twice those w.r.t. $t$, s.t. the norm is invariant under parabolic scaling.
Applying Hölder continuity estimate to (6.1), we deduce there exists a
universal constant $\gamma\in(0,1)$ s.t.

$\rho^{\gamma}[u]_{\gamma,Q(X_{*},\frac{1}{5}\rho)_{t_{0}}}\leq
C(p,q,l)\rho^{2\lambda_{l}+1}.$

According to the gradient Hölder continuity estimate, we may assume for the
same $\gamma$,

$\rho^{\gamma}[u^{\prime}]_{\gamma,Q(X_{*},\frac{1}{5}\rho)_{t_{0}}}\leq
C(p,q,l).$

Thus,

$|\frac{1}{1+u^{\prime
2}}|_{\gamma,Q(X_{*},\frac{1}{5}\rho)_{t_{0}}}^{(\rho)}+\rho|\frac{1}{x}P(\frac{u}{x})|_{\gamma,Q(X_{*},\frac{1}{5}\rho)_{t_{0}}}^{(\rho)}+\rho^{2}|\frac{1}{x^{2}}Q(\frac{u}{x})|_{\gamma,Q(X_{*},\frac{1}{5}\rho)_{t_{0}}}^{(\rho)}\leq
C(p,q,l).$

Applying Schauder’s estimate to (6.1),

$|u|_{2+\gamma,Q(X_{*},\frac{1}{6}\rho)_{t_{0}}}^{(\rho)}\leq
C(p,q,l)\rho^{2\lambda_{l}+1},$

which obviously implies the Lemma.

Below we will eventually prove Proposition 4.2, as well as the main existence
result Theorem 4.5:

######  Proposition 6.3.

If $\Lambda\gg 1$ (depending on $p,q,l$), $\rho\ll 1$, $\beta\gg 1$, $R\gg 1$
(depending on $p,q,l,\Lambda$), and $t_{0}\gg 1$ (depending on
$p,q,l,\Lambda,\beta,R$), then Proposition 4.2 holds.

Proof: Actually, (1) is exactly (5.14), and (2) is shown in Proposition 5.16,
(1). (4.11) is proved in Proposition 5.16, (2), and (4.12) is implied by
(4.19), provided $\beta\gg 1$ (depending on $p,q,l$). The statement “The
profile curve of
$\Sigma_{t}\cap(B(0,3\rho)-\bar{B}(0,\frac{1}{3}\beta(-t)^{\frac{1}{2}+\sigma_{l}}))$
can be parameterized by a single function as (3.3)” is a consequence of the
admissible condition (2), (3), Proposition 5.16, (2), and the $C^{0}$ and
$C^{1}$ estimates in (4.12), and the statement “The profile curve of
$\Sigma_{t}\cap B(0,3\rho)$ can be parameterized by a single function as
(3.2)” follows from the admissible condition (2), and Proposition 5.16, (2),
especially (4.11).
Now, it remains to prove (4.10). By Lemma 6.2, if $\Lambda\gg 1$ (depending on
$p,q,l$), then (4.10) holds for $\frac{3}{4}\rho\leq x\leq\rho$, $t_{0}\leq
t\leq\mathring{t}$.
By (3.18), for $x\geq\sqrt{-t}$, $i=0,1,2$,

$x^{i}|\partial_{x}^{i}(\frac{k}{c_{l}}(-t)^{\lambda_{l}+\frac{1}{2}}\varphi_{l}(\frac{x}{\sqrt{-t}}))|\leq
C(n,l)x^{2\lambda_{l}+1}.$

Thus by (4.14), if we choose $R\gg 1$, $\rho\ll 1$ (depending on
$p,q,l,\Lambda$), then

$x^{i}|\partial_{x}^{i}u(x,t)|\leq C(n,l)x^{2\lambda_{l}+1},\ R\sqrt{-t}\leq
x\leq\frac{3}{4}\rho,\ t_{0}\leq t\leq\mathring{t},\ i=0,1,2,$

and (4.10) holds for $(x,t)$ as above, provided $\Lambda\gg 1$ (depending on
$n,l$).
Again, by (3.18), for $y>0$, $i=0,1,2$,

$y^{i}|\partial_{y}^{i}(\frac{k}{c_{l}}e^{-\lambda_{l}s}\varphi_{l}(y))|\leq
C(n,l)e^{-\lambda_{l}s}(y^{\alpha}+y^{2\lambda_{l}+1}).$

Thus by (4.15), if we choose $s_{0}\gg 1$ (depending on
$p,q,l,\Lambda,\beta,R$), then

$y^{i}|\partial_{y}^{i}v(y,s)|\leq
C(n,l)e^{-\lambda_{l}s}(y^{\alpha}+y^{2\lambda_{l}+1}),\
e^{-\frac{1}{2}\sigma_{l}s}\leq y\leq R,\ s_{0}\leq s\leq\mathring{s},\
i=0,1,2,$

i.e.

$x^{i}|\partial_{x}^{i}u(x,t)|\leq
C(n,l)((-t)^{l}x^{\alpha}+x^{2\lambda_{l}+1}),\
(-t)^{\frac{1}{2}+\frac{1}{2}\sigma_{l}}\leq x\leq R\sqrt{-t},\ t_{0}\leq
t\leq\mathring{t},\ i=0,1,2,$

and (4.10) holds for $(x,t)$ as above, provided $\Lambda\gg 1$ (depending on
$n,l$).
By (2.13), for
$y>\hat{\psi}_{k}(0)\frac{\mu}{\sqrt{1+\mu^{2}}}e^{-\sigma_{l}s}$, $i=0,1,2$,

$y^{i}|\partial_{y}^{i}(e^{-\sigma_{l}s}\psi_{k}(e^{\sigma_{l}s}y))|\leq
C(p,q)e^{-\lambda_{l}s}y^{\alpha}.$

Thus by (4.17), if we choose $\beta\gg 1$ (depending on $p,q,l,\Lambda$), then

$y^{i}|\partial_{y}^{i}v(y,s)|\leq C(p,q)e^{-\lambda_{l}s}y^{\alpha},\
\frac{3}{2}\beta e^{-\sigma_{l}s}\leq y\leq e^{-\frac{1}{2}\sigma_{l}s},\
s_{0}\leq s\leq\mathring{s},\ i=0,1,2,$

i.e.

$x^{i}|\partial_{x}^{i}u(x,t)|\leq C(p,q)(-t)^{l}x^{\alpha},\
\frac{3}{2}\beta(-t)^{\frac{1}{2}+\sigma_{l}}\leq
x\leq(-t)^{\frac{1}{2}+\frac{1}{2}\sigma_{l}},\ t_{0}\leq t\leq\mathring{t},\
i=0,1,2,$

and (4.10) holds for $(x,t)$ as above, provided $\Lambda\gg 1$ (depending on
$p,q$).
By (2.14), for $\frac{\frac{1}{2}\beta}{\sqrt{1+\mu^{2}}}\leq
z\leq\frac{2\beta}{\sqrt{1+\mu^{2}}}$, $i=0,1,2$,

$z^{i}|\partial_{z}^{i}(\hat{\psi}_{k}(z)-\mu z)|\leq C(p,q)z^{\alpha}.$

Thus by (4.19), if we choose $\beta\gg 1$ (depending on $p,q,l$), then

$z^{i}|\partial_{z}^{i}(\hat{w}(z,\tau)-\mu z)|\leq C(p,q)z^{\alpha},\
\frac{\frac{1}{2}\beta}{\sqrt{1+\mu^{2}}}\leq
z\leq\frac{2\beta}{\sqrt{1+\mu^{2}}},\ \tau_{0}\leq\tau\leq\mathring{\tau},\
i=0,1,2.$

Using the linear transform

$T(x,y)=(\frac{x+\mu y}{\sqrt{1+\mu^{2}}},\frac{-\mu x+y}{\sqrt{1+\mu^{2}}}),$

we let

$T(z,\hat{w}(z,\tau))=(z_{1},w(z_{1},\tau)),$

then

$w(z_{1},\tau)=\frac{\hat{w}(z,\tau)-\mu z}{\sqrt{1+\mu^{2}}},\
w^{\prime}(z_{1},\tau)=\frac{\hat{w}^{\prime}(z,\tau)-\mu}{\mu\hat{w}^{\prime}(z,\tau)+1},\
w^{\prime\prime}(z_{1},\tau)=(1+\mu^{2})^{\frac{3}{2}}\frac{\hat{w}^{\prime\prime}(z,\tau)}{(\mu\hat{w}^{\prime}(z,\tau)+1)^{3}},$

and

$z^{i}|\partial_{z}^{i}w(z_{1},\tau)|\leq C(p,q)z^{\alpha},\
\frac{\frac{1}{2}\beta}{\sqrt{1+\mu^{2}}}\leq
z\leq\frac{2\beta}{\sqrt{1+\mu^{2}}},\ \tau_{0}\leq\tau\leq\mathring{\tau},\
i=0,1,2.$

If $\beta\gg 1$ (depending on $p,q$), then $z_{1}\approx\sqrt{1+\mu^{2}}z$,
and thus

$z^{i}|\partial_{z}^{i}w(z,\tau)|\leq C(p,q)z^{\alpha},\ \beta\leq
z\leq\frac{3}{2}\beta,\ \tau_{0}\leq\tau\leq\mathring{\tau},\ i=0,1,2.$

i.e.

$x^{i}|\partial_{x}^{i}u(x,t)|\leq C(p,q)(-t)^{l}x^{\alpha},\
\beta(-t)^{\frac{1}{2}+\sigma_{l}}\leq
x\leq\frac{3}{2}\beta(-t)^{\frac{1}{2}+\sigma_{l}},\ t_{0}\leq
t\leq\mathring{t},\ i=0,1,2,$

and (4.10) holds for $(x,t)$ as above, provided $\Lambda\gg 1$ (depending on
$p,q$). The proof is finished.

##  7\. Vanishing Theorems of Parabolic Equations on Lawson’s Cones and the
Related Minimal Hypersurfaces

In order to bound the mean curvature of the MCF solution we have constructed
above near the singularity, we need a blow up argument as in [sto], which
further requires some “vanishing theorems” for solutions to a kind of linear
parabolic equations like $\partial_{t}u=(\Delta+|A|^{2})u$ on the limit spaces
(Lawson’s cone $C_{p,q}$ and the related minimal hypersurface
$\mathcal{M}_{k}$), with certain growth control. Actually the right-hand side
of the equation is the so-called “Jacobi operator” of hypersurfaces, which is
closely related to the stability of such minimal hypersurfaces. We present the
following results, which can be proved similarly to those in [sto]:

######  Theorem 7.1.

Let $u=u(|\mathbf{x}|,t)$ be a smooth, radially symmetric, ancient solution to

$\partial_{t}u=\Delta_{\mathcal{M}_{k}}u+|A_{\mathcal{M}_{k}}|^{2}u,\
(\mathbf{x},t)\in\mathcal{M}_{k}\times(-\infty,0]$

for some $k>0$, where $\Delta_{\mathcal{M}_{k}}$, $A_{\mathcal{M}_{k}}$ denote
the Laplacian and the second fundamental form of $\mathcal{M}_{k}$
respectively. If there exists $C>0$ and $\delta>0$ s.t.

$|u(\mathbf{x},t)|\leq C(1+|\mathbf{x}|)^{\alpha-\delta},\
(\mathbf{x},t)\in\mathcal{M}_{k}\times(-\infty,0],$

then $u\equiv 0$.

######  Theorem 7.2.

Let $u=u(|\mathbf{x}|,t)$ be a smooth, radially symmetric, ancient solution to

$\partial_{t}u=\Delta_{C_{p,q}}u+|A_{C_{p,q}}|^{2}u,\
(\mathbf{x},t)\in(C_{p,q}-\\{0\\})\times(-\infty,0].$

If there exists $C>0$ and $0<\delta<n-3+2\alpha$ s.t.

$|u(\mathbf{x},t)|\leq C|\mathbf{x}|^{\alpha-\delta},\
(\mathbf{x},t)\in(C_{p,q}-\\{0\\})\times(-\infty,0],$

then $u\equiv 0$.

##  8\. Boundedness of the Mean Curvature

In this section, we will show the mean curvature of Velázquez’s solution
$\\{\Sigma_{t}^{\mathbf{a}}\\}=\\{\Sigma_{t}\\}$, obtained in Theorem 4.5,
remains bounded up to the singular time $t=0$, provided the parameter $l$ is
sufficiently large. In the outer region $|\mathbf{x}|\geq\sqrt{-t}$, the
boundedness of mean curvature is in fact a direct consequence of the estimates
obtained in Proposition 4.2.

######  Proposition 8.1.

If $\rho\ll 1$ (depending on $p,q,l,\Lambda$), then

$\sup_{\begin{subarray}{c}\mathbf{x}\in\Sigma_{t}-B(0,\sqrt{-t})\\\ t_{0}\leq
t<0\end{subarray}}|H_{\Sigma_{t}}(\mathbf{x})|<+\infty,$

where $H_{\Sigma_{t}}(\mathbf{x})$ is the mean curvature of $\Sigma_{t}$ at
$\mathbf{x}$.

Proof: First of all, by (2) of Proposition 4.2,

$\sup_{\begin{subarray}{c}\mathbf{x}\in\Sigma_{t}-\bar{B}(0,2\rho)\\\
t_{0}\leq t<0\end{subarray}}|H_{\Sigma_{t}}(\mathbf{x})|\leq
C(p,q,\rho)<+\infty.$

To estimate the remaining part, it’s not hard to compute directly that if a
planar curve has the form (3.3), then the mean curvature (w.r.t. the upward
unit vector) of the corresponding $O(p)\times O(q)$ invariant hypersurface at
$(\frac{x-\mu u(x,t)}{\sqrt{1+\mu^{2}}}\omega,\frac{\mu
x+u(x,t)}{\sqrt{1+\mu^{2}}}\phi)$ ($\omega\in\mathbb{S}^{p-1}$,
$\phi\in\mathbb{S}^{q-1}$), which is denoted by $H(u(x,t))$, is

$H(u(x,t))=\frac{1}{\sqrt{1+u^{\prime 2}}}(\frac{u^{\prime\prime}}{1+u^{\prime
2}}+(p-1)\frac{\mu+u^{\prime}}{x-\mu u}-(q-1)\frac{1-\mu u^{\prime}}{\mu
x+u})$ (8.1) $=\frac{1}{\sqrt{1+u^{\prime
2}}}(\frac{u^{\prime\prime}}{1+u^{\prime
2}}+P(\frac{u}{x})\frac{u^{\prime}}{x}+Q(\frac{u}{x})\frac{u}{x^{2}}),$

where $P,Q$ are defined in (6.2). For the part of the profile curve (3.3)
lying in $B^{2}(0,3\rho)$, if $x\geq\rho$, then (4.11) gives a bound of
$H(u(x,t))$. If $\frac{1}{2}\sqrt{-t}\leq x\leq\rho$, then by (4.10), we also
have $|\frac{u(x,t)}{x}|\leq\frac{1}{2}\min\\{\mu,\mu^{-1}\\}$, provided
$\rho\ll 1$ (depending on $p,q,l,\Lambda$), and thus
$P(\frac{u}{x}),Q(\frac{u}{x})$ are bounded. Then (4.10) also gives a bound of
$H(u(x,t))$. Since the part of the profile curve (3.3) lying in
$B^{2}(0,3\rho)-B^{2}(0,\sqrt{-t})$ is covered by the two cases above, the
proof is complete.

To bound the mean curvature in the intermediate and tip regions, we employ a
blow up argument as in [sto], which in addition requires some convergence
results of certain rescaled flows. They are presented in the following two
lemmas, which can be derived from the estimates in in Proposition 4.3. For two
sequences of real numbers $\\{a_{i}\\}$, $\\{b_{i}\\}$, we write $a_{i}\ll
b_{i}$ if $a_{i}=o(b_{i})$ as $i\rightarrow+\infty$.

######  Lemma 8.2.

For any sequence $t_{0}\leq t_{i}<0$, $t_{i}\nearrow 0$, and
$\Lambda_{i}=(-t_{i})^{-\frac{1}{2}-\sigma_{l}}$, the sequence of flows

$\tilde{\Sigma}_{\tau}^{i}:=\Lambda_{i}\Sigma_{t_{i}+\frac{\tau}{\Lambda_{i}^{2}}},\
(t_{0}-t_{i})\Lambda_{i}^{2}\leq\tau<-t_{i}\Lambda_{i}^{2}$

converges smoothly to $\mathcal{M}_{k}$ (a stationary MCF) in any bounded
space-time region in $\mathbb{R}^{n}\times\mathbb{R}$.

Proof: By Proposition 4.2, $\tilde{\Sigma}_{\tau}^{i}\cap
B(0,3\rho\Lambda_{i})$ can be parameterized as $(x,\hat{u}_{i}(x,\tau))$ (see
(3.2)), and
$\tilde{\Sigma}_{\tau}^{i}\cap(B(0,3\rho\Lambda_{i})-\bar{B}(0,\frac{1}{3}\beta(-t_{i}-\Lambda_{i}^{-2}\tau)^{\frac{1}{2}+\sigma_{l}}\Lambda_{i}))$
can be parameterized as $(\frac{x-\mu u_{i}(x,t)}{\sqrt{1+\mu^{2}}},\frac{\mu
x+u_{i}(x,t)}{\sqrt{1+\mu^{2}}})$ (see (3.3)). Thus it suffices to consider
the convergence of $\hat{u}_{i}$ and $u_{i}$.
By the definition of $\tilde{\Sigma}_{\tau}^{i}$ and (3.11),

$\hat{u}_{i}(x,\tau)=\Lambda_{i}\hat{u}(\Lambda_{i}^{-1}x,t_{i}+\Lambda_{i}^{-2}\tau)=\Lambda_{i}(-t_{i}-\Lambda_{i}^{-2}\tau)^{\frac{1}{2}+\sigma_{l}}\hat{w}(\Lambda_{i}^{-1}(-t_{i}-\Lambda_{i}^{-2}\tau)^{-(\frac{1}{2}+\sigma_{l})}x,\frac{1}{2\sigma_{l}}(-t_{i}-\Lambda_{i}^{-2}\tau)^{-2\sigma_{l}})$
$=(1-\tau(-t_{i})^{2\sigma_{l}})^{\frac{1}{2}+\sigma_{l}}\hat{w}((1-\tau(-t_{i})^{2\sigma_{l}})^{-(\frac{1}{2}+\sigma_{l})}x,\frac{1}{2\sigma_{l}}(-t_{i})^{-2\sigma_{l}}(1-\tau(-t_{i})^{2\sigma_{l}})^{-2\sigma_{l}}).$

According to (4.19), for any $M>0$,
$\hat{w}(x,\frac{1}{2\sigma_{l}}(-t_{i})^{-2\sigma_{l}}+\tau)$ converges
smoothly to $\hat{\psi}_{k}(x)$ on $0\leq
x\leq\frac{2\beta}{\sqrt{1+\mu^{2}}}$, $|\tau|\leq M$. Since
$1-\tau(-t_{i})^{2\sigma_{l}}$ and all its powers converge smoothly to 1 on
any bounded $\tau$-interval, and
$\frac{1}{2\sigma_{l}}(-t_{i})^{-2\sigma_{l}}(1-\tau(-t_{i})^{2\sigma_{l}})^{-2\sigma_{l}}-\frac{1}{2\sigma_{l}}(-t_{i})^{-2\sigma_{l}}$
converges smoothly to $\tau$ on any bounded $\tau$-interval, for any
$0<\epsilon<1<M$, $\hat{u}_{i}(x,\tau)$ converges smoothly to
$\hat{\psi}_{k}(x)$ on
$[0,\frac{2\beta}{\sqrt{1+\mu^{2}}}(1-\epsilon)]\times[-M,M]$.
Similarly,

$u_{i}(x,\tau)=(1-\tau(-t_{i})^{2\sigma_{l}})^{\frac{1}{2}+\sigma_{l}}w((1-\tau(-t_{i})^{2\sigma_{l}})^{-(\frac{1}{2}+\sigma_{l})}x,\frac{1}{2\sigma_{l}}(-t_{i})^{-2\sigma_{l}}(1-\tau(-t_{i})^{2\sigma_{l}})^{-2\sigma_{l}}).$

According to (4.17),

$z^{m+2r}|\partial_{z}^{m}\partial_{\tau}^{r}(w(z,\tau)-\psi_{k}(z))|\leq
C(p,q,l,\Lambda,m,r)\beta^{\tilde{\alpha}-\alpha}(\frac{\tau}{\tau_{0}})^{-\varrho}z^{\alpha}$

for $\frac{3}{2}\beta\leq z\leq(2\sigma_{l}\tau)^{\frac{1}{4}}$,
$\tau\geq\tau_{0}$, $m,r\geq 0$. Due to the same reason as above,
$u_{i}(x,\tau)$ converges smoothly to $\psi_{k}(x)$ on
$[\frac{3}{2}\beta(1+\epsilon),M]\times[-M,M]$. Therefore, the profile curve
of $\tilde{\Sigma}_{\tau}^{i}$ converges smoothly to the profile curve of
$\mathcal{M}_{k}$ in any bounded space-time region. Moreover, since the even
extensions of $\hat{u}_{i}(\cdot,\tau)$ and $\hat{\psi}_{k}$ are smooth, the
flow $\tilde{\Sigma}_{\tau}^{i}$ also converges to $\mathcal{M}_{k}$ in the
desired way.

######  Lemma 8.3.

For any sequence $t_{0}\leq t_{i}<0$, $t_{i}\nearrow 0$, and any sequence
$\\{\Lambda_{i}\\}$ s.t.
$(-t_{i})^{-\frac{1}{2}}\ll\Lambda_{i}\ll(-t_{i})^{-\frac{1}{2}-\sigma_{l}}$,
the sequence of flows

$\tilde{\Sigma}_{\tau}^{i}:=\Lambda_{i}\Sigma_{t_{i}+\frac{\tau}{\Lambda_{i}^{2}}},\
(t_{0}-t_{i})\Lambda_{i}^{2}\leq\tau<-t_{i}\Lambda_{i}^{2}$

converges smoothly to $C_{p,q}$ (a stationary MCF) in any compact space-time
region in $(\mathbb{R}^{n}-\\{0\\})\times\mathbb{R}$.

Proof: By Proposition 4.2,
$\tilde{\Sigma}_{\tau}^{i}\cap(B(0,3\rho\Lambda_{i})-\bar{B}(0,\frac{1}{3}\beta(-t_{i}-\Lambda_{i}^{-2}\tau)^{\frac{1}{2}+\sigma_{l}}\Lambda_{i}))$
can be parameterized as $(\frac{x-\mu u_{i}(x,t)}{\sqrt{1+\mu^{2}}},\frac{\mu
x+u_{i}(x,t)}{\sqrt{1+\mu^{2}}})$ (see (3.3)). Thus it suffices to consider
the convergence of $u_{i}$.
By (4.15) and (3.18), for $e^{-\frac{1}{2}\sigma_{l}s}\leq y\leq 1$, $s\geq
s_{1}\gg s_{0}$, $m,r\geq 0$,

$y^{m+2r}|\partial_{y}^{m}\partial_{s}^{r}v(y,s)|\leq
C(n,l,m,r)e^{-\lambda_{l}s}y^{\alpha}.$

Set $\bar{v}(y,s)=e^{-\sigma_{l}s}\psi_{k}(e^{\sigma_{l}s}y)$. According to
(2.13), in the domain of $\bar{v}$,
$y\geq\hat{\psi}_{k}(0)\frac{\mu}{\sqrt{1+\mu^{2}}}e^{-\sigma_{l}s}$, there
holds for all integers $m,r\geq 0:$ (note $k\approx 1$):

(8.2) $|\partial_{y}^{m}\partial_{s}^{r}\bar{v}(y,s)|\leq
C(p,q,m,r)e^{-\lambda_{l}s}y^{\alpha-m}.$

Now (4.17), (8.2) tell us, for $\frac{3}{2}\beta e^{-\sigma_{l}s}\leq y\leq
e^{-\frac{1}{2}\sigma_{l}s}$, $s\geq s_{0}$,

(8.3) $y^{m+2r}|\partial_{y}^{m}\partial_{s}^{r}v(y,s)|\leq
C(p,q,l,\Lambda,m,r)e^{-\lambda_{l}s}y^{\alpha}.$

Thus, (8.3) holds for $\frac{3}{2}\beta e^{-\sigma_{l}s}\leq y\leq 1$, $s\geq
s_{1}$. By the definition of $\tilde{\Sigma}_{\tau}^{i}$ and (3.7),

$u_{i}(x,\tau)=\Lambda_{i}u(\Lambda_{i}^{-1}x,t_{i}+\Lambda_{i}^{-2}\tau)=\Lambda_{i}(-t_{i}-\Lambda_{i}^{-2}\tau)^{\frac{1}{2}}v(\Lambda_{i}^{-1}(-t_{i}-\Lambda_{i}^{-2}\tau)^{-\frac{1}{2}}x,-\ln(-t_{i}-\Lambda_{i}^{-2}\tau))$
$=(-t_{i}\Lambda_{i}^{2}-\tau)^{\frac{1}{2}}v((-t_{i}\Lambda_{i}^{2}-\tau)^{-\frac{1}{2}}x,-\ln(-t_{i}-\Lambda_{i}^{-2}\tau)).$

Therefore, if
$\frac{3}{2}\beta(-t_{i}-\Lambda_{i}^{-2}\tau)^{\frac{1}{2}+\sigma_{l}}\Lambda_{i}\leq
x\leq(-t_{i}\Lambda_{i}^{2}-\tau)^{\frac{1}{2}}$,
$\tau\geq\Lambda_{i}^{2}(-e^{-s_{1}}-t_{i})$,

$x^{m+2r}|\partial_{x}^{m}\partial_{\tau}^{r}u_{i}(x,\tau)|\leq
C(p,q,l,\Lambda,m,r)(-t_{i}-\Lambda_{i}^{-2}\tau)^{(1-\alpha)(\frac{1}{2}+\sigma_{l})}\Lambda_{i}^{1-\alpha}x^{\alpha}.$

Since
$(-t_{i})^{-\frac{1}{2}}\ll\Lambda_{i}\ll(-t_{i})^{-\frac{1}{2}-\sigma_{l}}$,
on any bounded $\tau$-interval,
$(-t_{i}-\Lambda_{i}^{-2}\tau)^{(1-\alpha)(\frac{1}{2}+\sigma_{l})}\Lambda_{i}^{1-\alpha}\rightarrow
0$ uniformly, and
$(-t_{i}-\Lambda_{i}^{-2}\tau)^{\frac{1}{2}+\sigma_{l}}\Lambda_{i}\rightarrow
0$, $(-t_{i}\Lambda_{i}^{2}-\tau)^{\frac{1}{2}}\rightarrow+\infty$ uniformly.
Also, $\Lambda_{i}^{2}(-e^{-s_{1}}-t_{i})\rightarrow-\infty$, which implies
for any $M>1$, $u_{i}(x,\tau)\rightarrow 0$ smoothly on
$[M^{-1},M]\times[-M,M]$, and the flow $\tilde{\Sigma}_{\tau}^{i}$ also
converges to $C_{p,q}$ in the desired way.

The next proposition bounds the mean curvature in the intermediate and tip
regions:

######  Proposition 8.4.

If $-\alpha<a<1-\alpha$, and

(8.4) $\lambda_{l}(1-\frac{a}{1-\alpha})-\frac{1}{2}\geq 0,$

then

$\sup_{\begin{subarray}{c}\mathbf{x}\in\Sigma_{t}\cap\bar{B}(0,\sqrt{-t})\\\
t_{0}\leq
t<0\end{subarray}}(1+(-t)^{-(\frac{1}{2}+\sigma_{l})}|\mathbf{x}|)^{a}|H_{\Sigma_{t}}(\mathbf{x})|<+\infty.$

Proof: Suppose our assertion is not true, then there exists a sequence
$t_{0}\leq T_{i}\nearrow 0$ s.t.

$M_{i}:=\sup_{\begin{subarray}{c}\mathbf{x}\in\Sigma_{t}\cap\bar{B}(0,\sqrt{-t})\\\
t_{0}\leq t\leq
T_{i}\end{subarray}}(1+(-t)^{-(\frac{1}{2}+\sigma_{l})}|\mathbf{x}|)^{a}|H_{\Sigma_{t}}(\mathbf{x})|\rightarrow+\infty,$

and thus we can find $t_{i}\in[t_{0},T_{i}]$ and
$\mathbf{x}_{i}\in\Sigma_{t_{i}}\cap\bar{B}(0,\sqrt{-t_{i}})$ s.t.

(8.5)
$(1+(-t_{i})^{-(\frac{1}{2}+\sigma_{l})}|\mathbf{x}_{i}|)^{a}|H_{\Sigma_{t_{i}}}(\mathbf{x}_{i})|=M_{i}.$

After passing to a subsequence, there are several possibilities of the
behabior of the sequence $(\mathbf{x}_{i},t_{i})$ as $i\rightarrow+\infty$:

  1. (1)

$|\mathbf{x}_{i}|=O((-t_{i})^{\frac{1}{2}+\sigma_{l}})$.

  2. (2)

$(-t_{i})^{\frac{1}{2}+\sigma_{l}}\ll|\mathbf{x}_{i}|\ll\sqrt{-t_{i}}$.

  3. (3)

$|\mathbf{x}_{i}|\sim\sqrt{-t_{i}}$ (i.e. $|\mathbf{x}_{i}|=O(\sqrt{-t_{i}})$
and $\sqrt{-t_{i}}=O(|\mathbf{x}_{i}|)$).

When (1) happens, define a sequence of rescaled flows as

$\tilde{\Sigma}_{\tau}^{i}:=\Lambda_{i}\Sigma_{t_{i}+\frac{\tau}{\Lambda_{i}^{2}}},\
(t_{0}-t_{i})\Lambda_{i}^{2}\leq\tau\leq 0,$

where $\Lambda_{i}=(-t_{i})^{-\frac{1}{2}-\sigma_{l}}$. Then

$M_{i}=\sup_{\begin{subarray}{c}\mathbf{x}\in\Sigma_{t}\cap\bar{B}(0,\sqrt{-t})\\\
t_{0}\leq t\leq
t_{i}\end{subarray}}(1+(-t)^{-(\frac{1}{2}+\sigma_{l})}|\mathbf{x}|)^{a}|H_{\Sigma_{t}}(\mathbf{x})|$
$=\sup_{\begin{subarray}{c}\mathbf{y}\in\tilde{\Sigma}_{\tau}^{i}\cap\bar{B}(0,\sqrt{-t_{i}\Lambda_{i}^{2}-\tau})\\\
(t_{0}-t_{i})\Lambda_{i}^{2}\leq\tau\leq
0\end{subarray}}(1+(-t_{i}-\Lambda_{i}^{-2}\tau)^{-(\frac{1}{2}+\sigma_{l})}\Lambda_{i}^{-1}|\mathbf{y}|)^{a}\Lambda_{i}|H_{\tilde{\Sigma}_{\tau}^{i}}(\mathbf{y})|.$

Now, define functions $u_{i}:\tilde{\Sigma}_{\tau}^{i}\rightarrow\mathbb{R}$
as

$u_{i}(\mathbf{y},\tau)=\frac{\Lambda_{i}}{M_{i}}H_{\tilde{\Sigma}_{\tau}^{i}}(\mathbf{y})$

(the sign of $H$ can be chosen, e.g. corresponding to the outer unit normal
vector). According to [eh], Lemma 1.1, (iv), along the mean curvature flow
(whose velocity equals the mean curvature), $u_{i}$ satisfies

$\partial_{\tau}u_{i}=\Delta_{\tilde{\Sigma}_{\tau}^{i}}u_{i}+|A_{\tilde{\Sigma}_{\tau}^{i}}|^{2}u_{i},$

and

$(1+(-t_{i}-\Lambda_{i}^{-2}\tau)^{-(\frac{1}{2}+\sigma_{l})}\Lambda_{i}^{-1}|\mathbf{y}|)^{a}|u_{i}(\mathbf{y},\tau)|\leq
1,\
\mathbf{y}\in\tilde{\Sigma}_{\tau}^{i}\cap\bar{B}(0,\sqrt{-t_{i}\Lambda_{i}^{2}-\tau}),\
(t_{0}-t_{i})\Lambda_{i}^{2}\leq\tau\leq 0,$
$(1+|\mathbf{y}_{i}|)^{a}|u_{i}(\mathbf{y}_{i},0)|=1(\mathbf{y}_{i}=\Lambda_{i}\mathbf{x}_{i}).$

Since $\Lambda_{i}^{-1}\ll\sqrt{-t_{i}}$, as $i\rightarrow+\infty$,
$(t_{0}-t_{i})\Lambda_{i}^{2}\rightarrow-\infty$,
$\sqrt{-t_{i}\Lambda_{i}^{2}-\tau}\rightarrow+\infty$,
$(-t_{i}-\Lambda_{i}^{-2}\tau)^{-(\frac{1}{2}+\sigma_{l})}\Lambda_{i}^{-1}\rightarrow
1$ uniformly on compact $\tau$-intervals, and in particular $\\{u_{i}\\}$ are
uniformly bounded in any bounded space-time region. By Lemma 8.2, the sequence
of flows $\\{\tilde{\Sigma}_{\tau}^{i}\\}$ converges to the stationary flow
$\mathcal{M}_{k}$ locally smoothly, and thus, using Schauder’s estimate, we
may assume (by passing to a subsequence if necessary) that $\\{u_{i}\\}$
converges locally smoothly to a smooth ancient solution $u_{\infty}$ to

$\partial_{\tau}u_{\infty}=\Delta_{\mathcal{M}_{k}}u_{\infty}+|A_{\mathcal{M}_{k}}|^{2}u_{\infty},$

defined on $\mathcal{M}_{k}\times(-\infty,0]$, satisfying

(8.6) $|u_{\infty}(\mathbf{y},\tau)|\leq(1+|\mathbf{y}|)^{-a}$

on this domain. Since $\mathbf{y}_{i}=\Lambda_{i}\mathbf{x}_{i}$ are uniformly
bounded, we may also assume
$\mathbf{y}_{i}\rightarrow\mathbf{y}_{\infty}\in\mathcal{M}_{k}$, with

$|u_{\infty}(\mathbf{y}_{\infty},0)|=(1+|\mathbf{y}_{\infty}|)^{-a}>0.$

But (8.6) and Theorem 7.1 imply $u_{\infty}\equiv 0$ (note that $-a<\alpha$),
which is impossible.
When (2) happens, define a sequence of rescaled flows as

$\tilde{\Sigma}_{\tau}^{i}:=|\mathbf{x}_{i}|^{-1}\Sigma_{t_{i}+|\mathbf{x}_{i}|^{2}\tau},\
(t_{0}-t_{i})|\mathbf{x}_{i}|^{-2}\leq\tau\leq 0.$

Then

$M_{i}=\sup_{\begin{subarray}{c}\mathbf{x}\in\Sigma_{t}\cap\bar{B}(0,\sqrt{-t})\\\
t_{0}\leq t\leq
t_{i}\end{subarray}}(1+(-t)^{-(\frac{1}{2}+\sigma_{l})}|\mathbf{x}|)^{a}|H_{\Sigma_{t}}(\mathbf{x})|$
$=\sup_{\begin{subarray}{c}\mathbf{y}\in\tilde{\Sigma}_{\tau}^{i}\cap\bar{B}(0,\sqrt{-t_{i}|\mathbf{x}_{i}|^{-2}-\tau})\\\
(t_{0}-t_{i})|\mathbf{x}_{i}|^{-2}\leq\tau\leq
0\end{subarray}}(1+(-t_{i}-|\mathbf{x}_{i}|^{2}\tau)^{-(\frac{1}{2}+\sigma_{l})}|\mathbf{x}_{i}|\
|\mathbf{y}|)^{a}|\mathbf{x}_{i}|^{-1}|H_{\tilde{\Sigma}_{\tau}^{i}}(\mathbf{y})|$
$\geq\sup_{\begin{subarray}{c}\mathbf{y}\in\tilde{\Sigma}_{\tau}^{i}\cap\bar{B}(0,\sqrt{-t_{i}|\mathbf{x}_{i}|^{-2}-\tau})\\\
(t_{0}-t_{i})|\mathbf{x}_{i}|^{-2}\leq\tau\leq
0\end{subarray}}(\Lambda_{i}^{-1}(-t_{i}-|\mathbf{x}_{i}|^{2}\tau)^{-(\frac{1}{2}+\sigma_{l})})^{a}\Lambda_{i}^{a}|\mathbf{x}_{i}|^{a-1}|\mathbf{y}|^{a}|H_{\tilde{\Sigma}_{\tau}^{i}}(\mathbf{y})|,$

where $\Lambda_{i}=(-t_{i})^{-\frac{1}{2}-\sigma_{l}}$. Now, define functions
$u_{i}:\tilde{\Sigma}_{\tau}^{i}\rightarrow\mathbb{R}$ as

$u_{i}(\mathbf{y},\tau)=\frac{\Lambda_{i}^{a}|\mathbf{x}_{i}|^{a-1}}{M_{i}}H_{\tilde{\Sigma}_{\tau}^{i}}(\mathbf{y}).$

Again, along the mean curvature flow (whose velocity equals the mean
curvature), $u_{i}$ satisfies

$\partial_{\tau}u_{i}=\Delta_{\tilde{\Sigma}_{\tau}^{i}}u_{i}+|A_{\tilde{\Sigma}_{\tau}^{i}}|^{2}u_{i},$

and

$(\Lambda_{i}^{-1}(-t_{i}-|\mathbf{x}_{i}|^{2}\tau)^{-(\frac{1}{2}+\sigma_{l})})^{a}|\mathbf{y}|^{a}|u_{i}(\mathbf{y},\tau)|\leq
1,\
\mathbf{y}\in\tilde{\Sigma}_{\tau}^{i}\cap\bar{B}(0,\sqrt{-t_{i}|\mathbf{x}_{i}|^{-2}-\tau}),\
(t_{0}-t_{i})|\mathbf{x}_{i}|^{-2}\leq\tau\leq 0,$
$(1+|\mathbf{x}_{i}|^{-1}\Lambda_{i}^{-1})^{a}|u_{i}(\mathbf{y}_{i},0)|=1(\mathbf{y}_{i}=|\mathbf{x}_{i}|^{-1}\mathbf{x}_{i}).$

Since $\Lambda_{i}^{-1}\ll|\mathbf{x}_{i}|\ll\sqrt{-t_{i}}$, as
$i\rightarrow+\infty$, $(t_{0}-t_{i})|\mathbf{x}_{i}|^{-2}\rightarrow-\infty$,
$(1+|\mathbf{x}_{i}|^{-1}\Lambda_{i}^{-1})^{a}\rightarrow 1$,
$\sqrt{-t_{i}|\mathbf{x}_{i}|^{-2}-\tau}\rightarrow+\infty$,
$(\Lambda_{i}^{-1}(-t_{i}-|\mathbf{x}_{i}|^{2}\tau)^{-(\frac{1}{2}+\sigma_{l})})^{a}\rightarrow
1$ uniformly on compact $\tau$-intervals, and in particular $\\{u_{i}\\}$ are
uniformly bounded in any compact space-time region in
$(\mathbb{R}^{n}-\\{0\\})\times(-\infty,0]$. By Lemma 8.3, the sequence of
flows $\\{\tilde{\Sigma}_{\tau}^{i}\\}$ converges to the stationary flow
$C_{p,q}$ locally smoothly, and thus, using Schauder’s estimate, we may assume
(by passing to a subsequence if necessary) that $\\{u_{i}\\}$ converges
locally smoothly to a smooth ancient solution $u_{\infty}$ to

$\partial_{\tau}u_{\infty}=\Delta_{C_{p,q}}u_{\infty}+|A_{C_{p,q}}|^{2}u_{\infty},$

defined on $(C_{p,q}-\\{0\\})\times(-\infty,0]$, satisfying

(8.7) $|u_{\infty}(\mathbf{y},\tau)|\leq|\mathbf{y}|^{-a}$

on this domain. Since $u_{i},u_{\infty}$ are radially symmetric, for all
$\mathbf{y}_{\infty}\in C_{p,q}$, $|\mathbf{y}_{\infty}|=1$,

$|u_{\infty}(\mathbf{y}_{\infty},0)|=1>0.$

But (8.7) and Theorem 7.2 imply $u_{\infty}\equiv 0$ (note that
$\alpha-(n-3+2\alpha)\leq\alpha-1<-a<\alpha$), which is impossible.
When (3) happens, then there exists $0<\epsilon<1$ s.t. for all $i$ large
enough, $\mathbf{x}_{i}=(\frac{x_{i}-\mu
u(x_{i},t_{i})}{\sqrt{1+\mu^{2}}}\omega_{i},\frac{\mu
x_{i}+u(x_{i},t_{i})}{\sqrt{1+\mu^{2}}}\phi_{i})$, with
$\omega_{i}\in\mathbb{S}^{p-1}$, $\phi_{i}\in\mathbb{S}^{q-1}$,
$\epsilon\sqrt{-t_{i}}\leq x_{i}\leq\sqrt{-t_{i}}$. By the formula (8.1) and
the estimate (4.10),

$|H_{\Sigma_{t_{i}}}(\mathbf{x}_{i})|\leq
C(-t_{i})^{\lambda_{l}-\frac{1}{2}},$

where $C>0$ is independent of $i$. Thus,

$(1+(-t_{i})^{-(\frac{1}{2}+\sigma_{l})}|\mathbf{x}_{i}|)^{a}|H_{\Sigma_{t_{i}}}(\mathbf{x}_{i})|\leq
C(-t_{i})^{\lambda_{l}-\frac{1}{2}-a\sigma_{l}}\leq C<+\infty,$

since $\lambda_{l}-\frac{1}{2}-a\sigma_{l}\geq 0$, which contradicts to (8.5).
Because all the possible cases are impossible, our assertion follows.

######  Corollary 8.5.

If $n\geq 9$, $l\geq 3$, or $n=8$, $l\geq 4$, then

$\sup_{\begin{subarray}{c}\mathbf{x}\in\Sigma_{t}\cap\bar{B}(0,\sqrt{-t})\\\
t_{0}\leq t<0\end{subarray}}|H_{\Sigma_{t}}(\mathbf{x})|<+\infty.$

Proof: The condition (8.4) is equivalent to

$a\leq(1-\alpha)(1-\frac{1}{2\lambda_{l}}).$

Since we also require $-\alpha<a<1-\alpha$, such $a$ exists iff

$-\alpha<(1-\alpha)(1-\frac{1}{2\lambda_{l}})\Leftrightarrow\lambda_{l}>\frac{1-\alpha}{2}\Leftrightarrow
l>1-\alpha,$

which is true iff $n\geq 9$, $l\geq 3$, or $n=8$, $l\geq 4$. Choosing $a$
satisfying the conditions above and noting
$(1+(-t)^{-(\frac{1}{2}+\sigma_{l})}|\mathbf{x}|)^{a}\geq 1$, this corollary
is a direct consequence of Proposition 8.4.

The main Theorem 1.4 follows from Theorem 4.5, Proposition 8.1, and Corollary
8.5.

######  Remark 8.6.

Using the blow up argument in Proposition 8.4, one can prove for any
$\epsilon>0$,

$\sup_{\mathbf{x}\in\Sigma_{t},t_{0}\leq
t<0}(-t)^{\frac{1}{2}-\sigma_{l}+\epsilon}|H_{\Sigma_{t}}(\mathbf{x})|<+\infty$

if $l=2$, or

$\sup_{\mathbf{x}\in\Sigma_{t},t_{0}\leq
t<0}(-t)^{\epsilon}|H_{\Sigma_{t}}(\mathbf{x})|<+\infty$

if $n=8$, $l=3$. Actually, in order to derive a contradiction, the assertion
in Proposition 8.4 should be

$\sup_{\begin{subarray}{c}\mathbf{x}\in\Sigma_{t}\cap\bar{B}(0,\sqrt{-t})\\\
t_{0}\leq
t<0\end{subarray}}(-t)^{\frac{1}{2}-\sigma_{l}+\epsilon}(1+(-t)^{-(\frac{1}{2}+\sigma_{l})}|\mathbf{x}|)^{a}|H_{\Sigma_{t}}(\mathbf{x})|<+\infty$

with $-\alpha<a\leq(1-\alpha)(1-\lambda_{l}^{-1}(\sigma_{l}-\epsilon))$ if
$l=2$, or

$\sup_{\begin{subarray}{c}\mathbf{x}\in\Sigma_{t}\cap\bar{B}(0,\sqrt{-t})\\\
t_{0}\leq
t<0\end{subarray}}(-t)^{\epsilon}(1+(-t)^{-(\frac{1}{2}+\sigma_{l})}|\mathbf{x}|)^{a}|H_{\Sigma_{t}}(\mathbf{x})|<+\infty$

with $-\alpha<a\leq(1-\alpha)(1-\lambda_{l}^{-1}(\frac{1}{2}-\epsilon))$ if
$n=8$, $l=3$.

## Declarations

  * •

Conflict of interest: The author declares that he has no conflict of interest.

  * •

Data availability: No datasets were generated or analysed during the current
study.

## References


# Vis2Mus: Exploring Multimodal Representation Mapping for Controllable Music
Generation

###### Abstract

In this study, we explore the representation mapping from the domain of visual
arts to the domain of music, with which we can use visual arts as an effective
handle to control music generation. Unlike most studies in multimodal
representation learning that are purely data-driven, we adopt an analysis-by-
synthesis approach that combines deep music representation learning with user
studies. Such an approach enables us to discover interpretable representation
mapping without a huge amount of paired data. In particular, we discover that
visual-to-music mapping, for many people, has a nice property similar to
equivariant. In other words, we can use various image transformations, say,
changing brightness, changing contrast, style transfer, to control the
corresponding transformations in the music domain. In addition, we released
the Vis2Mus system as a controllable interface for symbolic music generation.
111Github Repo: https://github.com/ldzhangyx/vis2mus.

Index Terms—  Multimodal representation learning, Controllable music
generation

##  1 Introduction

Multimodal representation learning aims to bridge the heterogeneity gap
between different modalities [guo2019deep]. Recently, multimodal
representation learning has attracted widespread interest in applications such
as cross-modal retrieval [radford2021learning, wang2019camp], and cross-modal
generation [ramesh2021zero, hossain2019comprehensive]. For audio, some deep
learning-based work has progressed on tasks such as audio captioning
[chen2020audio], speech synthesis [valle2020flowtron], and music description
[manco2021muscaps, zhang2020butter].

In this paper, we follow this research path and aim to explore the
representation mapping from image to music. We believe that a better
understanding of visual-to-music mapping will not only shed light on the
theories of machine learning (especially transfer learning) and cognitive
science (especially synesthesia) but also have great practical values on music
information retrieval and controllable music generation. For most people,
music concepts, such as chords and texture, are quite abstract and it is
difficult to use these concepts to directly control music generation. On the
contrary, images are more “concrete” and can be used as an intuitive handle to
preview and guide music generation once a multimodal mappings is established.

Due to insufficient available paired image-music data, existing data-driven
approaches [tan2020automated, rivas2020pix2pitch] have difficulty in learning
a mapping directly from data and generate high-quality music from images. To
address this problem, we resort toanalysis-by-synthesis methodology [abs] and
develop a method that combines representation learning with user subjective
evaluations. In specific, we i) propose several transformations that can be
applied to both music and image representations, ii) synthesize image-music
pairs data using deep generative models by applying the proposed
transformations, and iii) conduct user studies to evaluate the synthesized
pairs in order to explore the properties of visual-to-image mappings.

We find that for many people, visual-to-music mapping has a nice property
analogous to equivariant, as shown in Figure 1. This property enables us to
use various image transformations (including changing brightness, changing
contrast and style transfer) to control the corresponding transformations in
the music domain. In addition, we release the Vis2Mus system, which applies
our approach to a large amount of paired image and music as a controllable
interface for symbolic music generation.

Fig. 1: The diagram shows the property of the mapping f between the visual
space and the music space, where g is a transformation can be applied to both
visual and music domains.

In summary, our contributions are:

  1. 1.

A data-efficient analysis-by-synthesis approach that combines deep
representation learning and human-computer interaction, which potentially can
be generalized to other multimodal tasks.

  2. 2.

We discovered an important property of visual-to-music mapping, which is
analogous to equivariant, between interpretable image and music
representations.

  3. 3.

We release the Vis2Mus system, a multimodal interface for controllable music
generation.

##  2 Related Work

Related research on image-to-music cross-modal generation is relatively
limited. As an early work, Wu et al. [wu2008study] correlate low-level
features of images with low-level features of music and generate music using a
rule-based approach. The main shortcoming of this early work is that it does
not take full advantage of advances in machine learning, resulting in limited
control flexibility. Tan et. al. [tan2020automated] propose an end-to-end
method with an ResNet as the image encoder and an RNN as the music decoder to
generate monophonic symbolic music. Ruzafa et al. [rivas2020pix2pitch] use a
U-Net to encode images and construct a GAN-based decoder to generate a mel
spectrogram corresponding to the music. These two data-driven methods cannot
yet generate high-quality music mainly because the scale of the current image-
music paired dataset is not sufficient for training. Also, they do not
explicitly model the image-music feature correspondence, which leads to low
model interpretability.

Our proposed Vis2Mus system takes into account the advantages of both types of
methods by combining deep music generation with user studies. This new
research paradigm enables us to discover interpretable representation mapping
without a huge amount of paired data.

##  3 Methodology

Given an image, the problem of “what is the ideal corresponding music” may
seem too difficult or intractable, and different people probably have
different answers. In this paper, we aim to solve a conditioned simplification
of the problem which is more accessible, in which we: 1) first assume an
image-music pair, 2) then modify the image by changing some features, and 3)
finally ask “how we should change the music accordingly to match the modified
image.”

Such problem setup is also graphically represented in Figure 2, in which we
can regard the given (conditioned) image-music pair as an “anchor” to control
or judge any modification or variation based on it. Formally, let $v$ denote
the visual input, $m$ denote the music output, and function $f$ be the
conceptual visual-to-music mapping. Our problem setting is that given a
specific pair $m=f(v)$ and a certain transformation $g$ such that
$v^{\prime}=g(v)$, what is the corresponding music $m’$, i.e., $f(v’)$?

Fig. 2: An illustration of problem setup: (a) unconditioned multimodal mapping
Vs. (b) conditional multimodal mapping. Our study focuses on the scenario of
(b)

We develop an analysis-by-synthesis method that combines deep music generation
with user studies. Inspired by Kandinsky [kandinsky2012concerning], our
hypothesis is that $f$ has an isomorphic property with respect to a certain
transformations that can be applied in both visual and music domains. In other
words, if $v^{\prime}=g(v)$ and $m^{\prime}=g(m)$, we can infer $m’$ without
knowing the exact form of $f$ given the pair $(v,m)$.

Under this hypothesis, we propose several concrete forms of $g$, then
synthesize the corresponding $m’$ via deep music generation, and finally
conduct user studies to see whether users can distinguish $m’$ from other
candidates. We introduce the deep generative model and how to apply $g$ on its
latent representation in Section 3.1. Then, we discuss several particular
forms of $g$ from Section 3.2 to Section 3.3, followed which we present the
user study and experimental results in Section 3.4 & Section 3.5.

###  3.1 The backend model

We build our backend model based on a state-of-the-art polyphonic
representation learning model [wang2020learning], which learns disentangled
chord and texture representations. Note that “texture”(as in English) can
refer to both visual and musical features, and our hypothetical cross-modal
transformation $g$ acts on the latent texture feature.

Fig. 3: An illustration of the polyphonic disentanglement model and how to
apply cross-modal transformation $g$ to the latent texture feature.

大改中！

Figure 3 shows the detailed model structure. For image (upper half of the
figure), we apply a transformation $g$ to the image pixels;

For music, the pre-trained polyphonic representation learning model (bottom
half of the figure) uses two separate encoders (the texture encoder and the
chord encoder) to learn two latent representations respectively: texture
vector $h_{\text{txt}}$ and chord vector $h_{\text{chd}}$. We apply a
transformation $g$ to the texture representation vector $h_{\text{txt}}$ to
manipulate the music representation in the latent space. From the changed
representation vectors, the model then uses a PianoTree decoder
[wang2020pianotree] to reconstruct the music. In specific, the texture encoder
takes a quasi-piano-roll input and then feeds it into a CNN layer followed by
a bi-directional GRU, and the chord encoder uses a bi-directional GRU. (For a
more detailed architecture, please refer to the original paper
[wang2020learning])

The rationale of such design is that the music feature map has an image-like
format, making it easy to perform various image transformations, say, changing
brightness and contrast.

###  3.2 Brightness and Contrast Transformation

####  3.2.1 Brightness Transformation

The brightness transformation $g_{b}$ can be regarded as a pixel-wise
function. Let $x$ be any channel (a matrix) of the $z_{m}$ or $v$, then for
every element (the pixel value) of $x_{i}$, its brightness transformation is
defined as:

$x_{i}^{\prime}=g_{b}\cdot x_{i}=x_{i}+\alpha_{b}/2\cdot\max(x)$ (1)

where $\alpha_{b}$ is a hyperparameter.

Figure 4 shows an example of synthesizing a new image-music pair $(v’,m’)$
based on an existing pair $(v,m)$ using brightness transformation.

Fig. 4: An illustration of applying cross-modal brightness transformation $g$
on image $v$ and music $m$, respectively, in which $(v,m)$ is a matched image-
music pair.

####  3.2.2 Contrast Transformation

Similarly to brightness transformation, the contrast transformation $g_{c}$ is
defined as:

$x_{i}^{\prime}=g_{c}\cdot
x_{i}=\overline{x}+(x_{i}-\overline{x})\cdot(1+\alpha_{c})$ (2)

where $\alpha_{c}$ is a hyperparameter.

Figure 5 shows an example of synthesizing a new image-music pair $(v’,m’)$
based on an existing pair $(v,m)$ using contrast transformation.

Fig. 5: An illustration of applying cross-modal contrast transformation $g$ on
image $v$ and music $m$, respectively, in which $(v,m)$ is a matched image-
music pair.

###  3.3 Style Transfer

Besides changing brightness and contrast, we also consider style transfer as a
form of transformation. In general, style transfer is achieved by fusing a
content element and a style element. In the image domain, people usually
regard image contour as content and texture as style [jing2019neural].
Correspondingly, we regard melody contour as content and polyphonic texture as
style in the music domain.

Formally, let $z=(z_{c},z_{\text{txt}})$ be the representation of an image or
a piece of music, where $z_{c}$ and $z_{\text{txt}}$ represent the
corresponding content and texture, respectively. For a pair of representations
of images or pieces of music $(z_{1},z_{2})$, the cross-modal style transfer
$g_{s}$ is defined as:

$(z_{1}^{\prime},z_{2}^{\prime})=g_{s}\cdot(z_{1},z_{2})=(z_{1,c},z_{2,txt},z_{2,c},z_{1,txt})$
(3)

Fig. 6: An illustration of applying cross-modal style transformation g on the
image group $[v_{1},v_{2}]$ and the music group $[m_{1},m_{2}]$, respectively,
in which $(v_{1},m_{1})$ and $(v_{2},m_{2})$ are two matched image-music
pairs.

Figure 6 shows an example of synthesizing a new image-music pair $(v’,m’)$
based on $([v_{1},v_{2}],[m_{1},m_{2}])$ using style transfer.

Here, we used the image style transfer algorithm from [zhang2017multistyle]
and music compositional style transfer method used in [zhao2021accomontage].
The compositional style transfer algorithm first adjusts the pitch register
and the tempo of the melody and accompaniment, respectively, to let them match
each other. Then, it applies chord style transfer [wang2020learning] to the
accompaniment by substituting the latent representation of original chords
with the latent representation of melody’s chords.

###  3.4 User Study Design

The user study contains three parts: 1) brightness transformation test, 2)
contrast transformation test, and 3) style transfer test. All parts are
repeated multiple times for each subject.

The first two parts of the experiment share the same three steps:

  1. 1.

Preview: The user is asked to view and listen to an image-music pair;

  2. 2.

Transformation: The image is transformed in terms of brightness or contrast,
while the music is altered both by the same transformation and the inverse
transformation, producing two new pieces of music. E.g. if the image is
transformed to be brighter, the inverse transformation means to make the
latent representation of the paired music dimmer;

  3. 3.

Test: The user is asked to select which one of the two new pieces better
matches the transformed image.

The third part of the experiment also has 3 steps:

  1. 1.

Preview:The user is asked to view and listen to four image-music pairs: two
sketch-melody pairs and two image-accompaniment pairs.

  2. 2.

Transformation: A synthesized image is shown, which is transformed from a
random sketch as the content and a random image as the style. At the same
time, four pieces of synthetic music are displayed, generated by all possible
combinations of the two melodies and the two accompaniments shown in the first
step using compositional style transfer.

  3. 3.

Test: The user is asked to select one synthetic music from the four that
perceptually best matches the synthesized image.

###  3.5 Experimental Results

Table 1 shows the detailed test accuracy for the three parts of the user
study, which are all significantly higher than baseline random guesses. For
the first part, a total of 12 participants were surveyed and 24 responses were
collected; for the second part, a total of 11 participants were surveyed and
22 responses were collected; for the third part, a total of 14 participants
were surveyed and 56 responses were collected.

Acc (%) | Brightness | Contrast | Style | Overall
---|---|---|---|---
Random | 50.0 | 50.0 | 50.0 | 25.0
Ours | 62.5∗ | 68.2∗ | 62.5∗ | 63.8∗
Table 1: Evaluation results of the transformation experiments. $\ast$: The
accuracy outperforms random guesses with $p<0.05$ on binomial test.

This result indicates that given an image-music pair $(m,v)$ and a transformed
image $v^{\prime}$ with respect to brightness, contrast or style, humans can
tell the corresponding music $m^{\prime}$ that is “brighter”/“dimmer”, with
more/less “contrast”, or style transformed without any training. The
commutative cross-modal transformations do exist, though not for everyone, yet
statistically significant. Such property also enables us to use image
transformations to control corresponding music transformations.

##  4 Vis2Mus: The User Interface

Based on the visual-to-music mapping, we developed a Vis2Mus system as a
controllable music generation interface. The interface enables a user to: 1)
select a sketch and retrieve the corresponding melody, 2) control music style
transfer (melody and accompaniment fusion) using image style transfer (sketch
and image fusion), and 3) change the brightness and contrast of the style-
transferred image to control the corresponding music segment.

Fig. 7: An overview of our Vis2Mus system.

As shown in Figure 7, the interface is divided into several parts. The input
image panel \raisebox{-0.9pt}{1}⃝ shows the current sketch and image; the
input music panel \raisebox{-0.9pt}{2}⃝ shows the current melody and texture;
\raisebox{-0.9pt}{5}⃝ is the selection panel that allows users to change the
input. The current interface allows user to retrieve and select among more
sketches to control melody. Once the input music and images are set, user can
use the button in stylize panel \raisebox{-0.9pt}{6}⃝ to generate a new
stylized image in the output image panel \raisebox{-0.9pt}{3}⃝ and the fused
new music in the output music panel \raisebox{-0.9pt}{4}⃝. Besides, user can
also use using the sliders in \raisebox{-0.9pt}{7}⃝ to change the brightness
and contrast of the style-transferred image and control the corresponding
music.

##  5 Conclusion and Future Work

In this study, we have contributed a novel analysis-by-synthesis method, which
combines deep generative modeling and user study to explore interpretable
visual-to-music mapping in a data effecient way. We discovered that different
people have different perferred latent representations for such multimodal
mapping, but their perferences are quite consistent and leads to predictable,
ranking preserved image-music pairs. Inspired by the discoveries, we have also
developed Vis2Mus, a controllable music generation interface using images and
image transformations as the control handlers.

We are well-awared that we have just explored a small portion of visual-to-
music mapping. In the future, we plan to study more forms of cross-modal
transformation acting on more possible latent factors. We also plan to conduct
a much larger scale user study, from which more interesting multimodal
patterns can potentially be discovered. We wish our study not only contribute
to visual-to-music mapping but also shed lights on multimodal learning in
other domains as well as cognitive science, especially studies on synesthesia
using data-driven methods.


# The hidden connection between Twisted Bilayer Graphene and the Quantum Hall
effect: Zero flat band electronic modes converge into coherent Landau level
states and its resultant 3/2 magic-angle quantization rule

Leonardo A. Navarro-Labastida and Gerardo G. Naumis<EMAIL_ADDRESS>Depto. de Sistemas Complejos, Instituto de Física,
Universidad Nacional Autónoma de México (UNAM)
Apdo. Postal 20-364, 01000, CDMX, México.

(October 2022)

###### Abstract

Flat band electronic modes in twisted graphene bilayers are responsible for
superconducting and other highly correlated electron-electron phases. Although
some hints were known of a possible connection between the quantum Hall effect
and zero flat band modes, it was not clear how such connection appears. Here
the electronic behavior in twisted bilayer graphene is studied using the
chiral model Hamiltonian. As a result it is proved that for high-order magic
angles, the zero flat band modes converge into coherent Landau states with a
dispersion $\sigma^{2}=1/3\alpha$, where $\alpha$ is a coupling parameter that
incorporates the twist angle and energetic scales. Then it is proved that the
square of the hamiltonian, which is a $2\times 2$ matrix operator, turns out
to be equivalent to a two-dimensional quantum harmonic oscillator. The
interlayer currents between graphene’s bipartite lattices are identified with
the angular momentum term while the confinement potential is an effective
quadratic potential. From there it is proved a limiting quantization rule for
high-order magic angles, i.e., $\alpha_{m+1}-\alpha_{m}=(3/2)m$ where $m$ is
the order of the angle. All these results are in very good agreement with
numerical calculations.

††preprint: APS/123-QED

##  I Introduction

In 2018 it was found experimentally that twisted bilayer graphene (TBG)
presents strongly correlated electron-electron quantum phases leading for
example to unconventional superconductivity and Mott insulator states [1].
More recently, trilayer twisted graphene has been found to be the most
strongly interacting correlated material [2, 3]. Such remarkable discoveries
presented a new paradigm in the so-called Moiré materials and unveiled the
importance of two-dimensional (2D) materials to understand unconventional
superconductivity in cuprates and heavy fermions systems, as they share
similar quantum phase diagrams [1, 2, 4]. TBG advantages are i) its
simplicity, as they are made from a single chemical element, and ii) they have
a high degree of manipulation that cuprates doesn’t have. In recent years,
there has been a significant interest in these phases of matter from a
fundamental point of view [5, 6, 7, 8, 9, 10, 11, 12, 13] but also because
they present a lot of possible electronic applications and quantum computing
advantages [14, 13]. There is also an interest connection between topological
phases, edge states, semimetals and fractional quantum Hall effect (FQHE) [15,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25]. A recently paper establishes a
connection between heavy fermion models and TBG [4], opening the prospect of
using heavy fermions physics to the superconducting physics of TBG and more
strongly correlated phases.

The discovery of such phases was proceeded by the Bistritzer-Mac Donald (BM)
theoretical observation that twisted bilayer graphene (TBG) develops flat
bands at certain twisting angles which are called magic [26]. BM considered a
continuum Dirac model in which the Moiré periodicity between layers produce
Moiré Bloch bands [26]. The model is continuum in the sense that the
interlayer potential between Carbon $\pi$ orbitals is a smooth function of the
spacial separation projected onto the graphene planes and also the hopping is
local and periodic, allowing to apply the Bloch´s theorem for any rotation
angle. For TBG it was demonstrated that non-Abelian gauge fields arise due to
the coupling between layers in the low-energy regime [27, 28].

Flat band modes which arise at magic angle, also known as zero energy modes,
have been investigated in many recent works [29, 30, 31, 32, 33, 34, 35, 36,
37, 38], and in particular, there were hints in the mathematics for a possible
connection with the quantum Hall effect (QHE) and the lowest Landau level [29,
32]. There are interesting properties of the zero mode wave function [32, 33,
34], in particular, the connection with the lowest Landau level reveals that
TBG presents topological phases [32, 39].

Importantly, the wave function is reminiscent of a quantum hall wave function
because is described in terms of Jacobi theta functions such as in the quantum
hall effect wave function [29, 30, 31]. This hidden wave function is important
to understand because leads to particular localization properties, orbital
current, density wave function distribution and symmetries of the pseudo-
magnetic gauge fields. Yet, exactly how this analogy arises was not clear, as
no connection between the quantum harmonic oscillator and the TBG hamiltonian
was ever found. Tarnopolsky et. al. also found that magic angles were
quantized but no explanation was provided for this fact [29]. Here we solve
both problems and moreover, we find that in fact the zero flat band modes
converge into coherent Landau levels. As we will discuss, this is done by
using boundary layer differential equations theory and squaring the
Hamiltonian [28, 40, 41, 42, 43].

##  II Chiral TBG and Squared TBG Hamiltonians

The chiral Hamiltonian of twisted bilayer graphene is a variant of the
original Bistritzer-MacDonald Hamiltonian in which the $AA$ tunneling is set
to zero [31]. Here we use as basis the wave vectors
$\Phi(\bm{r})=\begin{pmatrix}\psi_{1}(\bm{r}),\psi_{2}(\bm{r}),\chi_{1}(\bm{r}),\chi_{2}(\bm{r})\end{pmatrix}^{T}$
where the index $1,2$ represents each graphene layer and $\psi_{j}(\bm{r})$
and $\chi_{j}(\bm{r})$ are the Wannier orbitals on each inequivalent site of
the graphene’s unit cell. The chiral Hamiltonian is given [29, 44, 30],

$\begin{split}\mathcal{H}&=\begin{pmatrix}0&D^{\ast}(-\bm{r})\\\
D(\bm{r})&0\end{pmatrix}\\\ \end{split}$ (1)

where the zero-mode operator is defined as,

$\begin{split}D(\bm{r})&=\begin{pmatrix}-i\bar{\partial}&\alpha U(\bm{r})\\\
\alpha U(-\bm{r})&-i\bar{\partial}\end{pmatrix}\\\ \end{split}$ (2)

and,

$\begin{split}D^{*}(-\bm{r})&=\begin{pmatrix}-i\partial&\alpha
U^{*}(-\bm{r})\\\ \alpha U^{*}(\bm{r})&-i\partial\end{pmatrix}\\\ \end{split}$
(3)

with $\bar{\partial}=\partial_{x}+i\partial_{y}$,
$\partial=\partial_{x}-i\partial_{y}$. The potential is,

$U(\bm{\bm{r}})=e^{-i\bm{q}_{1}\cdot\bm{r}}+e^{i\phi}e^{-i\bm{q}_{2}\cdot\bm{r}}+e^{-i\phi}e^{-i\bm{q}_{3}\cdot\bm{r}}$
(4)

where the phase factor $\phi=2\pi/3$ and the Moiré lattice vectors are given
by $\bm{q}_{1}=k_{\theta}(0,-1)$,
$\bm{q}_{2}=k_{\theta}(\frac{\sqrt{3}}{2},\frac{1}{2})$,
$\bm{q}_{3}=k_{\theta}(-\frac{\sqrt{3}}{2},\frac{1}{2})$, the Moiré modulation
vector is $k_{\theta}=2k_{D}\sin{\frac{\theta}{2}}$ with
$k_{D}=\frac{4\pi}{3a_{0}}$ is the magnitude of the Dirac wave vector and
$a_{0}$ is the lattice constant of monolayer graphene. The model contains only
the parameter $\alpha$, defined as $\alpha=\frac{w_{1}}{v_{0}k_{\theta}}$
where $w_{1}$ is the interlayer coupling of stacking AB/BA with value
$w_{1}=110$ meV and $v_{0}$ is the Fermi velocity with value
$v_{0}=\frac{19.81eV}{2k_{D}}$. The operators $\partial$ and $\bar{\partial}$
are dimensionless as the Hamiltonian Eq. (1) is written in using units where
$v_{0}=1$, $k_{\theta}=1$. The twist angle only enters in the dimensionless
parameter $\alpha$. The combinations $\bm{b}_{1,2}=\bm{q}_{2,3}-\bm{q}_{1}$
are the Moiré Brillouin zone (mBZ) vectors and also
$\bm{b}_{3}=\bm{q}_{3}-\bm{q}_{2}$. Using this basis for the reciprocal space
lattice, some important high symmetry points of the Moiré Brillouin zone are
$\bm{K}=(0,0)$, $\bm{K^{\prime}}=-\bm{q}_{1}$, and $\bm{\Gamma}=\bm{q}_{1}$
(see ref. [40] for a diagram). For further use it is also convenient to define
a set of unitary vectors $\bm{\hat{q}}_{\mu}^{\perp}$ perpendicular to the set
$\bm{q}_{\mu}$ and given by
$\bm{\hat{q}}_{1}^{\perp}=(1,0),\bm{\hat{q}}_{2}^{\perp}=\big{(}-\frac{1}{2},\frac{\sqrt{3}}{2}\big{)},\bm{\hat{q}}_{3}^{\perp}=\big{(}-\frac{1}{2},-\frac{\sqrt{3}}{2}\big{)}$.

In a previous work we showed how, by taking the square of $H$, it is possible
to write the Hamiltonian as a $2\times 2$ matrix [28, 40],

$\begin{split}H^{2}&=\begin{pmatrix}-\nabla^{2}+\alpha^{2}|U(-\bm{r})|^{2}&\alpha
A^{\dagger}(\bm{r})\\\ \alpha
A(\bm{r})&-\nabla^{2}+\alpha^{2}|U(\bm{r})|^{2}\end{pmatrix}\end{split}$ (5)

where the squared norm of the potential is an effective trigonal confinement
potential,

$\begin{split}|U(\bm{r})|^{2}&=3+2\cos(\bm{b}_{1}\cdot\bm{r}-\phi)+2\cos(\bm{b}_{2}\cdot\bm{r}+\phi)\\\
&+2\cos(\bm{b}_{3}\cdot\bm{r}+2\phi)\end{split}$ (6)

and the off-diagonal term is,

$\begin{split}A^{\dagger}(\bm{r})&=-i\sum_{\mu=1}^{3}e^{-i\bm{q}_{\mu}\cdot\bm{r}}(2\bm{\hat{q}}_{\mu}^{\perp}\cdot\bm{\nabla}+1)\\\
\end{split}$ (7)

where $\bm{\nabla}^{\dagger}=-\bm{\nabla}$ with
$\bm{\nabla}=(\partial_{x},\partial_{y})$ and $\mu=1,2,3$.

##  III Zero-energy modes as coherent Landau states

Now we investigate the asymptotic limit $\alpha\rightarrow\infty$ by
numerically solving (see appendix) the Schrödinger equation
$\mathcal{H}\Psi(\bm{r})=E\Psi(\bm{r})$ where $E$ is the energy. As the
potential is periodic, it satisfies Bolch’s theorem and thus
$\psi_{\bm{k},j}(\bm{r})=e^{i\bm{k}\cdot\bm{r}}u_{\bm{k},j}(\bm{r})$ where
$u_{\bm{k},j}(\bm{r})$ has the periodicity of the lattice (see appendix). In
Fig. 1 we present the zero mode wave function, corresponding to $E=0$ at the
reciprocal space point $\bm{k}=\Gamma$ for the $m$-esim magic angles
($\alpha_{m}$) with $m=8$ and $m=9$. The electronic maxima of the density form
hexagons which are nearly localized at $\bm{r_{\mu}}\approx\pm\bm{q}_{\mu}$.
Such observation is detailed in Fig. 1. Moreover, the wave-function for other
$\bm{k}$ points follow the same behavior although the $\Gamma$ point best
captures the magic angle behavior [40]. In the limit of
$\alpha_{m}\rightarrow\infty$ we have verified that in fact, the electron
density is almost localized at $\bm{q}_{\mu}$. Notice that here we are working
with adimensional units but this suggests a connection with the QHE as
solutions seem self-dual [45], i.e., in real space are similar to those in
reciprocal space with renormalized parameters.

Although there are expressions for the wave-function [29, 46, 33] at any
$\bm{k}$ point that hinted a relationship with the lowest Landau levels, they
depend on the wave function at the $\bm{K}$ point, i.e.,

$\psi_{\bm{k},j}(\bm{r})=f_{\bm{k}}(z)\psi_{\bm{K},j}(\bm{r})$ (8)

where $z=x+iy$ and $f_{\bm{k}}(z)$ is an analytic function which satisfy the
boundary condition and turns out to be a Jacobi theta function. The form of
the $\psi_{\bm{K},j}(\bm{r})$ is not analytically known. Yet in Figs. 1 and 2
we see numerically that the electron wave function reaches an asymptotic limit
almost invariant as $\alpha_{m}\rightarrow\infty$ and does not depend much on
$\bm{k}$. This wave function tends to be localized in certain points of space
which are not the stacking points AA, AB and BA. In that sense, the solutions
are different from the first magic angle a fact that was explained elsewhere
[40].

Figure 1: Numerically obtained zero mode wave function localization for some
high-order magic angles at $\bm{k}=\bm{\Gamma}$. In Panels a) and b) we
present $\Re{\psi_{1}(\bm{r})}+\Im{\psi_{2}(\bm{r})}$ (orange curves) and
$\Im{\psi_{1}(\bm{r})}-\Re{\psi_{2}(\bm{r})}$ (purple curves) parts of a one
layer symmetrized wave function components
$\psi_{\pm}(\bm{r})=\psi_{1}(\bm{r})\mp i\mu_{\alpha}\psi_{2}(\bm{r})$ at the
symmetric line $(0,y)$ at magic angles $\alpha_{8}=11.345$ and
$\alpha_{9}=12.855$ respectively. Panels c) and d), contour plot of the global
electronic density $\rho_{1}(\bm{r})+\rho_{2}(\bm{r})$ for $\alpha_{8}=11.345$
and $\alpha_{9}=12.855$ respectively. The vertical line (yellow line) inside
the Wigner-Seitz indicates the cut along the $y$ axis used in panel a) and b).
The external hexagon is the Wigner-Seitz cell, where the AB (green), BA
(yellow) and AA (red) stacking points are indicated. For higher magic angles,
the wave-function density localizes in $6$ high density points, located at
$\bm{r}=\pm\bm{q}_{\mu}$, with $\mu=1,2,3$, forming the red spots of maximal
density.

To understand how this limiting wave function arises, let us discuss the zero-
mode equation
$D(\bm{r})\begin{pmatrix}\psi_{1}(\bm{r}),\psi_{2}(\bm{r})\end{pmatrix}^{T}=0$.
Although not essential for the analysis, it is easier to understand the
$\Gamma$ point solution. For this case we have that due to symmetry,
$\psi_{2}(\bm{r})=i\mu_{\alpha}\psi_{1}(-\bm{r})$ where $\mu_{\alpha}=\pm 1$
depending on the magic angle parity [29]. Therefore, we obtain,

$\bar{\partial}\psi_{1}(\bm{r})=\alpha\mu_{\alpha}U(\bm{r})\psi_{1}(-\bm{r})$
(9)
$\bar{\partial}\psi_{1}(-\bm{r})=-\alpha\mu_{\alpha}U(-\bm{r})\psi_{1}(\bm{r})$
(10)

To solve the equation in the limit $\alpha\rightarrow\infty$ we use the
boundary layer theory of differential equations [47], i.e., whenever the
gradients are small, we can neglect the derivative in Eqns. (9)-(10) when
compared to the potential term. Then our solution must satisfy
$\psi_{j}(r)\rightarrow 0$. The solution will be different from zero only
inside the boundary layer, i.e., whenever $\bar{\partial}\psi_{j}(\bm{r})$ is
of order $\alpha U(\pm\bm{r})\psi_{j}(\bm{r})$. Taken into account the
boundary layer we conclude that the solution must be strongly peaked around
certain regions of space. Then is natural to seek the solution within
continuous functions having a peak while keeping the form of Eq. (8). We then
propose a coherent Landau state ansatz for a given layer (and thus suppress
the subindex $j$),

$\begin{split}\psi(z,z^{*})&=f_{\lambda}(z)e^{-\frac{1}{4\sigma^{2}}|z|^{2}}\\\
\end{split}$ (11)

where $f_{\lambda}(z)$ is an analytic function [48],

$\begin{split}f_{\lambda}(z)&=\frac{1}{\sigma\sqrt{2\pi}}e^{\frac{1}{2\sigma^{2}}\lambda^{*}z}e^{-\frac{1}{4\sigma^{2}}\lambda\lambda^{*}}\end{split}$
(12)

The parameter $\lambda$ is the localization center (known as the guiding
coordinates in the QHE problem [49]) and $\sigma$ the standard deviation as
the electronic density is a Gaussian,

$\rho(\bm{r})=\frac{1}{2\pi\sigma^{2}}e^{\frac{-|z-\lambda|^{2}}{2\sigma^{2}}}$
(13)

Notice how the Gaussian envelope in Eq. (11) ensures the boundary layer
condition, i.e., the vanishing of the wave function whenever the gradient is
small.

However, still we need to make remarks. As the equation involves $\psi(r)$ and
$\psi(-r)$, the solutions can be written as a sum of a symmetrized and
antisymmetrized forms. Therefore, it will be a linear combination of the
symmetrized/antisymmetrized wavefunctions,

$\begin{split}\psi_{\pm}(z,z^{*})=e^{-\frac{1}{4\sigma^{2}}|z|^{2}}\frac{1}{\sqrt{2}}(f_{\lambda}(z)\pm
f_{-\lambda}(z))\\\ \end{split}$ (14)

provided that $\lambda$ is far away from $-\lambda$ to avoid overlap between
the Gaussians centered at $\lambda$ and $-\lambda$. In what follows we will
use our ansatz in the zero mode equation to prove how it satisfies the
equation and to obtain $\sigma$.

Before doing so, observe that $\psi(r)$ must transform according to the
$C_{3}$ symmetry group and this can be ensured by defining a $\lambda_{1}$
such that,

$\begin{split}\psi_{\pm}(z,z^{*})=\frac{1}{{\sqrt{6}}}\sum_{\mu=1}^{3}e^{-\frac{1}{4\sigma^{2}}|z|^{2}}(f_{\lambda_{\mu}}(z)\pm
f_{-\lambda_{\mu}}(z))\\\ \end{split}$ (15)

where $\lambda_{2}=e^{i\phi}\lambda_{1}$ and
$\lambda_{3}=e^{-i\phi}\lambda_{1}$ and the normalization constant was
modified to account for two layers and the three $\lambda_{\mu}$. Concerning
the boundary conditions, i.e., the Bloch theorem, we will discuss the subject
after testing the solution for a unit cell.

As the numerical simulation indicates that the electronic density is localized
on $\pm\bm{q}_{\mu}$, this suggests to propose
$\lambda_{\mu}=Q_{\mu}=q^{x}_{\mu}+iq^{y}_{\mu}$ where $q^{x}_{\mu}$ and
$q^{y}_{\mu}$ are the components of vector
$\bm{q}_{\mu}=(q^{x}_{\mu},q^{y}_{\mu})$. Finally, the parameter $\sigma$ will
be determined by imposing the ansatz to satisfy Eqns. (9)-(10).

Now we test our ansatz in the zero mode equation. Using complex numbers and
that $\bar{\partial}=2\frac{\partial}{\partial z^{*}}$, the zero mode equation
can be rewritten as,

$\begin{split}2\frac{\partial}{\partial
z^{*}}\psi_{\pm}(z,z^{*})=\mu_{\alpha}\alpha_{m}U(z,z^{*})\psi_{\pm}(-z,-z^{*})\end{split}$
(16)

where $\alpha_{m}$ indicate a magic angle and $U(z,z^{*})$ is the complex form
of the coupling layer potential $U(\bm{r})$.

In the limit $\alpha\rightarrow\infty$ we can expand $U(z,z^{*})$ locally
around $Q_{\mu}$ (see appendix) where the boundary layer lies, therefore,

$\begin{split}U(z,z^{*})\psi_{\pm}(-z,-z^{*})&\approx\frac{3z}{2}\psi_{\pm}(-z,-z^{*})\end{split}.$
(17)

Next we use that the anti-holomorphic derivative of an analytic function is
zero from where

$\begin{split}\bar{\partial}\psi_{\pm}(z,z^{*})&=2\frac{\partial\psi_{\pm}(z,z^{*})}{\partial
z^{*}}=-\frac{z}{2\sigma^{2}}\psi_{\pm}(z,z^{*})\end{split}$ (18)

Finally, we combine the left-side of the zero mode equation, Eq. (18), with
the right-hand side and use Eq. (17) to obtain,

$\begin{split}-\frac{z}{\sigma^{2}}\psi_{\mp}(z,z^{*})&\approx\mu_{\alpha}3\alpha_{m}z\psi_{\mp}(z,z^{*})\end{split}$
(19)

where we see that the equation imposes the need of a symmetric or
antisymmetric solution depending on the magic angle parity, given by the sign
of $\mu_{\alpha}$. This can be numerically verified in Fig. 1 where we plot
the symmetrized and antisymmetrized numerically obtained wavefunctions for
finite $\alpha$. Observe that for one of the layers
$\psi_{\pm}(\bm{r})=\psi_{1}(\bm{r})\pm\psi_{1}(\bm{-r})=\psi_{1}(\bm{r})\mp
i\mu_{\alpha}\psi_{2}(\bm{r})$, the other layer is obtained from
$\psi_{2}(\bm{r})\pm\psi_{2}(\bm{-r})$. The resulting symmetric/antisymmetric
components of $\psi_{\pm}(\bm{r})$ are purely real/imaginary respectively for
odd/even $m$. Moreover, from Eq. (19) we obtain the width of the coherent
Landau state,

$\lim_{m\rightarrow\infty}\sigma=\frac{1}{\sqrt{3\alpha_{m}}}$ (20)

To test these two results, in Fig. 2 we compare the evolution of the
electronic density as $\alpha\rightarrow\infty$ for several magic angles, in
this case for the axis $x=0$. The dashed line is the asymptotic solution given
by Eq. (15) which does not contain any free parameter.

In Fig. 3 we show a log-log plot of $\sigma$ versus $\alpha_{m}$ as obtained
by fitting Gaussians to the numerical results. The red line is the theoretical
prediction given by Eq. (20) giving a very good agreement with the numerical
data for higher order magic angles. In Fig. 3 b) we also plot the maximum
position of the numerically obtained wavefunctions ($|r_{m}|$), confirming the
tendency for localization seen in the inverse participation ratio [40]. This
is why the ansatz almost obey the Bloch theorem, i.e., zero modes are akin to
other confined states in which the overlap between wave functions at different
unitary cells is almost zero [28]. In fact, the set of coherent Landau levels
is overcomplete [50].

Figure 2: Coherent states limit for the wave function density along the
$y$-axis for layer $1$. The solid curves with shaded areas are the numerical
solutions of the zero mode Eqns. (9 and (10) for the indicated magic angles.
The dashed line is the theoretical result using a coherent state using the
ninth-magic angle ($\alpha_{9}$). The thick vertical lines (blue lines)
indicate the limiting localization points $\bm{q}_{1}$ and $-\bm{q}_{1}$.
Notice how as the magic angle goes from the second to the nine, the density
becomes sharply peaked. Figure 3: a) Log-log plot showing the standard
deviation $\sigma$ of the coherent Landau states vs $\alpha_{m}$. The red line
is the equation $\sigma=1/\sqrt{3\alpha}$ and the dots are values obtained
from a fit with Gaussians of the numerically obtained electronic density from
the zero mode equation. b) Distance from the origin of $|r_{m}|$ which is the
position of the maximum of the numerically obtained electronic density at the
magic angles. The blue horizontal line indicates the value $|q_{1}|=1$ when
$\alpha\rightarrow\infty$. For both panels, red vertical lines indicate magic
angles from the second to the nine, showing the
$\alpha_{m+1}-\alpha_{m}\approx 3/2$ rule for the magic angle separation.

##  IV Quantum Hall effect and squared twisted graphene hamiltonian:
quantization magic angle rule

Let us now prove why magic angles are quantized and the relation with the
quantum harmonic oscillator. Consider the square Hamiltonian $H^{2}$ for
energy zero,

$\begin{split}(-\nabla^{2}+\alpha^{2}|U(\bm{r})|^{2})\psi_{1}(\bm{r})+\alpha
A^{\dagger}(\bm{r})\psi_{2}(\bm{r})=0\end{split}$ (21)

By using complex notation for $H^{2}$, the symmetry relation between layer
components wave function at the $\Gamma$ point and Eq. (19) it follows that,

$\begin{split}\left(4p_{z}p^{*}_{z}+\left(\frac{3\alpha}{2}\right)^{2}|z|^{2}-3\alpha
L_{z}\right)\psi(z,z^{*})=0\end{split}$ (22)

where $L_{z}=i(zp_{z}-z^{*}p^{*}_{z})$ is the angular momentum and
$p_{z}=p_{x}-ip_{y}$ with $p_{j}=-i\hbar\partial_{j}$ is the momentum
operator. Moreover, we can define the creation/annihilation operator
associated,

$\begin{split}a_{z}=\sqrt{\frac{3\alpha}{4\hbar}}z+i\frac{2}{\sqrt{3\alpha\hbar}}p^{*}_{z}\end{split}$
(23)
$\begin{split}a^{\dagger}_{z}=\sqrt{\frac{3\alpha}{4\hbar}}z^{*}-i\frac{2}{\sqrt{3\alpha\hbar}}p_{z}\end{split}$
(24)

from where we obtain an effective two-dimensional quantum harmonic oscillator
Hamiltonian,

$\begin{split}H_{zz^{*}}\psi(z,z^{*})=\left(\hbar\omega
a^{\dagger}_{z}a_{z}+\omega L_{z}\right)\psi(z,z^{*})\end{split}$ (25)

with $\omega=3\alpha$. Defining the conjugate operators $a_{z^{*}}$ and
$a^{\dagger}_{z^{*}}$ (see appendix), it follows that the angular momentum is,

$\begin{split}L_{z}=\frac{\hbar}{2}(a^{\dagger}_{z^{*}}a_{z^{*}}-a^{\dagger}_{z}a_{z})\end{split}$
(26)

where $L^{\dagger}_{z}=L_{z}$ and therefore Eq. (25) is rewritten as,

$\begin{split}H_{zz^{*}}\psi(z,z^{*})=\hbar\omega\left(N_{zz^{*}}+1\right)\psi(z,z^{*})\end{split}$
(27)

where
$N_{zz^{*}}=\frac{1}{2}(a^{\dagger}_{z^{*}}a_{z^{*}}+a^{\dagger}_{z}a_{z})$ in
analogy to a 2D harmonic oscillator. The constant term $\hbar\omega$ is the
zero-point energy of the oscillator. Now is clear the $3/2$ quantization rule:
they are just different oscillator levels as, although in Eq. (27) the levels
are spaced by $\alpha=3m$, we must remember that we have two equations, one
for $m$ odd and $m$ even. Therefore, the $3\alpha m$ is the recurrence for
each parity sequence and thus magic angles, without distinction of parity,
occurs for $\alpha=(3/2)m$. Finally, we can compare Eq. (26) and the
interlayer currents between bipartite layers that were investigated in a
previous work [40]. Apart from a dimensional constant, both expressions
coincide from where we can identify the angular moment with such currents. As
explained elsewhere [40], magic angles have a precise kinetic, confinement
energies and current balance.

##  V Conclusions

In conclusion, we showed that the twisted bilayer hamiltonian can be reduced
to a quantum harmonic oscillator and thus magic angles are quantized. Zero
flat band modes converge into coherent Landau levels. Due to the multiple
mathematical and physical properties of coherent states, this opens many
exciting technological and physical possibilities in Moiré materials, as for
example, the possibility of controlling coherency by manipulation of the
twisting angle or the self-duality property of the wave functions [45].

This work was supported by UNAM DGAPA PAPIIT IN102620 (L.A.N.L. and G.G.N.)
and CONACyT project 1564464. We thank Patrick Ledwith, Eslam Kahalaf at
Harvard University, Jie Wang at Flatiron Institute (now at Harvard
University), F. Guinea and Pierre Pantaleon at IMDEA, Spain, for useful
comments on this project.

##  VI Appendix

###  VI.1 Wave-functions Fourier coefficients in reciprocal space

Here we use as basis the wave vectors
$\Phi(\bm{r})=\begin{pmatrix}\psi_{1}(\bm{r}),\psi_{2}(\bm{r}),\chi_{1}(\bm{r}),\chi_{2}(\bm{r})\end{pmatrix}^{T}$
where the index $1,2$ represents each graphene layer and $\psi_{j}(\bm{r})$
and $\chi_{j}(\bm{r})$ are the Wannier orbitals on each inequivalent site of
the graphene’s unit cell.

A general Bloch’s wave function with momentum $\bm{k}$ in the mBZ at each
sublattice has the form

$\begin{split}\Psi_{\bm{k}}(\bm{r})=\begin{pmatrix}\psi_{\bm{k},1}(\bm{r})\\\
\psi_{\bm{k},2}(\bm{r})\end{pmatrix}=\sum_{mn}\begin{pmatrix}a_{mn}\\\
b_{mn}e^{i\bm{q}_{1}\cdot\bm{r}}\end{pmatrix}e^{i(\bm{K}_{mn}+\bm{k})\cdot\bm{r}}\end{split}$
(28)

$\begin{split}\chi_{\bm{k}}(\bm{r})=\begin{pmatrix}\chi_{\bm{k},1}(\bm{r})\\\
\chi_{\bm{k},2}(\bm{r})\end{pmatrix}=\sum_{mn}\begin{pmatrix}c_{mn}\\\
d_{mn}e^{i\bm{q}_{1}\cdot\bm{r}}\end{pmatrix}e^{i(\bm{K}_{mn}+\bm{k})\cdot\bm{r}}\end{split}$
(29)

where $a_{mn}(b_{mn})$ are the Fourier coefficients of layer 1 (layer 2) for
sublattice $A$ and $c_{mn}(d_{mn})$ are the Fourier coefficients of layer 1
(layer 2) for sublattice $B$, and $\bm{K}_{mn}=m\bm{b}_{1}+n\bm{b}_{2}$ with
$\bm{b}_{1,2}$ are the two Moiré Brillouin zone vectors. If we substitute Eq.
(28) and Eq. (29) into
$\mathcal{H}\Phi_{\bm{k}}(\bm{r})=E\Phi_{\bm{k}}(\bm{r})$ we can calculate the
eigenfunctions of $\mathcal{H}$,

$\begin{split}(\bm{K}^{\prime x}_{mn}+i\bm{K}^{\prime
y}_{mn})a_{mn}+\alpha(b_{mn}&+e^{i\phi}b_{m+1,n}\\\
&+e^{-i\phi}b_{m,n+1})=Ea_{mn}\end{split}$ (30) $\begin{split}(\bm{K}^{\prime
x}_{mn}+i(\bm{K}^{\prime
y}_{mn}+\bm{q}_{1}))b_{mn}+&\alpha(a_{mn}+e^{i\phi}a_{m-1,n}\\\
&+e^{-i\phi}b_{m,n-1})=Eb_{mn}\end{split}$ (31) $\begin{split}(\bm{K}^{\prime
x}_{mn}-i\bm{K}^{\prime y}_{mn})c_{mn}+\alpha(d_{mn}&+e^{-i\phi}d_{m+1,n}\\\
&+e^{i\phi}d_{m,n+1})=Ec_{mn}\end{split}$ (32) $\begin{split}(\bm{K}^{\prime
x}_{mn}-i(\bm{K}^{\prime
y}_{mn}+\bm{q}_{1}))d_{mn}+&\alpha(c_{mn}+e^{-i\phi}c_{m-1,n}\\\
&+e^{i\phi}c_{m,n-1})=Ed_{mn}\end{split}$ (33)

where $\bm{K}^{\prime}_{mn}=\bm{K}_{mn}+\bm{k}$.

Here Eqns. (30-33) form a coupled linear system that can be solved to find the
corresponding eigenvalues. In general, there are $L=(2N+1)\times(2N+1)$
coefficients with $N$ the range of the matrix and $2N+1$ the elements in the
set, therefore, the Hamiltonian matrix has dimension $D=4L$. In a similar way
we can obtain the eigenfunctions of $H^{2}$ but the system is easier write as
the matrix is only of size $2\times 2$. Observe that all eigenfunctions of
$\mathcal{H}$ are always eigenfunctions of $H^{2}$ but the converse is not
true. This was discussed in detail elsewhere [28].

The system can be further reduced by using the $C_{3}$ symmetry. We denote the
corresponding rotational operators $\bm{R}_{\phi}$ and $\bm{R}_{2\phi}$ by the
angle $\phi$ and $2\phi$ respectively. Their matrix representations
$D_{\phi}$, $D_{2\phi}$ and the identity, have eigenvalues
$w=\big{\\{}1,e^{i\phi},e^{-i\phi}\big{\\}}$. Eigenfunctions of the
hamiltonian are also eigenfunctions of such operators, and thus we have the
relation
$\Psi_{\bm{k}}(\bm{R}_{2\phi}(\bm{r}))=e^{i\phi}\Psi_{\bm{k}}(\bm{r})$ and
$\Psi_{\bm{k}}(\bm{R}_{\phi}(\bm{r}))=e^{-i\phi}\Psi_{\bm{k}}(\bm{r})$, with
the property
$\bm{R}_{\phi}(\bm{q}_{\mu})\cdot\bm{r}=\bm{q}_{\mu}\cdot\bm{R}^{-1}_{\phi}(\bm{r})$.
One can obtain a relationship between coefficients using such rotations to
reduce the problem into a one trigonal sector.

To perform such calculation, it is very useful to have the Moiré reciprocal
basis vector transformation rules under the rotations. We reproduce below such
useful rules,

$\begin{split}\bm{R}_{\phi}(\bm{b}_{1})=\bm{b}_{2}-\bm{b}_{1},\\\
\bm{R}_{\phi}(\bm{b}_{2})=-\bm{b}_{1},\\\
\bm{R}_{2\phi}(\bm{b}_{1})=-\bm{b}_{2},\\\
\bm{R}_{2\phi}(\bm{b}_{1})=-\bm{b}_{2}+\bm{b}_{1}.\end{split}$ (34)

Figure 4: Moiré vectors rotational $C_{3}$ rules. The operator $\bm{R}_{\phi}$
rotate by $\phi=\frac{2\pi}{3}$ moiré vectors $\bm{b}_{1}$ ($\bm{b}_{2}$)
indicated by purple (green) arrows.

###  VI.2 Fourier coefficients in $\bm{\Gamma}$-point reciprocal

In the $\bm{\Gamma}$ point, using the symmetry of the Hamiltonian it can be
proved that the spinor components are related through
$\psi_{2,\bm{\Gamma}}(\bm{r})=i\mu_{\alpha}\psi_{1,\bm{\Gamma}}(-\bm{r})$.
Tarnoposky et. al [29] found that,

$\begin{split}\bar{\partial}\psi_{\bm{\Gamma},1}(\bm{r})\mp\alpha
U(\bm{r})\psi_{\bm{\Gamma},1}(-\bm{r})=E_{\Gamma}\psi_{\bm{\Gamma},1}(-\bm{x},\bm{y})\end{split}$
(35)

Following this analysis a reduced equation for the Fourier coefficients can be
written as,

$\begin{split}|\bm{K}_{mn}+\bm{q}_{1}|e^{i\theta(\bm{K}_{mn}+\bm{q}_{1})}a_{mn}\mp\alpha(a_{mn}+e^{i\phi}a_{-m,-n+1}\\\
\\\ +e^{-i\phi}a_{-m+1,-n})=E_{\Gamma}a_{mn}\end{split}$ (36)

From the $\bm{\Gamma}=\bm{q}_{1}$ point the symmetry of the component of the
spinors
$\psi_{2,\bm{\Gamma}}(\bm{r})=i\mu_{\alpha}\psi_{1,\bm{\Gamma}}(-\bm{r})$, we
found the following relation,

$\begin{split}\sum_{m,n}b_{mn}e^{i(\bm{K}_{mn}+2\bm{q}_{1})\cdot\bm{r}}=i\mu_{\alpha}\sum_{m^{\prime}n^{\prime}}a_{m^{\prime}n^{\prime}}e^{-i(\bm{K}_{m^{\prime}n^{\prime}}+\bm{q}_{1})\cdot\bm{r}}\end{split}$
(37)

or simplifying,

$\begin{split}\sum_{m,n}b_{mn}e^{i\bm{K}_{mn}\cdot\bm{r}}=i\mu_{\alpha}\sum_{m^{\prime},n^{\prime}}a_{m^{\prime}n^{\prime}}e^{i(-\bm{K}_{m^{\prime}n^{\prime}}-3\bm{q}_{1})\cdot\bm{r}}\end{split}$
(38)

note that $\bm{b}_{1}+\bm{b}_{2}=-3\bm{q}_{1}$ and Eq. (38) is rewritten as,

$\begin{split}\sum_{m,n}b_{mn}e^{i\bm{K}_{mn}\cdot\bm{r}}=i\mu_{\alpha}\sum_{m^{\prime},n^{\prime}}a_{m^{\prime}n^{\prime}}e^{i\bm{K}_{1-m^{\prime},1-n^{\prime}}\cdot\bm{r}}\end{split}$
(39)

change index $m=1-m^{\prime}\rightarrow m^{\prime}=1-m$ and
$n=1-n^{\prime}\rightarrow n^{\prime}=1-n$ in Eq. (39), therefore,

$\begin{split}\sum_{m,n}(b_{mn}-i\mu_{\alpha}a_{1-m,1-n})e^{i\bm{K}_{mn}\cdot\bm{r}}=0\end{split}$
(40)

Finally, the relation between Fourier coefficients of each layer in
$\bm{\Gamma}$ point is simply,

$\begin{split}b_{mn}=i\mu_{\alpha}a_{1-m,1-n}\end{split}$ (41)

with $\mu_{\alpha}=\pm 1$ and also follows that,

$\begin{split}\Re{b_{mn}}=-\mu_{\alpha}\Im{a_{1-m,1-n}}\end{split}$ (42)

and

$\begin{split}\Im{b_{mn}}=\mu_{\alpha}\Re{a_{1-m,1-n}}\end{split}$ (43)

Using the $C_{3}$ symmetry of the wave-function at the $\Gamma$-point,
$\bm{k}=\bm{q}_{1}$,

$\begin{split}\Psi{(\bm{R}_{\phi}(\bm{r}))}=\sum_{m,n}\begin{pmatrix}a_{mn}\\\
b_{mn}e^{i\bm{R}_{\phi}(\bm{q}_{1})\cdot\bm{r}}\end{pmatrix}e^{i\bm{R}_{\phi}(m\bm{b}_{1}+n\bm{b}_{2}+\bm{q}_{1})\cdot\bm{r}}\end{split}$
(44)

where
$m\bm{R}_{\phi}(\bm{b}_{1})+n\bm{R}_{\phi}(\bm{b}_{2})+\bm{R}_{\phi}(\bm{q}_{1})=m(\bm{b}_{2}-\bm{b}_{1})+n(-\bm{b}_{2})+\bm{q}_{2}$.
From the rotated vectors in Eq. (44) follows that,

$\begin{split}\sum_{m,n}a_{mn}e^{i(m\bm{b}_{1}+n\bm{b}_{2})\cdot\bm{r}}=e^{i\phi}\sum_{m,n}a_{mn}e^{i((-m-n+1)\bm{b}_{1}+m\bm{b}_{2})\cdot\bm{r}}\end{split}$
(45)

using new index $m^{\prime}=-m-n+1\rightarrow n=1-m^{\prime}-n^{\prime}$ and
$n^{\prime}=m$ follows that,

$\begin{split}\sum_{m,n}a_{mn}e^{i(m\bm{b}_{1}+n\bm{b}_{2})\cdot\bm{r}}=e^{i\phi}\sum_{m,n}a_{n^{\prime},1-m^{\prime}-n^{\prime}}e^{i(m^{\prime}\bm{b}_{1}+n^{\prime}\bm{b}_{2})\cdot\bm{r}}\end{split}$
(46)

therefore,

$\begin{split}e^{-i\phi}a_{mn}=a_{n,-m-n+1}\end{split}$ (47)

and,

$\begin{split}e^{i\phi}a_{mn}=a_{-m-n+1,m}\end{split}.$ (48)

using this last $C_{3}$ symmetry in (35) at the magic angles follows that,

$\begin{split}(i\frac{\sqrt{3}}{2}(m-n)&+1-\frac{3}{2}(m+n))a_{m,n}\mp\alpha_{m}(a_{1-m,1-n}\\\
&+a_{-n,m+n}+a_{m+n,-m})=0\end{split}.$ (49)

where
$K_{mn}=m\bm{b}_{1}+n\bm{b}_{2}=(\frac{\sqrt{3}}{2}(m-n),\frac{3}{2}(m+n)-1)$
is the Moiré reciprocal vector.

###  VI.3 Coherent Landau level solution in the limit
$\alpha\rightarrow\infty$

Consider a coherent state of the form,

$\begin{split}\psi(z,z^{*})&=f_{\lambda}(z)e^{-\frac{1}{4\sigma^{2}}|z|^{2}}\\\
&=\frac{1}{\sigma\sqrt{2\pi}}e^{\frac{1}{2\sigma^{2}}\lambda^{*}z}e^{-\frac{1}{4\sigma^{2}}\lambda\lambda^{*}}e^{-\frac{1}{4\sigma^{2}}|z|^{2}}\end{split}$
(50)

The complete form of wave function is obtained by summing over the
contributions for $\lambda$ and $-\lambda$,

$\begin{split}\psi_{\mp}(z,z^{*})&=\frac{1}{\sqrt{2}}(f_{\lambda}(z)e^{-\frac{1}{4\sigma^{2}}|z|^{2}}\mp
f_{-\lambda}(z)e^{-\frac{1}{4\sigma^{2}}|z|^{2}})\\\
&=\frac{1}{\sigma\sqrt{2}\sqrt{2\pi}}(e^{\frac{1}{2\sigma^{2}}\lambda^{*}z-\frac{1}{4\sigma^{2}}\lambda\lambda^{*}-\frac{1}{4\sigma^{2}}|z|^{2}}\\\
&\mp
e^{-\frac{1}{2\sigma^{2}}\lambda^{*}z-\frac{1}{4\sigma^{2}}\lambda\lambda^{*}-\frac{1}{4\sigma^{2}}|z|^{2}})\\\
\end{split}$ (51)

where $\lambda$ is the center of the Gaussians. Is important to note that Eq.
(51), satisfies the relation $\psi_{\mp}(-z,-z^{*})=\mp\psi_{\mp}(z,z^{*})$ as
we consider here the wavefunction as a sum of symmetrized/antisymmetrized
functions that can be treated separately. We also have,

$\begin{split}|\psi_{\mp}(z,z^{*})&|^{2}\approx\frac{1}{2\sigma^{2}(2\pi)}(e^{-\frac{1}{2\sigma^{2}}|z-\lambda|^{2}}+e^{-\frac{1}{2\sigma^{2}}|z+\lambda|^{2}})\end{split}$
(52)

where the overlap term between Gaussian’s is neglected in the limit
$\alpha\rightarrow\infty$. On the other hand, the coupling potential can be
written as,

$\begin{split}U(z,z^{*})=-2\frac{\partial}{\partial
z^{*}}S(z,z^{*})\end{split}$ (53)

where $\bar{\partial}=2\frac{\partial}{\partial z^{*}}$ and,

$\begin{split}S(z,z^{*})=\sum^{3}_{\mu=1}e^{-i(Q^{*}_{\mu}z+Q_{\mu}z^{*})/2}\end{split}$
(54)

where $Q_{\mu}=q^{x}_{\mu}+iq^{y}_{\mu}$. Therefore,

$\begin{split}\frac{\partial}{\partial
z^{*}}S(z,z^{*})&=\frac{1}{2}\sum^{3}_{\mu=1}(-iQ_{\mu})e^{-i(Q^{*}_{\mu}z+Q_{\mu}z^{*})/2}\end{split}$
(55)

Finally, $U(z,z^{*})$ is the complex form of the coupling layer potential,

$U(\bm{r})=U(z,z^{*})=\sum^{3}_{\mu=1}(iQ_{\mu})e^{-i(Q^{*}_{\mu}z+Q_{\mu}z^{*})/2}$
(56)

Notice that such result can be obtained straightforward from the definition of
$U(\bm{r})$ and $z$ and $z^{*}$, yet is illustrative to use the function
$S(z,z^{*})$ as this quantity appears in several conmmutators [28].

Substituting Eqns. (53)-(55) in Eq. (35) for magic angles,

$\begin{split}&2\frac{\partial}{\partial
z^{*}}\psi_{\mp}(z,z^{*})=\pm\alpha_{m}(-2\frac{\partial}{\partial
z^{*}}S(z,z^{*}))\psi_{\mp}(-z,-z^{*})\\\
&=\pm\alpha_{m}(\sum^{3}_{\mu=1}(iQ_{\mu})e^{-i(Q^{*}_{\mu}z+Q_{\mu}z^{*})/2})\psi_{\mp}(-z,-z^{*})\end{split}$
(57)

using the fact that in the limit $\alpha\rightarrow\infty$ the wave function
is localized in $Q_{\mu}$ and in other points is zero, we can expand
(LABEL:eq:GammaCOmplex) around $Q_{\mu}$ up to first order,

$\begin{split}2\frac{\partial}{\partial
z^{*}}\psi_{\mp}(z,z^{*})&\approx\pm\alpha_{m}\sum^{3}_{\mu=1}(iQ_{\mu})(1-i\frac{1}{2}(Q^{*}_{\mu}z\\\
&+Q_{\mu}z^{*}))\psi_{\mp}(-z,-z^{*})\end{split}$ (58)

and since $\sum_{\mu}Q_{\mu}=\sum_{\mu}Q^{2}_{\mu}=0$ and
$\sum_{\mu}|Q_{\mu}|^{2}=3$, it follows that,

$\begin{split}2\frac{\partial}{\partial
z^{*}}\psi_{\mp}(z,z^{*})\approx(\frac{3\alpha_{m}}{2})(-z)(\mp\psi_{\mp}(-z,-z^{*}))\end{split}$
(59)

since $\psi_{\mp}(z,z^{*})=\mp\psi_{\mp}(-z,-z^{*})$. Normalizing, the final
form of the ansatz wave function for $\alpha\rightarrow\infty$ is,

$\begin{split}\psi_{\pm}(z,z^{*})&=\frac{1}{\sqrt{6}}\sqrt{\frac{3\alpha}{2\pi}}\sum^{3}_{\mu=1}(f_{Q_{\mu}}(z)e^{-\frac{1}{4\sigma^{2}}|z|^{2}}\\\
&\pm f_{-Q_{\mu}}(z)e^{-\frac{1}{4\sigma^{2}}|z|^{2}})\end{split}$ (60)

From the square Hamiltonian at the zero flat band $H^{2}\Psi(\bm{r})=0$ it
follows that,

$\begin{split}(-\nabla^{2}+\alpha^{2}|U(-\bm{r})|^{2})\psi_{1}(\bm{r})+\alpha
A^{\dagger}(\bm{r})\psi_{2}(\bm{r})=0\end{split}$ (61)

rewritten in complex notation,

$\begin{split}(4p_{z}p^{*}_{z}+\left(\frac{3\alpha}{2}\right)^{2}|z|^{2})\psi(z,z^{*})+i\mu_{\alpha}\alpha
A^{\dagger}(z,z^{*})\psi(-z,-z^{*})=0\end{split}$ (62)

where
$A^{\dagger}(\bm{r})\approx-2i\sum^{3}_{\mu=1}e^{-i\bm{q}_{\mu}\cdot\bm{r}}\hat{\bm{q}}^{\perp}_{\mu}\cdot\bm{\nabla}$
for $\alpha\rightarrow\infty$ [40], therefore,

$\begin{split}A^{\dagger}(z,z^{*})\psi(-z,-z^{*})&=-2i\sum^{3}_{\mu=1}e^{-\frac{i}{2}(Q^{*}_{\mu}z+Q_{\mu}z^{*})}(Q^{\perp,*}_{\mu}\partial_{z^{*}}\\\
&+Q^{\perp}_{\mu}\partial_{z})\psi(-z,-z^{*})\end{split}$ (63)

now for $\alpha\rightarrow\infty$ we expand locally around $Q_{\mu}$ where the
boundary layer lies and considering that $Q^{\perp}_{\mu}=iQ_{\mu}$, it
follows that

$\begin{split}A^{\dagger}(z,z^{*})\psi(-z,-z^{*})&=-2i\sum^{3}_{\mu=1}(1-\frac{i}{2}(Q^{*}_{\mu}z+Q_{\mu}z^{*}))(iQ_{\mu}\partial_{z}\\\
&-iQ_{\mu}^{*}\partial_{z^{*}})\psi(-z,-z^{*})\end{split}$ (64)

with $\sum^{3}_{\mu=1}Q_{\mu}=0$ and $\sum^{3}_{\mu=1}|Q_{\mu}|^{2}=3$ we have
that,

$\begin{split}A^{\dagger}(z,z^{*})\psi(-z,-z^{*})&=-3i(z\partial_{z}-z^{*}\partial_{z^{*}})\psi(-z,-z^{*})\\\
&=3(zp_{z}-z^{*}p_{z^{*}})\psi(-z,-z^{*})\end{split}$ (65)

substituting Eq. (65) in Eq. (62),

$\begin{split}(4p_{z}p^{*}_{z}+\left(\frac{3\alpha}{2}\right)^{2}|z|^{2})&\psi(z,z^{*})+3i\mu_{\alpha}\alpha(zp_{z}\\\
&-z^{*}p_{z^{*}})\psi(-z,-z^{*})=0\end{split}$ (66)

however the angular momentum in complex notation is defined as
$L_{z}=i(zp_{z}-z^{*}p_{z^{*}})$ and we have that,

$\begin{split}(4p_{z}p^{*}_{z}+\left(\frac{3\alpha}{2}\right)^{2}|z|^{2})\psi(z,z^{*})+3\alpha
L_{z}\mu_{\alpha}\psi(-z,-z^{*})=0\end{split}$ (67)

for odd parity $\mu_{\alpha}=1$ the solution is anti-symmetric
$\psi(-z,-z^{*})=-\psi(z,z^{*})$ and for even parity $\mu_{\alpha}=-1$ the
solution is symmetric $\psi(-z,-z^{*})=\psi(z,z^{*})$. Therefore, the equation
is reduced as,

$\begin{split}(4p_{z}p^{*}_{z}+\left(\frac{3\alpha}{2}\right)^{2}|z|^{2}-3\alpha
L_{z})\psi(z,z^{*})=0\end{split}$ (68)

On the other hand, the quantum harmonic oscillator in complex notation is,

$\begin{split}H_{zz^{*}}\psi(z,z^{*})=\left(\frac{2p_{z}p^{*}_{z}}{m}+\left(\frac{m\omega^{2}}{2}\right)|z|^{2}\right)\psi(z,z^{*})\end{split}$
(69)

from where we identified by comparison with Eq. (68) that $m=\frac{1}{2}$ and
$\omega=3\alpha$, therefore we can define a creation/annihilation operators
associated as,

$\begin{split}a_{z}=\sqrt{\frac{3\alpha}{4\hbar}}z+i\frac{2}{\sqrt{3\alpha\hbar}}p^{*}_{z}\end{split}$
(70)
$\begin{split}a^{\dagger}_{z}=\sqrt{\frac{3\alpha}{4\hbar}}z^{*}-i\frac{2}{\sqrt{3\alpha\hbar}}p_{z}\end{split}$
(71)

from where,

$\begin{split}a^{\dagger}_{z}a_{z}=\frac{3\alpha}{4\hbar}|z|^{2}+4p_{z}p^{*}_{z}+i\frac{2}{\sqrt{3\alpha\hbar}}\sqrt{\frac{3\alpha}{4\hbar}}(z^{*}p^{*}_{z}-zp_{z})\end{split}$
(72)

multiplying both sides of Eq. (72) by $3\alpha$ follows that

$\begin{split}\hbar(3\alpha)a^{\dagger}_{z}a_{z}&=\left(\frac{3\alpha}{2\hbar}\right)^{2}|z|^{2}+4p_{z}p^{*}_{z}-3\alpha
L_{z}\\\ &=H_{zz^{*}}-3\alpha L_{z}\end{split}$ (73)

from where we obtain an effective two-dimensional quantum harmonic oscillator
Hamiltonian,

$\begin{split}H_{zz^{*}}\psi(z,z^{*})=\left(\hbar\omega
a^{\dagger}_{z}a_{z}+\omega L_{z}\right)\psi(z,z^{*})\end{split}$ (74)

We can also define the conjugate operators,

$\begin{split}a_{z^{*}}=\sqrt{\frac{3\alpha}{4\hbar}}z^{*}+i\frac{2}{\sqrt{3\alpha\hbar}}p_{z}\end{split}$
(75)
$\begin{split}a^{\dagger}_{z^{*}}=\sqrt{\frac{3\alpha}{4\hbar}}z-i\frac{2}{\sqrt{3\alpha\hbar}}p^{*}_{z}\end{split}$
(76)

and it follows that the angular momentum is,

$\begin{split}L_{z}=\frac{\hbar}{2}(a^{\dagger}_{z^{*}}a_{z^{*}}-a^{\dagger}_{z}a_{z})\end{split}$
(77)

where $L^{\dagger}_{z}=L_{z}$ is Hermitian and therefore Eq. (74) is rewritten
as,

$\begin{split}H_{zz^{*}}\psi(z,z^{*})=\hbar\omega\left(N_{zz^{*}}+1\right)\psi(z,z^{*})\end{split}$
(78)

with
$N_{zz^{*}}=\frac{1}{2}(a^{\dagger}_{z^{*}}a_{z^{*}}+a^{\dagger}_{z}a_{z})=\frac{1}{2}(N_{z}+N_{z^{*}})$
with $N_{z}=a^{\dagger}_{z}a_{z}$ and $N_{z^{*}}=a^{\dagger}_{z^{*}}a_{z^{*}}$
in analogy to a 2D harmonic oscillator where $\omega=3\alpha$. The constant
term $\hbar\omega$ is the zero-point energy of the oscillator.

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