# Topological edge states of massless fermion with non-quantized and zero
Berry phases

F. R. Pratama1,2<EMAIL_ADDRESS>Takeshi Nakanishi1
<EMAIL_ADDRESS>1Mathematics for Advanced Materials-OIL, AIST, 2-1-1
Katahira, Aoba, 980-8577 Sendai, Japan
2Advanced Institute for Materials Research, Tohoku University, 2-1-1 Katahira,
Aoba, 980-8577 Sendai, Japan

###### Abstract

We investigate the bulk-boundary correspondence for massless Dirac fermion in
$\alpha$-${T}_{3}$ lattice where the Berry phase can be continuously tuned
from $\pi$ (graphene) to $0$ ($T_{3}$ or dice lattice) without modifying the
energy dispersion. The topological origin of edge states is revealed by
unitary transform of the Hamiltonian, which maps $\alpha$-${T}_{3}$ zigzag
ribbon (ZR) into stub Su-Schrieffer-Heeger (SSH) chain. In the Majorana
representation of the eigenstates, the $\mathbb{Z}_{2}$ invariant is
manifested by azimuthal winding numbers on the Bloch sphere. Particularly,
$T_{3}$ ZR exemplifies a topologically nontrivial system with zero Berry
phase. Contrary to the conventional topological insulators, band gap closing
in the corresponding stub SSH chain is not accompanied by a topological phase
transition.

###### pacs:

72.20.Pa,72.10.-d,73.50.Lw

Topological matters host robust edge states Hasan and Kane (2010) that are
promising for applications in quantum computing and efficient electronic
devices Luo  _et al._ (2022). In the Chern insulators Thouless  _et al._
(1982); Haldane (1988), time-reversal symmetry (TRS) is broken and the number
of edge states is given by $\mathbb{Z}$ invariants. In systems with preserved
TRS, $\mathbb{Z}_{2}$ invariants Kane and Mele (2005); Bernevig  _et al._
(2006); Fu (2011) differentiate between topological insulators (TIs) and
trivial insulators. The simplest model for TI is the Su-Schrieffer-Heeger
(SSH) chain of polyacetylene Su  _et al._ (1979), in which the ratio between
intracell and intercell hopping integrals determine the Berry phase $\Gamma$
in one dimensional (1D) Brillouin zone (BZ), also known as the Zak phase Zak
(1989). Here, the presence and absence of edge states are indicated by
$\Gamma=\pi$ and $0$, respectively Delplace  _et al._ (2011). The existence of
edge states in graphene zigzag ribbon (ZR) can also be inferred from the Zak
phase Delplace  _et al._ (2011). By fixing a proper gauge for the Hamiltonian,
graphene is mapped into the SSH chain Ryu and Hatsugai (2002); Bellec  _et
al._ (2014); St-Jean  _et al._ (2021); Gong  _et al._ (2023). The mapping
establishes a relation between the conserved momentum of graphene ZR and the
hopping parameter of the SSH chain, and therefore, the system is reduced from
2D to 1D.

Consider a honeycomb lattice comprising $A$ and $B$ sites with hopping
parameter $t_{A}>0$. The $\alpha$-$T_{3}$ lattice Raoux  _et al._ (2014) is
constructed by connecting the $B$ sites to the additional $C$ sites at the
center of each hexagon with hopping parameter $t_{C}=\alpha t_{A}$ for
$\alpha\in[0,1]$, as depicted in Fig. 1(a). Here, $\Gamma$ varies continuously
between $\pi$ (graphene, pseudospin $S=1/2$) to $0$ ($T_{3}$ lattice, $S=1$)
for $\alpha=0$ and $1$, respectively, without altering the energy bands when
the energy is scaled by $t=\sqrt{{t_{A}}^{2}+{t_{C}}^{2}}$. A question that
naturally arises from this situation is whether the topological edge states
remain without the quantized $\Gamma$. However, the bulk-boundary
correspondence is yet to be formulated, even though the $\alpha$-$T_{3}$
lattice has been extensively studied due to its unconventional properties,
e.g. tunable orbital susceptibilities from dia- to paramagnetism Raoux  _et
al._ (2014), enhanced Klein tunneling Urban  _et al._ (2011); Fang  _et al._
(2016); Illes and Nicol (2017), and large Chern number $|\mathscr{C}|=2$ when
the TRS is broken Wang and Ran (2011); Dey and Ghosh (2019); Dey  _et al._
(2020); Mohanta  _et al._ (2023). Although the $\alpha$-$T_{3}$ and $T_{3}$
models were originally proposed Raoux  _et al._ (2014); Bercioux  _et al._
(2009) to be realized in optical lattices, the $\alpha$-$T_{3}$ Hamiltonian
can be employed to describe the optical properties of 2D
$\mathrm{Hg}_{1-x}\mathrm{Cd}_{x}\mathrm{Te}$ Orlita  _et al._ (2014); Malcolm
and Nicol (2015) at the critical doping $x\approx 0.17$. Furthermore, the
$T_{3}$ lattice is predicted to occur in perovskite-based heterostructuresWang
and Ran (2011); Köksal  _et al._ (2023) and strained blue-phosphorene oxide
Zhu  _et al._ (2016).

In this study, an example of $\alpha$-$T_{3}$ ZR [Fig. 1(a)] is mapped into
stub SSH chain Bartlett  _et al._ (2021); Cáceres-Aravena  _et al._ (2022)
[Fig. 1(b)] by unitary transform of the Hamiltonian. The stub SSH chain is
constructed by connecting an additional site to one of the sites in the SSH
dimer. In the Majorana representation of the eigenstates Majorana (1932);
Hannay (1998); Liu and Fu (2014), the $\mathbb{Z}_{2}$ invariant is defined
from the azimuthal winding number on the Bloch sphere Bartlett  _et al._
(2021). Thus, the origin of edge states in the $\alpha$-$T_{3}$ ZR can be
understood from the topological phase of the stub SSH chain. In particular,
$T_{3}$ ZR is topologically nontrivial despite $\Gamma=0$. In contrast to the
conventional TIs, the corresponding stub SSH chain is not subject to a
topological phase transition by band gap closing.

The $\alpha$-$T_{3}$ ZR in Fig. 1(a) is constructed by periodically
translating the unit cell (dashed hexagon) along the $x$ direction with
primitive vector $\bm{a}_{1}$. The width of the ZR is specified by $J$, which
is the number of trimer along the $y$ axis. The vector $\bm{L}$ connects the
missing $B$ and $C$ sites at $m=0$ and $3J+1$, respectively. The Hamiltonian
in the momentum space $\bm{k}=(k_{x},k_{y})$ is given by

$\displaystyle h_{\bm{k}}=-t\begin{pmatrix}0&\cos\vartheta f_{\bm{k}}&0\\\
\cos\vartheta f_{\bm{k}}^{*}&0&\sin\vartheta f_{\bm{k}}\\\ 0&\sin\vartheta
f_{\bm{k}}^{*}&0\end{pmatrix},$ (1)

where $\vartheta=\tan^{-1}\alpha$, and
$f_{\bm{k}}=\sum_{\ell=1}^{3}e^{-i\bm{k}\cdot\bm{b}_{\ell}}=|f_{\bm{k}}|e^{-i\varphi_{\bm{k}}}$.
$\bm{b}_{\ell}$’s are the nearest-neighbor vectors. The eigenvalues of
$h_{\bm{k}}$ include the energy bands of graphene
$\varepsilon_{s,\bm{k}}=st|f_{\bm{k}}|$, where $s=+1$ ($-1$) corresponds to
valence (conduction) band, and a flat band at zero energy
$\varepsilon_{0,\bm{k}}=0$.

By using the eigenstates $|\psi_{\bm{k},c}\rangle$ for the band
$c\in\\{s,0\\}$, the Berry phase $\Gamma_{c}$ is calculated by integration
along a path enclosing a Dirac point. The results are
$\Gamma_{s,\tau}=\pi\tau\cos(2\vartheta)$ and
$\Gamma_{0,\tau}=-2\pi\tau\cos(2\vartheta)$, where $\tau=+1$ ($-1$) for the
$K$ ($K^{\prime}$) valley Raoux  _et al._ (2014). Thus, $\Gamma_{c}$’s are not
quantized and differ for each valley when $\alpha\notin\\{0,1\\}$.

Since there are several ways to define the unit cell (each consists of the
unique configuration of trimer), distinct $\alpha$-$T_{3}$ ZRs can be
constructed by periodic translation of the unit cells along the direction of
$\bm{a}_{1}$. The ZR is characterized by the boundary conditions imposed on
the wavefunction
${\psi}_{s}=\begin{pmatrix}\psi_{s}^{A}&\psi_{s}^{B}&\psi_{s}^{C}\end{pmatrix}^{T}$
in the real space $\bm{r}=(x,y)$. ${\psi}_{s}(\bm{r})$ is given by a linear
combination of the Bloch state
$\psi_{s,\bm{k}}(\bm{r})=e^{i\bm{k}\cdot\bm{r}}|\psi_{s,\bm{k}}\rangle$. In
Fig. 1(a),

$\displaystyle{\psi}_{s}^{B}(\bm{O}+l\bm{a}_{1})$ $\displaystyle=0,$ (2)
$\displaystyle\alpha\left[{\psi}_{s}^{C}(\bm{L}+l\bm{a}_{1})+{\psi}_{s}^{C}(\bm{L}+(l-1)\bm{a}_{1})\right]+$
$\displaystyle{\psi}_{s}^{A}(\bm{L}-\bm{b}_{2}+l\bm{a}_{1})$
$\displaystyle=0,$ (3)

where $\bm{O}=(0,0)$ is the origin and $l\in\mathbb{Z}$. Eq. (2) describes the
missing $B$ sites at $m=0$. Eq. (3) describes the terminations of bonds
connecting the $B$ sites at $m=3J$ to the nearest $A$ and $C$ sites outside
the ZR. For the bulk states Delplace  _et al._ (2011), we adopt
${\psi}_{s}(\bm{r})=\mathcal{A}\psi_{s,\bm{k}}(\bm{r})+\mathcal{A}^{\prime}\psi_{s,\bm{k^{\prime}}}(\bm{r})$.
Eq. (2) implies $\mathcal{A}^{\prime}=-\mathcal{A}$, and
$\bm{k}\cdot\bm{a}_{1}=\bm{k^{\prime}}\cdot\bm{a}_{1}$ due to the conservation
of momentum in the $x$ direction. Additionally, the conservation of energy
$\varepsilon_{\bm{k}}=\varepsilon_{\bm{k^{\prime}}}$ gives
$\bm{k}\cdot\bm{a}_{1}=\bm{k^{\prime}}\cdot\bm{a}_{1}=(\bm{k}+\bm{k^{\prime}})\cdot\bm{a}_{2}$
and therefore $\bm{k^{\prime}}=(k_{x},-k_{y})$.

Now let us define a unitary matrix

$\displaystyle
M_{\bm{k}}=\begin{pmatrix}e^{i\bm{k}\cdot(\bm{b}_{2}-\bm{a}_{1}/2)}\cos\gamma&0&e^{-i\bm{k}\cdot\bm{b}_{3}}\sin\gamma\\\
0&1&0\\\
-e^{i\bm{k}\cdot(\bm{b}_{2}-\bm{a}_{1}/2)}\sin\gamma&0&e^{-i\bm{k}\cdot\bm{b}_{3}}\cos\gamma\end{pmatrix}.$
(4)

By setting $\gamma=\tan^{-1}(\alpha\beta)$ with $\beta=2\cos(k_{x}a/2)\geq 0$
for $k_{x}\in[-\pi/a,\pi/a]$, $h_{\bm{k}}$ is transformed into the Hamiltonian
of the stub SSH chain in Fig. 1(b) as follows:

$\displaystyle
H_{\bm{k}}=M_{\bm{k}}h_{\bm{k}}M_{\bm{k}}^{\dagger}=-t_{M}\begin{pmatrix}0&\mathcal{F}_{\bm{k}}&0\\\
\mathcal{F}_{\bm{k}}^{*}&0&v_{C}\\\ 0&v_{C}&0\end{pmatrix},$ (5)

where $t_{M}=t_{A}\cos\gamma$ and
$\mathcal{F}_{\bm{k}}=v_{A}+v_{A}^{\prime}e^{-i\sqrt{3}k_{y}a/2}=|\mathcal{F}_{\bm{k}}|e^{-i\Phi_{\bm{k}}}$.
In each unit cell (dashed rectangle), the positions of all sites in the $y$
axis are identically given by $Y_{j}=j\sqrt{3}a/2$.
$v_{A}=\beta(1+\alpha^{2})$ [$v_{A}^{\prime}=1+\alpha^{2}\beta^{2}$] is the
intracell [intercell] couplings between the $B$ and $A$ sites.
$v_{C}=\alpha(1-\beta^{2})$ is the coupling between the $B$ and $C$ sites.
Thus, the unitary transform reduces the $\alpha$-$T_{3}$ ZR into the stub SSH
chain by mapping $k_{x}$ into the hopping parameters. The eigenvalues of
$H_{\bm{k}}$ are
$E_{s,{\bm{k}}}=st_{M}\sqrt{|\mathcal{F}_{\bm{k}}|^{2}+{v_{C}}^{2}}$ and
$E_{0,\bm{k}}=0$. The case $v_{C}=0$ corresponds to the SSH chain (with
additional $E_{0,\bm{k}}=0$ originating from the uncoupled $C$ sites), where
the gap closing at $|k_{y}|=2\pi/\sqrt{3}a$ indicates a topological phase
transition when $v_{A}/v_{A}^{\prime}=1$. On the other hand, by denoting
$\Phi_{{\bm{k}}}$ at $k_{y}=\pm 2\pi/\sqrt{3}a$ as $\Phi_{\pm}$, it is noted
that regardless the value of $v_{C}$,

$\displaystyle\Phi_{\pm}=\begin{cases}\pm\pi~{}\mathrm{for}~{}v_{A}/v_{A}^{\prime}<1,\\\
0~{}\mathrm{for}~{}v_{A}/v_{A}^{\prime}>1.\end{cases}$ (6)

Eq. (6) will be employed to evaluate the $\mathbb{Z}_{2}$ invariant for the
stub SSH chain. By defining
$\Theta_{\bm{k}}=\tan^{-1}(v_{C}/|\mathcal{F}_{\bm{k}}|)$, the eigenstates are
given by

$\displaystyle|\Psi_{s,\bm{k}}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}\cos\Theta_{\bm{k}}e^{-i\Phi_{\bm{k}}}\\\
s\\\
\sin\Theta_{\bm{k}}\end{pmatrix},~{}|\Psi_{0,\bm{k}}\rangle=\begin{pmatrix}\sin\Theta_{\bm{k}}e^{-i\Phi_{\bm{k}}}\\\
0\\\ -\cos\Theta_{\bm{k}}\end{pmatrix}.$ (7)

Figure 1:  (a) The $\alpha$-$T_{3}$ ZR. The hopping parameter between the $B$
and $A$ ($C$) sites is $t_{A}$ ($t_{C}$), where $t_{C}/t_{A}=\alpha\in[0,1]$.
The dashed hexagon depicts the unit cell. $\bm{b}_{1}=-a(1/2,1/2\sqrt{3})$,
$\bm{b}_{2}=a(1/2,-1/2\sqrt{3})$, and $\bm{b}_{3}=a(0,1/\sqrt{3})$ are the
nearest-neighbor vectors. $\bm{a}_{1}=a(1,0)$ and
$\bm{a}_{2}=a(1/2,\sqrt{3}/2)$ are the primitive vectors. $m=1,2,\dots,3J$
indicate the position of site in the $y$ axis, where $J$ is the number of
trimers. $\bm{L}=(L_{x},L_{y})$ connects the missing $B$ and $C$ sites at
$m=0$ and $m=3J+1$, respectively, where $L_{y}=\sqrt{3}(J+1)a/2-a/\sqrt{3}$.
(b) The stub SSH chain. $t_{M}v_{A}$ ($t_{M}v_{A}^{\prime}$) is the intracell
(intercell) hopping parameters between the $B$ and $A$ sites. The hopping
parameter between the $B$ and $C$ sites is $t_{M}v_{C}$. The position unit
cell (dashed rectangle) is $Y_{j}=j\sqrt{3}a/2$.  Figure 2:  (a) Energy
spectra of the $\alpha$-$T_{3}$ ZR as a function of $k_{x}$ for $\alpha=0.8$
and $J=40$. The bold red lines indicate the edge states. The transitions from
bulk to edge states occur at $|k_{x}|\approx 0.3434\pi/a$ and $0.6991\pi/a$,
accordingly marked by the vertical dashed and dotted lines. Probability
density for the in-gap states per site $m$ for (b) $k_{x}=0.2\pi/a$, (c)
$0.5\pi/a$, (d) $0.7\pi/a$, and (e) $0.8\pi/a$.

The wavefunction of the chain is denoted by
$\Psi_{s}=\begin{pmatrix}\Psi_{s}^{A}&\Psi_{s}^{B}&\Psi_{s}^{C}\end{pmatrix}^{T}$.
As illustrated by Fig. 1(b), the missing $B$ ($A$) site at $Y_{0}$ ($Y_{J+1}$)
necessitates

$\displaystyle{\Psi}_{s}^{B}(Y_{0})$ $\displaystyle=0,$ (8)
$\displaystyle{\Psi}_{s}^{A}(Y_{J+1})$ $\displaystyle=0.$ (9)

The equivalence between the $\alpha$-$T_{3}$ ZR in Fig. 1(a) and the stub SSH
chain in Fig. 1(b) is established from the boundary conditions. It can be
shown that Eqs. (8) and (9) are identical to Eqs. (2) and (3), respectively,
because $\Psi_{s}(\bm{r})=M_{\bm{k}}\psi_{s}(\bm{r})$. Since
$\Psi_{s}(y)=\Psi_{s,\bm{k}}(y)-\Psi_{s,\bm{k^{\prime}}}(y)$, where
$\Psi_{s,\bm{k}}(y)=e^{ik_{y}y}|\Psi_{s,\bm{k}}\rangle$, the quantization
condition of $k_{y}$ is derived from Eq. (9) as follows:

$\displaystyle\sqrt{3}(J+1)k_{y}a/2-\Phi_{\bm{k}}=n\pi,~{}\mathrm{for}~{}n=1,2,\dots,J.$
(10)

Eq. (10) is solved graphically along $k_{y}\in(0,2\pi/\sqrt{3}a)$
[$\Psi_{s}(y)=0$ at $k_{y}=0$ and $2\pi/\sqrt{3}a$]. It is noted that Eq. (10)
also holds for the SSH chain because $\Phi_{\bm{k}}$ does not depend on
$v_{C}$. The missing solutions of Eq. (10) indicate the emergence of edge
states Delplace  _et al._ (2011) when

$\displaystyle v_{A}/v_{A}^{\prime}<1-1/(J+1).$ (11)

The ratio $(v_{A}/v_{A}^{\prime})_{J}=1-1/(J+1)$ is the critical value at
which the transition of bulk to edge states occur in a finite system Delplace
_et al._ (2011). On the other hand, the inversion symmetry is broken for
$v_{C}\neq 0$ Bartlett  _et al._ (2021). Subsequently, the existence of edge
states can not be predicted from the Zak phase, which is no longer quantized
to $\pi$ or $0$ Zak (1989).

The existence of edge states in the $k_{x}$ space of the $\alpha$-$T_{3}$ ZR
can be straightforwardly determined from Eq. (11). By inserting the
definitions of $v_{A}$ and $v_{A}^{\prime}$, we obtain

$\displaystyle\alpha^{2}\beta^{2}-\frac{1+\alpha^{2}}{1-(J+1)^{-1}}\beta+1>0.$
(12)

For $\alpha\neq 0$, the solutions of Eq. (12) are $\beta_{J}^{+}<\beta\leq 2$
and $0\leq\beta<\beta_{J}^{-}$, where
$\beta_{J}^{\pm}=2\cos(\chi_{J}^{\pm}a/2)$. Here, $\chi_{J}^{\pm}$ are the
critical values of $|k_{x}|\in[0,\pi/a]$ at which bulk states undergo
transitions to edge states appearing in the range

$\displaystyle
0\leq|k_{x}|<\chi_{J}^{+}~{}\mathrm{and}~{}\chi_{J}^{-}<|k_{x}|\leq{\pi}/{a}.$
(13)

In the limit $J\rightarrow\infty$, $\chi_{J}^{\pm}$ converge to

$\displaystyle\chi_{\infty}^{+}\sim({2}/{a})\cos^{-1}\left({1}/{2\alpha^{2}}\right),~{}\chi_{\infty}^{-}\sim{2\pi}/{3a}.$
(14)

Since $\chi_{\infty}^{+}$ is undefined for $\alpha\in(0,1/\sqrt{2})$, the
range of existence for the edge states is identical to that of the graphene ZR
Wakabayashi  _et al._ (2010); Delplace  _et al._ (2011). The edge states in
the $T_{3}$ ZR ($\alpha=1$) exist at all $|k_{x}|$ excluding
$\chi_{\infty}^{+}=\chi_{\infty}^{-}=2\pi/3a$.

The energy of the edge states $\tilde{\varepsilon}_{s}$ is deduced as follows.
In the SSH chain, $\tilde{E}_{s}\sim 0$ due to the chiral symmetry Ryu and
Hatsugai (2002); Delplace  _et al._ (2011). Accordingly for $v_{C}\neq 0$,
$\tilde{E}_{s}\sim st_{M}|v_{C}|$, where the state $s=-1$ ($+1$) is analogous
to the bonding (antibonding) molecular orbital. Since the eigenvalues are
invariant under the unitary transform,
$\tilde{\varepsilon}_{s}=\tilde{E}_{s}=st_{C}\cos\gamma|1-\beta^{2}|$. Thus,
the edge states in the $\alpha$-$T_{3}$ ZR are dispersive due to the
dependence on $\beta$. At $|k_{x}|\approx 2\pi/3a$ and $|k_{x}|=\pi/a$, we get
$\tilde{\varepsilon}_{s}\approx 0$ and $\tilde{\varepsilon}_{s}=st_{C}$,
respectively.

Fig. 2(a) shows the energy spectra as a function of $k_{x}$ for $\alpha=0.8$
and $J=40$. In addition to the dispersive bands, the spectra include a
$J$-fold degenerate flat band at $\varepsilon=0$. The vertical dashed and
dotted lines are located at $|k_{x}|=\chi_{40}^{+}\approx 0.3434\pi/a$ and
$\chi_{40}^{-}\approx 0.6991\pi/a$, respectively. The bold red lines depict
the edge states. Figs. 2(b)-(e) show the probability density $|\psi|^{2}$ of
the in-gap states at the site $m$ at several $k_{x}$’s. Figs. 2(b) and (e)
clearly show the spatial distribution of asymmetric edge states, where
$|\psi|^{2}$’s are concentrated around the top edge of the ZR and
exponentially decay towards the bottom edge. In Fig. 2(d), the decay of
$|\psi|^{2}$ is not pronounced because the value of $k_{x}$ is close to
$\chi_{40}^{-}$. Fig. 2(c) exhibits the profile of bulk states where
$|\psi|^{2}$ is extended throughout the ZR.

Figure 3:  Trajectories of the Majorana stars on the Bloch sphere (top panels)
and the $xy$ plane (bottom panels) for $\alpha=0.8$, at (a) $k_{x}=0.2\pi/a$,
(b) $\chi_{\infty}^{+}\approx 0.4292\pi/a$, (c) $0.5\pi/a$, (d)
$\chi_{\infty}^{-}=2\pi/3a$, (e) $0.7\pi/a$. The red and yellow [blue and
purple] curves correspond to the valence [conduction] band. The light- and
dark-green curves correspond to the flat band.

The edge states are topologically characterized using the Majorana
representation Majorana (1932); Liu and Fu (2014). By treating the eigenstate
$|\Psi\rangle$ of an $N\times N$ Hamiltonian as a pseudospinor for
$S=(N-1)/2$, $|\Psi\rangle$ can be represented by a set of the Pauli
pseudospinors
$|\zeta_{\mu}\rangle=\begin{bmatrix}\cos(\eta_{\mu}/2)&\sin(\eta_{\mu}/2)e^{i\xi_{\mu}}\end{bmatrix}^{T}$,
where $\eta$ and $\xi$ are the polar and azimuthal angles of the Bloch sphere.
The trajectory of each Majorana ’star’
$\bm{\zeta}_{\mu}=\langle\zeta_{\mu}|\bm{\sigma}|\zeta_{\mu}\rangle$ thus
visualizes the evolution of $|\Psi\rangle$.

In the Schwinger theory of angular momentum Schwinger (1952), $|\Psi\rangle$
can be constructed by the bosonic creation operators
$\hat{a}_{\uparrow,\downarrow}^{\dagger}$ acting on the vacuum state
$|\varnothing\rangle$ as follows:

$\displaystyle|\Psi\rangle=\frac{\mathcal{C}_{S}}{\sqrt{(2S)!}}\prod_{\mu=1}^{2S}\left[\hat{a}_{\uparrow}^{\dagger}+\lambda_{\mu}\hat{a}_{\downarrow}^{\dagger}\right]|\varnothing\rangle,$
(15)

where $\mathcal{C}_{S}$ is the amplitude of $|\Psi\rangle$, and
$\lambda_{\mu}=\tan(\eta_{\mu}/2)e^{i\xi_{\mu}}$ is obtained by solving the
Majorana polynomial Majorana (1932)

$\displaystyle\sum_{\mu=0}^{2S}(-1)^{\mu}\frac{\mathcal{C}_{S}-\mu}{\sqrt{(2S-\mu)!\mu!}}\lambda^{2S-\mu}=0.$
(16)

In particular for the stub SSH chain, $S=1$ and
$|\Psi\rangle=\begin{pmatrix}\mathcal{C}_{1}&\mathcal{C}_{0}&\mathcal{C}_{-1}\end{pmatrix}^{T}$
is given by Eq. (7). For $c=s$ and $c=0$, Eq. (16) accordingly reduces to

$\displaystyle\cos\Theta_{\bm{k}}e^{-i\Phi_{\bm{k}}}{\lambda^{2}}-\sqrt{2}s\lambda+{\sin\Theta_{\bm{k}}}=0,$
(17a)
$\displaystyle{\sin\Theta_{\bm{k}}}e^{-i\Phi_{\bm{k}}}{\lambda^{2}}-{\cos\Theta_{\bm{k}}}=0.$
(17b)

Let $\lambda_{s,\bm{k}}^{\mp}$ and $\lambda_{0,\bm{k}}^{\mp}$ be the roots of
Eqs. (17a) and (17b), respectively. From the coefficients of the quadratic
equations, it is clear that
$\Lambda_{s,\bm{k}}=\prod_{\nu=\mp}\lambda_{s,\bm{k}}^{\nu}=\tan\Theta_{\bm{k}}e^{i\Phi_{\bm{k}}}$,
and
$\Lambda_{0,\bm{k}}=\prod_{\nu=\mp}\lambda_{0,\bm{k}}^{\nu}=-\mathrm{cot}\Theta_{\bm{k}}e^{i\Phi_{\bm{k}}}$.

The $\mathbb{Z}_{2}$ topological index is defined by the azimuthal winding
number $W_{c}$, which counts for the total number of times
$\bm{\zeta}_{c,\bm{k}}^{\nu}$’s travel around the $z$ axis of the Bloch
sphere. $W_{c}$ is calculated as follows Bartlett  _et al._ (2021):

$\displaystyle{W}_{c}=\frac{1}{2\pi}\int_{\mathrm{BZ}}dk_{y}\frac{\partial}{\partial
k_{y}}\sum_{\nu=\mp}\xi_{c,\bm{k}}^{\nu}.$ (18)

Since
$\lambda_{c,\bm{k}}^{\nu}=\tan(\eta_{c,\bm{k}}^{\nu}/2)\mathrm{exp}({i\xi_{c,\bm{k}}^{\nu}})$
by definition, therefore $\sum_{\nu=\mp}\xi_{c,\bm{k}}^{\nu}=\Phi_{\bm{k}}$.
By inserting Eq. (6) into Eq. (18),

$\displaystyle{W}_{c}=\begin{cases}1~{}\mathrm{for}~{}v_{A}/v_{A}^{\prime}<1,\\\
0~{}\mathrm{for}~{}v_{A}/v_{A}^{\prime}>1.\end{cases}$ (19)

As $k_{y}$ traverses across the 1D BZ, the trajectories of
$\bm{\zeta}_{c,\bm{k}}^{\nu}$’s enclose the $z$ axis at $k_{x}$’s that carry
the edge states.

Figs. 3(a)-(e) show the trajectories of $\bm{\zeta}_{c,\bm{k}}^{\nu}$’s on the
Bloch sphere (top panels) and the $xy$ plane (bottom panels) for $\alpha=0.8$
at (a) $k_{x}=0.2\pi/a$, (b) $\chi_{\infty}^{+}\approx 0.4292\pi/a$, (c)
$0.5\pi/a$, (d) $\chi_{\infty}^{-}=2\pi/3a$, and (e) $0.7\pi/a$. The
red/yellow, light-green/dark-green, and blue/purple curves correspond to
$c=-1$, $0$, and $+1$, respectively. In Figs. 3(a) and (e), the red, purple,
and combined green loops wind around the $z$ axis once, thus $W_{c}=1$.
Conversely, in Fig. 3(c), none of the loops enclose the $z$ axis, so
$W_{c}=0$. At $\chi_{\infty}^{\pm}$, $W_{c}$ becomes ill-defined for different
reasons. In Fig. 3(b), the red, purple, and combined green curves intersect
the poles. This behavior indicates a topological phase transition without gap
closing Ezawa  _et al._ (2013). In Fig. 3(d), the trajectories of
$\zeta_{c}^{\nu}$’s are discontinuous due to the gap closing at
$|k_{y}|=2\pi/\sqrt{3}a$. Here, the system is reduced to the metallic SSH
chain because $\beta=1$ and consequently $v_{C}=0$, $v_{A}=v_{A1}^{\prime}$.
In the case of $T_{3}$ ZR, $W_{c}=1$ for all except at $|k_{x}|=2\pi/3a$.
Thus, the gap closing in the corresponding stub SSH chain is not accompanied
by a topological phase transition.

Previously, Liu and Wakabayashi Liu and Wakabayashi (2017) proposed a 2D SSH
model Liu and Wakabayashi (2017); Obana  _et al._ (2019) exhibiting a
topologically non-trivial phase with zero Berry curvature, due to the
conserved TRS and inversion symmetry. The $\mathbb{Z}_{2}$ invariant is given
by 2D Zak phase calculated by the integration of the Berry connection over the
BZ. In this study, we identify $T_{3}$ ZR as a topologically non-trivial
system with zero Berry phase, where the $\mathbb{Z}_{2}$ invariant is defined
by $W_{c}$.

Fig. 4 shows the topological phase diagram of the $\alpha$-$T_{3}$ ZR. The
dark (light) region indicates the non-trivial (trivial) phase, where the
presence (absence) of edge states are characterized by $W_{c}=1$ ($W_{c}=0$).
The topological phase transition with (without) gap closing is marked by the
brown (orange) line.

Since the Hamiltonians of the $\alpha$-$T_{3}$ lattice and stub SSH chain are
equivalent, the bulk topological property is conserved when the dimension is
compacted. Recently, Cáceres et al. Cáceres-Aravena  _et al._ (2022) realized
the stub SSH chain in an experiment using a photonic lattice. They
demonstrated that for $v_{A}/v_{A}^{\prime}<1$, the edge states appear at
$E_{s}=st|v_{c}|$ (consistent with the present result), and persist against
disorders that perturb $v_{A}$ and $v_{A}^{\prime}$. We analytically prove
that the edge states are characterized by $W_{c}$, which may explain the
topological origin of the robustness.

Figure 4:  Topological phase diagram of the $\alpha$-$T_{3}$ ZR. The dark
(light) region indicates the non-trivial (trivial) phase, where the edge
states are present (absent). The brown (orange) line marks the phase
transition without (with) the gap closing.

In summary, we formulate the bulk-boundary correspondence for the massless
Dirac fermion with a non-quantized Berry phase in the $\alpha$-$T_{3}$
lattice, which is mapped into the stub SSH chain by the unitary transform of
the Hamiltonian. In the Majorana representation of the eigenstates, the
$\mathbb{Z}_{2}$ topological index is defined from the azimuthal winding
number on the Bloch sphere. Our method also shows that the $T_{3}$ lattice is
topologically non-trivial despite the zero Berry phase. The existence of edge
states in distinct $\alpha$-$T_{3}$ ribbon configurations are discussed in
detail in the accompanying paper Pratama and Nakanishi (2024).

We thank R. Saito, N. T. Hung, S. Hayashi, and E. H. Hasdeo for helpful
discussions. This work is partly supported by Grant-in-Aid for Scientific
Research, JSPS KAKENHI (23K03293), and JST-CREST (JPMJCR18T1).

## References

  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045 (2010).
  * Luo  _et al._ (2022) H. Luo, P. Yu, G. Li, and K. Yan, “Topological quantum materials for energy conversion and storage,” Nat. Rev. Phys. 4, 611 (2022).
  * Thouless  _et al._ (1982) D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, “Quantized Hall conductance in a two-dimensional periodic potential,” Phy. Rev. Lett. 49, 405 (1982).
  * Haldane (1988) F. D. M. Haldane, “Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the” parity anomaly”,” Phys. Rev. Lett. 61, 2015 (1988).
  * Kane and Mele (2005) C. L. Kane and E. J. Mele, “Quantum spin Hall effect in graphene,” Phys. Rev. Lett. 95, 226801 (2005).
  * Bernevig  _et al._ (2006) B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, “Quantum spin Hall effect and topological phase transition in hgte quantum wells,” Science 314, 1757 (2006).
  * Fu (2011) L. Fu, “Topological crystalline insulators,” Phys. Rev. Lett. 106, 106802 (2011).
  * Su  _et al._ (1979) W.-P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698 (1979).
  * Zak (1989) J. Zak, “Berry’s phase for energy bands in solids,” Phys. Rev. Lett. 62, 2747 (1989).
  * Delplace  _et al._ (2011) P. Delplace, D. Ullmo, and G. Montambaux, “Zak phase and the existence of edge states in graphene,” Phys. Rev. B 84, 195452 (2011).
  * Ryu and Hatsugai (2002) S. Ryu and Y. Hatsugai, “Topological origin of zero-energy edge states in particle-hole symmetric systems,” Phys. Rev. Lett. 89, 077002 (2002).
  * Bellec  _et al._ (2014) M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Manipulation of edge states in microwave artificial graphene,” New J. Phys. 16, 113023 (2014).
  * St-Jean  _et al._ (2021) P. St-Jean, A. Dauphin, P. Massignan, B. Real, O. Jamadi, M. Milicevic, A. Lemaitre, A. Harouri, L. Le Gratiet, I. Sagnes, _et al._ , “Measuring topological invariants in a polaritonic analog of graphene,” Phys. Rev. Lett. 126, 127403 (2021).
  * Gong  _et al._ (2023) Z. Gong, F. Yang, and J. Yao, “Mapping topological states with compacted dimensions,” Phys. Rev. B 107, L081404 (2023).
  * Raoux  _et al._ (2014) A. Raoux, M. Morigi, J.-N. Fuchs, F. Piéchon, and G. Montambaux, “From dia-to paramagnetic orbital susceptibility of massless fermions,” Phys. Rev. Lett. 112, 026402 (2014).
  * Urban  _et al._ (2011) D. F. Urban, D. Bercioux, M. Wimmer, and W. Häusler, “Barrier transmission of Dirac-like pseudospin-one particles,” Phys. Rev. B 84, 115136 (2011).
  * Fang  _et al._ (2016) A. Fang, Z. Zhang, S. G. Louie, and C. T. Chan, “Klein tunneling and supercollimation of pseudospin-1 electromagnetic waves,” Phys. Rev. B 93, 035422 (2016).
  * Illes and Nicol (2017) E. Illes and E. Nicol, “Klein tunneling in the $\alpha$-${T}_{3}$ model,” Phys. Rev. B 95, 235432 (2017).
  * Wang and Ran (2011) F. Wang and Y. Ran, “Nearly flat band with Chern number ${C}=2$ on the dice lattice,” Phys. Rev. B 84, 241103 (2011).
  * Dey and Ghosh (2019) B. Dey and T. K. Ghosh, “Floquet topological phase transition in the $\alpha$-${T}_{3}$ lattice,” Phys. Rev. B 99, 205429 (2019).
  * Dey  _et al._ (2020) B. Dey, P. Kapri, O. Pal, and T. K. Ghosh, “Unconventional phases in a Haldane model of dice lattice,” Phys. Rev. B 101, 235406 (2020).
  * Mohanta  _et al._ (2023) N. Mohanta, R. Soni, S. Okamoto, and E. Dagotto, “Majorana corner states on the dice lattice,” Commun. Phys. 6, 240 (2023).
  * Bercioux  _et al._ (2009) D. Bercioux, D. Urban, H. Grabert, and W. Häusler, “Massless Dirac-Weyl fermions in $\alpha$-${T}_{3}$ optical lattice,” Phys. Rev. A 80, 063603 (2009).
  * Orlita  _et al._ (2014) M. Orlita, D. Basko, M. Zholudev, F. Teppe, W. Knap, V. Gavrilenko, N. Mikhailov, S. Dvoretskii, P. Neugebauer, C. Faugeras, _et al._ , “Observation of three-dimensional massless kane fermions in a zinc-blende crystal,” Nat. Phys. 10, 233 (2014).
  * Malcolm and Nicol (2015) J. D. Malcolm and E. J. Nicol, “Magneto-optics of massless Kane fermions: Role of the flat band and unusual Berry phase,” Phys. Rev. B 92, 035118 (2015).
  * Köksal  _et al._ (2023) O. Köksal, L. Li, and R. Pentcheva, “High Chern numbers in a perovskite-derived dice lattice $\mathrm{({L}a{X}{O}_{3})_{3}}$/$\mathrm{({L}a{A}l{O}_{3})_{3}(111)}$ with X= Ti, Mn and Co,” Sci. Rep. 13, 10615 (2023).
  * Zhu  _et al._ (2016) L. Zhu, S.-S. Wang, S. Guan, Y. Liu, T. Zhang, G. Chen, and S. A. Yang, “Blue phosphorene oxide: strain-tunable quantum phase transitions and novel 2d emergent fermions,” Nano Lett. 16, 6548 (2016).
  * Bartlett  _et al._ (2021) J. Bartlett, H. Hu, and E. Zhao, “Illuminating the bulk-boundary correspondence of a non-hermitian stub lattice with Majorana stars,” Phys. Rev. B 104, 195131 (2021).
  * Cáceres-Aravena  _et al._ (2022) G. Cáceres-Aravena, B. Real, D. Guzmán-Silva, A. Amo, L. E. F. Torres, and R. A. Vicencio, “Experimental observation of edge states in SSH-stub photonic lattices,” Phys. Rev. Res. 4, 013185 (2022).
  * Majorana (1932) E. Majorana, “Atomi orientati in campo magnetico variabile,” Il Nuovo Cimento (1924-1942) 9, 43 (1932).
  * Hannay (1998) J. Hannay, “The Berry phase for spin in the Majorana representation,” J. Phys. A: Math. Gen. 31, L53 (1998).
  * Liu and Fu (2014) H. Liu and L. Fu, “Representation of Berry phase by the trajectories of Majorana stars,” Phys. Rev. Lett. 113, 240403 (2014).
  * Wakabayashi  _et al._ (2010) K. Wakabayashi, K.-i. Sasaki, T. Nakanishi, and T. Enoki, “Electronic states of graphene nanoribbons and analytical solutions,” Sci. Technol. Adv. Mater. 11, 054504 (2010).
  * Schwinger (1952) J. Schwinger, “On angular momentum,” US Atomic Energy Commission, Report No. NYO-3071 (1952).
  * Ezawa  _et al._ (2013) M. Ezawa, Y. Tanaka, and N. Nagaosa, “Topological phase transition without gap closing,” Sci. Rep. 3, 2790 (2013).
  * Liu and Wakabayashi (2017) F. Liu and K. Wakabayashi, “Novel topological phase with a zero Berry curvature,” Phys. Rev. Lett. 118, 076803 (2017).
  * Obana  _et al._ (2019) D. Obana, F. Liu, and K. Wakabayashi, “Topological edge states in the Su-Schrieffer-Heeger model,” Phys. Rev. B 100, 075437 (2019).
  * Pratama and Nakanishi (2024) F. R. Pratama and T. Nakanishi, “Majorana representation for topological edge states of massless Dirac fermion with non-quantized Berry phase,” Submitted (2024).