# Nearly flat Chern band in periodically strained monolayer and bilayer
graphene

Xiaohan Wan  Theoretical Division, T-4 and CNLS, Los Alamos National
Laboratory, Los Alamos, New Mexico 87545, USA  Department of Physics,
University of Michigan, Ann Arbor, MI 48109, USA  Siddhartha Sarkar
Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA  Kai
Sun<EMAIL_ADDRESS>Department of Physics, University of Michigan, Ann
Arbor, MI 48109, USA  Shi-Zeng Lin<EMAIL_ADDRESS>Theoretical Division, T-4 and
CNLS, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
Center for Integrated Nanotechnologies (CINT), Los Alamos National Laboratory,
Los Alamos, New Mexico 87545, USA

###### Abstract

The flat band is a key ingredient for the realization of interesting quantum
states for novel functionalities. In this work, we investigate the conditions
for the flat band in both monolayer and bilayer graphene under periodic
strain. We find topological nearly flat bands with homogeneous distribution of
Berry curvature in both systems. The quantum metric of the nearly flat band
closely resembles that for Landau levels. For monolayer graphene, the strain
field can be regarded as an effective gauge field, while for Bernal-stacked
(AB-stacked) bilayer graphene, its role is beyond the description of gauge
field. We also provide an understanding of the origin of the nearly flat band
in monolayer graphene in terms of the Jackiw-Rebbi model for Dirac fermions
with sign-changing mass. Our work suggests strained graphene as a promising
platform for strongly correlated quantum states.

##  I Introduction

Electron correlation is at the center of condensed matter physics research and
is the driving mechanism for a variety of novel quantum states of matter.
There are several known routes to tune the system into a strongly correlated
region. In the Landau Fermi liquid, the Coulomb interaction can dominate the
electron kinetic energy by tuning the density of the electron to the dilute
region. The strongly correlated region can also be achieved using localized
electron orbitals, such as electrons in the $d$ shell or $f$ shell. This is
illustrated in two paradigmatic models in condensed matter physics: the
Hubbard model and the Anderson lattice model for $f$ electrons. In these
correlated systems, the physics connects to the atomic limit adiabatically.
For instance, by using more and more localized electron orbitals, the overlap
of the orbital wave function between neighboring sites can be progressively
reduced to approach the atomic limit. Therefore, the narrow electron band in
these systems generally has a trivial band topology. Another route is to take
advantage of the interference effect of the electron wavefunction when
electrons hop in the lattice. For certain lattices, such as the Lieb lattice
and Kagome lattice, there exists an exactly flat band. The condition for the
existence of exactly flat band was derived by Lieb PhysRevLett.62.1201. Nearly
flat bands have been achieved in carefully designed Hamiltonians.
PhysRevLett.106.236803; PhysRevLett.106.236802; PhysRevLett.106.236804
Recently, moiré superlattices have emerged as an exciting platform for
attaining a nearly flat band through the interference effect of the electron
wavefunction. bistritzer2011moire Moiré superlattice appears when two
incommensurate periodic structures are superposed together. This can be
achieved by twisting a two monolayer material with respect to each other or by
placing a layer of two-dimensional material on top of another material with a
different lattice parameter. As a result of incommensuration, a superlattice
with much longer lattice periodicity arises. (Rigorously speaking, the
superlattice is in general not periodic. But at the low-energy scale, the
incommensuration effect is not important, and the system can be treated as a
periodic lattice.) Here, we emphasize that the narrow band in the moiré
superlattice is not merely due to the reduced energy scale for a small
Brollouin zone in the superlattice with a long period. This can be seen by
checking the ratio between the band gap and the bandwidth, which can be large
under certain conditions. The single particle band is generally not exactly
flat in a moiré superlattice bistritzer2011moire, except for certain special
parameters tarnopolsky2019origins. Nevertheless, these nearly flat bands are
sufficient to stabilize a plathero of correlated states, such as
superconductivity, correlated insulating states, and Wigner crystals.
cao2018correlated; cao2018unconventional; lu2019superconductors;
yankowitz2019tuning; polshyn2019large; xie2019spectroscopic;
kerelsky2019maximized; cao2020strange; jiang2019charge; choi2019electronic;
zondiner2020cascade; wong2020cascade; nuckolls2020strongly; he2021symmetry;
liu2020tunable; regan2020mott; wang2020correlated; xie2020nature;
wu2020collective; su2020current; padhi2018doped; padhi2021generalized;
padhi2019pressure; stefanidis2020excitonic; bultinck2020mechanism; Regan_2020;
Xu_Liu_Mak_Shan_2020.

The third route is to apply a magnetic field which quenches the kinetic energy
of electrons in the plane perpendicular to the magnetic field. Electrons
occupy the Landau levels, which are excatly flat in the clean bulk. Moreover,
the system is topological as evidenced by an integer quantized Hall
conductance. The Landau levels have other unique properties, such as
saturation of the trace condition. These properties make Landau-level systems
important for stabilizing exotic quantum states, such as the fracitonal
quantum Hall state and many other states with nontrivial topological order.

It is natural to seek similar Landau level physics in material but without an
external magnetic field. This is indeed possible in systems where strain can
be regarded as a pseudo magnetic field. This idea has been explored
extensively in the context of graphene Vozmediano_Katsnelson_Guinea_2010, and
the formation of Landau levels has also been observed in experiments
Levy_Burke_Meaker_Panlasigui_Zettl_Guinea_Neto_Crommie_2010. Unlike the
physical magnetic field, the strain does not break the time-reversal symmetry.
This means that the pseudo-magnetic field must be opposite for Dirac fermions
in two different valleys in graphene. The pseudo magnetic field can also
stabilize nontrivial topological phase in graphene as was recently
demonstrated in Ref. PhysRevLett.128.176406. Experimentally, the narrow band
in graphene under periodic strain was reported Mao_Geim_2020 and was
investigated theoretically in Refs. PhysRevB.102.245427; gao2022untwisting.
Therefore, strain engineering represents an important direction to achieve
flat bands, parallel to the efforts to identify materials with intrinsic flat
bands. Regnault_Xu_Li2022

In this work, we study the occurrence and condition of a nearly flat band in
both monolayer and bilayer graphene under periodic strain. We identify nearly
flat bands with a nonzero valley resolved Chern number in monolayer graphene.
This flat band can be understood based on the Jackiw-Rebbi zero mode in 1+1
dimension Dirac fermion with spatially varying mass. We then go beyond the
simple Dirac fermion with winding number 1 to Dirac fermion with winding
number 2 stabilized in bilayer graphene. For Dirac fermions in graphene, the
strain field can only shift the position of the Dirac cone before the Dirac
cone annihilates with the other Dirac fermion in the opposite valley. In this
case, the strain can be regarded as a pseudo magnetic field. In bilayer
graphene, in addition to the shift of the position of the Dirac cone, the
Dirac fermion with winding number 2 can split into two Dirac fermion with
winding number 1 or other combinations such as 3 Dirac cone with winding 1 and
one cone with winding -1. In this case, the strain field decomposes into two
separate sectors: symmetric and antisymmetric (under inversion). In analogy to
strains in single layer systems, the symmetric sector results in an effective
gauge field. In contrast, the antisymmetric sector provides a distinctive
coupling between strain and electronic degrees of freedom, which can no longer
be treated as an effective gauge field, and results in a new family of moire
systems. We will show the appearance of nearly flat bands with almost ideal
quantum geometry as Landau levels in monolayer and bilayer graphene.

The remainder of the paper is organized as follows. In Sec. II, we review the
treatment of strain as a psuedo gauge field in monolayer graphene. In Sec.
III, we present results of nearly flat band in monolayer graphene. In Sec. IV,
we turn to the flat band in periodic strained bilayer graphene. In Sec. V, we
provide an understanding of the occurrence of flat bands in strained monolayer
graphene from the perspective of the Jackiw-Rebbi model. We end the paper with
a brief discussion and conclusion in Sec. VI.

##  II Strain

Figure 1: Schematic view of periodically strained graphene.
$\mathbf{a}^{m}_{1}=\sqrt{3}N_{n}a(1,0)$ and
$\mathbf{a}^{m}_{2}=\sqrt{3}N_{n}a(-\frac{1}{2},\frac{\sqrt{3}}{2})$ are the
superlattice lattice vectors. The color map is of the pseudo magnetic field
defined in Eq. (10).

Here we briefly review how the strain field can be modeled as a pseudo
magnetic field in graphene. The valley projected Hamiltonian of monolayer
graphene around $K_{+}$ is:

$H_{K_{+}}=\frac{3}{2}at\begin{pmatrix}0&q_{x}+iq_{y}\\\
q_{x}-iq_{y}&0\end{pmatrix}=\frac{3}{2}at\begin{pmatrix}0&-2i\partial_{\bar{z}}\\\
-2i\partial_{z}&0\end{pmatrix},$ (1)

where $\partial_{z}=\frac{1}{2}(\partial_{x}-i\partial_{y})$,
$\partial_{\bar{z}}=\frac{1}{2}(\partial_{x}+i\partial_{y})$, $a$ is the bond
length of graphene, $t$ is the nearest-neighbor hopping strength. After we
apply a deformation field $\vec{u}$ to monolayer graphene, the hopping
strength along $\vec{\delta}_{1}$, $\vec{\delta}_{2}$ and $\vec{\delta}_{3}$
(as shown in Fig. 1)are modulated by the strain field. Denote the hopping
strength along $\vec{\delta}_{1}$, $\vec{\delta}_{2}$ and $\vec{\delta}_{3}$
as $t_{1}$, $t_{2}$ and $t_{3}$ respectively. Then the Hamiltonian is modified
as (we use $\hbar=e=c=1$):

$H_{K_{+}}=\frac{3}{2}at\begin{pmatrix}0&-2i\partial_{\bar{z}}+e(A_{x}+iA_{y})\\\
-2i\partial_{z}+e(A_{x}-iA_{y})&0\end{pmatrix},$ (2)

where $ev_{F}A_{x}=\frac{1}{2}(\delta t_{2}+\delta t_{3}-2\delta
t_{1}),\,ev_{F}A_{y}=\frac{\sqrt{3}}{2}(\delta t_{3}-\delta t_{2})$ and
$v_{F}=\frac{3}{2}at$. The relationship between $\delta t_{i}$ and the strain
tensor $u_{ij}$ is:

$\begin{split}t_{i}&=t+\frac{\partial t}{\partial a}\delta a\\\
&=t+\frac{\partial\ln{t}}{\partial\ln{a}}\frac{t}{a}\hat{\delta_{i}}\cdot[\vec{u_{B}}(\vec{R_{A}}+\vec{\delta_{i}})-\vec{u_{A}}(\vec{R_{A}})],\end{split}$
(3)

In the long-wavelength limit,

$\vec{u_{B}}(\vec{R_{A}}+\vec{\delta_{i}})-\vec{u_{A}}(\vec{R_{A}})=\kappa(\vec{\delta_{i}}\cdot\nabla)\vec{u},$
(4)

where $\kappa$ is the reduction factor suzuura2002phonons. Then

$t_{i}=t+\kappa\frac{\partial\ln{t}}{\partial\ln{a}}\frac{t}{a^{2}}\vec{\delta_{i}}\cdot(\vec{\delta_{i}}\cdot\nabla)\vec{u}.$
(5)

Defining $\beta=-\frac{\partial\ln{t}}{\partial\ln{a}}$, which measures how
the bonds respond to being deformed, we have

$t_{i}=t-\frac{\kappa\beta
t}{a^{2}}\vec{\delta_{i}}\cdot(\vec{\delta_{i}}\cdot\nabla)\vec{u}.$ (6)

Plugging $\vec{\delta}_{1}=(0,a)$,
$\vec{\delta}_{2}=(-\frac{\sqrt{3}a}{2},-\frac{a}{2})$ and
$\vec{\delta}_{3}=(\frac{\sqrt{3}a}{2},-\frac{a}{2})$ into the above equation,
we obtain

$\displaystyle\delta t_{1}$ $\displaystyle=-\beta t\kappa u_{yy},$ (7)
$\displaystyle\delta t_{2}$ $\displaystyle=-\beta
t\kappa\left(\frac{3}{4}u_{xx}+\frac{1}{4}u_{yy}+\frac{\sqrt{3}}{2}u_{xy}\right),$
(8) $\displaystyle\delta t_{3}$ $\displaystyle=-\beta
t\kappa\left(\frac{3}{4}u_{xx}+\frac{1}{4}u_{yy}-\frac{\sqrt{3}}{2}u_{xy}\right),$
(9)

Then we have $A_{x}=-\frac{\beta\kappa}{2ea}(u_{xx}-u_{yy})$ and
$A_{y}=2\frac{\beta\kappa}{2ea}u_{xy}$.

##  III Nearly flat band in monolayer graphene

We set ${\delta t_{i}}/{t}=\delta t\sin(\vec{G}_{i}\cdot\vec{r})$, where
$\vec{G}_{1}=\frac{4\pi}{3N_{n}a}(0,1)$,
$\vec{G}_{2,3}=\frac{4\pi}{3N_{n}a}(\mp\sqrt{3}/2,-1/2)$ are reciprocal
lattice vectors. The pseudo magnetic field is

$\vec{B}(\vec{r})=B_{0}\sum_{i=1}^{3}\cos(\vec{G}_{i}\cdot\vec{r})\hat{z},$
(10)

where $B_{0}=\frac{4\pi t\delta t}{3ev_{F}N_{n}a}$. We define the normalized
$\delta t$ as $\alpha=\frac{\delta t}{\frac{3}{2}aG}$, where
$G=\frac{4\pi}{3N_{n}a}$ is the magnitude of the reciprocal lattice vector. We
obtain two nearly flat bands near the charge neutrality point when $\delta
t=-0.2$ and $N_{n}=70$, which corresponds to $\alpha=-2.22817$, as shown in
Fig. 2(a). Adding a $m_{0}\sigma_{z}$ term and setting $m_{0}=0.001$, we get
two separated flat bands. The valley Chern number is 1 for the bottom flat
band and 0 for the top flat band. Before adding $m_{0}\sigma_{z}$ term, the
ratio between the bandgap and the bandwidth of band 1/(-1) is around 3, as
shown in Fig. 2(b). Here band index $i$ means the $i$-th band counting from
charge neutrality point. After adding $m_{0}\sigma_{z}$ term, the ratio
between bandgap (the bandgap is between the band 1(-1) and 2(-2) for the
top(bottom) band.) and bandwidth of the top(bottom) are the same and around
18.3.

Figure 2: Topological flat band in buckled monolayer graphene. (a) Band
structure at $\alpha=-2.22817$. (b) Bandwidth, bandgap, ratio between bandgap
and bandwidth as a function of $|\alpha|$, respectively. Bandgap is the gap
between the middle two bands and the band 2, bandwidth is of band -1 or 1.
Here band index $i$ means the $i$-th band counting from charge neutrality
point, as shown in (a).

In addition to the band dispersion, we further characterize the bands using
the Fubini-Study metric $g_{ab}(\mathbf{k})$ and the Berry curvature
$F_{xy}(\mathbf{k})$, which are given by:

$\begin{split}g_{ab}(\mathbf{k})&=\Re\left(\eta_{ab}(\mathbf{k})\right),F_{xy}(\mathbf{k})=-2\Im\left(\eta_{xy}(\mathbf{k})\right),\\\
\text{where
}\eta_{ab}(\mathbf{k})&=\frac{\langle\partial_{a}u_{\mathbf{k}}|\partial_{b}u_{\mathbf{k}}\rangle}{||u_{\mathbf{k}}||^{2}}-\frac{\langle\partial_{a}u_{\mathbf{k}}|u_{\mathbf{k}}\rangle\langle
u_{\mathbf{k}}|\partial_{b}u_{\mathbf{k}}\rangle}{||u_{\mathbf{k}}||^{4}},\end{split}$
(11)

where $u_{\mathbf{k}}(\mathbf{r})$ is the periodic part of the Bloch function,
$\partial_{a}\equiv\partial_{k_{a}}$, and $||u_{\mathbf{k}}||^{2}=\langle
u_{\mathbf{k}}|u_{\mathbf{k}}\rangle$. In analogy to the Landau level physics,
the ideal candidate for the realization of fractional Chern insulators and
other correlated topological states should have an ideal quantum metric, i.e.
satisfying the trace condition
$\text{tr}(g(\mathbf{k}))=|F_{xy}(\mathbf{k})|$, and a very uniform Berry
curvature. To quantify the non-uniformity of the Berry curvature, we plot the
Berry curvature distribution of the bottom band in Fig. 3(a). The ratio
between the root-mean-square deviation of the Berry curvature and its average
value ${\Delta F_{xy}}/\bar{F}_{xy}=0.576179$. We also plot the violation of
the trace condition quantified by
$[{\text{tr}(g(\mathbf{k}))-|F_{xy}(\mathbf{k})|}]{|F_{xy}(\mathbf{k})|}^{-1}$
in Fig. 3(b). We find that the deviation of $\text{tr}(g(\mathbf{k}))$ from
$|F_{xy}(\mathbf{k})|$ is very small ($<0.18\%$) in the entire Brillouin zone
for the chosen value of $\alpha=-2.22817$.

Figure 3: (a) Berry curvature distribution of the topological band in
monolayer strained graphene, where
$\tilde{F}_{xy}(\mathbf{k})={F_{xy}(\mathbf{k})}/{\bar{F}_{xy}}$. (b)
Violation of the trace condition of the topological flat band. Here, we choose
$\alpha=-2.22817$ and $m_{0}=0.001$.

##  IV Nearly flat bands in bilayer graphene

Here, we consider Bernal-stacked (AB-stacked) bilayer graphene, which hosts
Dirac fermions with winding number 2. The generic valley projected Hamiltonian
of the strained bilayer graphene is:

$H_{K_{+}}=\frac{3}{2}at\begin{pmatrix}0&-2i\partial_{\bar{z}}+e\tilde{A}_{1}&0&0\\\
-2i\partial_{z}+e\tilde{A}_{1}^{*}&0&\frac{2\gamma}{3at}&0\\\
0&\frac{2\gamma}{3at}&0&-2i\partial_{\bar{z}}+e\tilde{A}_{2}\\\
0&0&-2i\partial_{z}+e\tilde{A}_{2}^{*}&0\\\ \end{pmatrix},$ (12)

where $\tilde{A}_{1}=A_{1x}+iA_{1y}$ and $\tilde{A}_{2}=A_{2x}+iA_{2y}$. We
define the symmetric and antisymmetric fields
$\tilde{A}^{(+)}=\frac{\tilde{A}_{1}+\tilde{A}_{2}}{2}$ and
$\tilde{A}^{(-)}=\frac{\tilde{A}_{1}-\tilde{A}_{2}}{2}$, respectively. In
Bernal stacked bilayer graphene, the aligned top layer $A$ sublattice and the
bottom layer $B$ sublattice hybridize strongly and are responsible for the
formation of bands away from zero energy. Therefore, we can project out these
two interlayer aligned sublattices and derive a low-energy Hamitlonian for the
unaligned sublattice. Then the 2 $\times$ 2 effective Hamiltonian of (12) is:

$H_{\text{eff}}=\frac{9a^{2}t^{2}}{4\gamma}\begin{pmatrix}0&H_{12}\\\
H_{12}^{*}&0\\\ \end{pmatrix},$ (13)

where

$\begin{split}H_{12}&=-\left(-2i\partial_{\bar{z}}+e\tilde{A}^{(+)}+e\tilde{A}^{(-)}\right)\left(-2i\partial_{\bar{z}}+e\tilde{A}^{(+)}-e\tilde{A}^{(-)}\right)\\\
&=4\partial_{\bar{z}}^{2}-e^{2}\left(\tilde{A}^{(+)}\right)^{2}+e^{2}\left(\tilde{A}^{(-)}\right)^{2}+4ei\tilde{A}^{(+)}\partial_{\bar{z}}\\\
&\phantom{=}+2ei\partial_{\bar{z}}\tilde{A}^{(+)}-2ei\partial_{\bar{z}}\tilde{A}^{(-)}.\end{split}$
(14)

It is evident from Eq. (13) that when the strain field is symmetric with
$\tilde{A}^{(-)}=0$ (i.e., the same for both layers), it acts as a gauge field
($-2i\partial_{\bar{z}}\rightarrow-2i\partial_{\bar{z}}+e\tilde{A}^{(+)}$).
However, the antisymmetric part of the strain field $\tilde{A}^{(-)}$ cannot
be treated as an effective gauge field. Note that the symmetric and
antisymmetric components of the strain field $\tilde{A}$ do not mix. In the
following, we discuss these two cases separately.

###  IV.1 Bilayer graphene with symmetric strain for two layers

When the two layers have the same strain, $\tilde{A}^{(-)}=0$. Setting
$\tilde{A}^{(+)}=A_{x}+iA_{y}$ and

$A_{x}=\frac{-1}{ev_{F}}t\delta
t[\sin(\vec{G_{1}}\cdot\vec{r})-\frac{1}{2}\sin(\vec{G_{2}}\cdot\vec{r})-\frac{1}{2}\sin(\vec{G_{3}}\cdot\vec{r})]$
(15) $A_{y}=\frac{-1}{ev_{F}}t\delta
t[\frac{\sqrt{3}}{2}\sin(\vec{G_{2}}\cdot\vec{r})-\frac{\sqrt{3}}{2}\sin(\vec{G_{3}}\cdot\vec{r})].$
(16)

We plot the band structure of the 4 $\times$ 4 Hamiltonian and the 2 $\times$
2 effective Hamiltonian and find that they agree well near the charge
neutrality point when the strength of the strain field is small,as shown for
$\alpha=0.159155$ in Fig. 4(a).

Figure 4: (a) 4 $\times$ 4 (Black line) and 2 $\times$ 2 (Blue line) band
structure for symmetric strained bilayer graphene with $\alpha=0.159155$. (b)
4$\times$ 4 (Black line) and 2$\times$ 2 (Blue line) band structure for
antisymmetric strained bilayer graphene with $\alpha=-0.159155$.

Because $\rho(C_{2}T)H_{\text{eff}}^{*}(\mathbf{r},\delta
t)\rho(C_{2}T)^{\dagger}=H_{\text{eff}}(C_{2}\mathbf{r},-\delta t)$, the band
structure for $\delta t$ and $-\delta t$ are the same. And both $\delta t>0$
and $\delta t<0$ lead to a nonzero bandgap between the middle two bands and
the other bands. Before adding any $\sigma_{z}$ term, the band 2 already has a
valley Chern number +2/-2, depending on the sign of $\delta t$. When $\delta
t<0$,after adding $\sigma_{z}$ term, the band -1, 1 and 2 have valley Chern
number -2,0,2, respectively. When $\delta t>0$, after adding $\sigma_{z}$
term, the bands -1,1,2 have valley Chern number 0,2,-2, respectively.

###  IV.2 Bilayer graphene with opposite strain for two layers

When the two layers have the opposite strain, we have $\tilde{A}^{(+)}=0$. We
set $\tilde{A}^{(-)}=A_{x}+iA_{y}$, where $A_{x}$ and $A_{y}$ are the same as
in Eq. (15) and (16). Unlike the case of symmetric strain, for antisymmetric
strain, positive and negative $\delta t$ lead to different behavior. When
$\delta t>0$, the middle two bands are gapless. When $\delta t<0$, the two
middle bands are separated from others, and the flatness of the two middle
bands increases monotonically as we increase $|\alpha|$, as shown in Fig. 5.

Figure 5: Bandgap and bandwidth plot of antisymmetric strained bilayer
graphene. (a) Bandwidth of band 1/(-1), bandgap between band 1(-1) and band
2(-2) and (b) the ratio between bandgap and bandwidth as a function of
$|\alpha|$. The plot is from the $4\times 4$ Hamiltonian. The normalized
$\delta t$ is defined as $\alpha=\frac{\delta t}{\frac{3}{2}aG}$. The energy
is shown in units of $\hbar v_{F}G$.

We calculate the valley Chern number of the middle two bands after adding
$\sigma_{z}$ term, the top flat band has valley Chern number -1 and the lower
one has valley Chern number +1. We plot the Berry curvature distribution and
the violation of the trace condition
$[{\text{tr}(g(\mathbf{k}))-|F_{xy}(\mathbf{k})|}]{|F_{xy}(\mathbf{k})|}^{-1}$
of the bottom band at $\alpha=-1.59155$. The ratio between bandgap and
bandwidth is greater than 1000 after adding $\sigma_{z}$, and we have an ideal
quantum geometry, as shown in Fig. 6(b).
$|\psi_{\bar{K_{+}}}(\mathbf{r}_{0}=\mathbf{0})|$ decrease exponentially as a
function of $|\alpha|$, as shown in Fig. 7. When $|\alpha|$ is large,
$|\psi_{\bar{K_{+}}}(\mathbf{r_{0}}=\mathbf{0})|$ is very close to 0,
suggesting that the wavefunction of the topological flat band is close to a
trial wavefunction
$\psi_{\mathbf{k}}(\mathbf{r})=f_{\mathbf{k}}(z)\psi_{\bar{K}_{+}}(\mathbf{r})$,
where $\psi_{\bar{K}_{+}}(\mathbf{r})$ has a zero at $\mathbf{r}=0$. This
trial wavefunction’s periodic part is holomophic in $k$, which gives rise to
the perfect quantum geometry of the band. As shown in Fig. 8, the decay rate
of
$[{\text{tr}(g(\mathbf{k}))-|F_{xy}(\mathbf{k})|}]{|F_{xy}(\mathbf{k})|}^{-1}$
as a function of $|\alpha|$ for the bilayer is much faster than that of the
monolayer, suggesting that the quantum geometry of the bilayer topological
flat band is closer to the ideal quantum geometry than the monolayer
topological flat band with the same $\alpha$.

Figure 6: (a) Distribution of the Berry curvature. (b) Violation of the trace
condition of the lower topological flat band for antisymmetric strained
bilayer graphene at $\alpha=-1.59155$ and $m_{0}=0.0001$. Figure 7: Plot of
$\ln{|\psi_{\bar{K_{+}}}(\mathbf{r_{0}}=\mathbf{0})|}$ as a function of
$|\alpha|$ for monolayer, bilayer 4 $\times$ 4 Hamiltonian and bilayer
2$\times$2 effective Hamiltonian. Red triangle is for monolayer, Black star is
for bilayer 2$\times$2, Green square is for bilayer 4$\times$4. Figure 8:
Trace condition violation for monolayer and bilayer 4 $\times$ 4 and 2
$\times$ 2 effective Hamiltonian as a function of $|\alpha|$ at a fixed $k$
point. We choose the middle point between $\bar{\Gamma}$ and $\bar{K_{-}}$ for
the monolayer case. We choose $\bar{\Gamma}$ for the bilayer case. Red
triangle is for monolayer, Black star is for bilayer 2$\times$2, Green square
is for bilayer 4$\times$4.

##  V Connection to the Jackiw-Rebbi zero mode

For Dirac fermions coupled to a gauge field, it was proved that there exist
zero modes, whose number is the same as the number of total flux minus one in
the system. PhysRevA.19.2461 This applies even to the situation where the
magnetic field is not uniform in space. However, in our case, the net magnetic
field is zero and the calculations of Ref. PhysRevA.19.2461 is no longer
applicable. Instead, we will show below that the nearly flat band can be
understood from the perspective of zero mode of Dirac fermion in 1+1 D with a
sign-changing mass term, the Jackiw-Rebbi (JR) zero mode PhysRevD.13.3398.

For a Dirac Hamiltonian in $1+1$ dimension,
$\mathcal{H}(x)=-i\partial_{x}\sigma_{x}+m(x)\sigma_{y}$ , with a sign change
mass term, i.e. $m(x\geq 0)\geq 0$ and $m(x<0)<0$, it supports a massless
fermionic mode localized at the domain wall $x=0$. To build intuition, we
first show that the zeroth Landau level in graphene can be understood in terms
of the JR mode. We take the gauge $A_{y}=B_{z}x$ and $A_{x}=0$, such that the
momentum $k_{y}$ in the $y$ direction is a good quantum number. For each
$k_{y}$, the Dirac Hamiltonian in one valley $K$ becomes
$\mathcal{H}(x)_{K}=-i\partial_{x}\sigma_{x}+(k_{y}-B_{z}x)\sigma_{y}$ which
is exactly the 1+1 Dirac Hamiltonian with a spatially dependent mass term,
$m(x)=k_{y}-B_{z}x$. For each $k_{y}$, there exists a JR zero mode at
$x_{JR}=k_{y}/B_{z}$. The degeneracy of the zeroth Landau level is equal to
the number of allowed $k_{y}$, which equals
$N_{y}=L_{x}/x_{JR}=L_{x}L_{y}B_{z}/(2\pi)=\Phi/\Phi_{0}$, where $L_{x}$ and
$L_{y}$ are the linear system sizes, and we take $k_{y}$ to be discretized in
a step of $2\pi/L_{y}$. This reproduces the well-known results that the
degeneracy is the same as the number of flux in units of flux quantum. In the
presence of an electric field $E_{y}$ along the $y$ direction, $k_{y}=E_{y}t$
from the semi-classical equation of motion for electrons. After one period,
the position of the zero mode in the $x$ direction is exactly shifted by one
period, which corresponds to quantized electrical Hall conductance
$\sigma_{xy}=1/2\pi$. Therefore, the zeroth Landua level is topological. The
JK mode reproduces the well-known facts about the zeroth Landau level of Dirac
fermions in the $2+1$ dimension. This understanding allows us to choose the
proper gauge field by designing the strain field to achieve a nearly flat
band. For a gauge field which corresponds to a uniform magnetic field, the
associated strain field diverges at large distances and thus cannot be
realized physically. To avoid divergence in the strain field, we choose a
sinusoidal modulated vector potential in the $x$ direction but uniform in the
$y$ direction, which can be realized using the deformation field
$u_{y}\propto\cos(\frac{2\pi}{N_{n}}\frac{x}{\sqrt{3}a})$ and $u_{x}=0$. Note
that there are infinitely many possible strain field configurations that give
the same pseudo magnetic field. The Hamiltonian is

$\displaystyle\mathcal{H}(x,k_{y})=v_{F}(-i\partial_{x}\sigma_{x}+(k_{y}+eA_{y}(x))\sigma_{y})$
(17)

where we take $A_{y}(x)=\frac{t}{ev_{F}}\delta
t\cos(\frac{2\pi}{N_{n}}\frac{x}{\sqrt{3}a})$. We set $t=1$. In the region
where $|k_{y}|<\delta t$, there exists a quasi-zero mode localized in the
region when $k_{y}+A_{y}(x)$ changes sign. Here “quasi-zero" means that these
localized modes generally hybridize, which lifts these modes away from zero
energy. The degree of hybridization depends on the ratio between the spread of
the wave function and the separation of the zeros of $k_{y}+A_{y}(x)$. As
$|k_{y}|$ approaches $\delta t$, the hybridization of the localized mode
becomes stronger and the energy of the modes starts to deviate from zero
energy. As a consequence, the band is nearly flat only in a certain range of
$k_{y}$ below $\delta t$ as illustrated in Fig. 9. We remark that the
partially flat band can also host correlated quantum states, when the Fermi
energy is parked in the flat band region.

Figure 9: Band structure of the band -1 as a function of $k_{x}$ and $k_{y}$
when $\delta t=1$.

Now we are in a position to discuss the periodic strain considered in Sec.
III. We take advantage of the gauge redundancy by choosing $A_{y}=0$ and
$A_{x}=-\frac{1}{ev_{F}}t\delta
t[\sin(\vec{G_{1}}\cdot\vec{r})-2\sin(\vec{G_{2}}\cdot\vec{r})-2\sin(\vec{G_{3}}\cdot\vec{r})]$.
Then $-\partial_{y}A_{x}(x,y)=B_{z}$ is the same pseudo magnetic field as in
Eq. (10). The Hamiltonian becomes

$H=-i\partial_{x}\sigma_{x}-i\partial_{y}\sigma_{y}-A_{x}\sigma_{x}$ (18)

Assuming that the variation in the $y$ direction is slow and its amplitude is
weak, we use the mode expansion

$A_{x,m}(y)\equiv\frac{1}{L_{x}}\int_{0}^{L_{x}}A_{x}(x,y)\exp(-imG_{x}x)dx,$
(19)

where $G_{x}=\frac{2\pi}{\sqrt{3}N_{n}a}$. Plug in (19) and
$\psi(x,y)=\sum_{m}e^{i(k_{x}+mG_{x})x}\psi_{m}(y)$ into the eigenvalue
equation of (18), we get

$[-i\partial_{y}\sigma_{y}+(k_{x}+mG_{x})\sigma_{x}]\psi_{m}-\sum_{m^{{}^{\prime}}}A_{x,m-m^{{}^{\prime}}}\sigma_{x}\psi_{m^{{}^{\prime}}}=E\psi_{m}$
(20)

For the $A_{x}(x,y)$ we use, only
$A_{x,0}=\frac{-3B_{0}N_{n}a}{4\pi}\sin(G_{1y}y)$,$A_{x,-1}=\frac{3B_{0}N_{n}a}{2\pi}\sin(G_{2y}y)$
and $A_{x,1}=\frac{3B_{0}N_{n}a}{2\pi}\sin(G_{3y}y)$ are nonzero. We consider
the zero mode and set $m=0$, then we rewrite the Hamiltonian near $E=0$ as

$H=-i\partial_{y}\sigma_{y}-M(y)\sigma_{x}$ (21)

where

$M(y)=k_{x}-A_{x,0}-A_{x,-1}A_{x,1}\left(\frac{1}{k_{x}+G_{x}-A_{x,0}}+\frac{1}{k_{x}-G_{x}-A_{x,0}}\right)$
(22)

Plug in the value of $A_{x,0}$, $A_{x,-1}$ and $A_{x,1}$, we obtain

$\begin{split}&M(y,k_{x},\delta t)=\frac{3}{2}atk_{x}+t\delta
t\sin(G_{1y}y)-4t^{2}\delta{t}^{2}\sin(G_{2y}y)\sin(G_{3y}y)\\\
&\left(\frac{1}{\frac{3}{2}at(k_{x}+G_{x})+t\delta
t\sin(G_{1y}y)}+\frac{1}{\frac{3}{2}at(k_{x}-G_{x})+t\delta
t\sin(G_{1y}y)}\right)\end{split}$ (23)

If we set $t=1$, we can plot $M(y)$ as a function of $y$ for certain $k_{x}$
and $\delta t$:

Figure 10: Zeros of $M(y)$ for different $k_{x}$ and $\delta t$. (a)-(c)
$k_{x}=0$,(d)-(f) $k_{x}=0.05$. (a) and (d) $\delta t=0.05$. (b) and (e)
$\delta t=0.2$. (c) and (f) $\delta t=0.4$.

There are two types of solution of zeros, so there should be two flat bands,
which is consistent with Fig. 2. Generally these localized modes hybridize and
therefore the band cannot be exactly flat.

##  VI Discussion and conclusion

First we discuss the existing results of the nearly flat band in graphene and
bilayer graphene under a periodic potential. Milovanović _et al._ studied band
flattening in buckled monolayer graphene numerically by modeling the buckling
effect as a psuedo magnetic field. Narrow bands with delocalized electronic
dynamics are found. PhysRevB.102.245427 Skurativska _et al._ calculated the
electronic structure of single layer graphene in the presence of periodic
adatoms within the framework of functional density theory, where relatively
flat bands with fragile topology are reported. PhysRevResearch.3.L032003 The
mechanism for the appearance of a nearly flat band in these two works is not
clear. In our work, we unravel the underlying machanism for the flat band
through the JR zero mode. In addition, we provide further characterization of
the bands using quantum metric. In Ref. ghorashi2022topological, both
topological and non-topological flat bands in the bilayer graphene under an
incommensurate potential are demonstrated. The incommensurate potential is
introduced through layer pseudo spin. While in our work, the strain modulates
the velocity of the Dirac fermion. We would also like to mention that closely
related systems of a single Dirac fermion on the surface of a topological
insulator in the presence of periodic potential, where flat bands and possible
correlated states are studied. PhysRevX.11.021024; PhysRevB.103.155157 The
flat band for Dirac fermions in the presence of a modulated on top a uniform
physical magnetic field is studied in Ref. dong2022dirac.

Experimentally, periodic strain can be introduced by placing graphene on top
of an array of dielectric nanodots. The out-phase strain for bilayer graphene
may be tricky to implement. Generally, one expects that the strain for
different layers can be different when the bilayer graphene is placed on a
periodic structure. This heterostrain, albeit weak, has been measured
experimentally in twisted bilayer graphene devices. Kerelsky_McGilly2019
Therefore there exists strain component that is antisymmetric with respect to
layer degree of freedom. To maximize the antisymmetric component, one can
sanwitch the bilayer graphene with periodic structure in both the bottom and
top. Although we restrict ourselves to periodic strain with $C_{3}$ symmetry
and strain modulated in one direction, the flat bands can also appear for
other periodic strain profiles, as can be inferred from the consideration of
the Jackiw-Rebbi zero modes. When the strain can be regarded as a gauge field,
because of the gauge redundancy, there exist infinitely many strain profiles
that produce the same energy spectrum. Therefore, one may take advantage of
this gauge redundancy to find a convenient strain profile for experimental
realization. As shown in Fig. 5, the band flattens as a function of the strain
strength. Compared to twisted bilayer graphene, which requires fine tuning of
the twist angle to achieve a flat band, strained graphene can achieve a nearly
flat band, and the flatness of the band can be improved by continuously tuning
the parameters.

As demonstrated in Sec. IV, the response of the system to a strain can be very
different depending on the winding nubmer of the Dirac fermion. For monolayer
graphene with Dirac fermion with winding number 1, the strain can be modeled
by an effective pseudo gauge field. For high winding numbers, the Dirac
fermion can split in the presence of strain, and the pattern of splitting can
be controlled by the symmetry of the strain. In this case, the strain
generally cannot be modeled by pseudo gauge field. For a higher winding
number, which can be realized in multilayer graphene, even richer physics can
be expected. Similarly, the many-body instability when electron interactions
are included is likely distinct for a Dirac fermion with different winding
numbers. The response to strain also depends on the position of the Dirac
dispersion in the Brillouin zone. Wan_Sarkar_Lin_Sun_2022

To summarize, we find topological nearly flat bands with a homogeneous
distribution of Berry curvature in both monolayer and Bernal-stacked bilayer
graphene under periodic strain. Depending on the winding number of the Dirac
fermion, the role of strain can either be treated as a pseudo gauge field as
in the monolayer graphene, or can be beyond the simple gauge field description
as in the case of bilayer graphene. The mechanism of these nearly flat bands
can be understood in terms of the Dirac fermion with spatially varying mass
terms as in the standard Jackiw-Rebbi model. We further show that the quantum
metric of the nearly flat band closely resembles that for Landau levels. These
topological flat bands are fertile playground for stabilization of interesting
many boday quantum states. Compared to other flat band systems
Ledwith2020Fractional; Repellin2020Chern; Liu2021Gate; Li2021Spontaneous;
Xie2021Fractional, it is easy to expect a valley polarized state and a
fractional Chern insulator at frational filling of the valley polarized bands.
It is also possible to host other non-abelian topological states due to the
nearly ideal quantum metric compared to Landau levels. We leave the
interaction effects for future study.

###### Acknowledgements.

Acknowledgments.— We thank Chunli Huang, Jacob Pettine, Houtong Chen and Filip
Ronning for very helpful discussions. The work at LANL (SZL) was carried out
under the auspices of the U.S. DOE NNSA under contract No. 89233218CNA000001
through the LDRD Program, and was supported by the Center for Nonlinear
Studies at LANL (XW), and was performed, in part, at the Center for Integrated
Nanotechnologies, an Office of Science User Facility operated for the U.S. DOE
Office of Science, under user proposals $\\#2018BU0010$ and $\\#2018BU0083$.
This work was supported in part by the Office of Navy Research MURI
N00014-20-1-2479 (XW, SS and KS) and Award N00014-21-1-2770 (XW and KS), and
by the Gordon and Betty Moore Foundation Award N031710 (KS).

Note added.— During the preparation of the manuscript, we are aware of the
work gao2022untwisting that overlaps in part with the band structure of
monolayer graphene under strain in Fig. 2.

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