# Edge state in AB-stacked bilayer graphene and its correspondence with SSH
ladder

Tixuan Tan<EMAIL_ADDRESS>Ci Li<EMAIL_ADDRESS>Wang Yao  Department of
Physics, The University of Hong Kong, Hong Kong, China  HKU-UCAS Joint
Institute of Theoretical and Computational Physics at Hong Kong, China

###### Abstract

We study edge states in AB-stacked bilayer graphene (BLG) ribbon where the
Chern number of the corresponding two-dimensional (2D) bulk Hamiltonian is
zero. The existence and topological features of edge states when two layers
ended with the same or different edge terminations (zigzag, bearded, armchair)
are discussed. The edge states (non-dispersive bands near the Fermi level) are
states localized at the edge of graphene nanoribbon that only exists in
certain range of momentum $k_{y}$. Their existence near the Fermi level are
protected by the chiral symmetry with topology well described by coupled Su-
Schrieffer-Heeger (SSH) chains model, i.e., SSH ladder, based on the bulk-edge
correspondence of one-dimensional (1D) systems. These zero-energy edge states
can exist in the whole $k_{y}$ region when two layers have zigzag and bearded
edges, respectively. Winding number calculation shows a topological phase
transition between two distinct non-trivial topological phases when crossing
the Dirac points. Interestingly, we find the stacking configuration of BLG
ribbon is important since they can lead to unexpected edge states without
protection from the chiral symmetry both near the Fermi level in armchair-
armchair case and in the gap within bulk bands that are away from Fermi level
in the general case. The influence of interlayer next nearest neighbor (NNN)
interaction and interlayer bias are also discussed to fit the realistic
graphene materials, which suggest the robust topological features of edge
states in BLG systems.

##  I Introduction

One of the most attractive phenomena in condensed matter physics is the
existence and behavior of edge states, whose wave function is localized at the
system’s edge, of two-dimensional (2D) systems. These states are different
from the bulk states in properties and play important roles in transport, e.g.
quantum Hall effect (QHE) and the quantum spin Hall effect (QSHE) Kane ; Ono ;
Wu ; Bern .On the other hand, the existence and properties of zero-energy edge
states near the Fermi level (flat bands) are usually connected with the non-
trivial topological phases of the bulk system by the bulk-edge correspondence
Kane1 ; Graf ; Rud ; Bern1 ; Asb , which can be distinguished by the specific
symmetry of the system and topological invariants such as the winding number
Bern1 ; Asb ; Chiu ; Gui .

After the progress in a decade, graphene, or the nanotube and nanoribbon, has
become one of the most active two-dimensional nanomaterial in condensed matter
physics due to excellent electrical and mechanical properties Sai1 ; Novo1 ;
Mey ; Che ; Uch ; Hua ; Zhou ; Neto ; Dres . The free standing monolayer
graphene (MLG) is a zero-gap semiconductor where the conduction and valance
band touch each other at the Dirac points Neto ; Dres ; Wal ; Gus ; Novo2 ;
Zhang ; Kat ; Novo3 . It has a trivial bulk topology as two inequivalent
valleys provide opposite topological charges, leading to a zero Chern number
Yao . However, edge states still exist in such graphene systems as non-
dispersive bands (flat bands) at the Fermi level Neto ; Dres ; Ryu ; Ryc ;
Yao1 ; Del ; Chiu1 , which are observed by supposing a semi-infinite system,
with quantized wave vectors $k_{y}$ in the infinite direction Gui ; Ryu ; Yao1
; Del ; Sai . The existence of these non-dispersive edge states and related
topology in the MLG can be further described by the bulk-edge correspondence
between winding number or Zak phase of one-dimensional (1D) SSH chain systems
and the existence of a localized state at the chain’s edge Gui ; Ryu ; Del ;
Chang-An ; SLZhang , as shown in Fig. 1.

On the other hand, due to the equivalence between the graphene system and the
honeycomb bosonic lattice system, i.e., the 2D magnon system which generally
results from the collinear Ferromagnet after Holstein-Primakoff transformation
Lie ; Kaw ; Yok , edge states similar to those observed in MLG can also be
found in both the related honeycomb bosonic lattices Pie ; Ser  and even non-
honeycomb bosonic lattices Mao . There are also both experimental and
theoretical study on other types of edges states in photonic honeycomb lattice
using different models. Plot ; Mil .

In this article, we focus on the existence and topology of edge states in AB-
stacked bilayer graphene (BLG) ribbon Novo ; Mc  by connecting them with
$k_{y}$-parameterized SSH ladder. Earlier works are mainly concentrated on the
existence of edge states of BLG under specific edge conditions Neto1 ; Mazo
and related equivalent bilayer magnon systems Sak ; Gha , or the behavior of
edge states when various symmetry-breaking terms are added Tse ; Wei . Here
the AB-stacked BLG ribbon we discuss involves three conventional edges
(zigzag, bearded, and armchair, as shown in Fig. 1), whose bulk 2D system
always has zero Chern number. Interlayer next nearest neighbor (NNN)
interaction and interlayer bias are considered in terms of their influence on
the topology of SSH ladder. A detailed topological classification based on
discrete symmetry Chiu  and topological invariants calculation for effective
1D bulk Hamiltonian of SSH ladder $H\left(k_{y},k\right)$ parameterized by
$k_{y}$ of AB-stacked BLG ribbon with various types of edge are performed, as
shown in Table 1. It shows the zero-energy edge states can only exist when
chiral symmetry is preserved for $H\left(k_{y},k\right)$ and can appear in the
whole $k_{y}$ region when two layers of BLG ribbon have zigzag and bearded
edge, respectively. On the other hand, straightforward calculation shows that
unexpected edge states can exist in the gap within bulk bands that are away
from the Fermi level. These edge states are unprotected by the chiral symmetry
and are dependent on the specific edge configurations of BLG ribbon.
Interlayer bias is included in our discussion as it explicitly breaks the
chiral symmetry responsible for the existence of zero-energy edge states.
However, edge states still exist after this chiral symmetry breaking as non-
zero energy states.

The rest of paper is organized as follows. We first give a brief review on the
existence of edge states and topology of MLG ribbon in Sec. II as a basis for
our discussion of AB-stacked BLG ribbon. In Sec. III, we turn to the behavior
and topology of edge states in AB-stacked BLG ribbon and their correspondence
with SSH ladder $H\left(k_{y},k\right)$. Then we discuss the geometrical
origin of the edge states appearing in the gap within bulk bands that are away
from the Fermi level in Sec. IV. Finally, we present our conclusions in Sec. V
as a summary.

Figure 1: (Color online) The schematic illustration of nanoribbons with
different edges and related effective SSH chain parameterized by $k_{y}$. The
primitive vectors are $\vec{a}_{1}$ and $\vec{a}_{2}$. The unit with
translational symmetry in the tight-binding Hamiltonians is emphasized by the
black box in each structures, respectively. The number pairs (1,1),(1,2),(2,1)
in each figure indicate the increasing direction of $m$ and $n$.

##  II Edge states in the MLG ribbon

To discuss the existence and topological features of edge states in AB-stacked
BLG ribbon, we first give a brief review of the edge states in the MLG ribbon.
In general, MLG tight-binding Hamiltonian with nearest neighbor (NN) hopping
energy $t$ and on-site potential $U_{i}$ can be written as Neto

$H=-t\sum_{\left\langle
i,j\right\rangle}c_{i}^{\dagger}c_{j}+\sum_{i}U_{i}c_{i}^{\dagger}c_{i},$ (1)

where $\sum_{\left\langle i,j\right\rangle}$ sums over only NN pairs. The
lattice primitive vectors are $\overrightarrow{a}_{1}$ and
$\overrightarrow{a}_{2}$, which are shown in Fig. 1. As examples and without
loss of generality, we mainly consider nanoribbons with three different types
of edges: zigzag, bearded, and armchair in $y$ direction and enforce the
periodic boundary condition (PBC) along this direction to see the edge states,
as shown in Fig. 1(a), (b), and (c), respectively.

###  II.1 MLG with bearded (zigzag) edges

The tight-binding Hamiltonian of a MLG with a bearded edge in $y$ direction,
as shown in Fig. 1(a), can be expressed as

$\displaystyle H_{\mathrm{bea}}$ $\displaystyle=$
$\displaystyle-t\sum_{m=1}^{M}\\{\sum_{n=1}^{N}[a_{m,n}^{\dagger}b_{m,n}+b_{m,n}^{\dagger}\left(a_{m,n+1}+a_{m+1,n+1}\right)]$
(2) $\displaystyle-
b_{m,N}^{\dagger}\left(a_{m,1}+a_{m+1,1}\right)+\mathrm{H.c.}\\},$

where $a_{m,n}\left(a_{m,n}^{\dagger}\right)$ annihilates (creates) an
electron on site $\left(m,n\right)$ on sublattice $A$ (an equivalent
definition is used for sublattice $B$). The system is assumed infinite along m
direction and finite along n direction. The minus term in the curly brackets
gives the open boundary condition (OBC) in the finite direction. The Fourier
transformation along the infinite direction is

$\displaystyle f_{m,n}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{M}}\sum_{k_{y}}e^{ik_{y}m}f_{k_{y},n,},f=a,b,$
(3) $\displaystyle k_{y}$ $\displaystyle=$
$\displaystyle\frac{2\pi\left(m-M/2\right)}{M},m=0,1,2,\ldots,M-1,$

and leads to

$\displaystyle H_{\mathrm{bea}}\left(k_{y}\right)$ $\displaystyle=$
$\displaystyle-t\\{\sum_{n=1}^{N-1}\left[a_{k_{y},n}^{\dagger}b_{k_{y},n}+\left(1+e^{ik_{y}}\right)b_{k_{y},n}^{\dagger}a_{k_{y},n+1}\right]$
(4) $\displaystyle+a_{k_{y},N}^{\dagger}b_{k_{y},N}+\mathrm{H.c.}\\},$

which is equivalent to an effective SSH chain parameterized by $k_{y}$ as
below. Notice that we have made a redefinition of basis by a phase such that
the hopping becomes real and it is easier to make association with the
original SSH chain model. Not doing such a redefinition would leave the
coupling complex, but all results in this paper are not affected.

$\displaystyle H_{\mathrm{bea}}\left(k_{y}\right)$ $\displaystyle\simeq$
$\displaystyle\sum_{n=1}^{N-1}\left(va_{n}^{\dagger}b_{n}+wb_{n}^{\dagger}a_{n+1}\right)+va_{N}^{\dagger}b_{N}+\mathrm{H.c.},$
$\displaystyle w$ $\displaystyle=$ $\displaystyle-2t\cos\frac{k_{y}}{2},v=-t.$
(5)

The bulk Hamiltonian of this equivalent chain is

$\displaystyle H_{\mathrm{bea}}\left(k_{y},k\right)$ $\displaystyle=$
$\displaystyle\eta^{\dagger}h_{\mathrm{b}}\left(k_{y},k\right)\eta,\eta=\left(a_{k},b_{k}\right)^{T},$
(6) $\displaystyle h_{\mathrm{b}}\left(k_{y},k\right)$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}0&v+we^{-ik}\\\
v+we^{ik}&0\end{array}\right],$ (9)

which belongs to the non-trivial topological class $\mathcal{BDI}$ (see Table.
1 for details).

We plot the band structure of bearded-edge graphene nanoribbon and winding
number for the bulk Hamiltonian $H_{\mathrm{bea}}\left(k_{y},k\right)$ (6) as
a function of parameter $k_{y}$ in Fig. 2(a), respectively. The winding number
we used in this paper is defined as Chiu

$W=-\frac{i}{4\pi}\int_{BZ}\mathrm{d}k\mathrm{Tr}\left(SQ^{-1}\partial_{k}Q\right),$
(10)

with

$Q\left(k\right)=\mathit{I}_{N}-2\mathcal{P}\left(k\right),\mathcal{P}\left(k\right)=\sum_{\alpha<0}\left|u_{\alpha}\right\rangle\left\langle
u_{\alpha}\right|,$ (11)

$\alpha<0$ refers to the occupied bands (eigenenstates of
$H\left(k_{y},k\right)$ below the Fermi level).
$Q^{-1}\left(k\right)=Q\left(k\right)$, as $Q^{2}=\mathit{I}_{N}$. Winding
number describes the topological properties near the zero energy (Fermi level)
of 1D bulk Hamiltonians $H\left(k_{y},k\right)$ with the chiral symmetry
operator $S$Chiu  . Here we would like to stress again one should not confuse
$k_{y}$ with $k$, since $k_{y}$ appearing in the bulk Hamiltonian
$H\left(k_{y},k\right)$ is a parameter of the system and $k$ is the wave
vector of the effective SSH chain when its length is taken to be infinite. The
integration appearing in the definition of $W$ is over $k$, with $W$ being a
function of $k_{y}$. A concrete example with some details omitted here is
given in the Appendix to make clearer the origin of $k_{y}$ and $k$. The zero-
energy edge states (flat bands) exist in the restricted region
$k_{y}\in\left[-\pi,-\frac{2}{3}\pi\right]\cup\left[\frac{2}{3}\pi,\pi\right]$
and correspond $W=1$, which agree with previous literatures Yao ; Del . The
effective 1D Hamiltonian $H_{\mathrm{zig}}\left(k_{y}\right)$ for a MLG with a
zigzag edge can be obtained by switching $w$ and $v$, $a_{n\text{ }}$and
$b_{n}$ in $H_{\mathrm{bea}}\left(k_{y}\right)$ as zigzag edge is related with
bearded edge through an exchange of basis and coupling (see Fig. 1(a) and
(b)). In terms of their corresponding SSH chain, zigzag nanoribbon and bearded
nanoribbon differ from each other by a switch between intercell coupling and
intracell coupling of the chain. It can be observed from Fig. 2(a) and (b)
that zero-energy edge states (flat bands) of
$H_{\mathrm{zig}}\left(k_{y}\right)$ and non-zero winding number of
$H_{\mathrm{zig}}\left(k_{y},k\right)$ appear in complementary region of
$k_{y}$ to the one with bearded edges.

Figure 2: (Color online) The band structure of MLG ribbon and related winding
number $W(k_{y})$ for effective $k_{y}$-parameterized SSH chain. (a) for the
MLG with bearded edges, (b) for the MLG with zigzag edges, and (c) for the MLG
with armchair edges, respectively. Here the nearest neighbor interaction is
chosen as $t=3.16\mathrm{eV}$ according to Ref. Kuz . The red solid lines
represent the edge states.

###  II.2 MLG with armchair edges

The tight-binding Hamiltonian of a MLG ribbon with an armchair edge is
different from the previous case, which is

$\displaystyle H_{\mathrm{arm}}$ $\displaystyle=$
$\displaystyle-t\sum_{n=1}^{N}[\sum_{m=1}^{M/2}(a_{2m,n}^{\dagger}b_{2m,n}+b_{2m,n}^{\dagger}a_{2m,n+1}$
$\displaystyle+a_{2m-1,n}^{\dagger}b_{2m-1,n}+b_{2m-1,n+1}^{\dagger}a_{2m-1,n})$
$\displaystyle+\sum_{m=1}^{M}a_{m+1,n}^{\dagger}b_{m,n}]$
$\displaystyle+\sum_{m=1}^{M/2}t\left(b_{2m,N}^{\dagger}a_{2m,1}+b_{2m-1,1}^{\dagger}a_{2m-1,N}\right)+\mathrm{H.c.},$

where $M/2\in\mathbb{N}$. Unit with translational symmetry of this ribbon in
Fig. 1(c) is constructed as
$\left(m,n\right)\simeq\left(b_{2m-1,n},a_{2m-1,n},a_{2m,n},b_{2m,n}\right)$.
The effective Hamiltonian parameterized by $k_{y}$ for this ribbon is no
longer a single SSH chain but two coupled uniform chains as shown in Fig.
1(c). The coupled chains have the bulk Hamiltonian:

$\displaystyle H_{\mathrm{arm}}\left(k_{y},k\right)$ $\displaystyle=$
$\displaystyle\eta^{\dagger}h_{\mathrm{a}}\left(k_{y},k\right)\eta,\eta=\left(b_{k}^{1},a_{k}^{1},a_{k}^{2},b_{k}^{2}\right)^{T},$
$\displaystyle h_{\mathrm{a}}\left(k_{y},k\right)$ $\displaystyle=$
$\displaystyle-t\left[\begin{array}[]{cc}h\left(k\right)&D\left(k_{y}\right)\\\
D^{\ast}\left(k_{y}\right)&h\left(k\right)\end{array}\right],$ (15)
$\displaystyle h\left(k\right)$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}0&1+e^{-ik}\\\
1+e^{ik}&0\end{array}\right],$ (18) $\displaystyle D\left(k_{y}\right)$
$\displaystyle=$ $\displaystyle\left[\begin{array}[]{cc}1&0\\\
0&e^{-ik_{y}}\end{array}\right],$ (21)

where the superscript $1/2$ distinguishes even and odd since there are two
sets of A/B in each unit of armchair MLG shown in Fig. 1(c):

$\begin{split}f_{k}^{1}&=\frac{1}{\sqrt{N}}\frac{1}{\sqrt{M/2}}\sum_{j=1}^{M/2}\sum_{n=1}^{N}e^{-ikn}e^{-ik_{y}j}f_{2j-1,n}\\\
f_{k}^{2}&=\frac{1}{\sqrt{N}}\frac{1}{\sqrt{M/2}}\sum_{j=1}^{M/2}\sum_{n=1}^{N}e^{-ikn}e^{-ik_{y}j}f_{2j,n}\end{split}$
(22)

$f=a,b$ as shown in Fig. 1. Notice, again, $k$ is the wave vector of the
coupled chains while $k_{y}$ is a parameter of its coupling. This bulk
Hamiltonian of the coupled chains belongs to the non-trivial topological class
$\mathcal{AIII}$ (see Table 1 for detail). Here we would like to point out
that although it belongs to the non-trivial topological class, the winding
number is zero in the whole region of $k_{y}$ and there are no edge states, as
shown in Fig. 2(c), which means this is a trivial case.

Figure 3: (Color online) The atomic structure of AB-stacked BLG in side view. Table 1: Topological classification for different effective 1D bulk Hamiltonians $H(k_{y},k)$ of SSH chain/ladder. The related winding number $W$ and number of zero-energy edge states $N_{ES}$ are also shown. The topological classification is based on the the presence or absence ($0$) of time-reversal ($T$), particle-hole ($C$), and chiral ($S$) symmetries Chiu , where all three symmetry operators are unitary, i.e., $O^{\dagger}O=1,O=T,C,S$ . They satisfy $T^{\dagger}H^{\ast}(k_{y},k)T=H(k_{y},-k)$, $C^{\dagger}H^{\ast}(k_{y},k)C=-H(k_{y},-k)$, $S^{\dagger}H(k_{y},k)S=-H(k_{y},k)$, respectively. $\pm$ in $T$ and $C$ comes from $T^{\ast}T=\pm 1$ and $C^{\ast}C=\pm 1$. $\mathit{I}_{N}$ is $N\times N$ identity matrix. $\sigma_{\alpha=x,y,z}$ represent Pauli matrices. “-” means that there is no well-defined winding number since the chiral symmetry is broken Chiu . Notice that when operator is written in direct product form, they can be understood as acting on different degrees of freedom (DOF). For example, $S=\mathit{I}_{2}\otimes\sigma_{z}$ for $H_{\mathrm{bea-bea}}$, where $I_{2}$ acts on layer DOF and $\sigma_{z}$ acts on sublattice DOF. Effective 1D bulk Hamiltonian | $T$ | $C$ | $S$ | Class | $W$ | $N_{ES}$
---|---|---|---|---|---|---
$H_{\mathrm{bea/zig}}(k_{y},k)$ (Eq. 6) | $\mathit{I}_{2}(+)$ | $\sigma_{z}(+)$ | $\sigma_{z}$ | $\mathcal{BDI}$ |  $1,0$ (Fig. 2) |  $2$ (Fig. 2)
$H_{\mathrm{arm}}(k_{y},k)$ (Eq. 21) | $0$ | $0$ | $\sigma_{z}\otimes\sigma_{z}$ | $\mathcal{AIII}$ |  $0$ (Fig. 2) |  $0$ (Fig. 2)
$H_{\mathrm{arm-arm}}^{\uparrow}(k_{y},k),U=0$ (Eq. 25) | $0$ | $0$ | $\mathit{I}_{2}\otimes\sigma_{z}\otimes\sigma_{z}$ | $\mathcal{AIII}$ | $0$ |  $0$ (Fig. 4)
$H_{\mathrm{arm-arm}}^{\uparrow}(k_{y},k),U\neq 0$ (Eq. 25) | $0$ | $0$ | $0$ | $\mathcal{A}$ | - |  $0$ (Fig. 4)
$H_{\mathrm{arm-arm}}^{\swarrow}(k_{y},k),U=0$ (Eq. 25) | $0$ | $0$ | $\mathit{I}_{2}\otimes\sigma_{z}\otimes\sigma_{z}$ | $\mathcal{AIII}$ | $0$ |  $0$ (Fig. 5)
$H_{\mathrm{arm-arm}}^{\swarrow}(k_{y},k),U\neq 0$ (Eq. 25) | $0$ | $0$ | $0$ | $\mathcal{A}$ | - |  $4$ (Fig. 5)
$H_{\mathrm{bea-bea/zig-zig}}^{\leftarrow(\swarrow)}(k_{y},k),U=0$ (Eq. 40) | $\mathit{I}_{4}(+)$ | $\mathit{I}_{2}\otimes\sigma_{z}(+)$ | $\mathit{I}_{2}\otimes\sigma_{z}$ | $\mathcal{BDI}$ |  $2,0$ (Fig. 7) |  $4$ (Fig. 7)
$H_{\mathrm{bea-bea/zig-zig}}^{\leftarrow(\swarrow)}(k_{y},k),U\neq 0$ (Eq. 40) | $\mathit{I}_{4}(+)$ | $0$ | $0$ | $\mathcal{AI}$ | - |  $4$ (Fig. 7)
$H_{\mathrm{bea-zig}}^{\swarrow(\nearrow)}(k_{y},k),U=0$ (Eq. 46) | $\mathit{I}_{4}(+)$ | $\sigma_{z}\otimes\sigma_{z}(+)$ | $\sigma_{z}\otimes\sigma_{z}$ | $\mathcal{BDI}$ |  $-1,1$ (Fig. 7) |  $2$ (Fig. 7)
$H_{\mathrm{bea-zig}}^{\swarrow(\nearrow)}(k_{y},k),U\neq 0$ (Eq. 46) | $\mathit{I}_{4}(+)$ | $0$ | $0$ | $\mathcal{AI}$ | - |  $2$ (Fig. 7)

##  III Edge states near the zero energy (Fermi level) in the AB-stacked BLG

Based on the three different graphene nanoribbons discussed in last section,
there are four types of AB-stacked BLG ribbon, i.e., arm-arm (both layers are
armchair edges), zig-zig/bea-bea (both layers are zigzag or bearded edges),
bea-zig (one layer is bearded edges and the other is zigzag edges), as listed
in Table. 1. They can be summarized by the tight-binding Hamiltonian Neto ; Mc
; Mc1

$H=\sum_{l}H_{\mathrm{edge}}^{l}+H_{\mathrm{int}}+H_{\mathrm{on-site}},$ (23)

where $l=1,2$ are labels of the bottom and top layers respectively.
$H_{\mathrm{on-site}}$ refers to the on-site energy of carbon atoms such as
interlayer bias $U$ as shown in Fig. 3. It includes contribution from both
layers. $H_{\mathrm{int}}$ describes the van der Waals interaction between two
layers Neto ; Mc ; Mc1 . The meanings of various interlayer couplings
$\gamma_{i}$ with $i=1,3,4$ are indicated in Fig. 3. Here, we take
$t=3.16\mathrm{eV}$, $\gamma_{1}=0.381\mathrm{eV}$ as typical experimental
values for AB-stacked BLGs Kuz , and $\gamma_{4}=0$ throughout this work since
it is pretty small compared with others in realistic systems Neto ; Mc1 ; Kuz
. We choose $\gamma_{3}=0.38\mathrm{eV}\approx\gamma_{1}$ when considering the
non-zero $\gamma_{3}$, which is close to most of the experimental observations
Neto ; Mc1 ; Kuz .

Figure 4: (Color online) Left panel: The atomic structure of an armchair-
armchair AB-stacked BLG ribbon in top view, corresponding to 1D effective
Hamiltonian $H_{\mathrm{arm-arm}}^{\uparrow}(k_{y},k)$ (Eq. 25). The edge
configuration is emphasized by the red box. Right panel: Related band
structure with different interlayer bias $U$ and NNN interaction $\gamma_{3}$.
Figure 5: (Color online) Right panel: The atomic structure of an armchair-
armchair AB-stacked BLG ribbon in top view, corresponding to 1D effective
Hamiltonian $H_{\mathrm{arm-arm}}^{\swarrow}(k_{y},k)$ (Eq. 25). The edge
configuration is emphasized by the red box, which is different from the one in
Fig. 4. Left panel: Related band structure with different interlayer bias $U$
and NNN interaction $\gamma_{3}$. The red solid lines are the edge states.
There are four of them. The wave function distribution in real space of a
typical edge state in red dashed box is shown on the right side. Vertical axis
is wave function’s amplitude, and the horizontal axis is the site index, which
increases along the finite direction of the ribbon.

###  III.1 AB-stacked BLG with armchair-armchair edges

Next we will discuss these four types of AB-stacked BLG structures in detail.
To form an AB-stacked BLG ribbon, armchair MLG can only be stacked with the
other armchair MLG, but not with zigzag/bearded MLG (see Fig. 1). Thus, we
first consider the Hamiltonian

$H_{\mathrm{arm-
arm}}=\sum_{l}H_{\mathrm{arm}}^{l}+H_{\mathrm{int}}+H_{\mathrm{on-site}},$
(24)

where both layers have armchair edge. We further consider two different edge
configurations as shown in left panel of Fig. 4 and 5, corresponding to
different $H_{\mathrm{int}}$. The band structure of $H_{\mathrm{arm-
arm}}^{\uparrow\left(\swarrow\right)}\left(k_{y}\right)$ parameterized by
$k_{y}$ is shown in right panel of Fig. 4 and 5 respectively, where the label
$\uparrow\left(\swarrow\right)$ represents the different edge configuration of
two setting. It denotes the direction to translate top layer if we effectively
consider the AB-stacked nanoribbon is formed from the relative translation
between two layers of AA-stacked nanoribbon. Similar calculation to that of
last section shows the related $k_{y}$-parameterized bulk Hamiltonian is

$\displaystyle H_{\mathrm{arm-
arm}}^{\uparrow\left(\swarrow\right)}\left(k_{y},k\right)$ $\displaystyle=$
$\displaystyle\eta^{\dagger}h_{\mathrm{aa}}^{\uparrow\left(\swarrow\right)}\left(k_{y},k\right)\eta,\eta=\left(\xi_{1},\xi_{2}\right)^{T},$
(25) $\displaystyle\xi_{l}$ $\displaystyle=$
$\displaystyle\left(b_{l,k}^{1},a_{l,k}^{1},a_{l,k}^{2},b_{l,k}^{2}\right),$
$\displaystyle
h_{\mathrm{aa}}^{\uparrow\left(\swarrow\right)}\left(k_{y},k\right)$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}h_{\mathrm{a}}^{1}\left(k_{y},k\right)&-H_{\mathrm{int}}^{\uparrow\left(\swarrow\right)}\left(k_{y},k\right)\\\
-\left[H_{\mathrm{int}}^{\uparrow\left(\swarrow\right)}\left(k_{y},k\right)\right]^{\dagger}&h_{\mathrm{a}}^{2}\left(k_{y},k\right)\end{array}\right]$
(28) $\displaystyle h_{\mathrm{a}}^{l}\left(k_{y},k\right)$ $\displaystyle=$
$\displaystyle\frac{(-1)^{l}U}{2}\mathit{I}_{4}+h_{\mathrm{a}}\left(k_{y},k\right),f_{k}=1+e^{ik}$
$\displaystyle H_{\mathrm{int}}^{\uparrow}\left(k_{y},k\right)$
$\displaystyle=$ $\displaystyle\left[\begin{array}[]{cccc}0&0&\gamma_{1}&0\\\
\gamma_{3}f_{k}&0&0&\gamma_{3}\\\
\gamma_{3}e^{ik_{y}}&0&0&\gamma_{3}f_{k}^{\ast}\\\
0&\gamma_{1}e^{ik_{y}}&0&0\end{array}\right],$ (33) $\displaystyle
H_{\mathrm{int}}^{\swarrow}\left(k_{y},k\right)$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{cccc}0&\gamma_{1}&0&0\\\
\gamma_{3}e^{ik}&0&0&\gamma_{3}f_{k}e^{-ik_{y}}\\\
\gamma_{3}f_{k}&0&0&\gamma_{3}\\\ 0&0&\gamma_{1}e^{ik}&0\end{array}\right],$
(38)

which belongs to the non-trivial topological class $\mathcal{AIII}$ when there
is no interlayer bias. When $U\neq 0$, it belongs to the trivial topological
class $\mathcal{A}$ (Table. 1).

For the BLG ribbon with the edge configuration shown in Fig. 4, there is no
edge state even if non-zero $U$ and $\gamma_{3}$ are considered. For the other
edge configuration as shown in Fig. 5, the edge states which are gapped and
flat appear when non-zero interlayer bias is added. Here we would like to
point out that these edge states are not topological for three reasons: (i)
They cannot be described by the bulk-edge correspondence we used before since
their energy is not at the Fermi level. (ii) They are not robust since they
disappear when including $\gamma_{3}$, the quantity that preserves original
chiral symmetry of the system and do not influence the existence of edge
states in the case of bilayer bearded-bearded (zigzag-zigzag) ribbon
(discussed below). (iii) Most importantly, they are not formed between two
Dirac points with different topological charges Yao ; Yao1 ; Del ; Man  in the
band structure. But this fact is still interesting since it indicates the
existence of edge states can be determined by interlayer bias.

Figure 6: (Color online) The atomic structure of bearded-bearded ((a) and (b))
and bearded-zigzag ((c) and (d)) AB-stacked BLG ribbon in top view. The
specific edge configuration is emphasized by the red box. The related
$k_{y}$-parameterized SSH ladders are shown at the bottom, where (a)-(b)
corresponds to $H_{\mathrm{ber-ber}}^{\leftarrow(\swarrow)}(k_{y},k)$ (40) and
(c)-(d) corresponds to $H_{\mathrm{ber-zig}}^{\swarrow(\nearrow)}(k_{y},k)$
(46), respectively. $w=-2tcos(k_{y}/2)$ and $v=-t$ are the same as used in
Fig. 1.

###  III.2 AB-stacked BLG with bearded-bearded (zigzag-zigzag) edges

The Hamiltonian that both layers have a bearded edge is analogous to the one
with a zigzag edge. We take the one with the bearded edge as an example, whose
Hamiltonian can be expressed as

$H_{\mathrm{bea-
bea}}=\sum_{l}H_{\mathrm{bea}}^{l}+H_{\mathrm{int}}+H_{\mathrm{on-site}}.$
(39)

There are two different edge configurations, corresponding to different forms
of $H_{\mathrm{int}}$. Here these two types of Hamiltonians are denoted as
$H_{\mathrm{bea-bea}}^{\longleftarrow\left(\swarrow\right)}\left(k_{y}\right)$
. Corresponding lattice structures are shown in Fig. 6(a) and (b),
respectively. Meanwhile, $H_{\mathrm{zig-
zig}}^{\longleftarrow\left(\swarrow\right)}\left(k_{y}\right)$ are obtained by
the substitution: $w\longleftrightarrow v$, $a_{l,n}\longleftrightarrow
b_{l,n}$ in Fig. 6(a) and (b). The bulk Hamiltonian of the
$k_{y}$-parameterized SSH ladder $H_{\mathrm{bea-
bea}}^{\longleftarrow\left(\swarrow\right)}\left(k_{y}\right)$ can be
expressed as

$\displaystyle H_{\mathrm{bea-
bea}}^{\longleftarrow\left(\swarrow\right)}\left(k_{y},k\right)$
$\displaystyle=$
$\displaystyle\eta^{\dagger}h_{\mathrm{bb}}^{\longleftarrow\left(\swarrow\right)}\left(k_{y},k\right)\eta,$
(40) $\displaystyle\eta$ $\displaystyle=$
$\displaystyle\left(a_{1,k},b_{1,k},a_{2,k},b_{2,k}\right)^{T},$
$\displaystyle
h_{\mathrm{bb}}^{\longleftarrow\left(\swarrow\right)}\left(k_{y},k\right)$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}h_{\mathrm{b}}^{1}\left(k_{y},k\right)&H_{\mathrm{int}}^{\longleftarrow\left(\swarrow\right)}\left(k_{y},k\right)\\\
\left[H_{\mathrm{int}}^{\longleftarrow\left(\swarrow\right)}\left(k_{y},k\right)\right]^{\dagger}&h_{\mathrm{b}}^{2}\left(k_{y},k\right)\end{array}\right],$
(43)

with

$h_{\mathrm{b}}^{l}\left(k_{y},k\right)=h_{\mathrm{b}}\left(k_{y},k\right)+\frac{(-1)^{l}}{2}U\mathit{I}_{2},l=1,2,$
(44)
$H_{\mathrm{int}}^{\longleftarrow}\left(k_{y},k\right)=-\left[\begin{array}[]{cc}0&\gamma_{1}\\\
\gamma_{3}e^{ik}\left(2\cos\frac{k_{y}}{2}+e^{ik}\right)&0\end{array}\right],$
$H_{\mathrm{int}}^{\swarrow}\left(k_{y},k\right)=-\left[\begin{array}[]{cc}0&\gamma_{3}\left(2\cos\frac{k_{y}}{2}+e^{-ik}\right)\\\
\gamma_{1}e^{ik}&0\end{array}\right].$

which belongs to non-trivial class $\mathcal{BDI}$ only when $U=0$, otherwise
it belongs to trivial class $\mathcal{AI}$ (Table. 1).

Unlike the armchair-armchair-edge case, the different geometry (distinguished
by arrows) of bearded-bearded (zigzag-zigzag) edges of BLGs do not influence
the band structure and edge states near the zero-energy (Fermi level). So we
only show the band structure of bearded-bearded nanoribbon corresponding to
$H_{\mathrm{bea-bea}}^{\longleftarrow}\left(k_{y}\right)$ in left panel of
Fig. 7 for simplicity. The non-trivial and trivial topological classification
straightly determines the existence of zero-energy edge states since $U$
breaks the chiral symmetry. The zero-energy edge states appear as flat
fourfold degenerate bands when $U=0$, belonging to non-trivial topological
class. The related winding number calculation tells that they correspond to a
winding number $W=2$ when they are flat and $W=0$ when they enter the bulk
bands. Flat bands from each layer are separated by a gap when $U\neq 0$, as
shown in Fig. 7. Bulk Hamiltonian of the SSH ladder $H_{\mathrm{bea-
bea}}(k_{y},k)(U\neq 0)$ belongs to topological trivial class (Table. 1). Here
the edge states for $U\neq 0$ are still topological since they connect two
topologically different Dirac points Man .

Figure 7: (Color online) Left panel: Band structure with different interlayer
bias $U$ and NNN interaction $\gamma_{3}$ corresponding to atomic structure
shown in Fig. 6(a). Right panel: Band structure with different interlayer bias
$U$ and NNN interaction $\gamma_{3}$ corresponding to atomic structure shown
in Fig. 6(d). The red solid lines show the edge states. Related winding number
$W(k_{y})$ is shown in the middle for $U=0$ cases in both panels, where the
chiral symmetry is preserved. The red dashed circle marks the energy band area
within bulk bands where the unexpected edge states may appear, which is
discussed in detail in Fig. 9. Figure 8: (Color online) Left panel: Band
structure with interlayer bias $U\approx 0$ and zero NNN interaction
$\gamma_{3}$. Corresponding SSH ladder parameterized by $k_{y}$ is shown at
the bottom. The reason for using $U\approx 0$ is explained in the text. Right
panel: The distribution of the wave function $|\varphi_{1}\rangle$ on
different edges and different layers as a function of $k_{y}$ and the
distribution of the wave function $|\varphi_{1}\rangle$ in real space for two
typical $k_{y}$ values with two different winding numbers. $n_{l=1,2}$ is site
index on the respective layer, which increases along the finite direction of
the ribbon. A transition between bottom-left and top-right is observed when
crossing the Dirac point.

###  III.3 AB-stacked BLG with bearded-zigzag edges

Armchair MLG ribbon can only be stacked with armchair MLG ribbon to form an
AB-stacked BLG ribbon. In contrast, zigzag MLG ribbon can be stacked with
bearded MLG ribbon to form an AB-stacked BLG ribbon, which is described by the
Hamiltonian

$H_{\mathrm{bea-zig(zig-
bea)}}=H_{\mathrm{bea(zig)}}^{1}+H_{\mathrm{zig(bea)}}^{2}+H_{\mathrm{int}}+H_{\mathrm{on-
site}},$ (45)

Here we choose $H_{\mathrm{bea-zig}}$ as the example. The situation is the
same as what we found in above subsection, where different edge configurations
lead to similar band structures and related edge states near the zero-energy.
Hamiltonians of different edge configurations can be denoted as
$H_{\mathrm{bea-zig}}^{\swarrow\left(\nearrow\right)}\left(k_{y}\right)$ as
shown in Fig. 6(c) and (d), respectively. The bulk Hamiltonian of the SSH
ladder $H_{\mathrm{bea-
zig}}^{\swarrow\left(\nearrow\right)}\left(k_{y}\right)$ can be expressed as

$\displaystyle H_{\mathrm{bea-
zig}}^{\swarrow\left(\nearrow\right)}\left(k_{y},k\right)$ $\displaystyle=$
$\displaystyle\eta^{\dagger}h_{\mathrm{bz}}^{\swarrow\left(\nearrow\right)}\left(k_{y},k\right)\eta,$
(46) $\displaystyle\eta$ $\displaystyle=$
$\displaystyle\left(a_{1,k},b_{1,k},b_{2,k},a_{2,k}\right)^{T},$
$\displaystyle
h_{\mathrm{bz}}^{\swarrow\left(\nearrow\right)}\left(k_{y},k\right)$
$\displaystyle=$
$\displaystyle\left[\begin{array}[]{cc}h_{\mathrm{b}}^{1}\left(k_{y},k\right)&H_{\mathrm{int}}^{\swarrow\left(\nearrow\right)}\left(k_{y}\right)\\\
\left[H_{\mathrm{int}}^{\swarrow\left(\nearrow\right)}\left(k_{y}\right)\right]^{\dagger}&h_{\mathrm{z}}^{2}\left(k_{y},k\right)\end{array}\right],$
(49)

with

$\displaystyle h_{\mathrm{b}}^{1}\left(k_{y},k\right)$ $\displaystyle=$
$\displaystyle h_{\mathrm{b}}\left(k_{y},k\right)-\frac{U}{2}\mathit{I}_{2},$
(50) $\displaystyle h_{\mathrm{z}}^{2}\left(k_{y},k\right)$ $\displaystyle=$
$\displaystyle h_{\mathrm{z}}\left(k_{y},k\right)+\frac{U}{2}\mathit{I}_{2},$
$\displaystyle H_{\mathrm{int}}^{\swarrow}\left(k_{y}\right)$ $\displaystyle=$
$\displaystyle-\left[\begin{array}[]{cc}\gamma_{3}\left(2e^{ik}\cos\frac{k_{y}}{2}+1\right)&0\\\
0&\gamma_{1}e^{ik}\end{array}\right],$ (53) $\displaystyle
H_{\mathrm{int}}^{\nearrow}\left(k_{y}\right)$ $\displaystyle=$
$\displaystyle-\left[\begin{array}[]{cc}\gamma_{1}&0\\\
0&\gamma_{3}\left(2\cos\frac{k_{y}}{2}+e^{ik}\right)\end{array}\right].$ (56)

When $U=0$, $H_{\mathrm{bea-zig}}^{\swarrow}\left(k_{y}\right)$ belongs to
non-trivial class $\mathcal{BDI}$. Otherwise, it belongs to trivial class
$\mathcal{AI}$ (Table. 1). The band structure of nanoribbon corresponding to
$H_{\mathrm{ber-zig}}^{\swarrow}\left(k_{y}\right)$ is plotted in right panel
of Fig. 7. The twofold degenerate zero-energy edge state (flat band) can exist
in the whole $k_{y}$ region when $U=0$. It corresponds to a topological phase
transition between two non-trivial topological phases characterized by $W=-1$
and $W=1$ when crossing the Dirac point, as shown in left panel in Fig. 7.
When $U\neq 0$, although the chiral symmetry is broken (i.e. no well-defined
winding number), topological edge state still exists in the whole $k_{y}$
region with different energies when crossing the Dirac point. Notice that when
$U\neq 0$, the bands are no longer twofold degenerate, as explained later.

This topological phase transition is not a numerical artifact. In contrast, it
is protected by interlayer coupling. Winding number describes the mapping of
Brillouin zone $(k)$ to $U(n)$ group, whose fundamental group is $Z$ Ling . In
BLG ribbon case, it is $U(2)$. This mapping is orientation sensitive. For
example, in calculating the winding number of BLG bea-bea ribbon, as presented
in Fig. 7. The non-zero winding number of it can be turned into $-2$ instead
of $2$ by reversing the direction of $k$. In the BLG bea-zig case, as in the
right panels of Fig. 7, it is the bearded layer/zigzag layer that is
responsible for the $1/-1$ part of winding number. If we can choose the
orientation of $k$ independently for each layer, we can make the winding
number always $1$, i.e. no phase transition is present. This simple conjecture
can be verified by setting $\gamma_{3}=\gamma_{1}=0$, i.e. two decoupled
ribbon. In that case, one can indeed find a chiral operator $S$ such that the
calculated winding number is always $1$(or always $-1$). For example, it can
be done by using $S=diag(1,-1,1,-1)$ for the Hamiltonian in Eq. (46) with
$\gamma_{3}=\gamma_{1}=0$. It is permissible for an uncoupled system to have
different orientation of Brillouin zone for each subsystem. However, the
presence of interlayer coupling, which is present in the real BLG ribbon
system, prohibits us from choosing the orientation of $k$ independently for
each layer. We are forced to choose the same orientation of $k$ for two layers
of ribbon. Otherwise, we are not able to write $H_{int}$ in Bloch form. More
specifically, this means that $S=diag(1,-1,1,-1)$ is no longer a chiral
symmetry operator for Eq. (46) with non-zero interlayer coupling. In this
sense, the observed topological phase transition is protected by interlayer
coupling.

The change in winding number can be seen from the behavior of the wave
function of edge states. For $W=1\left(\left|v\right|<\left|w\right|\right)$
region, one of the degenerate edge states can be approximately expressed as

$\displaystyle\left|\varphi_{1}\right\rangle$ $\displaystyle\approx$
$\displaystyle\frac{1}{\sqrt{\Omega_{1}}}\sum_{j=1}^{N}\left(-\frac{v}{w}\right)^{N-j}b_{1,j}^{\dagger}\left|0\right\rangle,$
(57) $\displaystyle\Omega_{1}$ $\displaystyle=$
$\displaystyle\frac{1-\left(v/w\right)^{2N}}{1-\left(v/w\right)^{2}}\approx\frac{w^{2}}{w^{2}-v^{2}},$

for $\gamma_{3}=U\approx 0$, which is equivalent to one of the edge states of
MLG ribbon with bearded edges (bottom layer), as shown in Fig. 8, where the
label $l=1,2$ in $a_{l,j}^{\dagger}\left(b_{l,j}^{\dagger}\right)$ represents
the bottom/top layer respectively. The other edge state is

$\displaystyle\left|\varphi_{2}\right\rangle$ $\displaystyle\approx$
$\displaystyle\frac{1}{\sqrt{\Omega_{2}}}\sum_{j=1}^{N}[\left(-\frac{v}{w}\right)^{j-1}a_{1,j}^{\dagger}\left|0\right\rangle$
$\displaystyle+\left(-1\right)^{j}j\frac{\gamma_{1}}{w}\left(\frac{v}{w}\right)^{j-1}a_{2,j}^{\dagger}\left|0\right\rangle],$
$\displaystyle\Omega_{2}$ $\displaystyle=$
$\displaystyle\frac{1-\left(v/w\right)^{2N}}{1-\left(v/w\right)^{2}}+\left(\frac{\gamma_{1}}{w}\right)^{2}\sum_{j=1}^{N}j^{2}\left(\frac{v}{w}\right)^{2j-2}$
$\displaystyle\approx$
$\displaystyle\frac{w^{2}}{w^{2}-v^{2}}+\left(\frac{\gamma_{1}}{w}\right)^{2}\sum_{j=1}^{N}j^{2}\left(\frac{v}{w}\right)^{2j-2}.$

Since $\gamma_{1}\ll t$ (one order smaller in our choice of parameters),
$\left|\varphi_{2}\right\rangle$ mainly distributes on $l=1$ layer. When
entering $W=-1\left(\left|v\right|>\left|w\right|\right)$ region,
$\left|\varphi_{1}\right\rangle$ and $\left|\varphi_{2}\right\rangle$
continuously change to

$\displaystyle\left|\varphi_{1}\right\rangle$ $\displaystyle\rightarrow$
$\displaystyle\frac{1}{\sqrt{\Omega_{1}}}\sum_{j=1}^{N}[\left(-\frac{w}{v}\right)^{j-1}b_{2,j}^{\dagger}\left|0\right\rangle$
$\displaystyle+\left(-1\right)^{j}j\frac{\gamma_{1}}{v}\left(\frac{w}{v}\right)^{j-1}b_{1,j}^{\dagger}\left|0\right\rangle],$
$\displaystyle\Omega_{1}$ $\displaystyle=$
$\displaystyle\frac{1-\left(w/v\right)^{2N}}{1-\left(w/v\right)^{2}}+\left(\frac{\gamma_{1}}{v}\right)^{2}\sum_{j=1}^{N}j^{2}\left(\frac{w}{v}\right)^{2j-2}$
$\displaystyle\approx$
$\displaystyle\frac{v^{2}}{v^{2}-w^{2}}+\left(\frac{\gamma_{1}}{v}\right)^{2}\sum_{j=1}^{N}j^{2}\left(\frac{w}{v}\right)^{2j-2},$

and

$\displaystyle\left|\varphi_{2}\right\rangle$ $\displaystyle\rightarrow$
$\displaystyle\frac{1}{\sqrt{\Omega_{2}}}\sum_{j=1}^{N}\left(-\frac{w}{v}\right)^{N-j}a_{2,j}^{\dagger}\left|0\right\rangle,$
(60) $\displaystyle\Omega_{2}$ $\displaystyle\approx$
$\displaystyle\frac{v^{2}}{v^{2}-w^{2}}.$

This indicates that when the winding number changes, the distribution of edge
states in real space will switch both edge and layer, as shown in Fig. 8.

We would like to stress that when $U$ is exactly zero, one can argue that
$\left|\varphi_{1}\right\rangle$ and $\left|\varphi_{2}\right\rangle$ are
exponentially degenerate. The degeneracy originates from the edge degree of
freedom. Thus, any linear combination of them are still edge states, with
$\left|\varphi_{1}\right\rangle$ and $\left|\varphi_{2}\right\rangle$ being
only one possible choice. Then it does not make sense to discuss the behaviour
of $\left|\varphi_{1}\right\rangle$ in real space when crossing the Dirac
point, e.g. from bottom-left to top-right, as a linear combination would mix
left edge and right edge. The naturalness of choosing
$\left|\varphi_{1}\right\rangle$ and $\left|\varphi_{2}\right\rangle$ is seen
as following: when a small positive bias U, e.g. $0.001eV$, is included as
perturbation $H_{\mathrm{per}}$, the cross term
$\left\langle\varphi_{1}\right|H_{\mathrm{per}}\left|\varphi_{2}\right\rangle=0$.
According to the first order degenerate perturbation theory Kato ,
$\left|\varphi_{1}\right\rangle$ and $\left|\varphi_{2}\right\rangle$ are the
good states to use in perturbation. Energy of state in Eq. (III.3) would
increase relatively to the energy of state in Eq. (57), as the former has
minor component on $l=2$ layer. Similarly, energy of state in Eq. (60) would
increase relatively to the energy of state in Eq. (III.3), as the latter has
minor component on $l=1$ layer.
Thus,$\left|\varphi_{1}\right\rangle$/$\left|\varphi_{2}\right\rangle$ would
continuously evolve into Eq. (III.3)/Eq. (60). Fig. 8 is plotted with
$U=0.001eV$ and $\gamma_{3}=0$. The behaviour of states is numerically smooth,
indicating the degeneracy is effectively lifted with a small bias.

Figure 9: (Color online) The band structures in the region marked in Fig. 7,
coming from different edge configurations of AB-stacked BLG ribbon with
different interlayer bias $U$ and NNN interaction $\gamma_{3}$. Only the bands
with $E<0$ are shown for simplicity since $E<0$ and $E>0$ bands are symmetric.
(a)-(d) correspond to the atomic structures shown in Fig. 6(a)-(d),
respectively. The red solid lines show the unexpected edge states appearing in
the gap within the bulk bands that are away from zero energy.

Figure 10: (Color online) Effective SSH ladder parameterized by $k_{y}$ when
$k_{y}\approx\pi$ for different AB-stacked BLG ribbon shown in Fig. 6. (a)-(d)
correspond to $H(k_{y})$ plotted in (a)-(d) in Fig. 6, respectively. The
equivalent lattice structures are shown in blue in each figure, the hopping
between sites are given in (a).

##  IV Edge states appearing in the gap away from the zero energy (Fermi
level)

In above sections, we have considered two edge configurations of BLG bea-zig
ribbon and two edge configurations of BLG bea-bea ribbon, denoted via
different arrows. It was shown that, for example, two edge configurations of
BLG bea-bea ribbon have the same topology. It turns out the difference between
these two edge configurations appears in the form of non-topological edge
states, as discussed below.

Besides the topological edge states we discussed in the last section, which
exist as gapless or gapped flat bands near the zero energy (Fermi level), some
unexpected edge states are found in the gap within bulk bands that are away
from the zero energy (Fermi level) in the AB-stacked BLGs with a bearded-
bearded (zigzag-zigzag) edge or a bearded-zigzag edge, as shown in Fig. 9. The
existence and number of these edge states are determined by the specific edge
configurations of AB-stacked BLGs. These unexpected edge states appear in the
region near $k_{y}=\pi$ in all situations, which reminds us their appearance
may be relevant to the effective Hamiltonians $H(k_{y}=\pi)$ for different
types of edge configurations.

Therefore, we show the lattice structure of the effective Hamiltonians
$H(k_{y}=\pi)$ for different types of edge configurations in Fig. 10, where
some couplings vanish since $\cos\left(\pi/2\right)=0$. $H_{\mathrm{bea-
bea}}^{\longleftarrow}(k_{y}=\pi)$ with $\gamma_{3}=0$, describes the simply
repeated decoupled $4$-site structures

$\displaystyle h_{4}$ $\displaystyle=$ $\displaystyle
w_{eq}c_{1}^{\dagger}c_{2}+v_{eq}c_{2}^{\dagger}c_{3}+w_{eq}c_{3}^{\dagger}c_{4}$
$\displaystyle+\mathrm{H.c.}+\sum_{j=1}^{2}\frac{\left(-1\right)^{j+1}U}{2}\left(c_{2j-1}^{\dagger}c_{2j-1}+c_{2j}^{\dagger}c_{2j}\right),$

as shown in Fig. 10(a). Each 4-site structure would provide 4 energy levels

$\displaystyle E$ $\displaystyle=$
$\displaystyle\pm\frac{1}{2}\sqrt{2\gamma_{1}^{2}+U^{2}+4t^{2}\pm
2\varepsilon},$ (62) $\displaystyle\varepsilon$ $\displaystyle=$
$\displaystyle\sqrt{4U^{2}t^{2}+4t^{2}\gamma_{1}^{2}+\gamma_{1}^{4}},$

and each of these four energy levels are highly degenerate since there are
many identical 4-site structures. When we consider the region close to
$k_{y}=\pi$, each of the four highly degenerate levels would split into many
bulk states according to first order degenerate perturbation theory. So there
are no edge states under this scenario in the neighbour of $k_{y}=\pi$ in the
bulk gap.

If $\gamma_{3}\approx\gamma_{1}$ as we discussed before, the energy bands of
$H_{\mathrm{bea-bea}}^{\longleftarrow}(k_{y}=\pi)$ is dominated by not only a
simply repeated decoupled $4$-site structure but also a newly formed SSH chain

$\displaystyle h_{\mathrm{SSH}}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}w_{eq}a_{n}^{\dagger}b_{n}+v_{eq}b_{n}^{\dagger}a_{n+1}+\mathrm{H.c.}$
$\displaystyle+\sum_{n=1}^{N}\frac{\left(-1\right)^{n+1}U}{2}\left(a_{n}^{\dagger}a_{n}+b_{n}^{\dagger}b_{n}\right),$

as shown in the Fig. 10(a). Because $t\gg\gamma_{1}$ in our choice of
parameters Kuz , which is general in most cases Neto ; Mc1 , this equivalent
SSH chain leads to no edge states. This is in accord with the results shown in
both Fig. 7 and Fig. 9(a).

However, the situation is different for $H_{\mathrm{bea-
bea}}^{\swarrow}\left(k_{y}\right)$. When $\gamma_{3}=0$, two isolated
$2$-site structures

$h_{2}=w_{eq}c_{1}^{\dagger}c_{2}+\mathrm{H.c.}\pm\frac{U}{2}\left(c_{1}^{\dagger}c_{1}+c_{2}^{\dagger}c_{2}\right),$
(64)

appear in addition to repeated decoupled $4$-site structures, as shown in Fig.
10(b). These isolated $2$-site structures provide the eigenstates with energy
$\pm U/2\pm t$, as shown in Fig. 9(b). When we consider the region close to
$k_{y}=\pi$, since the energy of 2-site structure is different from (well
separated) that of bulk 4-site structure, their eigenstates would
predominantly mix among themselves instead of mixing with states from those
4-site structures. Since these two 2-site only exist at the edge, the mixing
result would remain edge states. If $\gamma_{3}\approx\gamma_{1}$ , two
equivalent SSH chains exist as shown in Fig. 10(b). Each of these chains is
the same as the structure of $h_{\mathrm{SSH}}$, leading to no edge states
both near the zero energy (Fig. 7) and in the gap within bulk bands (Fig.
9(b)).

The condition is more complicated when we discuss the AB-stacked BLGs with a
bearded-zigzag edge. For $H_{\mathrm{ber-zig}}^{\swarrow}(k_{y}=\pi)$ with
$\gamma_{3}=0$, the same two isolated $2$-site structures at the end of chain
and repeated decoupled $4$-site structure appear as in previous cases.
Besides, there are also an isolated $1$-site
$h_{1}=\left(U/2\right)c_{1}^{\dagger}c_{1}$ and a $3$-site structure at two
ends of chain

$\displaystyle h_{3}$ $\displaystyle=$ $\displaystyle
w_{eq}c_{1}^{\dagger}c_{2}+v_{eq}c_{1}^{\dagger}c_{3}+\mathrm{H.c.}$
$\displaystyle-\frac{U}{2}\left(c_{1}^{\dagger}c_{1}+c_{2}^{\dagger}c_{2}-c_{3}^{\dagger}c_{3}\right),$

as shown in Fig. 10(c). The two edge states near zero energy are from $h_{1}$
and one of eigenstates of $h_{3}$ with close-to-zero energy. The three edge
states appearing in the gap within bulk bands shown in Fig. 9(c) are naturally
described by one eigenstate of $h_{3}$ with $E<0$ and two eigenstates of two
$h_{2}(U=0)$ with energy $-t$. Again, in the neighbourhood of $k_{y}=\pi$,
states from 1-site/2-site/3-site will mix among themselves instead of mixing
with bulk 4-site states due to energy difference. Thus these states would
remain edge states in this neighbourhood. When $\gamma_{3}\approx\gamma_{1}$,
two SSH chain structures appear with odd number of sites, each of them
provides one edge state near zero energy Han . All other states of odd-site
SSH chain are bulk states. Thus, there are no edge states in the gap within
bulk bands , as shown in Fig. 9(c).

At last we discuss $H_{\mathrm{ber-zig}}^{\nearrow}(k_{y}=\pi)$.
$\gamma_{3}=0$ leads to the repeated decoupled $4$-site structures, one
isolated $1$-site, and one $3$-site structures as shown in Fig. 10(d). The
edge state shown in Fig. 9(d) comes from one eigenstate of $h_{3}$ with $E<0$.
The two near-zero edge states are eigenstate of $h_{1}$ and one of eigenstates
of $h_{3}$. If $\gamma_{3}\approx\gamma_{1}$, two SSH chains structures appear
with odd number of sites. Together they provide two zero-energy edge states,
and no edge states in the gap within bulk bands, as shown in Fig. 9(d).

As a conclusion, number of edge states away from Fermi-energy is determined by
the number of isolated structures when $k_{y}=\pi$, which is in turn dependent
upon the edge configuration of AB-stacked BLGs.

##  V Conclusions and discussions

In this paper, we discussed the existence and topology of edge states in AB-
stacked BLG ribbon with various edge configurations. We illustrated the
correspondence between BLG ribbon and SSH ladder. A detailed topological
classification based on discrete symmetry and topological invariants
calculation for effective 1D bulk Hamiltonian of SSH ladder
$H\left(k_{y},k\right)$ parameterized by $k_{y}$ of AB-stacked BLG ribbon were
constructed,showing the bulk-edge correspondence between zero-energy edge
states of the ribbon and winding number of $H\left(k_{y},k\right)$.

In addition, we found the existence of bias-induced edge states in the
armchair-armchair BLG ribbon. This is not topologically protected, as
discussed in the text. However, this is still an interesting result, as it
implies that edge states can be produced via a mechanism that does not have
anything to do with the edge of systemsfootnote . We also found a topological
phase transition between two topologically non-trivial phases in zigzag-
bearded BLG ribbon, corresponding to a twofold degenerate zero-energy edge
state existing in the whole $k_{y}$ region. We demonstrated that when a edge
state crosses the phase transition point, it will switch both layer and edge.

Moreover, we pointed out that some non-topological edge states without the
protection of the chiral symmetry can be found in the gap within bulk bands
that are away from zero-energy (Fermi level). The existence and number of
these states are sensitive to the edge configurations of BLG ribbons even if
their bulk topologies are the same, which can be simply explained by effective
Hamiltonians $H(k_{y}=\pi)$ for different situations. Though we focus only on
the honeycomb lattice in this paper, it should be obvious that our study can
be generalized to lattices of different shapes, such as Kagome Tian ; Ni ; Nir
and triangular Kuru  lattices, and of higher dimensions, such as the
description of edge states and surface states in three dimensional topological
insulators Teo ; Shan . All of these provide potential directions for further
study.

## Acknowledgments

T. X. Tan would like thank Z. A. Hu and F. R. Fan for useful discussions. C.
Li would like to thank B. Fu for insightful suggestions. The work is support
by the University Grants Committee/Research Grant Council of the Hong Kong SAR
(AoE/P-701/20), the HKU Seed Funding for Strategic Interdisciplinary Research,
and the Croucher Senior Research Fellowship.

*

##  Appendix A The difference between wave vector number $k_{y}$ and $k$

The following example is provided to make the discussion in section II of the
main text more concrete. Consider the ribbon depicted in Fig. 1(a), the
Hamiltonian is given by Eq. (2) of the text:

$\displaystyle H_{bea}$ $\displaystyle=$
$\displaystyle-t\sum_{m=1}^{M}\\{\sum_{n=1}^{N}\left[a_{m,n}^{\dagger}b_{m,n}+b_{m,n}^{\dagger}(a_{m,n+1}+a_{m+1,n+1})\right]$
(66) $\displaystyle-b_{m,N}^{\dagger}(a_{m,1}+a_{m+1,1})+H.c.\\}$

For this specific ribbon, $N=5$. The value of $M$ is unimportant since the
ribbon is infinite (periodic) along that direction. It is a common practice to
assume that there is a periodicity in the operator, i.e.
$a_{i,j}^{(\dagger)}=a_{i+M,j+N}^{(\dagger)}$ and
$b_{i,j}^{(\dagger)}=b_{i+M,j+N}^{(\dagger)}$. Notice that under this
condition, the Hamiltonian given by Eq. (2) is invariant when adding $1$ to
all the first subscript of all operators. But it is not invariant when adding
$1$ to all operators’s second subscript. This difference is due to the term
$b_{m,N}^{\dagger}(a_{m,1}+a_{m+1,1})$, hence we say that this term gives the
open boundary condition.

By basic solid state physics, we are allowed to make Fourier transformation
along the invariant direction, which is the direction of the first subscript.
The Fourier transformation is given by Eq. (3) of the main text, while the
resulting Bloch form Hamiltonian is given by Eq. (4). Consider now doing the
following redefinition of operators, which is always allowed since it does no
affect the band structure.

$\displaystyle a_{k_{y},n}^{\dagger}$ $\displaystyle\rightarrow$
$\displaystyle a_{k_{y},n}^{\dagger}e^{ik_{y}n/2},$ (67) $\displaystyle
b_{k_{y},n}^{\dagger}$ $\displaystyle\rightarrow$ $\displaystyle
b_{k_{y},n}^{\dagger}e^{ik_{y}n/2}$

After this redefinition, the Hamiltonian in Eq. (4) will be transformed into
the form of Eq. (5), which is the following, where $a_{n}$/$b_{n}$ are
shorthand for $a_{k_{y},n}$/$b_{k_{y},n}$

$H_{bea}(k_{y})=\sum_{n=1}^{4}(va_{n}^{\dagger}b_{n}+wb_{n}^{\dagger}a_{n+1})+va_{N}^{\dagger}b_{N}+H.c.$
(68)

From this point on, we will leave MLG ribbon, and instead consider a $10$-site
SSH chain as illustrated in Fig. 11. Its intercell coupling $w$ and intracell
coupling $v$ are defined as following:

$v\equiv-t\quad w\equiv-2tcos(\frac{k_{y}}{2})$ (69)

Readers should refrain from associating $k_{y}$ appearing above with the one
obtained in the Fourier transformation of MLG ribbon, but instead should think
it just as a parameter on which $v$ and $w$ depend. The Hamiltonian of the
$10$-site SSH chain is:

$H_{\mathrm{dimer}}=\sum_{n=1}^{4}(va_{n}^{\dagger}b_{n}+wb_{n}^{\dagger}a_{n+1})+va_{N}^{\dagger}b_{N}+H.c.$
(70)

$H_{\mathrm{dimer}}$ is formally equivalent to $H_{\mathrm{bea}}(k_{y})$, i.e.
$H_{\mathrm{dimer}}\cong H_{\mathrm{bea}}(k_{y})$. However, it is more obvious
that this is an additional trick we can play with this $10$-site SSH chain. We
can make this chain infinite and obtain a Hamiltonian $H_{\mathrm{chain}}$
that can be put in Bloch form, with corresponding Bloch wave vector $k$, as
used in the main text.

$\begin{split}&H_{\mathrm{infinite\,chain}}=\sum_{k\in FBZ}H(k)\\\
&H(k)=\left(v+we^{-ik}\right)a_{k}^{\dagger}b_{k}+H.c.\\\
&=\begin{pmatrix}0&v+we^{-ik}\\\ v+we^{ik}&0\end{pmatrix}\end{split}$ (71)

This $H(k)$ is dependent on $k_{y}$ through $v$ and $w$. This $H(k)$ is what
is referred to as $h_{b}(k_{y},k)$ in Eq. (6) of the text, with which we can
calculate winding number $W$ for different $k_{y}$ using Eq. (7) and Eq. (8)
of the text. For this specific example, the chiral operator $S$ is

$S=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}$ (72)

This procedure can be easily generalized to bilayer case as we use in later
sections.

Figure 11: (color online)

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  * (64) The presence of these edge states is still dependent on edge configuration, since only one of the two edge configurations of BLG arm-arm ribbon show them. However, the mechnism itself, i.e. bias, does not treat the atoms on the edge differently from those deep inside the bulk. This is different from edge states induced by modulting only the edge onsite energy. More specifically, after we impose this bias, the effective $k_{y}-parameterized$ SSH ladder for BLG arm-arm ribbon still has a bulk Hamiltonian, on which we can discuss the topology. However, if we only modulate the onsite energy on the edge of BLG arm-arm ribbon, the effective SSH ladder does not have a bulk Hamiltonian.