Floquet Weyl semimetal phases in light-irradiated higher-order topological Dirac semimetals

2025-04-22 0 0 9.43MB 10 页 10玖币

Zi-Ming Wang,1, 2 Rui Wang,2, 3 Jin-Hua Sun,4, ∗Ting-Yong Chen,5, †and Dong-Hui Xu2, 3, ‡

1Department of Physics, Hubei University, Wuhan 430062, China

2Department of Physics and Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing 400044, China

3Center of Quantum Materials and Devices, Chongqing University, Chongqing 400044, China

4Department of Physics, Ningbo University, Ningbo 315211, China

5Shenzhen Institute for Quantum Science and Engineering,

Southern University of Science and Technology, Shenzhen 518055, China

(Dated: March 23, 2023)

Floquet engineering, the concept of tailoring a system by a periodic drive, is increasingly exploited to design

and manipulate topological phases of matter. In this work, we study periodically driven higher-order topological

Dirac semimetals associated with a k-dependent quantized quadrupole moment by applying circularly polarized

light. The undriven Dirac semimetals feature gapless higher-order hinge Fermi arc states which are the con-

sequence of the higher-order topology of the Dirac nodes. Floquet Weyl semimetal phases with hybrid-order

topology, characterized by both a k-dependent quantized quadrupole moment and a k-dependent Chern number,

emerge when illumining circularly polarized light. Such Floquet Weyl semimetals support both hinge Fermi arc

states and topological surface Fermi arc states. In addition, Floquet Weyl semimetals with tilted Weyl cones in

higher-order topological Dirac semimetals are also discussed. Considering numerous higher-order topological

Dirac semimetal materials were recently proposed, our ﬁndings can be testable soon.

Introduction. Understanding Dirac-like fermions has be-

come an imperative in modern condensed matter physics.

All across the research frontier, ranging from graphene to d-

wave high-temperature superconductors to topological insula-

tors and beyond, low-energy excitations in various electronic

systems can be well-described by the Dirac equation [1]. Of

particular interest are Dirac semimetals (DSMs) as they rep-

resent an unusual phase of quantum matter that hosts mass-

less Dirac fermions as quasiparticle excitations near bulk nodal

points. Graphene is a well-known two-dimensional (2D) DSM

protected by chiral (sublattice) symmetry [2], and stable three-

dimensional (3D) DSMs protected by crystalline symmetries

had been identiﬁed and realized in solid materials as well [3].

Due to the lack of bulk-boundary correspondence—the cor-

nerstone of topological phases of matter, the designation of

DSMs as a semimetallic topological phase was controver-

sial [4,5], whereas DSMs do serve as a parent phase for re-

alizing exotic topological states and topological phenomena.

For instance, breaking time-reversal symmetry (TRS) via mag-

netism in a DSM can result in the quantum anomalous Hall

state [6] or Weyl semimetal (WSM) states hosting massless

chiral fermions and surface Fermi arc states [7].

Floquet engineering is a versatile approach that uses time-

periodic driving of a quantum system to enable novel out-of-

equilibrium many-body quantum states [8–10]. Recent years

have witnessed intense eﬀorts toward exploiting Floquet en-

gineering to create topological phases in quantum materi-

als [11–50]. It is important to stress that circularly polarized

light (CPL) naturally breaks TRS, which provides an easy tun-

ing knob to induce dynamical topological phases such as Flo-

quet Chern insulators and Floquet WSMs in DSMs [16,17,

19–23,36,39,42,51–53]. Following the discovery of the

∗sunjinhua@nbu.edu.cn

†chenty@sustech.edu.cn

‡donghuixu@cqu.edu.cn

concept of higher-order topology that characterizes bound-

ary states with dimensions two or more lower than that of

the bulk system that accommodates them [54–60], there has

been a surge of interest in tailoring higher-order topological

phases by using Floquet engineering as well [61–78]. Mean-

while, numerous higher-order topological DSMs, which obey

the topological bulk-hinge correspondence and thus display

universal topological hinge Fermi arc states, have been pro-

posed [79–84]. The signature of hinge Fermi arc states was

recently observed in supercurrent oscillation experiments on

prototypical DSM material Cd3As2[85,86]. Cd3As2provides

a promising parent material for the realization of the higher-

order WSM [87–89] that supports hinge Fermi arc states in

addition to the usual surface Fermi arc states by using Floquet

engineering.

In this work, we explore tunable higher-order WSMs in

time-symmetric and P T -symmetric higher-order topological

DSMs under oﬀ-resonant CPL illumination. Without driving,

both types of higher-order topological DSMs have two bulk

Dirac nodes locating at the kz-axis and support gapless hinge

Fermi arc states. Meanwhile, the time-symmetric one has ad-

ditional closed surface Fermi rings. CPL drives each Dirac

node to split into a pair of Weyl nodes by symmetry breaking,

resulting in Floquet higher-order WSMs accommodating rich

topological boundary states. The coexistence of surface Fermi

arc and hinge Fermi arc states signals a hybrid-order topology

which can be captured by k-dependent quantized quadrupole

moment and Chern number. In addition, the surface Fermi

rings in the time-symmetric DSM are inherited in the Floquet

WSM. Moreover, we can achieve a type-II higher-order WSM

with overtilted Weyl cones by adjusting the incident direction

of CPL. Our proposal can be realized in DSM materials like

Cd3As2with current ultrafast experimental techniques.

DSM model and the Floquet theory. Undriven higher-order

topological DSMs are constructed based on a generic band in-

version DSM model on the cubic lattice. In reciprocal space,

arXiv:2210.01012v2 [cond-mat.mes-hall] 22 Mar 2023

the Hamiltonian matrix is

H(k) = 0(k) + λsin kxΓ1+λsin kyΓ2

+M(k)Γ3+G(k)Γ4,(1)

where the Γmatrices are Γ1=s3σ1,Γ2=s0σ2,Γ3=s0σ3,

Γ4=s1σ1, and Γ5=s2σ1, with sj=1,2,3and σj=1,2,3the

Pauli matrices labeling the spin and orbital degrees of free-

dom, respectively, and s0,σ0are 2×2identity matrices.

0(k) = t1(cos kz−cos K0

z) + t2(cos kx+ cos ky−2),

and M(k) = t(cos kx+ cos ky−2) + tz(cos kz−cos K0

z).

t1,2,t,tz, and λare the amplitudes of hoppings. The DSM

has two Dirac points locating at k0= (0,0,±K0

z).G(k)

is the coeﬃcient of Γ4term that gives birth to higher-order

topology in the present DSMs. Without the Γ4term, this

model describes an ordinary band-inversion DSM support-

ing helical surface Fermi arc states which are not topologi-

cally stable [4]. Equation (1) can describe the recently identi-

ﬁed higher-order DSM materials, including but not limited to

Cd3As2and KMgBi [80].

Treating kzas a parameter, then H(k)reduces to a 2D

Hamiltonian Hkz(kx, ky). The reduced Hamiltonian pos-

sesses higher-order topology which can be well characterized

by the quantized quadrupole moment. Furthermore, we can

use a kz-dependent quantized quadrupole moment Qxy(kz)

to capture the higher-order topology of the DSMs. Qxy in real

space [90,91] is

Qxy =1

2πIm[log hˆ

Uxyi],ˆ

Uxy =ei2πPriˆqxy (ri),(2)

where ˆqxy(ri) = xy

LxLyˆn(ri)is the quadrupole moment den-

sity per unit cell at site ri, and Lxand Lyare the length of the

system in the xand ydirections, respectively.

CPL is described by a time-periodic gauge ﬁeld

A(τ) = A(τ+T)with period T=2π

ωand the

frequency of light ω. In speciﬁc, the gauge ﬁeld

A=A(0, η sin ωτ, cos ωτ ), A(cos ωτ, 0, η sin ωτ ), and

A(ηsin ωτ, cos ωτ, 0), where η=±1labeling the hand-

edness, describe CPL propagating along the x,yand z

directions, respectively. In the main text, we mainly focus on

the case of CPL propagating along the z-direction, the case of

CPL propagating along the x-direction is also studied in the

Supplemental Material [92]. Electrons moving on a lattice

couple to the electromagnetic gauge ﬁeld via the Peierls substi-

tution: ˜

t→˜

texp[−iRrk

A(τ)·dr], where rjis the coordinate

of lattice site j. Thereby, in the presence of CPL, the DSM

Hamiltonian becomes periodic in time H(τ) = H(τ+T). In

the following, we use the natural units e=~=c= 1. Thanks

to Floquet’s theorem, we can transfer the time-dependent

Hamiltonian problem to a time-independent one [93,94].

In speciﬁc, the time-dependent Schr¨

odinger equation has a

set of solution |Ψ(τ)i=e−iτ |Φ(τ)i, where denotes the

Floquet quasienergy, and |Φ(τ)i=|Φ(τ+T)iis dubbed

the Floquet state. Expanding the Floquet state in a Fourier

series |Φ(τ)i=Pne−inωτ |Φni, we arrive at an inﬁnite

dimensional eigenvalue equation in the extended Hilbert (or

Sambe) space

(Hn−m−mωδmn)|Φm

αi=α|Φm

αi,(3)

X Y -Z Z X

-2

-1

- -0.5 00.5

-0.5

0.5

-0.5 00.5

-0.5

0.5

- -0.5 00.5

-0.5

0.5

1 20

- -0.5 00.5

0.5

(a)

(e)

(c)

(b)

(f)

(d)

FIG. 1. Electronic structures and bulk topology of the time-

symmetric DSM. (a) Bulk band structure along high-symmetry points

in the 3D Brillouin zone. The red dots mark the bulk Dirac points.

(b) The surface band dispersion versus kzdirection for ky= 0. The

open boundary condition (OBC) is imposed along the x-direction.

The solid blue lines show the gapless surface Dirac states. (c) The

surface spectral function on the ky-kzplane with the semi-inﬁnite

boundary along the x-direction when E= 0. The red dots at ±π/2

are the projection of Dirac points, and the red circle marks the surface

Fermi rings from the surface Dirac states shown in (b). (d) The energy

spectrum versus kzwith the OBC along both the xand ydirections.

The solid red lines represent the topologically protected hinge Fermi

arc states. The Dirac points at kz=±π/2are gapped due to the

size eﬀect. (e) The local density of states (LDOS) of the hinge Fermi

arc states at kz= 0.1π. (f) The kz-dependent quantized quadrupole

moment Qxy . The two quantized plateaus of Qxy correspond to two

segments of degenerate hinge Fermi arc states. The parameters are

chosen as t1= 0.3,t2= 0.2,λ= 0.5,t= 1,tz= 0.8,K0

z=π/2,

and g=−0.4.

where Hn−m=1

TRT

0dτei(n−m)ωτ H(τ). Throughout, we

focus on the case in the high frequency limit, where the reso-

nant interband transitions are very unlikely. This case yields

an eﬀective static Floquet Hamiltonian [95,96]

Heﬀ =H0+X

l6=0

[H−l, Hl]

lω +O(ω−2).(4)

In our calculations, the maximum value of lis determined by

checking whether the results converge.

Light-irradiated time-symmetric higher-order topological

DSM. First, we specify G(k) = g(cos kx−cos ky) sin kz,

which breaks the fourfold rotation symmetry C4zbut preserves

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摘要：

FloquetWeylsemimetalphasesinlight-irradiatedhigher-ordertopologicalDiracsemimetalsZi-MingWang,1,2RuiWang,2,3Jin-HuaSun,4,Ting-YongChen,5,yandDong-HuiXu2,3,z1DepartmentofPhysics,HubeiUniversity,Wuhan430062,China2DepartmentofPhysicsandChongqingKeyLaboratoryforStronglyCoupledPhysics,ChongqingUniversit...

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Floquet Weyl semimetal phases in light-irradiated higher-order topological Dirac semimetals

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