The $\mathbf{k}$ and $\mathbf{q}$ sums run over the Brillouin zones of the MBT
SL and BP layer, respectively. To deal with the remaining sums over $i$ and
$j$, we note that

$\displaystyle\sum\limits_{i\in\text{SL}_{6}}t_{b}\left(\mathbf{R}_{i}-\mathbf{R}_{j}+\boldsymbol{\tau}_{\mu}-\boldsymbol{\rho}_{\nu}\right)e^{-i\mathbf{k}\cdot\mathbf{R}_{i}}=\sum\limits_{(n_{1},n_{2})\in\mathbb{Z}^{2}}t_{b}\left(n_{1}\mathbf{a}_{1}^{(\text{MBT})}+n_{2}\mathbf{a}_{2}^{(\text{MBT})}-\mathbf{R}_{j}+\boldsymbol{\tau}_{\mu}-\boldsymbol{\rho}_{\nu}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times
e^{-i\mathbf{k}\cdot\left(n_{1}\mathbf{a}_{1}^{(\text{MBT})}+n_{2}\mathbf{a}_{2}^{(\text{MBT})}\right)}$
$\displaystyle=\sum\limits_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\int\limits_{-\infty}^{\infty}dx_{1}\int\limits_{-\infty}^{\infty}dx_{2}\,\delta(x_{1}-n_{1})\delta(x_{2}-n_{2})t_{b}\left(x_{1}\mathbf{a}_{1}^{(\text{MBT})}+x_{2}\mathbf{a}_{2}^{(\text{MBT})}-\mathbf{R}_{j}+\boldsymbol{\tau}_{\mu}-\boldsymbol{\rho}_{\nu}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times
e^{-i\mathbf{k}\cdot\left(x_{1}\mathbf{a}_{1}^{(\text{MBT})}+x_{2}\mathbf{a}_{2}^{(\text{MBT})}\right)}\text{
,}$ (S46)

where $x_{1}$ and $x_{2}$ are real dimensionless variables. Using the Poisson
summation formula, we can rewrite

$\sum\limits_{n=-\infty}^{\infty}\delta(x-n)=\sum\limits_{m=-\infty}^{\infty}e^{-i2\pi
mx}\text{ ,}$ (S47)

and then

$\displaystyle\sum\limits_{(n_{1},n_{2})\in\mathbb{Z}^{2}}\delta(x_{1}-n_{1})\delta(x_{2}-n_{2})$
$\displaystyle=\sum\limits_{(m_{1},m_{2})\in\mathbb{Z}^{2}}e^{-i2\pi\left(m_{1}x_{1}+m_{2}x_{2}\right)}$
$\displaystyle=\sum\limits_{(m_{1},m_{2})\in\mathbb{Z}^{2}}e^{-i\left(x_{1}\mathbf{a}_{1}^{(\text{MBT})}+x_{2}\mathbf{a}_{2}^{(\text{MBT})}\right)\cdot\left(m_{1}\mathbf{b}_{1}^{(\text{MBT})}+m_{2}\mathbf{b}_{2}^{(\text{MBT})}\right)}$
$\displaystyle=\sum\limits_{i}e^{-i\left(x_{1}\mathbf{a}_{1}^{(\text{MBT})}+x_{2}\mathbf{a}_{2}^{(\text{MBT})}\right)\cdot\mathbf{G}_{i}^{(\text{MBT})}}\text{
\,.}$ (S48)

Here,
$\mathbf{G}_{i}^{(\text{MBT})}=i_{1}\mathbf{b}_{1}^{(\text{MBT})}+i_{2}\mathbf{b}_{2}^{(\text{MBT})}$
denotes the unit cell position of the reciprocal MBT SL lattice, and the sum
over $m$ runs over the reciprocal MBT SL lattice unit cells.

Substituting Eq.(S48) into Eq.(S46), we obtain

$\sum\limits_{i\in\text{SL}_{6}}t_{b}\left(\mathbf{R}_{i}-\mathbf{R}_{j}+\boldsymbol{\tau}_{\mu}-\boldsymbol{\rho}_{\nu}\right)e^{-i\mathbf{k}\cdot\mathbf{R}_{i}}=\sum\limits_{i}\int\limits_{-\infty}^{\infty}dx_{1}\int\limits_{-\infty}^{\infty}dx_{2}\,t_{b}\left(x_{1}\mathbf{a}_{1}^{(\text{MBT})}+x_{2}\mathbf{a}_{2}^{(\text{MBT})}-\mathbf{R}_{j}+\boldsymbol{\tau}_{\mu}-\boldsymbol{\rho}_{\nu}\right)\\\
\times
e^{-i\left(\mathbf{k}+\mathbf{G}_{i}^{(\text{MBT})}\right)\cdot\left(x_{1}\mathbf{a}_{1}^{(\text{MBT})}+x_{2}\mathbf{a}_{2}^{(\text{MBT})}\right)}$
(S49)

Similarly

$\sum\limits_{j\in\text{BP}}t_{b}\left(\mathbf{R}_{i}-\mathbf{R}_{j}+\boldsymbol{\tau}_{\mu}-\boldsymbol{\rho}_{\nu}\right)e^{i\mathbf{q}\cdot\mathbf{R}_{j}}=\sum\limits_{j}\int\limits_{-\infty}^{\infty}dx_{3}\int\limits_{-\infty}^{\infty}dx_{4}\,t_{b}\left(\mathbf{R}_{i}-x_{3}\mathbf{a}_{1}^{(\text{BP})}-x_{4}\mathbf{a}_{2}^{(\text{BP})}+\boldsymbol{\tau}_{\mu}-\boldsymbol{\rho}_{\nu}\right)\\\
\times
e^{i\left(\mathbf{q}-\mathbf{G}_{j}^{(\text{BP})}\right)\cdot\left(x_{3}\mathbf{a}_{1}^{(\text{BP})}+x_{4}\mathbf{a}_{2}^{(\text{BP})}\right)}$
(S50)

where again $x_{3}$ and $x_{4}$ are real dimensionless variables and the $j$
sum runs over the unit cells of the reciprocal BP lattice
$\mathbf{G}_{j}^{(\text{BP})}=j_{1}\mathbf{b}_{1}^{(\text{BP})}+j_{2}\mathbf{b}_{2}^{(\text{BP})}$.

Combining Eqs. (S49) and (S50) and substituting them into Eq. (S45) yields

$\hat{H}_{\text{int}}^{(b)}=\frac{1}{\sqrt{N_{\text{SL}}N_{\text{BP}}}}\frac{1}{\Omega_{\text{SL}}\Omega_{\text{BP}}}\sum\limits_{\mu,\nu,\sigma}\sum\limits_{\mathbf{k},i}\sum\limits_{\mathbf{q},j}\biggl{[}\int
d^{2}r\int
d^{2}r^{\prime}\,t_{b}\left(\mathbf{r}-\mathbf{r}^{\prime}+\boldsymbol{\tau}_{\mu}-\boldsymbol{\rho}_{\nu}\right)\\\
\times
e^{i\left(-\mathbf{k}\cdot\boldsymbol{\tau}_{\mu}^{\parallel}+\mathbf{q}\cdot\boldsymbol{\rho}_{\nu}^{\parallel}\right)}e^{-i\left(\mathbf{k}+\mathbf{G}_{i}^{(\text{MBT})}\right)\cdot\mathbf{r}}e^{i\left(\mathbf{q}-\mathbf{G}_{j}^{(\text{BP})}\right)\cdot\mathbf{r}^{\prime}}\psi_{\mathbf{k}\mu,\sigma}^{{\dagger}}\phi_{\mathbf{q}\nu,\sigma}+\text{h.c.}\biggr{]}\text{
,}$ (S51)

where we applied the change of variables
$\mathbf{r}=x_{1}\mathbf{a}_{1}^{\text{(MBT)}}+x_{2}\mathbf{a}_{2}^{\text{(MBT)}}$
and
$\mathbf{r}^{\prime}=x_{3}\mathbf{a}_{1}^{\text{(BP)}}+x_{4}\mathbf{a}_{2}^{\text{(BP)}}$.
Recall that here the sum over $i$ ($j$) runs over the reciprocal unit cells of
MBT SL (BP), while the sum over $\mathbf{k}$ ($\mathbf{q}$) are restricted to
the first BZ of MBT SL (BP). Besides, $\Omega_{\text{SL}}$
($\Omega_{\text{BP}}$) denotes the area of the unit cell of the SL (BP)
Bravais lattices.

We focus now on the arguments of the hopping amplitude $t_{b}$. While
$\mathbf{r}$ and $\mathbf{r}^{\prime}$ are 2D vectors,
$\boldsymbol{\tau}_{\mu}$ and $\boldsymbol{\rho}_{\nu}$ are in general 3D, as
mentioned before. Defining

$s_{\mu\nu}\equiv\left(\boldsymbol{\tau}_{\mu}-\boldsymbol{\rho}_{\nu}\right)\cdot\hat{\mathbf{z}}=\tau^{\perp}_{\mu}-\rho^{\perp}_{\nu}\,,$
(S52)

we can rewrite
$\boldsymbol{\tau}_{\mu}-\boldsymbol{\rho}_{\nu}=\boldsymbol{\tau}_{\mu}^{\parallel}-\boldsymbol{\rho}_{\nu}^{\parallel}+s_{\mu\nu}\hat{\mathbf{z}}$.
Performing the change of variables
$\mathbf{y}=\mathbf{r}-\mathbf{r}^{\prime}+\boldsymbol{\tau}_{\mu}^{\parallel}-\boldsymbol{\rho}_{\nu}^{\parallel}$,
we obtain

$\hat{H}_{\text{int}}^{(b)}=\sqrt{\frac{N_{\text{SL}}}{N_{\text{BP}}}}\frac{1}{\Omega_{\text{BP}}}\sum\limits_{\mu,\nu,\sigma}^{(b)}\sum\limits_{i,\mathbf{k}}\sum\limits_{j,\mathbf{q}}\biggl{[}t_{\mu\nu}(\mathbf{k}+\mathbf{G}_{i}^{(\text{MBT})})e^{i\left(\mathbf{G}_{i}^{(\text{MBT})}\cdot\boldsymbol{\tau}_{\mu}^{\parallel}+\mathbf{G}_{j}^{(\text{BP})}\cdot\boldsymbol{\rho}_{\nu}^{\parallel}\right)}\\\
\times\psi_{\mathbf{k}\mu,\sigma}^{{\dagger}}\phi_{\mathbf{q}\nu,\sigma}\delta_{\mathbf{q},\mathbf{k}+\mathbf{G}_{i}^{(\text{MBT})}+\mathbf{G}_{j}^{(\text{BP})}}+\text{h.c.}\biggr{]}\,.$
(S53)

Here,

$t_{\mu\nu}^{(b)}(\mathbf{q})=\int
d^{2}y\,t_{b}\left(\mathbf{y}+s_{\mu\nu}\hat{\mathbf{z}}\right)e^{-i\mathbf{q}\cdot\mathbf{y}}\text{
,}$ (S54)

is the Fourier transform of the hopping amplitude, which we explicitly
calculate, and we used

$\int
d^{2}re^{i\mathbf{r}\cdot\left(-\mathbf{k}-\mathbf{G}_{i}^{(\text{MBT})}-\mathbf{G}_{j}^{(\text{BP})}+\mathbf{q}\right)}=N_{\text{SL}}\Omega_{\text{SL}}\,\delta_{\mathbf{q}-\mathbf{k},\mathbf{G}_{i}^{(\text{MBT})}+\mathbf{G}_{j}^{(\text{BP})}}\text{
.}$ (S55)

Since we here focus on a minimal model for the MBT-BP heterostructure that is
valid around the $\Gamma$ point and
$t_{\mu\nu}^{(b)}(\mathbf{k}+\mathbf{G}_{i})$ decays rapidly with
$|\mathbf{k}+\mathbf{G}_{i}|$ (on a scale determined by the size of the
orbitals), only $\mathbf{G}_{i}^{(\text{MBT})}=0$ contributes significantly in
the summation over $i$ in Eq. (S53). Momentum conservation in Eq.(S53) then
enforces $\mathbf{q}=\mathbf{G}_{j}^{(\text{BP})}+\mathbf{k}$, which can only
be satisfied for $\mathbf{G}_{j}^{\text{(BP)}}=0$ as $\mathbf{q}$ and
$\mathbf{k}$ lie in the first Brillouin zone. Therefore, the contribution to
the continuum model of the heterostructure around $\Gamma$ due to MBT-BP
coupling takes the final form

$\hat{H}_{\text{int}}^{(b)}=\sqrt{\frac{N_{\text{SL}}}{N_{\text{BP}}}}\frac{1}{\Omega_{\text{BP}}}\sum\limits_{\mu,\nu,\sigma}\sum\limits_{\mathbf{k}}\left[t_{\mu\nu}^{(b)}(\mathbf{k})\psi_{\mathbf{k}\mu,\sigma}^{{\dagger}}\phi_{\mathbf{k}\nu,\sigma}+\text{h.c.}\right]\text{
.}$ (S56)

In the following, we set $N_{\text{SL}}=N_{\text{BP}}$ and we recall that
$\Omega_{\text{BP}}$ denotes the are of the BP unit cell. This interaction
Hamiltonian involves the full $7\times 4$ dimensional hopping matrix
$t^{(b)}_{\mu\nu}$. However, since the minimum low-energy model for MBT SL
contains only four specific even and odd combinations of selected Bi and Te
orbitals [see low-energy MBT basis in Eq. (S17)], we only need to consider a
part of the matrix $t_{\mu\nu}^{(b)}(\mathbf{k})$ to construct
$\hat{U}_{b}(\mathbf{k})$. Projecting $\hat{H}_{\text{int}}^{(b)}$ into the
low-energy basis in Eq. (S17) yields

$\mathcal{P}\hat{H}_{\text{int}}^{(b)}\mathcal{P}^{-1}=\sum\limits_{\mathbf{k}}\left[\hat{\varphi}_{\mathbf{k}}^{{\dagger}}\hat{V}_{b,\uparrow}(\mathbf{k})\hat{\phi}_{\mathbf{k},\uparrow}+\hat{\varphi}_{\mathbf{k}}^{{\dagger}}\hat{V}_{b,\downarrow}(\mathbf{k})\hat{\phi}_{\mathbf{k},\downarrow}+\text{h.c.}\right]\text{\,.}$
(S57)

Here, $\hat{\varphi}_{\mathbf{k}}$ is a 4-dimensional spinor that creates
electrons in the MBT basis states in Eq. (S17) and

$\displaystyle\hat{V}_{b,\uparrow}(\mathbf{k})$
$\displaystyle=\frac{1}{\sqrt{2}\Omega_{\text{BP}}}\begin{pmatrix}t_{51}^{(b)}(\mathbf{k})+t_{61}^{(b)}(\mathbf{k}),&t_{52}^{(b)}(\mathbf{k})+t_{62}^{(b)}(\mathbf{k}),&t_{53}^{(b)}(\mathbf{k})+t_{63}^{(b)}(\mathbf{k}),&t_{54}^{(b)}(\mathbf{k})+t_{64}^{(b)}(\mathbf{k})\\\
0&0&0&0\\\
t_{11}^{(b)}(\mathbf{k})-t_{41}^{(b)}(\mathbf{k}),&t_{12}^{(b)}(\mathbf{k})-t_{42}^{(b)}(\mathbf{k}),&t_{13}^{(b)}(\mathbf{k})-t_{43}^{(b)}(\mathbf{k}),&t_{14}^{(b)}(\mathbf{k})-t_{44}^{(b)}(\mathbf{k})\\\
0&0&0&0\end{pmatrix}$ (S58) $\displaystyle\hat{V}_{b,\downarrow}(\mathbf{k})$
$\displaystyle=\frac{1}{\sqrt{2}\Omega_{\text{BP}}}\begin{pmatrix}0&0&0&0\\\
t_{11}^{(b)}(\mathbf{k})-t_{41}^{(b)}(\mathbf{k}),&t_{12}^{(b)}(\mathbf{k})-t_{42}^{(b)}(\mathbf{k}),&t_{13}^{(b)}(\mathbf{k})-t_{43}^{(b)}(\mathbf{k}),&t_{14}^{(b)}(\mathbf{k})-t_{44}^{(b)}(\mathbf{k})\\\
0&0&0&0\\\
t_{51}^{(b)}(\mathbf{k})+t_{61}^{(b)}(\mathbf{k}),&t_{52}^{(b)}(\mathbf{k})+t_{62}^{(b)}(\mathbf{k}),&t_{53}^{(b)}(\mathbf{k})+t_{63}^{(b)}(\mathbf{k}),&t_{54}^{(b)}(\mathbf{k})+t_{64}^{(b)}(\mathbf{k})\end{pmatrix}\,.$
(S59)

Here, we have set $N_{\text{SL}}=N_{\text{BP}}$. The relation between the
matrices $\hat{V}_{b,\sigma}$ and $\hat{U}_{b}$ becomes evident when
connecting Eq.(S57) with Eq.(S14): $\hat{U}_{b}$ is a $24\times 8$ matrix with
zero matrix elements everywhere other than its last four lines, which are
given by the $4\times 8$ matrix
$\left(\hat{V}_{b,\uparrow},\hat{V}_{b,\downarrow}\right)$. Note that
$t_{\mu,1}^{(b)}=t_{\mu,2}^{(b)}$ because the $z$-component of the fractional
positions of phosphorus $1$ and $2$ are the same, $\rho_{1,z}=\rho_{2,z}$ (see
Table S2 and Fig. S25). Similarly, $t_{\mu,3}^{(b)}=t_{\mu,4}^{(b)}$.
Morepver, the dominant matrix element of $\hat{V}_{b,\sigma}$ is
$t_{13}^{(b)}-t_{43}^{(b)}$, as shown in Fig. S28.

Similar steps apply for calculating the coupling between the top BP layer and
the top-most MBT SL (SL number one in, which is described by a Hamiltonian as
Eq.(S57) with $b\rightarrow t$. Due to inversion symmetry, the hopping
amplitudes $t_{\mu\nu}^{(t)}$ are related to $t_{\mu\nu}^{(b)}$. More
specifically, $t_{\mu\nu}^{(t)}=t_{\mu^{\prime}\nu^{\prime}}^{(b)}$, where
$\mu^{\prime}=4,1,6,5$ if $\mu=1,2,5,6$, respectively and
$\nu^{\prime}=3,4,1,2$ if $\nu=1,2,3,4$. We thus obtain

$\displaystyle\hat{V}_{t,\uparrow}(\mathbf{k})$
$\displaystyle=\frac{1}{\sqrt{2}\Omega_{\text{BP}}}\begin{pmatrix}t_{53}^{(b)}(\mathbf{k})+t_{63}^{(b)}(\mathbf{k})&t_{54}^{(b)}(\mathbf{k})+t_{64}^{(b)}(\mathbf{k})&t_{51}^{(b)}(\mathbf{k})+t_{61}^{(b)}(\mathbf{k})&t_{52}^{(b)}(\mathbf{k})+t_{62}^{(b)}(\mathbf{k})\\\
0&0&0&0\\\
-t_{13}^{(b)}(\mathbf{k})+t_{43}^{(b)}(\mathbf{k})&-t_{14}^{(b)}(\mathbf{k})+t_{44}^{(b)}(\mathbf{k})&-t_{11}^{(b)}(\mathbf{k})+t_{41}^{(b)}(\mathbf{k})&-t_{12}^{(b)}(\mathbf{k})+t_{42}^{(b)}(\mathbf{k})\\\
0&0&0&0\end{pmatrix}$ (S60) $\displaystyle\hat{V}_{t,\downarrow}(\mathbf{k})$
$\displaystyle=\frac{1}{\sqrt{2}\Omega_{\text{BP}}}\begin{pmatrix}0&0&0&0\\\
-t_{13}^{(b)}(\mathbf{k})+t_{43}^{(b)}(\mathbf{k})&-t_{14}^{(b)}(\mathbf{k})+t_{44}^{(b)}(\mathbf{k})&-t_{11}^{(b)}(\mathbf{k})+t_{41}^{(b)}(\mathbf{k})&-t_{12}^{(b)}(\mathbf{k})+t_{42}^{(b)}(\mathbf{k})\\\
0&0&0&0\\\
t_{53}^{(b)}(\mathbf{k})+t_{63}^{(b)}(\mathbf{k})&t_{54}^{(b)}(\mathbf{k})+t_{64}^{(b)}(\mathbf{k})&t_{51}^{(b)}(\mathbf{k})+t_{61}^{(b)}(\mathbf{k})&t_{52}^{(b)}(\mathbf{k})+t_{62}^{(b)}(\mathbf{k})\end{pmatrix}\text{
.}$ (S61)

Here, we have set $N_{\text{SL}}=N_{\text{BP}}$. The matrices
$\hat{V}_{t,\uparrow}$ and $\hat{V}_{t,\downarrow}$ define the blocks
$\hat{U}_{t}$ in Eq. (S14). Like $\hat{U}_{b}$, the matrix $\hat{U}_{t}$ is
also a $24\times 8$ matrix, but here the only non-zero elements are in the
first four lines of this matrix, which are formed by the $4\times 8$ matrix
$\left(\hat{V}_{t,\uparrow},\hat{V}_{t,\downarrow}\right)$.

####  IV.8.2 IV.8.2. MnBi2Te4-BP hopping integrals using Slater-Koster
approach

In this section we determine the hopping amplitude
$t_{\mu\nu}^{(b)}(\mathbf{q})$ that enters the final expression for the
coupling Hamiltonian in Eq. (S56). We obtain the real-space hopping amplitudes
between the $p_{z}$ orbitals of BP and its nearest-neighbor MBT SL using the
Slater-Koster approach, and then Fourier transform to obtain the desired
expression for $t_{b}(\mathbf{q})$ in Eq.(S54).

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}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})=V_{\sigma}(\leavevmode\resizebox{}{5.38193pt}{
\leavevmode\hbox to6.67pt{\vbox
to5.64pt{\pgfpicture\makeatletter\hbox{\hskip-0.5632pt\lower-1.95901pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@moveto{0.9pt}{-0.21529pt}\pgfsys@moveto{0.9pt}{-0.21529pt}\pgfsys@lineto{0.9pt}{3.48749pt}\pgfsys@lineto{6.1pt}{3.48749pt}\pgfsys@lineto{6.1pt}{-0.21529pt}\pgfsys@closepath\pgfsys@moveto{6.1pt}{3.48749pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}{}{}{{}}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.47362pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{1.34808pt}{0.75171pt}\pgfsys@curveto{2.4pt}{1.72221pt}{2.4pt}{1.72221pt}{3.19873pt}{2.97272pt}\pgfsys@curveto{3.36513pt}{3.23323pt}{3.27934pt}{3.32567pt}{3.00714pt}{3.17918pt}\pgfsys@curveto{2.9pt}{3.12152pt}{2.9pt}{3.12152pt}{2.83388pt}{3.03256pt}\pgfsys@curveto{2.64949pt}{2.78445pt}{2.74614pt}{2.63043pt}{3.04974pt}{2.68852pt}\pgfsys@curveto{7.00002pt}{3.44443pt}{0.8pt}{-1.72221pt}{5.0931pt}{0.83408pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.73497}{-0.6781}{0.6781}{-0.73497}{1.34808pt}{0.75171pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.8592}{0.51163}{-0.51163}{0.8592}{5.0931pt}{0.83408pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})\cos^{2}\theta+V_{\pi}(\leavevmode\resizebox{}{5.38193pt}{
\leavevmode\hbox to6.67pt{\vbox
to5.64pt{\pgfpicture\makeatletter\hbox{\hskip-0.5632pt\lower-1.95901pt\hbox
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@moveto{0.9pt}{-0.21529pt}\pgfsys@moveto{0.9pt}{-0.21529pt}\pgfsys@lineto{0.9pt}{3.48749pt}\pgfsys@lineto{6.1pt}{3.48749pt}\pgfsys@lineto{6.1pt}{-0.21529pt}\pgfsys@closepath\pgfsys@moveto{6.1pt}{3.48749pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}{}{}{{}}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.47362pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{1.34808pt}{0.75171pt}\pgfsys@curveto{2.4pt}{1.72221pt}{2.4pt}{1.72221pt}{3.19873pt}{2.97272pt}\pgfsys@curveto{3.36513pt}{3.23323pt}{3.27934pt}{3.32567pt}{3.00714pt}{3.17918pt}\pgfsys@curveto{2.9pt}{3.12152pt}{2.9pt}{3.12152pt}{2.83388pt}{3.03256pt}\pgfsys@curveto{2.64949pt}{2.78445pt}{2.74614pt}{2.63043pt}{3.04974pt}{2.68852pt}\pgfsys@curveto{7.00002pt}{3.44443pt}{0.8pt}{-1.72221pt}{5.0931pt}{0.83408pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.73497}{-0.6781}{0.6781}{-0.73497}{1.34808pt}{0.75171pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.8592}{0.51163}{-0.51163}{0.8592}{5.0931pt}{0.83408pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})\sin^{2}\theta\,.$
(S62)

Here, the subscripts $\sigma$ and $\pi$ denote the two orthogonal projections
of the $p_{z}$ orbitals along the direction of  ($\sigma$ and $\pi$ bonding).
The radial functions $V_{\sigma}(\leavevmode\resizebox{}{5.38193pt}{
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{}{{}}{}{}{}{}{{}}{}\pgfsys@moveto{0.9pt}{-0.21529pt}\pgfsys@moveto{0.9pt}{-0.21529pt}\pgfsys@lineto{0.9pt}{3.48749pt}\pgfsys@lineto{6.1pt}{3.48749pt}\pgfsys@lineto{6.1pt}{-0.21529pt}\pgfsys@closepath\pgfsys@moveto{6.1pt}{3.48749pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}{}{}{{}}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.47362pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{1.34808pt}{0.75171pt}\pgfsys@curveto{2.4pt}{1.72221pt}{2.4pt}{1.72221pt}{3.19873pt}{2.97272pt}\pgfsys@curveto{3.36513pt}{3.23323pt}{3.27934pt}{3.32567pt}{3.00714pt}{3.17918pt}\pgfsys@curveto{2.9pt}{3.12152pt}{2.9pt}{3.12152pt}{2.83388pt}{3.03256pt}\pgfsys@curveto{2.64949pt}{2.78445pt}{2.74614pt}{2.63043pt}{3.04974pt}{2.68852pt}\pgfsys@curveto{7.00002pt}{3.44443pt}{0.8pt}{-1.72221pt}{5.0931pt}{0.83408pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.73497}{-0.6781}{0.6781}{-0.73497}{1.34808pt}{0.75171pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.8592}{0.51163}{-0.51163}{0.8592}{5.0931pt}{0.83408pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})=A_{\sigma}e^{-\leavevmode\resizebox{}{3.76735pt}{
\leavevmode\hbox to5.07pt{\vbox to3.95pt{\pgfpicture\makeatletter\hbox{\hskip
0.0pt\lower-1.3713pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@moveto{0.63pt}{-0.1507pt}\pgfsys@moveto{0.63pt}{-0.1507pt}\pgfsys@lineto{0.63pt}{2.44125pt}\pgfsys@lineto{4.27pt}{2.44125pt}\pgfsys@lineto{4.27pt}{-0.1507pt}\pgfsys@closepath\pgfsys@moveto{4.27pt}{2.44125pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}{}{}{{}}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.33153pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}{{{\pgfsys@beginscope\pgfsys@setdash{}{0.0pt}\pgfsys@roundcap{}
{}
\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.16576pt}{0.0pt}\pgfsys@stroke\pgfsys@endscope}}
}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.94365pt}{0.5262pt}\pgfsys@curveto{1.68pt}{1.20555pt}{1.68pt}{1.20555pt}{2.2391pt}{2.08092pt}\pgfsys@curveto{2.35558pt}{2.26328pt}{2.29553pt}{2.32799pt}{2.105pt}{2.22545pt}\pgfsys@curveto{2.03pt}{2.18507pt}{2.03pt}{2.18507pt}{1.98372pt}{2.1228pt}\pgfsys@curveto{1.85464pt}{1.94913pt}{1.9223pt}{1.84131pt}{2.13483pt}{1.88196pt}\pgfsys@curveto{4.90001pt}{2.41112pt}{0.56pt}{-1.20555pt}{3.56516pt}{0.58386pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.73497}{-0.6781}{0.6781}{-0.73497}{0.94365pt}{0.5262pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.85922}{0.51161}{-0.51161}{0.85922}{3.56516pt}{0.58386pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}/a_{\sigma}}$ and
$V_{\pi}(\leavevmode\resizebox{}{5.38193pt}{ \leavevmode\hbox to6.67pt{\vbox
to5.64pt{\pgfpicture\makeatletter\hbox{\hskip-0.5632pt\lower-1.95901pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@moveto{0.9pt}{-0.21529pt}\pgfsys@moveto{0.9pt}{-0.21529pt}\pgfsys@lineto{0.9pt}{3.48749pt}\pgfsys@lineto{6.1pt}{3.48749pt}\pgfsys@lineto{6.1pt}{-0.21529pt}\pgfsys@closepath\pgfsys@moveto{6.1pt}{3.48749pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}{}{}{{}}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.47362pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{1.34808pt}{0.75171pt}\pgfsys@curveto{2.4pt}{1.72221pt}{2.4pt}{1.72221pt}{3.19873pt}{2.97272pt}\pgfsys@curveto{3.36513pt}{3.23323pt}{3.27934pt}{3.32567pt}{3.00714pt}{3.17918pt}\pgfsys@curveto{2.9pt}{3.12152pt}{2.9pt}{3.12152pt}{2.83388pt}{3.03256pt}\pgfsys@curveto{2.64949pt}{2.78445pt}{2.74614pt}{2.63043pt}{3.04974pt}{2.68852pt}\pgfsys@curveto{7.00002pt}{3.44443pt}{0.8pt}{-1.72221pt}{5.0931pt}{0.83408pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.73497}{-0.6781}{0.6781}{-0.73497}{1.34808pt}{0.75171pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.8592}{0.51163}{-0.51163}{0.8592}{5.0931pt}{0.83408pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})=A_{\pi}e^{-\leavevmode\resizebox{}{3.76735pt}{
\leavevmode\hbox to4.67pt{\vbox
to3.95pt{\pgfpicture\makeatletter\hbox{\hskip-0.39424pt\lower-1.3713pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@moveto{0.63pt}{-0.1507pt}\pgfsys@moveto{0.63pt}{-0.1507pt}\pgfsys@lineto{0.63pt}{2.44125pt}\pgfsys@lineto{4.27pt}{2.44125pt}\pgfsys@lineto{4.27pt}{-0.1507pt}\pgfsys@closepath\pgfsys@moveto{4.27pt}{2.44125pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}{}{}{{}}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.33153pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.94365pt}{0.5262pt}\pgfsys@curveto{1.68pt}{1.20555pt}{1.68pt}{1.20555pt}{2.2391pt}{2.08092pt}\pgfsys@curveto{2.35558pt}{2.26328pt}{2.29553pt}{2.32799pt}{2.105pt}{2.22545pt}\pgfsys@curveto{2.03pt}{2.18507pt}{2.03pt}{2.18507pt}{1.98372pt}{2.1228pt}\pgfsys@curveto{1.85464pt}{1.94913pt}{1.9223pt}{1.84131pt}{2.13483pt}{1.88196pt}\pgfsys@curveto{4.90001pt}{2.41112pt}{0.56pt}{-1.20555pt}{3.56516pt}{0.58386pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.73497}{-0.6781}{0.6781}{-0.73497}{0.94365pt}{0.5262pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.85922}{0.51161}{-0.51161}{0.85922}{3.56516pt}{0.58386pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}/a_{\pi}}$ decay
exponentially with the distance  between the orbitals. The parameters
$A_{\sigma}$, $A_{\pi}$, $a_{\sigma}$, and $a_{\pi}$ are adjustable parameters
in the Slater-Koster approach. While $a_{\sigma}$ and $a_{\pi}$ play the role
of a effective Bohr radii that set the characteristic radial decay range of
the MBT-BP coupling: larger $a_{\sigma}$ and $a_{\pi}$ results in a slower
$t(\leavevmode\resizebox{}{5.38193pt}{ \leavevmode\hbox to6.67pt{\vbox
to5.64pt{\pgfpicture\makeatletter\hbox{\hskip-0.5632pt\lower-1.95901pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{{}}{}{}{}{}{{}}{}\pgfsys@moveto{0.9pt}{-0.21529pt}\pgfsys@moveto{0.9pt}{-0.21529pt}\pgfsys@lineto{0.9pt}{3.48749pt}\pgfsys@lineto{6.1pt}{3.48749pt}\pgfsys@lineto{6.1pt}{-0.21529pt}\pgfsys@closepath\pgfsys@moveto{6.1pt}{3.48749pt}\pgfsys@clipnext\pgfsys@discardpath\pgfsys@invoke{
}{}{}{{}}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}{{}}{}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.47362pt}\pgfsys@invoke{
}{{}{}{{}}}{{}{}{{}}}{}{}{{}{}{{}}}{{}{}{{}}}{}{}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{1.34808pt}{0.75171pt}\pgfsys@curveto{2.4pt}{1.72221pt}{2.4pt}{1.72221pt}{3.19873pt}{2.97272pt}\pgfsys@curveto{3.36513pt}{3.23323pt}{3.27934pt}{3.32567pt}{3.00714pt}{3.17918pt}\pgfsys@curveto{2.9pt}{3.12152pt}{2.9pt}{3.12152pt}{2.83388pt}{3.03256pt}\pgfsys@curveto{2.64949pt}{2.78445pt}{2.74614pt}{2.63043pt}{3.04974pt}{2.68852pt}\pgfsys@curveto{7.00002pt}{3.44443pt}{0.8pt}{-1.72221pt}{5.0931pt}{0.83408pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-0.73497}{-0.6781}{0.6781}{-0.73497}{1.34808pt}{0.75171pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.8592}{0.51163}{-0.51163}{0.8592}{5.0931pt}{0.83408pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}})$ decays and
coupling functions $\hat{U}_{b}(\mathbf{k})$ and $\hat{U}_{t}(\mathbf{k})$
that are more localized in momentum space. In this work, we used
$a_{\sigma}=a_{\pi}=2$ Å.

The parameters $A_{\sigma}$ and $A_{\pi}$ have dimension of energy and
determine the hopping strength, which are physically given by the overlaps of
the orbital wavefunctions. Since the $\sigma$ projection of two orbitals is
generally larger than the $\pi$ projection, we choose $A_{\sigma}\gtrsim
A_{\pi}$. In this work, we used $A_{\sigma}=0.34$ eV and $A_{\pi}=0.0$.

In Eq. (S54), we take a partial Fourier transform of Eq.(S62) with respect to
the in-plane components, leaving the $z$ coordinate unchanged,

$t(\mathbf{q},z)=\int
d^{2}y\left[A_{\sigma}\frac{z^{2}}{y^{2}+z^{2}}e^{-a_{\sigma}^{-1}\sqrt{y^{2}+z^{2}}}+A_{\pi}\frac{y^{2}}{y^{2}+z^{2}}e^{-a_{\pi}^{-1}\sqrt{y^{2}+z^{2}}}\right]e^{-i\mathbf{q}\cdot\mathbf{y}}\text{
.}$ (S63)

We numerically calculate this integral and fit the result to a sum of two
decaying exponentials of the form

$t(\mathbf{q},z)=z^{2}\left[A_{\sigma}\gamma_{1}e^{-\sqrt{\gamma_{2}+\gamma_{3}q^{2}z^{2}}}+A_{\pi}\gamma_{4}\left(1-\gamma_{5}q^{2}z^{2}\right)e^{-\sqrt{\gamma_{6}+\gamma_{7}q^{2}z^{2}}}\right]\text{
.}$ (S64)

Here, $\gamma_{i}$ ($i=1,\ldots,6$) are real fitting constants that implicitly
depend on the Bohr radii $a_{\sigma}$ (for $i=1,2,3$) and $a_{\pi}$ (for
$i=4,5,6$). Substituting $z$ by the $z$ component distance of the orbitals
$s_{\mu\nu}=\tau_{\mu}^{\perp}-\rho_{\mu}^{\perp}$ in Eq.(S64) finally yields
$t_{\mu\nu}^{(b)}(\mathbf{q})$.

###  IV.9 IV.9. Parameter set used in the main text

The model for the heterostructure contains a total of 33 parameters. In Table
S3 we provide the numerical parameter values that we have used in the main
text to reproduce the experimental observations.

Fig. S1: The symmetry properties of even-layered MnBi2Te4, BP/MnBi2Te4 and
BP/MnBi2Te4/BP heterostructures at low temperature. (A) The symmetry of even-
layered MnBi2Te4. (B) The symmetry of BP/MnBi2Te4 heterostructure. (C) The
symmetry of BP/MnBi2Te4/BP heterostructure. The mirror planes of top BP and
bottom BP are aligned.

Fig. S2: The determination of crystalline directions of the MnBi2Te4 and BP.
(A) Top view of the MnBi2Te4 lattice. It shows three-fold rotational symmetry
$C_{3z}$. The three mirror planes are noted by the blue double arrow lines.
(B) The SHG signals of the MnBi2Te4 measured at room temperature. The largest
signals corresponding to the mirror planes of the MnBi2Te4. (C) Top view of
the BP lattice. It has a mirror plane $\mathcal{M}_{a}$ which is noted by the
red double arrow line. (D) The angular-resolved Raman signals of the BP
measured with a 532-nm laser. The largest signals corresponding to the mirror
plane (armchair direction) of the BP.

Fig. S3: The magneto-transport data of MnBi2Te4 with and without BP. (A to C)
The magneto-transport data of the MnBi2Te4 at charge neutral point. (D to F)
The magneto-transport data of the BP/MnBi2Te4 heterostructure at charge
neutral point. The BP/MnBi2Te4 heterostructure exhibits the same topological
phases as the MnBi2Te4, i.e. Chern insulator in FM phase and Axion insulator
in AFM phase.

Fig. S4: The typical four-probe resistance of BP and MnBi2Te4 as a function of
back gate voltages at 1.8 K. The thickness of BP is around 10 nm and MnBi2Te4
is 6SL. The contact geometry of BP and MnBi2Te4 is identical. The four-probe
resistance of BP is two orders of magnitude larger than that of MnBi2Te4.

Fig. S5: Identifying the current flowing in the BP and MnBi2Te4 layers. (A and
B) Optical microscopic image and schematic of the special BP/MnBi2Te4 device.
The contacts outlined by red dashed lines are deposited on the MnBi2Te4 layer
(which also contact with BP layer) while the contacts outlined by black dashed
lines are deposited on the BP layer. The measurement electrodes are
highlighted in yellow. The measurement circuit is shown in panel (B). A
constant AC current was applied on the drain contact and was collected from
two source electrodes S${}_{\textrm{{MBT}}}$ and S${}_{\textrm{{BP}}}$,
respectively. (C) $I_{\textrm{BP}}$ and $I_{\textrm{MBT}}$ as a function of
$V_{\textrm{BG}}$. (D) The percentage of current flowing in BP layer
($I_{\textrm{BP}}/I_{\textrm{total}}$) as a function of $V_{\textrm{BG}}$ and
$V_{\textrm{TG}}$. (E and F) The line-cut data in panel (D). The current
flowing in BP layer is always less than 2$\%$ of the total current.

Fig. S6: The demonstration of $\mathcal{C}_{3z}$ breaking by electrical
transport measurements. (A) The crystalgraphic bases are noted by $a$ and $b$.
The Cartesian bases are noted by $x$ and $y$. The angle difference between
crystalgraphic and Cartesian basis is noted by $\theta$. (B) A BP/MnBi2Te4
heterostructure with the circular-disc contacts. MnBi2Te4 was roughly
scratched into a circular shape with a tip in glovebox. Two-probe and four-
probe measurements configuration are shown in the panel. (C and D) The
longitudinal resistance ($R_{xx}$) of two-probe and four-probe measurements.
(E) The transverse resistance ($R_{yx}$) of four-probe measurements. The blue
and red curves correspond to the resistance of the same MnBi2Te4 sample before
and after interfacing with BP.

Fig. S7: The demonstration of $\mathcal{C}_{3z}$ breaking by optical SHG
measurements at room temperature. (A) One of the BP/MnBi2Te4 heterostructures
for optical SHG measurements. The MnBi2Te4 and BP are outline by red and cyan
dashed line. The measured area is marked by a blue circle. (B) The optical SHG
signals of BP/MnBi2Te4 heterostructure. (C) The same device as in (A) but
removed the BN and part of BP with Scotch tape. The laser shone on the same
area as in (A), noted by the red circle. (D) The optical SHG signals of the
MnBi2Te4.

Fig. S8: The electric field dependence of Berry curvature dipole and quantum
metric dipole. (A to C) The Berry curvature dipoles under the out of plane
electrical field $E_{z}>0$ and $E_{z}<0$. The bands for top and bottom layers
of MnBi2Te4 have the opposite Berry curvature. The electrical field $E_{z}>0$
and $E_{z}<0$ break the band degeneration in opposite ways gao2021layer .
Therefore, Berry curvature dipoles have different signs for opposite
electrical fields. (D to F) The quantum metric dipoles under the out of plane
electrical field $E_{z}>0$ and $E_{z}<0$. The bands for top and bottom layers
of MnBi2Te4 have the same quantum metric. Although the opposite electrical
field shift the bands of top and bottom layers of MnBi2Te4 in opposite ways,
the quantum metric distribution is identical. Therefore, the quantum metric
dipoles are symmetric as a function of $E_{z}$.

Fig. S9: The electric filed dependence of linear and nonlinear Hall signals.
(A) The linear Hall conductivity $\sigma_{xy}^{\omega}$ as a function of
vertical electric field $E_{z}$. (B) The nonlinear Hall conductivity
$\sigma_{yxx}^{2\omega}$ as a function of $E_{z}$. $\sigma_{xy}^{\omega}$ is
antisymmetric as a function of $E_{z}$, while $\sigma_{yxx}^{2\omega}$ is
symmetric as a function of $E_{z}$. At $E_{z}=0$, the $\mathcal{PT}$ confirms
the $\sigma_{xy}^{\omega}=0$ while $\sigma_{yxx}^{2\omega}\neq 0$.

Fig. S10: A mirror plane aligned 2L BP/6SL MnBi2Te4/2L BP heterostructure. (A)
The microscopic image of the BP/MnBi2Te4/BP heterostructure. The mirror planes
of top BP, MnBi2Te4 and bottom BP are all aligned along $x$ direction in
Cartesian axis. MnBi2Te4, top and bottom BP are outlined by pink, red and blue
dashed lines, respectively. The top and bottom BP are both bilayer. MnBi2Te4
is 6SL. (B) The polarized Raman signals of top (red curve) and bottom (blue
curve) BP at room temperature. (C) Room-temperature SHG signals of the 6SL
MnBi2Te4. (D) The schematic of aligned BP/MnBi2Te4/BP lattice structure. The
mirror plane is along $a$ ($x$). (E) The nonlinear Hall signals of the
heterostructure for driving current along ($V_{yxx}^{2\omega}$) and vertical
to ($V_{xyy}^{2\omega}$) the mirror direction.

Fig. S11: The sum frequency generation measurements. (A and B) The schematic
of $\sigma_{yxx}$ and $\sigma_{xyx}$ measurements. $\omega_{1}\gg\omega_{2}$.
(C) The experiment results of the $\sigma_{yxx}$ and $\sigma_{xyx}$ as a
function of carrier density $n_{e}$ on a BP/MnBi2Te4 device. It is notable
that $\sigma_{yxx}$ and $\sigma_{xyx}$ are antisymmetric, i.e.,
$\sigma_{yxx}=-\sigma_{xyx}$.

Fig. S12: The measurement setup of sum frequency generation. (A and B) The
circuit of $\sigma_{yxx}$ and $\sigma_{xyx}$ measurements. The time constant
$t$ for lock-in A and B are 30 ms and 3 s, respectively. $\omega_{1}=547$ Hz,
$\omega_{2}=1.37$ Hz. A 2.2 $\mu$F capacitor and 40 H inductor were chosen to
confirm the current flow in target direction.

Fig. S13: The temperature dependence of nonlinear Hall signals. (A and B) The
square of linear conductivity ($\sigma_{xx}^{\omega}$)2 and nonlinear Hall
conductivity $\sigma_{yxx}^{2\omega}$ as a function of temperature. The
scaling properties of $\sigma_{yxx}^{2\omega}$ with ($\sigma_{xx}^{\omega}$)2
for temperature lower the 15 K.

Fig. S14: The carrier density dependence of the nonlinear Hall signals. (A and
B) The experimental results of $\sigma_{yxx}^{2\omega}$ and
($\sigma_{xx}^{\omega}$)2 as a function of carrier density $n_{e}$.

Fig. S15: The summary of $\sigma_{yxx}^{2\omega}$ of 26 MnBi2Te4
heterostructures and 7 MnBi2Te4 devices.
The $\sigma_{yxx}^{2\omega}$ in MnBi2Te4 device is induced by nonlinear Drude
conductivity which is two orders of amplitude smaller then that in MnBi2Te4
heterostructures.

Fig. S16: The magnetic field dependence of nonlinear Hall signals in the
BP/MnBi2Te4/BP heterostructure. The nonlinear transverse voltages
$V_{yxx}^{2\omega}$ as a function of out of plane magnetic field $\mu_{0}H$.
At $\mu_{0}H=0$, the $V_{yxx}^{2\omega}$ have opposite signals for different
AFM states. The forward and backward scans and corresponding AFM states are
noted by the red and blue curves, respectively.

Fig. S17: The observation of antiferromagnetic nonlinear Hall in another
BP/MnBi2Te4/BP device. (A and B) The linear longitudinal resistance $R_{xx}$
as a function of back gate voltages for different AFM states. (C and E) The
linear longitudinal voltages $\sigma_{xx}^{\omega}$ and nonlinear Hall
voltages $\sigma_{yxx}^{2\omega}$ as a function of supply current for an AFM
state. Insets show the corresponding measurement circuits. (D and F) The same
as the panels (C) and (E) but for another AFM state.

Fig. S18: Additional temperature dependence data corresponding to Fig. 3A. (A)
The nonlinear Hall voltages $V_{yxx}^{2\omega}$ as a function of temperature.
(B) The nonlinear Hall conductivity $\sigma_{yxx}^{2\omega}$ as a function of
temperature. When temperature is lower than 15 K, $\sigma_{yxx}^{2\omega}$ is
almost independent of temperature.

Fig. S19: Additional temperature dependence data on another device (Device-
BM21). (A to C) The temperature dependence of the linear longitudinal
conductivity $\sigma_{xx}$, nonlinear Hall voltage $V_{yxx}^{2\omega}$ and
nonlinear Hall conductivity $\sigma_{yxx}^{2\omega}$. The nonlinear Hall
signals $V_{yxx}^{2\omega}$ and $\sigma_{yxx}^{2\omega}$ decrease to zero when
temperature is higher than Neel temperature ($T_{N}>21$ K). (D)
$\sigma_{yxx}^{2\omega}$ as a function of $\sigma_{xx}$.
$\sigma_{yxx}^{2\omega}$ is independent of $\sigma_{xx}$.

Fig. S20: Additional data of the nonlinear Hall signals on a BP/4SL MnBi2Te4
device. (A) The microscopic image of a BP/4SL MnBi2Te4 device. BP and 4SL
MnBi2Te4 are outlined by blue and red dashed lines. (B) The nonlinear Hall
conductivity $\sigma_{yxx}^{2\omega}$ as a function of carrier density
$n_{e}$. The $\sigma_{yxx}^{2\omega}$ in the BP/4SL MnBi2Te4 device shows the
same behavior as that in the BP/6SL MnBi2Te4 device (Fig. 4A).

Fig. S21: Microwave rectification measurements on a BP/6SL MnBi2Te4 device at
2 K. (A) The DC signals $V^{\textrm{DC}}$ as a function of carrier density
$n_{e}$ at 2.8 GHz. The inset shows the DC microwave rectification measurement
circuit. The electrical field direction of the microwave is not well defined.
Therefore, the recorded DC signals are only part of the the microwave
generated signals. (B) The power dependence of the $V^{\textrm{DC}}$. The
solid line is the fit of the $V^{\textrm{DC}}$. It scales quadratically with
$\sqrt{P}$ ($\sqrt{P}\propto I$). (C) The $V^{\textrm{DC}}$ as a function of
Frequency. The fluctuation is caused by the transmission losses of the RF
signal.

Fig. S22: An intuitive picture of quantum metric. (A) The Bloch sphere. The
north and south poles of the Bloch sphere correspond to the the basis vectors
of $\arrowvert\alpha\rangle$ and $\arrowvert\beta\rangle$, respectively. (B) A
trivial case ($g=0$) without band inversion. When electrons momenta change
from $k$ to $k+\delta k$, they occupy the same state
($\arrowvert\alpha\rangle$) and map on the same point on the Bloch sphere.
Therefore, the distance between the two states ($\arrowvert u_{k}\rangle$ and
$\arrowvert u_{k+\delta k}\rangle$) on the Bloch sphere is zero. (C) A
nontrivial case ($g\neq 0$) in which conducting band and valance band go
through a band inversion. In this case, each band becomes a
$\mathbf{k}$-dependent superposition of $\arrowvert\alpha\rangle$ and
$\arrowvert\beta\rangle$. The change of $\arrowvert u\rangle$ is dramatic near
$\mathbf{k}=0$ where band inversion occurs but weak at large $\mathbf{k}$.
When electrons momenta change from $k$ to $k+\delta k$, the state changes from
one to another. The nonzero distance between the two states is drawn on the
Bloch sphere.

Fig. S23: The DFT computed band structure of the BP/MnBi2Te4/BP
heterostructure. (A) The MnBi2Te4 (MBT) bands are marked in red, and the BP
bands are shown in blue. As evident from the plot, MBT and BP share similar
work functions, and as a result, the low-energy BP bands hybridize with MBT
bands around the Fermi level of MBT, forming a straddling gap (type-I)
arrangement. (B) The heterostructure considered here is created by a
rectangular supercell of 2SL-MBT and a 1x2 supercell of BP. The armchair
direction of the MBT is aligned along the zigzag direction of the BP.

Fig. S24: Quantum metric dipole dominated nonlinear Hall conductivity. Two-
band model (quantum metric dipole $D_{\textrm{Metric}}$ contribution) and
multiband model ($D_{\textrm{Metric}}+$ additional inter band contributions
AIC) calculated nonlinear Hall conductivity as a function of carrier density.
The nonlinear Hall signals are dominated by the quantum metric dipole
contribution.

Fig. S25: Real-space lattice structure of the MBT-BP heterostructure. (A)
Schematic drawing of the $\mathcal{PT}$ symmetric MBT-BP heterostructure
consisting of 6SL MBT encapsulated by BP monolayer. Red arrows indicate AFM
order of the MBT SLs. The direct electronic coupling between MBT and BP is
dominated by hopping processes between BP top and SL 1 $(\hat{U}_{t})$ and
between BP bottom and SL 6 $(\hat{U}_{b})$. (B) Lattice structure of a single
MBT SL with seven atomic layers labeled by index $\mu$. (C) Lattice structure
of a BP monolayer with four basis atoms labeled by index $\nu$. (D) Schematic
top view on an incommensurate bilayer structure consisting of bottom layer of
MBT (blue) and top P layer of BP (grey) with different lattice constants. (E)
Two-dimensional projection of MBT SL with primitive vectors
$\mathbf{a}^{\text{MBT}}_{1}$ and $\mathbf{a}^{\text{MBT}}_{2}$. Color coding
corresponds to that used in panel (B). (F) Two-dimensional projection of BP
monolayer with primitive vectors $\mathbf{a}^{\text{BP}}_{1}$ and
$\mathbf{a}^{\text{BP}}_{2}$. Color coding corresponds to that used in panel
(C).

Fig. S26: Energy band Fermi surface of MBT modeled by Eqs.(S15), (S30) and
(S31). (A) A cut along $k_{y}$ of the bands of a single MBT SL in the
paramagnetic state (dashed) and after the onset of the AFM order (solid line).
The band gap at $\Gamma$ ($\mathbf{k}=0$) in the paramagnetic state originates
from the hybridization of the surface states on the top and bottom of the SL
and is reduced after the onset of magnetism. The hybridization of the surface
states is suppressed by stacking multiple SLs along $\hat{\mathbf{z}}$, which
increases the system’s thickness. (B) The energy bands of a set of six SLs in
the paramagnetic state ($\gamma_{af}=0$), where the surface Dirac cone at
$\Gamma$ can be readily seen. (C) The onset of the AFM order open a gap in the
surface Dirac cone. (D to F) The Fermi surface of 6SL MBT when the chemical
potential is set to $\mu=0.06$ meV. The dashed (solid) line in (D) and (E)
refer to the Fermi surface without (with) the warping term. The red curve in
panel (F) is the same as in (E), while the blue curve is obtained by rotation
the red by $2\pi/3$ around $\hat{z}$ to highlight the break of $C_{3z}$
symmetry induced by strain. We set the parameters of the model to the values
listed in Table S3. Note that the choice of $k_{x}$ and $k_{y}$ axis is
rotated by $\pi/2$ with respect to the main text.

Fig. S27: Energy band and Fermi surface of monolayer BP. (A) Energy bands of
BP monolayer modeled by Eq.(S36) along a high-symmetry direction of the
Brillouin zone shown in blue as a inset. (B) A zoom-in view of the bands
around the $\Gamma$ point (e.g. $\mathbf{k}=0$). (C) The Fermi surface of BP
where we set the chemical potential to be $\mu=0.2$ eV (dashed gray line in
(B)). BP does not preserve 3-fold rotational symmetry around $\hat{z}$, which
is reflected in the elliptical shape of its Fermi surface. Therefore, similar
to the role of uniaxial strain, encapsulating the MBT SLs with BP monolayer
leads to a breaking of $C_{3z}$ symmetry of the heterostructure. We set the
parameters of the model to the values listed in Table S3.

Fig. S28: Matrix elements of $\hat{V}_{b,\uparrow}(\mathbf{k})$ along the
$k_{y}$ axis. We set the parameters of the model to the values listed in Table
S3.

Device type |  Device number |  Contact geometry |  Angle between top/bottom BP and MBT |  Angle between MBT and contacts |  top/bottom BP thickness |  $\sigma_{xx}^{\omega}$ ($\mu$S) |  $\sigma_{yxx}^{2\omega}$ (mA/V2)
---|---|---|---|---|---|---|---
|  BMB1 |  Hall bar |  0/0 |  0 |  2L/2L |  227 |  8.93
|  BMB2 |  Hall bar |  0/-1 |  0 |  2L/1L |  671 |  6.62
BP/MBT/BP |  BMB3 |  Hall bar |  -11/-6 |  1 |  2L/$\sim$10nm |  249 |  10.25
|  BMB4 |  Hall bar |  53/15 |  27 |  1L/$\sim$15nm |  585 |  10.5
|  BMB5 |  Hall bar |  -16/-17 |  16 |  2L/$\sim$10nm |  1466 |  15.68
|  BM1 |  Circular |  -5 |  0 |  $\sim$12nm |  154 |  13.97
|  BM2 |  Hall bar |  - |  - |  - |  500 |  6.24
|  BM3 |  Hall bar |  -30 |  25 |  - |  141 |  9.70
|  BM4 |  Hall bar |  - |  - |  - |  119 |  19.91
|  BM5 |  Hall bar |  - |  - |  - |  403 |  10.42
|  BM6 |  Hall bar |  - |  - |  - |  206 |  9.06
|  BM7 |  Hall bar |  -5 |  -2 |  - |  498 |  8.82
|  BM8 |  Hall bar |  -1 |  1 |  - |  608 |  10.26
BP/MBT |  BM9 |  Hall bar |  - |  - |  4L |  1070 |  8.37
|  BM10 |  Hall bar |  - |  - |  - |  2890 |  12.22
|  BM11 |  Hall bar |  - |  - |  - |  307 |  6.24
|  BM12 |  Hall bar |  - |  - |  5-7L |  398 |  7.82
|  BM13 |  Hall bar |  - |  - |  - |  450 |  5.95
|  BM14 |  Hall bar |  - |  - |  - |  943 |  8.75
|  BM15 |  Hall bar |  - |  - |  - |  483 |  7.97
|  BM16 |  Hall bar |  -26 |  -2 |  $\sim$8nm |  238 |  5.10
|  BM17 |  Hall bar |  - |  - |  - |  357 |  4.60
|  BM18 |  Hall bar |  -35 |  54 |  - |  500 |  7.00
|  BM19 |  Hall bar |  16 |  26 |  - |  357 |  8.70
|  BM20 |  Hall bar |  - |  - |  - |  700 |  5.86
|  BM21 |  Hall bar |  - |  - |  - |  930 |  5.69
Table S1: Device summary of 5 BP/MnBi2Te4/BP hetrostructures (BP/MBT/BP) and
21 BP/MnBi2Te4 hetrostructures (BP/MBT)

P basis atom | Components (Å)
---|---
$\hat{x}$ | $\hat{y}$ | $\hat{z}$
$\boldsymbol{\rho}_{1}$ | 2.71 | 1.65 | 15.05
$\boldsymbol{\rho}_{2}$ | 4.2 | 0 | 15.05
$\boldsymbol{\rho}_{3}$ | 0.41 | 0 | 17.15
$\boldsymbol{\rho}_{4}$ | 1.9 | 1.65 | 17.15
Table S2: Cartesian components of the position relative to the origin of the
unit cell of the four nonequivalent phosphorus atoms in the BP monolayer.

MBT |  |  |  |  |  |
---|---|---|---|---|---|---
$\gamma_{0}$ (eV) |  $\gamma$ (eV Å2) |  $m_{0}$ (eV) |  $\beta_{0}$ (eV Å2) |  $\beta_{1}$ (eV Å3) |  $\alpha$ (eV Å) |  $t_{1}$ (eV) |  $t_{2}$ (eV)
$-0.01$ | $17$ | $0.04$ | $9.4$ | $8000$ | $3.2$ | $-0.0533$ | $0.0463$
$\lambda$ (eV) |  $\alpha_{3}$ (eV Å2) | $\gamma_{af}$ |  $m_{1}$ (eV) |  $m_{2}$ (eV) | $\gamma_{s}$ |  $s_{1}$ (eV Å2) |  $s_{2}$ (eV Å2)
$0.0557$ | $0.0$ | $0.5$ | $0.05$ | $0.09$ | $1.0$ | $0.1$ | $0.1$
$s_{3}$ (eV Å2) |  $s_{4}$ (eV Å2) |  $s_{5}$ (eV Å) |  $s_{6}$ (eV Å) |  $s_{7}$ (eV Å) |  $s_{8}$ (eV Å) |  $s_{9}$ (eV) |
$50$ | $0.1$ | $0.1$ | $0.0$ | $0.1$ | $0.1$ | $0.0$ |
BP |  |  |  |  |  |
$\tilde{t}_{1}$ (eV) |  $\tilde{t}_{2}$ (eV) |  |  |  |  |  |
$-1.7575$ | $3.665$ |  |  |  |  |  |
MBT-BP coupling |  |  |  |  |  |
$A_{\sigma}/\Omega_{\text{BP}}$ (eV/Å2) |  $A_{\pi}/\Omega_{\text{BP}}$ (eV/Å2) |  $a_{\sigma}$ (Å) |  $a_{\pi}$ (Å) |  |  |  |
$0.34$ | $0.0$ | $2.0$ | $2.0$ |  |  |  |
Table S3: Numerical $k\cdot p$ model parameter values that are used in the
main text. The only other parameters that enter the model are the basis
positions of the atoms in the unit cell that enter via the parameters
$s_{\mu\nu}$. We provide the basis location of the P atoms in Table S2 and
refer to standard literature for the basis positions of the atoms in the MBT
SL. Note that the Bohr radii $a_{\sigma}$, $a_{\pi}$ enter the model via the
fit parameters $\gamma_{j}$ ($j=1,\ldots,6$). Finally, we note that
$\Omega_{\text{BP}}=14.5$Å2 such that $A_{\sigma}=4.9$ eV.