# High transmission in twisted bilayer graphene with angle disorder

Héctor Sainz-Cruz  $Imdea\ Nanoscience,\ Faraday\ 9,\ 28015\ Madrid,\ Spain$
Tommaso Cea  $Imdea\ Nanoscience,\ Faraday\ 9,\ 28015\ Madrid,\ Spain$ Pierre
A. Pantaleón  $Imdea\ Nanoscience,\ Faraday\ 9,\ 28015\ Madrid,\ Spain$
Francisco Guinea  $Imdea\ Nanoscience,\ Faraday\ 9,\ 28015\ Madrid,\ Spain$
$Donostia\ International\ Physics\ Center,\ Paseo\ Manuel\ de\ Lardizabal\ 4,\
20018\ San\ Sebastian,\ Spain$ $Ikerbasque\ Foundation,Maria\ de\ Haro\ 3,\
48013\ Bilbao,\ Spain$

(27th August 2024)

###### Abstract

Angle disorder is an intrinsic feature of twisted bilayer graphene and other
moiré materials. Here, we discuss electron transport in twisted bilayer
graphene in the presence of angle disorder. We compute the local density of
states and the Landauer-Büttiker transmission through an angle disorder
barrier of width comparable to the moiré period, using a decimation technique
based on a real space description. We find that barriers which separate
regions where the width of the bands differ by 50$\%$ or more lead to a minor
suppression of the transmission, and that the transmission is close to one for
normal incidence, which is reminiscent of Klein tunneling. These results
suggest that transport in twisted bilayer graphene is weakly affected by twist
angle disorder.

Introduction. Twisted bilayer graphene (TBG) is a superconductor [1, 2, 3] and
shows many more correlated phases [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,
16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30]. The microscopic
origins of superconductivity and almost all other phases are still unknown and
intensely debated [31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45,
46, 47, 48, 49, 50, 51, 52, 53, 54]. Recent highlights in the field are the
discovery of superconductivity in mirror-symmetric twisted trilayer graphene
[55, 56] and rhombohedral trilayer graphene [57], as well as the creation of
Josephson junctions made of different regions of single TBG crystals [58, 59].
The phase diagram of TBG strongly depends on the twist angle, which is a novel
tuning parameter but also an unavoidable source of disorder that creates a
unique landscape of domains with different angles in each sample. In devices
that show superconductivity and insulating behavior, the angle disorder can be
of order $0.1^{\circ}$ over micrometers [60]. On the other hand, certain
superconducting domes and insulating phases have been observed in samples with
very low angle disorder ($\lesssim 0.02^{\circ}$ throughout the device) [3,
29]. Angle disorder is a natural candidate to account for part of the
variability in phase diagrams, since it modulates the interlayer hopping, the
Fermi velocity and the bandwidth.

Early experiments on TBG observed moiré patterns of different periods through
scanning tunneling microscopy (STM) [4, 61, 62, 63] and several groups have
used STM to measure local twist angles from distortions in the moiré structure
[7, 9, 6, 8]. In parallel, transmission electron microscopy has enabled the
observation of lattice reconstruction [64, 65]. Recently, Raman spectroscopy
has emerged as a valuable probe of angle disorder [66] and lattice dynamics
[67]. Angle disorder was precisely characterized by Uri et. al. [60] who use a
SQUID to measure quantum Hall currents stemming from the Landau level
structure and which occur in the bulk of the sample across twist-angle equi-
contours. The angle maps of two superconducting samples reveal the coexistence
of smooth angle gradients together with stacking faults and dislocations.
Lattice distortions also occur due to strain, which is concomitant with long
range angle disorder. However, periodic tensions between layers develop always
and result in constant deformations of the unit cell, independent of disorder.
In particular, uniaxial heterostrain has a major impact on the physics of TBG
near the magic angle [68, 69, 70]. Together with angle disorder and strain,
TBG also has moderate charge disorder [71].

There are few theoretical works addressing angle disorder in TBG [72, 73, 74,
75]. In Ref. [72], the effect of angle disorder domains on the densities of
states (DOS) was studied. The bandwidth of the flat bands, the gaps to remote
bands and the sharpness of the van-Hove peaks were found to be sensitive to
angle disorder. So far, transport in the presence of angle disorder has only
been studied with a two-band version of the continuum model [73] and focused
on tunneling through narrow angle domain walls. Another paper [74] describes
the transmission of Dirac particles through quasi-1D angle domains, modelled
as a change in the Fermi velocity plus a gauge field that shifts the Dirac
points. In Ref. [75], a Landau-Ginzburg theory of strong correlations coupled
to angle disorder is developed in an attempt to explain the appearance of
semi-metallic phases near charge neutrality.

In this paper, we study electron transport in TBG across finite-width angle
disorder barriers using a tight binding model. To avoid the large moiré unit
cells associated to small twist angles, we perform a scaling approximation
[76, 77]. The transmission through the angle disorder barrier is calculated in
the Landauer-Büttiker formalism by means of a decimation technique that
obtains the dressed Green’s function at the barrier. We find that for disorder
of the magnitude seen in some superconducting samples ($\sim 0.1^{\circ}$),
the transmission through an angle disorder barrier is high, while the density
of states at the barrier shows van-Hove peaks significantly broadened.
Although conventional Klein tunneling [78] cannot occur in our system, the
transmission is complete ($>99\%$) for normal incidence. The transmission
within energy windows mixing ten channels, normal and oblique, depends
linearly on the bandwidth ratio of the angles on either side of the barrier.

Figure 1: (a) Sketch of the crystal lattice of a TBG nanotube. (b) Folding the
mini-Brilluoin zone (mBZ) of TBG onto the white rectangle and superimposing
the bands along the green lines, one obtains the 1D band structure of the TBG
nanotube. (c) Setup of transmission calculation. The section between vertical
lines is the planar projection of the unit cell of TBG nanotube
(56,2)@(-56-2), with N=4336 sites, where intralayer hopping disorder
$t^{\parallel}(r)$ mimics angle disorder $\theta(r)$ and which connects to two
semi-infinite pristine regions or ‘leads’. The rhombus shows a unit cell of
planar TBG.

Model. The system we study is a double-wall carbon nanotube made of two
nanotubes with opposite chiral angles, the 1D analog of TBG, see Fig. 1(a).
Such a system is entirely defined by the chiral vectors of its constituent
nanotubes $\vec{C}_{\pm}=\pm(n\vec{a}_{1}+m\vec{a}_{2})\equiv\pm(n,m)$, where
$\vec{a}_{1,2}=(\sqrt{3}/2,\pm 1/2)a$ are basis vectors of graphene and
$a=0.246$ nm is the lattice constant. The chiral vector specifies a possible
wrapping of the graphene lattice into a tube; it also determines the chiral
angle, which is equivalent to a twist angle and corresponds to the tilt angle
of the honeycomb lattice with respect to the tube’s axis. Imposing periodic
boundary conditions to close the nanotube quantizes the momentum parallel to
the barrier, and the only allowed values correspond to the green lines in Fig.
1(b). We will call the system a ‘TBG nanotube’, and denote it as
(n,m)@(-n,-m). Figure 1 shows a section of TBG nanotube (56,2)@(-56,-2), which
has a relative chiral angle (i.e. twist angle) of 3.48∘. The central unit cell
between lines is the angle disorder barrier that connects to semi-infinite
pristine leads. We focus on the large diameter limit, in which the TBG
nanotube reproduces the spectrum of TBG, just like a single-wall nanotube
becomes equivalent to graphene in that limit. The low diameter limit, in which
the 1D nature takes center stage, has recently been explored and shown to
enable strong interlayer coupling [79, 80]. Here, the main advantages in the
use of nanotubes are the knowledge of carriers’ momenta parallel to the
barrier and the avoidance of border effects. To model electronic properties,
we use a ‘minimum’ tight binding Hamiltonian [81],

${H=H_{\parallel}+H_{\bot}+H_{dip}=-\sum_{i\neq
j,m}\gamma_{ij}^{mm}(c^{\dagger}_{i,m}c_{j,m}+h.c.)}-\sum_{i,j,m}\gamma_{ij}^{m,m+1}(c^{\dagger}_{i,m}c_{j,m+1}+h.c.)+\sum_{i,m}V_{dip}(r)c^{\dagger}_{i,m}c_{i,m}\,.$
(1)

Here $i,j$ run over the lattice sites and $m$ is the layer index. The first
term in the Hamiltonian is an intralayer hopping to nearest-neighbours only
and the second an interlayer hopping that decays exponentially away from the
vertical direction,

$\gamma_{ij}^{mm}=t_{\parallel}\,,$ (2)
$\gamma_{ij}^{m,m+1}=t_{\perp}e^{-(\sqrt{r^{2}+d_{0}^{2}}-d_{0})/\lambda_{\perp})}\frac{d_{0}^{2}}{r^{2}+d_{0}^{2}}\,,$
(3)

where $d_{0}=0.335$ nm is the distance between layers, $t_{\parallel}=3.09$ eV
and $t_{\perp}=0.39$ eV are the intralayer and interlayer hopping amplitudes
and $\lambda_{\perp}=0.027$ nm is a cutoff for the interlayer hopping [81].
This model captures all the essential features of the band structure of magic-
angle TBG, i.e. flat bands separated by gaps from remote bands, van-Hove
singularities near the M points and Dirac cones near K and K’ points.

In experiments, one imposes a nearly uniform electron density through the
back-gate voltage. In regions with narrower bands this density translates into
a higher chemical potential. To avoid unbalanced charge, carriers redistribute
near the angle barrier and create a dipole potential $V_{dip}$ that
compensates the chemical potential mismatch, thus balancing the Fermi energies
on either side of the barrier. This leads to large unscreeened in-plane
electric fields [60]. The chemical potential difference corresponds to the
offset in Dirac point energies obtained in the band structure calculation, and
is $\Delta_{\mu}=3.67$ meV in case of 1.11∘ coupled to 1.21∘. Thus, at this
simplest level of description, $V_{dip}(r)$ is an on-site potential that
evolves smoothly as an hyperbolic tangent from 0 to $\Delta_{\mu}$ only within
the barrier, analogous to the function that introduces angle disorder
$\theta(r)$. A more rigorous dipole potential would extend far into the
pristine leads, but its self-consistent calculation is beyond the scope of
this research.

Using the continuum model, it can be shown that the bands of twisted bilayer
graphene depend, to first order, on a dimensionless parameter [33], which
involves the ratio of interlayer to intralayer hopping and the twist angle,

$\alpha=\frac{at_{\perp}}{2\hbar
v_{F}\sin(\theta/2)}\propto\frac{t_{\perp}}{t_{\parallel}\theta}\,.$ (4)

where $v_{F}$ is the Fermi velocity and $t_{\perp}$ is an average of the
interlayer hopping. We emphasise two important consequences. First, this
relation enables simulation of smaller angles with an appropriate reduction in
$t_{\parallel}$ [76, 77, 82]. Second, within this one-parameter model, there
is an equivalence between angle disorder $\theta(r)$ and intralayer hopping
disorder $t_{\parallel}(r)$ [82], since both result in identical
$\alpha(r)$-disorder. As a consequence, both angle disorder and intralayer
hopping disorder capture very well the pronounced changes in bandwidth near
the magic angle. In virtue of this, we have approximated angle disorder as
intralayer hopping disorder in the transmission calculation, which has the
advantage of preserving translational symmetry. Nevertheless, hopping disorder
does not reflect the small modifications in moiré period that angle disorder
would also provoke. Therefore, the approximation rests on the assumption that,
near the magic angle, moiré period variations due to a twist angle difference
of 0.1∘ (typically $\lesssim 10\%$) have a negligible impact on transport
compared to the more drastic bandwidth changes ($\gtrsim 150\%$ or more, see
Figure 2).

To study transport, we consider the standard setup of a conductor coupled to
semi-infinite leads [83]. We compute the transmission through an angle
disorder barrier of width one moiré unit cell (as in Figure 1) or two,
connected to left and right to semi-infinite leads which are pristine in twist
angle. The transmission is calculated in the Landauer-Büttiker formalism,
where it is proportional to the mesoscopic conductance and reads

$\mathcal{T}=\textrm{Tr}\big{\\{}\Gamma_{L}G^{r}\Gamma_{R}G^{a}\big{\\}}\,.$
(5)

Here $G^{r}$ and $G^{a}$ are the retarded and advanced dressed Green’s
functions at the angle disorder barrier.
$\Gamma_{\\{L,R\\}}=i\big{[}\Sigma^{r}_{\\{L,R\\}}-\Sigma^{a}_{\\{L,R\\}}\big{]}$
are the coupling functions of the barrier to the pristine leads. These in turn
depend on the self-energies of the leads $\Sigma_{\\{L,R\\}}$, i.e. the
cumulative corrections to the bare Green’s function at the barrier due to its
coupling with them. To obtain these quantities, we develop a decimation
technique based on Refs. [84, 85] (see also Ref. [82]). From the Green’s
function, the DOS also follows,

$\textrm{DOS}=\frac{1}{\pi}\textrm{Tr}\big{\\{}\textrm{Im}\\{G(\omega)\\}\big{\\}}\,.$
(6)

Figure 2: Low-energy band structures of TBG at 1.11∘ and 1.21∘. The bands are
obtained through exact diagonalization of a scaled tight binding model of TBG
nanotube (56,2)@(-56,-2) ($\theta=3.48^{\circ}$). Figure 3: (Left)
Transmission through an angle disorder barrier. Continuous lines are the bulk
transmissions through pristine leads 1.11∘ and 1.21∘. Only carriers that can
propagate on both angle domains may cross the angle disorder barrier, thus the
transmission through the barrier<EMAIL_ADDRESS>is bounded by
$min\\{\mathcal{T}_{1.11^{\circ}},\mathcal{T}_{1.21^{\circ}}$}. Crosses
(circles) correspond to transmission through a barrier of length one (two)
moiré periods ($L_{M})$. The transmission through the barrier is high all
across the flat band, specially for carriers near $\Gamma$, $K$ and
$K^{\prime}$. (Right) Local DOS at the disorder barrier<EMAIL_ADDRESS>compared
to the bulk DOS of its pristine leads. The van-Hove peaks are damped or erased
at the barrier.

Results. Figure 2 shows the band structures of TBG nanotube (56,2)@(-56-2),
($\theta=3.48^{\circ})$ scaled to simulate 1.11∘ and 1.21∘. The Brillouin zone
of a TBG nanotube results from a folding of the Brillouin zone of TBG. In the
case of (56,2)@(-56-2), the unit cell is four times larger than that of its
associated TBG and this results in eight flat bands, not counting spin or
valley degeneracy. More bands allow us to extract richer information from the
transmission calculation. The bandwidth of the flat bands for the 1.11∘ system
is approximately 16 meV, versus 44 meV for 1.21∘. The DOS and transmissions
through the bulk of the pristine leads can be extracted directly from the
bands; the number of transmission channels at a given energy is just number of
band crossings at that energy and the DOS can be obtained with an histogram of
the eigenvalues. The transmissions through the bulk of the pristine leads are
plotted in Figure 3 with continuous lines. For the disorder barrier, the bands
cannot be defined and using the decimation technique becomes a must. Figure 3
also shows the transmission through the angle barrier<EMAIL_ADDRESS>for a
barrier width of one or two moiré periods. Focusing first on the data points
near 35 meV, we see that their transmission is virtually complete. These are
normal incidence channels, but their perfect transmission is surprising since
they live near the $\Gamma$ point (see also Figure 2) and hence cannot
experience Klein tunneling [78]. Across the $\mathcal{T}=4$ plateau of 1.21∘,
the barrier<EMAIL_ADDRESS>allows transmissions in excess of 90%. These are four
Dirac-like channels, two normal and two oblique. Hence, we can infer that
these oblique channels near $K$ and $K^{\prime}$ have transmissions over 80% .
Finally we observe that in the narrow window near 26 meV, where both pristine
systems have ten transmission channels (two normal plus eight oblique), with
all $k_{\parallel}$ values allowed by the periodic boundary condition, the
transmission through the barrier still reaches 75%. The new channels have
noticeable electron-hole asymmetry, the transmission at the hole-like
$\mathcal{T}=10$ window near 26 meV is $10\%$ higher than the at the electron-
like window near 31 meV. Other angle combinations confirm this asymmetry in
oblique channel transmission. Figure 3 displays the bulk densities of states
of pristine leads 1.11∘ and 1.21∘, together with the local density of states
at the angle barrier. The pristine leads present several van-Hove peaks
corresponding to the various flat bands seen in Figure 2. At the angle
disorder barrier, some van-Hove peaks are erased while the rest suffer
pronounced damping.

The simulation of different angle combinations provides insight into the
factors that determine the transmission. For angle barriers<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>figures analogous to Figure 3 can be found in Ref. [82]. In view
of these, we conjecture that the bandwidth ratio of the connecting angles is
the main quantity governing the transmission through a disorder barrier. The
proximity to the magic angle and electron-hole asymmetry play secondary,
albeit significant, roles. To verify this picture, we perform two series of
simulations, shown in Figure 4. The idea is to start with a barrier (e.g.
<EMAIL_ADDRESS>and reduce the angle difference in small steps, i.e<EMAIL_ADDRESS><EMAIL_ADDRESS>etc., each time computing the transmission within the
$\mathcal{T}=10$ window at the same point in energy. The results are
compelling, the transmission grows linearly as the bandwidth ratio diminishes.

Discussion. The results presented here for twisted bilayer graphene show that
the transmission between regions with different twist angles is high,
specially for incidence normal to the barrier between the regions. On the
other hand, angle inhomogeneity significantly modifies the local density of
states, broadening and suppressing features such as van Hove singularities
(see also Ref. [72]). An interesting aspect is the complete transmission of
normal incidence carriers, which occurs even near $\Gamma$ and contrasts with
Ref. [73]. This behaviour is very robust, normal incidence channels have
virtually complete transmission ($\mathcal{T}\gtrsim 95\%$) even for the most
inhomogeneous barrier that we have studied, $1.11^{\circ}$@$3^{\circ}$,
whereas the transmission of the oblique Dirac channels falls to
$\mathcal{T}\lesssim 5\%$ already for<EMAIL_ADDRESS>A
straightforward description of this phenomenon in terms of Klein tunneling
cannot be carried out, although it suggests that the orbital characters of the
ingoing and outgoing waves are orthogonal. It is noteworthy that there is no
reduction in transmission when doubling the width of the angle disorder
barrier. This suggests that the transmission through angle disorder barriers
could decay sub-exponentially with the barrier width, differing from previous
predictions [73].

Figure 4: Normalized transmission through several angle disorder barriers
versus the bandwidth ratio of the connected angles. The values in each series
are computed at the same point in energy, within a $\mathcal{T}=10$ window.
The transmission decreases linearly with the bandwidth ratio.

The transmissions through a barrier between regions twisted by different
angles, and with ten channels is inversely proportional to the ratio between
the bandwidths in the two electrodes. Future research could explore if this
relation between bandwidth and transmission persists beyond our framework of
angle disorder, and clarify if disorder in TBG samples can indeed be described
as ‘bandwidth disorder’, encompassing the effects of angle disorder,
heterostrain and electron-electron interactions on the bandwidth. An
interesting feature of Figure 4 is the different slope of each series. It is
worth noting that the series starting with<EMAIL_ADDRESS>was taken at the hole
side of the spectrum, while the other series is from the electron side.
However, there is a deeper reason for the slope mismatch: the series starting
with<EMAIL_ADDRESS>is closer to the magic angle. Ultimately, it is the matching
of wavefunctions what determines the transmission through the barrier. As the
eigenfunctions change slowly near the magic angle, the transmission becomes
less correlated with the bandwidth ratio of the angles and hence the smaller
slope of the series. A toy model with single wall nanotubes reproduces all
main results: in a single-wall nanotube with markedly different hoppings
(hence bandwidths) in each lead, the transmission through the barrier is very
notable, despite the disappearance of van-Hove peaks, see Ref. [82]. We have
explored angle disorder within a single-particle picture. Hence a shortcoming
of the model is the absence of electron-electron interactions, thought to be
of crucial importance for the outstanding transport phenomena seen in TBG.
Moreover, our angle disorder is circumscribed to one or two moiré unit cells
connected to semi-infinite pristine leads, whereas real systems have angle
domains with many shapes and sizes. Also, due to computational constraints,
the disorder modelled here is one to two orders of magnitude sharper than
those seen in experiments, which have typical gradients of order
$0.01^{\circ}-0.05^{\circ}/\mu m$.

In conclusion, we have studied transport in TBG in the presence of angle
disorder. To access the magic angle regime, we have done a scaling
approximation. We have computed the Landauer-Büttiker transmission through an
angle disorder barrier connected to pristine leads, employing a decimation
technique. The van-Hove peaks of the local DOS at the barrier are very
sensitive and can disappear with angle disorder of $\sim 0.1^{\circ}$. In
contrast, the transmission through the barrier is remarkably high all across
the flat bands, and very close to one for normal incidence. Within
high-$\mathcal{T}$ windows, the transmission through an angle disorder barrier
is, to first order, inversely proportional to the bandwidth ratio of the
angles it connects. This work lays the foundations for future studies on the
effects of disorder on transport in TBG. A crucial step will be to understand
the interplay between electron interactions and disorder. Angle disorder is
omnipresent in moiré systems and is likely to have immediate as well as subtle
consequences on their phase diagrams, only future research will tell how far-
reaching are its ramifications in this new field of condensed matter physics.

Acknowledgements This work was supported by funding from the European
Commission, within the Graphene Flagship, Core 3, grant number 881603 and from
grants NMAT2D (Comunidad de Madrid, Spain), SprQuMat and SEV-2016-0686
(Ministerio de Ciencia e Innovación, Spain).

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## Supplementary material

## S1. Additional results

###  .1 Toy model of disorder

When TBG carriers pass from a region with 1.11∘ to another with 1.21∘ through
an angle disorder barrier, they see only a slight difference in moiré period.
In comparison, the bandwidth augments dramatically by a factor
$R_{\gamma}=2.7$. Therefore, we can try to capture the leading physical effect
with a toy model of a wide band system connected to another with a narrower
band. To that end, we consider a single-wall zig-zag nanotube with chiral
vector (90,0), one lead with nearest neighbour hopping $t_{\parallel}=3.09$ eV
and the other with $t_{\parallel}=1.14$ eV, connected through a hopping
disorder barrier with a width of ten unit cells. Figure S1 shows the resulting
transmission and DOS. Despite the sudden bandwidth reduction, there is
considerable transmission through the barrier when the narrow band activates.
Just like in TBG with angle disorder, van-Hove peaks are largely erased in the
presence of disorder, but carrier transmission remains remarkably high.
Furthermore, the right panel shows the normalized peak transmission through
various hopping disorder barriers versus the bandwidth ratio of their pristine
leads. The toy model reproduces well the linear relation between transmission
and bandwidth ratio of the leads.

Figure S1: Wide band – narrow band toy model. (Left) Transmission through zig-
zag nanotubes (90,0) with $t_{\parallel}=3.09$ eV and $t_{\parallel}=1.14$ eV,
compared to the transmission through a hopping-disorder barrier that connects
both. The width of the barrier is 10 unit cells. There is notable transmission
when the narrow band activates. (Centre) Bulk DOS of the pristine nanotubes
and local DOS at the hopping-disorder barrier. Disorder erases the van-Hove
peaks. (Right) Normalized peak transmission at several 10-cell hopping-
disorder barriers against the bandwidth ratio of their pristine leads, full
data sets in Figure S2.

Figure S2: Graphene as a toy model of disorder. Transmissions (left) and DOS
(right) through hopping disordered barriers $A@B$ that connect pairs of single
wall carbon nanotubes with different hoppings $A$ and $B$.

Figure S2 shows the DOS and transmissions through several toy model barriers,
including the one in Figure S1, 1.14 eV @ 3.09 eV. In the DOS, the original
van-Hove peaks of the systems that connect are eliminated at the barriers,
although smaller resonances appear and grow in number as disorder increases.
In the transmission, we can distinguish three regimes: for low disorder, there
is a single maximum in the disordered transmission which comes from the broad
band system (e.g. 2.29 eV @ 3.09 eV and 1.77 eV @ 3.09 eV); for medium
disorder, the disordered transmission has two maxima, inherited from both the
broad and narrow band systems (e.g. 1.40 eV @ 3.09 eV and 1.14 eV @ 3.09 eV);
for large disorder, the peak due to the broad band system is lost, and only
the one coming from the narrow system remains (0.91 eV @ 3.09 eV). The maxima
of each transmission curve provide the data for the right panel of Figure S1.
There we see that the relation between transmission and bandwidth ratio is
approximately linear, but not strictly so. This is because the existence of
several regimes results in two different slopes, one while the maximum comes
from the broad band system and other when it starts to originate from the
narrow band system.

###  .2 TBG with angle disorder: Klein tunneling

Normal incidence carriers transmit without back-scattering
($\mathcal{T}\approx 1$) through the angle barrier<EMAIL_ADDRESS>One naturally
wonders how robust this effect is. To answer this, we have looked at more
inhomogeneous barriers, up to<EMAIL_ADDRESS>For each barrier in the table
below, we compute the transmission of the two normal incidence channels near
$\Gamma$ (i.e. we recalculate the data point near 35 meV in Figure 3 in the
main text) and the transmission of the Dirac mixture (two normal plus two
oblique channels with $k_{\parallel}=\sqrt{3}k_{\perp}$) near the charge
neutrality point. With these, one can infer the transmission of the oblique
incidence channels. There is a stark difference between normal and oblique
incidence channels. Normal channels are almost impervious to disorder and
reach $\mathcal{T}\approx 1$ for all barriers. In contrast, the transmission
of oblique incidence channels is very low already for<EMAIL_ADDRESS>It is worth
noting that for abrupt barriers like this one, we cannot justify the
approximation that neglects changes in the moiré period and which allowed us
to model angle disorder as intralayer hopping disorder. Hence, for abrupt
barriers we need to bear in mind that the intralayer hopping disorder we
introduce can no longer faithfully mimic angle disorder. In any case, the
chiral nature of carriers in graphene, which enables Klein tunneling through
$n-p$ [86] and $n-p-n$ [78] potential barriers, could be forbidding back-
scattering of normal incidence carriers in TBG too. An immediate consequence
is that transport across twist angle domains leads to the collimation of
carrier momenta around normal incidence.

Transmision with angle disorder
---
Barrier $A@B$ |  Mini-gaps of $B$ |  Normal-channel $\mathcal{T}$ |  Dirac mixture $\mathcal{T}$ |  Oblique $k_{\parallel}=\sqrt{3}k_{\perp}$ $\mathcal{T}$
<EMAIL_ADDRESS>|  Two |  0.99 |  0.90 |  0.80
<EMAIL_ADDRESS>|  Two |  0.98 |  0.66 |  0.35
<EMAIL_ADDRESS>|  Two |  0.97 |  0.56 |  0.16
<EMAIL_ADDRESS>|  One |  0.96 |  0.50 |  0.03
<EMAIL_ADDRESS>|  None |  0.95 |  0.46 |  $<$0.01

Figure S3: (Left) $\theta=1.90^{\circ}$ bands. (Right) $\theta=3.00^{\circ}$
bands.

###  .3 TBG with angle disorder: Other barriers

In the main text we focus on the barrier<EMAIL_ADDRESS>111We have rounded 1.114∘
to 1.11∘ and 1.206∘ to 1.21∘ for clarity.. The transmission and densities of
states are obtained with the decimation technique outlined in section S4.
Figures S4 and S5 present the results of analogous computations for
<EMAIL_ADDRESS><EMAIL_ADDRESS>and<EMAIL_ADDRESS>and complement the discussion of the
main text about the factors that influence transmission. The similar values
obtained for barriers<EMAIL_ADDRESS><EMAIL_ADDRESS>and<EMAIL_ADDRESS>suggest that
the bandwidth ratio of the angles that the barrier connects is the most
important factor determining transmission through it. Other relevant
considerations are the proximity to the magic angle and the existence of
electron-hole asymmetry in favour of the holes. Together, these factors form a
nuanced picture, but the major effect of the bandwidth ratio can be distilled
from the rest by performing simulations in which larger angle is fixed and the
smaller increases until they match, see main text. During the procedure, the
bandwidth ratio changes fast while the other factors do not; the proximity to
the magic angle stays similar and the transmission is computed at the exact
same energy, avoiding $e^{-}-h^{+}$ asymmetry. The transmission is found to be
inversely proportional to the bandwidth ratio of the connecting angles.

Figure S4: (Left) Band structures of TBG nanotubes (56,2)@(-56,-2)
($\theta=3.48^{\circ}$) scaled to simulate 1.32∘ and 1.21∘. (Right)
Transmission through the angle disorder barrier<EMAIL_ADDRESS>compared to the
bulk transmission through its pristine leads.

Figure S5: (Left) Transmission through the angle disorder barrier<EMAIL_ADDRESS>compared to those of its pristine leads. (Right) Same for<EMAIL_ADDRESS>
The $e^{-}-h^{+}$ asymmetry appears in the transmission of oblique incidence
channels, within high-$\mathcal{T}$ windows. Studying the asymmetry
systematically is difficult because it usually makes such windows reach
different values on the electron and hole sides, until the coupled angles
become sufficiently similar and transmissions are too close to 1 for a
meaningful comparison. Nevertheless, comparing windows with different
$\mathcal{T}$ is still informative, bearing in mind that reaching higher
$\mathcal{T}$ implies adding more oblique incidence channels, which are in
principle less capable of crossing the barrier. For example, the transmission
is $60\%$ within the $\mathcal{T}=8$, $e^{-}$ window of<EMAIL_ADDRESS>(near
$\omega=33$ meV in Figure S4), while it reaches $72\%$ within the
$\mathcal{T}=10$, $h^{+}$ window near 20 meV. Because of the two extra
channels, the $\sim 12\%$ edge of the holes has more merit. In contrast, the
barrier<EMAIL_ADDRESS>has $82\%$ transmission at a $\mathcal{T}=8$, $h^{+}$
window and $72\%$ at a $\mathcal{T}=10$, $e^{-}$ window. For<EMAIL_ADDRESS>one
finds $\sim 82\%$ for both $\mathcal{T}=12$, $h^{+}$ and $\mathcal{T}=10$,
$e^{-}$ windows. Overall, these results suggest that there is less
$e^{-}-h^{+}$ asymmetry near the magic angle.

###  .4 Integrated charge map of TBG

Figure S6: (Bottom) 1.11∘ TBG charge map integrated up to the gap between
hole-like remote bands and flat bands. AA regions have a charge deficit of
$\sim 2\%$ relative to AB. (Top) Integrating further, up to the charge
neutrality point, the charge balances between AA and AB stacking regions
(relative deviations are $\lesssim 0.5\%$).

Finally, we looked at integrated charge maps. In Figure S6 we plot the
integrated charge map in the unit cell of a pristine system with
$\theta=1.11^{\circ}$, obtained from exact diagonalization. The bottom map is
integrated up to the gap between the hole-like remote bands and the flat
bands, and this results in AA regions having $\sim 2\%$ less accumulated
charge than AB regions. The top map shows that this deficit is compensated
when we include the hole-like flat bands, i.e. when we integrate all bands up
to the charge neutrality point. At the angle disorder barrier, the charge map
(not shown) approximately corresponds to a smooth interpolation of the maps on
its pristine leads.

## S2. Real lattice

The geometry of a carbon nanotube is uniquely defined by its chiral vector
$\vec{C}=n\vec{a}_{1}+m\vec{a}_{2}$, which specifies a possible wrapping of
the graphene lattice into tubular form. The chiral vector determines the
diameter of the nanotube and the number of atoms $N_{C}$ in its translational
unit cell. The chiral angle $\theta_{C}$ is defined with respect to the basis
vector $\vec{a}_{1}=(\sqrt{3}/2,1/2)a$, which forms $30^{\circ}$ with the
horizontal and is the chiral vector of zig-zag nanotubes. Figure S7 depicts a
way to construct a ‘twisted bilayer graphene nanotube’, a system of two
concentric nanotubes with opposite chiral vectors that we use to mimic TBG. It
should be noted that Figure S7(R) is a planar representation of the system’s
unit cell, and that the TBG nanotube results from imposing periodic boundary
conditions in the vertical direction.

Figure S7: Construction of a twisted bilayer graphene nanotube
(28,1)@(-28,-1), a system with a twist angle of $3.48^{\circ}$. (Left)
Graphene honeycomb lattice with highlighted units cells of nanotubes with
chiral vectors $\vec{C_{1}}=(n,m)=(28,1)$ (dark blue) and
$\vec{C_{2}}=(-28,-1)$ (red), the chiral angle is $\theta_{C}\approx
1.74^{\circ}$ in both cases. Also drawn is their translational vector
$\vec{T}.$ (Right) Unit cell of a twisted bilayer graphene nanotube
(28,1)@(-28-1), obtained from rotation of the cells in the left panel by
$120^{\circ}-\theta_{C}$ followed by reflection of the (-28,-1) cell with
respect to the x axis. The resulting twist angle is $\theta=2\cdot\theta_{C}$.

We include below the expressions of the two most relevant quantities for this
research that follow from the geometry. These are the twist angle $\theta$ and
the number of atoms N in the translational unit cell of a (n,m)@(-n,-m) TBG
nanotube, as a function of $n$ and $m$ [88],

$\theta=2\cdot arccos\Big{(}\frac{2n+m}{2\sqrt{n^{2}+nm+m^{2}}}\Big{)}\,,$
(S1)

$N=\frac{8(n^{2}+nm+m^{2})}{gcd(2m+n,2n+m)}\,.$ (S2)

It is worth mentioning that the system built in Figure S7 is one of the
smallest possible on which it is feasible to apply a scaling procedure plus a
decimation technique to calculate the transmission with disorder near the
magic angle ($\theta\sim 1^{\circ}$). In particular, (28,1)@(-28,-1) requires
a scaling factor $\lambda\sim 3.5$ and finite size effects appear beyond
$\lambda\sim 4.5$. However, scalings up to $\lambda\sim 7$ are still useful
for exact diagonalization methods that use many unit cells. For more details
see the sections on Scaling and Decimation below.

## S3. Reciprocal lattice

Because a TBG nanotube is a one-dimensional system, its Brillouin zone is one-
dimensional too. Indeed, it is the same Brillouin zone as the one of its
constituent single-wall nanotubes, namely the segment $X^{\prime}-\Gamma-X$ in
Figure S8(R), of length $2\pi/|T|$. More insightful is the observation that
the bands of a TBG nanotube can be obtained from a folding [88] of the bands
of planar TBG, bearing in mind momentum quantization. This is what we aim to
show in this section.

Figure S8: Construction of the 1D Brillouin zone of a TBG nanotube
(n,1)@(-n,-1). (Left) Construction of the mBZ of TBG, done with a large
$\theta$=$6.01^{\circ}$ for clarity, the blue and red hexagons are the
Brillouin zones of each graphene layer. The K point displacement determines
the size of the mBZ. When the twist angle is commensurate, one can do a tiling
of the mBZ. (Right) mBZ of TBG, BZ of an unrolled TBG nanotube (n,1)@(-n,-1)
(white) and allowed $k_{y}$ (green lines). To obtain the bands of the TBG
nanotube, one folds the mBZ onto the white rectangle and superimposes the TBG
bands along the green lines.

Consider the unit cell given by the dark blue lattice in Figure S7(L). Its
reciprocal basis vectors follow from the well known conditions
$\vec{a_{i}}\cdot\vec{b_{j}}=2\pi\delta_{ij}$, with $\vec{a_{1}}=\vec{C_{1}}$
and $\vec{a_{2}}=\vec{T}$. After rotation (plus reflection in the case of the
(-n,-m) lattice), the reciprocal unit cell of such lattice is in both cases
(before actually rolling them into nanotubes) the white rectangle in Figure
S8(L), with length $2\pi/|T|$ and width $2\pi/|C|$. For TBG nanotubes with
chiral angle $\theta$, such rectangle is inscribed in the mini Brillouin zone
(mBZ) of planar TBG with twist angle $\theta$.

Let the horizontal axis be the direction of propagation, rolling the lattice
to obtain the true nanotube unit cell imposes periodic boundary conditions
which quantize the momentum in the vertical direction. Therefore, starting
from the mBZ of planar TBG with twist angle $\theta$ in Figure S8(R), the 1D
Brillouin zone and band structure of the TBG nanotube with angle $\theta$ can
be obtained by folding the shaded corners of the mBZ onto the white rectangle
and superimposing the TBG energy bands along the green lines, which specify
the $k_{y}$ momenta allowed by the periodic boundary condition. It is worth
noting that the folding places the $K$ and $K^{\prime}$ points of the mBZ at
two thirds of $\Gamma-X$. It is straightforward to prove that any TBG nanotube
(n,1)@(-n,-1) will have allowed $k_{y}$ lines that pass through the center and
sides of the mBZ of the corresponding planar TBG. Here we set $a=1$. Starting
from the periodic boundary condition
$\psi(\vec{r}+|C|\hat{u}_{y})=e^{ik_{y}|C|}\psi(\vec{r})=\psi(\vec{r})$, one
obtains the spacing between permitted $k_{y}$,
$\Delta{k_{y}}=2\pi/|C|=2\pi/\sqrt{n^{2}+m^{2}+mn}$. On the other hand, the K
point relative displacement is $\frac{4\pi}{3}\sqrt{2-2cos(\theta)}$ and the
apothem of the mBZ is therefore $\frac{2\pi}{\sqrt{3}}\sqrt{2-2cos(\theta)}$.
Substituting the expression for $\theta$ in (1) in the previous formula and
using trigonometric identities the apothem reduces to
$2\pi\sqrt{m^{2}/(n^{2}+m^{2}+mn)}$, which for m=1 is equal to the $k_{y}$
spacing obtained before.

The band folding argument explains the connection between the bands of a TBG
nanotube and planar TBG. For practical purposes, the bandstructure of a TBG
nanotube can also be calculated by diagonalization of the Hamiltonian matrix
of the unit cell in Figure S7(R). One should impose periodic boundary
conditions from top to bottom (rolling the nanotube) and from left to right
and across the diagonals (reflecting the fact that the unit cell repeats in
the horizontal axis). Also, Bloch’s theorem dictates that each term in the
Hamiltonian should be multiplied by a phase factor $e^{ik_{x}\Delta x}$, where
$k_{x}$ is the continuum momentum along the propagation direction and $\Delta
x$ is the horizontal distance between the sites that the interact through that
particular term in the Hamiltonian.

## S4. Scaling

As already noted by Bistritzer and MacDonald [33] the bands of twisted bilayer
graphene depend on a single parameter, which involves the ratio of interlayer
to intralayer hopping and the twist angle. For small angles we can write:

$\alpha=\frac{at_{\perp}}{2\hbar
v_{F}\sin(\theta/2)}\propto\frac{t_{\perp}}{t_{\parallel}\theta}\,.$ (S3)

One can also understand this as a comparison between the time a carrier takes
to traverse the moiré unit cell versus the average time between interlayer
tunneling events. In virtue of (S3), it is possible to perform a scaling
approximation [76] that allows us to reproduce the spectrum at the magic angle
$\theta\sim 1^{\circ}$ using a larger angle $\theta^{\prime}$ with a reduction
in $t_{\parallel}$. The lattice should be magnified for consistency (e.g. so
that the moiré period of a $1^{\circ}$ system is $\approx 13$ nm). More
formally, scaling is possible because the Dirac equation that governs each
layer can be described in different ways, and our approach is a description
based on a lattice of ‘super-atoms’ with renormalized intralayer hopping and
also a renormalized lattice constant. Thus, the scaling transformations are,

$t_{\parallel}\rightarrow\frac{1}{\lambda}t_{\parallel}$ (S4)
$a\rightarrow\lambda a$ (S5) $d\rightarrow\lambda d$ (S6)

with

$\lambda=\frac{\sin(\frac{\theta^{\prime}}{2})}{\sin(\frac{\theta}{2})}\,.$
(S7)

Furthermore, Eq. (S3) invites us to model twist angle disorder as intralayer
hopping disorder, which is advantageous because it allows us to work with an
undistorted lattice in the calculation of the transmission. Next we include
some tests aimed at verifying this idea. Figure S9(L) shows the low energy
density of states (DOS) of a pristine magic angle system
($\theta=1.08^{\circ}$) simulated with $\theta^{\prime}=5.40^{\circ}$ i.e.
with a scaling factor $\lambda=5$, compared to the density of states of a
system in which intralayer hopping disorder simulates twist angle disorder in
the range $1.08^{\circ}$-$1.20^{\circ}$. Disorder damps and broadens the magic
angle van-Hove peaks. Next we compare intralayer hopping disorder to twist
angle disorder. Since angle disorder deforms the lattice, we need to
substitute the constant intralayer hopping in Eq. (2) in the main text with an
expression that accounts for distortions:
$\gamma_{ij}^{mm}=t_{\parallel}e^{-(r-a_{cc})/\lambda_{\parallel}}$, where
$a_{cc}=0.142$ nm is the carbon-carbon distance and
$\lambda_{\parallel}=0.042$ nm is a cutoff [89]. Figure S9(R) shows the
comparison between the system with intralayer hopping disorder and one with
twist angle disorder. There is good qualitative agreement between both types
of disorder.

Figure S9: (Left) DOS of a TBG system with uniform twist angle
$\theta=1.08^{\circ}$ (blue) compared to that of a system with intralayer
hopping disorder simulating $1.08^{\circ}$-$1.20^{\circ}$ twist angle disorder
(green). (Right) Intralayer hopping disorder DOS (green) compared to true
twist angle disorder (blue).

Disorder is introduced varying the twist angle (or the intralayer hopping)
with an hyperbolic tangent function over the sample; the function changes over
an extension large with respect to the moiré period. These and the following
results (Figures S9-S11) were obtained by exact diagonalization of a TBG
system with $\sim 44000$ lattice sites, which for these twist angles gives
rise to approximately 80 moirés. Because of the ensuing distortion of the
lattice in the system with twist angle disorder, it is simpler here to omit
the periodic boundary conditions, however we checked that for this system
size, their inclusion has a negligible effect.

We use the Inverse Participation Ratio (IPR) and its associated measure, the
localization length, as another measure of similitude between both types of
disorder. The IPR of a state ‘i’ is defined as the sum of the fourth powers of
its eigenvector components [90]:

$IPR_{i}=\sum_{k}{|\phi_{i,k}|^{4}}\,,$ (S8)

from which the localization length follows:

$\xi_{i}=\frac{1}{2}\sqrt{\frac{1}{IPR_{i}}}\,.$ (S9)

In Figure S10(L) we present a comparison of the localization length of states
in pristine TBG systems with $1.08^{\circ}$ and $1.20^{\circ}$. The latter
shows two well differentiated clusters corresponding to the van Hove peaks,
which in the former are merged into one, indicating the close proximity of
$1.08^{\circ}$ to the magic angle in our model. Figure S10(R) shows the
localization of states in systems with angle disorder in the range
($1.08^{\circ}$-$1.20^{\circ}$) and intralayer hopping disorder aiming to
mimic that angle disorder. Compared to the pristine systems, the average
localization decreases slightly and the two clusters merge. There is also a
considerable broadening of the bandwidth in comparison with the $1.08^{\circ}$
system. Hopping disorder leads to a slight redshift of the flat bands compared
to angle disorder and it also results in smaller localization lenght on
average, hence in a sense it is a more constraining type of disorder.

Figure S10: (Left) Localization length of states in pristine $1.08^{\circ}$
and $1.20^{\circ}$ TBG systems. (Right) Localization length of TBG systems
with twist angle disorder in the range $1.08^{\circ}$-$1.20^{\circ}$ and
equivalent intralayer hopping disorder.

Finally, we studied the real space charge localization maps of pristine, angle
disordered and hopping disordered systems. Some characteristic states are
plotted in Figure S11. Most of the states in a pristine systems tend to spread
over the whole sample, with charge accumulating at the AA stacking regions. In
contrast, systems with twist angle (or intralayer hopping) disorder lead to
states in which charge accumulates preferably at an angle (or hopping) domain,
which is consistent with their lower average localization length seen in
Figure S10.

Inspecting the charge maps of all states in the disordered systems, one sees
that within certain energy intervals, a type of state prevails. For example,
in the TBG with angle disorder, in the interval (41 meV, 45 meV) almost all
states show charge accumulation in the lower half of the sample (which has
larger twist angle $1.20^{\circ}$), whereas in the interval (47 meV, 50 meV)
the opposite happens, the upper region with smaller twist angle $1.08^{\circ}$
is preferred. Similarly, in the TBG with intralayer hopping disorder, in the
interval (37 meV, 42 meV) most states occupy the lower half (larger intralayer
hopping equivalent to $1.20^{\circ}$) whereas states in (45 meV, 49 meV) live
in the upper half (smaller intralayer hopping equivalent to $1.08^{\circ}$).
Therefore, this phenomenology is a further reassurance of the very good
qualitative agreement between both types of disorder.

Figure S11: (a,d) Charge maps of states in pristine $1.08^{\circ}$ and
$1.20^{\circ}$ TBG respectively, with charge distributed across the sample in
the AA stacking regions. (b,e) Maps of states in TBG with twist angle disorder
in the range ($1.08^{\circ}$-$1.20^{\circ}$) showing charge localized in
either angle domain. (c,f) Maps of states in TBG with intralayer hopping
disorder simulating twist angle disorder, charge accumulates in a hopping
domain.

It is worth mentioning that the scaling method ‘squeezes’ the entire spectrum
of TBG. It reduces the bandwidth of graphene from ($\Delta\nu\sim$ 18eV) to
$\Delta\nu^{\prime}\sim\Delta\nu/\lambda$. The graphene van-Hove singularities
thus appear at $\pm\Delta\nu^{\prime}/3$ and in principle the results are
reliable in an interval (-$\Delta\nu^{\prime}/6$,+$\Delta\nu^{\prime}/6$). In
practice, the scaling also leads to a slight blueshift of the spectrum and
band narrowing. As a consequence, we find $\lambda\sim 7$ to be the maximum
practicable scaling factor for these exact diagonalization simulations. For
the precise calculation of the transmission, which uses only a few unit cells,
the requirement will be more stringent ($\lambda\lesssim 4.5$), due to finite
size effects.

## S5. Decimation

We develop a framework for the calculation of the local Green’s function in a
disordered region, within the path integral formulation of quantum mechanics
and based on the technique of effective action. This allows us to compute the
self-energies that are added to the disordered cell Green’s function as a
consequence of its coupling to the semi-infinite pristine leads, these self-
energies are necessary for the calculation of the transmission through the
disordered cell [83]. The idea is to account for the effect of adjacent cells
by a renormalization procedure which obtains frequency-dependent coupling
matrices of the disordered cell to increasingly distant pristine cells, in the
spirit of the decimation techniques presented in Refs. [84, 85]. We first show
the derivation for a one dimensional chain of atoms, and then present the
general matrix formulas for unit cells that contain multiple atoms. In the 1D
case, the ‘disordered region’ is a single impurity atom, the Hamiltonian of
the full chain is

$H=\epsilon_{0}c_{0}^{\dagger}c_{0}+t_{R}(c_{0}^{\dagger}c_{1}+h.c.)+t_{L}(c_{0}^{\dagger}c_{-1}+h.c.)+\sum_{i=1}^{\infty}[\epsilon
c_{i}^{\dagger}c_{i}+t(c_{i}^{\dagger}c_{i+1}+h.c.)]+\sum_{-\infty}^{-1}[\epsilon
c_{i}^{\dagger}c_{i}+t(c_{i}^{\dagger}c_{i-1}+h.c.)]\,.$ (S10)

Here $\epsilon_{0}$ is the on-site energy of the impurity, $t_{R}$ ($t_{L}$)
the hopping between the impurity and the atom to its right (left). $t$ is the
hopping between atoms other than the impurity. Now consider the action of this
system,

$S=\int
d\omega\Big{[}\sum_{i}\omega\overline{c}_{i}(\omega)c_{i}(\omega)-H\Big{]}=\int
d\omega\Big{[}G_{0}^{-1}(\omega)\overline{c}_{0}(\omega)c_{0}(\omega)+\sum_{i\neq
0}G^{-1}(\omega)\overline{c}_{i}(\omega)c_{i}(\omega)-t_{R}[\overline{c}_{0}(\omega)c_{1}(\omega)+\overline{c}_{1}(\omega)c_{0}(\omega)]\\\
-t_{L}[\overline{c}_{0}(\omega)c_{-1}(\omega)+\overline{c}_{-1}(\omega)c_{0}(\omega)]-\sum_{i=1}^{\infty}t[\overline{c}_{i}(\omega)c_{i+1}(\omega)]+\overline{c}_{i+1}(\omega)c_{i}(\omega)]-\sum_{i=-\infty}^{-1}t[\overline{c}_{i}(\omega)c_{i-1}(\omega)]+\overline{c}_{i-1}(\omega)c_{i}(\omega)]\Big{]}\,,$
(S11)

where $c_{i}(\omega)$, $\overline{c}_{i}(\omega)$ are Grassmann numbers and
$G_{0}(\omega)=\frac{1}{\omega I-\epsilon_{0}}$, $G(\omega)=\frac{1}{\omega
I-\epsilon}$ are the local non-interacting Green’s functions at the impurity
and at a site $i\neq 0$, respectively. Next, we integrate the variables
$c_{\pm 1}(\omega)$, $\overline{c}_{\pm 1}(\omega)$ out of the action
performing gaussian integration. This results in the effective action

$S_{eff}^{(2)}=\int
d\omega\Big{[}\big{[}G_{0}^{(2)}(\omega)\big{]}^{-1}\overline{c}_{0}(\omega)c_{0}(\omega)+\sum_{i=\pm
2}\big{[}G_{i}^{-1}(\omega)\overline{c}_{i}(\omega)c_{i}(\omega)-t_{i}(\omega)[\overline{c}_{0}(\omega)c_{i}(\omega)+\overline{c}_{i}(\omega)c_{0}(\omega)]\big{]}+\\\
+\sum_{\mid
i\mid>2}G^{-1}(\omega)\overline{c}_{i}(\omega)c_{i}(\omega)-\sum_{i=2}^{\infty}t[\overline{c}_{i}(\omega)c_{i+1}(\omega)]+\overline{c}_{i+1}(\omega)c_{i}(\omega)]-\sum_{i=-\infty}^{-2}t[\overline{c}_{i}(\omega)c_{i-1}(\omega)]+\overline{c}_{i-1}(\omega)c_{i}(\omega)]\Big{]}\,,$
(S12)

where

$\big{[}G_{0}^{(2)}(\omega)\big{]}^{-1}=G_{0}^{-1}(\omega)-t_{R}^{2}G(\omega)-t_{L}^{2}G(\omega)$
(S13)

is the inverse of the local Green’s function at the impurity,

$G_{\pm 2}^{-1}(\omega)=G^{-1}(\omega)-t^{2}G(\omega)$ (S14)

that at sites $\pm 2$ and

$t_{\pm 2}(\omega)=tt_{R,L}G(\omega)$ (S15)

are the effective frequency-dependent hopping amplitudes between the site 0
and sites $\pm 2$. We can iterate this procedure by integrating out all the
degrees of freedom up to the sites $i=\pm(N-1)$. This leads to the effective
action

$S_{eff}^{(N)}=\int
d\omega\Big{[}\big{[}G_{0}^{(N)}(\omega)\big{]}^{-1}\overline{c}_{0}(\omega)c_{0}(\omega)+\sum_{i=\pm
N}\big{[}G_{i}^{-1}(\omega)\overline{c}_{i}(\omega)c_{i}(\omega)-t_{i}(\omega)[\overline{c}_{0}(\omega)c_{i}(\omega)+\overline{c}_{i}(\omega)c_{0}(\omega)]\big{]}+\\\
+\sum_{\mid
i\mid>N}G^{-1}(\omega)\overline{c}_{i}(\omega)c_{i}(\omega)-\sum_{i=N}^{\infty}t[\overline{c}_{i}(\omega)c_{i+1}(\omega)]+\overline{c}_{i+1}(\omega)c_{i}(\omega)]-\sum_{i=-\infty}^{-N}t[\overline{c}_{i}(\omega)c_{i-1}(\omega)]+\overline{c}_{i-1}(\omega)c_{i}(\omega)]\Big{]}\,.$
(S16)

Eq. (S16) is analogous to (S12) and describes a system in which the impurity
atom is coupled directly to sites $\pm N$ via frequency-dependent hopping
amplitudes $t_{\pm N}$. The recursive formulas for the Green’s functions
$G_{0}^{N}$, $G_{\pm N}$ and the hopping functions $t_{\pm N}$ are,

$\big{[}G_{0}^{(N)}(\omega)\big{]}^{-1}=\big{[}G_{0}^{N-1}(\omega)\big{]}^{-1}-t_{N-1}^{2}(\omega)G_{N-1}(\omega)-t_{-(N-1)}^{2}(\omega)G_{-(N-1)}(\omega)\,,$
(S17)

$G_{\pm N}^{-1}(\omega)=G^{-1}(\omega)-t^{2}G_{\pm(N-1)}(\omega)\,,$ (S18)

$t_{\pm N}(\omega)=tt_{\pm(N-1)}G_{\pm(N-1)}(\omega)\,.$ (S19)

Eqs. (S13-S15) set the step $N=2$ and the input to Eqs. (S17-S19). In the
limit $N\rightarrow\infty$, $G_{0}^{(N)}$ converges to the local Green
function at the impurity, including the effect of its environment. For
practical purposes, one can compute the retarded Green’s function, doing
analytical projection on the upper half-plane, by substituting everywhere
$\omega\rightarrow\omega+i\delta$, with $\delta$ a finite broadening. As an
example, we consider an impurity with $\epsilon_{0}=2$ eV, the rest of atoms
with $\epsilon=1$ eV, $t=t_{L}=t_{R}=\frac{1}{2}$ and $\delta=0.05$ eV. Figure
S12 shows the imaginary and real parts of $G_{0}^{(\infty)}$ for this set of
parameters. Due to the environment, the impurity’s Green function experiences
a weak blueshift and damping at frequencies smaller than $\epsilon_{0}$.

Figure S12: Minus the imaginary part (left) and real part (right) of the
impurity’s retarded Green’s function, before and after accounting for its
coupling to the environment. Here $\epsilon_{0}=2$ eV, $\epsilon=1$ eV,
$t=t_{L}=t_{R}=\frac{1}{2}$, $\delta=0.05$ eV.  Figure S13: Sketch of the
geometry involved in the decimation technique. All cells to the left of the
central one $H_{ii},i<0$ are pristine with uniform intralayer hopping
$t^{\parallel}_{1}$. Cells to the right $H_{ii},i>0$ are pristine with a
different hopping $t^{\parallel}_{2}$. These two semi-infinite leads are
connected through the central disordered cell ‘0’, whose Hamiltonian $H_{00}$
has intralayer hopping disorder ($t^{\parallel}_{1}$ evolving smoothly into
$t^{\parallel}_{2}$ as an hyperbolic tangent). One needs to consider
separately the coupling matrices from the disordered cell to its adjacent
pristine cells ($T_{L}$ and $T_{R}$) and the coupling between pristine cells
($T_{EL}$ and $T_{ER}$). The goal is the calculation of the Green’s function
at the disordered cell, accounting for the effect of the semi-infinite
pristine leads. Periodic boundary conditions are applied in the vertical
direction.

For multi-atomic cells, we start with the unperturbed Green’s functions,
$G_{0}(\omega)=\frac{1}{\omega I-H_{00}}$, $G(\omega)=\frac{1}{\omega
I-H_{ii}}$, where $H_{00}$ is the Hamiltonian of the disordered cell and
$H_{ii}$ the Hamiltonians of pristine unit cells, equal for all $i>0$ and for
all $i<0$. We also define $T_{R}=H_{0,1}$ and $T_{ER}=H_{i,i+1}$ with $i>0$ as
well as $T_{L}=H_{0,-1}$ and $T_{EL}=H_{i,i-1}$ with $i<0$ (see Figure S13)
the recursive formulas to obtain the Green’s functions and coupling matrices
are:

$\displaystyle\big{[}G_{0}^{(2)}(\omega)\big{]}^{-1}$
$\displaystyle=G_{0}^{-1}(\omega)-T_{R}G(\omega)T_{R}^{\dagger}-T_{L}G(\omega)T_{L}^{\dagger}$
(S20) $\displaystyle G_{2}^{-1}(\omega)$
$\displaystyle=G^{-1}(\omega)-T_{ER}^{\dagger}G(\omega)T_{ER}$ $\displaystyle
G_{-2}^{-1}(\omega)$
$\displaystyle=G^{-1}(\omega)-T_{EL}^{\dagger}G(\omega)T_{EL}$ $\displaystyle
T_{02}(\omega)$ $\displaystyle=T_{R}G(\omega)T_{ER}$ $\displaystyle
T_{20}(\omega)$ $\displaystyle=T_{ER}^{\dagger}G(\omega)T_{R}^{\dagger}$
$\displaystyle T_{0,-2}(\omega)$ $\displaystyle=T_{L}G(\omega)T_{EL}$
$\displaystyle T_{-2,0}(\omega)$
$\displaystyle=T_{EL}^{\dagger}G(\omega)T_{L}^{\dagger}$

$\displaystyle\big{[}G_{0}^{(N+1)}(\omega)\big{]}^{-1}$
$\displaystyle=\big{[}G_{0}^{(N)}\big{(}\omega)]^{-1}-T_{0N}(\omega)G_{N}(\omega)T_{N0}(\omega)-T_{0,-N}(\omega)G_{-N}(\omega)T_{-N,0}(\omega)$
(S21) $\displaystyle G_{N+1}^{-1}(\omega)$
$\displaystyle=G^{-1}(\omega)-T_{ER}^{\dagger}G_{N}(\omega)T_{ER}$
$\displaystyle G_{-N-1}^{-1}(\omega)$
$\displaystyle=G^{-1}(\omega)-T_{EL}^{\dagger}G_{-N}(\omega)T_{EL}$
$\displaystyle T_{0,N+1}(\omega)$
$\displaystyle=T_{0N}(\omega)G_{N}(\omega)T_{ER}$ $\displaystyle
T_{N+1,0}(\omega)$ $\displaystyle=T_{ER}^{\dagger}G_{N}(\omega)T_{N0}(\omega)$
$\displaystyle T_{0,-N-1}(\omega)$
$\displaystyle=T_{0,-N}(\omega)G_{-N}(\omega)T_{EL}$ $\displaystyle
T_{-N-1,0}(\omega)$
$\displaystyle=T_{EL}^{\dagger}G_{-N}(\omega)T_{-N,0}(\omega)$

To obtain the transmission one also needs to compute advanced versions of all
the quantities in (S20) and (S21), by doing analytical continuation with
$\omega\rightarrow\omega-i\delta$ ($\delta$ a small broadening) instead of
$\omega\rightarrow\omega+i\delta$ as in the retarded case. Once the matrices
in (S21) have converged, the transmission can be readily calculated. First one
computes the left and right ‘coupling functions’ of the impurity cell to the
semi-infinite leads [83]:

$\Gamma_{\\{L,R\\}}=i\big{[}\Sigma^{r}_{\\{L,R\\}}-\Sigma^{a}_{\\{L,R\\}}\big{]}\,.$
(S22)

Here $\Sigma^{r}_{L}=\sum_{i}T_{0,-i}G_{-i}T_{-i,0}$ and
$\Sigma^{r}_{R}=\sum_{i}T_{0,i}G_{i}T_{i,0}$ are the sums of all the terms
subtracted from $[G_{0}(\omega)]^{-1}$ to reach the convergent value
$[G^{\infty}_{0}(\omega)]^{-1}$, as a consequence of the coupling to the left
and right pristine leads respectively. $\Sigma^{a}_{\\{L,R\\}}$ are the
advanced versions. With these we can write the transmission,

$\mathcal{T}=\textrm{Tr}\big{\\{}\Gamma_{L}G_{0}^{\infty,r}\Gamma_{R}G_{0}^{\infty,a}\big{\\}}\,.$
(S23)

Note that $G^{\infty}_{0}(\omega)$ also gives us the local density of states
at the disordered region,

$\textrm{LDOS}=\frac{1}{\pi}\textrm{Tr}\big{\\{}\textrm{Im}\\{G^{\infty}_{0}(\omega)\\}\big{\\}}\,.$
(S24)

Matrix inversion is the computational bottleneck of the algorithm, using
standard BLAS libraries it scales with the size of the matrix as $N^{2}$.
Initial simulations were attempted with TBG nanotube (19,1)@(-19,-1) (N=1016
and $\lambda\sim 5$ for a magic angle simulation). Later we became aware that
this system has finite size effects that lead to fine energy splittings of the
flat bands. Therefore we recommend that calculations that demand high
precision and employ one or few unit cells are done with scaling factor
$\lambda\lesssim 4.5$. However, exact diagonalization calculations that use
many cells work well up to $\lambda\lesssim 7$. The main results of the paper
were obtained with (56,2)@(-56,-2), which like (28,1)@(-28,-1) has a scaling
factor of $\lambda\sim 3.5$ and better resembles planar TBG since it fits two
moirés in the direction perpendicular to the propagation.