# Polarization textures in crystal supercells with topological bands

Wojciech J. Jankowski  Theory of Condensed Matter, Cavendish Laboratory,
University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, UK  Daniel
Bennett  John A. Paulson School of Engineering and Applied Sciences, Harvard
University, Cambridge, Massachusetts 02138, USA  Aneesh Agarwal  Theory of
Condensed Matter, Cavendish Laboratory, University of Cambridge, J. J. Thomson
Avenue, Cambridge CB3 0HE, UK  Gaurav Chaudhary  Theory of Condensed Matter,
Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge
CB3 0HE, UK  Robert-Jan Slager  Theory of Condensed Matter, Cavendish
Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE,
UK

###### Abstract

Two-dimensional materials are a highly tunable platform for studying the
momentum space topology of the electronic wavefunctions and real space
topology in terms of skyrmions, merons, and vortices of an order parameter.
Such textures for electronic polarization can exist in moiré heterostructures.
A quantum-mechanical definition of local polarization textures in insulating
supercells was recently proposed. Here, we propose a definition for local
polarization that is also valid for systems with topologically non-trivial
bands. We introduce semilocal hybrid polarizations, which are valid even when
the Wannier functions in a system cannot be made exponentially localized in
all dimensions. We use this definition to explicitly show that nontrivial
real-space polarization textures can exist in topologically non-trivial
systems with non-zero Chern number under (1) an external superlattice
potential, and (2) under a stacking-induced moiré potential. In the latter, we
find that while the magnitude of the local polarization decreases
discontinuously across a topological phase transition from trivial to
topologically nontrivial, the polarization does not completely vanish. Our
findings suggest that band topology and real-space polar topology may coexist
in real materials.

##  I Introduction

The understanding and control of exotic electronic states of matter is one of
the central aims of condensed matter physics. One notable avenue in this
regard is the study of topological materials, hosting anomalous bulk and
boundary effects and protected edge currents Qi and Zhang (2011); Hasan and
Kane (2010). Topological insulators and semimetals are promised to affect
technological advancements, with applications ranging from spintronics to
possibly providing platforms for quantum computing Armitage  _et al._ (2018);
Nayak  _et al._ (2008). This field was arguably initiated by the observation
that even without a net magnetic field, Hall responses can be achieved in the
form of quantum anomalous Hall effects (QAHE) Haldane (1988). In such QAH
systems, wavefunctions exhibit a nontrivial winding characterized by a
topological invariant known as a Chern class, which is an archetypal example
of a characteristic class associated with complex vector bundles.

On a seemingly different note, there has been a lot of recent interest in
engineering exotic states via stacking engineering of layered materials.
Combining layers with relative twist angles or lattice mismatches to form
superlattice structures known as moiré materials Bistritzer and MacDonald
(2011) can lead to interesting phenomena such as superconductivity Cao  _et
al._ (2018a); Lu  _et al._ (2019); Saito  _et al._ (2020), Mott-insulating
behaviour Cao  _et al._ (2018b), ferroelectricity Li and Wu (2017); Yasuda
_et al._ (2021); Bennett and Remez (2022); Bennett (2022); Ko  _et al._
(2023), nontrivial topology, both of bands Koshino  _et al._ (2018); Po  _et
al._ (2018); Song  _et al._ (2019); Po  _et al._ (2019); Wu  _et al._ (2021)
and real space quantities including polarization Bennett  _et al._ (2023a, b),
twist fields Engelke  _et al._ (2023) and magnetic fields Guerci  _et al._
(2022). A favorable aspect of such stacking-engineered phases is that they can
in principle be tuned through the supercell period (twist angle or lattice
mismatch), number of layers, and chemistry (changing the materials).
Topological states can be engineered with constituent materials which are
ordinarily trivial, such as transition-metal dichalcogenides (TMDs), e.g.
$\mathrm{MoS_{2}}$, $\mathrm{WSe_{2}}$, etc. Angeli and MacDonald (2021);
Zhang  _et al._ (2021); Li  _et al._ (2021), where fractional Chern states at
zero magnetic field have also been predicted Ledwith  _et al._ (2020) and
recently experimentally observed Xie  _et al._ (2021); Cai  _et al._ (2023);
Park  _et al._ (2023). Moreover, because of the additional length scale of the
superlattice potential, locally nonzero Chern numbers can be found in
different stacking domains within the moiré superlattice Efimkin and MacDonald
(2018); Guerci  _et al._ (2023a, b); Xia  _et al._ (2023). The idea that such
a topological invariant can be attributed to a regions in real space, which we
refer to as “Chern domains”, is very intriguing for applications. For example,
knowledge of such domains, and the ability to engineer domains with different
Chern numbers implies that edge currents can be induced and controlled on the
domain walls separating them. The topological nature of a Chern domain is
locally reflected by the presence of QAHE at the domain walls Efimkin and
MacDonald (2018) and they can be computationally characterized by Chern
markers Resta (1998); Bianco and Resta (2011).

In moiré heterostructures of non-elemental compounds, the crystalline
superlattices can very naturally break the inversion symmetry $\mathcal{I}$
within a domain, offering a natural platform for the development of
polarization textures, which also can support topological features therein.
Such topological polarization textures realizing merons or skyrmions,
corresponding topologically to the $\pi_{2}[S^{2}]\cong\mathbb{Z}$ homotopy,
were predicted in stacked bilayers of hexagonal boron nitride (hBN) under
twist or strain Bennett  _et al._ (2023a, b). Similar topological polar
textures are also commonly observed in perovskite nanostructures Junquera  _et
al._ (2023), and were recently also realized in perovskites layered under
moiré geometry Sánchez-Santolino  _et al._ (2024). Currently, it is not clear
whether topological polarization textures can coexist with momentum-space band
topological features. For example, the notion of localizability breaks down in
topologically non-trivial bands in two or higher dimensions, where one cannot
describe the electronic states using a basis of exponentially localized
Wannier functions. As a result, the definition of local polarization textures,
as applied to a trivial insulator Bennett  _et al._ (2023b), is no longer
applicable.

In this work, we address this problem by proposing a definition of local
polarization in a Chern insulator, and showing that the real-space polar
topology can coexist with band topology. Our formulation is a natural
extension to the definition of local polarization in a crystal supercell
Bennett  _et al._ (2023b), which is not straightforward, as the Berry phase is
a global property of the system Vanderbilt and King-Smith (1993); King-Smith
and Vanderbilt (1993); Resta (1994); Vanderbilt (2018). We formulate the local
polarization by decomposing the Berry phase in terms of semilocal hybrid
polarizations (SHPs), while also making a connection to Chern topology Coh and
Vanderbilt (2009). In particular, we consider the evolution of the local
polarization in a crystal superlattice and elucidate the correspondences
between the local polarization textures, local polarization jumps Yoshida  _et
al._ (2023a), and the changes of the bulk state topology realized in
minibands. We show that, across a topological phase transition (TPT), the
local polarization in a texture, although decreasing in magnitude, does not
vanish entirely.

Figure 1: Benchmarking the gauge-independent formulation of the local
polarization in crystal supercells with topological bands. The local
polarization $P_{\beta}({\bf r}_{j})$ is defined componentwise as a sum of the
semilocal hybrid polarizations (SHPs) $P^{h}_{\beta}({\bf r}_{j},k_{\perp})$
over the perpendicular quasimomenta $k_{\perp}$. SHPs are constituted by the
shifts of the hybrid Wannier charge center (HWCCs), subject to local
depolarizing perturbations ${\bf x}({\bf r}_{j})$. In particular, the shifts
in HWCCs at every fixed $k_{\perp}$ can be induced by a superlattice potential
$V_{\text{SL}}$. (a) A HWCC localized in the $x$ direction, shifting across
the supercell on applying a perturbation to the central subcell, and defining
$P_{\beta}({\bf r}_{j}={\bf 0})$. (b) The Wilson loop representing the winding
of Berry phase $\phi(k_{\perp})$ in two-dimensional supercell. The Berry phase
is equivalent to the total sum of HWCCs corresponding to the occupied bands,
which indicates a $total$ polarization in the supercell. On the other hand,
its winding reflects the non-vanishing Chern number in the occupied
topological bands. As captured by Eq. (II); for supercells with topological
bands there is indeed a net winding of the total hybrid polarization
$P^{h}_{\beta}(k_{\perp})$ as a function of $k_{\perp}$, when all occupied
HWCCs are summed.

##  II Results

Since our aim is to define local polarization in a periodic solid, the most
natural setting is to consider a system experiencing a superlattice potential
(via moiré engineering or external potential), such that within the supercell
the polarization can acquire spatial dependence and its local definition is
meaningful. The local polarization in a crystal supercell can be defined as
the total change in the Berry phase of the supercell, subject to a local
depolarizing perturbation in a given subcell starting from a non-polar
reference cell configuration Bennett  _et al._ (2023b). Equivalently, the
corresponding local polarization can be computed by integrating the Born
effective charges along a path of phonon displacements which connect the
atomic configurations in each cell. Alternatively, the local polarization can
be recast as the change in all the Wannier centers in the system with respect
to the local perturbations in a given cell, as long as the Wannierized bands
are topologically trivial. For topologically trivial systems, the Wannier
functions can be made exponentially localized, and the Wannier centers for
each band are essentially the Berry phases, but with units of length
Vanderbilt and King-Smith (1993); King-Smith and Vanderbilt (1993); Marzari
and Vanderbilt (1997); Souza  _et al._ (2001); Marzari  _et al._ (2012). We
briefly review these definitions of local polarization in App. A. It should be
stressed that, contrary to the first two approaches of Ref. Bennett  _et al._
(2023b), the third way via Wannier functions is not directly applicable to
topological systems, which is an issue that we resolve in this work.

As mentioned above, for topologically nontrivial systems, there is an
obstruction to obtaining exponentially localized Wannier functions, and the
Wannier centers cannot be obtained Coh and Vanderbilt (2009). However, we can
describe the winding of the Bloch states, which is equivalent to the Berry
phase, using hybrid Wannier charge centers (HWCCs) or Wilson loops (see Fig.
1). The HWCCs can be obtained as the expectation values of a single component
of position operator $\hat{r}_{\beta}$:

$\bar{w}^{h}_{n,\beta}(k_{\perp})\equiv\bra{w^{h}_{n}(k_{\perp})}\hat{r}_{\beta}\ket{w^{h}_{n}(k_{\perp})}\;,$
(1)

using a basis of hybrid Wannier functions $\ket{w^{h}_{n}(k_{\perp})}$ (HWFs),
which are obtained by Fourier transforming the Bloch states only in the
direction $\beta$ (for more details, see App. B). In the case of more than two
spatial dimensions; to deduce the local polarization, the Fourier transform in
only one direction to obtain HWFs is similarly required. Here, $k_{\perp}$ are
the wavenumbers in the direction orthogonal to $\beta$. The total hybrid
polarization in a supercell can be defined in terms of HWCCs summed over the
occupied band indices $n$,

$P^{h}_{\beta}(k_{\perp})=-\frac{ef}{\Omega_{\text{sc}}}\sum_{n}^{\text{occ}}\bar{w}^{h}_{n,\beta}(k_{\perp})\;,$
(2)

where $f$ is an occupation factor, $\Omega_{\text{sc}}$ is the supercell
volume, and the HWCCs are localized in the direction $k_{\perp}$. Analogously,
for the purposes of defining the local polarization in a topological, non-
Wannierizable crystal supercell, it is useful to introduce

$P^{h}_{\beta}({\bf
r}_{j},k_{\perp})=-\frac{ef}{\Omega_{0}}\sum_{n}^{\text{occ}}\int_{0}^{{\bf
x}({\bf
r}_{j})}\partial_{x^{\prime}_{\kappa,\alpha}}\bar{w}^{h}_{n,\beta}(k_{\perp})\mathop{}\\!\mathrm{d}x^{\prime}_{\kappa,\alpha}\;,$
(3)

which we define as the semilocal hybrid polarization (SHP), where $\Omega_{0}$
is a subcell volume (see App. B for more details). In the spirit of Ref.
Bennett  _et al._ (2023b); here, the integral represents the change of hybrid
polarizations on introducing local displacements/reparametrizations: ${\bf
x}({\bf r}_{j})=\\{x_{\kappa,\alpha}\\}$ where $\kappa$ labels the atoms and
$\alpha$ specifies the perturbation direction. Additionally, the Einstein
summation convention was assumed. In order to deduce $P^{h}_{\beta}({\bf
r}_{j},k_{\perp})$, the local perturbations are imposed only in a subcell
${\bf r}_{j}$ of a supercell, bringing its configuration to the non-polar
reference state ${\bf x}({\bf r}_{j})={\bf 0}$. We propose that the
introduction of the semilocal hybrid polarizations allows us to evaluate the
local polarization in a topological supercell as

$P_{\beta}({\bf r}_{j})=\oint_{\mathcal{C}}P^{h}_{\beta}({\bf
r}_{j},k_{\perp})\mathop{}\\!\mathrm{d}k_{\perp}\;,$ (4)

where $\mathcal{C}$ is a closed loop in the BZ, starting from $k_{\perp}^{0}$
and of length of the superlattice reciprocal vector ${\bf b}_{\perp}$. By
substituting Eq. (3) into Eq. (4), we obtain

$P_{\beta}({\bf
r}_{j})=-\frac{ef}{\Omega_{0}}\sum_{n}^{\text{occ}}\oint_{\mathcal{C}}\int_{0}^{{\bf
x}({\bf
r}_{j})}\partial_{x^{\prime}_{\kappa,\alpha}}\bar{w}^{h}_{n,\beta}(k_{\perp})\mathop{}\\!\mathrm{d}x^{\prime}_{\kappa,\alpha}\mathop{}\\!\mathrm{d}k_{\perp}\;,$
(5)

which is a natural extension to the method of computing the local polarization
in terms of the Wannier functions. However, contrary to the previous
definition Bennett  _et al._ (2023b) (see also App. A), Eq. $\eqref{P-local}$
is valid for both topological and trivial bands. The above relation states
that local polarization in a system with topologically non-trivial bands can
be obtained componentwise i.e. a certain polarization component is simply the
projection of the flow of hybrid Wannier center that is exponentially
localized along the same direction. This definition is motivated by the fact
that change in polarization is the physical quantity that is fundamentally
related to the polarization currents
$\textbf{j}_{\text{P}}\equiv\mathop{}\\!\mathrm{d}\textbf{P}/\mathop{}\\!\mathrm{d}t$
flowing through the system as it is adiabatically evolved from an initial to
the final state
($\Delta\textbf{P}\equiv\int\textbf{j}_{\text{P}}\mathop{}\\!\mathrm{d}\text{t}=\textbf{P}_{f}-\textbf{P}_{i}$).
This relation can be resolved componentwise, allowing us to contruct hybrid
Wannier functions that are maximally localized in only one direction and
observing their flow as the polariozation currents (See App. B for more
details).

We note that for the two-dimensional case of e.g. Chern insulators,
$\beta=x,y$ specifies the in-plane directions. Here, $k^{0}_{\perp}$ should be
chosen consistently for finding polarization changes, e.g. $P^{h}_{\beta}({\bf
r}_{j},k_{\perp}^{0})=0$, when ${\bf x}({\bf r}_{j})={\bf 0}$, which, upon
choosing a maximally-smooth gauge, should ensure a vanishing polarization for
nonpolar configurations Bennett  _et al._ (2023b). Importantly, the point
$(k^{0}_{x},k^{0}_{y})$ needs to be chosen consistently for the evaluation of
$\textbf{P}({\bf r}_{j})=(P_{x}({\bf r}_{j}),P_{y}({\bf r}_{j}))$, with the
real-space integration limits of Eq. (5), which define initial and final
states with respect to which the local polarization is computed as a change
($\textbf{P}\equiv\Delta\textbf{P}=\textbf{P}_{f}-\textbf{P}_{i}$) Resta
(1992). If the $k$-space integral is performed inconsistently in the initial
and final real-space states, the resulting polarizations acquire an erroneous
term depending on the shift in the integration endpoints and $C$, as was
pointed out for arbitrary Chern insulators in Ref. Coh and Vanderbilt (2009).
The above definition is analogous to the Berry-phase formulation of the total
polarization in Chern insulators Coh and Vanderbilt (2009) as a global
quantity. Indeed, upon relating HWCCs to Berry phases

$\phi_{n}(k_{\gamma})=i\oint_{k_{\perp}=k_{\gamma}}\braket{u_{n{\bf
k}}}{\partial_{k_{\beta}}u_{n{\bf k}}}\mathop{}\\!\mathrm{d}k_{\beta}\;,$ (6)

our definition is consistent with the previous formulations of electric
polarization in Chern insulators Coh and Vanderbilt (2009) that obtains the
$total$ polarization of a topological system, without partitioning into any
local contributions to the net electric dipole moment present in a supercell.

Furthermore, we can relate the SHPs to band topology in crystal supercells,
therefore settling whether any information about the topological character of
the minibands can be inferred from $P^{h}_{\beta}({\bf r}_{j},k_{\perp})$. It
is known that the Chern number $C$ of a system can be calculated from the
winding of HWCCs Gresch  _et al._ (2017), or equivalently, hybrid
polarization, along the quasimomentum component (here $k_{\gamma}$).
Essentially, it is the winding of Berry phases $\phi_{n}(k_{\gamma})$ across a
Wilson loop,

$\displaystyle
C=\sum_{n}^{\text{occ}}\frac{1}{L_{\beta}}\Big{[}\bar{w}^{h}_{n,\beta}(k_{\perp}=2\pi)-\bar{w}^{h}_{n,\beta}(k_{\perp}=0)\Big{]}$
$\displaystyle\hskip
7.11317pt=-\frac{1}{efa_{\beta}}\Big{[}P^{h}_{\beta}(k_{\perp}=2\pi)-P^{h}_{\beta}(k_{\perp}=0)\Big{]},$
(7)

where we impose $L_{\perp}=1$ for simplicity. We propose a further variant of
this correspondence for the local perturbations in a supercell, namely,

$\displaystyle\Delta C(\\{{\bf
r}_{j}\\})=-\frac{1}{efL_{\beta}}\sum_{j\in\\{{\bf
r}_{j}\\}}\left[P^{h}_{\beta}({\bf r}_{j},k_{\perp}=2\pi)-P^{h}_{\beta}({\bf
r}_{j},k_{\perp}=0)\right]$ $\displaystyle\hskip
39.83368pt=-\frac{1}{efL_{\beta}}\sum_{j\in\\{{\bf
r}_{j}\\}}\left.P^{h}_{\beta}({\bf
r}_{j},k_{\perp})\right|_{k_{\perp}^{0}}^{k_{\perp}^{0}+{\bf b}_{\perp}}.$ (8)

The natural interpretation of $\Delta C(\\{{\bf r}_{j}\\})$ is the change of
the total Chern number in the supercell minibands, as induced by a
depolarizing perturbation imposed in the chosen cells $\\{{\bf r}_{j}\\}$. In
particular, to induce a non-trivial change $\Delta C(\\{{\bf r}_{j}\\})$, the
gap between minibands close to the Fermi level must be very small, in order to
admit topological phase transitions (TPTs) that cross an intermediate metallic
state. Realizations of such band gaps can be naturally achieved by applying a
superlattice potential to a Chern insulator, bringing it close to the critical
point associated with a TPT. Under such circumstances, local displacements
induced by finite-size probes, or an additional local depolarizing potential,
could change the Chern number of the supercell ground state; see Fig. 2(d) for
reference.

Due to the local nature of this close-to-critical setup, it is also
interesting to compare it with the local Chern markers $C({\bf r}_{j})$ Bianco
and Resta (2011), which are a real space decomposition of the total Chern
number present in the minibands, reflecting the local anomalous Hall
conductivity (see App. C for a review). Contrary to the conventional Chern
marker $C({\bf r}_{j})$, $\Delta C(\\{{\bf r}_{j}\\})$ is necessarily
quantized, although restricted to supercells, i.e. it cannot be efficiently
applied to amorphous, or arbitrarily disordered systems, unless a finite size
of systems supercell is assumed. The reason for quantization is that
$k_{\perp}=0$ and $k_{\perp}=2\pi$ physically correspond to the same value
over a compact BZ, thus also identifying the corresponding states. Hence, the
HWCC flows captured by $P^{h}_{\beta}({\bf r}_{j},k_{\perp}=2\pi)$ need to be
integer in $L_{\beta}$, or otherwise $k_{\perp}=0$ and $k_{\perp}=2\pi$ would
be physically distinguishable.

##  III Model realizations

We utilize the above theory and illustrate our findings using two examples:
(i) a Chernful supercell in the presence of a superlattice potential, and (ii)
a twisted moiré system with Chern topology, realizing a supercell with
spatially modulated interlayer tunnelling. In the first case, we consider a
simple Chern insulator, namely the Haldane model (see Fig. 2(a)), subject to
an addition of a superlattice potential with magnitude $V_{\rm SL}$ (see Fig.
2(b)), which we refer to as the super-Haldane model. We first consider a
honeycomb lattice with nearest ($t_{1}$) and second-nearest ($\pm it_{2}$)
neighbor hoppings, and onsite mass $\pm m$ on a bipartite lattice of atoms
$A,B$. Further to this, within the orbital basis $A,B$, we impose the
superlattice potential ${V_{Aj}=-V_{Bj}=V_{\rm SL}\cos\Big{(}\frac{\pi
x}{N}\Big{)}\cos\Big{(}\frac{\pi y}{N}\Big{)}}$, where the unit cell $j$
resides at fractional coordinates $0<x,y<N$. The Hamiltonian for the super-
Haldane model is given by

$\displaystyle H=-t_{1}\sum_{\langle
I,i^{\prime}\rangle}(c^{\dagger}_{Ai}c_{Bi^{\prime}}+\textnormal{h.c.})-i|t_{2}|\sum_{\langle\langle
I,i^{\prime}\rangle\rangle}(c^{\dagger}_{Ai}c_{Ai^{\prime}}+c^{\dagger}_{Bi}c_{Bi^{\prime}}+\textnormal{h.c.})$
$\displaystyle\hskip
28.45274pt+\sum_{j}[(V_{Bj}+m)c^{\dagger}_{Bj}c_{Bj}+(V_{Aj}-m)c^{\dagger}_{Aj}c_{Aj}],$
(9)

where $c^{\dagger}_{A/Bj}$ and $c_{A/Bj}$ are the creation and annihilation
operators for electron in orbital $A/B$ in the cell located at ${\bf r}_{j}$.
Here, $\langle...\rangle$ denotes first neighbors, and
$\langle\langle...\rangle\rangle$ denotes second neighbors, see also Fig.
2(a). We find that the model realizes a polarization texture [Fig. 2(c)], and
there is a sharp change of the local polarization across the boundary, where
the superlattice potential combined with onsite mass $m$ approaches the value
of the topological mass imposed with $t_{2}$. The texture in Fig. 2(c) was
obtained using Eq. (5) for the parametrization
$(t_{1},t_{2},m,V_{SL})=(1,1,0.5,1)$ of the model introduced in Eq. (III),
with the supercell size, $N=51$. Notably, the texture demonstrates that the
local polarization discontinuously flips on moving away from the supercell
centre, as the Haldane mass $t_{2}$ dominates $m$ combined with the modulated
$V_{\text{SL}}$. Hence, the local polarization forms a circular domain
surrounded by a visible ring, consistently with the discontinuities found
between Chern and trivial insulators realized without supercells Yoshida  _et
al._ (2023a); Vaidya  _et al._ (2024). Furthermore, we find that when
superlattice potential dominates the hopping locally – effectively as a local
onsite, or Semenoff Semenoff (1984), mass term – a reduction of the local Hall
conductivity occurs. We support this finding by calculating the local Chern
marker $C({\bf r}_{j})$ (Fig. 2(d)), which we contrast with the quantized
$\Delta C(\\{{\bf r}_{j}\\})$ (Fig. 2(e)) introduced in the previous section.
The changes in local Chern markers are in close correspondence with the
trivialization of the SHP indicated by $\Delta C(\\{{\bf r}_{j}\\})$,
corresponding to topological transitions in response to depolarizing
perturbations.

Figure 2:  (a) Sketch of the Haldane model. A bipartite lattice with nearest
neighbor hopping $t_{1}$ and second nearest neighbor hopping $\pm it_{2}$. (b)
Illustration of the superlattice potential $V_{\rm SL}$. (c) Local
polarization field in the $N=51$ super-Haldane model calculated using Eq. (5).
The local polarization is defined as a change determined by the integration
limits of SHPs, and is expressed in the units of the quantum of polarization
Bennett  _et al._ (2023b). (d) Quantized $\Delta C(\\{{\bf r}_{j}\\})$ numbers
for the $N=51$ super-Haldane model, calculated using Eq. (II) for depolarizing
perturbations of size $11\times 11$ cells around each cell $j$. (e)
Conventional Chern markers $C({\bf r}_{j})$ for the $N=51$ super-Haldane
model.

Furthermore, we examine the connection between SHPs and band topology in
another example, by stacking two monolayer copies of the Haldane model and
introducing a relative twist between the layers. We refer to this system as
twisted bilayer Haldanium, see App. D for more details. The effective tight-
binding model adapted for the studied twisted Haldanium bilayer can be
compactly written as

$\displaystyle H_{\rm tHB}=[m-t_{2}\sum^{3}_{i=1}\sin({\bf k}\cdot{\bf
b_{i}})]\tau_{z}\otimes\mathsf{1}_{2}+[t_{{\bf
k}}\tau_{+}\otimes\mathsf{1}_{2}+\text{h.c.}]$ $\displaystyle\hskip
85.35826pt+[T({\bf k},{\bf x})\otimes\sigma_{+}+\text{h.c.}].$ (10)

Here, $\tau_{i}$ are Pauli matrices acting in the single-layer orbital basis
($A$,$B$), whereas the Pauli matrices $\sigma_{i}$ act in the top/bottom layer
basis ($l=t,b$), with $\tau_{+}=\frac{1}{2}(\tau_{x}+i\tau_{y})$, and
analogously for $\sigma_{+}$. Additionally, $\otimes$ denotes a Kronecker
product, and ${\bf b_{i}}$ correspond to the second-neighbour hopping vectors.
Importantly, $t_{\bf k}$ represents the nearest-neighbour intra-layer hopping,
while $T({\bf k},{\bf x})$ is a $2~{}\times~{}2$ local stacking/configuration-
dependent interlayer hopping matrix representing the tunnelling of electrons
between the layers, as expressed explicitly in App. D. Finally, in addition to
the adapted tight-binding model, for more general possible studies of low-
energy physics associated with topological fermions on a bilayer consisting of
honeycomb lattice an effective continuum model for twisted Haldanium reads

$\begin{split}H_{\text{moir\'{e}}}=\sum_{l=\text{t,b}}\int\psi^{\dagger}_{l}\Bigg{[}\biggl{(}m+b\Big{(}\partial_{r_{\beta}}+\frac{\partial\mathcal{D}_{l,\gamma}}{\partial
r_{\beta}}\partial_{r_{\gamma}}\Big{)}^{2}\biggl{)}\tau_{3}\\\
-iv\left(\tau^{\beta}+\frac{\partial\mathcal{D}_{l,\beta}}{\partial
r_{\gamma}}\tau^{\gamma}\right)\partial_{r_{\beta}}+v({\bf
K}\cdot\partial_{r_{\beta}}\mathcal{D}_{l})\tau^{\beta}\Bigg{]}\psi_{l}\mathop{}\\!\mathrm{d}^{2}{\bf
r}\\\
+\int\psi^{\dagger}_{t}T(\mathcal{D}_{\text{t}}-\mathcal{D}_{\text{b}})\psi_{b}\mathop{}\\!\mathrm{d}^{2}{\bf
r}\ +\ \text{h.c.}\end{split}\;,$ (11)

per each valley; here, without loss of generality, ${\bf K}$ (see also App.
D). Consistently with Refs. Balents (2019); Bennett  _et al._ (2023b), we
introduce $\mathcal{D}_{\text{l}}$ as a deformation field in layer $l$,
$\psi^{\dagger}_{l}({\bf r})$, $\psi_{l}({\bf r})$ are the fermion
creation/annihilation operators, $m$/$b$ are the trivial and topological
masses, while $T(\mathcal{D}_{\text{t}}-\mathcal{D}_{\text{b}})$ represents
the interlayer tunnelling. The model holds beyond the configuration space
approximation Carr  _et al._ (2018); Bennett  _et al._ (2023a), hence as with
the super-Haldane model, the polarization texture can be obtained by
generalizing Eq. (III) to the continuum model, see App. D for details.

Figure 3:  (a) Phase diagram of the twisted Haldanium bilayer with topological
winding numbers $Q$ of polarization textures. For $|m|<3\sqrt{3}t_{2}$, each
of the layers acquires a Chern number, giving a total of $|C|=2$. (b) In-plane
local polarization ($\Delta P_{||}({\bf r})$) along the $s$ direction
connecting $AA$ and $AB$ stackings in twisted Haldanium bilayer with
$\theta\approx 5^{\circ}$, as indicated in plots (c)–(d). The TPT occurs at
$|t_{2}|\approx 0.43$, abruptly changing the magnitude of the polarization,
while preserving the real-space topology of the texture. For an additional
plot of $\Delta P_{||}({\bf r})$ vs. $t_{2}$, see App. D. (c) Polarization
texture in twisted Haldanium bilayer for topologically-trivial phase
($|C|=0$), realized with the parameters ${(m,t_{2})=(2.25,0)}$. The emergent
polarization merons are consistent with the findings in twisted hBN bilayers
Bennett  _et al._ (2024) with trivial bands. (d) Twisted Haldanium bilayer
with topological bands ($|C|=2$), at the point ${(m,t_{2})=(2.25,0.8)}$ of the
phase diagram. The polarization texture preserves the winding, but the
magnitude of the local polarization is significantly reduced across the TPT,
upon the addition of next-nearest neighbour hoppings experiencing staggered
fluxes. The local polarization was expressed in the units of the quantum of
polarization Bennett  _et al._ (2023b).

As shown in Fig. 3, by tuning the Haldane mass Haldane (1988) ($t_{2}$) and
the Semenoff mass Semenoff (1984) ($m$), we find that the non-trivial band
topology can modify the polarization texture. Correspondingly, the local
polarization constituting the polarization texture can be discontinuously
reduced, while preserving the topological character, i.e. the winding number.
In other words; despite a significant change in the magnitude of the
polarization across the TPT, the vorticity of the polarization texture $Q$ is
preserved. Here, the winding of the polarization texture is given by Bennett
_et al._ (2023a)

$Q=\frac{1}{4\pi}\int\mathop{}\\!\mathrm{d}^{2}{\bf x}~{}\hat{{\bf P}}({\bf
x})\cdot\Big{(}\partial_{x}\hat{{\bf P}}({\bf x})\times\partial_{y}\hat{{\bf
P}}({\bf x})\Big{)}\;,$ (12)

where $\hat{{\bf P}}$ is the normalized local polarization, and the
integration is performed over an individual polar domain. Correspondingly,
$Q=\pm 1/2$ indicates a presence of the merons/antimerons in the triangular
domain span between $AA$ stacking points. Instead, it should be noted that
rather than trivializing the merons across TPTs, such real-space topological
polarization features survive discontinuous jumps, and are retrieved across a
metallic critical point. In particular, at every stacking configuration, apart
from the non-polar AA, where the local polarization is always identically
zero, the polarization approximately retains its direction respecting the
stacking geometry, see Fig. 3.

##  IV Discussion

Our findings, supported by analytical arguments and numerical model
validation, not only offer a well-defined way of capturing local polarization
in crystal supercells with topological bands, but also provide natural
connection to the band topology of the supercell, while also going beyond
configuration the space approximation used in the previous works Bennett  _et
al._ (2023a, b). We stress that, despite the reference to the notion of a
local configuration, the computation of local polarization $P({\bf r}_{j})$ or
SHP $P^{h}_{\beta}({\bf r}_{j},k_{\perp})$ does not require the configuration
space approximation. This is a crucial distinction, given that the topological
states obtained under such approximations might be Wannierizable in
configuration space, despite the non-Wannierizability of the minibands in real
space. In particular, such scenario arguably occurs in twisted TMDs such as
$t$-MoTe2 with Chern bands, while the commensurate homobilayers MoTe2 are
deemed topologically trivial (in the 1T phase). As we show, our formulations
do not suffer from such kind of ambiguities, and furthermore allow to
explicitly study TPTs which may occur in crystal supercells. While the links
between topological phase transitions and associated changes in polarization
captured by geometric Berry phases according to the modern theory of
polarization King-Smith and Vanderbilt (1993); Coh and Vanderbilt (2009) have
been established in simple systems without supercells Yoshida  _et al._
(2023b, a), we report an analogous effect in crystal supercells, e.g. provided
by polar heterostructures supporting topologically non-trivial polarization
textures in real space. It is important to note that for the other types of
band topologies, i.e. upon the inclusion of additional symmetries, such as
time-reversal in quantum spin Hall insulators, the bands are completely
Wannierizable, if a gauge is chosen to $not$ respect the symmetry protecting
the invariant. However, if a gauge satisfying the symmetry is chosen, the non-
Wannierizability issue for defining the local polarization can be tackled
similarly to the framework proposed here for the Chernful supercells. The
study of polarization textures in the context of other band topologies is left
for a subject of future research.

In the context of the topology of Chernful supercells, it should be stressed
that our definition of $\Delta C(\\{{\bf r}_{j}\\})$, Eq. (II), captures how
the total Chern number changes with respect to local perturbations in the
individual parts of a supercell. It is naturally quantized, quantifying
changes of the total anomalous Hall response of an insulating supercell, and
hence is well-defined. We note that such quantum electronic transitions, as
induced in the presence of a superlattice potential, may be of technological
interest, given that it shows that the Chern topology, partial or local in the
form of a domain in a supercell, can be controlled with an external potential,
thus changing the anomalous Hall conductivity locally. A change in the
magnitude of the polarization texture is associated with this type of
trivialization. This is consistent with the finding of the polarization jumps
on trivializing topology by changing the Hamiltonian parameters in the Haldane
model without a superlattice potential Yoshida  _et al._ (2023a).

Finally, we note that our findings are not limited to the Haldane model, but
are expected in any Chern insulator with additional supercell lenghtscales and
with local inversion symmetry breaking. It should be noted, however, that the
Haldane model is of particular relevance for the real materials, and was
realized experimentally in monolayer hBN Mitra  _et al._ (2024), most
recently. Therefore, the polar twisted Haldanium heterostructures considered
in this work can be in principle engineered in real material setup.
Furthermore, we note that the presence of additional symmetries such as time
reversal Kane and Mele (2005) may lead to invariants beyond Chern numbers, as
captured by the tenfold way Kitaev (2009); Chiu  _et al._ (2016), or by
further taking into account the role of crystalline symmetries Fu (2011);
Slager  _et al._ (2013); Kruthoff  _et al._ (2017); Po  _et al._ (2017);
Slager (2019); Bradlyn  _et al._ (2017), possibly culminating in multi-gap
topologies Bouhon  _et al._ (2020a); Ahn  _et al._ (2019); Bouhon  _et al._
(2020b); Slager  _et al._ (2024); Jiang  _et al._ (2021); Ahn and Yang (2019).
The interplay of such symmetries within the above context of polarization
textures presents indeed an interesting future pursuit in itself.

##  V Conclusions

In this work, we show how local polarization textures can be defined in
crystal supercells with topologically nontrivial bands. We introduce the
concept of semilocal hybrid polarization, the winding of which captures the
quantized Chern numbers across TPTs within supercells. We demonstrate our
findings with models for Chern insulators under superlattice potentials
imposed externally, or internally, by an adequate stacking of a moiré
structure. We verify these concepts using two examples, namely a Chern
insulator in a superlattice potential, and two Chern insulators with a
stacking mismatch, forming a moiré superlattice. By calculating the
polarization textures on both sides of a TPT, we find that the magnitude of
the local polarization decreases when going from a trivial to a nontrivial
phase, but it does not vanish completely. Our findings show that local
polarization textures may persist in systems with nontrivial band topology,
and that band topology and polar topology in real space may coexist.

Additionally, we show that one can change the band topology of a supercell,
purely by the local perturbations imposed in its subsystems. As a consequence,
one could also control the presence of associated edge currents by the use of
an external superlattice potential combined with local probes, which may be of
interest for applications of novel electronics involving Chern insulators.

Our theoretical results are of relevance for real polar materials with Chern
bands, such as twisted MoTe2 heterostructures. Engineering the parameter
tuning to manipulate polarization textures with external superlattice
potentials, or within moiré materials with nontrivial band topology, may be of
potential use in optical or electronic devices. Finally, our theoretical
framework is generalizable to other topological multilayers.

###### Acknowledgements.

W. J. J. thanks Prof. Shuichi Murakami for helpful discussions. W. J. J.
acknowledges funding from the Rod Smallwood Studentship at Trinity College,
Cambridge. D. B. acknowledges the US Army Research Office (ARO) MURI project
under grant No. W911NF-21-0147 and from the Simons Foundation award No.
896626. R.-J. S. and G. C. acknowledge funding from a New Investigator Award,
EPSRC grant EP/W00187X/1. R.-J.S. also acknowledges funding from a EPSRC ERC
underwrite grant EP/X025829/1 as well as Trinity College, Cambridge.

##  Appendix A Local polarization in Wannierizable supercells

Here, we review the formalism of gauge-invariant local polarization in crystal
superlattices, introduced in our previous work Bennett  _et al._ (2023b). We
start with the equivalent definitions in terms of Wannier charge centers (WCC)
and Born effective charges, both of which were crucial for studying the local
polarization in moiré polar heterostructures Bennett and Remez (2022); Bennett
(2022); Bennett  _et al._ (2023b, a), and were based on the introduction of
the local displacements ${\bf x}({\bf r}_{j})=\\{x_{\kappa,\alpha}\\}$, which
correspond to perturbing atoms $\kappa$ in the directions $\alpha$.
Accordingly, for the local polarization ${\bf P}({\bf r}_{j})$ in the unit
cell at ${\bf r}_{j}$, we could write

${\bf P}({\bf
r}_{j})=-\frac{ef}{\Omega_{0}}\sum_{n}^{\text{occ}}\int_{0}^{{\bf x}({\bf
r}_{j})}\partial_{x^{\prime}_{\kappa,\alpha}}\bar{{\bf
w}}_{n}\mathop{}\\!\mathrm{d}x^{\prime}_{\kappa,\alpha}\;,$ (13)

where the Einstein summation convention for the indices $\kappa$ and $\alpha$
was used, $n\in\text{occ}$ are the band indices of occupied bands, and WCC are
defined as ${\bar{{\bf w}}_{n}\equiv\braket{w_{n,0}}{\hat{{\bf r}}}{w_{n,0}}}$
in terms of the Wannier functions represented by the states Vanderbilt and
King-Smith (1993); King-Smith and Vanderbilt (1993); Marzari and Vanderbilt
(1997); Souza  _et al._ (2001); Marzari  _et al._ (2012),

$\ket{w_{n,{\bf
R}}}=\frac{\Omega_{\text{sc}}}{(2\pi)^{3}}\oint_{\text{scBZ}}e^{-i{\bf
k}\cdot{\bf R}}\ket{\psi_{n,{\bf k}}}\mathop{}\\!\mathrm{d}{\bf k}.$ (14)

Here, $\Omega_{\text{sc}}$ denotes the real-space supercell volume, scBZ is
the corresponding Brillouin zone associated with the superlattice, and ${\bf
R}$ is a supercell position vector. Equivalently, we can express the local
polarization in an alternative form, using dynamical Born effective charges.
Componentwise, it reads

$P_{\beta}({\bf r}_{j})=\frac{1}{\Omega_{0}}\int_{0}^{{\bf x}({\bf
r}_{j})}Z^{*}_{\kappa,\alpha\beta}({\bf
x}^{\prime})\mathop{}\\!\mathrm{d}x^{\prime}_{\kappa,\alpha}\;,$ (15)

with Born charges defined as
$Z^{*}_{\kappa,\alpha\beta}=\Omega_{\text{sc}}\frac{\partial
P_{\beta}}{\partial x_{\kappa,\alpha}}$, which in terms of the bands and
phonon displacements of atoms $\kappa$ in direction $a$, $x_{\kappa,\alpha}$,
can be expressed as Gonze and Lee (1997); Ghosez  _et al._ (1998)

$Z^{*}_{\kappa,\alpha\beta}({\bf x}({\bf
r}_{j}))=\frac{-2ief\Omega_{\text{sc}}}{(2\pi)^{3}}\sum_{n}^{\text{occ}}\oint_{\text{scBZ}}\braket{\partial_{x_{\kappa,\alpha}}u_{n,{\bf
k}}}{\partial_{k_{\beta}}u_{n,{\bf k}}}\mathop{}\\!\mathrm{d}{\bf k}.$ (16)

Hence, consistently with the previous expression, in the Wannierizable
systems, we retrieve,

$Z^{*}_{\kappa,\alpha\beta}({\bf
x})=-\frac{ef}{\Omega_{0}}\sum_{n}^{\text{occ}}\partial_{x_{\kappa,\alpha}}\bar{{\bf
w}}_{n,\beta}.$ (17)

Here, $\Omega_{0}$ is the volume of a unit cell, with ${\bf x}({\bf r}_{j})$
corresponding to a set of displacements of cores in unit cell ${\bf r}_{j}$.
Last, we note that in the context of real materials, the Born charge
definition can be naturally extended by the use of the non-adiabatic Born
effective charges (NABECs) introduced in Ref. Dreyer  _et al._ (2022). Here,
under the implementation of NABECs to deduce the local polarization at ${\bf
r}_{j}$, an analogous integration to the one adapted for the regular Born
charges in Ref. Bennett  _et al._ (2023b) could be performed, i.e.

$P_{\beta}({\bf r}_{j})=\frac{1}{\Omega}\int^{{\bf x}({\bf
r}_{j})}_{0}Z^{*}_{\kappa,\alpha\beta}(\omega\rightarrow 0,~{}{\bf x}({\bf
r}_{j}))\mathop{}\\!\mathrm{d}x_{\kappa,\alpha},$ (18)

which for insulators, in the $\omega\rightarrow 0$ limit, coincides with the
Eq. (15). Here, the NABECs at the frequency $\omega$ are given by
$Z_{\kappa,\alpha\beta}^{*}(\omega)$ Dreyer  _et al._ (2022); in the context
of a two-dimensional, as relevant to this work, topologically nontrivial
system, amounting to

$\displaystyle
Z^{*}_{\kappa,\alpha\beta}(\omega)=-\text{Im}~{}\text{lim}_{\eta\rightarrow
0^{+}}\int_{\text{BZ}}\frac{\mathop{}\\!\mathrm{d}^{2}{\bf
k}}{(2\pi)^{2}}\sum_{n\neq m}\frac{f_{n{\bf k}}-f_{m{\bf k}}}{E_{n{\bf
k}}-E_{m{\bf k}}+\omega+i\eta}$ $\displaystyle\hskip
56.9055pt\times\braket{u_{n{\bf k}}}{\partial_{k_{a}}u_{m{\bf
k}}}\bra{u_{m{\bf k}}}\partial_{x_{\kappa,\alpha}}H\ket{u_{n{\bf k}}}.$ (19)

To obtain the NABECs, the bands $\ket{u_{n{\bf k}}}$ with energies $E_{n{\bf
k}}$ and Fermi-Dirac occupation factors $f_{n{\bf k}}$ are used, and the
derivatives of the Hamiltonian subject to the local phonon displacements
$\partial_{x_{\kappa,\alpha}}H$ are evaluated. It should be noted that here,
as in the rest of the work, only the electronic contribution to the
polarization is considered, while the ion (core) contribution (which trivially
obtains a dipole moment of core charges) is not included.

##  Appendix B Semilocal hybrid polarization

In this section, we extend our definition of local polarization to non-
Wannierizable systems, such as Chern insulators, in further detail. As Wannier
functions are not exponentially localized in such case, the polarization in
terms of WCC is ill-defined for arbitrary gauges Coh and Vanderbilt (2009);
Yoshida  _et al._ (2023a). However, hybrid Wannier functions (HWFs)
exponentially-localized in one direction, which we denote as $||$, can be
defined,

$\ket{w^{h}_{n,{\bf
R}}(k_{\perp})}=\frac{\Omega_{\text{sc}}}{(2\pi)^{2}}\oint_{\text{scBZ}}e^{-ik_{||}({\bf
R})_{||}}\ket{\psi_{n,{\bf k}}}~{}\mathop{}\\!\mathrm{d}k_{||}.$ (20)

Here, $({\bf R})_{||}$ is the component of a superlattice vector ${\bf R}$,
which is parallel to the direction, in which the polarization component is to
be deduced. Analogously, the hybrid Wannier charge centers (HWCC) can be
introduced. On introducing a shortcut notation,
$\ket{w^{h}_{n}(k_{\perp})}\equiv\ket{w^{h}_{n,{\bf 0}}(k_{\perp})}$, we have,

$\bar{w}^{h}_{n,\beta}(k_{\perp})\equiv\braket{w^{h}_{n}(k_{\perp})}{\hat{r}_{\beta}}{w^{h}_{n}(k_{\perp})},\\\
$ (21)

with $\hat{r}_{\beta}$, the position operator components. Before defining the
semilocal version of the hybrid polarization with the introduced HWCCs, we
define the $total$ hybrid polarization itself. Componentwise, it reads,

$P^{h}_{\beta}(k_{\perp})=-\frac{ef}{\Omega_{sc}}\sum^{\text{occ}}_{n}\bar{w}^{h}_{n,\beta}(k_{\perp}).$
(22)

On introducing the notion of local configuration for defining the local
polarization, consistently with Ref. Bennett  _et al._ (2023b), we can now
define the semilocal hybrid polarizations (SHPs), see also Fig. 4. Namely,
using phonon displacements, or equivalently, depolarizing perturbations (e.g.
in the super-Haldane model – equivalent to setting the vanishing onsite
potential), which directly encode the local configuration ${\bf x}({\bf
r}_{j})$, we write,

$P^{h}_{\beta}({\bf
r}_{j},k_{\perp})=-\frac{ef}{\Omega_{0}}\sum_{n}^{\text{occ}}\int_{0}^{{\bf
x}({\bf
r}_{j})}\partial_{x^{\prime}_{\kappa,\alpha}}\bar{w}^{h}_{n}(k_{\perp})~{}\mathop{}\\!\mathrm{d}x^{\prime}_{\kappa,\alpha}.$
(23)

It should be noted that, with $P^{h}_{\beta}({\bf r}_{j},k_{\perp})$
introduced as a change in the gauge-invariant sum of the HWCCs over occupied
band indices (or equivalently, change of the Berry phase), the SHPs are
definitionally gauge-invariant objects. Furthermore, we know that physically,
on adding up electric dipole moments associated with local polarizations, one
obtains the total polarization $\textbf{P}_{\text{tot}}$,

$\textbf{P}_{\text{tot}}=\frac{1}{N_{\text{tot}}}\sum^{N_{\text{tot}}}_{j}{\bf
P}({\bf r}_{j}),$ (24)

which is also consistent with the additivity of phonon displacements in the
integral limits ${\bf x}({\bf r})$. $N_{\text{tot}}$ is the total number of
subcells contained in a supercell ($N_{\text{tot}}=N_{x}N_{y}$, for two
spatial dimensions). By an analogous argument, we have

$P^{h}_{\beta}(k_{\perp})=\frac{1}{N_{\text{tot}}}\sum^{N_{\text{tot}}}_{j}P^{h}_{\beta}({\bf
r}_{j},k_{\perp}).$ (25)

which, on summing over $k_{\perp}$ as detailed in the main text, provides an
adequate decomposition of the chosen total polarization component into
contributions associated with distinct unit subcells. Here, it should be noted
that all polarizations, in this work considered under periodic boundary
conditions, are defined modulo a quantum of polarization respecting the
superlattice vector R. In the context of local polarizations within crystal
supercells, such modular character, intrinsically due to the gauge ambiguity,
was in fact discussed in detail in Ref. Bennett  _et al._ (2023b).

Figure 4: (a) HWCC in the reference configuration (${\bf x}({\bf
r}_{j})=\textbf{0}$) under the consistently choosen gauge (fixing ${\bf
k}_{0}$). (b) Flow of the HWCC subject to the sliding (${\bf x}({\bf
r}_{j})\neq\textbf{0}$), which allows to reconstruct the local polarization in
a subcell of a given configuration.

Next, we remark on the relations between the hybrid polarizations (or
equivalently HWCCs), the polarization currents ${\bf j}_{P}={\bf\dot{P}}$, and
Berry phases $\phi_{n}(k_{\perp})$ defined in the main text, which further
justify the proposed construction of the local polarization definition
utilizing SHPs. Within an independent-particle picture, the time-dependent
polarization current can generally be decomposed in a two-dimensional system
into individual contributions as Vanderbilt (2018),

${\bf
j}_{P}(t)=f\sum^{\text{occ}}_{n}\int\frac{\mathop{}\\!\mathrm{d}^{2}\textbf{k}}{(2\pi)^{2}}~{}{\bf
j}_{n{\bf k}}(t),$ (26)

where $f$ is the occupation factor in the valence bands (${f=2}$ for spin-
degenerate systems in zero-temperature limit). In terms of the individual
contributions under an adiabatic current-inducing perturbation, we obtain to
first order

$\displaystyle{\bf j}_{n{\bf k}}(t)=-e\bra{u_{n,{\bf k}}}\hat{{\bf
v}}\ket{u_{n,{\bf k}}}$ $\displaystyle\hskip 28.45274pt-ie\hbar\sum_{m\neq
n}\Big{[}\frac{\braket{\dot{u}_{m{\bf k}}}{u_{n,{\bf k}}}\bra{u_{n,{\bf
k}}}\hat{{\bf v}}\ket{u_{m,{\bf k}}}}{E_{m{\bf k}}-E_{n{\bf
k}}}-\text{c.c.}\Big{]},$ (27)

where $\hat{{\bf v}}=\frac{1}{\hbar}\nabla_{\bf k}H({\bf k})$ is the velocity
operator, and $H({\bf k})$ is the Bloch Hamiltonian. On further recognizing
that for $n\neq m$, $\braket{u_{n,{\bf k}}}{\nabla_{\bf k}u_{m,{\bf
k}}}=\frac{\bra{u_{n,{\bf k}}}\nabla_{\bf k}H\ket{u_{m,{\bf k}}}}{E_{n{\bf
k}}-E_{m{\bf k}}}$, one obtains,

${\bf j}_{n{\bf k}}(t)=-e\bra{u_{n,{\bf k}}}\hat{{\bf v}}\ket{u_{n,{\bf
k}}}-ie\Big{[}\braket{\dot{u}_{n{\bf k}}}{\nabla_{\bf k}u_{n,{\bf
k}}}-\text{c.c.}\Big{]}.$ (28)

Furthermore, on substituting to
$\Delta\textbf{P}\equiv\int\textbf{j}_{\text{P}}\mathop{}\\!\mathrm{d}\text{t}$,
which was introduced in the main text, the Eq. (B) results in

$\Delta
P_{\beta}=-ef\int^{1}_{0}\mathop{}\\!\mathrm{d}\lambda\sum^{\text{occ}}_{n}\int\frac{\mathop{}\\!\mathrm{d}^{2}\textbf{k}}{(2\pi)^{2}}i\Big{[}\braket{\partial_{\lambda}u_{n,{\bf
k}}}{\partial_{k_{\beta}}u_{n,{\bf k}}}-\text{c.c.}\Big{]},$ (29)

consistently with the seminal formula of Ref. Vanderbilt and King-Smith
(1993). Here, the variables were changed from $t$ to $\lambda$, with $\lambda$
parametrizing the time-dependent adiabatic switching of the perturbation which
induces the polarization. We quote the general result from Ref. Vanderbilt
(2018): $\bar{w}^{h}_{n,\beta}(k_{\perp})=\frac{1}{2\pi}\phi_{n}(k_{\perp})$,
which we combine with Eq. (6), $\phi_{n}(k_{\gamma})\equiv
i\oint_{k_{\perp}=k_{\gamma}}\braket{u_{n{\bf
k}}}{\partial_{k_{\beta}}u_{n{\bf k}}}\mathop{}\\!\mathrm{d}k_{\beta}$, and a
choice of the adiabatic perturbations $\lambda\equiv\\{x_{\kappa,\alpha}\\}$,
equivalent to the local displacements ${\bf x}({\bf r}_{j})$. Upon direct
insertion of these identities, we finally obtain Eq. (5),

$\Delta
P_{\beta}=-\frac{ef}{\Omega_{0}}\sum_{n}^{\text{occ}}\oint_{\mathcal{C}}\int_{0}^{{\bf
x}({\bf
r}_{j})}\partial_{x^{\prime}_{\kappa,\alpha}}\bar{w}^{h}_{n,\beta}(k_{\perp})\mathop{}\\!\mathrm{d}x^{\prime}_{\kappa,\alpha}\mathop{}\\!\mathrm{d}k_{\perp}.$
(30)

This concludes the derivation, which we include to expose the correspondences
between polarization currents, Berry phases, and HWCCs that were used to
define the hybrid polarizations. Manifestly, we note that the Eq. (5), which
directly captures the local polarization in the supercells with Chern bands,
can be partitioned into the semilocal hybrid polarizations, as was explicitly
presented in the Eq. (4) of the main text.

##  Appendix C Conventional Chern markers

Importantly, a superlattice potential (or even more generically, a random
potential disorder, which defines a supercell of infinite size)
changes/removes the periodicity of the system. In the case of the systems
realizing crystalline superlattices, the topological invariants may become
computationally costly to evaluate, or in the latter case, may be no longer
possible to deduce as ${\bf k}$-integrals over a well-defined BZ. Therefore,
under the settings of such kinds: local, real-space indicators (markers) for
bulk topology are in demand.

For the Chern topology central to this work, the Chern markers satisfying
these conditions can be defined Kitaev (2006); Bianco and Resta (2011); Baù
and Marrazzo (2024) under both periodic and open boundary conditions. To
achieve this goal, we follow the derivation by Bianco and Resta Bianco and
Resta (2011), that starts by recognizing that

$C=-\frac{1}{\pi}\mathfrak{Im}\sum^{N_{\text{occ}}}_{n=1}\sum^{\infty}_{m=N_{\text{occ}+1}}\int_{\textnormal{BZ}}\mathop{}\\!\mathrm{d}^{2}\textbf{k}~{}\braket{\partial_{k_{x}}u_{n,{\bf
k}}}{u_{m,{\bf k}}}\braket{u_{m,{\bf k}}}{\partial_{k_{y}}u_{n,{\bf k}}},$
(31)

where $N_{\text{occ}}$ denotes the number of occupied bands, and on
substituting the identity for $n\neq m$, with
$\ket{\psi_{n,\textbf{k}}}=e^{i{\bf k}\cdot{\bf r}}\ket{u_{n,{\bf k}}}$,

$\braket{u_{n,{\bf k}}}{\nabla_{\textbf{k}}u_{m,{\bf
k}}}=-i\braket{\psi_{n,\textbf{k}}}{\hat{\textbf{r}}}{\psi_{m,\textbf{k}}},$
(32)

the expression for the Chern number can be rearranged into

$\displaystyle
C=-\frac{A}{4\pi^{3}}\mathfrak{Im}\sum^{N_{\text{occ}}}_{n=1}\sum^{\infty}_{m=N_{\text{occ}}+1}\int_{\textnormal{BZ}}\mathop{}\\!\mathrm{d}^{2}\textbf{k}\mathop{}\\!\mathrm{d}^{2}\textbf{k}^{\prime}$
$\displaystyle\hskip
56.9055pt\times\braket{\psi_{n,\textbf{k}}}{\hat{x}}{\psi_{m,\textbf{k}^{\prime}}}\braket{\psi_{m,\textbf{k}^{\prime}}}{\hat{y}}{\psi_{n,\textbf{k}}}.$
(33)

Here, $A$ represents the unit cell area of the system, and the vanishing of
the matrix elements for $\textbf{k}^{\prime}\neq\textbf{k}$ is exploited. On
defining the projectors onto occupied and unoccupied states
($\hat{P}+\hat{Q}=1$), which can be written as:

$\hat{P}=\frac{A}{(2\pi)^{2}}\sum^{N_{\text{occ}}}_{n=1}\int_{\textnormal{BZ}}\mathop{}\\!\mathrm{d}^{2}\textbf{k}\ket{\psi_{n,\textbf{k}}}\bra{\psi_{n,\textbf{k}}},$
(34)
$\hat{Q}=\frac{A}{(2\pi)^{2}}\sum^{\infty}_{m=N_{\text{occ}}+1}\int_{\textnormal{BZ}}\mathop{}\\!\mathrm{d}^{2}\textbf{k}^{\prime}\ket{\psi_{m,\textbf{k}^{\prime}}}\bra{\psi_{m,\textbf{k}^{\prime}}},$
(35)

one finally arrives at the Chern marker formula, after having inserted the
resolution of the identity in the localized orbital basis
($1=\sum_{j,a}\ket{{\bf r}_{j,a}}\bra{{\bf r}_{j,a}}$),

$C({\bf
r}_{j})=-\frac{4\pi}{A}\mathfrak{Im}\sum_{\textnormal{a}\in\textnormal{cell}}\braket{\textbf{r}_{j,a}}{\hat{P}\hat{x}\hat{Q}\hat{y}}{\textbf{r}_{j,a}}\equiv-\frac{4\pi}{A}\mathfrak{Im}\textnormal{Tr}_{\textnormal{cell}}\\{\hat{P}\hat{x}\hat{Q}\hat{y}\\}.$
(36)

In particular, under the periodic boundary conditions,
${C=\frac{1}{N}\sum_{j}C({\bf r}_{j})}$, which in a continuum limit can be
written as ${C=\frac{1}{A_{\text{tot}}}\int\mathop{}\\!\mathrm{d}^{2}{\bf
r}~{}C({\bf r}_{j})}$ with $A_{\text{tot}}=\int\mathop{}\\!\mathrm{d}^{2}{\bf
r}$. Numerically, one can average Chern markers over multiple unit cells,
obtaining local Chern numbers (LCN), which should converge to the Chern
numbers on inclusion of a sufficient number of cells. We adapt an
implementation of the projectors used for evaluating the markers, consistently
with Ref. Varjas  _et al._ (2020). It should be noted that while definition
Eq. (36) appears to be ill-behaved in systems under periodic boundary
conditions, which we consider in this work, it can be manifestly recasted in a
well-defined way Baù and Marrazzo (2024),

$C({\bf
r}_{j})=-\frac{4\pi}{A}\mathfrak{Im}\sum_{\textnormal{a}\in\textnormal{cell}}\braket{\textbf{r}_{j,a}}{\hat{P}[\hat{x},\hat{P}][\hat{y},\hat{P}]}{\textbf{r}_{j,a}},$
(37)

where it is recognized that the commutators $[\hat{x},\hat{P}]$ and
$[\hat{y},\hat{P}]$ are well-behaved. Ultimately, for the evaluation included
in Fig. 2(e), we apply the Chern markers under open boundary conditions, with
the superpotential applied to a bulk subsystem of size $51\times 51$ subcells,
within a slab of size $71\times 71$. The boundary of the $71\times 71$ system
hosts values of the marker, which cancel the bulk contributions after a
complete summation over the entire system, consistently with the general
expectation of the method Bianco and Resta (2011).

##  Appendix D Effective models for twisted Haldanium bilayer

We here elaborate on the models for twisted Haldanium bilayer realizing
topological bands. First, we introduce an effective tight-binding (TB) model,
which is based on the configuration space approximation picture Bennett  _et
al._ (2023a). Finally, we conclude by introducing a continuum Bistritzer-
MacDonald (BM) model for the low-energy physics of the model.

###  D.1 Tight binding model

The local polarization in moiré bilayers can be obtained from a simple TB
model, as shown in Ref. Yu  _et al._ (2023). However, the simplicity comes at
the cost of the configuration space approximation. In case of moiré hBN
bilayers, effectively, one can model insulator with 4 bands
$(\ket{u_{v_{t},{\bf k}}},\ket{u_{v_{b},{\bf k}}},\ket{u_{c_{t},{\bf
k}}},\ket{u_{c_{b},{\bf k}}})$, i.e. valence and conduction bands of two
uncoupled layers. Following the Ref. Yu  _et al._ (2023), the monolayer gap
can be approximated as $\Delta E_{\bf k}=E_{c_{t},{\bf k}}-E_{v_{t},{\bf
k}}\approx E_{c_{b},{\bf k}}-E_{v_{b},{\bf k}}$, and the interlayer tunnelling
can be treated perturbatively, to second order, hybridizing bands as

$\ket{\tilde{u}_{v_{t/b},{\bf
k}}}\approx\Big{(}1-\frac{1}{2}\Big{|}\frac{t_{v_{t/b}c_{b/t},{\bf k}}}{\Delta
E_{\bf k}}\Big{|}^{2}\Big{)}\ket{u_{v_{t/b},{\bf
k}}}-\frac{t_{v_{t/b}c_{b/t},{\bf k}}}{\Delta E_{\bf k}}\ket{u_{c_{b/t},{\bf
k}}},$ (38) $\ket{\tilde{u}_{c_{t/b},{\bf
k}}}\approx\Big{(}1-\frac{1}{2}\Big{|}\frac{t_{v_{b/t}c_{t/b},{\bf k}}}{\Delta
E_{\bf k}}\Big{|}^{2}\Big{)}\ket{u_{c_{t/b},{\bf
k}}}+\frac{t^{*}_{v_{b/t}c_{t/b},{\bf k}}}{\Delta E_{\bf
k}}\ket{u_{c_{b/t},{\bf k}}}.$ (39)

Here, $t_{v_{t/b}c_{b/t},{\bf k}}$ are matrix elements providing inter-gap
interlayer coupling and the convention for the interlayer coupling (that
satisfies the above perturbation relations) is $t_{v_{t/b}c_{b/t},{\bf
k}}=\bra{u_{c_{b/t},{\bf k}}}H\ket{u_{v_{t/b},{\bf k}}}$. Due to the Fermi
occupation of states, the effects of $t_{{c/v}_{t/b}{c/v}_{t/b},{\bf k}}$ are
negligible at second order. Crucially, $t_{v_{t/b}c_{b/t},{\bf k}}$ varies
between configurations described by different sliding vectors ${\bf x}$ and
can be computed from a TB Hamiltonian $H_{\text{moir\'{e}},\text{TB}}$:

$H_{\text{moir\'{e}},\text{TB}}=\begin{pmatrix}\tilde{m}_{\bf k}&t_{\bf
k}&t_{A_{t}A_{b},{\bf k}}&t_{A_{t}B_{b},{\bf k}}\\\ t^{*}_{\bf
k}&-\tilde{m}_{\bf k}&t_{B_{t}A_{b},{\bf k}}&t_{B_{t}B_{b},{\bf k}}\\\
t^{*}_{A_{t}A_{b},{\bf k}}&t^{*}_{B_{t}A_{b},{\bf k}}&\tilde{m}_{\bf k}&t_{\bf
k}\\\ t^{*}_{A_{t}B_{b},{\bf k}}&t^{*}_{B_{t}B_{b},{\bf k}}&t^{*}_{\bf
k}&-\tilde{m}_{\bf k}\\\ \end{pmatrix},$ (40)

where the off-diagonal $2~{}\times~{}2$ blocks define the orbital and
stacking-dependent tunnelling $T({\bf k},{\bf x})$. In particular, $T({\bf
k},{\bf x})$ corresponds to the top-right $2~{}\times~{}2$ block, while the
bottom-left block constitutes $T^{\dagger}({\bf k},{\bf x})$. Here, we label
atoms in top and bottom layers as $(A_{t},B_{t},A_{b},B_{b})$ (which defines
the basis of the Bloch states of the two atomic species), and contrary to Ref.
Yu  _et al._ (2023), we consider a combined Semenoff $m$ and Haldane mass
$t_{2}$ in the form of $\tilde{m}_{\bf
k}=m-2\sum_{i}t_{2}\sin(\textbf{k}\cdot{\bf b}_{i})$, with ${\bf b}_{i}$ the
vectors corresponding to the second-neighbour hoppings. On the other hand, the
in-plane $nearest$-neighbour hoppings read $t_{\bf
k}=\bra{u_{A}}H\ket{u_{B}}=t_{1}\sum^{3}_{i=1}e^{i{\bf k}\cdot\Delta{\bf
R}_{i}}$, where $\Delta{\bf R}_{i}$ label nearest-neighbour displacements
(from the A to B species according to the convention used for $t_{\bf k}$).
Out-of-plane displacements are given by $\Delta{\bf R}_{A_{b}A_{t}}=\Delta{\bf
R}_{B_{b}B_{t}}={\bf x}$, $\Delta{\bf R}_{A_{b}B_{t},i}={\bf x}+\Delta{\bf
R}_{i}$, $\Delta{\bf R}_{B_{b}A_{t},i}={\bf x}-\Delta{\bf R}_{i}$. Note that
here, ${\bf x}$ can be chosen to lie in the Wigner-Seitz unit cell of a B
atom. On the same order of approximation as the intra-layer nearest-neighbour
coupling series truncation, it is sufficient to consider coupling with atoms
connected by the aforementioned out-of-plane displacements; since only these
atoms can lie inside the Wigner-Seitz unit cell for any ${\bf x}$. From
solving the monolayer problem first, one obtains unperturbed bands
$\ket{u_{{c/v}_{t/b},{\bf k}}}$ in terms of $\ket{u_{A_{t/b}}}$ and
$\ket{u_{B_{t/b}}}$ (the periodic parts of the A and B atomic monolayer Bloch
states),

$\displaystyle\ket{u_{c_{t/b},{\bf k}}}$ $\displaystyle=c_{A{\bf
k}}\ket{u_{A_{t/b}}}+c_{B{\bf k}}\ket{u_{B_{t/b}}},$ (41)
$\displaystyle\ket{u_{v_{t/b},{\bf k}}}$ $\displaystyle=(c^{*}_{B{\bf
k}})\ket{u_{A_{t/b}}}-c_{A{\bf k}}\ket{u_{B_{t/b}}},$ (42)

where from a single-layer problem unperturbed by tunnelling, one obtains the
corresponding coefficients:

$c_{A{\bf k}}=\frac{\tilde{m}_{\bf k}+\sqrt{\tilde{m}_{\bf k}^{2}+|t_{\bf
k}|^{2}}}{\sqrt{\Big{(}\tilde{m}_{\bf k}+\sqrt{\tilde{m}_{\bf k}^{2}+|t_{\bf
k}|^{2}}\Big{)}^{2}+|t_{\bf k}|^{2}}},$ (43) $c_{B{\bf k}}=\frac{t^{*}_{\bf
k}}{\sqrt{\Big{(}\tilde{m}_{\bf k}+\sqrt{\tilde{m}_{\bf k}^{2}+|t_{\bf
k}|^{2}}\Big{)}^{2}+|t_{\bf k}|^{2}}}.$ (44)

Furthermore, these yield $t_{v_{t/b}c_{b/t},{\bf k}}({\bf
x})=\bra{w_{c_{b/t}}}H\ket{w_{v_{t/b}}}$, where $\ket{w_{c_{b/t}}}$ is the
Wannier function obtained from $\ket{u_{c_{b/t}}}$,

$\displaystyle t_{v_{t}c_{b},{\bf k}}({\bf x})=e^{i{\bf k}\cdot{\bf
x}}\Big{[}c_{A{\bf k}}c^{*}_{B{\bf k}}(t_{A_{b}A_{t}}({\bf
x})-t_{B_{b}B_{t}}({\bf x}))$ $\displaystyle+(c^{*}_{B{\bf
k}})^{2}\sum^{3}_{i=1}\left(t_{B_{b}A_{t},i}({\bf x})e^{-i{\bf
k}\cdot\Delta{\bf R}_{i}}\right)-c^{2}_{A{\bf
k}}\sum^{3}_{i=1}\left(t_{A_{b}B_{t},i}({\bf x})e^{i{\bf k}\cdot\Delta{\bf
R}_{i}}\right)\Big{]},$ (45) $\displaystyle t_{v_{t}c_{b},{\bf k}}({\bf
x})=e^{i{\bf k}\cdot{\bf x}}\Big{[}c_{A{\bf k}}c^{*}_{B{\bf
k}}(t_{A_{b}A_{t}}({\bf x})-t_{B_{b}B_{t}}({\bf x}))$
$\displaystyle+(c^{*}_{B{\bf k}})^{2}\sum^{3}_{i=1}\left(t_{B_{b}A_{t},i}({\bf
x})e^{-i{\bf k}\cdot\Delta{\bf R}_{i}}\right)-c^{2}_{A{\bf
k}}\sum^{3}_{i=1}\left(t_{A_{b}B_{t},i}({\bf x})e^{i{\bf k}\cdot\Delta{\bf
R}_{i}}\right)\Big{]},$ (46) $\displaystyle t_{v_{b}c_{t},{\bf k}}({\bf
x})=e^{-i{\bf k}\cdot{\bf x}}\Big{[}c_{A{\bf k}}c^{*}_{B{\bf
k}}(t^{*}_{A_{b}A_{t}}({\bf x})-t^{*}_{B_{b}B_{t}}({\bf x}))$
$\displaystyle+(c^{*}_{B{\bf
k}})^{2}\sum^{3}_{i=1}\left(t^{*}_{A_{b}B_{t},i}({\bf x})e^{-i{\bf
k}\cdot\Delta{\bf R}_{i}}\right)-c^{2}_{A{\bf
k}}\sum^{3}_{i=1}\left(t^{*}_{B_{b}A_{t},i}({\bf x})e^{i{\bf k}\cdot\Delta{\bf
R}_{i}}\right)\Big{]}.$ (47)

Here, $t_{A_{b}B_{t},i}({\bf x})=\bra{\phi_{A_{b}}}H\ket{\phi_{B_{t}}}$ and
$A_{b}$ and $B_{t}$ are separated by $\Delta R_{A_{b}B_{t},i}$ (as earlier,
$\ket{\phi_{A_{b/t}}}$ is the Wannier function for $\ket{u_{B_{b/t}}}$).
Writing in the explicit dependence of $t_{A_{b}B_{t},i}({\bf x})$ on ${\bf
x}$,

$t_{{A}_{b}{B}_{t},i}({\bf
x})=\begin{cases}t^{0}_{A_{b}B_{t},i}e^{-|\Delta{\bf R}_{A_{b}B_{t},i}({\bf
x})|/\lambda_{AB}}&\text{if }|\Delta{\bf R}_{A_{b}B_{t},i}({\bf
x})|<|\Delta{\bf R}_{i}|,\\\ 0&\text{otherwise},\end{cases}$ (48)

with $\lambda_{AB}$ a layer-separation dependent regularization of the hopping
amplitudes, reflecting the overlaps of the orbitals between which the hopping
occurs. The stacking-dependent interlayer hoppings for the other three orbital
flavour combinations were regularized analogously. To obtain the results in
Fig. 3, the values chosen for the various tight-binding parameters are as
follows: $m=2.25$, $t_{1}=2.4$,
$t^{0}_{A_{b}B_{t},i}=t^{0}_{B_{b}A_{t},i}=1.28$, $t^{0}_{A_{b}A_{t},i}=0.8$,
$t^{0}_{B_{b}B_{t},i}=0.6$, $\lambda_{AB}=\lambda_{BA}=1.32$,
$\lambda_{AA}=1.36$, and $\lambda_{BB}=1.27$, providing a faithful
representation of the twisted hBN bilayer Yu  _et al._ (2023), subject to an
addition of the second neighbor hoppings, which experience staggered magnetic
fluxes.

With the evaluated interlayer coupling constants, the perturbed bands
$\ket{\tilde{u}_{v/c_{t/b},{\bf k}}}$ can be found within the perturbation
theory, as described earlier. Finally, to deduce the in-plane polarization,
the Berry connection can be furthermore obtained. Namely, the resulting Berry
connection in the calculated bands reads

$\begin{split}A_{cc/vv}({\bf k})=-i\braket{\tilde{u}_{{c/v}_{t/b},{\bf
k}}}{\nabla_{\bf k}\tilde{u}_{{c/v}_{t/b},{\bf k}}},\end{split}$ (49)

which can be further integrated over ${\bf k}$-space to obtain the Berry phase
and polarization, consistently with the main text, Eqs. (5), (6). Crucially,
our model goes beyond the configuration space tight-binding adaptation of Ref.
Yu  _et al._ (2023). In our effective model, the hoppings, which are stacking-
dependent, are regularized by $both$ out-of-plane and in-plane distances. This
modification more accurately reflects an overlap of the corresponding basis
orbitals, which electrons experience on hopping.

Figure 5: The in-plane local polarization $\Delta P_{||}({\bf r})$ at
different stackings ${\bf x}$ as a function of $t_{2}$. The chosen stackings
${\bf x}\equiv{\bf x}({\bf r})=\\{1/6,1/2,3/4\\}$ correspond to the fractional
coordinates along the $s$ direction, with $s=0$ and $s=1$ corresponding to the
AA bilayer stacking, see Fig. 3 for further reference. We emphasize that the
model breaks down in the proximity of the critical points $(|t_{2}|\approx
0.4)$, where the gap becomes too small for the second order perturbation
theory to yield reliable values of the local electric polarization.

For completeness, we accordingly include a plot (Fig. 5) of the in-plane local
polarization $\Delta P_{||}({\bf r})$ as a function of $t_{2}$, which was
realized and computed within the model described above. We show $\Delta
P_{||}({\bf r})$ for different stackings ${\bf x}$, that were defined along
the direction $s$; see also Fig. 3 for reference. We note that close to
$t_{2}=0.43$ corresponding to the metallic critical point, the local
polarization is ill-defined and diverges, as the perturbative model considered
here breaks down for small gaps $\Delta E_{\bf k}$. On the contrary, Fig. 3(c)
and Fig. 3(d) where obtained on both sides away from the critical point, where
the gap is well-preserved.

###  D.2 Continuum model

Last, we elaborate on the continuum model for topological fermions, which
captures the low-energy physics, including the electric polarization, in the
twisted Haldanium bilayer. Following the continuum formulation of the local
polarization introduced in our previous work Ref. Bennett  _et al._ (2023b),
we start by recognizing that, if the two layers are twisted, with $\theta$
denoting the angle of the twist, this change can be described with a
deformation field given by

$\mathcal{D}_{\text{t}}({\bf r})=-\mathcal{D}_{\text{b}}({\bf
r})=\frac{\theta}{2}\mathbf{\hat{z}}\times{\bf r}.$ (50)

Here, each layer experiences a deformation field $\mathcal{D}_{l}$, and
${l=\text{t,b}}$ is the layer index. As a result of a small deformation in
each individual layer, the $\psi$ electron field, introduced in the main text,
is correspondingly modified as Balents (2019),

$\psi_{l}({\bf r})=\left(1-\bm{\nabla}\cdot\mathcal{D}_{l}({\bf
r})\right)^{1/2}\psi_{l}({\bf x}({\bf r}))e^{-i{\bf
K}\cdot\mathcal{D}_{l}({\bf r})}\;,$ (51)

in addition to the consistent transformation of the integration measure. Here,
${\bf K}$ is the momentum at the Dirac point, corresponding to an individual
valley. The continuum Hamiltonian of the decoupled topological bilayers can be
obtained as Balents (2019),

$\displaystyle
H_{\text{BL}}=\sum_{l=\text{t,b}}\int\psi^{\dagger}_{l}\Bigg{[}\biggl{(}m+b\Big{(}\partial_{r_{\beta}}+\frac{\partial\mathcal{D}_{l,\gamma}}{\partial
r_{\beta}}\partial_{r_{\gamma}}\Big{)}^{2}\biggl{)}\tau_{3}$ $\displaystyle-
iv\left(\tau^{\beta}+\frac{\partial\mathcal{D}_{l,\beta}}{\partial
r_{\gamma}}\tau^{\gamma}\right)\partial_{r_{\beta}}+v({\bf
K}\cdot\partial_{r_{\beta}}\mathcal{D}_{l})\tau^{\beta}\Bigg{]}\psi_{l}\mathop{}\\!\mathrm{d}^{2}{\bf
r},$ (52)

where Einstein summation convention is implied and $\tau^{\gamma}$ are the
Pauli matrices. Here, we kept terms only linear in the deformation field and
introduced $v$ as the Fermi velocity of the Dirac fermions, which were further
gapped by the trivial and topological masses $m$ and $b$. In particular, in
the context of the Haldane model, we recognize that $b=3\sqrt{3}t_{2}$ for
each valley. Having defined a continuum theory for an uncoupled Haldanium
bilayer, we further introduce an interlayer coupling under a twisted stacking.
In that case, an additional interlayer tunnelling term described by

$H_{\text{tun}}=\int\psi^{\dagger}_{t}T(\mathcal{D}_{\text{t}}-\mathcal{D}_{\text{b}})\psi_{b}\mathop{}\\!\mathrm{d}^{2}{\bf
r}\ +\ \text{h.c.},$ (53)

needs to be included, where
$T(\mathcal{D}_{\text{t}}-\mathcal{D}_{\text{b}})=\sum_{{\bf G}}T_{{\bf
G}}e^{i{\bf G}\cdot(\mathcal{D}_{\text{t}}-\mathcal{D}_{\text{b}})}$, ${\bf
G}$ are the reciprocal lattice vectors of the monolayer, and $T_{{\bf G}}$ are
the tunnelling amplitudes, which depend on the layer separation. Hence,
ultimately we arrive at the final expression, introduced in the main text, on
combining two terms,

$\displaystyle H_{\text{moir\'{e}}}=H_{\text{BL}}+H_{\text{tun}}.$ (54)

The complete continuum Hamiltonian $H_{\text{moir\'{e}}}$, yields the moiré
bands $\ket{u^{n}_{\mathcal{D},{\bf G}}({\bf k})}$ as eigenfunctions, which
within the parametrization by the deformation field
$\mathcal{D}=(\mathcal{D}_{t},\mathcal{D}_{l})$, fully encode the local
polarization. In particular, the local polarization is heavily-dependent on
the stacking-induced deformation field $\mathcal{D}$, influencing the Berry
phases in the topological bands obtained from the bilayer Hamiltonian. Namely,

$\displaystyle{\bf P}(\mathcal{D}({\bf
r}))=\frac{-2ief}{(2\pi)^{2}}\sum^{\text{occ}}_{n}\int^{\mathcal{D}({\bf
r})}_{0}\oint_{\text{mBZ}}\braket{\partial_{\mathcal{D}}u^{n}_{\mathcal{D},{\bf
G}}({\bf k})}{\partial_{{\bf k}}u^{n}_{\mathcal{D},{\bf G}}({\bf
k})}d\mathcal{D},$ (55)

which is expected to change correspondingly across the topological phase
transitions controlled by the mass parameters $m$ and $b$, where $b$ combined
with the Laplacian act effectively as the further second-neighbor hopping
$t_{2}$.

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