# Chirality probe of twisted bilayer graphene in the linear transport regime

Dario A. Bahamon  School of Engineering, Mackenzie Presbyterian University,
São Paulo - 01302-907, Brazil<EMAIL_ADDRESS>Guillermo Gómez-
Santos  Departamento de Física de la Materia Condensada, Instituto Nicolás
Cabrera and Condensed Matter Physics Center (IFIMAC), Universidad Autónoma de
Madrid, E-28049 Madrid, Spain<EMAIL_ADDRESS>Dmitri K. Efetov
Fakultät für Physik, Ludwig-Maximilians-Universität, Schellingstrasse 4,
München 80799, Germany<EMAIL_ADDRESS>Tobias Stauber  Departamento de
Teoría y Simulación de Materiales, Instituto de Ciencias de Materiales de
Madrid, CSIC, E-28049, Madrid, Spain<EMAIL_ADDRESS>
###### Abstract

We propose minimal transport experiments in the coherent regime that can probe
the chirality of twisted moiré structures. We show that only with a third
contact and in the presence of an in-plane magnetic field (or other time-
reversal symmetry breaking effect), a chiral system may display non-reciprocal
transport in the linear regime. We then propose to use the third lead as a
voltage probe and show that opposite enantiomers give rise to different
voltage drops on the third lead. Additionally, in the scenario of layer-
discriminating contacts, the third lead can serve as a current probe, capable
of detecting different handedness even in the absence of a magnetic field. In
a complementary configuration, applying opposite voltages on the two layers of
the third leads gives rise to a chiral (super)current in the absence of a
source-drain voltage whose direction is determined by its chirality.

###### keywords:

Quantum Transport; Chirality; Twisted Bilayer Graphene; Reciprocity Relations.

MackGraphe Graphene and Nanomaterials Research Institute, Mackenzie
Presbyterian University, São Paulo -01302-907, Brazil Departamento de Teoría y
Simulación de Materiales, Instituto de Ciencias de Materiales de Madrid, CSIC,
E-28049, Madrid, Spain Munich Center for Quantum Science and Technology
(MCQST), München, Germany

##  1 Introduction

The idea of chirality permeates many branches of science, 1, 2, 3 and also in
condensed matter physics the concept is used to describe electronic properties
in reciprocal and real space. In reciprocal space, chirality defines the
handedness of the spin of the electron to its momentum,4, 5, 6 in real space,
chirality depicts the handedness of molecules and solids that cannot be
superimposed onto their mirror images.1, 7 Most generally, chirality always
emerges when discrete symmetries such as refection, time-reversal or particle-
hole symmetry are broken.

Independent of the setting, chiral systems offer opportunities to observe new
phenomena as well as challenges regarding their detection. 6, 8, 7 The
signature of topological insulators, e.g., is the existence of chiral
dissipationless states at the boundaries of the sample,4, 6, 8 and the
electrical detection of these states requires non-local transport
measurements. 9 On the real space side, chiral molecules spin polarize the
electric current passing through them,10 and the chiral-induced spin
selectivity (CISS) effect is detected in a two terminal configuration as a
non-linear I-V characteristic.7 Also, electrical magneto-chiral anisotropy in
a classical four terminal configuration has been observed, where the
longitudinal enantio-selective magneto-resistance is non-linear in the
current.11, 12, 13 Finally, in non-centrosymmetric materials, chirality
manifests itself in various non-reciprocal response phenomena 14, 15, 16, 17,
18 where the resistance depends linearly on the current causing a non-linear
voltage drop in a two terminal setup. 19, 20, 21

Van der Waals moiré materials offer new opportunities to engineer certain
geometric structures that can lead to novel properties.21, 22, 23 For example,
twisted bilayer graphene24, 25, 26, 27, 28, 29 (TBG), which has recently
attracted great attention due to discovery of novel electronic phases,30, 31,
32, 33, 34 can be rotated clock- or anti-clockwise. Due to the finite
interlayer separation, TBG heterostructures with opposite twist angles can
only be superimposed onto each other after performing a mirror reflection with
respect to the $xy$-plane. Thus, TBG is intrinsically chiral which has been
experimentally demonstrated by observing optical dichroism without breaking
time-reversal symmetry.35 The effect becomes largest for frequencies which
induce transitions between states that are maximally delocalized between the
two layers such that the misalignment between layers is most effective.36, 37
But also in the dc-limit, the chirality of TBG is manifested by intrinsic
magnetic-electric coupling.38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51 In the dc-limit with broken time reversal symmetry, an electrical
magnetochiral anisotropy is anticipated to emerge. 52 Nevertheless, in
conventional transport experiments where the net current in voltage leads is
maintained at zero,53, 54 the influence of chirality has not been a
significant factor thus far.

In this work, we precisely fill in this gap by investigating electronic
transport through TBG in the linear regime within the Landauer-Büttiker
formalism. Using general symmetry arguments, we point out that in order to
distinguish different enantiomers, i.e., samples with twist angles $\theta$
and $-\theta$, respectively, it is crucial to have three leads. This is
contrary to the zero-field superconducting diode effect where two leads are
sufficient.16, 55, 56, 57, 58, 59, 60, 61 The third lead can now either be
used as a voltage probe that detects the chirality if a magnetic field in
applied parallel to the layer. Or, in the case of layer-discriminating
contacts, it can be used as current probe even in the absence of a magnetic
field. We thus show that it is not necessary to break time-reversal symmetry
with a magnetic field in order to observe the layer-discriminating transverse
currents effect in a typical transport experiment.

##  2 Landauer formalism and general symmetries

Figure 1:  (Color online) (a) and (b): Proposed setup for voltage probe
detection of chirality in the presence of a magnetic field
(${\bm{B}}=B\,{\bm{e}}_{x}$). The reading of the voltage $V_{3}$ is opposite
for the opposite chiralities depicted. The Green color stands for a layer
twisted by a positive angle, while the violet means a negative twisted angle
layer. (c) Schematic representation of the twisted bilayer graphene (TBG)
three-terminal device, the yellow and red dots represent the atomic sites in
contact with leads 1, 2 and 3 respectively.

We consider two systems with opposite twist angles $\pm\theta$ or chiralities,
related to each other by a mirror symmetry with respect to a horizontal plane
midway between the layers. Notice that in this symmetry operation we include
the leads, which we consider, for the moment, equally coupled to both layers.
The reason for this layer-symmetric attachment of leads is twofold: first,
hopefully easier experimental realization, and second, simplified chiral
analysis for, otherwise, a right lead attached to the bottom layer in the
$\theta$ flake would become a right lead, but in the top layer for the mirror
flake with opposite chirality $-\theta$. Let us also assume that there are $N$
leads attached to the system, and that the current $I_{p}$ in lead $p$ is 54

$\displaystyle I_{p}=\sum_{q}G_{pq}[V_{p}-V_{q}],$ (1)

where $V_{q}$ is the potential of lead $q$ over ground level and the
conductance $G_{pq}=\frac{2e^{2}}{h}\overline{T}_{pq}$ is proportional to the
total transmission $\overline{T}_{pq}$ between lead $q$ lead $p$. Because of
the inherent symmetries of the transmission matrices, the following
reciprocity relation holds54

$\displaystyle G_{pq}(\theta,{\bm{B}})=G_{qp}(\theta,-{\bm{B}})\;,$ (2)

where $\bm{B}$ is the magnetic field, see the Supplementary Information (SI).
In our case, it will denote an in-plane magnetic field.

We now perform a z-reflection by mapping $z$ to $-z$ for each piece of matter,
including sources if external fields are present. This is the composition of
space-inversion, $(x,y,z)\to(-x,-y,-z)$, followed by a $\pi$-rotation around
the z-axis, see Fig. 1-(c), where the origin is placed in the center and
midway between the planes. By this, one maps $\theta\to-\theta$, $p\to p$,
$q\to q$, and, crucially important here, ${\bm{B}}\to-{\bm{B}}$, i.e.,
${\bm{B}}$ is an axial vector, first unchanged by space-inversion but later
changing sign upon the $\pi$-rotation around the z-axis. In twisted systems,
we thus also have the chiral reciprocity relation

$\displaystyle G_{pq}(\theta,{\bm{B}})=G_{pq}(-\theta,-{\bm{B}})\;,$ (3)

valid for in-plane magnetic fields and layer-symmetric attachments. This
correlation shows that there is no way to detect the sign of the chirality
without magnetic fields in twisted arrangements with layer-symmetric leads,
because $G_{pq}(\theta,{\bm{B}}={\bm{0}})=G_{pq}(-\theta,{\bm{B}}={\bm{0}})$.
On the other hand, the correspondence expressed in Eq. (12) implies a locking
of chirality and field.

To exploit this relation, let us now fix the direction, ${\bm{e}}_{\bm{n}},$
of the in-plane magnetic field, ${\bm{B}}=B{\bm{e}}_{\bm{n}}$ and expand the
conductance to linear order in $B$,

$\displaystyle G_{pq}(\theta,B)=G_{pq}(\theta,0)+\Delta_{pq}(\theta)B\;,$ (4)
$\displaystyle G_{qp}(\theta,B)=G_{qp}(\theta,0)+\Delta_{qp}(\theta)B\;.$ (5)

In conjunction with Eqs. (11) and (12) and knowing, as stated earlier, that
$G_{pq}(\theta,0)=G_{qp}(\theta,0)=G_{qp}(-\theta,0)$ we arrive at

$\displaystyle\Delta_{pq}(\theta)=-\Delta_{qp}(\theta)=-\Delta_{pq}(-\theta)\;.$
(6)

The above equation indicates an electrical mechanism to detect chirality of
twisted devices through conductance measurements even with layer-symmetric
leads.

##  3  Including the third lead as voltage probe for chirality

Let us now focus on the minimal system that can detect chiral properties in
the linear regime. The device with three leads is depicted in Fig. 1(a)-(b).
All leads couple equally to both layers, as previously assumed and lead 1 and
2 have equal number of channels. In the absence of a magnetic field, the
currents can be obtained from Eq. 1

$\left(\begin{matrix}I_{1}\\\ I_{2}\\\
I_{3}\end{matrix}\right)=\left(\begin{matrix}G_{12}^{0}+G_{13}^{0}&-G_{12}^{0}&-G_{13}^{0}\\\
-G_{12}^{0}&G_{12}^{0}+G_{13}^{0}&-G_{13}^{0}\\\
-G_{13}^{0}&-G_{13}^{0}&2G_{13}^{0}\end{matrix}\right)\left(\begin{matrix}V_{1}\\\
V_{2}\\\ V_{3}\end{matrix}\right)\;,$ (7)

where it is defined that
$G_{pq}(\pm\theta,{\bm{B}}={\bm{0}})=G_{pq}^{0}=G_{qp}^{0}$ as well as
assumed, based on the symmetry of the problem, that $G_{13}^{0}=G_{23}^{0}$.
When ${\bm{B}}\neq{\bm{0}}$, the additional contribution to the conductance
matrix is:

$\left(\begin{matrix}\delta I_{1}\\\ \delta I_{2}\\\ \delta
I_{3}\end{matrix}\right)=\left(\begin{matrix}0&-\delta G&+\delta G\\\ +\delta
G&0&-\delta G\\\ -\delta G&+\delta
G&0\end{matrix}\right)\left(\begin{matrix}V_{1}\\\ V_{2}\\\
V_{3}\end{matrix}\right)\;,$ (8)

where the corrections are characterized by a single parameter to linear order
in $|{\bm{B}}|$, $\delta G=\Delta(\theta)B$. This is a consequence of the
conductance sum rule $\sum_{q\neq p}G_{pq}(+{\bm{B}})=\sum_{q\neq
p}G_{pq}(-{\bm{B}})$ which implies that $\sum_{q\neq p}\Delta_{pq}=0$. Now, we
can use the third lead as a voltage probe, imposing $I_{3}=0$ while applying a
source-drain voltage drop, $V_{1}=V/2,~{}V_{2}=-V/2$. Then to linear order in
the $B$-field, the voltage probe, $V_{3}$, yields

$\displaystyle\frac{V_{3}}{V}=\frac{\delta
G}{2G_{13}^{0}}=\frac{\Delta(\theta)}{2G_{13}^{0}}B.$ (9)

Previously, it was demonstrated that $\Delta(\theta)=-\Delta(-\theta)$, this
ensures that the voltage probe becomes also a probe of the chirality sign
($V_{3}(\theta)=-V_{3}(-\theta)$).

###  3.1 Numerical implementation

Having established the basic relations that allow the detection of handedness
through electric measurements, our attention turns to their numerical
calculation in a TBG region of dimensions $W\times L_{x}$, where both $W$ and
$L_{x}$ are set to 50 nm (the system is sketched in Fig. 1(e)). The
computation is performed employing the tight-binding model and Green’s
functions. It’s crucial to note that we opt for multiple neighbors for each
site to replicate the characteristics of TBG. As a consequence, the charge
neutrality point (CNP) emerges at approximately $E\approx 0.2961t_{0}$, where
$t_{0}$ represents the nearest neighbor hopping. Consequently, all numerical
outcomes presented in this study are offset by this value and
$E_{F}=E-0.2961t_{0}$. For detail, we refer to the supplemental information.
There, we also show that the reciprocity relations as delineated in Eqs. (11)
and (12) are correctly implemented in our numerical calculations.

Figure 2:  (Color online) (a), (d)-(f) $V_{3}$ for the different twist angles
as function of ${\bm{B}}$ parallel to the $x$-axis. (b) Value and (c) full
width at half maximum (FWHM) of the DOS peak without magnetic field. The Roman
numerals I, II, and III, along with the shaded regions, serve as visual guides
to differentiate various chirality behaviors. In the panels where $V_{3}/V$ is
presented, the Fermi energy is set to zero, $E_{F}=0$.

##  4 Detecting chirality by voltage probe with a magnetic field

We now consider the case where
$V_{1}=V/2,~{}V_{2}=-V/2~{}\text{and}~{}I_{3}=0$. Using this condition,
$V_{3}/V=(G_{31}-G_{32})/(2~{}G_{31}+2~{}G_{32})$ can be calculated as
function of the magnetic field for different twist angles and energies. For
$\theta=\pm 1.29^{\circ}$ and Fermi energy $E_{F}=0$ meV, Fig. 2(a) shows that
$V_{3}/V$ is linear in B with opposite slope for opposite twist angles,
confirming the prediction of Eq. (9) that lead 3 becomes also a probe for
chirality and that quantum transport is sensitive to chirality. However, it is
not possible to relate in advance the sign nor the strength of the linear
field-dependence to a certain twist angle. This is similar to previous
observations for the infinite system,37, 38, 40 where the chiral component of
the conductivity was shown to exhibit highly non-monotonous behavior with
filling factor and twist angle.

For large angles, the effect of the handedness of the junction on transport is
negligible and $V_{3}/V_{1}\approx 0$. This already indicates that the effect
is mainly electronically and not configurationally driven, for both layers are
highly decoupled for large angles. For small angles, the density of states
(DOS) is presented in Fig. 2(b)-(c) in terms of the the largest value as well
as the full width at half maximum (FWHM) of the main peak. From there, we
infer that the magic angle is located around $\sim 1.23^{\circ}$ where the
highest and narrowest DOS peak appears and serves to diagnose the magic-angle
regime.

Additionally, three regions are shaded and identified by the Roman numbers I,
II and III conforming to the behavior of $V_{3}$. For $\theta>1.26^{\circ}$
(region III in red), a linear-in-$B$ regime is perfectly defined with negative
slopes for positive twist angles as shown in Fig. 2(d). In the blue shaded
region I ($\theta<1.20^{\circ}$), the signal of the linear response is
inverted and positive slopes appear for positive twist angles; characteristic
lineshapes of $V_{3}$ for angles in these region are presented in Fig. 2(f).

Around the magic angle, the linear relation on the magnetic field becomes
weaker and the non-linear $B$-dependence dominate the response as shown in
Fig. 2(e). However, even for twist angles in this region, the chirality obeys
the symmetry relation $V_{3}(+\theta)=-V_{3}(-\theta)$) as discussed in the
SI, where also the effect of different coupling strengths to lead 3 are
presented.

Let us finally emphasize that our approach utilizes the Landauer-Büttiker
formalism within the linear regime. In this regime, the conductance $G$ is
computed in equilibrium, thus avoiding any dependence on current density. This
distinction ensures that in our setup, the relationship between the voltage
probe reading, $V_{3}$, and the source-drain voltage (and current) remains
linear.

##  5 Detecting chirality by current probe without a magnetic field

Figure 3:  (Color online) (a) and (b): Proposed setup for chirality detection
by current probe without magnetic field. Top and bottom transverse currents
reverse direction for the opposite chiralities depicted. The Green color
stands for a layer twisted by a positive angle, while the violet means a
negative twisted angle layer. The detection of chirality by current probe
requires now four leads (1, 2 3t and 3b), because the layer index top and
bottom of lead 3 is now discriminated. (c) and (d): Alternative set-up showing
the same electro-magnetic coupling without magnetic field. Now, there is no
source-drain voltage, i.e., the potential $V_{1}$ and $V_{2}$ are equal, but a
potential difference between the top and the bottom layer induces a current
whose direction depends on the chirality.

Let us now discuss how to detect the chirality of a system without a magnetic
field. In the infinite TBG, the transverse conductivity is equal in magnitude,
but of opposite direction with respect to the two layers due to basic symmetry
constrains. 37

###  5.1 Inducing a transverse current

In the setup proposed in Figs 3(a)-(b), we expect to see layer-discriminating
transverse currents without the presence of a magnetic field as they are
allowed by the same symmetry principles. This implies that a source-drain
voltage $V=V_{1}-V_{2}$ between lead 1 and lead 2 will be accompanied by
transverse currents in both layers, flowing in opposite directions. As
$V_{3t}=V_{3b}$, one can interpret this as a vertical "supercurrent" flowing
from one layer to the other. In an infinite system, this effect implies that a
net current in the $x$ direction is accompanied by layer-opposite currents in
the $y$ direction, which can be thought of as an in-plane magnetic moment
along the net flow, a hallmark of chirality37, 38, 62, 63, 64.

We need first to generalize Eq. (12) by including the layer index via $q\to
q\nu$ with $\nu=t,b$. Performing the reflection and assuming the same coupling
between the two layers as schematically shown in Figs. 3(a)-(b), we then have

$\displaystyle
G_{q\nu,p\mu}(\theta)=G_{q{\bar{\nu}},p{\bar{\mu}}}(-\theta)\;,$ (10)

where $\bar{\nu},\bar{\mu}$ denote the opposite layer of $\nu,\mu$. We can now
extend Eq. (33) to effectively four leads: leads one and two remain layer-
symmetric as before, while the original third lead splits into top ($3t$) and
bottom ($3b$), see Fig. 1(c)-(d) and SI for reference. With the above in mind,
in Fig. 4(a) we show the conductance from lead 1 to lead 3 in the top layer,
$G_{3t1}$, and the conductance from lead 1 to the lead 3 in the bottom layer,
$G_{3b1}$. Since the contacts are symmetric, there would be no difference in
the two conductances if the system were achiral without a magnetic field.
However, a clear difference is seen giving rise to a chiral current probe in
the absence of a magnetic field. Note that reversing the angle maps the
conductance $G_{3b1}(\theta)$ to $G_{3t1}(-\theta)$ and vice versa, as
dictated by the general symmetry relations of Eq. (10). Notice that there is
no need for dealing separately with lead 2 for, in our geometry,
$G_{3t\,2}=G_{3b\,1}$ and $G_{3t\,1}=G_{3b\,2}$.

Figure 4:  (Color online) (a) Conductance (in units of $2e^{2}/h$) from lead 1
to lead 3 bottom layer ($G_{3b\,1}$) and from lead 1 to lead 3 top layer
($G_{3t\,1}$) for $\theta=\pm 1.20^{\circ}$. (b)
$G_{3t\,1}-G_{3b\,1}=(I_{3b}-I_{3t})/V$ (in units of $2e^{2}/h$) for
$\theta=\pm 1.20^{\circ},\pm 1.30^{\circ}$. From the formalism sign
conventions, the finite value of the plotted magnitude implies currents of
equal magnitude, $|I_{3b}|=|I_{3t}|$, running opposite in opposite leads, and
reversing direction upon chirality reversal, as depicted in Fig. 3(a)-(b).

Now, we can consider the arrangement schematized in Figs. 3(a)-(b),
$V_{1}=V/2$, $V_{2}=-V/2$, and $V_{3b}=V_{3t}=0$, in which no net current
flows between system and the two $3t$ and $3b$ reservoirs: $I_{3b}+I_{3t}=0$.
Yet, a finite current from the upper lead to the lower lead can then be
deduced from the finite value of $\delta
G(\theta)=(I_{3b}-I_{3t})/(V_{1}-V_{2})=G_{3t\,1}(\theta)-G_{3b\,1}(\theta)$.
Remember that, from the sign conventions of the formalism, this implies
currents of equal magnitude, $|I_{3b}|=|I_{3t}|$, running in opposite
directions in top and bottom leads, despite the terminals $3t$ and $3b$ having
equal potentials, as illustrated in Figs. 1(c)-(d). Reversing the chirality
reverses the layer currents, $I_{3b(t)}(\theta)=-I_{3b(t)}(-\theta)$,
therefore the experimental detection of these currents becomes a probe for
chirality without magnetic field ($\delta G(\theta)=-\delta G(-\theta)$) as
shown in Fig. 4(b).

The detection of this vertical "supercurrent" can be difficult with real,
lossy leads. Alternatively, one could use the top and bottom third leads as
independent voltage probes, whose different readings would then reveal the
chirality, as explained in the SI.

###  5.2 Inducing a longitudinal current

A complementary configuration based on the same symmetry principles is
presented in Figs 3(c)-(d), where the same phenomenon also gives rise to a
"supercurrent" from reservoirs 1 to 2 without any voltage drop between them,
$V_{1}=V_{2}=0$, when now the transverse currents are driven by a
corresponding voltage drop, $V_{3t}=-V_{3b}=V$.

A finite current of magnitude $|I_{1}|=I_{2}|$ from the lead 1 to lead 2 (or
viceversa), can then be deduced from the finite value of $\delta
G(\theta)=(I_{1}-I_{2})/(V_{3t}-V_{3b})=G_{3t\,1}(\theta)-G_{3b\,1}(\theta)$.
Note that $\delta G(\theta)$ is given by the same expression as in the
previous subsection and which is shown in Fig. 4(b). The electro-magnetic
coupling is thus the same in both cases as already discussed in Ref. 37.
Further discussion can be found in the SI, where it is also shown how to,
instead, use terminals 1 and 2 as voltage probes to bypass the difficulties of
lossy leads in the "supercurrent" detection.

##  6 Summary and conclusions.

In summary, we have proposed two linear transport experiments that can detect
the intrinsic handedness of chiral systems. First, exact symmetry arguments
were provided to show that a voltage probe can become a chirality probe in the
presence of a in-plane magnetic field for external leads that couple
symmetrically to both layers. Secondly, we also demonstrated that a current
probe discriminating between the two layers can become a chirality probe even
in the absence of a magnetic field. Different enantiomers can thus be
distinguished by minimal transport experiments that will hopefully shed more
light on this intriguing symmetry, also present in various organic molecules.

Using the Landauer-Büttiker formalism, the voltage reading of the third lead
and the layer-discriminating, opposite currents, between system and third
reservoir(s) were explicitly calculated for a finite sample of twisted bilayer
graphene (TBG), confirming our predictions. In both situations, when
approaching the magic angle, there is change of chirality measured through
electronic means (voltage and current probes) that does not correspond to an
inversion of the actual twist angle between the layers.40 In the case of
layer-discriminating leads, we showed that, in the presence of chirality, a
net current flow between source and drain reservoirs is accompanied by
transverse currents, opposite in each layer. This provides a realization in a
Landauer-Büttiker scenario, more accessible experimentally, of predictions
previously made37, 38, 40 on the basis of linear response for infinite
systems, that in-plane magnetic moments should accompany net current flows in
chiral bilayers.

Our one-particle formalism does not allow for symmetry broken ground-states.
However, for a time-reversal symmetry (TRS) broken ground-state $|GS\rangle$
we expect an alternative chiral reciprocity relation even in the absence of a
magnetic field, involving only the ground-state
$G_{pq}(\theta,|GS\rangle)=G_{pq}(-\theta,\mathcal{T}|GS\rangle)$ where
$\mathcal{T}$ denotes a the anti-unitary time-reversal operator. This relation
allows for the detection of a TRS broken ground-state by measuring a non-zero
voltage at the third lead at ${\bf B}=0$.

Our discussion should also be relevant for other chiral systems, e.g., for
those displaying the planar Hall effect.65, 66 Then, the continuous variable
$\theta$ denoting the twist angle is simply replaced by a discrete variable
$\chi=\pm 1$ denoting the different enantiomers. Even the emergence of a magic
angle seems to be more general since it can be observed in Weyl semimetals,
too.67 Let us finally note that the third lead can also be realized by a local
probe, especially in the layer-discriminating case. Real-space mapping of the
handedness and thus small angle-deviations around the magic angle should be
detectable.

This work has been supported by project nos. PGC2018-096955-B-C42,
PID2020-113164GB-i00, and CEX2018-000805-M financed by MCIN/ AEI/
10.13039/501100011033 as well as by the CSIC Research Platform on Quantum
Technologies PTI-001. DAB acknowledges support from the Brazilian Nanocarbon
Institute of Science and Technology (INCT/Nanocarbon), CAPES-PRINT (grants
nos. 88887.310281/2018-00 and 88887.899997$/$2023-00), CNPq (309835/2021-6)
and Mackpesquisa. DAB is also grateful to the hospitality of the ICMM where
this work was completed.

Supplementary Information

##  7 Green´s functions and tight binding method

Employing the Green’s functions we have
$G_{pq}=\frac{2e^{2}}{h}\text{Tr}[\Gamma_{p}\mathcal{G}\Gamma_{q}\mathcal{G}^{\dagger}]$,
where $\mathcal{G}=[E-H_{TBG}-\Sigma_{1}-\Sigma_{2}-\Sigma_{3}]^{-1}$ is the
Green’s function of the central region, $\Sigma_{1(2)(3)}$ are the self-
energies of the leads and
$\Gamma_{1(2)(3)}=i[\Sigma_{1(2)(3)}-\Sigma_{1(2)(3)}^{\dagger}]$ are the
couplings of the central region to the leads. Without loss of generality we
can assume a wide band model for the leads, that is a constant density of
states (DOS) around the Fermi energy, which is typical in metallic contacts.
In that case, the self-energy term can be written as
$\Sigma_{1(2)(3)}=-i\pi\rho|t|^{2}$,54 were $\rho$ is de DOS of the contact
and $t$ is the hopping parameter between the leads and the central twisted
region. To guarantee a large number of injected modes we set $\rho_{L(R)}=(\pi
t)^{-1}$ were $t$ is the nearest neighbor hopping of graphene. 62

The Hamiltonian of the central region ($H_{TBG}$) is described by a tight-
binding model where the hopping amplitudes between sites $i$ and $j$,
$t_{ij}(d_{ij})=V_{pp\sigma}(d_{ij})\cos^{2}(\phi)+V_{pp\pi}(d_{ij})\sin^{2}(\phi)$,
where the bond length $d_{ij}=|{\bm{d}}_{ij}|=|{\bm{R}}_{j}-{\bm{R}}_{i}|$ and
$\phi$ denotes the angle formed by ${\bm{d}}_{ij}$ and the z-axis. The value
of the inter-atomic matrix elements is a function of the bond length:68, 27
$V_{pp\sigma}=V_{pp\sigma}^{0}e^{-\frac{d_{ij}-d_{0}}{\delta}},~{}V_{pp\pi}=V_{pp\pi}^{0}e^{-\frac{d_{ij}-a}{\delta}}$
where $V_{pp\sigma}^{0}=t_{\perp}^{0}=0.48~{}\text{eV}$,
$V_{pp\pi}^{0}=t_{0}=-2.7~{}\text{eV}$, $a=0.142~{}\text{nm}$,
$d_{0}=0.335~{}\text{nm}$ and $\delta=0.184\sqrt{3}a$. To accurately describe
the electronic properties of TBG, for each site $i$, the neighbours $j$ are
chosen inside a disc of radius $d_{ij}\leq 4a$. To include the effect of the
in-plane magnetic field
${\bm{B}}=B\left(\cos\theta_{B}{\bm{e}}_{x}+\sin\theta_{B}{\bm{e}}_{y}\right)$
in the tight-binding Hamiltonian, we use the Peierls substitution where the
hopping parameters are modified to $t_{ij}=t_{ij}e^{i\phi_{ij}}$ where
$\phi_{ij}=ie{\bm{A}}\cdot\left({\bm{R}}_{i}-{\bm{R}}_{j}\right)/\hbar$. In
this expression the vector potential
${\bm{A}}=zB\left(\sin\theta_{B}{\bm{e}}_{x}-\cos\theta_{B}{\bm{e}}_{y}\right)$
is evaluated at $\left({\bm{R}}_{i}+{\bm{R}}_{j}\right)/2$. Furthermore, we
use a symmetric arrangement where top and bottom layer are located at $z=\pm
d/2$ being $d=0.335$ nm the interlayer distance.

Figure 5:  (Color online) Illustration of field dependence and symmetry
properties of conductance matrix entries. (a) Conductance $G_{21}(\pm
1.16^{\circ},B)$ in units of $2e^{2}/h$ as a function of the Fermi energy
$E_{F}$, showing the effect of the in-plane magnetic field
${\bm{B}}=B{\bm{e}}_{x}$. (b) The reciprocity relations,
$G_{21}(+{\bm{B}})=G_{12}(-{\bm{B}})$. (c)
$G_{21}(+\theta,{\bm{B}}={\bm{0}})=G_{21}(-\theta,{\bm{B}}={\bm{0}})$. (d)-(f)
The chiral reciprocity relations,
$G_{pq}(\theta,{\bm{B}})=G_{pq}(-\theta,-{\bm{B}})$.

###  7.1 Numerical validation of reciprocity relations

In the main text, we derived the standard and chiral reciprocity relation

$\displaystyle G_{pq}(\theta,{\bm{B}})=G_{qp}(\theta,-{\bm{B}})\;,$ (11)
$\displaystyle G_{pq}(\theta,{\bm{B}})=G_{pq}(-\theta,-{\bm{B}})\;.$ (12)

We will now numerically validate the reciprocity relations Eqs. (11) and (12)
in order to show the reliability of our numerical approach. To this end, we
distinguish the effect of an in-plane magnetic field on the conductance and
then change twist angle and the field orientation. The black line in Fig. 5(a)
corresponds to $G_{21}$ for a TBG junction with $\theta=+1.16^{\circ}$ without
magnetic field, a peak originated by Van Hove singularities around the CNP is
clearly appreciated. 69, 70 When the magnetic field is switched on along the
$+x$-direction, the effect is mainly observed for low energies as a splitting
of the conductance peak. This is produced by the separation of the Dirac
points 71, 72, 73 as function of the strength and orientation of the field,
that when projected onto the transport direction, appear at different energies
(momentum) of the incoming electrons. Inverting the direction of the field,
i.e., the field pointing to the negative $x$-direction, we observe that the
traditional reciprocity relations, Eq. (11), are fulfilled. In Fig. 5(b), we
only present the case
$G_{21}(+1.16^{\circ},+20~{}T)=G_{12}(+1.16^{\circ},-20~{}T)$ to keep the
discussion more transparent.

Without a magnetic field, it is not possible to detect the handedness of the
TBG junction for layer-symmetric leads. To check this numerically, we again
select the specific case
$G_{21}(+1.16^{\circ},0~{}T)=G_{12}(-1.16^{\circ},0~{}T)$ in Fig. 5(c).
However, this relation is no longer valid if an in-plane magnetic field is
present as already anticipated. In Figs. 5(d)-(f), we recognize that the
chiral reciprocity relations, Eq. (12), are fulfilled, i.e.,
$G_{21}(+1.16^{\circ},+20~{}T)=G_{21}(-1.16^{\circ},-20~{}T)$,
$G_{31}(+1.16^{\circ},+20~{}T)=G_{31}(-1.16^{\circ},-20~{}T)$ and
$G_{23}(+1.16^{\circ},+20~{}T)=G_{23}(-1.16^{\circ},-20~{}T)$. Although, in
Fig. 5 we only focus on specific twist angles ($\theta=\pm 1.16^{\circ}$) and
fields (${\bm{B}}=\pm 20{\bm{e}}_{x}~{}T$) the reciprocity relations, Eqs.
(11) and (12), are valid irrespective of the twist angle and field intensity.

Figure 6:  (Color online) (a)-(c) Density of states for different twist angles
without magnetic field. (d)-(f) $V_{3}/V$ for the angles indicated in the DOS.

##  8 Magic angle regime and density of states

In the main text, we presented the highest value and the full width half
maximum (FWHM) of the density of states (DOS). Based on that, three regimes
were identified and we labeled the second regime as magic angle or flat band
regime as it displayed the largest peaks and the smallest FWHMs. The twist
angles around $1.23^{\circ}$ are slightly larger than the one found in
experiments of $\sim 1.1^{\circ}$ due to the finite system size considered
here.

The DOS was calculated by summing the diagonal elements of the spectral
function

$\displaystyle\rho(E)=\frac{1}{2\pi}\text{Tr}[A],$ (13)

which is defined as

$\displaystyle
A=\mathcal{G}^{R}(\Gamma_{1}+\Gamma_{2}+\Gamma_{3})\mathcal{G}^{A}.$ (14)

The couplings
$\Gamma_{1(2)(3)}=i[\Sigma_{1(2)(3)}-\Sigma_{1(2)(3)}^{\dagger}]$ and the
retarded Green’s function
$\mathcal{G}=[E-H_{TBG}-\Sigma_{1}-\Sigma_{2}-\Sigma_{3}]^{-1}$ were defined
in the main text. Note that we neglect the inclusion of an infinitesimal
complex value in the retarded Green’s to avoid any possible effect in the
chiral response of the system.52 This fact has no effect on the transport
properties because the self-energies ($\Sigma_{i}$) have their own imaginary
part. However, they are non-zero only for sites in the vicinity of the leads.
As a result, the peaks of the spectral function and consequently the DOS are
not perfectly smoothed around the CNP where a large number of states is
present.54

In Fig. 6, we present the full DOS and also the voltage probe $V_{3}$ for a
large range of twist angles between $\theta=1.47^{\circ}-1.10^{\circ}$. The
left panel shows the results for twist angles $\theta>1.26^{\circ}$ where
negative slopes in the linear regime are found for positive magnetic fields.
The central panel presents the DOS and $V_{3}$ for twist angles around the
magic angles ($1.20^{\circ}\leq\theta\leq 1.26^{\circ}$); in this regime, no
definite slope can be assigned to the voltage probe $V_{3}$ around $B\approx
0$. Finally, the right panel shows the same information for twist angles
$\theta<1.20^{\circ}$, where the slope is reversed and thus positive in the
linear regime.

This result show that there is a change of chirality around the magic angle as
found in the extended system.40 The observed voltage probe is thus an
indication of an electronic chirality which is not directly linked with the
real space chirality which does not change for positive twist angles.

Figure 7:  (Color online) (a)-(c) $V_{3}$ in units of $V$ for the positive and
negative twist angles with $E_{F}=0$ and $\bm{B}$ parallel to the $x$-axis.
(d) Angular dependence of $V_{3}$, where $\theta_{B}$ is the angle between the
in-plane magnetic field and the $x$-axis, for $\theta=\pm 1.16^{\circ}$ and
$|{\bm{B}}|=1$ T. (e) Effect of the Fermi energy on $V_{3}$ for
$\theta=1.20^{\circ}$. (f) $V_{3}$ changing the coupling between the central
region and lead 3 ($t_{3D}=0.1t$).

##  9 Dependence of the voltage probe on magnetic field direction and coupling

Let us now discuss further results, i.e., the dependence of $V_{3}$ on
negative magnetic fields, on the angle of the magnetic field with respect to
the symmetry axis, on a backgate, as well as on the coupling to lead 3. In all
cases we choose a symmetric source-drain voltage drop,$V_{1}=-V_{2}=V/2$.

###  9.1 Behavior for negative magnetic fields

The symmetry arguments of the main text indicate that the relation,
$V_{3}(+{\bm{B}},\theta)=-V_{3}(-{\bm{B}},\theta)$, is valid to linear order
in $\bm{B}$, in principle. Yet, the results seen in Figs. 7(a)-(c) point to an
exact relation. This is true because, in those Figs., the field is along the
$x$ direction of Fig. 1 of the main text, where an exact symmetry of our
geometric setup holds, as explained later when considering non-linear effects.
It is also obviously that the behavior is independent of whether the twist
angle is above, around, or below the twist angle.

###  9.2 Angle-dependence of the magnetic field on the conductance

For the same reason, we have $V_{3}(+\theta,\bm{B})=-V_{3}(-\theta,\bm{B})$
only in the linear field regime, in principle. As before if the magnetic field
is along the $x$ direction, this relation also holds in the non-linear regime.
For arbitrary directions, however, this symmetry is slightly broken as seen in
Fig. 7(d) that shows the angular dependence of $V_{3}$. In this Figure,
$\theta_{B}$ denotes the angle between the in-plane magnetic field and the
$x$-axis, for $\theta=\pm 1.16^{\circ}$ and $|{\bm{B}}|=1$ T. $V_{3}$ is thus
not symmetric with respect to $\theta_{B}=\pi$ and the phenomenological
correction term ${\bm{I}}\cdot{\bm{B}}=|{\bm{I}}||{\bm{B}}|\cos\theta_{B}$
that was discussed in a different context for macroscopic devices, see e.g.
Refs. 19, 1, 14, does not match our results.

###  9.3 Gate dependence

All results so far were obtained for a neutral twisted graphene sample. In Fig
7(e), we show results also for Fermi energies $E_{F}=\pm 1.2$ and $E_{F}=\pm
2.4$ meV. As can be seen, the linear regime becomes more evident for
$\theta=1.20^{\circ}$ at finite gate voltage. For $E_{F}=\pm$1.2meV, we also
have $V_{3}(E_{F})=-V_{3}(-E_{F})$. This might have been expected as also the
chiral part of the conductance is an odd function of the chemical potential.40

However, this relation is not valid for $E_{F}=\pm 2.4$ meV. We believe this
to be a finite size effect as also the DOS of the whole system as function of
the twist angle changes, but more detailed simulations are needed in order to
understand the gate dependence of the voltage probe.

###  9.4 Chiral engineering

Let us investigate the possibility of enhancing the chiral probe in the linear
field regime as suggested by Eq. (9) of the main text. In Fig. 7(f), we show
$V_{3}$ for $\theta=1.20^{\circ}$ with a reduced coupling between the third
lead and the central region ($t_{3D}=0.1t$). As anticipated, the reduction of
the coupling enhances the linear effect showing a possible route to engineer
the chiral response.

##  10 Reciprocity relations for the two-terminal conductance

A dependence of the conductivity on the direction of a magnetic field is
forbidden by the general Onsager relations. These state that for any linear
response function $\mathcal{K}_{AB}$ one has
$\mathcal{K}_{A,B}({\bm{B}})=\mathcal{K}_{\bar{B},\bar{A}}(-{\bm{B}})$ with
$A,B$ arbitrary operators and $\bar{A},\bar{B}$ the time-reversed operators.
For $A=B=j_{x}$, we have $\bar{A}=\bar{B}=-j_{x}$ and
$\mathcal{K}_{j_{x},j_{x}}({\bm{B}})=\mathcal{K}_{-j_{x},-j_{x}}(-{\bm{B}})=\mathcal{K}_{j_{x},j_{x}}(-{\bm{B}})$.

This result also holds within the Landauer formalism in a two-terminal setup,
because the unitarity of the S matrix imposes left-right reciprocity for any
system attached to $M_{T}$ modes, grouped into left and right leads,
irrespective of any symmetry (or lack of) of the Hamiltonian. To show this, we
first discuss the following relation for the S matrix:

$\displaystyle\sum_{m=1}^{M_{T}}|s_{mn}|^{2}=1\;\;\;,\forall n\;,$ (15)

see Datta (3.1.3b), where $m(n)$ run over all channels attached to our system.
Using that $|s_{mn}|^{2}=|s_{m\leftarrow n}|^{2}=T_{m\leftarrow n}$ (Datta
(3.1.1)) and grouping these channels into left(L) and right(R) leads, one has

$\displaystyle\overline{T}_{LL}+\overline{T}_{R\leftarrow L}$
$\displaystyle=N_{L}\;,$ (16)
$\displaystyle\overline{T}_{RR}+\overline{T}_{L\leftarrow R}$
$\displaystyle=N_{R}\;.$ (17)

where $N_{L(R)}$ is the number of channels in the $L(R)$ lead, with
$N_{R}+N_{L}=M_{T}$ , and

$\displaystyle\overline{T}_{R\leftarrow L}=\sum_{m\in R}\sum_{n\in
L}T_{m\leftarrow n}\;,$ (18)

with similar relations for the other pairings, see Datta (3.1.2).

Unitarity also implies the "not so obvious relation" in Datta’s own words, see
Datta (3.1.3b) again,

$\displaystyle\sum_{m=1}^{M_{T}}|s_{nm}|^{2}=1\;\;\;,\forall n\;,$ (19)

which leads to

$\displaystyle\overline{T}_{LL}+\overline{T}_{L\leftarrow R}$
$\displaystyle=N_{L}\;,$ (20)
$\displaystyle\overline{T}_{RR}+\overline{T}_{R\leftarrow L}$
$\displaystyle=N_{R}\;.$ (21)

The combination of Eqs. (16) and (20) implies reciprocity:

$\displaystyle\overline{T}_{R\leftarrow L}=\overline{T}_{L\leftarrow R}\;,$
(22)

irrespective of any other consideration, except unitarity, always given in the
non-dissipative, linear regime.

##  11 Relations beyond linear order

Let us finally discuss extensions to the linear approach presented in the main
text. Quite generally, one can always write

$\begin{bmatrix}I_{1}\\\ I_{2}\\\
I_{3}\end{bmatrix}=(\bm{M}^{s}+\bm{M}^{a})\begin{bmatrix}V_{1}\\\ V_{2}\\\
V_{3}\end{bmatrix}\;,$ (23)

where $\bm{M}^{s}$ is a symmetric matrix, even in chirality and field, which
we can denote as $\bm{M}^{s}(\theta^{2},\bm{B}^{2})$. $\bm{M}^{a}$ is an
asymmetric matrix that, from unitarity, depends on a single entry

$\bm{M}^{a}=\begin{bmatrix}0&-\delta G&+\delta G\\\ +\delta G&0&-\delta G\\\
-\delta G&+\delta G&0\end{bmatrix},$ (24)

with

$\delta
G=\frac{G_{12}-G_{21}}{2}=-\frac{G_{13}-G_{31}}{2}=\frac{G_{23}-G_{32}}{2},$
(25)

where now, $\delta G$ is odd in chirality and field, which we can describe as

$\delta G(\theta,\bm{B})=\sigma\sigma^{\prime}\delta
G(\sigma\theta,\sigma^{\prime}\bm{B}),$ (26)

with $\sigma(\sigma^{\prime})=\pm 1$. In the main text, we restricted $\delta
G$ to lowest (linear) order in $\bm{B}$, but it can be any odd function of
field and chirality, generalizing Eq. (9) of the main text.

The voltage probe in the case of $I_{3}=0$ can be written as

$V_{3}=\frac{V_{1}+V_{2}}{2}+\delta\tilde{V}_{3}\frac{V_{1}-V_{2}}{2},$ (27)

with

$\delta\tilde{V}_{3}=\delta\tilde{V}^{e}_{3}+\delta\tilde{V}^{o}_{3},$ (28)

where $\delta\tilde{V}^{e}=\delta\tilde{V}^{e}(\theta^{2},\bm{B}^{2})$ is an
even function of field and chirality, given explicitly by

$\delta\tilde{V}^{e}_{3}=\frac{(G_{31}+G_{13})-(G_{32}+G_{23})}{2(G_{31}+G_{32})}\;.$
(29)

It vanishes for $\bm{B}=0$ in our left-right symmetrically chosen attachment
for the third lead as $G_{31}=G_{32}$.

On the other hand, $\delta\tilde{V}^{o}$, with expression

$\delta\tilde{V}^{o}_{3}=\frac{(G_{31}-G_{13})-(G_{32}-G_{23})}{2(G_{31}+G_{32})}=\frac{\delta
G}{(G_{31}+G_{32})},$ (30)

is an odd function of field and chirality, formally:

$\delta\tilde{V}^{o}(\theta,\bm{B})=\sigma\sigma^{\prime}\tilde{V}^{o}(\sigma\theta,\sigma^{\prime}\bm{B}),$
(31)

with $\sigma(\sigma^{\prime})=\pm 1$. Notice that to linear order in $\bm{B}$,
one has
$\delta\tilde{V}_{3}(\theta,\bm{B})=-\delta\tilde{V}_{3}(-\theta,\bm{B})$, as
asserted in the main text. This relation needs not hold beyond linear order,
but, nevertheless, this does not compromise the chiral sensitivity of the
voltage probe for, in general,
$\delta\tilde{V}_{3}(\theta,\bm{B})\neq\delta\tilde{V}_{3}(-\theta,\bm{B})$.

In spite of the above generalizations, our numerical calculations show that,
when the field is in the $x$-direction, one always has

$\delta\tilde{V}_{3}(\theta,B)=-\delta\tilde{V}_{3}(-\theta,B),\;\;\bm{B}=B{\bm{e}}_{x}\;,$
(32)

as seen e.g. in Fig. 2 a,b,c. This is no accident, because in our geometry, a
$\pi$ rotation around the $y$-axis reverses $B$ and exchanges leads 1 and 2,
leading necessarily to Eq. (32) as an exact statement - beyond linear order.
This symmetry is also responsible for the vanishing of $\delta\tilde{V}_{3}$
when the field is oriented along the $y$-direction, see Fig. 2 d.

As said, Eq. (32) no longer holds beyond linear order when the field is
oriented in an arbitrary direction, as illustrated in Fig. 2 d. Nevertheless,
non-linear effects are rather small, at least for $B=1T$, as can be seen in
that figure. Moreover, as previously asserted, the chiral sensitivity of the
probe remains such that
$\delta\tilde{V}_{3}(\theta,\bm{B})\neq\delta\tilde{V}_{3}(-\theta,\bm{B})$.

##  12 Landauer-Büttiker description of the layer-discriminating setup without
magnetic field

Now we have four leads: 1, 2, 3t, and 3b. Therefore,

$\left(\begin{matrix}I_{1}\\\ I_{2}\\\ I_{3t}\\\
I_{3b}\end{matrix}\right)=\left(\begin{matrix}G_{21}+G_{3t1}+G_{3b1}&-G_{21}&-G_{3t1}&-G_{3b1}\\\
-G_{21}&G_{21}+G_{3t1}+G_{3b1}&-G_{3b1}&-G_{3t1}\\\
-G_{3t1}&-G_{3b1}&G_{3b3t}+G_{3t1}+G_{3b1}&-G_{3b3t}\\\
-G_{3b1}&-G_{3t1}&-G_{3b3t}&G_{3b3t}+G_{3t1}+G_{3b1}\end{matrix}\right)\left(\begin{matrix}V_{1}\\\
V_{2}\\\ V_{3t}\\\ V_{3b}\end{matrix}\right)\;,$ (33)

where we have made use of the reciprocity in the absence of magnetic field,
$G_{ij}=G_{ji}$, and the additional symmetry of our geometric arrangement,
$G_{3t\,2}=G_{3b1}$ and $G_{3t1}=G_{3b2}$.

###  12.1 Detecting chirality with transverse currents: layer contrasting Hall
effect

If the terminals $3t$ and $3b$ are kept at the same potential,
$V_{3t}=V_{3b}$, then

$\frac{I_{3b}-I_{3t}}{V_{1}-V_{2}}=G_{3t1}-G_{3b1},$ (34)

as asserted in the main text, with a finite value of this magnitude implying
chirality.

For $V_{1}=V/2$, $V_{2}=-V/2$, and $V_{3b}=V_{3t}=0$, in addition to the
standard source-drain current between 1 and 2 reservoirs, one gets transverse
currents,

$I_{3b}+I_{3t}=0,$ (35)

and, therefore, no net current flows from the system to reservoirs $3t+3b$,
kept at the same potential $V_{3}=0$. Yet a finite, $|I_{3t}|=|I_{3b}|$, and
layer-opposite current in the transverse leads emerges due to chirality, as
depicted in Fig. 3 (a) and (b) of the main text.

The experimental detection of these transverse currents could be problematic,
for real leads are always resistive. Nevertheless, the same physics can be
exposed with an alternative measurement, where the transverse leads are used
as independent voltage probes, and chirality manifests as different voltage
readings, $V_{3t}\neq V_{3b}$. This voltage difference can be interpreted
physically as the chemical potential difference due to the carrier
accumulation-depletion of the "frustrated supercurrent" at the edges. The new
conditions would be $V_{1}=V/2$, $V_{2}=-V/2$, and $I_{3b}=I_{3t}=0$. Eq. 33
then gives

$V_{3t}-V_{3b}=\frac{G_{3t1}-G_{3b1}}{G_{3t1}+G_{3b1}+2G_{3b3t}}V.$ (36)

Notice that the denominator of Eq. 36 is even in chirality and, therefore, a
finite value of $(G_{3t\,1}-G_{3b\,1})$, opposite for opposite chiralities
$(\pm\theta)$, reveals the chirality, as in the case of current detection.

###  12.2 Complementary setup for chirality detection

Guided by results for the conductivity in an infinite system37, 38, in the
main text, we discussed a complementary setup based on the same symmetry
properties. It consists of forcing the opposite transverse currents by means
of a voltage drop in the corresponding reservoirs,
$V_{3t}=+V/2,\;V_{3b}=-V/2$, and then observing as chiral probe the emergence
of a net current between reservoirs 1 and 2, without any voltage drop between
the terminals, $V_{1}=V_{2}=0$, a "supercurrent" in some sense, see Fig. 3 (c)
- (d) of main text. Under the specified conditions,
$V_{3t}=+V/2,\;V_{3b}=-V/2,\,V_{1}=0,\,V_{2}=0$, Eq. 33 leads to

$I_{1}=-I_{2}=(G_{3b\,1}-G_{3t\,1})V/2,$ (37)

that is, the appearance of a net longitudinal current between reservoirs 1 and
2. To avoid potential misunderstandings, notice that the minus sign of $I_{2}$
in Eq. 37 comes from usual convention in the Büttiker formalism that a
positive $I_{i}$ means current injected from reservoir $i$ to the system, and
viceversa. Therefore, Eq. 37 describes the same amount of current entering to
the flake from reservoir 1 and exiting to reservoir 2, again, without any
voltage drop between them, $V_{1}=V_{2}=0$.

As in the original arrangement of the manuscript, the effect requires
chirality manifested as a finite value of $(G_{3t\,1}-G_{3b\,1})$, opposite
for opposite chiralities $(\pm\theta)$. The value of $(G_{3t\,1}-G_{3b\,1})$
as function of Fermi level in twisted bilayer graphene was plotted in the main
manuscript.

As before, experimental detection of this "supercurrent" with real resistive
leads can be problematic. We can then opt for using leads 1 and 2 as voltage
probes, as in the previous case. The new conditions would be $V_{3t}=+V/2$,
$V_{3b}=-V/2$, and $I_{1}=I_{2}=0$. Eq. 33 then gives

$V_{1}-V_{2}=\frac{G_{3t1}-G_{3b1}}{G_{3t1}+G_{3b1}}\,V.$ (38)

As in Eq. 36, the denominator of Eq. 38 is even in chirality and, once again,
it is the finite value of $(G_{3t\,1}-G_{3b\,1})$, opposite for opposite
chiralities $(\pm\theta)$, what reveals the chirality.

##  13 Change in chirality in the infinite system with time-reversal symmetry

Figure 8:  (Color online) Left panel: The chiral Drude weight for large twist
angles. Right panel: The chiral Drude weight below and above the magic angle
$\theta_{m}=1.08^{\circ}$.

In Ref. 40, it was shown that electronic chirality is a Fermi-surface property
and given by the chiral Drude weight

$\displaystyle
D_{xy}=\frac{1}{2A}\sum_{\bm{k},n}\bm{e}_{z}\cdot(\bm{j}_{\bm{k},n}^{1}\times\bm{j}_{\bm{k},n}^{2})\delta(\epsilon_{\bm{k},n}-E_{F})\;.$
(39)

It is defined by the sheet currents of layer $\ell$ denoted by
$\bm{j}_{\bm{k},n}^{\ell}$ and sum over all states labeled by the band index
$n$ and Bloch states $\bm{k}$ with energies around the Fermi energy $E_{F}$.
Furthermore, $A$ denotes the area of the sample and the eigenenergies
$\epsilon_{\bm{k},n}$ are given in units of the carbon-carbon hopping
amplitude $t\sim 2.7$eV.

The continuum model of twisted bilayer graphene shows an approximate particle-
hole symmetry. Then the chiral Drude weight is in an odd function of the Fermi
energy with $D_{xy}(E_{F})=-D_{xy}(-E_{F})$. The change in chirality at the
neutrality point can be seen on the left and right panel of Fig. 8. However,
for large angles the chirality is always positive for $E_{F}<0$. This changes
when the twist angle is below the magic angle $\theta_{m}=1.08^{\circ}$ and
the chirality for e.g. $\theta=0.96^{\circ}$ is negative for $E_{F}<0$.

Interestingly, the shape of the curves is similar to the one of the right
panel for positive twist angle of Fig. 4 of the main text. In contrary to the
case of the infinite system, these curves do not change around zero due to a
residual chirality which is due to finite size effects and not related to the
band structure.

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