Hybrid-order topological odd-parity superconductors via Floquet engineering Hong Wu1 2and Jun-Hong An1 2 1Key Laboratory of Quantum Theory and Applications of MoE Lanzhou University Lanzhou 730000 China

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Hybrid-order topological odd-parity superconductors via Floquet engineering

Hong Wu 1, 2 and Jun-Hong An 1, 2, ∗

1Key Laboratory of Quantum Theory and Applications of MoE, Lanzhou University, Lanzhou 730000, China

2Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical

Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China

Having the potential for performing quantum computation, topological superconductors have

been generalized to the second-order case. The hybridization of diﬀerent orders of topological

superconductors is attractive because it facilitates the simultaneous utilization of their respective

advantages. However, previous studies found that they cannot coexist in one system due to the

constraint of symmetry. We propose a Floquet engineering scheme to generate two-dimensional

(2D) hybrid-order topological superconductors in an odd-parity superconductor system. Exotic

hybrid-order phases exhibiting the coexisting gapless chiral edge states and gapped Majorana corner

states not only in two diﬀerent quasienergy gaps but also in one single quasienergy gap are created

by periodic driving. The generalization of this scheme to the 3D system allows us to discover a

second-order Dirac superconductor featuring coexisting surface and hinge Majorana Fermi arcs.

Our results break a fundamental limitation on topological phases and open a feasible avenue to

realize topological superconductor phases without static analogs.

I. INTRODUCTION

As one of the central ﬁelds in modern physics, topo-

logical phases of matter not only enrich the paradigm

of condensed matter physics and stimulate the discov-

ery of many novel quantum materials, but also generate

various fascinating applications in quantum technologies

[1–5]. In the family of topological phases, the topologi-

cal superconductor has attracted wide attention. It suc-

cessfully simulates the mysterious Majorana fermion in a

condensed matter system [6]. It also may be a promising

application in realizing quantum computation due to its

non-Abelian statistics [7–9]. Recently, there is an intense

interest in extending the traditional topological phases to

higher-order ones [10–13]. Being parallel to the theoreti-

cal proposals [14–19] and experimental observations [20–

29] on second-order topological insulators, second-order

topological superconductors (SOTSCs) have been pro-

posed [30–42]. This opens another avenue toward topo-

logical quantum computation [43–46].

Explorations of physical systems supporting exotic

topological features and of eﬃcient ways to control these

features are not only in the mainstream of condensed

matter physics, but are also a demand of quantum tech-

nologies. Symmetry plays an important role in realizing

SOTSCs [30,32–42]. It is conventionally believed that

SOTSCs are achieved by applying an even-parity term

in an odd-parity superconductor to break the symme-

try [43,47–49]. Recently, a general approach to real-

ize SOTSCs in odd-parity superconductors was proposed

[50]. An interesting question is whether the ﬁrst-order

and SOTSCs can coexist in one system. First, this kind

of hybrid-order topological superconductor (HOTSC) fa-

cilitates the simultaneous utilization of the advantages

of both corner states and gapless chiral edge states in

∗anjhong@lzu.edu.cn

designing bifunctional devices [27]. Second, it enriches

the family of topological phases and leads to a diﬀer-

ent band theory. Nevertheless, according to the general

view, ﬁrst- and second-order topologies exist in diﬀer-

ent systems [48,51,52] due to their substantially diﬀer-

ent features in the energy spectrum at the boundaries

[48,51–54]. Therefore, it seems that a HOTSC is im-

possible to realize in a system under the framework of

traditional topological energy-band theory. On the other

hand, coherent control via periodic driving, referred to

as Floquet engineering, has become a versatile tool in

artiﬁcially creating novel topological phases in systems

of ultracold atoms [55,56], photonics [57,58], super-

conductor qubits [59], and graphene [60]. Many exotic

phases absent in static systems have been controllably

generated by periodic driving [61–74]. One of the in-

teresting ﬁndings is that periodic driving can cause dif-

ferent high-symmetry points to have diﬀerent topological

charges [75], which lays the foundation to realize diﬀerent

types of topological phases [76]. Is it possible to realize

HOTSCs by periodic driving?

Addressing these questions, we propose a general

scheme to create a two-dimensional (2D) HOTSC in a

two-band odd-parity superconductor system by Floquet

engineering. A complete topological characterization to

such a Floquet quasienergy band structure is established.

We discover two kinds of HOTSCs, where in one of

them the coexisting gapless chiral edge states and the

gapped Majorana corner states reside in two diﬀerent

quasienergy gaps, and in the other they reside in one com-

mon gap. The generalization to the three-dimensional

(3D) odd-parity superconductor system reveals another

exotic phase, i.e., a second-order Dirac superconductor,

which features coexisting surface and hinge Majorana

Fermi arcs. Breaking the conventional constraint on re-

alizing hybrid-order topology, our result reveals Floquet

engineering as a useful way to explore diﬀerent topologi-

cal phases.

arXiv:2210.12439v2 [cond-mat.supr-con] 16 Jun 2023

II. STATIC SYSTEM

Conventional studies suggested that superconductors

with appropriate mixed-parity pairings are candidates for

SOTSCs [47,48]. Reference [50] presented a scheme to re-

alize SOTSCs in purely odd-parity pairing superconduc-

tors. We use this static system as the starting point of our

Floquet engineering. We consider a p-wave superconduc-

tor whose Hamiltonian reads ˆ

H1=1

2Pkˆ

Ψ†

kH1(k)ˆ

Ψk,

with ˆ

Ψ†

k= (ˆc†

k,ˆc−k) and

H1(k) = ϵ(k) ∆(k)

∆∗(k)−ϵ(k),(1)

where ϵ(k) is the kinetic energy of the normal state

and ∆(k) is the pairing potential of the superconductor.

Equation (1) has spatial inversion symmetry P=σz.

We focus on the inversion symmetric normal state ϵ(k)

and the odd-parity pairings satisfying ∆(k) = −∆(−k).

Via a Hopf map [77], a SOTSC model which is trivial in

the ﬁrst-order topology in this odd-parity superconduc-

tor is constructed as ∆(k) = 2(cos kx+λx+icos ky+

iλy)(sin kx−isin ky) and ϵ(k) = Pj=x,y[(cos kj+λj)2−

sin2kj]. The SOTSC is formed when |λx,y|<1,

where four gapped zero-mode Majorana corner states are

formed [see Figs. 1(a) and 1(b)]. Diﬀerent from conven-

tional topological phases, this SOTSC is not caused by

the closing and reopening of the bulk-band gap. The en-

ergy spectra under the x-direction open boundary con-

dition in Figs. 1(b)-1(d) indicate that, when the param-

eter runs from the SOTSC to the topologically trivial

regimes, the bulk-band gap is persistently open, how-

ever, the boundary-state gap occurs a closing and re-

opening. Therefore, we cannot ﬁnd a topological invari-

ant of the bulk bands to characterize the SOTSC. This

type of SOTSC is referred to as the boundary obstructed

topological phase [54]. A counterexample arises when

λx=λy, where the system has a mirror-rotation sym-

metry σyH1(kx, ky)σy=−H1(ky, kx). Its topology can

be described by the mirror-graded winding number de-

ﬁned in the bulk bands as W=i

2πR2π

0⟨u(k)|∂k|u(k)⟩dk,

where |u(k)⟩is the eigenstate of H1(k), along the high-

symmetry line kx=ky≡k[see Fig. 1(a)].

III. FLOQUET ENGINEERING

To generate HOTSC, we apply a periodic driving

H(k, t) = H1(k) + H2(k)δ(t/T −n),(2)

where H2=mσz,Tis the driving period, and nis

an integer. The periodic system H(k, t) = H(k, t +

T) does not have an energy spectrum because its

energy is not conserved. According to the Floquet

theorem, the one-period evolution operator U(T) =

Texp[−iRT

0H(k, τ)dτ] deﬁnes Heﬀ(k)≡iT −1ln U(T)

whose eigenvalues are called quasienergies [75]. The

FIG. 1. (a) Phase diagram in the λx-λyplane. The red and

white lines mark the regimes with W= 1 and 0, respectively.

Energy spectra under the x-direction open boundary condi-

tion when (b) λy= 0.8, (c) 1.0, and (d) 1.2. The inset in (a)-

open boundary condition. We use λx= 0.3 and lattice sizes

Lx=Ly= 50.

topological phases of our system are deﬁned in such a

quasienergy spectrum. Applying the Floquet theorem to

Eq. (2), we have Heﬀ(k) = iT −1ln[e−iH2(k)Te−iH1(k)T]

[78]. Diﬀerent from the static case, our topological phase

transition occurs not only at zero quasienergy but also

at π/T due to the periodicity of the quasienergies, which

may cause the Floquet anomaly in pure-order topological

phases [79].

We ﬁrst focus on Floquet engineering when λx=λy≡

λ, where the phases are characterized by the bulk bands.

First, we can derive from Heﬀ(k) that the phase transi-

tion occurs when the parameters satisfy either

2(1 −λ2)−m=nπ/T (3)

or 2[1 + λ2+λ(eiαx+eiαy)] + m=nαx,αyπ/T (4)

at the quasienergy zero (π/T ) for even (odd) nand

nαx,αy, with αx,y = 0 or π, (see Appendix A). Giv-

ing the phase boundaries of our system, Eqs. (3) and

(4) oﬀer us suﬃcient space to artiﬁcially synthesize ex-

otic topological phases absent in the static system (1)

by periodic driving. Second, we can establish a com-

plete topological characterization on Heﬀ(k) to the rich

emergent topological phases in our periodic system. Be-

sides the intrinsic SOTSC in the static case, our peri-

odic driving also induces a ﬁrst-order Chern supercon-

ductor phase. We have to ﬁnd appropriate topological

invariants to characterize the ﬁrst- and second-order su-

perconductor phases occurring at both zero and π/T

quasienergies. The overall topology of the ﬁrst- and

second-order phases at the α/T quasienergy, with α= 0

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Hybrid-ordertopologicalodd-paritysuperconductorsviaFloquetengineeringHongWu1,2andJun-HongAn1,2,∗1KeyLaboratoryofQuantumTheoryandApplicationsofMoE,LanzhouUniversity,Lanzhou730000,China2LanzhouCenterforTheoreticalPhysics,KeyLaboratoryofTheoreticalPhysicsofGansuProvince,LanzhouUniversity,Lanzhou730000,...

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Hybrid-order topological odd-parity superconductors via Floquet engineering Hong Wu1 2and Jun-Hong An1 2 1Key Laboratory of Quantum Theory and Applications of MoE Lanzhou University Lanzhou 730000 China

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