Scattering theory of delicate topological insulators

2025-04-15 0 0 5.23MB 15 页 10玖币

侵权投诉

Penghao Zhu,1Jiho Noh,2Yingkai Liu,1and Taylor L. Hughes1

1Department of Physics and Institute for Condensed Matter Theory,

University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA

2Department of Mechanical Science and Engineering,

University of Illinois at Urbana–Champaign, Urbana, IL 61801 USA

(Dated: October 7, 2022)

We study the scattering theory of delicate topological insulators (TIs), which are novel topological

phases beyond the paradigm of the tenfold way, topological quantum chemistry, and the symmetry

indicator method. We demonstrate that the phase of the reﬂection amplitude can probe the delicate

topology by capturing a characteristic feature of a delicate TI. This feature is the returning Thouless

pump, where an integer number of charges are pumped forward and backward in the ﬁrst and second

half of the adiabatic cycle respectively. As a byproduct of our analysis we show that requiring

additional symmetries can stable the boundary states of a delicate TI beyond the conventional

requirement of a sharply deﬁned surface. Furthermore, we propose a photonic crystal experiment

to implement a delicate TI and measure its reﬂection phase which reveals the delicate topology.

I. INTRODUCTION

Delicate topology has been introduced recently as a

ﬁne-grained classiﬁcation of phases [1,2] that were pre-

viously considered trivial in the tenfold way [3,4], topo-

logical quantum chemistry [5–7], and symmetry indica-

tor [8–19] classiﬁcation schemes. The characteristic fea-

ture of a class of delicate topological insulators (TIs) is

a 2π-quantized diﬀerence between two Berry phases de-

ﬁned over a pair of high symmetry lines in the Brillouin

zone (BZ), and it is delicate in the sense that this quan-

tization can be nulliﬁed by adding trivial bands to either

the occupied or unoccupied subspace.

As we review below, since nontrivial Chern numbers

are precluded in a delicate TI by assumption, the 2π-

quantized diﬀerence in the Berry phase does not indicate

an adiabatic pumping of charge [20], but instead gener-

ates a returning Thouless pump (RTP). Indeed, if we re-

gard the momentum in the direction perpendicular to the

high symmetry lines as an adiabatic parameter, an RTP

indicates that an integer number of charges are pumped

toward one direction by an integer number of unit cells in

the ﬁrst half of the cycle, and are pumped back to their

starting point in the second half of the cycle [1,2,21]. A

non-vanishing RTP in a delicate TI guarantees that gap-

less surface states will be localized on a sharp boundary

with no deformations/reconstructions, which is weaker

than the conventional TI bulk-boundary correspondence.

Such a sharp boundary is perhaps diﬃcult to achieve nat-

urally in solid-state materials, but could be engineered in

metamaterials such as photonic or acoustic crystals.

Besides the gapless surface states on a sharp bound-

ary and previous work by some of us on surface mag-

netism [22], few robust experimental observables of del-

icate TIs have been proposed. Here we plan to leverage

the scattering theory of topological phases to provide a

practical route to extract delicate topological data from

a reﬂection matrix [23–26]. Previous studies have com-

prehensively discussed the scattering theory for stable

topological phases protected by internal symmetries, i.e.,

phases in the tenfold way periodic table[24]. Recently,

the study of scattering theory has been extended to a

wider range of topological phases, e.g., 2D higher-order

TIs that have gapped edges but gapless corners [27,28].

However, the scattering theory of delicate TIs, which may

provide new insight and experimental observables for del-

icate topology, is still absent. To address this, here we

study the scattering theory of delicate TIs protected by

rotation/mirror symmetry and propose a photonic exper-

iment to measure our predicted reﬂection-phase observ-

able.

The remainder of the article is organized as follows. In

Sec. II, we give a brief review of delicate TIs. Then, in

Sec. III, we investigate the relationship between the num-

ber of pumped charges during an adiabatic process and

the reﬂection phase. We show that the pumped charges

that lead to an RTP in a delicate TI can be detected by

the phase of the reﬂection amplitude on a sharp bound-

ary. We also show, for the ﬁrst time, that if there are

extra symmetries on the high symmetry lines in the BZ,

then the boundary modes, and the nontrivial behavior

of the reﬂection phase, can remain robust even when the

requirement of a sharp boundary is relaxed. Finally, in

Sec. IV we propose a photonic experiment to implement

a delicate TI in a synthetic dimension where measure-

ments of the reﬂection phase are possible and can be used

to detect the bulk RTP and hence delicate topology. We

end our article with conclusions and remarks about open

questions for future research in Sec. V.

II. REVIEW OF DELICATE TOPOLOGY, RTP,

AND SURFACE STATES

Before discussing the scattering theory, we shall ﬁrst

review delicate topology and its characteristic RTP pro-

tected by rotation/mirror symmetry. For clarity, let us

ﬁrst set up our notation. We separate 3D momentum k

in the BZ into k⊥and kkthat are respectively perpen-

dicular or parallel to the direction of the symmetry axis

arXiv:2210.02459v1 [cond-mat.mes-hall] 5 Oct 2022

(i.e., the rotation axis in 3D and/or mirror axis in 2D).

We call the subspace spanned by k⊥at each given kkthe

reduced Brillouin zone (rBZ). Then, we consider tight-

binding Hamiltonians written in a basis {|φli(R)i} where

Rrepresents the unit cell coordinates, and i= 1,2, . . .

labels the orbitals with symmetry eigenvalues liin each

unit cell (i.e., eigenvalues of the rotation/mirror opera-

tor). We assume that all |φlii’s with diﬀerent liin one

unit cell have their center localized on the same symme-

try axis, with respect to which the symmetry eigenvalues

are deﬁned.

If a point in the rBZ remains invariant under a rota-

tion/mirror transformation up to a reciprocal lattice vec-

tor, we then call this point a high symmetry point and

denote it as Λ. At each high symmetry point, we deﬁne

the Berry phase of the Bloch bands in sector lalong the

direction of the symmetry axis as

γl(Λ) = Z2π

dkkX

ihul

n(kk,Λ)|∂kkul

n(kk,Λ)i,(1)

where ul

n(kk,Λ)is the n-th eigenband with ro-

tation/mirror eigenvalue lat k= (kk,Λ), and

ihul

n(kk,Λ)|∂kkul

n(kk,Λ)i ≡ Al(k) is the Berry connec-

tion, and nis summed over the whole Hilbert space, i.e.,

occupied and unoccupied states. The Berry phase γl(Λ)

has the physical meaning of polarization along the sym-

metry axis contributed by all states with momentum Λ

and symmetry eigenvalue l. We note that in this work

we always assume the Bloch Hamiltonian is periodic in

momentum space, i.e., H(k+G) = H(k) where Gis a

reciprocal lattice vector. For simplicity we also ignore the

internal structure of the unit cell, i.e., we assume all or-

bitals in one unit cell have the same position coordinates

within the unit cell, which we take to be the position ref-

erence point. More detailed discussions which relax this

assumption can be found in Ref. 2.

With γl(Λ) deﬁned in Eq. (1), let us now discuss the

2π-quantized diﬀerence between γlat a pair of high sym-

metry points in the rBZ. We denote the set of sym-

metry eigenvalues of all occupied (unoccupied) bands

at Λas lv(Λ) (lc(Λ)). If the intersection of lv(Λ)

and lc(Λ) is empty, i.e., the mutually disjoint condition

lv(Λ)∩lc(Λ) = ∅is satisﬁed, then: (i) the Berry phase

γv(Λ) of all occupied bands at a high symmetry point

Λcan be expressed as γv(Λ) = Pl∈lvγl(Λ), and (ii)

γl(Λ)≡2πn with nan integer for any l. Point (i) is

obviously true, so let us now explain point (ii). At a high

symmetry point Λin the rBZ, the Bloch Hamiltonian

is block-diagonal. Each of its blocks, Hl(kk,Λ), can be

viewed as a 1D tight-binding Hamiltonian in the symme-

try sector labeled by l, i.e., the Hilbert space spanned by

all |φlii’s with li=l. The Berry phase of all Bloch bands

(i.e., both occupied and unoccupied Bloch bands) of such

a 1D tight-binding model is always an integer multiple of

2π, i.e., 2πn(Λ) where n(Λ) is the integer multiple at

Λ. This is because the Wannier orbitals constructed by

inverse-Fourier transforming all Bloch bands are just the

basis orbitals centered at the reference point up to an

integer number of lattice constants.

After establishing points (i) and (ii), we can conclude

that if at all Λ’s, the mutually disjoint condition is sat-

isﬁed, then the diﬀerence between two Berry phases at

a pair of high symmetry points, i.e., γv(Λ1)−γv(Λ2),

should be quantized to 2πm with man integer. Given

that the polarization along the symmetry axis should

change continuously from Λ1to Λ2in the rBZ, the

2πm quantization of γv(Λ1)−γv(Λ2) necessarily indi-

cates an RTP if the system has vanishing Chern num-

bers, as will be the case for delicate topology1. Explicitly,

the Berry phase diﬀerence indicates that mcharges are

pumped toward one direction along the symmetry axis

as we move from Λ1to Λ2, and they are then pumped

back when we complete the other half the loop in the

rBZ from Λ2to Λ1. In the following, we say the sys-

tem has an RTP malong the path Λ1→Λ2→Λ1if

γv(Λ1)−γv(Λ2)=2πm. The RTP is protected as long

as the bulk energy gap, the rotation/mirror symmetry,

and the Hilbert space restriction lv(Λ)∩lc(Λ) = ∅are

preserved. This kind of topology is delicate in the sense

that adding extra trivial bands (in either the occupied or

unoccupied subspaces) that make the intersection of lv

and lcat any Λnon-empty, can destabilize the RTP.

If we apply Stokes theorem we ﬁnd

γv(Λ1)−γv(Λ2) = ZdkkZΛ2

Λ1

dk⊥Tr Ωv(k),(2)

where Ωv(k) is the Berry curvature in the occupied sub-

space, and the integral dk⊥is on a 1D path from Λ1

to Λ2.Similarly, we can also deﬁne the Berry phase

along the direction of the symmetry axis of all unoc-

cupied bands as γc(Λ) = Pl∈lcγl(Λ). We denote the

Berry curvature in the unoccupied subspace as Ωc(k).

For any Bloch Hamiltonian Tr Ωv(k) = −Tr Ωc(k), so

we can conclude that the RTP in the unoccupied sub-

space is opposite to that in occupied subspace, i.e.,

γv(Λ1)−γv(Λ2) = −(γc(Λ1)−γc(Λ2)).

We also recall that delicate TIs with nonzero RTP gen-

erate gapless boundary modes on a sharp boundary. A

sharp boundary means that the Hamiltonian of a system

with boundaries is almost the same as the corresponding

periodic Hamiltonian except that all hopping matrix el-

ements across the boundaries are turned oﬀ. Intuitively,

the bulk RTP implies that states with speciﬁc symmetry

eigenvalues protrude from the bulk at the boundary, and

these states must be “compensated” by surface states

with complementary symmetry eigenvalues to keep the

balance of states in diﬀerent symmetry sectors in each

layer. These boundary states are gapless on only a sharp

boundary because the sharpness of the boundary con-

strains the energy of the surface states and forces them

1Remember that to discuss the delicate topology in a system, the

nontrivial Chern number is precluded by assumption.

to cross the insulating bulk gap. This intuitive picture

will become more transparent in the following example.

To illustrate all the concepts and conclusions dis-

cussed above, let us take the C4-symmetric two-band

tight-binding model of the Hopf insulator constructed by

Moore, Ran, and Wen (MRW) [29]. The Hamiltonian

of the MRW model is constructed from the well-known

Hopf map:

z= (z1+iz2, z3+iz4)T,

d=z†σz, σ= (σx, σy, σz),

HMRW(k) = d·σ,

(3)

where σx, σy, σzare Pauli matrices, and

z1= sin kx, z2= sin ky, z3= sin kz,

z4=u−cos kx−cos ky−cos kz.(4)

The Hamiltonian HMRW(k) has a C4rotation sym-

metry along the z-axis with rotation operator C4z=

exp (iπσz/4). For u6=±1,±3, HMRW (k) is gapped

and always has lv= exp(iπ/4) and lc= exp(−iπ/4)

at both high symmetry points Γ (kx=ky= 0) and

M(kx=ky=π). This satisﬁes the mutually disjoint

Hilbert space restriction discussed above. We ﬁnd that

γv(Γ) and γv(M) always have a 2π-quantized diﬀerence

when 1 <|u|<3 where the bulk Hopf invariant is −12,

which indicates a non-vanishing RTP along M→Γ→M

as shown in Fig. 1(a).

Let us now focus on the MRW model with u= 1.5

where γv(Γ) −γv(M) = 2π, and γc(Γ) −γc(M) = −2π.

This RTP implies that in the bulk, the occupied (unoc-

cupied) states at Γ are one layer higher (lower) in the

z-direction than the occupied (unoccupied) states at M,

as illustrated by red (orange) lines in Fig. 1(b). Thus,

on the top surface near z=Nthere will be one extra oc-

cupied (unoccupied) state with l=lv(l=lc) protruding

from the bulk to the surface at Γ (M). If we count both

occupied and unoccupied states there should always be

states with l=lvand lcat both Γ and Min each layer

along the z-direction. Thus, the extra state (protruding

from the bulk) with l=lv(l=lc)atΓ(M) in the sur-

face layer should be “compensated” by a surface state

with l=lcat Γ (l=lvat M).

While the previous argument is generally true, the en-

ergies of the compensating surface states are not gener-

ically required to cross the energy gap, i.e., the states

localized on the surface could only have energies in the

bulk band regions. If we make a further assumption of a

sharp boundary then we can make more deﬁnitive state-

ments about surface state energies. The sharpness of the

boundary guarantees that each diagonal block Hl(Λ) of

2When |u|<1, the bulk Hopf number is 2 and γv(Γ)−γv(M) = 0.

However, there is a 2π-quantized diﬀerence between γv(Γ) and

γv(X) that is protected by the two-fold rotation symmetry. [2]

v/(2⇡)



rBZ



(a)

(b)

0.5

u=3.5

u=1.5

u=0.5

u=1.5

12N

N-1

…



(c)

0.5

<latexit sha1_base64="JvqtmKjOjSVklRLxk1NH64mmVWA=">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</latexit>

<latexit sha1_base64="NqZs9GdUqI7nTYh3jcwcOSA+A9I=">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</latexit>

<latexit sha1_base64="ZXxvlidlDw5wHoY9un4wfhl6vIg=">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</latexit>

<latexit sha1_base64="BD/37BD8T2mRoTT28kBfRxYJB+Q=">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</latexit>

<latexit sha1_base64="7koVPzmarykQGM2ASKToUhzCLXU=">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</latexit>

0 50 100 150 200

-4

-2

FIG. 1. The left panel of (a) shows the RTP (change

of γv/(2π)) along the dotted-line path shown in the right

panel. We include plots for the MRW model with u=

−3.5,−1.5,0,1.5. The right panel of (a) shows C4rotation

invariant points Γ and Min the rBZ. (b) Illustration showing

bulk states protruding into the surface (green (orange) color

for occupied (unoccupied) states). The red (blue) line rep-

resents surface states localized on the top (bottom) surface

with the compensating angular momentum at Γ and M. (c)

shows an exact diagonalization calculation of the gapless sur-

face states of the MRW model at u= 1.5 with 10 unit cells

along the open z-direction (with sharp boundary), and peri-

odic boundaries in x, y. The surface band localized on the top

(z= 10) and the bottom (z= 1) surface are highlighted by

red and blue colors, respectively.

the eﬀective 1D tight-binding Hamiltonian at Mor Γ

is a block Toeplitz matrix, i.e., the hopping matrix ele-

ments between unit cells at Rand R0depend on only

R−R0and satisfy Hl,RR0≡Hl,R−R0= (Hl,R0−R)†.

The spectrum theorem of block Toeplitz matrices tells us

that in the thermodynamic limit, the spectrum of Hl(Λ)

is bounded by the spectrum of its corresponding Bloch

Hamiltonian Hl(kz,Λ) (i.e., the corresponding Hamilto-

nian under periodic boundary conditions) [2]. Thus, the

surface state at Γ (M) in the l=lc(l=lv) sector must

lie in the bulk unoccupied (occupied) band of the energy

spectrum. This then guarantees that the surface bands

must cross the bulk gap as k⊥goes from Γ to Mshown in

Fig. 1(c). The same arguments can also be made for the

bottom surface. In summary, if we relax the sharpness

condition, the state-compensation argument is still valid,

but the surface states are no longer guaranteed to have

energies in the insulating gap unless additional symme-

tries are imposed as we show below.

III. SCATTERING THEORY FOR DELICATE

TOPOLOGY

A. Pumped charge and reﬂection phase

From the review in Sec. II, we see that the RTP over

a loop in the rBZ describes an adiabatic pump in an ef-

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ScatteringtheoryofdelicatetopologicalinsulatorsPenghaoZhu,1JihoNoh,2YingkaiLiu,1andTaylorL.Hughes11DepartmentofPhysicsandInstituteforCondensedMatterTheory,UniversityofIllinoisatUrbana-Champaign,Urbana,Illinois61801,USA2DepartmentofMechanicalScienceandEngineering,UniversityofIllinoisatUrbana{Champaig...

展开>> 收起<<

Scattering theory of delicate topological insulators.pdf

共15页,预览3页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Scattering theory of delicate topological insulators

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: