††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Supplemental material

Alex Thomson  Department of Physics, California Institute of Technology,
Pasadena, CA 91125, USA  Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA 91125, USA  Walter Burke
Institute for Theoretical Physics, California Institute of Technology,
Pasadena, CA 91125, USA  Ina Sorensen  Department of Physics, California
Institute of Technology, Pasadena, CA 91125, USA  Institute for Quantum
Information and Matter, California Institute of Technology, Pasadena, CA
91125, USA  Stevan Nadj-Perge  Institute for Quantum Information and Matter,
California Institute of Technology, Pasadena, CA 91125, USA  T. J. Watson
Laboratory of Applied Physics, California Institute of Technology, Pasadena,
CA 91125, USA  Jason Alicea  Department of Physics, California Institute of
Technology, Pasadena, CA 91125, USA  Institute for Quantum Information and
Matter, California Institute of Technology, Pasadena, CA 91125, USA  Walter
Burke Institute for Theoretical Physics, California Institute of Technology,
Pasadena, CA 91125, USA

###### Contents

  1. I Continuum model without spin orbit coupling
    1. I.1 Continuum model definition
    2. I.2 Interactions
    3. I.3 Symmetry action on microscopic operators
  2. II Spin orbit coupling in twisted bilayer graphene
    1. II.1 Monolayer graphene with induced spin orbit coupling
    2. II.2 Twisted bilayer graphene with induced spin orbit coupling
    3. II.3 Multiple TMDs and mirror symmetry ${\mathcald{M}}_{y}$
      1. II.3.1 Ising SOC
      2. II.3.2 Rashba SOC
  3. III Proximity-coupled wire at $\nu=\pm 4$
    1. III.1 Projection to flat bands in 2$d$ without SOC
      1. III.1.1 2$d$ flat band Hamiltonian without SOC
    2. III.2 Effective flat band Hamiltonian in 2$d$ with SOC
      1. III.2.1 2$d$ flat band Rashba Hamiltonian
      2. III.2.2 2$d$ flat band Ising Hamiltonian
    3. III.3 Trivial wire Hamiltonian
      1. III.3.1 Wire Hamiltonian without SOC
      2. III.3.2 Wire Hamiltonian for Rashba SOC
      3. III.3.3 Wire Hamiltonian for Ising SOC
    4. III.4 Analysis in the absence of ${\mathcald{M}}_{y}$ or ${\mathcald{C}}_{3}$
  4. IV IVC proximity-coupled wire at $\nu=0$ and $\nu=\pm 2$
    1. IV.1 Outline of methodology
    2. IV.2 $\nu=0$ IVC order parameters
      1. IV.2.1 Effective Hamiltonian with proximity-induced IVC singlet order
      2. IV.2.2 Effective Hamiltonian with proximity-induced IVC triplet order
    3. IV.3 $\nu=\pm 2$ IVC order parameters
      1. IV.3.1 Effective Hamiltonian with proximity-induced FM IVC order
      2. IV.3.2 Effective Hamiltonian with proximity-induced AFM IVC order
    4. IV.4 Relation of internal SOC parameters and wire direction
  5. V Wire proximity-coupled to IVC and superconducting states
  6. VI Five band model
    1. VI.1 Basic definitions
    2. VI.2 Symmetries
      1. VI.2.1 120${}^{\,\circ}$ rotation, ${\mathcald{C}}_{3}$
      2. VI.2.2 Mirror, ${\mathcald{M}}_{y}$
      3. VI.2.3  180${}^{\,\circ}$ rotation + time reversal, ${\mathcald{C}}_{2}{\mathcald{T}}$
      4. VI.2.4 Time reversal
    3. VI.3 Five band model without SOC
      1. VI.3.1 Hopping between $\mathrm{hex}_{A}$ and $\mathrm{hex}_{B}$ orbitals
      2. VI.3.2 Hopping between $\mathrm{tri}_{\alpha}$ orbitals
      3. VI.3.3 Hopping between $\mathrm{tri}_{\alpha}$ and $\mathrm{hex}_{A}$ orbitals
      4. VI.3.4 Hopping between $\mathrm{tri}_{\alpha}$ and $\mathrm{hex}_{B}$ orbitals
      5. VI.3.5 Complete Hamiltonian
    4. VI.4 Spin-orbit coupling
      1. VI.4.1 Rashba SOC
      2. VI.4.2 Ising SOC
      3. VI.4.3 Total SOC contribution
    5. VI.5 IVC order
      1. VI.5.1 Singlet IVC order
      2. VI.5.2 Antiferromagnetic IVC order
    6. VI.6 Wire setup and simulation
      1. VI.6.1 Trivial wire simulations
      2. VI.6.2 sIVC-coupled wire simulations
      3. VI.6.3 AFM IVC-couple wire simulations
    7. VI.7 Discussion

## I. Continuum model without spin orbit coupling

###  Continuum model definition

We begin by giving a short summary of the continuum model [1, 2] of twisted
bilayer graphene (TBG) in the absence spin orbit coupling (SOC). We let the
operators $d_{t/b}({\bm{k}})$ denote electron annihilation operators in the
momentum basis residing on the top ($t$) and bottom ($b$) graphene sheets,
which we take to be rotated by an angle $\theta\ll 1$ from one another. In
defining $d_{t/b}$, both sublattice and spin indices have been suppressed. The
regime of interest occurs at very low energies, and it is therefore sufficient
to restrict our study to the low-energy excitations of each of the graphene
monolayers. These states take the form of Dirac fermions at the Brillouin zone
(BZ) corners $K$ and $K^{\prime}=-K$, and we therefore define
$\Psi_{K^{(\prime)},t/b}({\bm{k}})=d_{t/b}({\bm{k}}+{\bm{K}}^{(\prime)}_{t/b})$,
where $K^{(\prime)}_{t}$ and $K^{(\prime)}_{b}$ differ by a small amount as a
result of the twist angle offset between the two layers.

The continuum model Hamiltonian may be expressed as

$\displaystyle H_{\mathrm{cont}}$
$\displaystyle=H_{t}+H_{b}+H_{\mathrm{tun}}.$ (1)

The first two terms on the right-hand side respectively denote the Dirac
Hamiltonian of the top and bottom layers in the absence of tunnelling:

$\displaystyle H_{t/b}$
$\displaystyle=\int_{\bm{k}}\Psi^{\dagger}_{t/b}({\bm{k}}){\mathcalb{h}}_{t/b}({\bm{k}})\Psi_{t/b}({\bm{k}}),$
(2)

where

$\displaystyle{\mathcalb{h}}_{t}({\bm{k}})$
$\displaystyle=-v_{0}e^{i\theta\tau^{z}\sigma^{z}/4}(k_{x}\tau^{z}\sigma^{x}+k_{y}\sigma^{y})e^{-i\theta\tau^{z}\sigma^{z}/4},$
$\displaystyle{\mathcalb{h}}_{b}({\bm{k}})$
$\displaystyle=-v_{0}e^{-i\theta\tau^{z}\sigma^{z}/4}(k_{x}\tau^{z}\sigma^{x}+k_{y}\sigma^{y})e^{i\theta\tau^{z}\sigma^{z}/4}.$
(3)

Here, $\tau^{x,y,z}$ and $\sigma^{x,y,z}$ act on the (suppressed) valley and
sublattice indices, respectively. The Fermi velocity is approximately
$v_{0}\sim 10^{6}\,\mathrm{\text{m/s}}$ [3]. The layers tunnel through

$\displaystyle H_{\mathrm{tun}}$
$\displaystyle=\sum_{\ell=1,2,3}\sum_{v=K,K^{\prime}}\int_{\bm{k}}\Psi^{\dagger}_{v,t}({\bm{k}})T^{(v)}_{\ell}\Psi_{v,t}({\bm{k}}+\eta_{v}{\bm{q}}_{\ell})+h.c.,$
(4)

where $\eta_{K}=+1$, $\eta_{K^{\prime}}=-1$. All other indices are suppressed.
The momenta exchanged are defined as

$\displaystyle{\bm{q}}_{\ell}$
$\displaystyle=k_{\theta}\left(-\sin\left[\frac{2\pi}{3}\left(\ell-1\right)\right]\hat{\bm{x}}+\cos\left[\frac{2\pi}{3}\left(\ell-1\right)\right]\hat{\bm{y}}\right),$
$\displaystyle k_{\theta}$ $\displaystyle=\frac{4\pi}{3a}2\sin(\theta/2),$ (5)

while the tunnelling matrices themselves take the form

$\displaystyle T^{(\pm)}_{\ell}$ $\displaystyle=w_{0}+w_{1}\left(e^{\mp
2\pi(\ell-1)i/3}\sigma^{+}+e^{\pm 2\pi i(\ell-1)/3}\sigma^{-}\right).$ (6)

The tunnelling parameters are typically taken to be
$(w_{0},w_{1})=(85,108)\,\mathrm{\text{meV}}$ [2, 4, 5, 6].

Note that this model allows no tunnelling between valleys $K$, $K^{\prime}$.
This assumption is valid when the twist angle is very small, $\theta\ll 1$, as
considered here, since it then follows that the momentum ${\bm{q}}_{\ell}$ is
much smaller than the magnitude of the momentum separating the two valleys.
Below, this absence of inter-valley tunnelling is formulated in terms of an
emergent U(1) symmetry.

In order to diagonalize $H_{\mathrm{cont}}$, we first observe that because the
momentum arguments of $\Psi_{K^{(\prime)},t/b}({\bm{k}})$ are measured
relative to distinct momenta $K_{t}^{(\prime)}\neq K_{b}^{(\prime)}$, the
interlayer tunnelling term in Eq. (4) does not in fact involve a momentum
transfer of $\pm{\bm{q}}_{\ell}$ between layers. It’s useful to move to a
representation expressed in terms of operators
$\tilde{\Psi}_{K^{(\prime)},t/b}(\bm{k})$ identical to the previous notation
save that the momentum arguments of these new operators are now measured
relative to a common point. In term of the $\tilde{\Psi}$ operators, the
Hamiltonian $H_{\mathrm{cont}}$ may be written in a form where it’s manifestly
apparent that it only involves momentum exchanges by a set of Bravais lattice
vectors $\bm{G}$ defined by basis vectors
${\bm{G}}_{1}={\bm{q}}_{1}-{\bm{q}}_{2}$ and
${\bm{G}}_{2}={\bm{q}}_{2}-{\bm{q}}_{3}$. As an example, we might define, for
valley $K$,

$\displaystyle\tilde{\Psi}_{K,t}({\bm{k}})$
$\displaystyle=\Psi_{K,t}({\bm{k}}),$
$\displaystyle\tilde{\Psi}_{K,b}({\bm{k}})$
$\displaystyle=\Psi_{K,b}({\bm{k}}-{\bm{q}}_{1}).$ (7)

Regardless of the specific conventions chosen, the Hamiltonian may then be
written as

$\displaystyle H_{\mathrm{cont}}$
$\displaystyle=\sum_{v=K,K^{\prime}}\sum_{{\bm{G}},{\bm{G}}^{\prime}}\int_{{\bm{k}}\in{\mathcald{m}\mathrm{BZ}}}\tilde{\Psi}^{\dagger}_{v,i}({\bm{k}}+{\bm{G}}){\mathcalb{h}}^{(v)}_{i,{\bm{G}};j,{\bm{G}}^{\prime}}({\bm{k}})\tilde{\Psi}_{v,j}({\bm{k}}+{\bm{G}}^{\prime}),$
(8)

where $v$ labels the $K$-valley, $i,j$ are combined indices including layer
and sublattice, and the spin index is suppressed. The momentum integration
over ${\bm{k}}$ only includes values within the moiré Brillouin zone (denoted
‘${\mathcald{m}\mathrm{BZ}}$’) defined by the Bravais lattice vectors
${\bm{G}}$.

Diagonalizing $H_{\mathrm{cont}}$ one obtains an infinite set of bands both
above and below charge neutrality. Our focus will be the bands closest to
charge neutrality. It can be shown that these bands will also possess Dirac
cones located at the ${\mathcald{m}\mathrm{BZ}}$ corners, which we denote
$\kappa$ and $\kappa^{\prime}$ to distinguish them from the Brillouin zone
corners of the graphene monolayers. By considering the limit of
infinitesimally small interlayer tunnelling ($w_{0,1}\to 0$ in Eq. (6)), one
can show that for valley $K$, the Dirac cone at $\kappa$ descends from the top
layer, $K_{t}$, while the Dirac cone at $\kappa^{\prime}$ descends from the
bottom layer, $K_{b}$. Similarly, for valley $K^{\prime}$, the Dirac cone at
$\kappa$ descends from the bottom layer, $K_{b}^{\prime}$, while the Dirac
cone at $\kappa^{\prime}$ descends from the top layer, $K_{t}^{\prime}$.

At the magic angle, $\theta\sim 1.1^{\circ}$, the bands above and below charge
neutrality become nearly completely flat, allowing interactions to dominate.
Further, provided $w_{0}<w_{1}$, as is believed to be the case, the flat bands
are also isolated from the ‘dispersive’ bands at higher or lower energy by a
gap $\Delta_{\mathrm{disp}}$. The filling of the flat bands is expressed via
the filling factor $\nu$: all flat bands are empty at $\nu=-4$, all flat band
are filled at $\nu=+4$, and $\nu=0$ corresponds to charge neutrality.

###  Interactions

The primary source of interactions believed to be relevant to TBG is the
Coulomb interaction:

$\displaystyle H_{\mathrm{int}}$ $\displaystyle=\frac{1}{2}\int
d^{2}{\bm{q}}\,\rho^{\dagger}({\bm{q}})V({\bm{q}})\rho({\bm{q}}),$ (9)

where the Coulomb potential satisfies $V({\bm{q}})\propto
1/\left|{\bm{q}}\right|$ (possibly with additional taking screening into
account). In terms of the microscopic graphene operators introduced at the
beginning of the previous section, the density operator takes the form
$\rho({\bm{q}})=\int_{\bm{k}}d^{\dagger}({\bm{k}})d({\bm{k}}+{\bm{q}})$. As
stated in the previous section, it is sufficient to focus on the low-energy
states of the graphene monolayers, which is equivalent to restricting the
momentum arguments of $d({\bm{k}})$ to values close to $K$, $K^{\prime}$, or,
equivalently, writing everything in terms of the $\Psi$ operators defined
above. As a result only density operators whose arguments are either very
small or are very close to $K$, $K^{\prime}$ are important:

$\displaystyle\rho({\bm{q}})$
$\displaystyle\cong\sum_{v=K,K^{\prime}}\int_{\bm{k}}\Psi^{\dagger}_{v}({\bm{k}})\Psi_{v}({\bm{k}}+{\bm{q}}),$
$\displaystyle\rho({\bm{q}}+{\bm{K}})$
$\displaystyle\cong\int_{\bm{k}}\Psi^{\dagger}_{K}({\bm{k}})\Psi_{K^{\prime}}({\bm{k}}+{\bm{q}}).$
(10)

In line with the reasoning above, the integration over ${\bm{k}}$ is
restricted to momenta that are small compared to the magnitude of the momentum
difference $\left|K-K^{\prime}\right|$. These expressions imply that the
interaction term $H_{\mathrm{int}}$ may be separated into two pieces:
$H_{\mathrm{int}}\cong H_{C}+H_{J}$ with

$\displaystyle H_{C}$
$\displaystyle=\frac{1}{2}\int_{\bm{q}}\rho^{\dagger}({\bm{q}})V({\bm{q}})\rho({\bm{q}}),$
$\displaystyle H_{J}$
$\displaystyle=\int_{\bm{q}}\rho^{\dagger}({\bm{q}}+{\bm{K}})V({\bm{q}}+{\bm{K}})\rho({\bm{q}}+{\bm{K}}).$
(11)

Again, ${\bm{q}}$ is restricted to small momenta. Given the form of
$V({\bm{q}})$ quoted above, it is easy to demonstrate that the magnitude of
the second term, which we will denote the Hund’s term, is suppressed by a
factor of $\sim\theta\ll 1$. As discussed in Ref. 7, it is reasonable to
consider a perturbative expansion with $H_{C}$ as the dominant contribution;
this reasoning is what ultimately leads the authors to propose that TBG
possesses intervalley coherent (IVC) order at $\nu=0$ (charge neutrality) and
at $\nu=\pm 2$, as discussed in Sec. IV.

The Hund’s term $H_{J}$ is only relevant in selecting among ground states that
otherwise are degenerate at the level of $H_{C}$ and $H_{\mathrm{cont}}$.
However, given its small magnitude, $H_{J}$ is susceptible to substantial
renormalization; in particular, phonon interactions my potentially alter the
sign of the coupling $V_{\mathrm{eff}}({\bm{K}}+{\bm{k}})\cong
V_{\mathrm{eff}}({\bm{K}})$ [8]. Given the inherent uncertainty is such
calculations, we take the sign of the Hund’s coupling as an unknown, which
leads us to consider two distinct (but related) many-body ground states for
the fillings under consider in Sec. IV below.

###  Symmetry action on microscopic operators

We now summarize the symmetries that are present in the system. We work with
the operators $\Psi({\bm{k}})$, but note that the momentum-shifted operators
$\tilde{\Psi}({\bm{k}})$ defined in the previous section transform in an
identical fashion.

In addition to the usual $\mathrm{U_{c}}(1)$ symmetry associated with charge
conservation, TBG without SOC possesses two continuous symmeties, a
$\mathrm{U_{v}}(1)$ valley symmetry and the SU(2) spin symmetry:

$\displaystyle\mathrm{U}_{\mathrm{v}}(1)$
$\displaystyle:\qquad\Psi({\bm{k}})\to e^{i\phi\tau^{z}}\Psi({\bm{k}}),$
$\displaystyle\mathrm{SU}(2)$ $\displaystyle:\qquad\Psi({\bm{k}})\to
e^{i\theta(\hat{\bm{n}}\cdot\bm{s})/2}\Psi({\bm{k}}),$ (12)

where $\tau^{x,y,z}$ ($s^{x,y,z}$) Pauli matrices act on the valley $K$
indices (spin indices) and $\hat{\bm{n}}$ is a unit vector. The valley
symmetry is a direct consequence of the (physically correct) omission of
tunnelling terms between fermions originating in valley $K$ and those of
valley $K^{\prime}$ in Sec. I.1. In fact, both $H_{\mathrm{cont}}$ and $H_{C}$
are invariant under a much larger continuous symmetry group,
$\mathrm{U(2)\times
U(2)}\cong\mathrm{U_{c}}(1)\times\mathrm{U_{v}}(1)\times\mathrm{SU(2)_{K}\times\mathrm{SU}(2)_{K^{\prime}}}$,
where $\mathrm{SU}(2)_{K/K^{\prime}}$ correspond to _independent_ spin
rotations in valleys $K/K^{\prime}$:

$\displaystyle\mathrm{SU}(2)_{K}\times\mathrm{SU}(2)_{K^{\prime}}$
$\displaystyle:\qquad\Psi({\bm{k}})\to\left(\mathcal{P}_{K}e^{i\phi_{+}\hat{\bm{n}}_{K}\cdot\bm{s}/2}+\mathcal{P}_{K^{\prime}}e^{i\phi_{-}\hat{\bm{n}}_{K^{\prime}}\cdot\bm{s}/2}\right)\Psi({\bm{k}}).$
(13)

Here, $\hat{\bm{n}}_{K^{(\prime)}}\in S^{2}$ are arbitrary unit vectors and
$\mathcal{P}_{K}=(\mathds{1}+\tau^{z})/2$ and
$\mathcal{P}_{K^{\prime}}=(\mathds{1}-\tau^{z})/2$ project onto the $K$ and
$K^{\prime}$ valleys, respectively. This symmetry is only broken once the
effects of the Hund’s interaction in Eq. (11) is taken into account.

There are a number of additional discrete symmetries, as well as time
reversal. While all of these symmetry transformations should be composed with
spin rotations, it is convenient in this appendix to separate the internal
degrees of freedom deriving from spin from those of the discrete symmetry
operations. We therefore consider the action of “spinless” versions of the
discrete symmetries:

$\displaystyle{\mathcald{T}}$
$\displaystyle:\qquad\Psi({\bm{k}})\to\tau^{x}\Psi(-{\bm{k}}),\qquad i\to-i,$
$\displaystyle{\mathcald{C}}_{2}$
$\displaystyle:\qquad\Psi({\bm{k}})\to\tau^{x}\sigma^{x}\Psi(-{\bm{k}}),$
$\displaystyle{\mathcald{C}}_{2}{\mathcald{T}}$
$\displaystyle:\qquad\Psi({\bm{k}})\to\sigma^{x}\Psi({\bm{k}}),\qquad i\to-i,$
$\displaystyle{\mathcald{C}}_{3}$ $\displaystyle:\qquad\Psi({\bm{k}})\to
e^{-2\pi i\tau^{z}\sigma^{z}/3}\Psi(R_{3}{\bm{k}}),$
$\displaystyle{\mathcald{M}}_{y}$
$\displaystyle:\qquad\Psi({\bm{k}})\to\mu^{x}\sigma^{x}\Psi(R_{y}{\bm{k}}),$
(14)

where $\Psi({\bm{k}})$ are the 16 component electron annihilation operators of
the two graphene layers and the Pauli matrices $\tau^{x,y,z}$,
$\sigma^{x,y,z}$, and $\mu^{x,y,z}$ act on $K$-valley, sublattice, and layer
indices respectively. The matrices $R_{3}$ and $R_{y}$ are given by

$\displaystyle R_{3}$ $\displaystyle=\begin{pmatrix}-1/2&-\sqrt{3}/2\\\
\sqrt{3}/2&-1/2\end{pmatrix},$ $\displaystyle R_{y}$
$\displaystyle=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$ (15)

We have redundantly included the composite symmetry
${\mathcald{C}}_{2}{\mathcald{T}}$ as it commutes with the $\mathrm{U_{v}}(1)$
symmetry, making it especially useful when considering only a single
$K$-valley.

The symmetries listed above are not the true physical symmetries of the
problem, and they will clearly no longer be preserved when SOC is included
below. The physical symmetries may be expressed as

$\displaystyle\mathcal{T}$ $\displaystyle=is^{y}{\mathcald{T}},$
$\displaystyle C_{2}$ $\displaystyle=is^{z}{\mathcald{C}}_{2},$ $\displaystyle
C_{2}\mathcal{T}$ $\displaystyle=is^{x}{\mathcald{C}}_{2}{\mathcald{T}},$
$\displaystyle C_{3}$ $\displaystyle=e^{-2\pi is^{z}/3}{\mathcald{C}}_{3},$
$\displaystyle M_{y}$ $\displaystyle=is^{x}{\mathcald{M}}_{y}.$ (16)

In reality, it is the ‘spinful’ symmetries above that should be viewed as
fundamental. The ‘spinless’ symmetries of Eq. (I.3) are then more
appropriately obtained by appending an additional spin rotations using the
SU(2) spin degree of freedom. For bookkeeping purposes we nevertheless largely
discuss symmetries in terms of the ‘spinless’ symmetries.

## II. Spin orbit coupling in twisted bilayer graphene

In this section, we outline how spin-orbit coupling (SOC) is introduced to the
continuum model of twisted bilayer graphene. We begin by discussing a
monolayer of graphene coupled to a transition metal dichalcogenide (TMD),
before considering what happens in twisted bilayer graphene.

###  Monolayer graphene with induced spin orbit coupling

We begin by describing the induced spin-orbit felt by monolayer graphene
adjacent to a TMD. In the absence of SOC, the low-energy Hamiltonian for the
monolayer is

$\displaystyle H_{\mathrm{MLG},0}$
$\displaystyle=-v_{0}\int_{\bm{k}}\psi^{\dagger}({\bm{k}})\Big{(}k_{x}\tau^{z}\sigma^{x}+k_{y}\sigma^{y}\Big{)}\psi({\bm{k}}),$
(17)

where $\psi({\bm{k}})$ is an eight component spinor with sublattice, valley,
and spin indices. The Pauli matrices $\sigma^{x,y,z}$ act on sublattice
indices of the spinor, while $\tau^{x,y,z}$ act on the valley indices. The
proximate TMD induces both Ising and Rashba terms, which may be included by
taking $H_{\mathrm{MLG},0}\to H_{\mathrm{MLG},0}+H_{\mathrm{MLG,SO}}$, where
[9, 10, 11]

$\displaystyle H_{\mathrm{MLG,SO}}$
$\displaystyle=\int_{\bm{k}}\psi^{\dagger}({\bm{k}})\Bigg{(}\frac{\lambda_{I}}{2}\tau^{z}s^{z}+\frac{\lambda_{R}}{2}\left(\tau^{z}\sigma^{x}s^{y}-\sigma^{y}s^{x}\right)\Bigg{)}\psi({\bm{k}}).$
(18)

Here, $s^{x,y,z}$ act on the spin indices. The parameters $\lambda_{I}$ and
$\lambda_{R}$ quantify the strength of the Ising and Rashba terms
respectively. Their values, as expected from numerical calculations, vary
widely: $\lambda_{I}\sim 1-5\,\mathrm{\text{meV}}$, and $\lambda_{R}\sim
1-15\,\mathrm{\text{meV}}$ [12, 9, 10, 13, 14]. We note that only Ref. 14
considered the effects of the twist angle between the graphene sheet and the
TMD monolayer, and that the Rashba coupling they obtain is consequently
significantly smaller than predicted previously: $\lambda_{R}\lesssim
4\,\mathrm{\text{meV}}$. The presence of SOC has also been confirmed
experimentally [15, 16, 17, 18], but extracting the magnitudes of
$\lambda_{I}$ and $\lambda_{R}$ is difficult.

###  Twisted bilayer graphene with induced spin orbit coupling

We now consider what occurs when a TMD is placed adjacent to one or both of
the graphene monolayers that compose twisted bilayer graphene. We assume for
the moment the most physically relevant scenario in which a TMD monolayer is
adjacent to a single layer of graphene, as shown in the Fig. 1 of the main
text. In contrast to that figure, we first assume the TMD is alongside the top
layer. The quadratic portion of the Hamiltonian is thus modified to

$\displaystyle H_{\mathrm{cont,SO}}$
$\displaystyle=H_{\mathrm{cont}}+H_{\mathrm{SO}}$ (19)

where

$\displaystyle H_{\mathrm{SO}}$
$\displaystyle=\int_{\bm{k}}\Psi^{\dagger}({\bm{k}})\big{(}{\mathcalb{h}}_{I}+{\mathcalb{h}}_{R}\big{)}\Psi({\bm{k}}).$
(20)

Here, ${\mathcalb{h}}_{I}$ and ${\mathcalb{h}}_{R}$ are simply the terms from
Eq. (18):

$\displaystyle{\mathcalb{h}}_{I}$
$\displaystyle=\frac{\lambda_{I}}{2}\mathcal{P}_{t}\tau^{z}s^{z},$
$\displaystyle{\mathcalb{h}}_{R}$
$\displaystyle=\frac{\lambda_{R}}{2}\mathcal{P}_{t}\left(\tau^{z}\sigma^{x}s^{y}-\sigma^{y}s^{x}\right),$
(21)

where $\mathcal{P}_{t}=(\mathds{1}+\mu^{z})/2$ projects onto the top layer:
only the operators $\Psi_{t}$ are present in Eq. (20). We note that both
${\mathcalb{h}}_{R}$ and ${\mathcalb{h}}_{I}$ are rotationally invariant,
which explains the absence of the rotational matrices present in the Dirac
parts of the Hamiltonian in Eq. (3). The influence of SOC in TBG has been
observed experimentally in Ref. 19.

The addition of ${\mathcalb{h}}_{I}$ and ${\mathcalb{h}}_{R}$ to the
Hamiltonian breaks the SU(2) spin as well as a number of other symmetries. In
Table 1, the symmetries preserved by the introduction of Rashba and Ising are
separately listed. Equivalently, the table may be interpreted as showing the
symmetries preserved by the Hamiltonian $H_{\mathrm{cont,SO}}$ when
$\lambda_{R}\neq 0$, $\lambda_{I}=0$ and when $\lambda_{R}=0$,
$\lambda_{I}\neq 0$, respectively.

Table 1 will serve as the basis for the body of this appendix and so we
describe the information it contains in detail. The top row of the table
provides both the continuous subgroups and the generators of the discrete
symmetry operations that are preserved by the interacting continuum model
$H_{\mathrm{cont}}+H_{C}$ without SOC and in the absence of the Hund’s
coupling. The symmetry group represented by the top row of Table 1 is the
largest group our theory can realize—all of the symmetry groups represented in
the rows below are contained within the group defined by the top row. (The
table does not explicitly reference the emergent
$\mathrm{SU}(2)_{K}\times\mathrm{SU}(2)_{K^{\prime}}$ symmetry of Eq. (13).
When the terms of interest break this symmetry in a way that the existence of
the $\mathrm{SU}(2)_{K}\times\mathrm{SU}(2)_{K^{\prime}}$ parent symmetry
matters, the resulting preserved symmetries are expressed across both
columns.) In the columns labelled by the two continuous symmetries, SU(2) and
$\mathrm{U_{v}}(1)$, the corresponding entry of the Rashba and Ising shows the
subgroup of the symmetry still preserved when Rashba or Ising SOC is present
or, if no subgroup survives, an ‘✗’ is written instead. The SOC terms also
break a number of the discrete symmetries of Eq. (I.3). When one or more of
the continuous symmetries has also been broken, a residual composite symmetry
in which a discrete symmetry operation is followed by a continuous symmetry
operation may survive. In this case, that composite symmetry is listed
instead. For instance, for Rashba SOC, none of the ‘spinless’ versions of the
symmetries survive—${\mathcalb{h}}_{R}$ is only invariant under their action
when they are composed with additional spin rotation transformations, as seen
in the first row of Table 1. We see that the discrete symmetries preserved by
Rashba are in fact the physical space group operations and electronic time
reversal provided in Eq. (I.3). We note that the generators chosen in each
instance are not unique, _e.g._ , we could equally well have added a
$\mathrm{U_{v}}(1)$ transformation to each of the operations for both Rashba
and Ising SOC.

When both Rashba and Ising SOC are simultaneously present (_i.e._ ,
$H_{\mathrm{cont,SO}}$ with $\lambda_{R},\lambda_{I}\neq 0$), an even smaller
set of symmetries remains. As an example, the Rashba term is preserved under
the physical inversion operation, $is^{z}{\mathcald{C}}_{2}=C_{2}$, while the
Ising term instead requires a spin flipped version,
$is^{x}{\mathcald{C}}_{2}$. These symmetries are not compatible and thus we
conclude that in the presence of both Ising and Rashba SOC, inversion is no
longer present. As we argue below, we are able to neglect such effects.

###  Multiple TMDs and mirror symmetry ${\mathcald{M}}_{y}$

Inclusion of $H_{\mathrm{SO}}$ in Eqs. (19) and (20) breaks the mirror
symmetry ${\mathcald{M}}_{y}$ since the latter interchanges the layers, only
one of which possesses proximity-induced SOC. It is useful below to consider
the most physically relevant scenario in which both graphene monolayers
“symmetrically” possess SOC. More precisely, we will construct Rashba and
Ising terms that preserve a spin rotated mirror symmetry,
$is^{x}{\mathcald{M}}_{y}=M_{y}$, and the spinless mirror,
${\mathcald{M}}_{y}$, respectively. To do so we need to couple _both_ graphene
monolayers to a TMD, so that
$H_{\mathrm{cont,SO}}=H_{\mathrm{cont}}+H_{\mathrm{SO}}+H_{\mathrm{SO}}^{\prime}$
with $H_{\mathrm{SO}}^{\prime}$ defined in direct analogy to $H_{\mathrm{SO}}$
except using operators originating from the bottom graphene sheet with
corresponding SOC strengths $\lambda_{R}^{\prime}$ and $\lambda_{I}^{\prime}$.
Here, we assume that $\lambda_{I/R}^{\prime}$ possess the same magnitudes as
$\lambda_{I/R}$, but allow them to take different signs. The choice most
relevant to the physical scenario in which a single monolayer of TMD is
present is determined below.

#### II.3.1. Ising SOC

We begin by outlining what occurs to first order when Ising SOC of strength
$\lambda_{I}$ is added to the Dirac Hamiltonian of the top layer only (in the
absence of Rashba). We focus on the physics occurring close to charge
neutrality at the Dirac cones at $\kappa$, $\kappa^{\prime}$, for the moment
specifying to the $K$ valley ($\tau^{z}=+1$). The analysis for the
$K^{\prime}$ valley follows directly. Using first order perturbation theory
similar to the analysis used to ‘derive’ the magic angle in Ref. 2, one can
show that the presence of $\lambda_{I}\neq 0$ induces _effective_ Ising SOC
terms for the Dirac cones at both $\kappa=K_{t}$ and $\kappa^{\prime}=K_{b}$.
More precisely, in the low energy theory of the Dirac cones of the moiré
system, to first order in $\lambda_{I}$, the Dirac cone at $\kappa$ has an
effective SOC parameter $\tilde{\lambda}_{I,\kappa}=c_{1}\lambda_{I}$ while
the Dirac cone at $\kappa^{\prime}$ has SOC parameter
$\tilde{\lambda}_{I,\kappa^{\prime}}=c_{2}\lambda_{I}$, where $c_{1}$ and
$c_{2}$ are real numbers. Importantly, one finds that $c_{1}$ and $c_{2}$ have
the same sign, meaning that $\tilde{\lambda}_{I,\kappa}$ and
$\tilde{\lambda}_{I,\kappa^{\prime}}$ do as well. These results are further
supported by numerics. The preceding scenario is what we want the setup with
two TMD monolayers to resemble.

When Ising SOC is also present in the Dirac Hamiltonian of the bottom layer,
it will similarly induce effective Ising SOC into the Dirac cones at $\kappa$,
$\kappa^{\prime}$:
$\tilde{\lambda}_{I,\kappa}^{\prime}=c_{2}\lambda^{\prime}_{I}$ and
$\tilde{\lambda}^{\prime}_{I,\kappa^{\prime}}=c_{1}\lambda_{I}^{\prime}$,
where $c_{1}$ and $c_{2}$ are the same constants appearing above and their
assignment is determined by symmetry. Within the first order analysis we
consider here, the total effective Ising parameters are given by the sum,
_i.e._
$\tilde{\lambda}_{I,\kappa}^{\mathrm{tot}}=\tilde{\lambda}_{I,\kappa}+\tilde{\lambda}_{I,\kappa}^{\prime}$
and
$\tilde{\lambda}_{I,\kappa^{\prime}}^{\mathrm{tot}}=\tilde{\lambda}_{I,\kappa^{\prime}}+\tilde{\lambda}_{I,\kappa^{\prime}}^{\prime}$.
It’s clear that in order for $\tilde{\lambda}^{\mathrm{tot}}_{I,\kappa}$ to
have the same sign as $\tilde{\lambda}^{\mathrm{tot}}_{I,\kappa^{\prime}}$, we
must have $\mathrm{sgn}(\lambda_{I})=\mathrm{sgn}(\lambda_{I}^{\prime})$. As
mentioned, we further specify to the situation in which they have the same
magnitude. We conclude that the appropriate symmetry-enhanced version of
${\mathcalb{h}}_{I}$ is

$\displaystyle{\mathcalb{h}}_{I}^{*}$
$\displaystyle=\frac{1}{2}(\lambda_{I}\mathcal{P}_{t}+\lambda_{I}\mathcal{P}_{b})\tau^{z}s^{z}=\frac{\lambda_{I}}{2}\tau^{z}s^{z},$
(22)

where $\mathcal{P}_{t/b}=(\mathds{1}\pm\mu^{z})/2$ project onto the top or
bottom graphene monolayer. In Table 1, in the column labelled
‘${\mathcald{M}}_{y}$,’ we write ${\mathcald{M}}_{y}^{*}$ in the Ising column
to indicate that the mirror symmetry ${\mathcald{M}}_{y}$ is only truly
preserved when the first-quantized Hamiltonian ${\mathcalb{h}}_{I}^{*}$ is
used in place of ${\mathcalb{h}}_{I}$. The Hamiltonian
${\mathcalb{h}}_{I}^{*}$ is otherwise invariant under the same symmetries as
${\mathcalb{h}}_{I}$.

#### II.3.2. Rashba SOC

An identical analysis to the one described above for the Ising SOC indicates
that the presence of Rashba SOC in the Dirac cone descending from one of the
graphene layers (say, the top at $\kappa=K_{t}$) should be accompanied by
_opposite sign_ Rashba SOC in the Dirac cone descending from the other
monolayer (bottom, say, at $\kappa^{\prime}=K_{b}$). It follows that the two-
layer extension of the Rashba SOC-coupled theory that most resembles the
single layer case (_i.e._ , the SOC contribution is not cancelled at linear
order in $\lambda_{R}$), is obtained through

$\displaystyle{\mathcalb{h}}_{R}^{*}$
$\displaystyle=\frac{1}{2}(\lambda_{R}\mathcal{P}_{t}-\lambda_{R}\mathcal{P}_{b})(\tau^{z}\sigma^{x}s^{y}-\sigma^{y}s^{x})=\frac{\lambda_{R}}{2}\mu^{z}(\tau^{z}\sigma^{x}s^{y}-\sigma^{y}s^{x}).$
(23)

The action of all the symmetries is the same for ${\mathcalb{h}}_{R}^{*}$ as
for ${\mathcalb{h}}_{R}$ save for the mirror symmetry ${\mathcald{M}}_{y}$. It
of course requires a spin flip operation—in line with our comment above that
Rashba preserves the physical version of the symmetry operations of Eq. (I.3).
An asterisk is added in Table 1 to indicate that $is^{x}{\mathcald{M}}_{y}$ is
only preserved by the modified Rashba term ${\mathcalb{h}}^{*}_{R}$. That is,
we write $is^{x}{\mathcald{M}}_{y}^{*}$ in the appropriate column.

Order | SU(2) | Uv(1) | ${\mathcald{T}}=\tau^{x}{\mathcal{K}}$ | ${\mathcald{C}}_{2}$ | ${\mathcald{C}}_{2}{\mathcald{T}}$ | ${\mathcald{C}}_{3}$ | ${\mathcald{M}}_{y}$ |  | Mixed
---|---|---|---|---|---|---|---|---|---
Rashba (R) | ✗ | Uv(1) | $is^{y}{\mathcald{T}}$ | $is^{z}{\mathcald{C}}_{2}$ | $is^{x}{\mathcald{C}}_{2}{\mathcald{T}}$ | $e^{-2\pi is^{z}/3}{\mathcald{C}}_{3}$ | $is^{x}{\mathcald{M}}_{y}^{*}$ |  |
Ising (I) | $\mathrm{U}_{z}(1)$ | Uv(1) | $is^{y}{\mathcald{T}}$ | $is^{x}{\mathcald{C}}_{2}$ | ${\mathcald{C}}_{2}{\mathcald{T}}$ | ${\mathcald{C}}_{3}$ | ${\mathcald{M}}_{y}^{*}$ |  |
sIVC | SU(2) | ✗ | $i\tau^{y}{\mathcal{K}}$ | $\tau^{z}{\mathcald{C}}_{2}$ | ${\mathcald{C}}_{2}{\mathcald{T}}$ | ${\mathcald{C}}_{3}$ | $i\tau^{z}{\mathcald{M}}_{y}$ |  |
sIVC + R | ✗ | ✗ | $\tau^{y}s^{y}{\mathcal{K}}$ | $\tau^{z}s^{z}{\mathcald{C}}_{2}$ | $is^{x}{\mathcald{C}}_{2}{\mathcald{T}}$ | $e^{-2\pi is^{z}/3}{\mathcald{C}}_{3}$ | $\tau^{z}s^{x}{\mathcald{M}}_{y}^{*}$ |  |
sIVC + I | $\mathrm{U}_{z}(1)$ | ✗ | $\tau^{y}s^{y}{\mathcal{K}}$ | $\tau^{z}s^{y}{\mathcald{C}}_{2}$ | ${\mathcald{C}}_{2}{\mathcald{T}}$ | ${\mathcald{C}}_{3}$ | $i\tau^{z}{\mathcald{M}}_{y}^{*}$ |  |
tIVC | $\mathrm{SU(2)_{tIVC}}$ | $is^{y}{\mathcald{T}}$ | $is^{x}{\mathcald{C}}_{2}$ | ${\mathcald{C}}_{2}{\mathcald{T}}$ | ${\mathcald{C}}_{3}$ | $i\tau^{z}{\mathcald{M}}_{y}$ |  |
tIVC + R | ✗ | ✗ | $is^{y}{\mathcald{T}}$ | $\tau^{z}s^{z}{\mathcald{C}}_{2}$ | $\tau^{z}s^{x}{\mathcald{C}}_{2}{\mathcald{T}}$ | $e^{-2\pi is^{z}/3}{\mathcald{C}}_{3}$ | $is^{x}{\mathcald{M}}_{y}^{*}$ |  |
tIVC + I | $\mathrm{U}_{z}(1)$ | ✗ | $is^{y}{\mathcald{T}}$ | $is^{x}{\mathcald{C}}_{2}$ | ${\mathcald{C}}_{2}{\mathcald{T}}$ | ${\mathcald{C}}_{3}$ | $i\tau^{z}{\mathcald{M}}_{y}^{*}$ |  |
FM IVC | $\mathrm{U}_{z}(1)$ | ✗ | $i\tau^{y}{\mathcal{K}}$ | $i\tau^{z}{\mathcald{C}}_{2}$ | ${\mathcald{C}}_{2}{\mathcald{T}}$ | ${\mathcald{C}}_{3}$ | $i\tau^{z}{\mathcald{M}}_{y}$ |  |
FM IVC + R | ✗ | ✗ | ✗ | $\tau^{z}s^{z}{\mathcald{C}}_{2}$ | ✗ | $e^{-2\pi is^{z}/3}{\mathcald{C}}_{3}$ | ✗ |  | $is^{z}{\mathcald{M}}_{y}^{*}{\mathcald{T}}$
FM IVC + I | $S^{z}$ | ✗ | ✗ | ✗ | ${\mathcald{C}}_{2}{\mathcald{T}}$ | ${\mathcald{C}}_{3}$ | $i\tau^{z}{\mathcald{M}}_{y}^{*}$ |  |
AFM IVC |  $\tilde{\mathrm{U}}_{z}(1)$, $\mathrm{U}_{\mathrm{v}\cdot z}(1)$ | $is^{y}{\mathcald{T}}$ | $is^{y}{\mathcald{C}}_{2}$ | ${\mathcald{C}}_{2}{\mathcald{T}}$ | ${\mathcald{C}}_{3}$ | $i\tau^{z}{\mathcald{M}}_{y}$ |  |
AFM IVC + R | ✗ | ✗ | $is^{y}{\mathcald{T}}$ | ✗ | ✗ | $e^{-2\pi i(s^{z}+\tau^{z})/3}{\mathcald{C}}_{3}$ | ✗ |  | $\tau^{z}s^{y}{\mathcald{C}}_{2}{\mathcald{M}}_{y}^{*}$
Table 1:  Symmetries preserved in 2$d$ in the presence of the various order
parameters and SOC terms listed. The title row provides a list the generators
of the symmetry group preserved by the Hamiltonian $H_{\mathrm{cont}}+H_{C}$
in the absence of SOC or IVC order. The entries in the columns corresponding
to the continuous symmetries, SU(2) and ${\mathrm{U_{v}}(1)}$, show the
subgroup preserved by the order labelling the row. Notably, when the $z$
component of spin is conserved, the subgroup $\mathrm{U}_{z}(1):c({\bm{k}})\to
e^{i\theta s^{z}/2}c({\bm{k}})$ remains a good symmetry. When no subgroup is
preserved, an ‘✗’ is written instead. While we do not explicitly indicate
reference the $\mathrm{SU}(2)_{K}\times\mathrm{SU}(2)_{K^{\prime}}$ symmetry
in the top row, we do record the scenarios in which a preserved continuous
symmetry descends from this larger group, _i.e._ the preserved subgroup or
subgroups “mix” the SU(2) spin and $\mathrm{U_{v}}(1)$ symmetries. In this
case, the preserved subgroups are listed across both columns. This scenario is
relevant for the tIVC state, which preserves the
$\mathrm{SU}(2)_{\mathrm{tIVC}}$ symmetry with generators
$\\{s^{z},\tau^{z}s^{x},\tau^{z}s^{y}\\}$, and for the AFM IVC state, which
preserves the two U(1) symmetries $\tilde{\mathrm{U}}_{z}(1):\;c({\bm{k}})\to
e^{i\theta(\tau^{z}+s^{z})/2}c({\bm{k}})$ and $\mathrm{U}_{\mathrm{v}\cdot
z}(1):c({\bm{k}})\to e^{i\theta\tau^{z}s^{z}/2}c({\bm{k}})$. The entries in
the columns corresponding to the discrete time reversal and lattice symmetries
show the generator of a conserved symmetry either equal to the discrete
symmetry or obtained by composing it with an element of a broken continuous
symmetry group. When no such combination exists, ‘✗’ is present instead. We
note that $is^{y}{\mathcald{T}}=\mathcal{T}$ is the physical time reversal
symmetry and that $i\tau^{y}\mathcal{K}={\mathcald{T}}_{\mathrm{sIVC}}$ is the
non-unitary symmetry preserved by the sIVC state and discussed in the main
text. The relation of the other generators to the physical symmetries of the
system is provided in Eq. (I.3). When relevant, the final column, ‘Mixed,’
lists any conserved generators arising out of the composition of multiple
discrete symmetries in addition to possible continuous symmetry operations.

## III. Proximity-coupled wire at $\nu=\pm 4$

In this section, we illustrate our derivation of the “trivial” wire
Hamiltonian. We derive the effective Hamiltonians in two dimensions for the
flat bands about the moiré BZ centre, denoted the $\gamma$ point (Secs. III.1
and III.2), which allows us to extract useful information on the scaling of
the parameters, as we describe in Sec. III.3. We finish with a discussion in
Sec. III.4 of how our analysis would differ in the absence of the
${\mathcald{C}}_{3}$ and/or mirror symmetries.

###  Projection to flat bands in 2$d$ without SOC

The foundation of our analysis is the flat band Hamiltonian without SOC or IVC
order—these pieces will be added in a perturbative fashion in subsequent
sections using the basis and symmetry action described here.

As mentioned at the end of Sec. I.1, in order to diagonalize the continuum
model, it is convenient to write it in terms of moiré lattice vectors
${\bm{G}}$ and momenta ${\bm{k}}$ restricted to the moiré BZ. We reproduce
here Eq. (24), the continuum model Hamiltonian in the absence of SOC:

$\displaystyle H_{\mathrm{cont}}$
$\displaystyle=\sum_{v=K,K^{\prime}}\int_{{\bm{k}}\in{\mathcald{m}\mathrm{BZ}}}\tilde{\Psi}^{\dagger}_{v,i}({\bm{k}}+{\bm{G}}){\mathcalb{h}}^{(v)}_{i,{\bm{G}};j,{\bm{G}}^{\prime}}({\bm{k}})\tilde{\Psi}_{v,j}({\bm{k}}+{\bm{G}}^{\prime}),$
(24)

where $i,j$ are combined indices including layer and sublattice, and
${\bm{G}}$, ${\bm{G}}^{\prime}$ are moiré reciprocal lattice vectors. The spin
index is suppressed. Recall that the $\tilde{\Psi}$ operators are related to
the $\Psi$ operators through a simple momentum shift (_e.g._ , Eq. (7)), and
that they therefore transform in the same way under the symmetry action
detailed in Sec. I.3. The Hamiltonian $H_{\mathrm{cont}}$ is diagonalized
through a unitary transformation

$\displaystyle\tilde{\Psi}_{v,i}({\bm{k}}+{\bm{G}})$
$\displaystyle=\sum_{\alpha}\tilde{U}^{(v)}_{i,{\bm{G}};\alpha}({\bm{k}})\tilde{c}_{v,\alpha}({\bm{k}}),$
$\displaystyle\tilde{c}_{v,\alpha}({\bm{k}})$
$\displaystyle=\sum_{i,{\bm{G}}}\tilde{U}^{(v){\dagger}}_{\alpha;i,{\bm{G}}}({\bm{k}})\tilde{\Psi}_{v,i}({\bm{k}}+{\bm{G}}),$
(25)

satisfying

$\displaystyle\sum_{i,{\bm{G}},j,{\bm{G}}^{\prime}}\tilde{U}^{(v){\dagger}}_{\alpha;i,{\bm{G}}}{\mathcalb{h}}_{i,{\bm{G}};j,{\bm{G}}^{\prime}}^{(v)}({\bm{k}})\tilde{U}_{j,{\bm{G}}^{\prime};\beta}^{(v)}({\bm{k}})=\delta_{\alpha\beta}\epsilon^{(v)}_{\alpha}({\bm{k}}).$
(26)

Here, $\epsilon_{\alpha}^{(v)}({\bm{k}})$ represents the energy of band
$\alpha$ at momentum ${\bm{k}}$ for valley $v$. Time reversal requires that
$\epsilon^{(K)}_{\alpha}({\bm{k}})=\epsilon^{(K^{\prime})}_{\alpha}(-{\bm{k}})$.

As mentioned at the end of Sec. I.1, the flat bands, $\alpha\in\mathit{fl}$,
comprise the band above and below charge neutrality for each spin and valley,
meaning that the Fermi energy intersects these states for fillings
$-4<\nu<+4$. It turns out that the basis defined in Eq. (26) is not the most
convenient for describing the flat bands. We instead choose a basis in which
the operators we work with transform in a certain way under the symmetries of
the Hamiltonian. This basis change is accomplished simply through a rotation

$\displaystyle\tilde{c}_{v,\alpha}({\bm{k}})$
$\displaystyle=V^{(v)}_{\alpha\beta}({\bm{k}}){c}_{v,\beta}({\bm{k}}),$
$\displaystyle c_{v,\alpha}({\bm{k}})$
$\displaystyle=V^{(v)\dagger}_{\alpha\beta}({\bm{k}})\tilde{c}_{v,\beta}({\bm{k}}),$
(27)

where $V_{\alpha\beta}^{(v)}({\bm{k}})$ is a $2\times 2$ unitary matrix. It
was demonstrated in Ref. zou_band_2018mbultinck_ground_2020 that there exists
a basis (_i.e._ , a set of matrices $V_{\alpha\beta}^{(v)}({\bm{k}})$) in
which the flat band operators $c({\bm{k}})$ transform as

$\displaystyle{\mathcald{T}}$ $\displaystyle:\quad
c({\bm{k}})\to\tau^{x}c(-{\bm{k}}),\;i\to-i,$
$\displaystyle{\mathcald{C}}_{2}$ $\displaystyle:\quad c({\bm{k}})\to
e^{i\theta({\bm{k}})}\tau^{x}\sigma^{x}c(-{\bm{k}}),$
$\displaystyle{\mathcald{C}}_{2}{\mathcald{T}}$ $\displaystyle:\quad
c({\bm{k}})\to e^{i\theta({\bm{k}})}\sigma^{x}c({\bm{k}}),\;i\to-i.$ (28)

Here, $\tau^{x,y,z}$ Pauli matrices continue to represent transformations
acting on the valley indices of the operators. Conversely, the
$\sigma^{x,y,z}$ Pauli matrix is no longer acting on the sublattice indices
(since sublattice is typically not a good quantum number), but are instead
acting on an additional ‘band index’ in our basis111In certain approximations,
these flat band indices acted on by $\sigma^{x,y,z}$ in fact coincide with the
actual sublattice index of the UV theory—that is not the case here, but
explains the nomenclature.. Close to the ${\gamma}$ point, $\theta({\bm{k}})$
may be set to zero. Further, we find that, when applicable, the transformation
of $c({\bm{k}})$ under ${\mathcald{C}}_{3}$ and ${\mathcald{M}}_{y}$ can be
written

$\displaystyle{\mathcald{C}}_{3}$ $\displaystyle:\quad c({\bm{k}})\to
c(R_{3}{\bm{k}}),$ $\displaystyle{\mathcald{M}}_{y}$ $\displaystyle:\quad
c({\bm{k}})\to\sigma^{x}c(R_{y}{\bm{k}}),$ (29)

within an open region containing the $\gamma$ point. The matrices $R_{3}$ and
$R_{y}$ are provided in Eq. (15). (See Sec. III.4 for a discussion of what in
the absence of ${\mathcald{M}}_{y}$ and/or ${\mathcald{C}}_{3}$.) The SU(2)
spin and Uv(1) symmetries act on the operators $c({\bm{k}})$ in exactly the
same fashion as they act on the operators $\Psi$ in Eq. (I.3).

The Hamiltonian itself is generically no longer diagonal in this basis. It
takes the form

$\displaystyle H^{(0)}_{\mathit{fl}}$
$\displaystyle=\int_{{\bm{k}}\in{\mathcald{m}\mathrm{BZ}}}c^{\dagger}({\bm{k}})h^{(0)}({\bm{k}})c({\bm{k}}),$
(30)

where we have suppressed all indices, including the valley index $v$. That is,
$c({\bm{k}})$ is an 8-component vector, while $h^{(0)}({\bm{k}})$ is an
$8\times 8$ dimensional matrix. We will keep with the convention that flat
band effective Hamiltonians are written in normal, serifed script, whereas
calligraphic script will continue to be used for first-quantized Hamiltonians
in the basis of the monolayers operators $\Psi$, $\tilde{\Psi}$. Although
determining $h^{(0)}({\bm{k}})$ analytically is a completely intractable task,
there is nevertheless a great deal that can be said only using the continuous
symmetries $\mathrm{U_{v}}(1)$ and $\mathrm{SU}(2)$ of Eq. (I.3) as well as
the discrete symmetries of Eqs. (29) and (III.1).

#### III.1.1. 2$d$ flat band Hamiltonian without SOC

We start by considering the form the Hamiltonian in the gauge just described
takes in the absence of spin orbit coupling, _i.e._ no explicit spin terms in
the Hamiltonian. The SU(2) spin symmetry thus prohibits the presence of spin
Pauli matrices $s^{x,y,z}$, while the Uv(1) symmetry limits the valley Pauli
matrices to $\tau^{0,z}$. Imposing ${\mathcald{C}}_{2}{\mathcald{T}}$ requires
that terms proportional to $\sigma^{z}$ or $\tau^{z}\sigma^{z}$ vanish.
Finally, of the terms remaining, time reversal indicates that they are either
even or odd under ${\bm{k}}\to-{\bm{k}}$ according to

$\displaystyle h_{e}^{(0)}({\bm{k}})$
$\displaystyle=h^{(0)}_{0,0,0}({\bm{k}})+{h^{(0)}_{0,x,0}({\bm{k}})\sigma^{x}}+{h^{(0)}_{z,y,0}({\bm{k}})\tau^{z}\sigma^{y}},$
$\displaystyle h_{o}^{(0)}({\bm{k}})$
$\displaystyle={h^{(0)}_{0,y,0}({\bm{k}})\sigma^{y}}+{h^{(0)}_{z,0,0}({\bm{k}})\tau^{z}}+{h^{(0)}_{z,x,0}({\bm{k}})\tau^{z}\sigma^{x}},$
(31)

where the subscripts ‘$e$’ and ‘$o$’ indicate the parity (even and odd) of the
two terms. The total effective Hamiltonian is
$h^{(0)}({\bm{k}})=h^{(0)}_{e}({\bm{k}})+h^{(0)}_{o}({\bm{k}})$.

As explained below, we are specifically interested in the physics of the
lower, hole-doped flat band close to the $\gamma=(0,0)$ point at the
${\mathcald{m}\mathrm{BZ}}$ centre. The action of the mirror symmetry
${\mathcald{M}}_{y}$ implies that at the $\gamma$ point, $h^{(0)}_{z,y,0}$
must vanish. As a result,
$h^{(0)}({\bm{k}}=0)=h^{(0)}_{0,0,0}(0)+h_{0,x,0}^{(0)}(0)\sigma^{x}$. Since
SOC and the proximity-induced IVC order will both be considered in a
perturbative limit below, it is appropriate to further project the
Hamiltonians onto the $\sigma^{x}=\pm 1$ basis (the actual sign of
$\sigma^{x}$ will not matter). Doing so yields

$\displaystyle\bar{h}^{(0)}_{e}({\bm{k}})$
$\displaystyle=t_{0,0}^{(0)}({\bm{k}}),$
$\displaystyle\bar{h}_{o}^{(0)}({\bm{k}})$
$\displaystyle=t_{z,0}^{(0)}({\bm{k}})\tau^{z}.$ (32)

In the left equation $t_{0,0}$ can be expressed as

$\displaystyle t_{0,0}^{(0)}({\bm{k}})=\frac{{\bm{k}}^{2}}{2m}-\mu.$ (33)

On the other hand, the ${\mathcald{C}}_{3}$ symmetry requires that

$\displaystyle{t}_{z,0}^{(0)}({\bm{k}})=\tilde{t}_{o}^{(0)}k_{x}(k_{x}^{2}-3k_{y}^{2})+\tilde{t}_{o}^{(0)\prime}k_{y}(3k_{x}^{2}-k_{y}^{2}).$
(34)

The terms we derived are recorded in Table 5. Mirror symmetry
${\mathcald{M}}_{y}$ further sets $\tilde{t}_{o}^{(0)\prime}=0$. Nevertheless,
since ${\mathcald{M}}_{y}$ is technically not a good symmetry of the problem,
${\mathcald{M}}_{y}$-breaking perturbations may be generated. We therefore
have included the ${\mathcald{M}}_{y}$-breaking couplings in Table 5,
delineated by curly braces ‘$\\{\\}$’ to distinguish them from
${\mathcald{M}}_{y}$-breaking terms allowed within our perturbative expansion.
We emphasize that the applicability of mirror symmetry in this context is
independent of whether SOC is ultimately included through
${\mathcalb{h}}_{I/R}$ or the mirror-symmetrized ${\mathcalb{h}}_{I/R}^{*}$.

###  Effective flat band Hamiltonian in 2$d$ with SOC

We treat the addition of SOC perturbatively, most notably in the sense that we
assume that it induces minimal mixing between the flat and non-flat/dispersive
bands at $\nu<-4$, $+4<\nu$. Our goal is therefore to derive the effective
Hamiltonian in terms of the operators $c({\bm{k}})$ defined in Eq. (27). Such
a perturbative expansion is well-defined provided
$\lambda_{R/I}\ll\Delta_{\mathrm{disp}}$, where $\Delta_{\mathrm{disp}}$ is
the gap separating the flat and dispersive bands. Since
$\Delta_{\mathrm{disp}}\sim 30\,\mathrm{\text{meV}}$ is the typically obtained
experimentally close to the magic angle [20, 21] while $\lambda_{I/R}\lesssim
5\,\mathrm{\text{meV}}$ [12, 9, 10, 13, 14], this assumption is reasonable.

We will specifically restrict our analysis to a _first order_ approximation.
We envision obtaining our effective Hamiltonian by projecting
$H_{\mathrm{SO}}$ onto the flat bands through the identification

$\displaystyle\Psi_{v,i}({\bm{k}}+{\bm{G}})$
$\displaystyle\sim\sum_{\beta\in\mathit{fl}}{U}^{(v)}_{i,{\bm{G}};\beta}({\bm{k}})c_{v,\beta}({\bm{k}}),$
$\displaystyle{U}^{(v)}_{i,{\bm{G}};\beta}({\bm{k}})$
$\displaystyle=\sum_{\alpha\in\mathit{fl}}\tilde{U}^{(v)}_{i,{\bm{G}};\alpha}V_{\alpha\beta}^{(v)}({\bm{k}}),$
(35)

where $\tilde{U}^{(v)}_{i,{\bm{G}};\beta}({\bm{k}})$ and
$V_{\alpha\beta}^{(v)}({\bm{k}})$ are defined in Eqs. (25) and (26). It
therefore follows that

$\displaystyle H_{\mathrm{SO}}$
$\displaystyle=\sum_{{\bm{G}}}\int_{{\bm{k}}\in{\mathcald{m}\mathrm{BZ}}}\Psi^{\dagger}({\bm{k}}){\mathcalb{h}}_{\mathrm{SO}}\Psi({\bm{k}}+{\bm{G}})\sim\int_{{\bm{k}}\in{\mathcald{m}\mathrm{BZ}}}c^{\dagger}({\bm{k}})\underbrace{U^{\dagger}({\bm{k}}){\mathcalb{h}}_{\mathrm{SO}}U({\bm{k}})}_{h_{\mathrm{SO}}({\bm{k}})}c({\bm{k}}).$
(36)

Consistent with the conventions above, the effective flat band Hamiltonian,
$h_{\mathrm{SO}}({\bm{k}})$, is written in normal font, while the full first-
quantized Hamiltonian in the basis of the microscopic graphene operators,
${\mathcalb{h}}_{\mathrm{SO}}$, is expressed in calligraphic font. Note that
although ${\mathcalb{h}}_{\mathrm{SO}}$ is momentum independent, the effective
flat band Hamiltonian $h_{\mathrm{SO}}$ depends on ${\bm{k}}$.

A straightforward consequence of the approximation of Eq. (36) is that, like
${\mathcalb{h}}_{\mathrm{SO}}$, the effective Hamiltonian
$h_{\mathrm{SO}}({\bm{k}})$ may be divided into a strictly Rashba and a
strictly Ising part:

$\displaystyle h_{\mathrm{SO}}({\bm{k}})$
$\displaystyle=h_{R}({\bm{k}})+h_{I}({\bm{k}}),$ (37)

where $h_{R}({\bm{k}})\propto\lambda_{R}$ and
$h_{I}({\bm{k}})\propto\lambda_{I}$. Further, because ${\mathcalb{h}}_{R}$ and
${\mathcalb{h}}_{I}$ are both proportional to Pauli matrices acting on the
spin indices, $s^{x,y,z}$, and the unitary transformations
$U^{(v)}_{i,{\bm{G}};\alpha}({\bm{k}})$ do not act on the spin, the effective
spin-orbit contributions must also be proportional to spin Pauli matrices.

As in the previous section, it is not necessary to derive
$h_{\mathrm{SO}}({\bm{k}})$ directly—we instead acquire its form by imposing
the symmetries listed in Table 1. Further, given our restriction to first
order, terms prohibited by a symmetry that is only broken when _both_ Rashba
and Ising SOC are present will not be present in $h_{\mathrm{SO}}({\bm{k}})$
in Eq. (37) (_e.g._ , ${\mathcald{C}}_{2}$). This simplification accounts for
the absence of a row in Table 1 listing the symmetries preserved when both
Rashba and Ising SOC are present.

#### III.2.1. 2$d$ flat band Rashba Hamiltonian

We now use the symmetries outlined in Table 1 to restrict the form of the
Rashba effective Hamiltonian. From $is^{x}{\mathcald{C}}_{2}{\mathcald{T}}$,
we find that

$\displaystyle h^{(R)}({\bm{k}})$
$\displaystyle=\sum_{\ell=0,z}\bigg{(}\sum_{i=0,x,y}\sum_{a=0,x,y}h^{(R)}_{\ell,i,a}({\bm{k}})\tau^{\ell}\sigma^{i}s^{a}+h^{(R)}_{\ell,z,z}({\bm{k}})\tau^{\ell}\sigma^{z}s^{z}\bigg{)}.$
(38)

From the action of $is^{y}{{\mathcald{T}}}$, we separate these terms into even
and odd components, _i.e._ $h^{(R)}_{e}(-{\bm{k}})=h^{(R)}_{e}({\bm{k}})$,
$h^{(R)}_{o}(-{\bm{k}})=-h^{(R)}_{o}({\bm{k}})$:

$\displaystyle h^{(R)}_{e}({\bm{k}})$
$\displaystyle=h^{(R)}_{0,0,0}({\bm{k}})+h^{(R)}_{0,x,0}({\bm{k}})\sigma^{x}+\sigma^{y}(h^{(R)}_{0,y,x}({\bm{k}})s^{x}+h^{(R)}_{0,y,y}({\bm{k}})s^{y})$
$\displaystyle\quad+\tau^{z}\Big{[}\beta_{z,0,x}({\bm{k}})s^{x}+h^{(R)}_{z,0,y}({\bm{k}})s^{y}+\sigma^{x}(h^{(R)}_{z,x,x}({\bm{k}})s^{x}+h^{(R)}_{z,x,y}({\bm{k}})s^{y})+h^{(R)}_{z,y,0}({\bm{k}})\sigma^{y}+h^{(R)}_{z,z,z}({\bm{k}})\sigma^{z}s^{z}\Big{]},$
$\displaystyle h^{(R)}_{o}({\bm{k}})$
$\displaystyle=h^{(R)}_{0,0,x}({\bm{k}})s^{x}+h^{(R)}_{0,0,y}({\bm{k}})s^{y}+\sigma^{x}(\beta_{0,x,x}({\bm{k}})s^{x}+h^{(R)}_{0,x,y}({\bm{k}})s^{y})+h^{(R)}_{0,y,0}({\bm{k}})\sigma^{y}+h^{(R)}_{0,z,z}({\bm{k}})\sigma^{z}s^{z}$
$\displaystyle\quad+\tau^{z}\big{[}\beta_{z,0,0}({\bm{k}})+\beta_{z,x,0}({\bm{k}})\sigma^{x}+\sigma^{y}(\beta_{z,y,x}({\bm{k}})s^{x}+\beta_{z,y,y}({\bm{k}})s^{y})\big{]}.$
(39)

To derive the effective one-band Hamiltonian close to the $\gamma$ point, we
start by projecting onto states with either $\sigma^{x}=+1$ or
$\sigma^{x}=-1$, as outlined in Sec. III.1.1. Doing so, we obtain

$\displaystyle\bar{h}^{(R)}_{e}({\bm{k}})$
$\displaystyle=t^{(R)}_{0,0}({\bm{k}})+\big{(}t^{(R)}_{z,x}({\bm{k}})\tau^{z}s^{x}+t^{(R)}_{z,y}({\bm{k}})\tau^{z}s^{y}\big{)},$
$\displaystyle\bar{h}^{(R)}_{o}({\bm{k}})$
$\displaystyle=t^{(R)}_{0,x}({\bm{k}})s^{x}+t^{(R)}_{0,y}({\bm{k}})s^{y}+t^{(R)}_{z,0}({\bm{k}})\tau^{z}.$
(40)

We next expand the functions $t^{(R)}_{a,n}({\bm{k}})$ in powers of ${\bm{k}}$
with the $e^{-2\pi is^{z}/3}{\mathcald{C}}_{3}$ symmetry as a restriction.
Both $t_{0,0}^{(R)}$ and $t_{z,0}^{(R)}$ will take identical forms as in the
case without SOC, and so these terms need not be considered separately from
the others. Moreover, as indicated in the discussion below Eq. (37), the
Rashba SOC term is only able to generate such spin-diagonal terms at higher
orders in perturbation theory; in the first order approximation used here,
$t_{0,0}^{(R)}$ and $t_{z,0}^{(R)}$ thus vanish. It follows that the addition
of mirror-symmetry-breaking Rashba SOC does _not_ generate the valley-orbit
term $\tilde{t}^{(0)\prime}_{o}k_{y}(3k_{x}^{2}-k_{y}^{2})\tau^{z}$ (Eq. (34))
that we argued was forbidden by mirror symmetry in the spin symmetric portion
of the Hamiltonian. For the remaining, non-zero terms, rotation symmetry
implies that they take the form

$\displaystyle t^{(R)}_{z,x}({\bm{k}})\tau^{z}s^{x}+t^{(R)}_{z,y}({\bm{k}})$
$\displaystyle=\tilde{t}^{(R)}_{e}\tau^{z}\left(\frac{1}{2}(k_{x}^{2}-k_{y}^{2})s^{x}-k_{x}k_{y}s^{y}\right)+\tilde{t}^{(R)\prime}_{e}\tau^{z}\left(k_{x}k_{y}s^{x}+\frac{1}{2}(k_{x}^{2}-k_{y}^{2})s^{y}\right),$
$\displaystyle t^{(R)}_{0,x}({\bm{k}})s^{x}+t^{(R)}_{0,y}({\bm{k}})s^{y}$
$\displaystyle=\tilde{t}^{(R)}_{o}\big{(}k_{x}s^{x}+k_{y}s^{y}\big{)}+\tilde{t}^{(R)\prime}_{o}\big{(}k_{y}s^{x}-k_{x}s^{y}\big{)},$
(41)

where $\tilde{t}^{(R)}_{e}$, $\tilde{t}^{(R)\prime}_{e}$,
$\tilde{t}^{(R)}_{o}$, and $\tilde{t}^{(R)\prime}_{o}$ are real numbers whose
values would have to be obtained numerically. Since the correction is linear
in $\lambda_{R}$, we trivially conclude that
$\tilde{t}^{(R)}_{e/o},\tilde{t}^{(R)\prime}_{e/o}\propto\lambda_{R}$. Each
independent term in Eq. (III.2.1) is recorded in Table 5.

If we further impose mirror symmetry, we see that both
$\tilde{t}^{(R)\prime}_{e}=\tilde{t}^{(R)\prime}_{0}=0$ as well. However,
unlike the situation without SOC, there is no reason for these terms to vanish
in our scheme when the (physically relevant) Hamiltonian ${\mathcalb{h}}_{R}$
is used instead of ${\mathcalb{h}}_{R}^{*}$—the expressions that
$\tilde{t}^{(R)\prime}_{e}$ and $\tilde{t}^{(R)\prime}_{0}$ multiply are thus
included in Table 5 without curly braces.

#### III.2.2. 2$d$ flat band Ising Hamiltonian

Again, we use the symmetry action provided in Table 1 to obtain an effective
Hamiltonian for the Ising SOC. The preservation of both
${\mathcald{C}}_{2}{\mathcald{T}}$ and $\mathrm{U}_{z}(1)$, the symmetry
responsible for spin rotations about the $z$-axis, implies

$\displaystyle h_{I}({\bm{k}})$
$\displaystyle=\sum_{\ell=0,z}\sum_{i=0,x,y}\tau^{\ell}\sigma^{i}\Big{(}h^{(I)}_{\ell,i,0}({\bm{k}})+h^{(I)}_{\ell,i,z}({\bm{k}})s^{z}\Big{)}.$
(42)

From the action of time-reversal, ${\mathcald{T}}$, we separate these
expressions into even, ‘$e$,’ and odd, ‘$o$,’ components like in the previous
sections:

$\displaystyle h_{I,e}({\bm{k}})$
$\displaystyle=h^{(I)}_{0,0,0}({\bm{k}})+h^{(I)}_{0,x,0}({\bm{k}})\sigma^{x}+h^{(I)}_{0,y,z}({\bm{k}})\sigma^{y}s^{z}+\tau^{z}\Big{[}h^{(I)}_{z,0,z}({\bm{k}})s^{z}+h^{(I)}_{z,x,z}({\bm{k}})\sigma^{x}s^{z}+h^{(I)}_{z,y,0}({\bm{k}})\sigma^{y}\Big{]},$
$\displaystyle h_{I,o}({\bm{k}})$
$\displaystyle=h^{(I)}_{0,0,z}({\bm{k}})s^{z}+h^{(I)}_{0,x,z}({\bm{k}})\sigma^{x}s^{z}+h^{(I)}_{0,y,0}({\bm{k}})\sigma^{y}+\tau^{z}\Big{[}h^{(I)}_{z,0,0}({\bm{k}})+h^{(I)}_{z,x,0}({\bm{k}})\sigma^{x}+h^{(I)}_{z,y,z}({\bm{k}})\sigma^{y}s^{z}\Big{]}.$
(43)

As before, we project onto a single flat band close to $\gamma$, which is
equivalent to setting $\sigma^{x}=\pm 1$:

$\displaystyle\bar{h}^{(I)}_{e}({\bm{k}})$
$\displaystyle=t^{(I)}_{0,0}({\bm{k}})+t^{(I)}_{z,z}({\bm{k}})\tau^{z}s^{z},$
$\displaystyle\bar{h}^{(I)}_{o}({\bm{k}})$
$\displaystyle=t^{(I)}_{0,z}({\bm{k}})s^{z}+t^{(I)}_{z,0}({\bm{k}})\tau^{z}.$
(44)

Both $t^{(I)}_{0,0}$ and $t^{(I)}_{z,0}$ take an identical form to the SU(2)
symmetric Hamiltonian of Eq. (32), and, further, as these terms are
independent of spin, they are only generated by the Ising SOC at higher orders
in $\lambda_{I}$. We now use the ${\mathcald{C}}_{3}$ symmetry respected by
Ising SOC to expand the remaining terms generated by the Ising SOC at linear
order in $\lambda_{I}$, $t^{(I)}_{z,z}$ and $t^{(I)}_{0,z}$, in powers of
${\bm{k}}$ about the $\gamma$ point:

$\displaystyle t^{(I)}_{z,z}({\bm{k}})$ $\displaystyle=\tilde{t}^{(I)}_{e},$
$\displaystyle t^{(I)}_{0,z}({\bm{k}})$
$\displaystyle=\tilde{t}^{(I)}_{o}k_{x}(k_{x}^{2}-3k_{y}^{2})+\tilde{t}^{(I)\prime}_{o}k_{y}(3k_{x}^{2}-k_{y}^{2}),$
(45)

where $\tilde{t}^{(I)}_{e}$, $\tilde{t}^{(I)}_{o}$ and
$\tilde{t}^{(I)\prime}_{o}$ are constants whose values depend on the details
of the theory. Analogous to the Rashba case, by explicit construction these
constants are proportional to the Ising SOC coupling constant, $\lambda_{I}$:
$\tilde{t}^{(I)}_{e},\tilde{t}^{(I)}_{o},\tilde{t}^{(I)\prime}_{o}\propto\lambda_{I}$.
In the above, only the lowest non-zero power of ${\bm{k}}$ is kept. When we
choose to work with ${\mathcalb{h}}_{I}^{*}$ in place of the (physically
relevant) ${\mathcalb{h}}_{I}$, $t^{(I)\prime}_{o}=0$.

###  Trivial wire Hamiltonian

Proximate Order | Wire Symmetries | Wire Perturbations | Parameter Scaling
---|---|---|---
trivial wire |  $\mathcal{T}^{2}=-1$, $\mathrm{U}_{\mathrm{v}}(1)$ |  $\frac{k^{2}}{2m}-\mu+ck\tau^{z}+k\bm{\alpha}_{1}\cdot\mathbf{s}+\tau^{z}\bm{\alpha}_{2}\cdot\mathbf{s}$ |  $c\propto 1/W^{2}$,
$\alpha_{1}^{x,y}\propto\lambda_{R}$,
$\alpha_{2}^{x,y}\propto\lambda_{R}/W^{2}$,
$\alpha_{1}^{z}\propto\lambda_{I}/W^{2}$, $\alpha_{2}^{z}\propto\lambda_{I}$
$\nu=0$ singlet IVC | $\mathcal{T}_{\rm sIVC}^{2}=+1$ |  $a_{1}k\tau^{x}+\tau^{x}\left(\beta_{1}^{x}s^{x}+\beta_{1}^{y}s^{y}\right)+\beta_{2}\tau^{x}s^{z}$ |  $a_{1}\propto 1/W^{2}$, $\beta_{1}^{x,y}\propto\lambda_{R}/W^{2}$,
$\beta_{2}\propto\lambda_{I}$
$\nu=0$ triplet IVC | $\mathcal{T}^{2}=-1$ |  $a_{1}^{\prime}k\tau^{x}s^{z}+k\tau^{y}\left(\beta_{1}^{\prime x}s^{x}+\beta_{1}^{\prime y}s^{y}\right)+\beta_{2}^{\prime}\tau^{x}$ |  $a_{1}^{\prime}\propto 1/W^{2}$, $\beta_{1}^{\prime x,y}\propto\lambda_{R}$,
$\beta_{2}^{\prime}\propto\lambda_{I}$
$\nu=\pm 2$ FM IVC | none |  $a_{1}k\tau^{x}+a_{1}^{\prime}k\tau^{x}s^{z}+a_{1}^{\prime\prime}k\tau^{z}s^{z}+a_{2}^{\prime\prime}s^{z}$
$+\tau^{x}(\beta_{1}^{x}s^{x}+\beta_{1}^{y}s^{y})+\beta_{2}\tau^{x}s^{z}$
$+k\tau^{y}(\beta_{1}^{\prime x}s^{x}+\beta_{1}^{\prime
y}s^{y})+\beta_{2}^{\prime}\tau^{x}$
$+\beta_{1}^{\prime\prime}k+\beta_{2}^{\prime\prime}\tau^{z}$ |  $a_{1},a_{1}^{\prime},a_{1}^{\prime\prime}\propto 1/W^{2}$,
$\beta_{1}^{x,y}\propto\lambda_{R}/W^{2}$, $\beta_{1}^{\prime
x,y}\propto\lambda_{R}$,
$\beta_{2},\beta_{2}^{\prime},\beta_{2}^{\prime\prime}\propto\lambda_{I}$,
$\beta_{1}^{\prime\prime}\propto\lambda_{I}/W^{2}$
$\nu=\pm 2$ AFM IVC | $\mathcal{T}^{2}=-1$ |  $a_{1}^{\prime\prime\prime}k(\tau^{x}s^{x}+\tau^{y}s^{y})+a_{2}^{\prime\prime\prime}ks^{z}+a_{3}^{\prime\prime\prime}\tau^{z}s^{z}$
$+k\tau^{x}\bm{\beta}^{\prime\prime\prime}_{1}\cdot\mathbf{s}+k\tau^{y}\bm{\beta}^{\prime\prime\prime}_{2}\cdot\mathbf{s}+\beta_{3}^{\prime\prime\prime}\tau^{x}+\beta_{4}^{\prime\prime\prime}\tau^{y}$ |  $a_{1,2}^{\prime\prime\prime}\propto 1/W^{2}$, $|\bm{\beta}_{1,2}^{\prime\prime\prime}|\propto\lambda_{R}$,
$\beta_{3,4}^{\prime\prime\prime}\propto\lambda_{R}/W^{2}$
Table 2:  Reproduction of Table 1 from the main text with an additional column
providing the scaling of the parameters listed. Wire Hamiltonian terms for the
trivial wire (first row) and perturbations generated by proximate IVC orders
(subsequent rows) are shown. The valley-orbit coupling $c$ persists in the
absence of both IVC order and SOC; ${\bm{\alpha}}_{1,2}$ denote SOC terms that
appear already for the trivial wire; couplings labelled by $a$’s survive with
IVC order in the absence of SOC; and ${\beta}$ terms can be viewed as
additional IVC order-parameter components generated due to SOC, akin to the
spin-orbit-induced admixture of singlet and triplet pairing in inversion-
asymmetric superconductors [22]. For $\nu=0$ triplet and $\nu=\pm 2$ IVC
orders, we assumed that, without SOC, the spins orient in the out-of-plane
($\pm s^{z}$) direction. The rightmost column lists the parameter scaling as a
function of the SOC strengths, $\lambda_{I/R}$, and the wire width $W$. The
$1/W^{2}$ dependence signals that ${\mathcald{C}}_{3}$ symmetry must be broken
for the relevant terms to take nonzero values, which we assume in the majority
of this appendix only occurs through the wire’s presence. However, in the
presence of strain or interaction-induced nematic order, the $1/W^{2}$ scaling
should be replaced by functions quantifying such effects.

We are finally in a position to discuss the effective Hamiltonian for the
trivial wire presented in Table 1 of the main text and reproduced here in
Table 2 (with an additional column). As depicted in Fig. 1(b) of the main
text, the wire is constructed by electrostic confinement of the 2$d$ system,
which allows us to extract estimates of the wire Hamiltonian from the
effective expressions we just derived. Specifically, we assume that within a
region around $\sim~{}10$ moiré unit cells wide, the chemical potential
intersects the lower flat band band bottom, which is situated near the
$\gamma$ point in momentum space; the filling in this region is
$\nu_{\mathrm{wire}}=-4+\delta$, for some small $\delta$. Outside the wire, in
the bulk region, the chemical potential is tuned so that the flat bands are
completely empty with $\nu_{\mathrm{bulk}}=-4$. (Our analysis would proceed in
a nearly identical fashion for $\nu_{\mathrm{wire}}=+4-\delta$ and
$\nu_{\mathrm{bulk}}=+4$).

We remark that within this approximation scheme, the Hamiltonian we derive is
equivalent to what would be obtained for a strip of finite width (provided one
ignores details related to the edges of the ‘wire’). The primary difference is
that instead of tuning the chemical potential in the bulk region to lie within
the gap separating the flat and dispersive bands, as done in our setup, in a
finite-width strip the existence of any other bands is not considered and the
‘outer’ regions are taken to be vacuum. In terms of the symmetry-obtained
Hamiltonians, however, the result is identical.

#### III.3.1. Wire Hamiltonian without SOC

We begin by considering the wire Hamiltonian when SOC is absent. It is also
convenient for the moment to ignore the ‘valley-orbit’ coupling defined in Eq.
(34): $\tilde{t}^{(0)}_{0,z},\tilde{t}^{(0)\prime}_{z,0}\to 0$. It is re-
introduced below. This simplification leaves us with the rotationally
invariant quadratic Hamiltonian of Eq. (33). We assume for the moment that the
wire extends in the $x$ direction. It follows that $k_{x}$ remains a good
quantum number, allowing us to write the wavefunction as
$\Phi_{k_{x}}(x,y)=e^{ik_{x}x}\phi_{k_{x}}(y)$. The $y$-dependent portion of
the wavefunction is obtained through the standard quantum mechanical particle
in a potential well problem. In the simplest case, one considers a box
potential of width $W$. The solution proceeds by first solving the Schrodinger
equation in the three regions, $-W/2<y$, $-W/2<y<W/2$, and $W/2<y$ and
subsequently implementing the appropriate boundary conditions. Regardless of
the wire profile details, the negative energy bound states are confined to the
wire region, $-W/2\lesssim y\lesssim W/2$, and, further, assuming an even wire
profile, the lowest energy states will possess even parity:
$\phi_{k_{x}}(y)=\phi_{k_{x}}(-y)$. These wavefunctions form the foundation of
the analysis that follows here and in subsequent sections.

We now re-introduce the valley-orbit terms by projecting them to the space
spanned by the confined wavefunctions, $\phi_{k_{x}}(y)$. In particular, the
odd portion of the wire Hamiltonian is obtained via

$\displaystyle{h}^{(0)}_{\mathrm{wire},o}(k_{x}=k)$
$\displaystyle=\tau^{z}\int
dy\,\phi_{k}^{\dagger}(y)\left[\tilde{t}_{z,0}^{(0)}k\big{(}k^{2}-3(-i\partial_{y})^{2}\big{)}+\tilde{t}_{z,0}^{(0)\prime}(-i\partial_{y})\big{(}3k^{2}-(-i\partial_{y})^{2}\big{)}\right]\phi_{k}(y)$
$\displaystyle=\left[\tilde{t}_{z,0}^{(0)}k\big{(}k^{2}-3\braket{k_{y}^{2}}\big{)}+\tilde{t}_{z,0}^{(0)\prime}\big{(}3\braket{k_{y}}k^{2}-\braket{k_{y}^{3}}\big{)}\right]\tau^{z},$
(46)

where

$\displaystyle\Braket{k_{y}^{n}}$ $\displaystyle=\int
dy\,\phi_{k}^{\dagger}(y)(-i\partial_{y})^{n}\phi_{k}(y).$ (47)

The $k$-dependence of the correlation function is left implicit. A consequence
of the even parity of $\phi_{k}(y)$ is that only even powers of $n$ return
non-zero values. The total trivial wire Hamiltonian we find is

$\displaystyle h_{\mathrm{wire}}^{(0)}(k)$
$\displaystyle=\frac{k^{2}}{2m}-\mu-3\tilde{t}_{o}^{(0)}\Braket{k_{y}^{2}}k\tau^{z},$
(48)

where a constant has been absorbed into the chemical potential and higher
powers of $k$ are ignored. Up to shifts that can be easily accounted for by
the back gate voltage, the mass term and chemical potential are simply the
1$d$ version of the flat band Hamiltonian, _i.e._ , what one would get by
ignoring all $k_{y}$ dependence. By contrast, we see that the breaking of
${\mathcald{C}}_{3}$ symmetry allows a linearly dependent valley-orbit term
proportional to $\tau^{z}$, whose coefficient $c$ in the first row of Table 2
is equal to $-3\tilde{t}_{o}^{(0)}\Braket{k_{y}^{2}}$. One generally expects
$\Braket{k_{y}^{2}}\sim(2\pi/W)^{2}$, and it follows that $c\propto 1/W^{2}$,
as indicated in the fourth column of Table 2. Importantly, the scaling of $c$
is itself independent of the wire direction: we always find
$c\propto\braket{k^{2}_{\perp}}\propto 1/W^{2}$ ($k_{\perp}$ is the transverse
momentum). Note, however, that had we instead chosen our wire to lie along the
$y$-direction, we would have found
$c=3\tilde{t}_{o}^{(0)\prime}\braket{k_{x}^{2}}$. This relation satisfies the
scaling $c\propto 1/W^{2}$, but within our perturbative expansion, the mirror
symmetry should be imposed upon the spin symmetric portion of the trivial wire
Hamiltonian—in which case $\tilde{t}^{(0)\prime}_{o}=0$. In formulating Table
2, we assume that the wire is oriented in an arbitrary direction that does not
possess these cancellations.

As we discuss in the next section, explicit breaking of the
${\mathcald{C}}_{3}$ symmetry either through strain or interaction-induced
nematicity would also contribute to the coefficient $c$. This observation
follows from noting that it was the ${\mathcald{C}}_{3}$ symmetry that
prohibited the linear terms from appearing in the expression for
$t_{0,z}^{(0)}({\bm{k}})$ in Eq. (34).

Finally, one can ask what would occur had we derived the confined
wavefunctions $\phi_{k}$ starting with the valley-orbit-coupled Hamiltonian.
The result is analogous save for an additional phase:
$\tilde{\phi}_{k_{x}}(y)\sim e^{i\theta y}\phi_{k_{x}}(y)$ with $\theta\sim
2m\cdot 3\tilde{t}^{(0)}_{o}\braket{k_{y}^{2}}$. Our assumption that this term
is small is supported by the factor of $1/W^{2}$—any errors resulting from the
fact that we study the problem using the functions $\phi_{k_{x}}$ instead of
$\tilde{\phi}_{k_{x}}$ will only manifest at higher orders in $1/W^{2}$, which
is equivalent to our truncation in powers of ${\bm{k}}$.

#### III.3.2. Wire Hamiltonian for Rashba SOC

As in the case without SOC, we extract the wire Hamiltonian directly from the
results of Sec. III.2.1 (or, alternatively, from Table 5) under the assumption
that the wire points along the $x$-direction. We find

$\displaystyle h^{(R)}_{\mathrm{wire}}(k)$
$\displaystyle=-\frac{1}{2}\braket{k_{y}^{2}}\tau^{z}\big{(}\tilde{t}^{(R)}_{e}s^{x}+\tilde{t}^{(R)\prime}_{e}s^{y}\big{)}+k\big{(}\tilde{t}^{(R)}_{o}s^{x}-\tilde{t}^{(R)\prime}_{o}s^{y}\big{)}$
(49)

where $k\equiv k_{x}$ and the correlations are defined via Eq. (47), with
correlation functions for all odd powers of $k_{y}$ assumed to vanish.
Comparing with the notation of the main text reproduced in Table 2, we
conclude that

$\displaystyle\alpha_{1}^{x}$ $\displaystyle=\tilde{t}_{0}^{(R)},$
$\displaystyle\alpha_{1}^{y}$ $\displaystyle=-\tilde{t}_{0}^{(R)\prime},$
$\displaystyle\alpha_{2}^{x}$
$\displaystyle=-\frac{1}{2}\Braket{k_{y}^{2}}\tilde{t}^{(R)}_{e},$
$\displaystyle\alpha_{2}^{y}$
$\displaystyle=-\frac{1}{2}\Braket{k_{y}^{2}}\tilde{t}^{(R)\prime}_{e}.$ (50)

While it’s clear that all four parameters above are proportional at leading
order to the UV Rashba coupling constant, $\lambda_{R}$ (per Eq. (21)), both
$\alpha_{2}^{x}$ and $\alpha_{2}^{y}$ are also proportional to
$\Braket{k_{y}^{2}}\sim 1/W^{2}$. These scalings are noted in Table 2.

#### III.3.3. Wire Hamiltonian for Ising SOC

We follow the procedure of the previous two sections, now with Eqs. (44) and
(45) (or the row labelled ‘Ising’ in Table 5) as our starting point. We find

$\displaystyle h^{(I)}_{\mathrm{wire}}(k)$
$\displaystyle=\tilde{t}^{(I)}_{e}\tau^{z}s^{z}3\tilde{t}^{(I)}_{o}\Braket{k_{y}^{2}}k,$
(51)

implying that

$\displaystyle\alpha_{1}^{z}$
$\displaystyle=-3\tilde{t}^{(I)}_{o}\Braket{k_{y}^{2}},$
$\displaystyle\alpha_{2}^{z}$ $\displaystyle=\tilde{t}^{(I)}_{e}.$ (52)

Once more, we conclude that $\alpha_{1}^{z}\propto\lambda_{I}$ and
$\alpha_{2}^{z}\propto\lambda_{I}/W^{2}$ as shown in Table 2.

###  Analysis in the absence of ${\mathcald{M}}_{y}$ or ${\mathcald{C}}_{3}$

It is worth emphasizing which aspects of the analysis above hold when
${\mathcald{M}}_{y}$ and/or ${\mathcald{C}}_{3}$ are not present in even the
continuum model $H_{\mathrm{cont}}$ without SOC. In particular, we explain
that provided ${\mathcald{T}}$, ${\mathcald{C}}_{2}$, and the relevant
continuous symmetries are preserved, the expressions derived in Eqs. (32),
(III.2.1), and (44) remain valid, including the parity assignments.

We first consider the case where ${\mathcald{M}}_{y}$ is not present in the
SOC-free Hamiltonian. The basis chosen in Eq. (27) is no longer relevant, and
it is more convenient to keep to the operators $\tilde{c}({\bm{k}})$ of Eq.
(25). We restrict this operator to a single band,
$\tilde{c}_{v,n,\alpha=1}({\bm{k}})\to\tilde{c}_{v,n}({\bm{k}})$, where $v$,
and $n$ are valley and spin indices respectively, and $\alpha=1$ specifies the
band of interest. Importantly, because the Hamiltonian $H_{\mathrm{cont}}$
preserves ${\mathcald{T}}$ and ${\mathcald{C}}_{2}{\mathcald{T}}$, these
symmetries may be chosen to act as

$\displaystyle{\mathcald{T}}$
$\displaystyle:\quad\tilde{c}({\bm{k}})\to\tilde{c}(-{\bm{k}}),\quad i\to-i,$
$\displaystyle{\mathcald{C}}_{2}{\mathcald{T}}$
$\displaystyle:\quad\tilde{c}({\bm{k}})\to
e^{i\tilde{\theta}({\bm{k}})}c({\bm{k}}),\quad i\to-i,$ (53)

which is identical to what you obtain by projecting onto $\sigma^{x}=\pm 1$ in
Eq. (III.1), as argued in Sec. III.1.1. Although it is now clear that the
projection procedure in the sections above could have been avoided, the basis
Eq. (27) will prove useful in the following section.

When ${\mathcald{C}}_{3}$ is preserved, its action on the
$\tilde{c}({\bm{k}})$ operators also remains unchanged, and the analysis
follows as described above with the minor difference that certain terms
previously prohibited by the mirror symmetry are now present _i.e._ , the
curly braced terms in Table 5 are now equally valid. Conversely, when
${\mathcald{C}}_{3}$ is broken, the Hamiltonians of Eqs. (32), (III.2.1), and
(44) may be directly translated to the one dimensional wire limit by simply
taking ${\bm{k}}\to k$ and expanding in even or odd powers of $k$. Returning
to Table 5, this step is equivalent to noting only which matrices
$\tau^{a}s^{n}$ are allowed in each scenario as well as the parity of the term
that multiplies it. While no information is gained on the scaling of the
parameters in the theory except for the $\lambda_{R/I}$ dependence, by
comparison with the ${\mathcald{C}}_{3}$ symmetric analysis, it is clear that
all terms described as scaling with $1/W^{2}$ in Sec. III.3 and Table 2 must
now depend on the strain or nematic order parameter responsible for the
${\mathcald{C}}_{3}$ breaking.

## IV. IVC proximity-coupled wire at $\nu=0$ and $\nu=\pm 2$

In this section, we detail our derivation of the 1$d$ wire Hamiltonians
proximity-coupled to various IVC insulating states at charge neutrality and
$\nu=\pm 2$. We first outline the methodology we use before describing our
analysis and the reasoning behind it. The results of this section appear in
Table 2, and are based on the information provided in Tables 1 and 5.

###  Outline of methodology

As discussed, the effective Hamiltonian derived in Sec. III.3 could equally
well describe a ‘wire’ in the form of a nanoribbon with physical (though
assumed unimportant) boundaries in addition to the electrostatically defined
‘wires’ considered here. The derivation of the proximity-coupled wire could
then proceed by first coupling the degrees of freedom of the trivial wire
Hamiltonian to the gapped bulk degrees of freedom of the adjoining IVC phase
under consideration. Since the IVC phases are gapped, an effective Hamiltonian
for the proximity-coupled wire could be obtained by systematically integrating
out the bulk fermions. The chemical potential necessary to obtain the
adjoining IVC phases of interest and whether or not the wire was obtained via
electrostic gating or in some nanoribbon scenario are not expected to alter
the universal, symmetry-constrained physics of interest here. However, even
with the assumptions already in place, this integrating-out procedure remains
impractical. We instead consider a closely related companion problem whose
qualitative features are expected to be the same.

In the scenario we consider, we start with two 2$d$ TBG systems stacked atop
one another and allowed to tunnel _very_ weakly (much more weakly than the
graphene sheets of each TBG system are coupled). We denote these the primary
system and the auxiliary system. In the primary system, the chemical potential
is tuned to lie close to the flat band bottom around the $\gamma$ point. By
constrast, the chemical potential of the auxiliary system is chosen to lie
either at $\nu=0$ (charge neutrality) or at $\nu=\pm 2$. In both situations,
we assume that as a result of interactions, the auxiliary system spontaneously
breaks $\mathrm{U_{v}}(1)$ and possibly other symmetries, resulting in a
intervalley coherent (IVC) insulator whose nature is described in the sections
below. Because the auxiliary system is gapped, it only has a perturbative
effect on the gapless degrees of freedom in the primary system. In particular,
in second order perturbation theory, it alters the energy of the primary
system’s Hamiltonian at $\sim\mathcald{O}(t^{2}/\Delta_{\mathrm{IVC}})$ where
$t$ is the tunnelling strength between the primary and auxiliary systems and
$\Delta_{\mathrm{IVC}}$ is the gap of the auxiliary system—set, of course, by
the IVC order. We can then derive the effective Hamiltonian of the primary
system in the 2$d$ limit using symmetry arguments in a manner directly
analogous to the analysis of Secs. III.1.1, III.2.1, and III.2.2. Importantly,
even though the order parameter may fundamentally alter the structure of the
flat bands within the auxiliary system, it does not do so in the primary
system in the sense that the basis and electron operators defined in Eq. (27)
continue to provide a good description of the system. Finally, we imagine
obtaining our wire Hamiltonian by reducing the primary system to a narrow
strip, allowing the methodology of Sec. III.3 to be applied. That is, we
envision projecting the IVC corrections onto the wire wavefunctions
$\phi_{k}(y)$ defined through Eqs. (III.3.1) and (47). The primary and
auxiliary systems are now more correctly denoted the wire and bulk systems,
respectively. Similary, the ‘interlayer’ tunnelling constant $t$ is identified
as the tunnelling strength of the narrow wire region to the adjoining bulk
states.

We consider two possible IVC order parameters for the auxiliary/bulk system at
both $\nu=0$ and $\nu=\pm 2$, the details of which are provided below. For
each IVC order, we break our study into three scenarios, which we consider in
order:

  1. 1.

IVC order without SOC

  2. 2.

IVC with Rashba SOC

  3. 3.

IVC with Ising SOC

(Following the arguments of Sec. III.2, terms arising from the simultaneous
presence of Rashba and Ising SOC are neglected.) For each of these scenarios,
we perform the following analysis:

  * •

We note which symmetries remain and record the results in Table 1.

  * •

For these symmetries, we determine all terms allowed in the 2$d$ effective
Hamiltonian of the primary system in an expansion about the $\gamma$ point.

  * •

Terms that were prohibited in the effective Hamiltonian without IVC order
(first three rows of Table 5) or in a previous scenario (_i.e._ , terms
allowed with IVC but without SOC) are typically allowed for each new scenario.
These new terms are recorded in Table 1. Note that this step implies a built-
in hierarchy, in which more symmetric scenarios are considered first.

  * •

Following Sec. III.3, we use the 2$d$ Hamiltonian we just derived to obtain
the corresponding 1$d$ wire Hamiltonian along with the scaling of the
parameters within the theory. These results are shown in Table 2.

While the terms generated in scenarios 2 and 3 will be proportional to
$\lambda_{R}$ and $\lambda_{I}$, we cannot conclude that any of the additional
terms derived in this fashion are proportional to $\Delta_{\mathrm{IVC}}$ or
even $t^{2}/\Delta_{\mathrm{IVC}}$. The confounding factor is that the second
order perturbative expansion necessarily alters the quasiparticle weight of
the effective flat-band operators,
$Z\sim[1+t^{2}/\Delta_{\mathrm{IVC}}]^{-1}$, which in turn requires that the
effective Hamiltonian pick up an additional factor, $h_{\mathrm{eff}}\to
Zh_{\mathrm{eff}}$. The result is an effective IVC coupling
$\tilde{\Delta}_{\mathrm{IVC}}\sim Z\cdot
t^{2}/\Delta_{\mathrm{IVC}}=-(1-Z)\Delta_{\mathrm{IVC}}$. The terms generated
accordingly depend on $\Delta_{\mathrm{IVC}}$ and $t$ in a complicated and
nonuniversal fashion that may not prove relevant to the physical situation of
interest.

While the steps outlined above all assume ${\mathcald{C}}_{3}$ and mirror
symmetry at the level of the trivial Hamiltonian, the discussion in Sec. III.4
is equally valid here. Without mirror, the terms included in Table 5 are
allowed. When ${\mathcald{C}}_{3}$ is absent, the information contained in
Table III.4 should be interpreted in a drastically simpler fashion: only the
matrices allowed and the parity of the terms they multiple are relevant to the
wire problem.

###  $\nu=0$ IVC order parameters

Reference 7 argued that the ground state described by the dominant terms in
the Hamiltonian, $H_{\mathrm{cont}}+H_{C}$ (see Secs. I.1 and I.2), should be
an IVC insulator. Following the treatment given in their appendix, we now
describe these candidate IVC ground states. They may be defined through a
projection operator $\mathcal{P}_{\nu=0}$: all occupied (empty) states in the
IVC insulator satisfy $\mathcal{P}_{\nu=0}\Ket{\phi}=\Ket{\phi}$
($\mathcal{P}_{\nu=0}\Ket{\phi}=0$). In a given basis, the projector can
generally be written in terms of what we will refer to as the _order
parameter_ $Q_{\nu=0}$:

$\displaystyle\mathcal{P}_{\nu=0}({\bm{k}})$
$\displaystyle=\frac{1}{2}\big{(}\mathds{1}+Q_{\nu=0}({\bm{k}})\big{)},$
$\displaystyle Q^{2}_{\nu=0}({\bm{k}})$ $\displaystyle=\mathds{1},$ (54)

where both $Q_{\nu=0}$ and $\mathcal{P}_{\nu=0}$ are $8\times 8$ matrices. In
the basis satisfying Eq. (III.1), the matrix $Q_{\nu=0}({\bm{k}})$
corresponding to an IVC state may be expressed in a momentum independent
fashion as

$\displaystyle Q_{\nu=0}$
$\displaystyle=\sigma^{y}\otimes\begin{pmatrix}0&V\\\
V^{\dagger}&0\end{pmatrix}_{\\!\tau},$ $\displaystyle V$
$\displaystyle=e^{i\phi}e^{i\frac{\theta}{2}\hat{\bm{n}}\cdot\bm{s}},$ (55)

where the $\sigma^{a}$ Pauli matrices act on the “band” indices, $\tau^{a}$
Paulis act on the valley indices, and $s^{a}$ Paulis act on the spin. As
discussed, although not true across the full moiré Brillouin zone, close to
the $\gamma$ point, we can set $\theta({\bm{k}})=0$ in Eq. (III.1) and
describe the action of ${\mathcald{C}}_{3}$ and ${\mathcald{M}}_{y}$ via Eq.
(29).

As per the perturbative scheme outlined in Ref. 7, the leading order behaviour
is controlled by $H_{C}+H_{\mathrm{cont}}$. These terms both possess an
effective symmetry under which the spins and phases of states originating in
each valley to be rotated separately,
$\mathrm{U}(2)_{K}\times\mathrm{U}(2)_{K^{\prime}}\cong\mathrm{U_{v}(1)}\times\mathrm{U_{c}(1)}\times\mathrm{SU}(2)_{K}\times\mathrm{SU}(2)_{K^{\prime}}$,
where $\mathrm{U_{v}}(1)$ ($\mathrm{U_{c}}(1)$) is the U(1) valley (charge)
symmetry and $\mathrm{SU}(2)_{K}$ ($\mathrm{SU}(2)_{K^{\prime}}$) correspond
to spin rotations in the $K$ ($K^{\prime}$) valley (see Eq. (I.3)). The order
parameter $Q_{\nu=0}$ breaks this symmetry down to the usual charge symmetry
$\mathrm{U_{c}}(1)$ as well as a residual $\mathrm{SU}(2)$. That is, for all
choices of $V$, there exists a subgroup
$\mathrm{SU}(2)\subset\mathrm{U_{v}}(1)\times\mathrm{SU}(2)_{K}\times\mathrm{SU}(2)_{K^{\prime}}$
whose action leaves $Q_{\nu=0}$ invariant.

At this order, ground states defined by $Q_{\nu=0}$ for all functions $V$ are
energetically equivalent. In the absence of spin orbit coupling, the states
lying within this SU(2) manifold are distinguished by the so-called valley
Hunds term. Depending on its sign, it either prefers a singlet state in which
$V=e^{i\phi}$ or a triplet state in which
$V=e^{i\phi}e^{i\pi\hat{\bm{n}}\cdot\bm{s}/2}=e^{i\phi}i\hat{\bm{n}}\cdot\bm{s}$.
To ensure that ${\mathcald{C}}_{3}$ symmetry is preserved in the triplet case,
we restrict our study to $\hat{\bm{n}}\propto\hat{\bm{z}}$.

#### IV.2.1. Effective Hamiltonian with proximity-induced IVC singlet order

We focus now on the singlet IVC (sIVC) state, defined by $V=e^{i\phi}$ in Eq.
(55). The order parameters are equivalent for all values of $\phi$ and so we
select $\phi=0$. This choice returns the order parameter

$\displaystyle Q_{\mathrm{sIVC}}$ $\displaystyle=\tau^{x}\sigma^{y}.$ (56)

It is clear that the residual SU(2) symmetry preserved by this state is the
usual spin rotation symmetry. The discrete symmetries preserved by
$Q_{\mathrm{sIVC}}$ without SOC and with Rashba and with Ising SOC
(separately) in 2$d$ are listed in Table 1. As detailed above, we use these
symmetries to restrict the form of an effective 2$d$ Hamiltonian close to the
$\gamma$ point. The additional terms generated by the presence of sIVC order
and subsequently by the pairwise presence of sIVC order and Rashba/Ising are
shown in Table 5. These in turn directly lead to the wire Hamiltonian shown in
Table 2.

We note that one of the terms obtained in the manner just described is
necessarily second order in the SOC coupling constants. In particular, when
the sIVC order above is present alongside Rashba SOC, a constant term
proportional to $\tau^{y}s^{z}$ is allowed by symmetry. However, as explained
in Sec. III.2, the leading order contributions of the Rashba term are all
proportional to the spin Pauli matrices present in the UV Rashba SOC induced
from the TMD monolayer ${\mathcalb{h}}_{R}$—namely, they must be proportional
to $s^{x}$ or $s^{y}$. Since the sIVC order parameter can itself not
contribute any additional powers of spin, we conclude that the $\tau^{y}s^{z}$
term necessarily arises at order $\lambda_{R}^{2}$ or possibly an even higher
power. For this reason, the $\tau^{y}s^{z}$ term is not included in the 1$d$
wire Hamiltonian of Table 2.

#### IV.2.2. Effective Hamiltonian with proximity-induced IVC triplet order

The triplet IVC (tIVC) state is obtained by setting
$V=e^{i\phi}i\hat{\bm{n}}\cdot\bm{s}$. As mentioned, to preserve the
${\mathcald{C}}_{3}$ symmetry, the spin must point in the out-of-plane
direction, $\hat{\bm{n}}\propto\hat{\bm{z}}$. We use the $\mathrm{U_{v}}(1)$
degrees of freedom to set $\phi=-\pi/2$, resulting in the order parameter

$\displaystyle Q_{\mathrm{tIVC}}$ $\displaystyle=\tau^{x}\sigma^{x}s^{z}.$
(57)

While clearly not invariant under the physical SU(2) symmetry, as discussed,
this order parameter nonetheless preserves an SU(2) subgroup of
$\mathrm{U_{v}}(1)\times\mathrm{SU}(2)_{K}\times\mathrm{SU}(2)_{K^{\prime}}$.
In particular, the action of the group $\mathrm{SU(2)_{tIVC}}$ generated by
the matrices $\\{s^{z},\tau^{z}s^{x},\tau^{z}s^{y}\\}$ leaves
$Q_{\mathrm{tIVC}}$ unchanged. Note that the $\mathrm{U}(1)$ subgroup of the
spin symmetry responsible for $z$-axis spin rotations, which we denote
$\mathrm{U}_{z}(1)$, is a subset of this conserved symmetry, _i.e._ ,
$\mathrm{U}_{z}(1)\subset\mathrm{SU(2)_{tIVC}}$. The lattice symmetries
preserved by $Q_{\mathrm{sIVC}}$ without SOC are shown in the row labelled
‘tIVC’ in Table 1. Below, the symmetries preserved when either Rashba SOC or
Ising SOC is also present are given as well. We follow the procedure outlined
above to obtain the terms of Table 5, leading to the wire Hamiltonian of Table
2.

As with the singlet IVC order, one of the terms allowed by symmetry when tIVC
order is present alongside Rashba SOC necessarily only occurs at higher orders
in $\lambda_{R}$ and is therefore not included in the wire Hamiltonian shown
in Table 2. Specifically, the term $\tau^{y}$ contains no powers of spin. The
Rashba contribution must therefore have contributed a power of $s^{z}$ or a
power of the identity—neither of which occur at linear order in $\lambda_{R}$.

###  $\nu=\pm 2$ IVC order parameters

The IVC states at $\nu=\pm 2$ are defined analogously to the $\nu=0$ order
parameters of Sec. IV.2 [7]. For concreteness, we focus on the $\nu=-2$
scenario. The primary difference between $\nu=-2$ and $\nu=0$ is that
additional states must be projected away. That is, the projection operator
satisfies $\mathrm{tr}\,\mathcal{P}_{\nu=-2}=2$, compared to the $\nu=0$
projection operator, which satisfied $\mathrm{tr}\,\mathcal{P}_{\nu=0}=4$. One
may therefore view the $\nu=-2$ order parameter as a combination of two
commuting order parameters, $Q_{1,2}$:

$\displaystyle Q_{\nu=-2}$
$\displaystyle=\mathcal{P}_{+}+\mathcal{P}_{-}Q_{1},$
$\displaystyle\mathcal{P}_{\pm}$ $\displaystyle=\frac{1}{2}(\mathds{1}\pm
Q_{2}),$ $\displaystyle Q_{1,2}^{2}$ $\displaystyle=\mathds{1},$
$\displaystyle[Q_{1},Q_{2}]$ $\displaystyle=0.$ (58)

The actual projection operator defining the $\nu=-2$ IVC insulator is then
$\mathcal{P}_{\nu=-2}=(\mathds{1}-Q_{\nu=-2})/2$.

The authors of Ref. 7 demonstrate that within the perturbative scheme they
consider, the lowest energy insulators at $\nu=\pm 2$ are the so-called spin-
polarized IVC states. These statesmay be obtained starting from an order
parameter defined by the choice $Q_{1}=\tau^{x}\sigma^{y}$ and $Q_{2}=s^{z}$.
As for $\nu=0$, to first order in the limit studied in Ref. 7 (only
$H_{\mathrm{cont}}+H_{C}$), the system possesses a $\mathrm{U(2)\times
U(2)}\cong\mathrm{U_{c}(1)}\times\mathrm{U_{v}}(1)\times\mathrm{SU}(2)_{K}\times\mathrm{SU}(2)_{K^{\prime}}$
symmetry, and so acting on $Q_{\nu=-2}$ by any element of this group returns
an energetically equivalent ground state. Doing so, one finds that the generic
form of the spin polarized IVC state is

$\displaystyle Q_{\nu=-2}$
$\displaystyle=\begin{pmatrix}\frac{1}{2}(\mathds{1}+\hat{\bm{n}}_{+}\cdot\bm{s})&V\sigma^{y}\\\
V^{\dagger}\sigma^{y}&\frac{1}{2}(\mathds{1}+\hat{\bm{n}}_{-}\cdot\bm{s})\end{pmatrix}_{\\!\tau},$
(59)

where

$\displaystyle\hat{\bm{n}}_{\pm}$
$\displaystyle=\frac{1}{2}\mathrm{tr}\big{(}U_{\pm}^{\dagger}s^{z}U_{\pm}\bm{s}\big{)},$
$\displaystyle V$
$\displaystyle=U_{+}^{\dagger}\frac{1}{2}(\mathds{1}-s^{z})U_{-},$ (60)

with $U_{\pm}$ $2\times 2$ unitary matrices acting on the spin indices. It can
be shown that $Q_{\nu=-2}$ is invariant under the action of a
$\mathrm{U}(1)\times\mathrm{U}(1)\times\mathrm{U}(1)$ symmetry (one of which
is $\mathrm{U_{c}}(1)$) [7], implying that it takes values in the manifold
$\mathrm{U}(2)\times\mathrm{U}(2)/[\mathrm{U}(1)\times\mathrm{U}(1)\times\mathrm{U}(1)]\cong\mathrm{U_{v}}(1)\times
S^{2}\times S^{2}$. Here, $\mathrm{U_{v}}(1)$ is the usual IVC phase we set to
zero and the two factors of the unit sphere, $S^{2}$, correspond to the spin
directions in either valley, $\hat{\bm{n}}_{\pm}$.

Just as for the IVC orders at $\nu=0$, the large degeneracy of $Q_{\nu=-2}$ is
broken by the valley Hunds term. Depending on its sign, either a ferromagnetic
(FM) state given by $U_{+}=e^{i\phi}U_{-}$ or an antiferromagnetic (AFM) IVC
state given by $U_{+}=s^{x}e^{i\phi}U_{-}$ is preferred. These choices
accordingly select $\hat{\bm{n}}_{+}=\hat{\bm{n}}_{-}$ and
$\hat{\bm{n}}_{+}=-\hat{\bm{n}}_{-}$ for the FM IVC and AFM IVC states,
respectively. As was the case at $\nu=0$, in order to preserve
${\mathcald{C}}_{3}$ symmetry, the spins must point in the out-of-plane
direction, $\hat{\bm{n}}_{\pm}\propto\hat{\bm{z}}$, which we assume to be the
case.

#### IV.3.1. Effective Hamiltonian with proximity-induced FM IVC order

The first state we consider is the ferromagnetic (FM) IVC state which is
obtained by choosing $\hat{\bm{n}}_{+}=\hat{\bm{n}}_{-}=\hat{\bm{z}}$ in Eq.
(59). An equivalent definition of this state follows by selecting $Q_{1}$ and
$Q_{2}$ in Eq. (58) equal to two distinct elements of
$\\{s^{z},\tau^{x}\sigma^{y},\tau^{x}\sigma^{y}s^{z}\\}$ up to some signs; the
unassigned element is then equal to $\pm Q_{1}Q_{2}$. Regardless, the
resulting order parameter is

$\displaystyle Q_{\mathrm{FM\,IVC}}$
$\displaystyle=\frac{1}{2}(\mathds{1}+s^{z})+\frac{1}{2}\tau^{x}\sigma^{y}(\mathds{1}-s^{z}).$
(61)

We can check that the three residual U(1) symmetries preserved by this order
parameter are generated by $\mathds{1},s^{z},\tau^{z}(\mathds{1}-s^{z})$,
which respectively correspond to $\mathrm{U_{c}}(1)$, $\mathrm{U}_{z}(1)$, and
a restricted version of $\mathrm{U_{v}}(1)$ that only acts on the $\downarrow$
spins. While the first two are good symmetries, the second is an artifact of
the simplistic, fine-tuned representation implied by Eq. (61). Essentially,
$Q_{\mathrm{FM\,IVC}}$ has degenerate eigenvalues equal to $-1$ for the two
states with both $s^{z}=\downarrow$ and $\tau^{x}\sigma^{y}=-1$ and
eigenvalues equal to $+1$ for the remaining six states. However, although
$Q_{\mathrm{FM\,IVC}}$ is a good description of the ground state, it is
unlikely to describe the effective mean field Hamiltonian one would actually
obtain in an interacting system. Instead, one would expect something like
$H_{\mathrm{FM\,IVC}}\sim
as^{z}+b\tau^{x}\sigma^{y}-c\tau^{x}\sigma^{y}s^{z}$, with, say, $a,b,c>0$.
While $(\mathds{1}-Q_{\mathrm{FM\,IVC}})/2$ indeed projects onto the filled
states of $H_{\mathrm{FM\,IVC}}$ at $\nu=-2$, the Hamiltonian
$H_{\mathrm{FM\,IVC}}$ itself only preserves the U(1) symmetry generated by
$\tau^{z}(\mathds{1}-s^{z})$ when $b=c$, which is not required by any
symmetry. We therefore do not include this symmetry in our analysis.

All of the terms present in the effective Hamiltonian for a wire proximity-
coupled to a $\nu=0$ sIVC insulator and for a wire proximity-coupled to a
$\nu=0$ tIVC insulator (Secs. III.3, IV.2.1, and IV.2.2) will also be
generated here. In Table 5, these terms are summarized in rows 1-8. Additional
terms are obtained by considering the FM IVC order by itself and alongside the
Rashba and Ising SOC individually. The symmetry group for each of these cases
is provided in Table 1 and the additional terms present in each case are
listed in Table 5. From the results of this table, we follow the prescription
outlined in Sec. III.3 to obtain the effective 1$d$ Hamiltonian and the
corresponding parameter scaling shown in Table 2.

#### IV.3.2. Effective Hamiltonian with proximity-induced AFM IVC order

The AFM IVC order parameter is given by selecting $U_{+}=\mathds{1}$ and
$U_{-}=s^{x}$ in Eqs. (59) and (60):

$\displaystyle Q_{\mathrm{AFM\,IVC}}$
$\displaystyle=\frac{1}{2}\big{(}\mathds{1}+\tau^{z}s^{z}+\tau^{x}\sigma^{y}s^{x}+\tau^{y}\sigma^{y}s^{y}\big{)}$
(62)

Comparing against Eq. (58), we see that that $Q_{\mathrm{AFM\,IVC}}$ could
equivalently have been obtained by choosing $Q_{1}$, $Q_{2}$ equal to two out
of $\tau^{z}s^{z}$, $\tau^{x}\sigma^{y}s^{x}$, and $\tau^{y}\sigma^{y}s^{y}$
(as always, up to $\mathrm{U_{v}}(1)$ rotations). We note that the mean field
generation of a $\tau^{z}s^{z}$ term is equivalent to the spontaneous
generation of Ising SOC; thus, when we consider the effect of adding SOC to
the system, we will not need to consider Ising SOC as we did for the other
cases. We recognize the other two order parameters, $\tau^{x}\sigma^{y}s^{x}$
and $\tau^{y}\sigma^{y}s^{y}$, as in-plane versions of the IVC triplet order
parameter $Q_{\mathrm{tIVC}}=\tau^{x}\sigma^{y}s^{z}$ considered in Sec.
IV.2.2.

As written, $Q_{\mathrm{AFM\,IVC}}$ preserves the three U(1) symmetries
generated by $\mathds{1}$, $\tau^{z}s^{z}$, and $\tau^{z}+s^{z}$, and, unlike
in the previous case, there is a physical reason to expect that the AFM IVC
may be invariant under all of these U(1) symmetries. The primary issue is the
symmetry generated by $\tau^{z}+s^{z}$. It appears to be similar to the U(1)
symmetry that we argued in Sec. IV.3.1 was a non-physical artifact, the result
of fine-tuning. Here, however, the symmetry generated by $\tau^{z}+s^{z}$ can
be interpreted as a residual version of
$\mathrm{U_{v}}(1)\times\mathrm{U}_{z}(1)$, in which a spin rotation about the
$z$ axis is followed by a valley rotation:
$\tilde{\mathrm{U}}_{z}(1):c({\bm{k}})\to
e^{i\theta(\tau^{z}+s^{z})/2}c({\bm{k}})$. Although the introduction of SOC
ultimately breaks both the $\mathrm{U}_{z}(1)$ and ‘residual’
$\tilde{\mathrm{U}}_{z}(1)$ symmetry, its presence allows the existence of a
residual version of the spinful 120∘ rotation symmetry even when SOC is
included: $\tilde{C}_{3}=e^{-2\pi i\tau^{z}/3}C_{3}=e^{-2\pi
i(s^{z}+\tau^{z})/3}{\mathcald{C}}_{3}$. Since the extra valley rotation that
has been appended occurs only at the scale of the microscopic graphene
lattice, this ‘residual’ $\tilde{C}_{3}$ symmetry is largely indistinguishable
from the ‘physical’ $C_{3}$ symmetry we started with—all lattice symmetries
are after all only well-defined in a low-energy limit, at length scales much
larger than the microscopic lattice constant. As a result, provided
$\tau^{z}+s^{z}$ is preserved in the AFM IVC state without SOC, the AFM IVC
state with SOC will _not_ be nematic. While some experiments show evidence for
nematicity around $\nu=\pm 2$ [20, 23, 24, 25], for simplicity, we restrict
ourselves to the fully symmetric case. This assumption is equivalent to
requiring that the two in-plane IVC order parameters occur with identical
coefficients, _i.e._ , in the form
$\tau^{x}\sigma^{y}s^{x}+\tau^{y}\sigma^{y}s^{y}$.

The symmetries preserved by this order parameter are listed in Table 1 in the
limit just discussed. In turn, its presence induces the terms listed in Table
5 in the 2$d$ effective Hamiltonian. Following the prescription outlined in
Sec. III.3, we arrive at the effective wire Hamiltonian provided in Table 2
with the parameter scalings shown.

###  Relation of internal SOC parameters and wire direction

One aspect of the above analysis that may not be immediately clear is the
relation it implies between the spin and wire direction. This point is best
illustrated by direct comparison with a more conventional spin-orbit coupled
wire. In a typical $2d$ SOC-coupled material, the Rashba term takes the form
$\sim k_{x}s^{y}-k_{y}s^{x}$. Reducing to a wire scenario (directed along the
$x$ direction) then returns a $1d$ Rashba contribution $\sim k_{x}s^{y}$. In
the presence of an in-plane Zeeman field, different behaviour should be
observed depending on whether the magnetic field points parallel to the wire,
$\sim\frac{1}{2}g\mu_{B}Bs^{x}$, or perpendicular to the wire,
$\sim\frac{1}{2}g\mu_{B}Bs^{y}$.

By contrast, our symmetry analysis does not fix the relation between the
spatial wire direction and the internal spin directions favored by SOC. In
particular, we repeat here the second line of Eq. (III.2.1), which gives the
odd parity Rashba SOC terms generated at first order:

$\displaystyle\bar{h}^{(R)}_{o}({\bm{k}})$
$\displaystyle=\tilde{t}^{(R)}_{o}\big{(}k_{x}s^{x}+k_{y}s^{y}\big{)}+\tilde{t}^{(R)\prime}_{o}\big{(}k_{y}s^{x}-k_{x}s^{y}\big{)}.$
(63)

Generically both $\tilde{t}^{(R)}_{o}$ and $\tilde{t}^{(R)\prime}_{o}$ are
non-zero. In the $1d$ limit we therefore obtain a sum $\sim
k_{x}\left(\tilde{t}^{(R)}_{o}s^{x}-\tilde{t}^{(R)\prime}_{o}s^{y}\right)$,
indicating that the spin axis favored by Rashba SOC depends on non-universal
details. Since to a good approximation an applied magnetic field couples only
to the wire’s spin degrees of freedom, the non-universal relation between the
wire orientation and spin directions implies a non-universal relation between
the wire and magnetic field directions.

This non-universality is the primary motivation for our introduction of the
fine-tuned mirror-symmetric Hamiltonians of Sec. II.3. When
${\mathcalb{h}}^{*}_{R}$ is present instead of ${\mathcalb{h}}_{R}$, the
${\mathcald{M}}_{y}^{*}$ (or, more correctly, the
$is^{x}{\mathcald{M}}_{y}^{*}$) symmetry sets $\tilde{t}^{(R)\prime}_{o}$ to
zero, in which case we can definitively ‘tie’ the wire and internal SOC
directions to one another, even at the level of the symmetry analysis. Moving
beyond this fine-tuned limit, although we cannot say that
$\tilde{t}^{(R)\prime}_{o}$ vanishes in the physical situation, we may
nevertheless predict that because of the strong interlayer mixing that occurs
between the two layers, the mirror symmetry is only weakly broken, implying
that
$\left|\tilde{t}^{(R)\prime}_{o}\right|\ll\left|\tilde{t}^{(R)}_{o}\right|$.
(A caveat to this reasoning is that mirror symmetry is most strongly broken in
the region of interest, close to the $\gamma$ point.)

In selecting parameters for Fig. 2 and 3 of the main text, this weak breaking
of the mirror symmetry was imposed. Within this set of assumptions, the
magnetic field direction chosen is consistent with being perpendicular to the
wire. We further assumed that the wire was directed in such a way that the
1$d$ projection of neither $k_{x}(k_{x}^{2}-3k_{y}^{2})$ nor
$k_{y}(3k_{x}^{3}-k_{y}^{2})$ vanished.

## V. Wire proximity-coupled to IVC and superconducting states

Figure 1:  Plots of excitation gap in a wire proximity-coupled to both an sIVC
insulator and a superconductor that contributes pairing terms of the form
provided in Eq. (4) of the main text. Topological (topo), trivial, and gapless
phases are noted. The same parameters used in Figs. 2(c) and 3(a) are used
above. The inset in the centre reproduces Fig. 2(c) of the main text. The two
odd-channel energy windows marked with grey boxes are labelled
$\epsilon_{1,2}$ and $\bar{\epsilon}$ is defined as their average. (a)
Excitation gap as a function of $\Delta_{1}/\bar{\epsilon}$ and $\mu$ for zero
magnetic field. The vertical line on the left, marked $\Delta_{1}^{*}$,
corresponds to the pairing used in Fig. 3(a) of the main text. As in that
figure, $\Delta_{2}=-0.4\Delta_{1}$. (b) Excitation gap as a function of
magnetic field for $\Delta_{1}$ equal to $\Delta_{1}^{**}$, shown in (a) by
the vertical line on the right. $\Delta_{2}$ is again equal to
$-0.4\Delta_{1}$. Although the lower topological phase is absent at zero
magnetic field, it re-emerges at finite field. (c) Excitation gap as a
function of $\Delta_{1}/\bar{\epsilon}$ and $\mu$ at zero magnetic field with
$\Delta_{2}=0$. The upper topological phase that arises out of the three-
channel regime shown in the inset is now absent. (d) Excitation gap as a
function of $\Delta_{2}/\bar{\epsilon}$ for zero magnetic field with
$\Delta_{1}=0$. Both topological phases are clearly present. Furthermore, the
one-channel phase extends to larger values of the externally induced pairing
term $\Delta_{2}$.

Figures 1 and 2 build on the information in Fig. 3 of the main text.

Through the excitation gap, Fig. 1 presents the phase diagram of a wire
proximity coupled to both the sIVC insulator and the superconducting state in
a number of scenarios. In (a), the excitation gap is plotted as a function of
$\Delta_{1}$ and $\mu$ at zero magnetic field. As in the main text, we assume
$\Delta_{2}=-0.4\Delta_{1}$. Two distinct topological states are observed at
precisely the location of the odd channel regimes, as shown in Fig. 2(c) of
the main text and reproduced in the central inset of Fig. 1. The white line on
the left of (a), labelled $\Delta_{1}^{*}$, indicates the pairing strength
used to generate Fig. 3(a) in the main text. Figure 1(b) plots the excitation
gap as a function of magnetic field for the fixed pairing strength indicated
by the line labelled $\Delta_{1}^{**}$ in (a). Again,
$\Delta_{2}=-0.4\Delta_{1}^{**}$. Notably, since $\Delta_{1}^{**}$ does not
intersect the lower topological lobe of (a) (corresponding to the one-channel
regime), there is topological phase at $B=0$ in (b). At finite $B$, however,
the topological phase re-emerges. Figures 1(c) and (d) are similar to (a) in
that they also plot the excitation gap as a function of external pairing in
zero magnetic field. In (c), the triplet pairing is absent, $\Delta_{2}=0$,
and we see that $\Delta_{1}$ is unable to open a gap in the three channel
regime—only the lower topological lobe is present. In (d), the singlet pairing
is instead set to zero, $\Delta_{1}=0$, and topological phases are observed in
both the one and three channel regimes as a function of the triplet pairing
$\Delta_{2}$.

The plots in Fig. 2 similarly expand upon the image shown Fig. 3(b) of the
main text. In (a), the excitation gap is plotted as a function of the chemical
potential $\mu$ and the pairing strength $\Delta_{1}$ at a fixed magnetic
field, $\bar{B}$. The value of $\bar{B}$ is chosen such that
$\frac{1}{2}g\mu_{B}\bar{B}/\tilde{\Delta}_{1}=2.0$, where
$\tilde{\Delta}_{1}$ is the singlet pairing strength used to generate Fig.
3(b) of the main text and indicated in (a) with a vertical dashed white line.
Figure 2(b) is directly analogous to Fig. 3(b) of the main text, save that
$\Delta_{2}=+0.4\tilde{\Delta}_{1}$. Two distinct lobes are now apparent at
finite magnetic field; they are, however, much smaller than topological region
shown in (a).

Figure 2:  Plots of the excitation gap in a wire proximity-coupled to both an
AFM IVC insulator and a superconductor that contributes pairing terms of the
form provided in Eq. (4) of the main text. Topological (topo), trivial, and
gapless phases are noted. The same parameters used in Fig. 2(i) as well as
Fig. 3(b) are used above. (a) Excitation gap as a function of
$\Delta_{1}/(\frac{1}{2}g\mu_{B}\bar{B})$ and $\mu$ for fixed magnetic field
$\bar{B}$ chosen such that
$\frac{1}{2}g\mu_{B}\bar{B}/\tilde{\Delta}_{1}=2.0$, where
$\tilde{\Delta}_{1}$ is the fixed pairing strength used to generate the Fig.
3(b) of the main text and shown with the white dashed line. (b) Excitation gap
in the wire as a function of magnetic field
$\frac{1}{2}g\mu_{B}B/\tilde{\Delta}_{1}$ and $\mu$ for fixed singlet pairing
strength $\tilde{\Delta}_{1}$—the same value used to obtain Fig. 3(a) of the
main text and plotted (a). Unlike Fig. 3(b) of the main text, the triplet
pairing strength is set to equal $\Delta_{2}=+0.4\tilde{\Delta}_{1}$. There
are now two distinct topological lobes, although they are quite a bit smaller
in size.

## VI. Five band model

As a proof of concept, we perform numerical simulations of a wire proximity-
coupled to various IVC phases within the context of the five model for the
flat bands of TBG described in Ref. 26. In Secs. VI.1, VI.2, and VI.3, we
describe the model setup, symmetry actions, and the inter-orbital hopping
terms first provided in Ref. 26. We subsequently use symmetries in conjunction
with the information contained in Table 1 to incorporate SOC and IVC orders
into the five-band model—focusing in particular on the sIVC insulator at
$\nu=0$ and the AFM IVC insulator at $\nu=\pm 2$. We describe how the wire is
realized and present out results in Sec. VI.6. We finish in Sec. VI.7 by
discussing some limitations of this study.

###  Basic definitions

While the two flat bands (per spin per valley) would ideally be describable
using a microscopic two band model, the bands themselves possess a ‘fragile
topology’ [27, 26, 28] that prohibits such a description. Instead, the minimal
description one may formulate involves five orbitals centered at different
locations within the moiré unit cell. Here, we study a model with $p_{z}$,
$p_{+}$, and $p_{-}$ orbitals on the triangular lattice sites at the centre of
each moiré unit cell (where the graphene sheets have AA stacking) and $s$
orbitals on the A and B sublattices of the honeycomb lattice (where the
graphene sheets have AB and BA stacking). Below, the three $p$ orbitals on the
triangular lattice are sometimes denoted $\mathrm{tri}_{z,\pm}$, while the two
$s$ orbitals on the hexagon sites are sometimes denoted $\mathrm{hex}_{A/B}$.
See Fig. 3(a) for details. We arrange the corresponding electron annihilation
operators as

$\displaystyle f_{v,{\bm{r}}}$
$\displaystyle=\Big{(}f_{v,z,{\bm{r}}},f_{v,+,{\bm{r}}},f_{v,-,{\bm{r}}},f_{c,A,{\bm{r}}},f_{v,B,{\bm{r}}}\Big{)}^{T},$
$\displaystyle v$ $\displaystyle=K,K^{\prime}.$ (64)

A spin index is suppressed in the above. The position ‘${\bm{r}}$’ is defined
via the primitive vectors shown in Fig. 3. That is,
${\bm{r}}=n_{1}{\bm{a}_{1}}+n_{2}{\bm{a}_{2}}$ where $n_{1,2}\in\mathds{Z}$
and

$\displaystyle{\bm{a}_{1}}$
$\displaystyle=\left(\frac{\sqrt{3}}{2}\mathbin{\raisebox{2.15277pt}{,}}-\frac{1}{2}\right),$
$\displaystyle{\bm{a}_{2}}$ $\displaystyle=\left(0,1\right).$ (65)

The moiré lattice constant has been set to unity.

Figure 3:  (a) Unit cell of the five band model with Bravais lattice vectors
${\bm{a}_{1}}$ and ${\bm{a}_{2}}$ shown in green. Note that the hexagon shown
is the _moiré_ unit cell, which itself comprises thousands of graphene unit
cells. The location of the three $p$ orbitals, $p_{z,\pm}$, at the centre of
the moiré hexagon is shown in purple, the $s$ orbital at the A sublattice of
the hexagon corners is shown in magenta, and the $s$ orbital at the B
sublattice of the hexagon corner is shown in blue. The corresponding electron
annihilation operators are written alongside each orbital. (b) Inter-orbital
hopping terms for $K$ valley orbitals as described in Sec. VI.3.

###  Symmetries

In this section, we outline the action of the symmetries on the orbitals of
the five band model. With some minor differences in presentation, we again use
the ‘spinless’ version of the symmetries, which are related to the physical
symmetries via Eq. (I.3). We opt to only explicitly describe
${\mathcald{C}}_{2}{\mathcald{T}}$ and ${\mathcald{T}}$, from which the
${\mathcald{C}}_{2}$ transformation may be derived. We further only explicitly
relate the action of the symmetries on the operators originating from valley
$K$; the action on valley $K^{\prime}$ may be obtained using time reversal
symmetry. The continuous internal symmetries act on the spin and valley
indices as they do in Eq. (I.3).

#### VI.2.1. 120${}^{\,\circ}$ rotation, ${\mathcald{C}}_{3}$

The ${\mathcald{C}}_{3}$ symmetry acts as

$\displaystyle\bm{a}_{1}$ $\displaystyle\to{\bm{a}_{2}},$
$\displaystyle{\bm{a}_{2}}$ $\displaystyle\to-{\bm{a}_{1}}-{\bm{a}_{2}},$ (66)

and the electron operators transform as

$\displaystyle{\mathcald{C}}_{3}:\quad f_{K,{\bm{r}}}$
$\displaystyle\to\begin{pmatrix}f_{K,z,{\bm{r}}^{\prime}}\\\
\omega^{*}f_{K,+,{\bm{r}}^{\prime}}\\\ \omega f_{K,-,{\bm{r}}}\\\
f_{K,A,{\bm{r}}^{\prime}-{\bm{a}_{1}}-{\bm{a}_{2}}}\\\
f_{K,B,{\bm{r}}^{\prime}-{\bm{a}_{1}}}\end{pmatrix}$ (67)

where ${\bm{r}}=r_{1}{\bm{a}_{1}}+r_{2}{\bm{a}_{2}}$,
${\bm{r}}^{\prime}=-r_{2}{\bm{a}_{1}}+(r_{1}-r_{2}){\bm{a}_{2}}$ for
$r_{1,2}\in\mathds{Z}$.

#### VI.2.2. Mirror, ${\mathcald{M}}_{y}$

Under ${\mathcald{M}}_{y}$, the primitive vectors transform as

$\displaystyle{\bm{a}_{1}}$ $\displaystyle\to{\bm{a}_{1}}+{\bm{a}_{2}},$
$\displaystyle{\bm{a}_{2}}$ $\displaystyle\to-{\bm{a}_{2}}.$ (68)

The real space electron operator transform as

$\displaystyle{\mathcald{M}}_{y}:\quad f_{K,{\bm{r}}}$
$\displaystyle\to\begin{pmatrix}-f_{K,z,{\bm{r}}^{\prime}}\\\
f_{K,-,{\bm{r}}^{\prime}}\\\ f_{K,+,{\bm{r}}^{\prime}}\\\
f_{K,A,{\bm{r}}^{\prime}-{\bm{a}_{2}}}\\\
f_{K,B,{\bm{r}}^{\prime}}\end{pmatrix},$ (69)

where ${\bm{r}}=r_{1}{\bm{a}_{1}}+r_{2}{\bm{a}_{2}}$,
${\bm{r}}^{\prime}=r_{1}{\bm{a}_{1}}+(r_{1}-r_{2}){\bm{a}_{2}}$ for
$r_{1,2}\in\mathds{Z}$.

#### VI.2.3. 180${}^{\,\circ}$ rotation + time reversal,
${\mathcald{C}}_{2}{\mathcald{T}}$

${\mathcald{C}}_{2}{\mathcald{T}}$ acts on the primitive vectors as

$\displaystyle{\mathcald{C}}_{2}{\mathcald{T}}$ $\displaystyle:$
$\displaystyle{\bm{a}_{1}}$ $\displaystyle\to-{\bm{a}_{1}},$
$\displaystyle{\bm{a}_{2}}$ $\displaystyle\to-{\bm{a}_{2}},$ $\displaystyle
i\to-i.$ (70)

Its action on the real space electron operators is

$\displaystyle{\mathcald{C}}_{2}{\mathcald{T}}:\quad
f_{K,{\bm{r}}}\to\begin{pmatrix}f_{K,z,-{\bm{r}}}\\\ f_{K,-,-{\bm{r}}}\\\
f_{K,+,-{\bm{r}}}\\\ f_{K,B,-{\bm{r}}-{\bm{a}_{1}}-{\bm{a}_{2}}}\\\
f_{K,A,-{\bm{r}}-{\bm{a}_{1}}-{\bm{a}_{2}}}\end{pmatrix},\qquad i\to-i.$ (71)

#### VI.2.4. Time reversal

Finally, time reversal acts as

$\displaystyle{\mathcald{T}}:\quad
f_{K,{\bm{r}}}\to\begin{pmatrix}f_{K^{\prime},z,{\bm{r}}}\\\
f_{K^{\prime},-,{\bm{r}}}\\\ f_{K^{\prime},+,{\bm{r}}}\\\
f_{K^{\prime},A,{\bm{r}}}\\\ f_{K^{\prime},B,{\bm{r}}}\end{pmatrix},\qquad
i\to-i.$ (72)

###  Five band model without SOC

We describe the model as provided in Ref. 26, focusing on valley $K$. All
terms corresponding to the $K^{\prime}$ valley may be obtained via time
reversal, Eq. (72). We divide the terms considered into the tunnelling types
shown in Fig. 3(b): hopping between the honeycomb lattice orbitals
($\mathrm{hex}_{A/B}\to\mathrm{hex}_{B/A}$), hopping between the three
orbitals on the triangular lattice sites
($\mathrm{tri}_{z,\pm}\to\mathrm{tri}_{z,\pm}$), and hopping between the
triangular lattice orbitals and the A and B sublattice of the honeycomb
lattice ($\mathrm{tri}_{z,\pm}\to\mathrm{hex}_{A}$ and
$\mathrm{tri}_{z,\pm}\to\mathrm{hex}_{B}$). The equations below are
parametrized by real numbers $a,b,c,d,$ and $\alpha$ (which should not to be
confused with the effective wire parameters of Table 2); an energy scale
$t_{0}$; and a set of orbital potentials.

#### VI.3.1. Hopping between $\mathrm{hex}_{A}$ and $\mathrm{hex}_{B}$
orbitals

Nearest-neighbour hopping between $s$ orbitals on the honeycomb lattice is
schematically shown on the top left of Fig. 3(b) and may be expressed using
the notation of Sec. VI.1 as

$\displaystyle H_{hh}^{(K)}$
$\displaystyle=-t_{0}d^{2}e^{-2i\alpha}\sum_{\bm{r}}f_{K,A,{\bm{r}}}^{\dagger}\Big{(}f_{K,B,{\bm{r}}}+f_{K,B,{\bm{r}}+{\bm{a}_{2}}}+f_{K,B,{\bm{r}}-{\bm{a}_{1}}}\Big{)}+h.c.$
(73)

#### VI.3.2. Hopping between $\mathrm{tri}_{\alpha}$ orbitals

The hopping between the $p_{z}$, $p_{+}$ and $p_{-}$ orbitals on triangular
lattice sites is depicted on the top right of Fig. 3(b). The corresponding
$K$-valley Hamiltonian is

$\displaystyle H_{tt}^{(K)}$
$\displaystyle=-t_{0}\sum_{\alpha,\beta=z,\pm}\sum_{\bm{r}}f_{\alpha,{\bm{r}}}^{\dagger}\Big{(}t_{\alpha\beta}^{(1)}f_{\beta,{\bm{r}}+{\bm{a}_{2}}}+t_{\alpha\beta}^{(2)}f_{\beta,{\bm{r}}-{\bm{a}_{1}}-{\bm{a}_{2}}}+t_{\alpha\beta}^{(3)}f_{\beta,{\bm{r}}+{\bm{a}_{1}}}\Big{)}+h.c.,$
(74)

where 1, 2, 3 label the different tunnelling directions and the inter-
triangular lattice hopping parameters are $3\times 3$ matrices given by

$\displaystyle t^{(1)}$ $\displaystyle=\begin{pmatrix}-a^{2}&-iab&-iac\\\
-iac&bc&c^{2}\\\ -iab&b^{2}&bc\end{pmatrix},$ $\displaystyle t^{(2)}$
$\displaystyle=U_{r}^{\dagger}t^{(1)}U_{r},$ $\displaystyle t^{(3)}$
$\displaystyle=U_{r}t^{(1)}U_{r}^{\dagger}.$ (75)

Above we defined $U_{r}=\mathrm{diag}(1,\omega^{*},\omega)$; here and below
$\omega=e^{2\pi i/3}$. The matrices $t^{(\ell)}$, $\ell=1,2,3$ and $U_{r}$ are
expressed in a basis with the $p$ orbital ordering given in Eq. (64).

#### VI.3.3. Hopping between $\mathrm{tri}_{\alpha}$ and $\mathrm{hex}_{A}$
orbitals

As shown to the bottom left of Fig. 3(b), there are three types of nearest-
neighbour tunnelling directions connecting orbitals on the A honeycomb
sublattice to those on the triangular lattice. We label these directions 1, 2,
3. The Hamiltonian expressing these terms is

$\displaystyle H_{th_{A}}^{(K)}$
$\displaystyle=-t_{0}\sum_{\mu=\pm}\sum_{\bm{r}}f_{K,A,{\bm{r}}}^{\dagger}\Big{(}x_{\mu}^{(1)}f_{K,\mu,{\bm{r}}}+x_{\mu}^{(2)}f_{K,\mu,{\bm{r}}+{\bm{a}_{1}}+{\bm{a}_{2}}}+x_{\mu}^{(3)}f_{K,\mu,{\bm{r}}+{\bm{a}_{2}}}\Big{)}+h.c.$
(76)

The $x_{\ell}^{(\mu)}$ parameters are constrained by symmetry and can be
decomposed according to

$\displaystyle x_{+}^{(1)}$ $\displaystyle=d\,e^{-i\alpha}(\omega^{*}b+c),$
$\displaystyle x_{-}^{(1)}$ $\displaystyle=d\,e^{-i\alpha}(b+\omega c),$
$\displaystyle x_{+}^{(2)}$ $\displaystyle=\omega^{*}x_{+}^{(1)},$
$\displaystyle x_{-}^{(2)}$ $\displaystyle=\omega x_{-}^{(1)},$ $\displaystyle
x_{+}^{(3)}$ $\displaystyle=\omega x_{+}^{(1)},$ $\displaystyle x_{-}^{(3)}$
$\displaystyle=\omega^{*}x_{-}^{(1)}.$ (77)

Tight-binding model parameters
---
$a$ |  $b$ |  $c$ |  $d$ |  $\alpha$ |  $\bar{\mu}_{z}$ |  $\bar{\mu}_{+}$ |  $\bar{\mu}_{-}$ |  $\bar{\mu}_{A}$ |  $\bar{\mu}_{-}$ |  $t_{0}$
$0.25$ |  $0.2$ |  $0.1$ |  $0.67$ |  $0$ |  $-0.043$ |  $0$ |  $0$ |  $0.05$ |  $0.05$ |  $80$ meV
Table 3: Parameters defining five band model without SOC as per Ref. 26.

#### VI.3.4. Hopping between $\mathrm{tri}_{\alpha}$ and $\mathrm{hex}_{B}$
orbitals

Finally, we consider the Hamiltonian describing the nearest-neighbour
tunnelling between the B honeycomb orbitals and the orbitals on the triangular
lattice sites. As in the previous section, there are three directions and
corresponding tunnelling constants labelled 1, 2, 3; see the image on the
bottom right of Fig. 3(b). We find

$\displaystyle H_{th_{B}}^{(K)}$
$\displaystyle=-\sum_{\mu=\pm}\sum_{\bm{r}}f_{K,B,{\bm{r}}}^{\dagger}\Big{(}y_{\mu}^{(1)}f_{K,\mu,{\bm{r}}}+y_{\mu}^{(2)}f_{K,\mu,{\bm{r}}+{\bm{a}_{1}}}+y_{\mu}^{(3)}f_{K,\mu,{\bm{r}}+{\bm{a}_{1}}+{\bm{a}_{2}}}\Big{)}+h.c.$
(78)

with

$\displaystyle y_{+}^{(1)}$ $\displaystyle=d\,e^{i\alpha}(\omega^{*}b+\omega
c),$ $\displaystyle y_{-}^{(1)}$
$\displaystyle=d\,e^{i\alpha}(\omega^{*}b+\omega c),$ $\displaystyle
y_{+}^{(2)}$ $\displaystyle=\omega^{*}y_{+}^{(1)},$ $\displaystyle
y_{-}^{(2)}$ $\displaystyle=\omega y_{-}^{(1)},$ $\displaystyle y_{+}^{(3)}$
$\displaystyle=\omega y_{+}^{(1)},$ $\displaystyle y_{-}^{(3)}$
$\displaystyle=\omega^{*}y_{-}^{(1)}.$ (79)

#### VI.3.5. Complete Hamiltonian

Figure 4:  Electronic structure of five band model without SOC defined in Sec.
VI.4 along the linecut shown in the inset of (a). The parameters used are
given in Table 3. Blue and magenta lines respectively plot the energies of
valley $K$ and $K^{\prime}$. (a) The flat bands. (b) All bands represented
within the five band band model.

The full Hamiltonian is obtained by assembling the terms of the previous
sections and including some orbital dependent chemical potential shifts:

$\displaystyle H_{0}$ $\displaystyle=H_{0}^{(K)}+H_{0}^{(K^{\prime})},$
$\displaystyle H_{0}^{(v)}$
$\displaystyle=H_{hh}^{(v)}+H_{th_{A}}^{(v)}+H_{th_{B}}^{(v)}+H_{tt}^{(v)}+H_{dg,1}^{(v)}+H_{dg,2}^{(v)},$
(80)

where $v=K,K^{\prime}$. The $H_{dg,1}^{(v)}$ and $H_{dg,2}^{(v)}$ pieces read

$\displaystyle
H_{dg,\ell}^{(v)}=\sum_{\bm{r}}f_{v,{\bm{r}}}^{\dagger}h_{dg,\ell}f_{v,{\bm{r}}}.$
(81)

The matrices are the same for both valleys and given by

$\displaystyle h_{dg,1}$
$\displaystyle=-t_{0}\mathrm{diag}\Big{(}6a^{2},\,3(b^{2}+c^{2}),\,3(b^{2}+c^{2}),\,3d^{2},\,3d^{2}\Big{)},$
$\displaystyle h_{dg,2}$
$\displaystyle=t_{0}\mathrm{diag}\Big{(}\bar{\mu}_{z},\,\bar{\mu}_{+},\,\bar{\mu}_{-},\,\bar{\mu}_{A},\,\bar{\mu}_{B}\Big{)}.$
(82)

Table 3 lists the parameter values that we use—which are taken from Ref.
26—and Fig. 4 shows the resulting band structure for valley $K$. Once again,
the Hamiltonian $H_{0}^{(K^{\prime})}$ may be obtained directly from
$H_{0}^{(K)}$ via Eq. (72). By construction, the flat bands are _completely_
flat when
$\bar{\mu}_{z}=\bar{\mu}_{+}=\bar{\mu}_{-}=\bar{\mu}_{A}=\bar{\mu}_{B}=0$,

###  Spin-orbit coupling

We add spin-orbit coupling by finding terms that satisfy the symmetries of
Rashba and Ising SOC given in Table 1. Since the weight of the two flat bands
resides predominantly on the $p_{\pm}$ orbitals at the hexagon centres [26],
we focus on terms involving these orbitals.

#### VI.4.1. Rashba SOC

The simplest Rashba term involving the $p_{\pm}$ orbitals that satisfies the
same symmetries as in Table 1 is

$\displaystyle H_{R,0}^{(K)}$
$\displaystyle=\frac{\lambda_{R,0}}{2}\sum_{\bm{r}}e^{i\varphi_{0}}f_{K,+,{\bm{r}}}^{\dagger}\big{(}s^{x}-is^{y}\Big{)}f_{K,-,{\bm{r}}}+h.c.$
(83)

The Hamiltonian corresponding to the other valley $H^{(K^{\prime})}_{R,0}$ as
usual follows from time reversal. In order to preserve mirror symmetry
(specifically, $is^{x}{\mathcald{M}}_{y}^{*}$), the phase must vanish:
$\varphi_{0}=0$.

We further consider two symmetry-allowed Rashba terms involving nearest-
neighbour hopping between the $p_{+}$ and $p_{-}$ orbitals:

$\displaystyle H_{R,1}^{(K)}$
$\displaystyle=-\frac{\lambda_{R,1}}{3}\sum_{\bm{r}}e^{i\varphi_{1}}f_{K,+,{\bm{r}}}^{\dagger}s^{x}\Big{[}f_{K,-,{\bm{r}}+{\bm{a}_{2}}}+\omega^{*}e^{2\pi
is^{z}/3}f_{K,-,{\bm{r}}-{\bm{a}_{1}}-{\bm{a}_{2}}}+\omega e^{-2\pi
is^{z}/3}f_{K,-,{\bm{r}}+{\bm{a}_{1}}}\Big{]}+h.c.$ $\displaystyle
H_{R,2}^{(K)}$
$\displaystyle=-\frac{\lambda_{R,2}}{3}\sum_{\bm{r}}e^{i\varphi_{2}}f_{K,-,{\bm{r}}}^{\dagger}s^{x}\Big{[}f_{K,+,{\bm{r}}+{\bm{a}_{2}}}+\omega^{*}e^{2\pi
is^{z}/3}f_{K,+,{\bm{r}}-{\bm{a}_{1}}-{\bm{a}_{2}}}+\omega e^{-2\pi
is^{z}/3}f_{K,+,{\bm{r}}+{\bm{a}_{1}}}\Big{]}+h.c.$ (84)

These terms also satisfy the symmetries of Table 1, and the term corresponding
to the other valley may similarly be obtained by applying time reversal;
$is^{x}{\mathcald{M}}_{y}^{*}$ is only satisfied when
$\varphi_{1}=\varphi_{2}=0$.

#### VI.4.2. Ising SOC

Figure 5:  Bulk band structures of trivial state (a), sIVC ordered insulator
(b), and AFM IVC-ordered insulator (c). The parameters used are provided in
Table 3 and the appropriate columns of Table 4. The dotted and dash-dotted
lines respectively denote the bulk and wire chemical potentials
($\mu_{\mathrm{bulk}}$ and $\mu_{\mathrm{wire}}$) used in the wire simulation.
The momentum linecut used to plot the energies is shown in the inset of (a).

At the level of the microscopic graphene lattice, Ising SOC acts simply as a
chemical potential whose sign depends on the spin and valley. It can therefore
be translated in a straightforward manner to the five band model as

$\displaystyle H_{I}$
$\displaystyle=H_{I}^{(K)}+H_{I}^{(K^{\prime})}=\frac{\lambda_{I}}{2}\sum_{\bm{r}}\big{(}f_{K,{\bm{r}}}^{\dagger}s^{z}f_{K,{\bm{r}}}-f_{K^{\prime},{\bm{r}}}^{\dagger}s^{z}f_{K^{\prime},{\bm{r}}}\big{)},$
(85)

where the orbital and spin indices have both been suppressed. Unlike the
Rashba terms, this expression involves all five orbitals, a choice based
largely on convenience. The Ising contribution as written preserves the mirror
symmetry ${\mathcald{M}}_{y}^{*}$. A mirror-breaking Ising term involving the
$p_{+}$ and $p_{-}$ orbitals does exist, but requires inter-site hopping
(similar to Eq. (VI.4.1)); we do not consider it here.

#### VI.4.3. Total SOC contribution

The total contribution of the SOC terms to the five band Hamiltonian that we
consider is

$\displaystyle H_{\mathrm{SO}}$
$\displaystyle=\sum_{v=K,K^{\prime}}\Big{(}H_{R,0}^{(v)}+H_{R,1}^{(v)}+H_{R,2}^{(v)}+H_{I}^{(v)}\Big{)}.$
(86)

Figure 5(a) shows the flat band energies with the choice of SOC parameters
given on the left side of Table 4. As expected, the twofold spin degeneracy of
Fig. 4 is lifted by the SOC.

###  IVC order

The IVC terms may be obtained in exactly the same fashion as the SOC terms
considered above: we find the simplest expressions involving $p_{\pm}$ that
satisfy the appropriate symmetries of Table 1. For simplicity, here we focus
on the singlet IVC (sIVC) case at $\nu=0$ and antiferromagnetic IVC (AFM IVC)
case at $\nu=\pm 2$.

#### VI.5.1. Singlet IVC order

As was the case for Rashba and Ising SOC, there exists a simple onsite term
involving the $p_{\pm}$ orbitals and satisfying the correct symmetries:

$\displaystyle H_{\mathrm{sIVC}}$
$\displaystyle=\frac{\Delta_{\mathrm{sIVC}}}{2}\sum_{{\bm{r}}}e^{i\phi_{0}}\Big{[}f_{K,+,{\bm{r}}}^{\dagger}f_{K^{\prime},+,{\bm{r}}}-f_{K,-,{\bm{r}}}^{\dagger}f_{K^{\prime},-,{\bm{r}}}\Big{]}+h.c.$
(87)

Notably, this term breaks the $\mathrm{U_{v}}(1)$ symmetry and time reversal,
but preserves $\mathcal{T}_{\mathrm{sIVC}}=i\tau^{y}\mathcal{K}$. While the
Hamiltonian is equivalent by definition for every value of $\phi_{0}$, its
value alters which $\mathrm{U_{v}}(1)$ valley operations must be composed with
the physical discrete symmetries. The choice consistent with Table 1 is
$\phi_{0}=0$.

Figure 5(b) shows the electronic structure with
$\Delta_{\mathrm{sIVC}}=3.0\,\mathrm{\text{meV}}$ and the SOC parameters
defined on the left hand side of Table 4. A clear gap at $\nu=0$ is observed.

Spin-orbit coupling | sIVC | AFM IVC
---|---|---
$\lambda_{R,0}$ | $\varphi_{0}$ | $\lambda_{R,1}$ | $\varphi_{1}$ | $\lambda_{R,2}$ | $\varphi_{2}$ | $\lambda_{I}$ | $\Delta_{\mathrm{sIVC}}$ | $\Delta_{\mathrm{tIVC},xy}$ | $\lambda_{I}^{\mathrm{(MF)}}$ | $\frac{1}{2}\mu_{B}gB$
$1.5$ meV | $\pi/8$ |  $0.1$ meV | $\pi/6$ |  $-0.25$ meV | $-\pi/7$ |  $-0.5$ meV |  $3.0$ meV |  $4.0$ meV |  $-2.5$ meV |  $0.23$ meV
Table 4: Spin orbit parameters (Sec. VI.4) and IVC mean field strengths (Sec.
VI.5) chosen for five band model simulations of the wire. The resulting band
structures and wire profiles are shown in Fig. 7.

#### VI.5.2. Antiferromagnetic IVC order

As discussed in Sec. IV.3.2, the AFM IVC state is obtained by considering
_two_ order parameters. The first is simply the Ising Hamiltonian of Eq. (85).
To distinguish this interaction-induced term from the TMD-induced term, we
denote the strength of the former mean field Ising SOC strength by
$\lambda_{I}^{\mathrm{(MF)}}$. The second order parameter needed to describe
the AFM IVC insulator corresponds to two in-plane IVC triplet orders. We
consider

$\displaystyle H_{\mathrm{tIVC},xy}$
$\displaystyle=\frac{\Delta_{\mathrm{tIVC},xy}}{4}\sum_{{\bm{r}}}e^{i\phi_{xy}}\Big{[}f_{K,+,{\bm{r}}}^{\dagger}(s^{x}-is^{y})f_{K^{\prime},+,{\bm{r}}}-f_{K,-,{\bm{r}}}^{\dagger}(s^{x}-is^{y})f_{K^{\prime},-,{\bm{r}}}\Big{]}+h.c.$
(88)

The phase choice consistent with the analysis of Sec. IV.3.2 is obtained when
$\phi_{xy}=0$. This state preserves the physical time reversal symmetry
$\mathcal{T}=is^{y}{\mathcald{T}}$ and thus necessarily possesses degeneracy
at $k=0$, as in Fig. 2(i) of the main text. We lift this degeneracy through
the application of an in-plane magnetic field
$\bm{B}=B(\cos\theta_{B},\sin\theta_{B})$ whose primary effect is as a Zeeman
field:

$\displaystyle H_{Z}$
$\displaystyle=\frac{1}{2}g\mu_{B}B\sum_{v=K,K^{\prime}}\sum_{\bm{r}}f_{v,{\bm{r}}}^{\dagger}\big{(}\cos\theta_{B}s^{x}+\sin\theta_{B}s^{y}\big{)}f_{v,{\bm{r}}}.$
(89)

We restrict our study to $\theta_{B}=0$. In Fig. 5(c), the electronic
structure with $\lambda_{I}^{\mathrm{(MF)}}=-2.5\,\mathrm{\text{meV}}$,
$\Delta_{\mathrm{tIVC},xy}=4.0\,\mathrm{\text{meV}}$, and
$\frac{1}{2}g\mu_{B}B=0.23\,\mathrm{\text{meV}}$ is shown. A clear gap is seen
at $\nu=+2$. These parameters are recorded in Table 4 and are used in Sec.
VI.6 in the wire simulations.

###  Wire setup and simulation

Figure 6: (a) Real space representation of wire set up. The orbitals within
the moiré lattice are shown using the same colour scheme as in Fig. 3(a). To
simulate the wire, a unit cell that was doubled in the $x$ direction was used;
they are outlined with dashed lines. The two halves of the unit cell,
distinguished by even or odd $n_{x}$, are divided by a dash-dotted line and
two such subcells are highlighted in orange and blue for $(n_{x},n_{y})$ and
$(n_{x}+1,n_{y})$, respectively. (b) The wire profile chosen for the different
varying parameters as a function of position. Periodic boundary conditions in
which $n_{x}=71$ is identified with $n_{x}=1$. Here, $X_{\mathrm{wire/bulk}}$
may refer to the chemical potential $\mu$ or the IVC mean field parameters.
For the trivial wire, $\mu_{\mathrm{bulk}}<\mu_{\mathrm{wire}}$ and so the
true profile would require flipping the vertical axis direction. Although the
different sites within a unit cell $n_{x}$ are situated a different $x$
coordinates, these fine details were not taken into account: the value of $X$,
$X=\mu,\Delta_{\mathrm{sIVC}},\Delta_{\mathrm{tIVC},xy}$, and/or
$\lambda_{I}^{\mathrm{(MF)}}$ used was determined solely by $n_{x}$.

A real-space illustration of the wire setup is shown in Fig. 6(a). We
arbitrarily chose to consider a wire orienting along the $y$-direction, which
implies that the Hamiltonian can be written in terms of conserved momenta
$k_{y}$. This wire configuration requires that we double the unit cell in the
$x$ direction. As a result, a single unit cell is defined by integers $n_{x}$
and $n_{x}+1$. Full unit cells are outlined by dashed lines in Fig. 6(a),
while dash-dotted lines divide the two subcells. We emphasize again that the
hexagonal unit cells depicted in Fig. 6(a) represent _moiré_ unit cells that
are themselves composed of $\sim 10\,000$ graphene unit cells.

The wire is simulated in a fashion directly analogous to the experimentally
proposed setup: we impose a spatially varying chemical potential $\mu(n_{x})$
whose values interpolate between the fillings under study. Note that the
spatial separation of the different orbitals within a unit subcell labelled by
$n_{x}$ are not taken into account—any features dependent on such small scale
distinctions cannot be trusted, as the five band model fails precisely in such
situations (see Sec. VI.7). Our simulations below assume a system with total
length $70$ in the $x$ direction, with periodic boundary conditions. Figure
6(b) displays how the chemical potential varies between its bulk and wire
values, $\mu_{\mathrm{bulk}}$ and $\mu_{\mathrm{wire}}$, in our simulations.
The precise values of the chemical potential are tuned to return the desired
results, as would be done experimentally.

While interactions in the bulk are assumed to generate spontaneous breaking of
symmetries defining the sIVC and AFM IVC phases, we input these orders by hand
at a mean field level. The generation of these orders is then treated
analogously to the chemical potential. For instance, for the sIVC phase,
$\Delta_{\mathrm{sIVC}}(n_{x})$ interpolates between its mean field value (see
Table 4) in the bulk and zero within the wire using the same profile shown in
Fig. 6(b). This setup differs from a scenario in which the two IVC phases
adjoining the wire are not connected in any way. Arguably, it is possible that
the arbitrary $\mathrm{U_{v}}(1)$ phases could differ on either side of the
wire. We assume that a valley Josephson effect locks those two phases
together, resulting in the system considered here.

#### VI.6.1. Trivial wire simulations

The trivial wire is obtained by tuning the chemical potential so that the flat
bands within the bulk are completely empty, $\nu_{\mathrm{bulk}}=-4$, while
the chemical potential in the wire region sits just above the band bottom,
yielding a filling $\nu_{\mathrm{wire}}=-4+\delta$ for $\delta$ small. The
values of $\mu_{\mathrm{bulk}}$ and $\mu_{\mathrm{wire}}$ considered here are
shown relative to the corresponding bulk band structure in Fig. 5(a). Figures
7(a) and (b) plot the resulting the energy levels of the trivial wire as a
function of the conserved momentum $k_{y}$. The dashed line at zero energy in
both images displays the Fermi energy $E_{F}$. In Fig. 7(a), all flat band
energy states are shown. To see the confined wire states, Fig. 7(b) zooms in