# Moiré polar vortex, flat bands and Lieb lattice in twisted bilayer BaTiO3

Seungjun Lee<EMAIL_ADDRESS>Department of Electrical and Computer
Engineering, University of Minnesota, Minneapolis, Minnesota 55455, USA  D. J.
P. de Sousa<EMAIL_ADDRESS>Department of Electrical and Computer
Engineering, University of Minnesota, Minneapolis, Minnesota 55455, USA
Bharat Jalan  Department of Chemical Engineering and Materials Science,
University of Minnesota, Minneapolis, Minnesota 55455,USA  Tony Low
<EMAIL_ADDRESS>Department of Electrical and Computer Engineering, University of
Minnesota, Minneapolis, Minnesota 55455, USA  Department of Physics,
University of Minnesota, Minneapolis, Minnesota 55455, USA

###### Abstract

Advances in material fabrication techniques and growth methods have opened up
a new chapter for twistronics, in the form of twisted freestanding three-
dimensional material membranes. Through first-principles calculations based on
density functional theory, we investigate the crystal and electronic
structures of twisted bilayer BaTiO3. Our findings reveal that large stacking
fault energy leads to chiral in-plane vortex pattern that was recently
observed in experiments. Moreover, we also found non-zero out-of-plane local
dipole moments, indicating that the strong interlayer interaction might offer
promising strategy to stabilize ferroelectric order in the two-dimensional
limit. Remarkably, the vortex pattern in the twisted BaTiO3 bilayer support
localized electronic states with quasi-flat bands, associated with the
interlayer hybridization of oxygen $p_{z}$ orbitals. We found that the
associated band width reaches a minimum at $\sim$19∘ twisting, configuring the
largest magic angle in moiré systems reported so far. Further, the moiré
vortex pattern bears a striking resemblance to two interpenetrating Lieb
lattices and corresponding tight-binding model provides a comprehensive
description of the evolution the moiré bands with twist angle and reveals the
topological nature of these states.

##  I Introduction

The ability to precisely control the twist angle between van der Waals (vdW)
materials has opened up an emerging field of research, called ’‘twistronics”.
[1, 2] The first major breakthrough arose from the discovery of magic-angle
twisted bilayer graphene, [3, 4, 5] followed by other vdW materials, [6, 7, 8]
and further open up tremendous opportunities in the exploration of new
emergent phenomena. Advances in material fabrication technologies have further
expanded the possibilities, enabling the design of entirely new types of moiré
materials. Utilizing vdW materials [9, 10, 11, 12, 13] or a sacrificial layer
[14, 15, 16, 17] , it is now possible to realize atomically thin and
freestanding membrane consisting of three-dimensional (3D) oxide materials.
Recently, moiré perovskites bilayer membranes were realized experimentally.
[18, 19, 20] Due to the interlayer interaction in these moiré oxides, unique
vortex patterns that possess strong local polarization were also observed,
[20] a signature of the broken symmetry due to the chiral interlayer coupling.
These developments thus ushered in a new chapter in oxide twistronics.

Despite these first experimental observations, [20] little is known about the
origin and character of the polar vortex, the Moire electronic structures, the
magic angle and associated flat band physics in this material. In this work,
we investigated the crystal and electronic structure of twisted 2L BaTiO3, by
means of first-principles calculations based on density functional theory
(DFT) [21, 22]. Our findings reveal a substantial generalized stacking fault
energy (GSFE) in 2L BaTiO3 with a BaO-BaO type interface. This results in
significant intralayer relaxation and induces chiral in-plane moiré vortex
patterns at finite twist angles $\theta_{t}$, in full agreement with recent
experimental observations [20]. Notably, the chirality of the vortex patterns
are opposing for the two layers. We also find that interlayer interactions
give rise to non-zero local out-of-plane dipole moments and strong
electrostatic surface potentials. Thus, twisted 2L perovskites can serve as
“smart” substrates for epitaxial growth, or to imprint electrostatic
superlattices on vdW materials.

Pertaining to the electronic structures, our results indicate that the moiré
vortex pattern hosts localized electrons states with quasi-flat bands over a
wide range of twist angles. We found that the maximum band flatness occurs at
${\theta_{t}}{\approx}19^{\circ}$, configuring the largest magic angle
reported in literature, to the best of our knowledge. Remarkably, the low
energy bands formed from these localized states are well described by a tight-
binding model on two interpenetrating Lieb lattices, whose sites correspond to
the distinct local vortex patterns. Thus, moiré states in twisted 2L BaTiO3
hold the potential to enable the first realization of an inorganic electronic
Lieb lattice.

##  II Results and discussion

###  II.1 Crystal and electronic structures of untwisted bilayer BaTiO3

Figure 1: Giant stacking energy fault and electronic structures of untwisted
bilayer BaTiO3 (a) Crystal structures of bilayer (2L) BaTiO3 with AA and AB
stacking configurations, and (b) generalized stacking fault energy (GSFE) map
for 2L BaTiO3. (c) Orbital-resolved electronic structure of monolayer BaTiO3.
(d) Electronic structures of 2L BaTiO3 with AA, AB, AA′ stacking
configurations. AA(3.6 Å) stacking has the same in-plane atomic arrangement to
AA stacking, but has 1 Å smaller interlayer distance compared to the AA
stacking. Inset visualizes the band structure near the valence band maximum.

In the ABO3 perovskites, there are two possible types of interfaces, related
to the AO and BO2 surfaces. Therefore, in twisted 2L BaTiO3, we have a total
of three different interface configurations, namely BaO-BaO, TiO2-TiO2, and
BaO-TiO2. Our DFT calculations indicate that both TiO2-TiO2 and BaO-TiO2
interfaces give rise to complex and unsaturated chemical bonds between the
interfaces, resulting in many defect-like states in their electronic
structures. (See Figs LABEL:figs_Tio2 and LABEL:figs_Bao_Tio2 in Supplementary
Information (SI)) On the other hand, BaO-BaO interface has vdW-like interlayer
interaction. On the experimental front, various surface termination have been
reported in ABO3 perovskites. [23] Among them, one recent experimental study
observed the topmost BaO surface on the BaTiO3 thin film, suggesting that the
fabrication of BaO-BaO interfaces is possible in experiments. [24] Since
membranes mediated by vdW-like forces are more amenable to “twistronics”
approaches, we shall focus exclusively on membranes with BaO termination.

We begin by discussing the untwisted bilayer structures. With BaO-BaO surface
termination, the chemical formula of a BaTiO3 membrane can be described as
Ban+1TinO3n+1 with $n$ being infinity. As a first step, we performed DFT
calculations for $n$=1 case corresponding to 2 layers of BaTiO3 sandwiching
one layer of BaO and made two different stacking configurations, AA and AB, as
shown in Fig. 1(a). For more detailed description of DFT calculations, see
Method section. Due to the opposite charge states in Ba and O, the interlayer
distance in the AA configuration is much larger than AB. To understand the
stacking energy landscape, we calculated generalized stacking fault energy
(GSFE) [25, 26] as function of the in-plane sliding, as shown in Fig. 1(b).
The GSFE corresponding to the AA stacking is almost 1 eV, which is orders of
magnitude larger than that of typical vdW materials, such as graphene (20
meV/cell) and MoS2 (60 meV/cell). [26] On the other hand, some layered
materials exhibit comparable GSFE to 2L BaTiO3, such as SnS/SnSe
heterostructure (0.88 eV/cell). [27] It means that the interlayer bonding
characteristics of 2L BaTiO3 is in the range of vdW-like rather than covalent
and ionic. Hence, substantial atomic relaxation is expected to take place in
the twisted configurations, which will be addressed in upcoming sections.

Figure 1(c) shows the orbital-resolved electronic structure of monolayer (1L)
BaTiO3. Its bandgap is calculated to be 1.78 eV within PBE XC functional,
which is slightly larger than that of bulk BaTiO3 (1.62 eV) calculated by the
same computational method. While the character of the parabolic conduction
band is mainly due to Ti $d$ orbitals, O $p$ orbitals dominates the valence
band. Also, we can find sizable contributions of $p_{z}$ states of O in BaO
plane in valence band maximum (VBM, along the $\Gamma-X$ path in the Brillouin
zone). Figure 1(d) shows the electronic structures of bilayer BaTiO3 with AA
and AB stacking configurations. Their electronic structures are very similar
with that of 1L BaTiO3, implying a weak interlayer interaction in both
configurations. To understand the role of the O $p_{z}$ states, we constructed
an artificial structure with an AA(3.6 Å) stacking configuration, which has an
interlayer distance reduced by 1 Å as compared to the true AA stacking
configuration, and found a higher VBM as shown in grey color in Fig. 1(d).
Thus, it suggest that the interlayer O-O interaction controls the
hybridization between the valence bands of the constituents layers. Note that,
interlayer coupling does not affect conduction bands due to its Ti $d$
orbital.

Figure 2: In-plane polar vortex in twisted bilayer BaTiO3 (a) Schematics of
the supercell construction method with non-zero twist angle
(${\theta}_{\rm{t}}$) in a square lattice. Green sphere indicates Ba
sublattice and blue square and $a_{0}$ indicate unitcell and lattice constant
of BaTiO3. The size of the supercell and corresponding ${\theta}_{\rm{t}}$ can
be determined by four lattice vectors (red arrows), which are ($n$,$m$),
($m$,$-n$), ($-n$,$m$), and ($-m$,$-n$), respectively, where ($n$,$m$)
indicate lattice vector defined as $na_{0}\hat{x}+ma_{0}\hat{y}$ and $a_{0}$
indicates the lattice constant of BaTiO3. Then, ${\theta}_{\rm{t}}$ was
evaluated to be ${\theta}_{\rm{t}}=2~{}\rm{arctan}$($n$/$m$). (b) Top and side
views of the relaxed crystal structure of 2L BaTiO3 with
${\theta}_{\rm{t}}=28.07^{\circ}$. The lattice vector of this supercell can be
described as (1,4). (c-g) In-plane Ti displacement vector
($\delta{\mathbf{r}}^{\rm{Ti}}$) maps for supercell structures with
${\theta}_{\rm{t}}=$28.07, 22.62, 18.92, 16.26, and, 14.25∘, respectively. The
$\delta{\mathbf{r}}^{\rm{Ti}}$ was amplified by a factor of 50 for easy
visualization. Blue and red arrows indicate $\delta{\mathbf{r}}^{\rm{Ti}}$ of
upper (u) and lower (l) layers, and grey dahsed lines show supercell
structures. In (c), we overlaid atomic structure of TiO2 sublayer for better
understanding. (h) Maximum magnitude of in-plane $\delta{\mathbf{r}}$ of
twisted 2L BaTiO3, graphene, [28] and MoS2 [29] as function of
${\theta}_{\rm{t}}$.

###  II.2 In-plane polar vortex in twisted bilayer BaTiO3

Due to the two-dimensional square lattice structure of 1L BaTiO3, we can
systematically construct commensurate supercell structures of 2L BaTiO3 with
finite twist angle (${\theta}_{\rm{t}}$) without any additional strain, as
illustrated in Fig. 2(a). The size of the supercell and corresponding
${\theta}_{\rm{t}}$ can be determined by four lattice vectors, corresponding
to the combinations ($n$,$m$), ($m$,$-n$), ($-n$,$m$), and ($-m$,$-n$),
respectively, where ($n$,$m$) indicate lattice vector defined as
$na_{0}\hat{x}+ma_{0}\hat{y}$ and $a_{0}$ indicates the lattice constant of 1L
BaTiO3. Then, ${\theta}_{\rm{t}}$ was evaluated to be
${\theta}_{\rm{t}}=2~{}\rm{arctan}$($n$/$m$). Note that, when $n$+$m$=even, we
can generally find a smaller supercell structure with the same
${\theta}_{\rm{t}}$, but characterized by the lattice vector of
($(n+m)/2$,$(m-n)/2$). However, in this work, we also utilize the lattice
vector ($n$,$m$) when $n$+$m$=even, for consistent visualization purposes.
Using this approach, we constructed a total of five supercell structures with
${\theta}_{\rm{t}}$ ranging from 12.7∘ to 28.07∘. (See Fig.
LABEL:figs_structure in SI for their relaxed atomic structures) Figure 2(b)
shows the top and side views of one of twisted 2L BaTiO3 structure with 28.07∘
twist angle. The averaged interlayer distance was 3.173 $\rm{\AA}$, which is
intermediate value between AA and AB, indicating a vdW-type interlayer
interaction. The obtained averaged interlayer distance decreases slowly as
${\theta}_{\rm{t}}$ is decreased. For example, the interlayer distance was
calculated to be 3.131 Å for 18.92∘, and 3.122 Å for 14.25∘. Within
${\theta}_{\rm{t}}$ ranging from 28.07∘ to 14.25∘, we cannot observe a large
variation of the interlayer distance between AA and AB sites.

To understand the origin of polar vortex pattern observed in the recent
experiments [20], we calculated in-plane Ti displacement vector
($\delta{\mathbf{r}}^{\rm{Ti}}$) defined as
$\delta{\mathbf{r}}^{\rm{Ti}}=\mathbf{r}^{\rm{Ti}}_{\rm{relaxed}}-\mathbf{r}^{\rm{Ti}}_{\rm{ini}}$,
where $r^{\rm{Ti}}_{\rm{relaxed}}$ is the relaxed position of Ti and
$r^{\rm{Ti}}_{\rm{ini}}$ the initial position with cubic symmetry. Then, we
represented $\delta{\mathbf{r}}^{\rm{Ti}}$ for upper (u) and lower (l) layers,
as shown in Fig. 2(c-g). Our results indicate the emergence of chiral vortex
displacement patterns for all twist angles. Notably, the vortex patterns are
opposite for upper and lower layers. In addition, the in-plane displacement is
larger for smaller twist angles (large morié structure), as shown in Fig.
2(h), similar to the twisted 2L graphene and MoS2 [28, 29] systems. More
interestingly, the observed in-plane displacement are much larger than those
observed in other 2D materials at similar twist angles, which we attribute to
the substantial GSFE. It is worth noting that the physical origin of our
observed polar vortex in twisted perovskites is fundamentally distinct from
the recently reported pattern in twisted hexagonal boron nitride. [30] The
former originates from intralayer relaxation due to the interlayer
interaction, while the later originates from the stacking order.

###  II.3 Out-of-plane local dipole moments emerging in twisted bilayer BaTiO3

In addition to in-plane polar vortex structure, our first principles
calculations also indicate the emergence of local out-of-plane Ti
displacements ($\delta{\mathbf{z}^{\rm{Ti}}}$), which can generate non-zero
local dipole moment. Here, we defined $\delta{\mathbf{z}^{\rm{Ti}}}$ using the
z positions of two O atoms which are placed on top ($\rm{O_{t}}$) and bottom
($\rm{O_{b}}$) of Ti, described as
$\delta{\mathbf{z}}^{\rm{Ti}}=z^{\rm{Ti}}-(z^{\rm{O_{t}}}+z^{\rm{O_{b}}})/2$,
as shown in Fig. 3(a). Then, we calculated $\delta{\mathbf{z}}$ for
${\theta}_{\rm{t}}=$18.92∘ structure, as shown in Fig. 2(b). We found non-zero
$\delta{\mathbf{z}}$ in twisted 2L BaTiO3, implying emergence of non-zero out-
of-plane dipole moment even in 1L thickness. We have also calculated
$\delta{\mathbf{z}}^{\rm{Ti}}$ for all other structures, which is presented in
Fig. 3(c) along with their maximum values as function of ${\theta}_{\rm{t}}$.
Besides non-zero $\delta{\mathbf{z}}^{\rm{Ti}}$ for all structures, we found a
negative correlation between $\delta{\mathbf{z}}^{\rm{Ti}}$ and
${\theta}_{\rm{t}}$. In perovskites thin film, out-of-plane dipole moment is
strongly suppressed below a critical thickness due to the depolarization
fields from the dangling bonds at the surface [31, 32]. Our results imply that
the moiré interlayer interaction might offer a promising strategy to stabilize
ferroelectric order in the two-dimensional limit.

Figure 3: Out-of-plane local dipole moments in twisted bilayer BaTiO3 (a) Side
view of the relaxed crystal structure of 2L BaTiO3 with
${\theta}_{\rm{t}}=18.92^{\circ}$ and detailed atomic arrangement of at the
center of the upper layer. Out-of-plane Ti displacement vector
($\delta{\mathbf{z}^{\rm{Ti}}}$) was defined as
$\delta{\mathbf{z}^{\rm{Ti}}}=z^{\rm{Ti}}-(z^{\rm{O_{t}}}+z^{\rm{O_{b}}})/2$.
(b) Calculated $\delta{\mathbf{z}^{\rm{Ti}}}$ of 2L BaTiO3 with
${\theta}_{\rm{t}}=18.92$. Blue and red arrows indicate
$\delta{\mathbf{z}^{\rm{Ti}}}$ of upper (u) and lower (l) layers, and
$\delta{\mathbf{z}^{\rm{Ti}}}$ was amplified by a factor of 50 for easy
visualization. (c) Maximum magnitude of $\delta{\mathbf{z}^{\rm{Ti}}}$ of
twisted 2L BaTiO3 as function of ${\theta}_{\rm{t}}$ (d, e) Surface potential
profiles of ${\theta}_{\rm{t}}=18.92$ evaluated at distance 3.1 Å and 5.7 Å
from the top BaO surface (zBaO, marked in (a)), respectively.

We further analyze the out-of-plane displacement through visualizing the
surface potential variations of ${\theta}_{\rm{t}}=18.92$ evaluated at
distance 3.1 Å and 5.7 Å from the top BaO surface (zBaO, marked in Fig. 3(a)),
as shown in Fig. 3(d) and (e), respectively. At relatively short distances
(3.1 Å), a strong atomic potential difference between Ba and O directly maps
into surface potential distribution. As a results, higher (lower) potential,
red (blue) color, directly describes position of O (Ba) atoms, as shown in
Fig. 3(e). At larger distances (Fig. 2(h)), an underlying long-range order
with sizable potential differences of up to 60 meV can be clearly visualized
(See Fig. LABEL:figs_zz_detail in SI for more details). We note that this
long-range order resembles the in-plane polar vortex pattern shown in Fig.
3(f). In addition, the higher potential magnitudes (red) correspond to the
local regions where the out-of-plane displacements
$\delta{\mathbf{z}^{\rm{Ti}}}$ point inward, represented in Fig. 2(f),
suggesting that the local out-of-plane dipole moments play a significant role
in generating surface potential map. We also found a similar long-range order
in other twist angles, as shown in Fig. LABEL:figs_zz_others in SI. It was
recently reported that potential variations emerging from moiré pattern can be
utilize an effective substrate that can modulate physical properties of two-
dimensional materials. [33] Therefore, the twisted BaTiO3 can be a promising
substrate material to manipulate the physical properties of adjacent
functional layers.

###  II.4 Topological Lieb lattice in twisted bilayer BaTiO3

Figure 4: Morié flat bands in twisted bilayer BaTiO3 Electronic structure of
twisted 2L BaTiO3. (a-e) Evolution of the moiré bands for 28.07∘ \- 14.25∘
twisted structures. The insets show the detailed structure of a few states at
the top of the valence moiré bands. (f) The evolution of moiré band widths of
twisted 2L BaTiO3. We also represented band widths of twisted 2L graphene,
[34] MoS2, [35] and In2Se3, [36] for comparison.

Figure 4(a-e) shows the electronic spectrum of five twisted 2L BaTiO3
structures corresponding to different ${\theta}_{\rm{t}}$. The conduction
bands are parabolic in all cases, which bears similarities to the untwisted
bilayer apart from the zone folding. This indicates that ${\theta}_{\rm{t}}$
does not affect the conduction states in a significant manner.

To understand the evolution of the valence band with ${\theta}_{\rm{t}}$, we
zoom into a smaller energy window in the associated insets. Remarkably, our
results indicate the presence of a single detached quasi-flat band at the
valence band maximum, when 28.07∘$\leq{\theta}_{\rm{t}}\leq$16.26∘, although
the lattice constants of the supercell structures are relatively small
compared to other moiré system. To put in perspective, we summarized in in
Fig. 4(f) the bandwidth of twisted 2L BaTiO3 and other moiré materials, such
as twisted bilayer graphene, [34] MoS2, [35] and In2Se3. [36] Upon decreasing
${\theta}_{\rm{t}}$, the bandwidth of 2L BaTiO3 initially decreases, reaches a
minimum near ${\theta}_{\rm{t}}$=18.92∘, and then increases again. As
${\theta}_{\rm{t}}$ decreases further, the gap between the flat band and the
rest of the valence bands closes, giving rise to a multi-fold degeneracy at M
point, as apparent in the ${\theta}_{\rm{t}}$=14.25∘ twisted structure. In
other words, the “magic” angle of 2L BaTiO3, where the band flatness is
maximized, is predicted to be around 18.92∘. More interestingly, the bandwidth
of the flat band at ${\theta}_{\rm{t}}$=18.92∘ is evaluated to be 5.5 meV,
which is comparable with those of the magic angle 2L graphene ($\sim$10 meV at
${\theta}_{\rm{t}}$$\sim$1∘) [4, 5, 34], 2L MoS2 ($\sim$11 meV at
${\theta}_{\rm{t}}$$\sim$2∘) [35], and 2L In2Se3 ($\sim$0.9 meV at
${\theta}_{\rm{t}}$=9.43∘). These results indicate that strong correlation
effects might become relevant in twisted oxide membranes.

To better analyse the electron valence states in twisted 2L BaTiO3, we
investigated the associated charge density distribution in real space. We show
the charge density of the topmost valence band for the
${\theta}_{\rm{t}}$=14.25∘ twisted structure in the top panel of Fig. 5(a). We
found the existence of localized charges at the local stacking regions, “A”
(0,0) , “B” (0.5,0), “C” (0,0.5), and “D” (0.5,0.5) sites as shown in the
bottom panel Fig. 5(a), within the moiré supercell. These regions are
equivalent to the four possible vortex configurations shown, for instance, in
Fig. 2(g) and associated with distinct potential strengths shown in Fig. 3(e),
indicating the intimate connection between the moiré vortex pattern and
localized electronic states.

Figure 5: Topological Lieb lattice in twisted bilayer BaTiO3 Lattice model
describing the dispersion of states localized at unequal vortex sites. (a) The
charge density of top of valence band states and related tight-binding lattice
are displayed. The tight-binding sites are labelled A, B, C and D, as shown in
the upper panel. (b-f) Evolution of the tight-binding bands with twist angle.
At this level of description, the twist angle dependence is assumed to mainly
impact the onsite energies of the tight-binding sites. The onsite energies are
qualitatively estimated from the O-O interlayer distances at the vortex sites,
as shown in the colormaps. Grey dashed boxes show the primitive unitcell
structure in each twist angle.

The above considerations motivate us to construct a tight-binding model for
the localized states, in a lattice set by these four “atomic sites”, to
provide qualitative insights in regards to some of the features of the topmost
valence bands shown in Fig. 4. The model Hamiltonian describing such electron
states becomes

$\displaystyle
H(\textbf{k})=\left[\begin{tabular}[]{cccc}$\epsilon_{A}$&$t_{AB}f_{y}(\textbf{k})$&$t_{AC}f_{x}(\textbf{k})$&$t_{AD}g(\textbf{k})$\\\
$t_{AB}f_{y}(\textbf{k})$&$\epsilon_{B}$&$t_{BC}g(\textbf{k})$&$t_{BD}f_{x}(\textbf{k})$\\\
$t_{AC}f_{x}(\textbf{k})$&$t_{BC}g(\textbf{k})$&$\epsilon_{C}$&$t_{CD}f_{y}(\textbf{k})$\\\
$t_{AD}g(\textbf{k})$&$t_{BD}f_{x}(\textbf{k})$&$t_{CD}f_{y}(\textbf{k})$&$\epsilon_{D}$\end{tabular}\right],$
(5) (6)

where
$\epsilon_{j}(\textbf{k})=\epsilon_{j}-2t^{j}_{NNN}\cos(k_{x}a)-2t^{j}_{NNN}\cos(k_{y}a)$,
with $j=A,B,C,D$ and $\epsilon_{j}$ refers to the onsite energies while
$t_{ij}$ ($t^{j}_{NNN}$) describes nearest-neighbor (next nearest-neighbor)
hopping processes between neighboring sites. The structure factors
corresponding to the lattice configuration are,
$f_{x,y}(\textbf{k})=2\cos(k_{x,y}a/2)$ and
$g(\textbf{k})=2\cos[(k_{x}+k_{y})a/2]+2\cos[(k_{x}-k_{y})a/2]$, where $a$ is
the lattice constant of the moiré unit cell. Because sites “B” and “C” are
related by a $C_{4z}$ operation, we have $\epsilon_{B}=\epsilon_{C}$,
$t_{AB}=t_{AC}$ ($t_{BD}=t_{CD}$) throughout. All model parameters are given
in Note LABEL:note1 in SI.

The surface potential map shown in Figure 3(e) gives a hint on how to
constrain the onsite energies of our tight-binding model. In that case, the
$18.92^{\circ}$ twisted structure is characterized by O regions with different
onsite energies, which roughly obey
$|\epsilon_{A}|>|\epsilon_{BC}|>\epsilon_{D}$. We note that such spatially
dependent potential profile is connected to the O-O interlayer distances
within the moiré unit cell, which is twist angle dependent. We found that
regions with larger potentials are associated with smaller O-O interlayer
distances. Therefore, we quantitatively interpreted the O-O interlayer
distances as being related to the onsite energies in our tight-binding
approach, which we address next.

The tight-binding electronic structures, corresponding to onsite energy
configurations qualitatively similar to surface potential profiles, are shown
in Fig. 5(b-f), along with the interlayer O-O distance maps. We observe that
distinct onsite energies induced by varying local stacking orders in the
twisted BaTiO3 bilayers will naturally cause the upper band, describing
localized states at the “A” site, to separate from the lower bands while
reducing its band width, for all cases except at $\theta_{t}<14.25^{\circ}$
twisted lattices. This is consistent with the features observed in our first
principles calculations, where a detached quasi-flat band appears in the
spectrum whenever the local $A$ site energy exceeds $\epsilon_{B}$,
$\epsilon_{C}$ and $\epsilon_{D}$, as apparent from the minimum O-O interlayer
distances.

Note that the remaining set of tight-binding bands are in good qualitative
agreement with our first principles calculations as well, e.g., band crossing
at $M$ and dispersion throughout the moiré Brillouin zone. As the twist angle
is decreased further below $16.92^{\circ}$, the onsite energy $\epsilon_{A}$
approaches $\epsilon_{B,C,D}$ from above. Hence, we assume next an onsite
energy configuration qualitatively similar to that of the $14.25^{\circ}$
structure, i.e. $|\epsilon_{A}|\approx|\epsilon_{BC}|>\epsilon_{D}$. We note
that a multi-fold degeneracy at the $M$-point emerges naturally in this
configuration with two linearly crossing bands, in good qualitative agreement
with our first principles results shown in Fig. 4(f). Such feature on a 2D
square lattice is a hallmark of electron states in a topological Lieb lattice.
[37, 38, 39] In fact, Hamiltonian (6) describes two interpenetrating Lieb
lattices sharing “edge” sites ($B,C$), but with distinct central sites ($A$ or
$D$). An “ideal” Lieb lattice can be obtained from Hamiltonian (6) by taking,
for instance, $|\epsilon_{A}|\gg|\epsilon_{B,C,D}|$, in which case the $A$
site can be “factored” out through Löwdin partitioning, leaving behind a
three-site Lieb lattice configuration. Hence, electron states in twisted
BaTiO3 bilayers configure distorted interpenetrating Lieb lattice with twist
angle-dependent hopping and onsite energies, a crucial step towards the
realization of alternative topological electronic states in condensed matter.

##  III Conclusion

In summary, using first-principles calculations based on density functional
theory, we investigated crystal and electronic structures of twisted BaTiO3
bilayer. Among three possible interface configurations in twisted 2L BaTiO3,
we found that BaO-BaO interface resembles the vdW interface but exhibits a one
to two orders of magnitude larger GSFE compared to other morié systems. While
the conduction bands are insensitive to interlayer interactions, the valence
bands are greatly influenced by the interlayer hopping between oxygen $p$
orbitals, highlighting the importance of interlayer coupling.

With a non-zero ${\theta}_{\rm{t}}$, the huge GSFE induces substantial
relaxation within a layer, resulting in chiral in-plane vortex patterns that
were recently observed experimentally, with opposite chirality in the two
layers. The interlayer interactions also gave rise to local out-of-plane
dipole moments and robust surface electrostatic potentials. More intriguingly,
we found an isolated flat band at the valence band maximum over a wide range
of ${\theta}_{\rm{t}}$. The bandwidth of the flat bands near
${\theta}_{\rm{t}}$$\sim$19∘ is comparable with those of magic-angle bilayer
graphene, indicating strong correlation effects in our system.

We further constructed a tight-binding lattice model based on the localized
states associated with the topmost valence bands. The localized state sites
correspond to the four inequivalent local in-plane vortex sites. Our model
provides qualitative insights as to the origins of the detached quasi-flat
bands observed in our first principles calculations. These bands are well
described by two interpenetrating Lieb lattice, with twist angle-dependent
onsite energies. These observations indicate that twisted 2L BaTiO3 host
unique electronic behavior, offering a new perspective towards what might be
the first realization of an inorganic and electronic Lieb lattice.

##  IV method

The first-principles DFT calculation [21] was carried out using Vienna ab
initio simulation package (VASP). [22] We employed the plane wave basis to
expand the electronic wavefunctions with a kinetic energy cutoff of 500 eV.
The projector-augmented wave pseudopotentials [40, 41] were used for the
valence electrons, and the exchange-correlation (XC) functional was treated
within the generalized gradient approximation of Perdew–Burke–Ernzerhof (PBE).
[42] The van der Waals interaction between two adjacent layer was treated by
Grimme D3 methods [43]. A sufficiently large vacuum region
($>15~{}\textrm{\AA}$) was included in the unitcell to avoid any spurious
interlayer interactions. The atomic basis was carefully relaxed until the
Helmann-Feynman force acting on every atom was smaller than 0.01 eV/Å. The
15$\times$15$\times$1 $k$-mesh was used to sample the Brillouin zone for
untwisted structures and $\Gamma$ point sampling was used for all other
twisted structures. The equilibrium lattice constant of monolayer BaTiO3 was
calculated to be 4.002 Å, and the same value was used for all other
structures.

Acknowledgments S. L. is supported by Basic Science Research Program through
the National Research Foundation of Korea funded by the Ministry of Education
(NRF-2021R1A6A3A14038837). S. L. and T. L. are partially supported by NSF
DMREF-1921629. D. S and T. L. acknowledge partial support from Office of Naval
Research MURI grant N00014-23-1-2567. B. J. acknowledges support by the Air
Force Office of Scientific Research (AFOSR) through Grant Nos.
FA9550-21-1-0025 and FA9550-21-0460. We thank Richard D. James for helpful
comments.

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$\bigvee_{a^{\prime}\in\mathbf{a}^{\prime}}\operatorname{\downarrow}f^{-1}(a^{\prime})=\bigvee_{a^{\prime}\in\mathbf{a}^{\prime}}g^{\prime}(\operatorname{\downarrow}\\{a^{\prime}\\})=\bigvee_{a^{\prime}\in\mathbf{a}^{\prime}}\bigcup_{b^{\prime}\in\operatorname{\downarrow}\\{a^{\prime}\\}}f^{-1}(b^{\prime})=\bigcup_{a^{\prime}\in\mathbf{a}^{\prime},b^{\prime}\preceq
a^{\prime}}f^{-1}(b^{\prime})=\bigcup_{a^{\prime}\in\mathbf{a}^{\prime}}f^{-1}(b^{\prime})=g^{\prime}(\mathbf{a}^{\prime})$,
by using Lemma 12, (70b), exploiting that $\mathbf{a}^{\prime}$ is a down-set
(65a), then (70b) again, respectively. ∎

Although it will not be used, the following lemma shows that the whole
construction in this section works also if up-sets are used instead of down-
sets.

######  Lemma 15.

Lemmas 8, 9, 10, 11, 12, 13 and 14 hold also if up-sets are used instead of
down-sets in the construction (67).

###### Proof.

All the proofs can be repeated with slight modifications. ∎

###  A.4 Application to the set and integer partitions

Now we turn to the case of the main text, for which the machinery developed in
the previous sections is applied. That is, for the roles of $(P,\preceq)$ and
$(P^{\prime},\preceq^{\prime})$ we have the poset $(P_{\text{I}},\preceq)$ of
set partitions (2b) with the refinement relation (2c), and the poset
$(\hat{P}_{\text{I}},\preceq)$ of integer partitions (18b) with the refinement
relation (18c).

First, we have the $\sigma\in\mathrm{S}(L)$ permutations of the elementary
subsystems $L$, acting naturally on the partitions
$\xi=\\{X_{1},X_{2},\dots,X_{{\lvert\xi\rvert}}\\}\in P_{\text{I}}$ as
$\sigma(\xi)=\bigl{\\{}\\{\sigma(i)\;|\;i\in
X\\}\;\big{|}\;X\in\xi\bigr{\\}}\in P_{\text{I}}$. We will make use of the
obvious observations that all the permutations of a partition $\xi\in
P_{\text{I}}$ are of the same type,
$s(\sigma(\xi))=s(\xi)\in\hat{P}_{\text{I}}$; and all partitions of a given
type $\hat{\xi}\in\hat{P}_{\text{I}}$ can be transformed into one another by
suitable permutations, that is, for all $\xi_{1},\xi_{2}\in
s^{-1}(\hat{\xi})$, there exists at least one $\sigma\in\mathrm{S}(L)$ by
which $\xi_{2}=\sigma(\xi_{1})$. Also, for all $\sigma\in\mathrm{S}(L)$
permutations, $\upsilon\preceq\xi$ if and only if
$\sigma(\upsilon)\preceq\sigma(\xi)$. Using these, we can show that (63)-like
conditions hold for this case.

######  Lemma 16.

In the setting of the main text, the following conditions hold

$\displaystyle\begin{split}&\forall\hat{\xi},\hat{\upsilon}\in\hat{P}_{\text{I}},\;\;\text{if}\;\;\hat{\upsilon}\preceq\hat{\xi}\;\;\text{then}\;\;\\\
&\qquad\forall\xi\in s^{-1}(\hat{\xi}),\;\exists\upsilon\in
s^{-1}(\hat{\upsilon}),\;\;\text{s.t.}\;\;\upsilon\preceq\xi;\end{split}$
(72a)
$\displaystyle\begin{split}&\forall\hat{\xi},\hat{\upsilon}\in\hat{P}_{\text{I}},\;\;\text{if}\;\;\hat{\upsilon}\preceq\hat{\xi}\;\;\text{then}\;\;\\\
&\qquad\forall\upsilon\in s^{-1}(\hat{\upsilon}),\;\exists\xi\in
s^{-1}(\hat{\xi}),\;\;\text{s.t.}\;\;\upsilon\preceq\xi;\end{split}$ (72b)
$\displaystyle\begin{split}&\forall\xi,\upsilon,\zeta\in
P_{\text{I}},\;\;\text{if}\;\;\zeta\preceq\upsilon\preceq\xi\\\
&\qquad\;\;\text{and}\;\;s(\zeta)=s(\xi),\;\;\text{then}\;\;s(\upsilon)=s(\xi).\end{split}$
(72c)

###### Proof.

Recall that $\hat{\upsilon}\preceq\hat{\xi}$ by definition if there exist
$\upsilon_{1}\in s^{-1}(\hat{\upsilon})$ and $\xi_{1}\in s^{-1}(\hat{\xi})$,
such that $\upsilon_{1}\preceq\xi_{1}$, see (18c), in accordance with
(LABEL:eq:relp). Then, for all $\xi_{2}\in s^{-1}(\hat{\xi})$ we can construct
$\upsilon_{2}$ which is $\upsilon_{2}\preceq\xi_{2}$, simply as
$\upsilon_{2}:=\sigma(\upsilon_{1})$ by any of the permutations implementing
$\xi_{2}=\sigma(\xi_{1})$, leading to (72a).
(72b) can be proven analogously.
(72c) can be proven by noting that for all $\xi,\upsilon\in P$ we have that if
$\upsilon\preceq\xi$ then ${\lvert\upsilon\rvert}\leq{\lvert\xi\rvert}$, by
(2c); and ${\lvert\xi\rvert}={\lvert s(\xi)\rvert}$, by (17). Then
$\zeta\preceq\upsilon\preceq\xi$ and $s(\zeta)=s(\xi)$ lead to
${\lvert\zeta\rvert}\leq{\lvert\upsilon\rvert}\leq{\lvert\xi\rvert}$ with
${\lvert\zeta\rvert}={\lvert\xi\rvert}$, form which we have
${\lvert\upsilon\rvert}={\lvert\xi\rvert}$. Now we need that
${\lvert\upsilon\rvert}={\lvert\xi\rvert}$ and $\upsilon\preceq\xi$ lead to
$\upsilon=\xi$ (then $s(\upsilon)=s(\xi)$), which holds, because (2c) must be
fulfilled with the same number of parts: let us number them accordingly as
$Y_{i}\subseteq X_{i}$ for $\xi=\\{X_{1},X_{2},\dots,X_{k}\\}$ and
$\upsilon=\\{Y_{1},Y_{2},\dots,Y_{k}\\}$, with the constraints of being
disjoint and $\bigcup_{i=1}^{k}X_{i}=\bigcup_{i=1}^{k}Y_{i}=L$, which leads to
$Y_{i}=X_{i}$. ∎

Recall that Lemma 1 states that if a poset $(P,\preceq)$ is transformed by a
function $f$ as given in (LABEL:eq:relp), then
$(P^{\prime},\preceq^{\prime})=(f(P),f(\preceq))$ is a poset if the conditions
(63) hold. Now Lemma 16 shows that for the poset $(P_{\text{I}},\preceq)$, if
transformed by $s$ as given in (18b) and (18c) in the main text, the
corresponding conditions (72) hold, so
$(\hat{P}_{\text{I}},\preceq)=(s(P),s(\preceq))$ is a poset.

Recall also that if we form the down-set lattices (66), then Lemma 4 and Lemma
5 together state that (67) commutes, Lemma 6 states that the conditions in
(63) are inherited. Then with Lemma 7, we have that in the main text, the
three-level construction is well-defined as follows.

######  Corollary 17.

The diagram (14) in the main text commutes.

With Lemma 12 and (71), we also have that for the first two levels of the
construction in the main text, the following identities concerning the
principal ideals hold.

######  Corollary 18.

In the setting of the main text, for all $\hat{\xi}\in\hat{P}_{\text{I}}$ and
$\hat{\boldsymbol{\xi}}\in\hat{P}_{\text{II}}$, we have

$\displaystyle\vee s^{-1}(\operatorname{\downarrow}\\{\hat{\xi}\\})$
$\displaystyle=\operatorname{\downarrow}s^{-1}(\hat{\xi}),$ (73a)
$\displaystyle\vee s^{-1}(\hat{\boldsymbol{\xi}})$
$\displaystyle=\bigvee_{\hat{\xi}\in\hat{\boldsymbol{\xi}}}\operatorname{\downarrow}s^{-1}(\hat{\xi}).$
(73b)

With Lemma 13 and Lemma 10, we also have that for the first two levels of the
construction in the main text, the following order isomorphisms hold.

######  Corollary 19.

In the setting of the main text, for all
$\hat{\xi},\hat{\upsilon}\in\hat{P}_{\text{I}}$ and
$\hat{\boldsymbol{\xi}},\hat{\boldsymbol{\upsilon}}\in\hat{P}_{\text{II}}$, we
have

$\displaystyle\hat{\upsilon}\preceq\hat{\xi}$
$\displaystyle\;\;\Longleftrightarrow\;\;\vee
s^{-1}(\operatorname{\downarrow}\\{\hat{\upsilon}\\})\preceq\vee
s^{-1}(\operatorname{\downarrow}\\{\hat{\xi}\\}),$ (74a)
$\displaystyle\hat{\boldsymbol{\upsilon}}\preceq\hat{\boldsymbol{\xi}}$
$\displaystyle\;\;\Longleftrightarrow\;\;\vee
s^{-1}(\hat{\boldsymbol{\upsilon}})\preceq\vee
s^{-1}(\hat{\boldsymbol{\xi}}).$ (74b)

##  Appendix B Miscellaneous proofs

###  B.1 Monotonicity of correlation measures

Here we show the $\xi$-LO monotonicity of the $\xi$-correlation (5a), from
which the LO monotonicity of the $\boldsymbol{\xi}$-correlation (10a) follows,
because the latter one is a minimum of some of the former ones. A quantum
channel (completely positive trace preserving map [6, 7]) $\Phi$ is $\xi$-LO,
if it can be written as $\Phi=\bigotimes_{X\in\xi}\Phi_{X}$ with quantum
channels $\Phi_{X}:\mathcal{D}_{X}\to\mathcal{D}_{X}$ for all $X\in\xi$. With
this,

$\begin{split}C_{\xi}\bigl{(}\Phi(\varrho)\bigr{)}&\overset{\eqref{eq:CI}}{{\;\;=\;\;}}\min_{\sigma\in\mathcal{D}_{\xi\text{-unc}}}D\bigl{(}\Phi(\varrho)\big{\|}\sigma\bigr{)}\\\
&\overset{\eqref{eq:free}}{{\;\;\leq\;\;}}\min_{\sigma^{\prime}\in\Phi(\mathcal{D}_{\xi\text{-unc}})}D\bigl{(}\Phi(\varrho)\big{\|}\sigma^{\prime}\bigr{)}\\\
&{\;\;=\;\;}\min_{\sigma\in\mathcal{D}_{\xi\text{-unc}}}D\bigl{(}\Phi(\varrho)\big{\|}\Phi(\sigma)\bigr{)}\\\
&\overset{\eqref{eq:contr}}{{\;\;\leq\;\;}}\min_{\sigma\in\mathcal{D}_{\xi\text{-unc}}}D\bigl{(}\varrho\big{\|}\sigma\bigr{)}\\\
&\overset{\eqref{eq:CI}}{{\;\;=\;\;}}C_{\xi}(\varrho),\end{split}$

where the first inequality comes from

$\Phi(\mathcal{D}_{\xi\text{-unc}})\subseteq\mathcal{D}_{\xi\text{-unc}},$
(75)

which holds for $\xi$-LOs; and the second inequality comes from the
monotonicity of the relative entropy (1b),

$D\bigl{(}\Phi(\varrho)\big{\|}\Phi(\sigma)\bigr{)}\leq D(\varrho\|\sigma),$
(76)

which holds for all quantum channels [6, 7].

Note that the monotonicity shown here is a particular case of a much more
general property of monotone distance based _geometric measures_ [65] in
resource theories [75]: the monotonicity with respect to free maps, by which
the set of free states is mapped onto itself.

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