in 3D, and the resonance splitting which occurs when
the eigenenergy nears the pair-breaking threshold. For a more realistic
description of plasmons in cuprates, our study should be extended to 2D
superconductors Millis2020 , with possibly Josephson interlayer couplings
Yamada1998 ; Wu2021 . Our work may also be applied to superfluids of ultracold
fermions Zwerger2012  where different kind of long-range interactions can be
engineered with dipolar atoms Ferlaino2018 .

###### Acknowledgements.

We acknowledge financial support from the Research Foundation-Flanders (FWO-
Vlaanderen) Grant No. G.0618.20.N, and from the research council of the
University of Antwerp.

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