1  

Supporting Information Appendix

Photonic topological insulator with broken time-reversal symmetry, by Cheng He et al.

Part A: Effect Constitutive Relation of Piezoelectric (PE) and Piezomagnetic (PM) Superlattice

The coupling effect between the electromagnetic (EM) waves and lattice vibrations of PE and PM

superlattice can be described by the classical PE and PM equations as follows:

0

0

( ) ( )

( ) ( , 1,2,3; , 1,2,...6)

( )

I IJ J Ij j Ij j

i iJ J ij j

i iJ J ij j

T C S e z E m z H

D e z S E i j I J

B m z S H

, (1SA)

where

,

00000

00000

00000

000

000

000

44

44

44

331313

131112

131211

C

C

C

CCC

CCC

CCC

CIJ

,

000000

00000

00000

14

14

e

e

eiJ and

000

00000

00000

333131

15

15

mmm

m

m

miJ.

In Eq.(1SA), TI, SJ, Ej (Hi), Di (Bi) and ij ( ij ) are the stress, strain, electric field (magnetic field),

electric displacement (magnetic flux density) and permittivity (permeability) respectively. The elastic

coefficient is iCC IJEIJ with the damping coefficient γ. The periodically modulated PE (PM)

coefficients are )()( zfeze iJiJ [ )()( zgmzm iJiJ ], where the modulation functions is f(z) = 0,1

[g(z) =1, 0] in the corresponding domains. i(j) =1,2,3 represents the x-, y- and z- axis respectively, I(J)

=1,2,3,4,5,6 is the abbreviation subscript of the tensor. It should be noticed that to realize pseudospin

degenerate meta-atom, the point groups of PE and PM materials are chosen to be 622 and 6mm (or 422

2  

and 4mm) with nonzero PE coefficient 14e and PM coefficient 15m , respectively. The effective

parameters of the PE and PM superlattice are summarized in Tab. 1. Note that for simplicity,

impedance matching condition (average ) is assumed. The values for density, elastic coefficient

and PM coefficient are based on the BaTiO3-CoFe2O4 superlattice (1, 2). Considering the fact that the

PE coefficient of point group 422 (or 622) is not as large as 6mm BaTiO3 ( 211.6 C m ),

214 1.512e C m is assumed.

Density Elastic coefficient Permittivity permeability PE/PM coefficient

PE 35700 Kg m 33 162C Gpa

44 43C Gpa 11 22.9

33 24.7 11 9

33 10 2

14 1.512e C m

PM 35000 Kg m 33 269.5C Gpa

44 45.3C Gpa 11 9

33 10 11 22.9

33 24.7 1 1

15 550m N A m

Effective 35350 Kg m 33 215.75C Gpa

44 44.15C Gpa _______ _______ _______

Tab. 1 Parameters of PE and PM superlattice. Impedance matching condition is

assumed. The parameters of density, elastic coefficient and PM coefficient are based on

the BaTiO3-CoFe2O4 superlattice (1, 2). PE coefficient 214 1.512e C m is used.

Meanwhile, the Newton’s equation, which describes the lattice vibrations, could be

expressed as

2

2

Ji i J

iJ J i

u S

T ut

, (2SA)

where

000

000

000

xyz

xzy

yzx

iJ

.

In Eq. (2SA), ui is the displacement or saying the generated acoustic fields in superlattice, and ρ is the

material density. The divergence operator is TiJJi .

3  

By using the divergence operator on both side and substitution of Eq. (2SA) into Eq. (1SA), and

replacing SJ by ui, the dynamic equations of the acoustic waves are obtained as:

.)(

,)(

,)()(

0

0

2

2

kikkJkiJi

kikkJkiJi

kJkiJkJkiJkJkKJiKi

HuzgmB

EuzfeD

HzgmEzfeuCut

(3SA)

Here, we can omit the differential coefficients in xy-plane as no modulation. Eq. (3SA) reveals that the

vibrations in the superlattice under EM incidence could be related to the typical forced vibration mode,

where the electric (magnetic) fields Ek (Hk) represent the external force and ρ denotes the mass density.

The electric and magnetic fields of incident EM waves can excite phonons due to vibration of

superlattice via PE and PM effect, and such excited phonons will eventually be couple to incident EM

wave to form phononic polaritons (3-6).

Here, we assume the contrast ratio of PE and PM superlattice is 1:1. Then we expand the

modulation function in Eq. (3SA) through the Fourier transformation:

dnzdnen

niegzg

dnznden

niefzf

ziGn

ziGnn

ziGn

ziGnn

nn

nn

)22()12(,)cos1(

2

1

2

1)(

)12(2,)cos1(

2

1

2

1)(

0

0

, （4SA）

where )(2

Nnd

nnG n

, n is the order of the reciprocal vectors ranging from -∞ to +∞.

Considering the periodicity, the electric, magnetic and acoustic fields could be written in the forms of

Bloch wave:

)( tzkykxki

jzyxeEE

, ( )x y zi k x k y k z t

jH H e , )(

)(tzqi

iqzequu

. (5SA)

Here we expand the waves into Bloch modes by the Fourier transformation, which reveals that there

needs enough number of periods along the x- or y- axis. Generally, the number of periods only has the

influence on the impedance mismatching and density of output power, little on frequency shift. To

realize the coupling between the incident EM wave and the superlattice vibration, the

4  

quasi-phase-match condition needs to be satisfied, which is nGkq

. As the EM wavelength is

much larger than the period of superlattice, the long-wavelength approximation of 0 kGn

is

applicable, which finally leads to the quasi-phase-match condition of nGq

.

With nGq

and the substitution of Eq. (4SA) and (5SA) into Eq. (3SA), we will get:

,)()( 22kjknnkjknnnkjknjk HZgiGEYfiGGuXG (6SA)

where

44 14

44 14

33

0 0 0 0

0 0 , 0 0 ,

0 0 0 0 0

jk jk

C e

X C Y e

C

and 15

15

0 0

0 0

0 0 0

jk

m

Z m

.

With known frequency and the direction of incident EM waves, we could obtain )( ni Gu

. Then, we can

get the field distributions of u from Eq. (6SA). Furthermore, the resonant frequencies could also be

calculated from Eq. (6SA).

By the substitution of )( nGu

into Eq. (3SA), the constitutive relation can be described as:

.))(

())(

()(

,))(

())(

()(

22

2

022

2

22

2

22

2

0

k

jknjk

jkiknnnikk

jknjk

jkiknnni

k

jknjk

jkiknnnk

jknjk

jkiknnniki

HXG

ZZgzgGE

XG

YZfzgGzB

HXG

ZYgzfGE

XG

YYfzfGzD

(7SA)

Due to the continuous condition at boundary ( PMPEPMPE HHEE 2,12,12,12,1 , , PMPEPMPE BBDD 3333 , ), the

periodical average electric displacement and magnetic flux density can be described as

d

dzzDd

D2

02,12,1 )(

2

1 , d

dzzBd

B2

02,12,1 )(

2

1 , and the z component electric and magnetic fields are

d

dzzEd

E2

033 )(

2

1 , d

dzzHd

H2

0 33 )(2

1 . According to long wave-length approximation, such PE

and PM superlattices can be treated as a kind of Tellegen metamaterials, whose magneto-electric

coupling breaks the Tb symmetry. Finally, the effective constitutive relation is

0

0

ˆˆ

ˆ ˆ

D E H

B E H

, (8SA)

where

5  

,)2(2

,2

,2

ˆ

,2

,2

,2

ˆ

3333

23323333

0

21511111

0

21511111

3333

3333

0

21411111

0

21411111

PMPE

PMPEPMPEPMPE

PMPE

PMPEPMPEPMPE

mAmAmAdiag

eAeAdiag

,

000

00

00

ˆˆ15141

15141

meA

meAT

./,/,)(

2

,/,/,)(

2

33332

3312222

44442

4412221

CCGid

A

CCGid

A

LLL

LL

TTT

TT

In Eq. (8SA), there are two types of polaritons corresponding to transverse vibration frequency T

(be coupled to xy-plane EM wave) and longitudinal vibration frequency L (be coupled to z-axis

EM wave). And, in lossless condition 0T and 0L , T ˆˆ only have real tensor elements

that indicate this superlattice is a kind of Tellegen metamaterials, whose magneto-electric effect breaks

the nonreciprocity. Flipping the polarization of PE or PM direction (head to head polarization) would

flip the magneto-electric coupling coefficient ̂ .

Here, we set the thickness of each layer of PE and PM material d=500 nm. Transverse and

longitudinal vibration frequency are GHzT 05.18 and GHzL 901.39 , respectively. Focus on

the frequency region near T , we plot effective permittivity, permeability and magneto-electric

parameter in Fig. 1SA, where a typical loss T 0.001 is considered. In xy-plane ( yyxx and

yyxx ), permittivity, permeability and magneto-electric coupling exhibit a resonance at the

frequency of T . Thus, the loss near such resonance frequency is very large. The z components of

permittivity and permeability ( zzzz ) are nondissipative and nondispersive because the

longitudinal polariton frequency L is far away from transverse one T .

 

F

ma

eigen

wher

The

Simp

param

zz

Figure 1SA: E

agneto-electric

With nonreci

n equations ca

re x(

//

parameters in

ple without

meters at

1 zz

Effect paramet

c coupling in x

t

iprocal constit

an be describe

xx

xyxxxx

)2 ,

n Eq. (9SA) ar

losing genera

the freque

2.14 to carry

ters of PE and

xy-plane. d, p

the real and im

tutive relation

ed as,

(

(

20

20

zz

zz

k

k

xx

(//

re shown in F

ality, we cho

ncy of 1.

y out our next

6 

d PM superlatt

permeability al

maginary part

n in Eq. (8SA

)1

)1

//

//

xx

xyxxx

)2

,

Fig. 2SA, whe

oose nondiss

T.076 ):

t simulations.

tice. a, permit

long z-axis. O

s, respectively

A) and Maxwe

z

z

H

E

and (

ere //

sipative and

3.0

tivity, b, perm

Open and solid

y.

ll equations, a

.z

z

E

H

,

)2xyxxxx

xy

// and the lo

nondispersive

, //

meability and c

d circles repre

a set of orthog

(

).

oss can be om

e parameters

5.2//

c,

sent

gonal

(9SA)

mitted.

(the

and

 

Fig

rect

Acco

to th

wher

O

insul

Gene

break

a go

we

gure 2SA: Dis

tangular frequ

ording to Eq.

he following se

re k 200 H

On the other

lator”. It is ne

erally, optical

king systems.

od candidate

plot the tran

spersive a, uency area aro

solid circles

(9SA), a set o

et of equation

H

xzz //

1

hand, the th

ecessary for u

l isolators ar

. Our design o

to construct L

nsmission in

// and c, /ound frequencs represent the

of orthogonal

s,

0

RCP

LCP

H

yx //

1

hought of “iso

us to explain t

e a class of

of metamateri

LCP/RCP pol

Figs. 3SA.

7 

// according

cy of interest ae real and ima

eigen equatio

0 0

10 i

H

HH

y//

1 and H

olator” was o

the difference

optical devic

ial breaks tim

larization filte

Such metam

to Eq. (9SA),

are shown in baginary parts, r

ons for LCP an

0

10

RC

LC

i

H

yx 1H

one of potent

e in details be

ces based on

e-reversal sym

er or isolator [

aterial breaki

in which zoo

b and d, respecrespectively.

nd RCP states

0

CP

CP ,

xy .

tial applicatio

etween these t

nonreciproca

mmetry, which

[see Fig. 2C in

ing time-reve

om-in view of

ctively. Open

s can be simp

(1

on of “topolo

two terminolo

al or time-rev

h might be us

in main text].

ersal symmet

f the

and

plified

0SA)

ogical

ogies.

versal

sed as

Here

try is

 

origi

with

time

I

comp

part

the f

form

an ex

this p

wher

I

cont

spec

Usin

inated from th

hout external

-reversal sym

It should be n

plex Hermitia

doesn’t mean

fields in a two

m to determine

xample, the im

point in detail

re 0 x H

It seems that it

ains operators

cific, with Pla

ng Bloch wave

he real magne

magnetic bia

mmetry media

Figure

noticed that su

an system due

n non-Hermiti

o-dimension sy

e whether it is

maginary eige

ls as follows. T

x y y H

t is a non-Her

s ,x y . For

ane-Wave-Exp

e expansion of

eto-electric cou

as. But, those

(7-11) cannot

e 3SA. LCP/R

uch model is

to the Hermit

ian because w

ystem [notice

Hermitian or

en equations a

Taking the for

0H iH

1 x y H

rmitian Hamilt

r this reason,

pansion metho

f periodic para

8 

upling, which

e previous re

t really realize

RCP isolator tr

a close syste

tian constituti

we use a reduc

the term i(∂xκ

r not. Taking m

are also Herm

rm of lower p

1 RCPH

y x ,

tonian. But fo

it final form m

od, the Hamil

ameter (G is B

h can realize s

eported photo

e such nonrecip

ransmission sp

em and witho

ve relation. Th

ced set of equ

κ∂y - ∂/∂yκ∂/∂x

magneto-optic

mitian and has

art of Eq. 10S

20zz RCPk ,

or our problem

may be affect

ltonian can b

Bloch wave ve

some non-reci

nic topologic

procal phenom

pectra.

out considerati

he eigen equa

uations for the

x)]. One shoul

cal materials o

s real eigen va

SA for exampl

m, the imagina

ed by Bloch w

e transformed

ector):

iprocal pheno

cal insulators

mena.

tion of loss. I

ation has imag

e z componen

ld recall the m

or Faraday effe

alues. We des

le, we can obt

(1

ary part of Eq.

wave vector. T

d to be Herm

mena

with

t is a

ginary

nts of

matrix

fect as

scribe

ain

1SA)

. (R1)

To be

mitian.

9  

exp

exp , exp

i k rRCP k k k

G

G G

r u r e G i k G r

G iG r G iG r

(12SA)

Substituting Eq. (12SA) into Eq. (11SA), we get a complex form

0 0 0

20

0 0 0

+x x y y

k kG

x y y x

G G k G k G k G k GG G

i G G k G k G k G k G

(13SA)

The eigenvalue term has been simplified as 2 . With the nth order of G

and 0G

, we can make

up Eq. (13SA) as a n n matrix. Take the element of the mth row, the nth column for example

+

mn mn mn

m n m n m nx x y y

m n m n m nx y y x

H m

G G k G k G k G k G

i G G k G k G k G k G

Here, and are all real number

*G G and *G G

It is easy to prove that

*

*

= +

= +

=

mn

m n m n m nx x y y

n m n m n mx x y y

nm

G G k G k G k G k G

G G k G k G k G k G

and

*

* *

*

mn

m n m n m nx y y x

n m n m n mx y y x

n m n m n mx y y x

nm

i G G k G k G k G k G

i G G k G k G k G k G

i G G k G k G k G k G

Finally, we get the conclusion *mn nmH H , and vice versa for the upper part of Eq. 10SA. Therefore,

our Hamiltonian is a complex Hermitian system.

 

Part

Fig.

to Fi

negl

wave

Fig

T

RCP

the E

Fw

t B: Clockwis

1SB shows th

ig. 3a in main

ect the x com

e approximati

gure 1SB: SchG

The one-way

P wave around

EM propagatin

Figure 2SB: onwaves (lower p

resp

se Propagatio

he schematic

text. The x-ax

mponent of LC

ion. A little dis

ematic of oneGreen and red s

clockwise pr

d the whole bo

ng at the boun

ne-way clockwpanel) in U-shapectively. Colo

on for LCP w

of one-way p

xis is the prop

CP and RCP w

stortion is cau

e-way a, LCP stars represent

ropagation for

oundary are s

ndary.

wise and anti-ape model. Gr

or blue and red

10 

while Anti-cloc

propagation fo

pagation direc

waves to show

used by the inf

and b, RCP prnt the LCP and

r LCP wave

shown in Fig.

-clockwise proreen and red sd represent ne

ckwise for RC

or two pseudo

ction along the

w their oppos

fluence of bou

ropagation acd RCP inciden

and one-way

2SB. U-shap

opagations forstars representegative and po

CP

o-spins LCP a

e boundary. In

site circulation

undary (12).

cording to Figce, respective

anti-clockwis

e cladding lay

r LCP (upper pt the LCP and

ositive field va

and RCP acco

n the schemati

n due to the

g. 3a in main tely.

se propagatio

yer is used to

panel) and RC RCP incidencalues.

ording

ic, we

plane

text.

on for

limit

CP ce,

 

Part

As w

exten

frequ

likel

=0.0

oper

the l

Figu

a

para

Here

magn

(9SA

lowe

t C: Loss Con

we all know, o

nt. In our cas

uency range)

ly to possess h

001 for the su

rating frequen

oss at the ope

ure 3SC. (a) F

are considered

ameters as ε|| =

e, we consid

neto-electric p

A). As shown

er than 0.5 dB

ndition.

optical loss is

e, the materia

and compare

huge loss. In

uperlattice as

ncy determined

erating frequen

Field distribut

d by introduci

= 2.5-0.025i, μ

der 1% loss,

parameters as

in Fig. 3, the

B/lattice, whil

fundamental

al is piezoelec

ed with the co

fact, we have

indicated in

d by the requ

ncy can thus b

ion of LCP in

ing imaginary

μ|| = 2.5-0.025i

frequency

, introducing

ε|| = 2.5-0.025

robust transm

le the backwa

11 

and almost a

ctric and piezo

ommon metal

e analyzed ou

the supplem

uired dispersio

be omitted (se

n lossy case. (b

y part of permi

i and κ= 0.3-0

y is chosen at 0

g by imagina

5i, μ|| = 2.5-0.

mission still w

ard propagatio

ll the optical

omagnetic sup

l-based metam

ur system with

mentary section

on is far away

e Fig. 2SA an

b) LCP transm

ittivity, perme

0.003i in supp

0.6 (2πc/a).

ary part of

.025i and κ =

works well. Th

on is exponen

devices will b

perlattice (wor

materials this

h a loss term

n, and we ha

y from resonan

d Fig. 2SB in

mission versus

ability and ma

lementary Eq

permittivity,

0.3-0.003i in

he loss of forw

ntial decay. W

bear loss to ce

orking at giga

superlattice i

represented b

ave found tha

ance frequency

n supplementar

s position. 1%

agneto-electri

q. (9SA). Oper

permeability

supplementar

ward propagati

We have added

ertain

hertz

is not

by γ

at the

y and

ry).

loss

ic

rating

y and

ry Eq.

ion is

d this

 

figur

and t

edge

Part

Take

one u

The

L

xp

inter

betw

wher

repre

our s

defin

re in the suppl

the loss is app

e to confine th

t D: Theoretic

e LCP mode f

unit cell.

basis for TBA

and L

yp

ractions conta

ween Lx

p an

re 0

is the

esent the elec

system, the in

ned as:

lementary par

proximately 0

he field locatin

cal Model via

for instance. A

Fig.

A model can t

only contain

in both the on

nd Ly

p can

2p

iiV

d

frequency of

ctric(magnetic

nteraction inv

rt. Such metam

.33 dB/cm (20

ng in the air ga

a Tight-Bindi

As shown in F

1SD: Differen

thus be repres

ns the on-site

n-site coupling

n be calculated

0

2

33 xp

ds

ds E

f the resonanc

c) fields of

volving both th

12 

materials is de

0 GHz for exa

ap, getting low

ing Approxim

Fig. 1SD, the

nt three LCP s

sented as Ls

coupling, whi

g and nearest

d as (13)

*

2

33

x

x

p p

p

z E H

H d

ce in the abse

Lx

p and p

the on-site cou

esigned workin

ample). Furthe

wer loss [7].

mation (TBA)

ere are three p

states in a unit

L , Lx

p and

ich is caused

neighboring c

*

2

33

y x y

y

p p p

p

H E

ds E

ence of , an

Ly

p , respectiv

upling and ne

ng at giga her

ermore, we ca

) Approach.

possible states

t cell.

Ly

p . The in

by term,

coupling. The

2

33 ypH ,

nd x xp pE H

ely. Based on

earest neighbo

rtz frequency r

an also truncat

s for LCP mo

nteraction bet

while all the

on-site intera

(

and ypE H

n the symmet

or coupling ca

range,

te the

ode in

tween

other

action

1SD)

ypH

try of

an be

13  

, ,

, , , ,

, , , ,

, , , ,

, , , ,

, , , ,

, , , , 0,

L L Ls

L L L L Lx x y y p

L L L L Ls

L L L L Lx x y y p

L L L L Lx x y y p

L L L L Lx x sp

L L L Lx y

s m H s m

p m H p m p m H p m

s m H s m x s m H s m y t

p m H p m x p m H p m y t

p m H p m y p m H p m x t

s m H p m x s m H p m x t

s m H p m y s m H p m x

, (2SD)

where m , m x and m y represent the neighboring sites. The corresponding Hamiltonian on the

basis Ls , Lxp and L

yp is given by

2 sin 2 sin

2 sin

2 sin

s sp x sp y

L sp x px p

sp y p py

E k it k it k

H it k E k iV

it k iV E k

, (3SD)

where

2 cos cos

2 cos 2 cos

2 cos 2 cos

s s s x y

px p p x p y

py p p x p y

E t k k

E t k t k

E t k t k

. (4SD)

Note that superscript “L” is omitted in the above equations.

The superposition of the two p orbits can be described as 1

2L L L

x yp p i p . On the basis

of Ls , Lxp and L

yp , the Hamiltonian would take the form of

2 sin sin 2 sin sin

1 12 sin sin

2 21 1

2 sin sin2 2

s sp x y sp x y

L sp x y px py p px py

sp x y px py px py p

E it k i k it k i k

H it k i k E E V E E

it k i k E E E E V

(5SD)

Using the second-order perturbation theory and keeping just the lowest order term, we can expand

the above Hamiltonian linearly as a function of k vector. The perturbation theory is outlined here. A

two-by-two matrix Hamiltonian H can be expressed as:

14  

†A AB

AB B

H HH

H H

, (6SD)

where the two elements AH and BH are coupled to each other via ABH . Assuming wave function

,T

A B and eigen-value E, the eigen-equations become: A A AB B AH H E and

†AB A B B BH H E . By combining these two equations and eliminating B , the effective

Hamiltonian of A can be expressed as:

1 †, ,,A eff A A A eff A AB B ABH E H H H E H H . (7SD)

Eq. (7SD) is a self-consistent equation because the effective Hamiltonian ,A effH also depends on the

eigen energy E. According to the perturbation theory and assuming that the eigen energies of AH and

BH have considerable difference, we can approximate the energy E in ,A effH by the zero-order eigen

energy of AH , denoted as ,0AE . This is equivalent to the second order perturbation theory. Such an

approximation can be used to simplify our theoretical model.

By converting Eq. (5SD) into a two-by-two matrix as indicated in Eq. (6SD), AH , BH and ABH

are given by

2 sin sin

12 sin sin

21

2

2 sin sin

1

2

s sp x y

A

sp x y px py p

B px py p

sp x y

AB

px py

E it k i kH

it k i k E E V

H E E V

it k i kH

E E

. (8SD)

Near the point of origin, 0x yk k , and 1 1

2 2s px py p px py pE E E V E E V . We further

make a reasonable assumption 1 1~

2 2s px py p px py pE E E V E E V , in which case the energy

dominator 1 1

2Bp

E HV

and effective Hamiltonian ,A effH is thus given by

15  

1 †

2 2 2

2

22 sin sin sin sin

2

2 1sin sin

2 4

AB B AB

sp x y sp px py x y

sp px py x y px py

H E H H

t k k it E E k i k

it E E k i k E E

(9SD)

Notice that all the terms in the second order perturbation corresponding to the long range hopping can

be neglected in our first-order analysis. The final result shown in the manuscript can be obtained after

expanding this effective Hamiltonian ,A effH linearly with respect to the linear k vector.

As a result, we rewrite the Hamiltonian shown in Eq. (5SD) by using mixed basis Lp instead of

two individual Lp and Lp basis (the rewritten Hamiltonian is not shown here) and linearly

expand it. The obtained Hamiltonian is

0 0 0

0 2

12 0 0

2

s sp x y

L

sp x y px py p

z f y x x y

E it k ikH

it k ik E E V

m v k k H

, (10SD)

where 0

1 10 0 0

2 2

s px px pE E E V , 0

1 10 0 0

2 2s px px pm E E E V

,

2f spv t , and x , y and z are Pauli matrices. Notice that the Hamiltonian now takes the form

of Dirac equation. It can also be written in the form of second quantization

† † † † † † † †0 00 s s p p s s p p f y s p p s f y s p p sc c c c m c c c c v k c c c c i v k ck cH c c

. (11SD)

The artificial pseudo-time-reversal symmetry operator defined in the manuscript is p yT i K , which

is just the time-reversal symmetry operator in electronic system. Using this operator, the relationship

between LCP and RCP mode can be expressed as † †

, ,p ps L R s R LT c c T ,

† †

, ,p pp L R p R LT c c T ,

and † †

, ,p pp L R p R LT c c T . The total Hamiltonian can then be written as

0

*0

,H k

HH k

(12SD)

where 0

H is defined in Eq. (10SD). This representation of total Hamiltonian has the exact same form

16  

as the Hamiltonian in BHZ model of TI in a 2D electron system (14). Therefore the physics

interpretation of Eq. (12SD) will be similar to that of electrons.

A convenient method to solve the Eq. (10SD) at the boundary is to introduce a second order term,

after which the Hamiltonian becomes

20 0 z f y x x yH k m Bk v k k

. (13SD)

In the semi-infinite configuration (x>0) with the two boundary conditions 0 0x and

0x , the eigen-equation LCPH x E x can be separated into two equations:

2 20 0y x z f x y

f y x

m Bk B x i v x

v k x E x

. (14SD)

Using a trial solution ~ xx e and multiplying z to the first equation of Eq. (14SD), we obtain

2 20 y f xm Bk B v . (15SD)

Compared with the second equation in Eq. (14SD), the two-component wave function can be

treated as the eigen-state of x , x s ss s , in which case Eq. (15SD) has the solutions

2 2, 0

14

2s f f ys v v B m BkB

. (16SD)

The complete form of trial solution becomes

, ,s sx xs s s

s

x c e c e . (17SD)

Now after applying boundary condition 0 0x , Eq. (17SD) becomes

, ,s sx xs s

s

x c e e . (18SD)

Notice that the relationship , ,s s is used in deriving Eq. (18SD). The other boundary condition

0x of Eq. (18SD) requires , 0s or , 0s . Using Viere theorem, Eq. (18SD) can be

described as

1,x xx e e

N

(19SD)

 

wher

Subs

the b

B

Ham

Fina

The

As fo

distr

Fig.

Fig.

The

re 1

2B

stituting Eq. (

boundary state

By applying

miltonian, *0H

ally, the disper

dispersion rel

for the 2Z inv

ributions and t

2SD.

2SD: The Blo

parities of the

f fv v

(19SD) into th

e:

the same pro

k

, a simila

rsion relation o

sE

lations of edge

variant, wheth

the directions

och field distr

e four points s

2

04B m

he second equ

ocedure as ou

r dispersion re

of bulk states

2

pE E E

e states becom

E

her the topolog

of energy flow

ibutions and t

how non-trivi

17 

2yBk

and

uation in Eq. (

f yE v k

utlined above

elation can be

f yE v k .

can be derive

2 2 2p pV E

me

,f yv k

gical states ar

w of four high

the directions

ial property.

x .

(14SD), we ca

to the other

e obtained

ed from Eq. (1

2

p fE v k

1 .

re trivial or no

h symmetric p

of energy flow

an get the dis

r diagonal ele

0SD),

2 0k .

on-trivial depe

points (15), wh

w of four high

spersion relati

(2

ement of the

(2

(2

(2

ends on the fie

hich are show

h symmetric p

on of

0SD)

total

1SD)

2SD)

3SD)

eld

n in

points.

18  

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