# A Generic Topological Criterion for Flat Bands in Two Dimensions

Alireza Parhizkar  Joint Quantum Institute, University of Maryland, College
Park, MD 20742, USA  Victor Galitski  Joint Quantum Institute, University of
Maryland, College Park, MD 20742, USA  Center for Computational Quantum
Physics, The Flatiron Institute, New York, NY 10010, United States

###### Abstract

Mutually distorted layers of graphene give rise to a moiré pattern and a
variety of non-trivial phenomena. We show that the continuum limit of this
class of models is equivalent to a $(2+1)$-dimensional field theory of Dirac
fermions coupled to two classical gauge fields. We further show that the
existence of a flat band implies an effective dimensional reduction in the
field theory, where the time dimension is “removed.” The resulting two-
dimensional Euclidean theory contains the chiral anomaly. The associated
Atiyah-Singer index theorem provides a self-consistency condition for the
existence of flat bands. In particular, it reproduces a series of quantized
magic angles known to exist in twisted bilayer graphene in the chiral limit
where there is a particle-hole symmetry. We also use this criterion to prove
that an external magnetic field splits this series into pairs of magnetic
field-dependent magic angles associated with flat moiré-Landau bands. The
topological criterion we derive provides a generic practical method for
finding flat bands in a variety of material systems including but not limited
to moiré bilayers.

Moiré phenomena in twisted bilayer graphene [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
and other systems [11, 12] have been an active topic of research in recent
years. Of particular interest for the former are flat bands, where localized
electrons exhibit a variety of strongly-correlated phases. In this Letter we
draw a universal picture for the occurrence of the flat bands through emergent
gauge fields and the anomaly.

Consider a bilayer graphene system with a small twist and/or strain applied to
the layers. As emphasized in Ref. [11], both can be described in terms of a
diffeomorphism:
$\bm{\mathbf{\xi}}_{\omega}\equiv\epsilon\omega(\mathbf{r})\mathbf{r}$ for
strain and
$\bm{\mathbf{\xi}}_{\theta}\equiv\epsilon\theta(\mathbf{r})\hat{z}\times\mathbf{r}$
for a twist (parameterized by $\epsilon\ll 1$). With
$\bm{\mathbf{\xi}}_{\omega}$ and $\bm{\mathbf{\xi}}_{\theta}$ being orthogonal
to each other and thus forming a basis, all possible flows in two dimensions
can be described as a linear combination of these two fields. Therefore, a
bilayer system with a general infinitesimal deformation can be achieved by
applying the diffeomorphism
$\mathbf{r}\rightarrow\mathbf{r}+\bm{\mathbf{\xi}}_{\omega,\theta}/2$ to one
layer and
$\mathbf{r}\rightarrow\mathbf{r}-\bm{\mathbf{\xi}}_{\omega,\theta}/2$ to the
other, as in Fig. 1. A finite deformation is reached by successively applying
such diffeomorphisms. The time dependent moiré fields $\omega(t,\mathbf{r})$
and $\theta(t,\mathbf{r})$ can be further looked at as bosonic phonon-like
degrees of freedom for the bilayer system.

Figure 1: Each blue hexagon designates the Brillouin zone of a single layer of
graphene that has gone under either a strain, $\bm{\mathbf{\xi}}_{\omega}$, or
a twist, $\bm{\mathbf{\xi}}_{\theta}$. In (a) the Brillouin zones are
expanded/shrunken with respect to the undeformed Brillouin zone (green dotted
hexagon), where as in (b) they are rotated with respect to each other. Each
pair gives rise to its moiré reciprocal lattice (shown by red hexagonals).
Also $K$ points are designated by black dots.

Focusing on only one valley, $K$, in the Brillouin zone, an electron that hops
from the $K$-point of one layer to that of the other, can do so along three
different momentum vectors that will form the moiré reciprocal lattice vectors
of the bilayer system. This is because the deformation has separated the
equivalent $K$-points in three different ways (along
$\mathbf{q}^{\theta}_{1,2,3}$ for twist and $\mathbf{q}^{\omega}_{1,2,3}$ for
strain as depicted in Fig. 1) that are related to each other by a $2\pi/3$
rotation. For the general deformation these vectors are given by
$\mathbf{q_{1}}\equiv
2K_{D}(\sinh\frac{\omega}{2}\hat{x}-\sin\frac{\theta}{2}\hat{y})$ with
$\mathbf{q}_{2,3}$ derived by successive $2\pi/3$ rotations of
$\mathbf{q}_{1}$, while $K_{D}$ is the distance between $K$-points of the
undeformed Brillouin zone and its center.

The characteristic length of the superstructure, i.e. the moiré lattice
constant, $L$, can be read off of the magnitude of the moiré reciprocal
lattice vectors,
$|\mathbf{q}|=2K_{D}(\sinh^{2}\frac{\omega}{2}+\sin^{2}\frac{\theta}{2})^{1/2}$,
which is equal to the distance between the corresponding $K$-points (denoted
by $k$ and $k^{\prime}$ in Fig. 1): $L=4\pi/3|\mathbf{q}|$. If the electron
hops from one layer to the other without changing its position, it acquires a
phase determined only by the moiré reciprocal lattice vectors. The dynamics of
this electron is given by the following Hamiltonian density [8],

$\displaystyle
h(\mathbf{r})\\!=\\!\\!\left[\\!\begin{array}[]{cc}-i\nu\not{\partial}&T(\mathbf{r})\\\
T^{\dagger}(\mathbf{r})&-i\nu\not{\partial}\end{array}\\!\right]\\!,\
T(\mathbf{r})\\!\equiv\\!\\!\left[\\!\begin{array}[]{cc}vV(\mathbf{r})&uU^{*}(-\mathbf{r})\\\
uU(\mathbf{r})&vV(\mathbf{r})\end{array}\\!\right]\\!,$ (5)

where $-i\nu\not{\partial}$ is the Hamiltonian density for a single layer of
graphene with $\not{\partial}\equiv\sigma^{i}\partial_{i}$, and
$T(\mathbf{r})$ is the interlayer tunneling matrix and encodes the periodic
profile of the moiré pattern, Fig. 3.
$V(x)=\sum_{j}e^{-i\mathbf{q}_{j}\cdot\mathbf{r}}$ takes the electron from one
sublattice (either $A$ or $B$) to the same sublattice on the other layer,
while $U(x)=\sum_{j}e^{-i\mathbf{q}_{j}\cdot\mathbf{r}}e^{i(j-1)2\pi/3}$ takes
it to the opposing sublattice, with $v$ and $u$ being the corresponding
tunneling amplitudes and $j=1,2,3$.

Using the algebra of gamma matrices,
$\\{\gamma^{\mu},\gamma^{\nu}\\}=2\eta^{\mu\nu}$, with $\eta_{\mu\nu}$ as the
metric, and the unitary transformation,

$\displaystyle h(\mathbf{r})\rightarrow\Omega h(\mathbf{r})\Omega^{\dagger}\ \
\ \text{with}\ \ \
\Omega=\frac{1}{\sqrt{2}}\left[\begin{array}[]{cc}-\mathds{1}&\mathds{1}\\\
\sigma^{z}&\sigma^{z}\end{array}\right]\,,$ (8)

we can write the Hamiltonian density (5) in terms of Dirac fermions coupled to
non-fluctuating gauge fields as follows

$\displaystyle
H=\int\\!d^{2}x\Big{[}\,\bar{\psi}i\nu\gamma^{a}\left(\partial_{a}+i\mathcal{A}_{a}\gamma_{5}+i\mathcal{S}_{a}i\gamma_{3}\right)\psi$
$\displaystyle-\bar{\psi}\gamma^{0}\mathcal{A}_{0}\gamma_{5}\psi-\bar{\psi}\gamma^{0}\mathcal{S}_{0}i\gamma_{3}\psi\Big{]}\,,$
(9)

with $\bar{\psi}\equiv\psi^{\dagger}\gamma^{0}$. Or in the even tidier action
formulation,

$S=\int\\!d^{3}x\,\bar{\psi}i\not{D}\psi,\quad\\!\not{D}\equiv\gamma^{\mu}\left(\partial_{\mu}+i\mathcal{A}_{\mu}\gamma_{5}+i\mathcal{S}_{\mu}i\gamma_{3}\right),$
(10)

where $a=1,2$, $\mu=0,1,2$ and the field components are,

$\displaystyle\mathcal{A}_{0}=-\frac{v}{\nu}\text{Re}[V(\mathbf{r})],\ \
\mathcal{S}_{0}=\frac{v}{\nu}\text{Im}[V(\mathbf{r})],$ (11)
$\displaystyle\mathcal{A}_{1}=\frac{u}{2\nu}\text{Re}[U(\mathbf{r})+U(-\mathbf{r})],\
\mathcal{A}_{2}=\frac{u}{2\nu}\text{Im}[U(\mathbf{r})+U(-\mathbf{r})],$
$\displaystyle\mathcal{S}_{1}=\frac{u}{2\nu}\text{Im}[U(\mathbf{r})-U(-\mathbf{r})],\
\mathcal{S}_{2}=\frac{u}{2\nu}\text{Re}[U(-\mathbf{r})-U(\mathbf{r})].$

The total field strength associated with $\not{D}$ is given by
$F_{\mu\nu}=\gamma_{5}F^{\mathcal{A}}_{\mu\nu}+i\gamma_{3}F^{\mathcal{S}}_{\mu\nu}+2i\gamma_{5}\gamma_{3}\mathcal{A}_{\mu}S_{\nu}$,
with $F_{\mu\nu}^{\mathcal{A},\mathcal{S}}$ being the field strengths
generated by $\mathcal{A}_{\mu}$ and $\mathcal{S}_{\mu}$ respectively.

Looking at the action (10), we see that the bilayer problem has transformed
into that of Dirac fermions moving in a $(2+1)$ dimensional spacetime and
acted upon by two axial-vector fields: A chiral gauge field,
$\mathcal{A}_{\mu}$, and a spin field, $\mathcal{S}_{\mu}$. Note that
$\gamma_{3}$ measures the spin along the direction normal to the material
plane. The gauge fields, (11), are periodic with periodicity $\sqrt{3}L$,
hence their corresponding field strengths are proportional to $1/L$ and also
periodic with the same period, while the distance between each minimum and its
neighboring maximum is $L$ (see Fig. 2 for example). In particular the spatial
part of the field strength
$F^{\mathcal{A}}_{12}=\partial_{2}\mathcal{A}_{1}-\partial_{1}\mathcal{A}_{2}$
is given by

$\displaystyle B(\mathbf{r})\equiv
F^{\mathcal{A}}_{12}=\frac{u}{\nu}\sum_{j}\hat{\mathbf{q}}^{\theta}_{j}\cdot\bm{\mathbf{\nabla}}(\mathbf{q}_{j}\cdot\mathbf{r})\,\sin\\!{\left(\mathbf{q}_{j}\cdot\mathbf{r}\right)}\,,$
(12)

where $\hat{\mathbf{q}}^{\theta}_{j}\cdot\bm{\mathbf{\nabla}}$ is the
derivative along the unit vector
$\hat{\mathbf{q}}^{\theta}_{j}\equiv\mathbf{q}^{\theta}_{j}/|q^{\theta}_{j}|$.
That of $\mathcal{S}_{a}$ is given by
$\sum_{j}\hat{\mathbf{q}}^{\theta}_{j}\cdot\bm{\mathbf{\nabla}}(\mathbf{q}_{j}\cdot\mathbf{r})\,\cos\\!{\left(\mathbf{q}_{j}\cdot\mathbf{r}\right)}$.

Figure 2: (a) The magnetic field $B$ created by $\mathcal{A}_{a}$ felt by
$\psi_{+}$ while its negative is exerted upon the $\psi_{-}$. (b) The field
strength associated with the spin field. (c) Vector fields $\mathcal{A}_{a}$
(blue) and $\mathcal{S}_{a}$ (red). The black line designates two magnetic
regions related by parity. Parallel sides are identified with each other and
the whole system can be reconstructed by sewing these together. The
$\mathcal{S}_{a}$ field is zero everywhere on the green dashed hexagon. (d)
The chiral scalar potential $\mathcal{A}_{0}$ as experienced by $\psi_{+}$ and
(e) that of $\mathcal{S}_{0}$. Except for (c) the fields are zero on the black
curves. Note how $\mathcal{S}_{0}$ coincides with the magnetic field in (a)
and also how $\mathcal{A}_{0}$ coincides with field strength in (b), in
particular $\mathcal{S}_{0}$ vanishes on edge of each magnetic region. Also
note that a fermion configuration localized on the edge of each magnetic
region will be perpendicular to $\mathcal{S}_{a}$ (see (c) for example).

Consider the following chiral transformations associated with action (10)

$\displaystyle\left\\{\\!\begin{array}[]{cc}\bar{\psi}\rightarrow\bar{\psi}e^{i\alpha\gamma_{5}}\\\
\psi\rightarrow e^{i\alpha\gamma_{5}}\psi\end{array}\\!\right\\}\\!,\
\left\\{\\!\begin{array}[]{cc}\bar{\psi}\rightarrow\bar{\psi}e^{i\alpha
i\gamma_{3}}\\\ \psi\rightarrow e^{i\alpha
i\gamma_{3}}\psi\end{array}\\!\right\\}\\!,\
\left\\{\\!\begin{array}[]{cc}\bar{\psi}\rightarrow\bar{\psi}e^{i\alpha\Gamma}\\\
\psi\rightarrow e^{i\alpha\Gamma}\psi\end{array}\\!\right\\}\\!,$ (19)

with $\Gamma\equiv\gamma_{0}\gamma_{3}\gamma_{5}$. Each one of the above
chiral transformations can become a symmetry of the action (10) under
additional constraints: $\psi\rightarrow e^{i\alpha\gamma_{5}}\psi$,
$\psi\rightarrow e^{i\alpha i\gamma_{3}}\psi$, and $\psi\rightarrow
e^{i\alpha\Gamma}\psi$ are promoted to symmetries if $\mathcal{S}_{\mu}=0$,
$\mathcal{A}_{\mu}=0$, and $\mathcal{A}_{0}=\mathcal{S}_{0}=0$ respectively.
Chiral particles can be defined with respect to each of these symmetries by
using projection operators, e.g.
$\psi_{\pm}\equiv\frac{1}{2}(1\pm\gamma_{5})\psi$. Since $\gamma_{5}$ and
$\gamma_{3}$ do not commute we cannot simultaneously create fermions with
definite $\gamma_{5}$ and $\gamma_{3}$ handedness, in contrast to $\Gamma$
which commutes with both. So we can, for example, have
$\psi_{{\uparrow\\!\downarrow}}^{\circlearrowright,\circlearrowleft}\equiv\frac{1}{4}(1\pm
i\gamma_{3})(1\pm\Gamma)\psi$.

Our interest here is focused, more than anything else, on flat bands, which
can be looked at as a class of modes covering the whole Brillouin zone at
constant energy,
$\bar{\psi}_{k}\gamma^{0}\partial_{0}\psi_{k}=\mu\psi^{\dagger}_{k}\psi_{k}$,
within which the electrons are therefore localized $\partial E_{k}/\partial
k=0$. If we only consider this class, we eliminate the time dependence from
the action entirely and reduce the $(2+1)$ dimensional theory to its $(2+0)$
dimensional version. Specifically, in the case of $v=0$, which supports exact
flat bands [4], we have

$I=\int\\!\mathcal{D}\bar{\psi}\psi\exp\\!\left\\{\int\\!d^{2}x\,\bar{\psi}i\gamma^{a}\left(\partial_{a}+i\mathcal{A}_{a}\gamma_{5}+i\mathcal{S}_{a}i\gamma_{3}\right)\psi\right\\}\\!.$
(20)

Using any of the chiral projections,
$\psi^{\circlearrowright,\circlearrowleft}_{\pm\ \text{or}\
{\uparrow\\!\downarrow}}$, we can break the above path-integral further down,
for instance, to

$\displaystyle
I=\int\\!\mathcal{D}\bar{\psi}_{\pm}\psi_{\pm}\exp\\!\Bigg{\\{}\int\\!d^{2}x\Big{[}\bar{\psi}_{\pm}i\gamma^{a}\left(\partial_{a}\pm
i\mathcal{A}_{a}\right)\psi_{\pm}$ $\displaystyle\quad\
+\bar{\psi}_{\mp}\gamma^{a}\epsilon_{ab}\mathcal{S}^{b}\psi_{\pm}\Big{]}\Bigg{\\}}\,,$
(21)

where the path-integral is over the four field variables $\bar{\psi}_{\pm}$
and $\psi_{\pm}$, while $\psi_{\pm}$ fermions are coupled to
$\pm\mathcal{A}_{a}$.

In this form the anomaly residing in the theory given by the path integral
(20) takes the familiar shape of the chiral anomaly in two dimensional
Euclideanized spacetime. In a path integral such as above the gauge field
$\mathcal{A}_{a}$ has an index associated with it that is directly given by
the chiral anomaly [13, 14]. But the index must be an integer number which as
we will see is only possible for certain values of $\theta$ and $\omega$ that
coincide with the magic angle. This consistency condition can therefore tell
us whether a flat band exists or not. Before proceeding to a more detailed
investigation, it is worth mentioning that this reasoning, following the
reduction from Eq. (10) to Eq. (21), is generalizable to other more
complicated systems such as multilayer graphene in which case we should use
higher dimensional gamma matrices to accommodate for the additional layers.

Since the gauge potentials are periodic the path integral can be divided into
equivalent patches sewn together by an integration over all field
configurations on the boundaries.

$\displaystyle
I=\int\prod_{\triangledown}\mathcal{D}\bar{\psi}_{\partial\triangledown}\psi_{\partial\triangledown}\,I_{\triangledown}\left[\bar{\psi}_{\partial\triangledown},\psi_{\partial\triangledown}\right]\,,$
(22) $\displaystyle\text{with}\quad
I_{\triangledown}\left[\bar{\psi}_{\partial\triangledown},\psi_{\partial\triangledown}\right]=\int_{\bar{\psi}_{\partial\triangledown},\psi_{\partial\triangledown}}\mathcal{D}\bar{\psi}_{\triangledown}\psi_{\triangledown}e^{iS_{\triangledown}}\,,$
(23)

where ${\partial\triangledown}$ designates the boundary of the patches and the
configuration residing on it, while $\triangledown$ designates the patch
itself. $I_{\triangledown}$ is the path-integral over all configurations on
one patch, $[\bar{\psi}_{\triangledown},\psi_{\triangledown}]$, that go to
$[\bar{\psi}_{\partial\triangledown},\psi_{\partial\triangledown}]$ on the
boundary. Also $S_{\triangledown}$ is the same action before but with an
integral only over $\triangledown$. The patches are chosen so that the action
$S_{\triangledown}$ is the same in all $I_{\triangledown}$. While satisfying
this property, we choose $\triangledown$ so that $\mathcal{S}_{a}$ will either
vanish on or be perpendicular to ${\partial\triangledown}$. This way the edge
configurations along ${\partial\triangledown}$ will have an additional chiral
symmetry with respect to $e^{i\alpha\gamma_{5}}$ and the edge mode residing on
${\partial\triangledown}$ will obtain no phase from $\mathcal{S}_{a}$.

If the fermions remain confined within one patch, which should be the case
when they are localized, we can exclude from the path-integration those
configurations that connect different patches together. The probability
density and current are here given by $\psi^{\dagger}\psi$ and
$j^{a}\equiv\bar{\psi}\gamma^{\mu}\psi$ respectively. We expect the excluded
configurations to be those with a nonzero flow of probability current,
$j^{a}$, out of $\triangledown$:
$\int_{\triangledown}\partial_{0}\psi^{\dagger}\psi=\int_{\triangledown}\partial_{a}j^{a}=\int_{\partial\triangledown}\hat{n}_{\partial\triangledown}^{a}\cdot
j_{a}\neq 0$. In that case the problem is reduced, from the initial path-
integral $I$, to segregated path-integrals of
$I_{\triangledown}=\int\mathcal{D}\bar{\psi}_{\triangledown}\psi_{\triangledown}\exp\\{iS_{\triangledown}\\}$.
If, moreover, there is a flat-band then the transition amplitude and therefore
the path-integral, from any state to any other state within the flat-band
would be time independent

$\left<\zeta\right|e^{iHt}\left|\chi\right>\equiv\int^{\zeta}_{\chi}\mathcal{D}\bar{\psi}\psi
e^{iS}=\left<\zeta\
\vphantom{\chi}\right|\left.\chi\vphantom{\zeta}\right>\,.$ (24)

This allows us to reduce the theory to $(2+0)$ dimensions as in Eq. (10) to
Eq. (21):

$\displaystyle
I_{\triangledown}=\int_{\triangledown}\\!\mathcal{D}\bar{\psi}_{\pm}\psi_{\pm}\exp\\!\Bigg{\\{}\int_{\triangledown}\\!d^{2}x\Big{[}\bar{\psi}_{\pm}i\gamma^{a}\left(\partial_{a}\pm
i\mathcal{A}_{a}\right)\psi_{\pm}$ $\displaystyle\quad\
+\bar{\psi}_{\mp}\gamma^{a}\epsilon_{ab}\mathcal{S}^{b}\psi_{\pm}\Big{]}\Bigg{\\}}\,,$
(25)

where we have removed the $\triangledown$ sign from the fermionic field
variables (and brought it under the path-integral sign instead) in order to
avoid clutter.

Focusing only on $I_{\triangledown}$, we see however, that not all gauge field
configurations fit within the boundaries ${\partial\triangledown}$; only those
with a complete integer index,
$n_{\circlearrowright}-n_{\circlearrowleft}\in\mathbb{Z}$. One way to observe
this is first to notice that a continuous chiral rotation of
$\psi_{+}\rightarrow e^{i2\pi\Gamma}\psi_{+}=\psi_{+}$ takes the spinor field
to itself while leaving the action unchanged. However, the theory (25) is
subject to chiral anomaly, namely, the Jacobian of our chiral transformation,
$J_{5}$, is non-trivial,

$\displaystyle I_{\triangledown}=$
$\displaystyle\int_{\triangledown}\\!\left[\mathcal{D}\bar{\psi}_{+}\psi_{+}\right]\mathcal{D}\bar{\psi}_{-}\psi_{-}e^{iS_{\triangledown}}\longrightarrow$
(26)
$\displaystyle\int_{\triangledown}\\!\left[\mathcal{D}\bar{\psi}_{+}\psi_{+}J_{5}\right]\mathcal{D}\bar{\psi}_{-}\psi_{-}e^{iS_{\triangledown}}=I_{\triangledown}e^{i2\pi(n_{\circlearrowright}-n_{\circlearrowleft})}\,.$

The last equality above comes from knowing that the Jacobian of chiral
transformation is connected to the Atiyah-Singer index [15, 13, 14]. Chiral
transformation discriminates between right and left handed modes,
$n_{\circlearrowright}-n_{\circlearrowleft}$, hence the path-integral (which
yields the determinant of the Dirac operator) obtains a phase associated with
this difference. This phase encodes the winding number of the gauge field
associated with the Dirac operator—an integer number which in two dimensional
spacetime is written as,

$\frac{1}{2\pi}\int_{\triangledown}d^{2}x\,\epsilon^{ab}\partial_{a}\mathcal{A}_{b}=n_{\circlearrowright}-n_{\circlearrowleft}\,.$
(27)

But since the field variables do not change by a complete rotation, in Eq.
(26), the path integral must also remain the same:
$I_{\triangledown}=I_{\triangledown}e^{i2\pi(n_{\circlearrowright}-n_{\circlearrowleft})}$.
Thus, if $e^{i2\pi(n_{\circlearrowright}-n_{\circlearrowleft})}\neq 1$, then
the only way that the initial and the transformed path-integrals can be equal
is for them to vanish, $I_{\triangledown}=0$. This zero valued partition
function means that the state is unrealizable. In contrast, the flat-band can
be realized if $n_{\circlearrowright,\circlearrowleft}\in\mathbb{Z}$.

What we have discussed so far applies generally to all deformations of any
bilayer system that shares the symmetries of graphene. Let us now focus on
uniform twist for which
$\mathbf{q}_{1}=\mathbf{q}_{1}^{\theta}=q^{\theta}(-1,0)$ and
$\mathbf{q}_{2,3}=\mathbf{q}_{2,3}^{\theta}=q^{\theta}(\pm\sqrt{3}/2,1/2)$
with $q^{\theta}=2K_{D}\sin(\theta/2)=4\pi/3L$. The gauge fields
$\mathcal{A}_{a}$ and $\mathcal{S}_{a}$ generated by uniform twist are
divergenceless with their corresponding field strengths proportional to $u/\nu
L$. The gauge potentials and field strengths are shown in Fig. 2. We need to
calculate the minimum value of $L$ corresponding to $n_{\circlearrowright}=1$.
Using Fujikawa’s method of calculating anomalies [13] we find,

$\displaystyle
n_{\circlearrowright}\\!=\\!\frac{1}{4\pi}\int_{\triangledown}\\!d^{2}x\,\epsilon^{ab}\partial_{a}\mathcal{A}_{b}=\frac{1}{4\pi}\int_{\triangledown}\\!d^{2}x\,B=\frac{3\sqrt{3}u}{4\pi\nu}L\,,$
(28)

which is equal to $1$ for $L=L_{0}\equiv 4\pi\nu/3\sqrt{3}u$. Considering
$L=a/2\sin(\theta/2)$, where $a=2.46\text{\AA}$ is the graphene lattice
constant, the first magic angle is given by $\theta\approx 3\sqrt{3}au/4\pi
v\approx 1.1^{\circ}$ with $u=0.11\text{eV}$ and $\nu K_{D}=9.9\text{eV}$.

To develop semi-classical intuition, let us first assume that the spin current
term $\mathcal{S}_{a}\bar{\psi}\gamma^{a}\gamma_{3}\psi$ is disregardable.
Then we are left with only a magnetic field with strength proportional to
$u/\nu L$ acting with opposite signs on $\psi_{\pm}$ fermions that are
completely decoupled from each other. If the magnetic field was constant
across the material then the electrons would have been subject to Landau
localization rotating around a fixed center, in a semi-classical picture, and
forming Landau levels. Since the moiré effective magnetic field is
inhomogeneous, the semi-classical picture changes to that of drifting
fermions—rotating around a cycling center. See Fig. 3. For localization to be
possible, the drifting fermion should be able to fit into one magnetic region.
The smallest rotating fermion, according to the uncertainty principle, has a
size of $\ell_{B}\propto 1/\sqrt{\bar{B}}$ and, since here the average
magnetic field $\bar{B}$ is proportional to $1/L$, it expands with $\sqrt{L}$.
But the size of the magnetic regions, or $\triangledown$, is proportional to
$L$ which grows faster than $\ell_{B}\propto\sqrt{L}$. Thus $\triangledown$
gets bigger faster than the smallest possible electron and eventually can
catch one, at which moment one electron has just been trapped inside the
magnetic region and can complete a cycle there without getting out of it.

Figure 3: On the left, a generic electron (red trajectory) drifting in the
magnetic region while another electron (green trajectory) is moving close to
the edge of one magnetic region. A right or left handed fermion belongs only
to either of the magnetic regions but since the sides are identified a right
handed fermion in one region is the left handed fermion in the other. Having
one electron in both regions is similar to having two electrons in one region
and forgetting about the other. On the right the twisted bilayer graphene at
first magic angle with blue dots denoting AA stacking and yellow/green dots
AB/BA stacking. The distance between the neighboring equivalent dots is equal
to $L$.

At that exact moment we expect to have an edge mode on the boundary of the
magnetic region. Appropriately, for this mode the spin current term is indeed
disregardable since $\mathcal{S}_{a}$ and $\bar{\psi}\gamma^{a}\gamma_{3}\psi$
are perpendicular to each other at ${\partial\triangledown}$. But if there is
an edge mode it means that the number of right or left handed fermions that
reside in the magnetic region must at least be one,
$n_{\circlearrowright,\circlearrowleft}=1$. This again leads us to Eq. (28)
and the magic angle. This also can be seen in terms of unit of flux and a
restricted type of Landau quantization. The size of the unit flux is given by
$\ell_{B}=\sqrt{L\nu/u}$. Therefore, $n=L^{2}/2\pi\ell_{B}^{2}=uL/2\pi\nu$
will be the degeneracy of the restricted Landau level, at least for large
$n\in\mathbb{Z}$. Thus we expect a series of magic angles connected to each
other by steps of $\delta L\approx 2\pi\nu/u$.

Reformulating the theory in terms of Dirac fermions (10) has other merits as
well. For example, the application of an external electromagnetic field yields
the same theory (10) but now with the Dirac operator carrying an additional
external gauge field, $A_{\mu}$,

$\not{D}\equiv\gamma^{\mu}\left(\partial_{\mu}+iA_{\mu}+i\mathcal{A}_{\mu}\gamma_{5}+i\mathcal{S}_{\mu}i\gamma_{3}\right)\,.$

Upon projecting the Dirac fermions into
$\psi_{\pm}\equiv\frac{1}{2}(1\pm\gamma_{5})\psi$ as before, we see that the
chiral fermions are now coupled to the shifted gauge fields
$A_{a}\pm\mathcal{A}_{a}$. For example, if $A_{a}$ is due to a constant
magnetic field $H$, Eq. (28) reads

$\displaystyle
n_{\circlearrowright}=\frac{1}{4\pi}\int_{\triangledown}\\!d^{2}x\,\left(H\pm
B\right)=\frac{H}{4\pi}\frac{3\sqrt{3}}{4}L^{2}\pm\frac{3\sqrt{3}u}{4\pi\nu}L\,.$
(29)

For $u^{2}/\nu^{2}>4\pi H/3\sqrt{3}$, the requirement
$n_{\circlearrowright}=\pm 1$ has more than one solution in contrast to Eq.
(28). Thus, an external magnetic field splits each magic angle into

$L=\pm\frac{u}{\nu}\frac{2}{H}\pm\sqrt{\left(\frac{u}{\nu}\frac{2}{H}\right)^{2}\pm\frac{16\pi}{3\sqrt{3}H}}\,,$
(30)

with the magic angles given by $\theta=2\arcsin(L/2)$, while each combination
of pluses and minuses above yields a solution to Eq. (29). For a small
external magnetic field $H\ll(3\sqrt{3}/4\pi)u^{2}/\nu^{2}\approx
140\text{mT}$ the positive solutions can be written as,

$L_{1}^{\pm}=L_{0}\pm\frac{4\pi^{2}}{27}\frac{\nu^{3}}{u^{3}}H\,,\ \
L_{2}^{\pm}=\frac{4}{H}\frac{u}{\nu}\pm
L_{0}-\frac{4\pi^{2}}{27}\frac{\nu^{3}}{u^{3}}H\,.$ (31)

with $L_{0}\equiv 4\pi\nu/3\sqrt{3}u$, being the magic angle in the absence of
the external magnetic field. In the limit $H\rightarrow 0$ we regain the
previous magic angle from $L_{1}^{\pm}$, in addition to having
$L_{2}^{\pm}\rightarrow\infty$ that happens when the bilayer is untwisted,
$\theta\rightarrow 0$, and the moiré reciprocal lattice, which would have a
vanishing size, is in fact flat.

Let us also consider applying a uniform strain on top of the already existing
uniform twist. The moiré pattern will rotate by
$\arccos(q^{\theta}/|\mathbf{q}|)$ and its length, $L$, will shrink
accordingly. In this case, using Eq. (12), the magnetic field generated by
$\mathcal{A}_{a}$ and its corresponding index are given by
$B=q^{\theta}\frac{u}{\nu}\sum_{j}\sin(\mathbf{q}_{j}\cdot\mathbf{r})$ and
$n_{\circlearrowright}=(q^{\theta}/|\mathbf{q}|)L/L_{0}$ respectively.
Comparing this with Eq. (28) we see that the flat-band now happens at,

$L=L_{0}\sqrt{1+\frac{\sinh^{2}(\omega/2)}{\sin^{2}(\theta/2)}}\,.$ (32)

Now consider the case of finite $AA$ hoppings by gradually increasing $v$ from
zero. This is equivalent to reintroducing $\mathcal{S}_{0}$ and
$\mathcal{A}_{0}$ to the Lagrangian. The former vanishes on the edges of
magnetic regions (see Fig. 2) and therefore will have no effect on the edge
mode. On the other hand, the $\mathcal{A}_{0}$ term acts as an electric
potential for either $\psi_{\pm}$. An edge mode trajectory is perpendicular to
constant $\mathcal{A}_{0}$ curves, so even though $\mathcal{A}_{0}$ might
redistribute the density of the mode along the edge, it will do little to
deviate it out of the edge. Therefore, the first magic angle is robust against
a non-vanishing $v$. Like a mass term, $\gamma^{0}\mathcal{A}_{0}$ does not
anti-commute with the generator of the chiral transformation, $\Gamma$. Since
we have $\\{\not{D},\Gamma\\}\propto v$ the $v\rightarrow 0$ ensures that the
action is invariant under $\psi\rightarrow e^{i\alpha\Gamma}\psi$ and that for
each positive eigenvalue of $\not{D}$ there is a negative one with the same
magnitude. Deviating form the $v=0$ limit breaks particle-hole symmetry,
shifts the eigenvalues of $\not{D}$, disconnects the number of left- and
right-handed zero modes $n_{\circlearrowright,\circlearrowleft}$ from the
Jacobian of the transformation, and gives the formerly flat-band a curvature.
In this situation we can define the magic angle, where the flatness of the
band is approximate, through the flux felt by $\psi_{\pm}$, e.g.
$\int_{\triangledown}F_{12}/4\pi=\int_{\triangledown}\mathcal{A}_{a}\bar{\psi}_{\partial\triangledown}\gamma^{a}\psi_{\partial\triangledown}=1$,
but this implies an exact flat-band upon the existence of a well-defined
$n_{\circlearrowright,\circlearrowleft}$ which can be found only at $v=0$.

As we have seen in this Letter, although anomalies are not present in all
dimensions, it is still possible to conjure them in specific situations. In
particular, a flat band can be described through an anomaly in the timeless
version of its hosting theory. We saw that the dimensionally reduced theory
and thus the flat band are not always realizable. In the case of bilayer
graphene, the obstruction comes from the chiral anomaly and the need to
satisfy an index condition, which in turn confirms the topological nature of
the flat band. The Dirac field theory form, Eq. (10), of the bilayer moiré
lattice problem allows many generalizations including the finite temperature
case, the presence of complex inhomogeneous external fields and general
deformations, and interaction effects in the spirit of Refs. [16, 17] where
the interplay of anomaly with interactions are discussed. Of particular
interest are quantum Hall phenomena and unconventional superconducting pairing
associated with the moiré gauge fields.

This work was supported by the National Science Foundation under Grant No.
DMR-2037158, the U.S. Army Research Office under Contract No. W911NF1310172,
and the Simons Foundation.

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# If our aim is to build morality into an artificial agent, how might we begin
to go about doing so?

Reneira Seeamber  Department of Computing, Imperial College London, London,
SW7 2BX, United Kingdom  Cosmin Badea  Department of Computing, Imperial
College London, London, SW7 2BX, United Kingdom

###### Abstract

As Artificial Intelligence (AI) becomes pervasive in most fields, from
healthcare to autonomous driving, it is essential that we find successful ways
of building morality into our machines, especially for decision-making.
However, the question of what it means to be moral is still debated,
particularly in the context of AI. In this paper, we highlight the different
aspects that should be considered when building moral agents, including the
most relevant moral paradigms and challenges. We also discuss the top-down and
bottom-up approaches to design and the role of emotion and sentience in
morality. We then propose solutions including a hybrid approach to design and
a hierarchical approach to combining moral paradigms. We emphasize how
governance and policy are becoming ever more critical in AI Ethics and in
ensuring that the tasks we set for moral agents are attainable, that ethical
behavior is achieved, and that we obtain good AI.

If our aim is to build morality into an artificial agent, or machine, how
might we begin to go about doing so? This is a crucial question that is of
particular relevance at present, in the context of the flurry of recent
activity in the field, including recent but already famous applications such
as ChatGPT and Bard.

Let us begin by defining key terms in this question, namely ‘morality’ and
‘artificial agent’. Morality can be defined as the differentiation of
intentions, decisions, and actions into those that are good (or right) and
those that are bad (or wrong) [1]. An artificial agent, or machine, we will
consider to be a system that can intelligently perform certain tasks by making
decisions.

Deciding whether or not we should build morality into artificial intelligence
(AI) is determined by many factors, including whether we want the AI to act as
a moral agent or adviser and whether we want to build autonomy into it. Going
down the latter route brings with it an obvious need for morality to be
implemented, but the fundamentals of what it is to be ’good’ have long been
debated and no clear consensus has been reached[6].

To begin addressing this dilemma, we can look at the three main philosophical
theories of morality in AI Ethics: virtue ethics, utilitarianism, and
deontology[6]. Deontology emphasizes the imperative moral value of certain
actions or rules, rather than their consequences. Virtue ethics, on the other
hand, focuses on the character of the individual who is acting, rather than on
specific rules or actions. It emphasizes the importance of developing virtuous
traits, such as honesty and courage and developing a virtuous character, which
allows individuals to make morally good choices. Utilitarianism is a
consequentialist theory that focuses on actions that maximize utility,
commonly measured as happiness or pleasure, for the greatest number of people
[6].

Existing research has investigated different approaches to developing good AI
and important obstacles and potential solutions have been identified. For
instance, the Interpretation Problem, introduced by Badea and Artus (2022)[2]
is the idea that any symbolic representation we might use for AI necessarily
gives rise to a plethora of possible interpretations, and thus ambiguity and
mistakes by the AI. There are ways to mitigate its effects, such as building
character into the agents, thus creating virtuous agents by using values as a
tether between what the AI understands and what we mean to communicate to it
[2]. There are also more practical questions, such as those around what role
agents would play in a clinical context, with proposals including them acting
alongside the human experts, perhaps relying on such experts as moral
exemplars. [3]. Additionally, there are arguments around the use of particular
moral paradigms (such as utilitarianism, deontology and virtue ethics) in
building ethical AI and resolving contemporary moral dilemmas like breaking
bad news to a patient [4] or antimicrobial resistance and use [5].
Furthermore, the variety of potential moral paradigms and lack of agreement
has also led to a perceived necessity for a moral paradigm-agnostic decision-
making framework, which Badea (2022) proposes as MARS [6]. However, building
decision-making agents is itself a difficult process, and deciding on how to
do it, or Meta-decision-making for AI [7], is a new and very promising field
of research, which we aim to contribute to here. An ontology for this process
has been proposed before, based on three steps, in Badea and Gilpin (2022). In
this paper, we continue previous attempts to identify methods and challenges
for building good AI, and to propose solutions that overcome these challenges.
With this work, we examine canonical approaches to AI to provide a broader
perspective on implementing morality. Having identified many different
challenges and potential ways of dealing with them, we found ourselves in the
position of wanting to take a step back and focus on the techniques of
building ethical AI. Thus, we examined existing methodology from the
literature and identified top-down and bottom-up approaches from classical AI
[9]. Having identified these canonical approaches, we aimed to apply them to
the task of mitigating the issues discussed above. Hence we set out with the
goal of coming up with and categorizing such methods, and continuing this line
of thought in attempting to examine the potential technical approaches to
implementing the suggestions above and constructing working ethical AI.

##  1 THE TOP-DOWN APPROACH ALONE MIGHT BE INSUFFICIENT

An important aspect to consider in designing AI is the influence of culture
and how it affects morals. Massachusetts Institute of Technology (MIT) created
the Moral Machine, an online simulation where the public can decide for
themselves which is the ‘right’ action to take given variations of the famous
‘Trolley dilemma’ [10]. The dilemma originally involved choosing between
allowing a group of five people to die or choosing to pull a lever and
sacrificing one person instead, while the Moral Machine creates variations on
this with different demographics in each of the two groups. Global data from
the MIT study revealed variations in how people respond to such moral
dilemmas, given their demographic. For example, people in Western countries
were more likely to favor saving younger people than the elderly compared with
people in Eastern countries. This casts doubt on the possibility of using a
top-down approach alone because implementing such approaches (utilitarianism,
deontology) might be nigh impossible, given that there may never be a global
consensus between humans on developing a universal set of ‘rules’ or ‘duties’.

Noothigattu et al. (2018) proposed a voting-based system, whereby data from
the moral machine is collected and used to train multiple models to learn
societal preference for alternative outcomes across multiple ethical dilemmas.
These models are then aggregated to form a model that considers the collective
outcome of all voters, thus satisfying the utilitarian approach. This method
is limited in that self-reported preferences have been shown to vary from
real-life decisions [11]. Additionally, it may be argued that few AI
developers would have to deal with ethical dilemmas, such as the Trolley
problem, in their AI application and the majority of humans have not or will
not have to make such a decision. Instead, we should be looking towards more
realistic dilemmas, such as social media advertising and whether it could lead
to accessibility to violence or compulsive behavior formation. It is up for
debate whether a set of deducible principles could be created and whether a
model needs more than simply a set of internal coherence constraints, as it
could easily satisfy constraints but still produce undesirable outcomes. In
reality, multiple goals and rules come into conflict with each other,
producing complexity. Developing rules that a machine will understand in the
way we desire is another issue in itself, which may stem from us not knowing
explicitly what it is we want them to understand [2].

In Plato’s dialogue Euthyphro, Socrates asked Euthyphro, “Is the pious loved
by the Gods because it is pious or is it pious because it is loved by the
Gods?”. This is proving very relevant to AI morality as we, humans, may begin
to see ourselves as the Gods, or creators, of AI. Thus, is it what we
(subjectively) believe morality to be that AI should consider as moral, or
should we leave it to the AI to determine what is moral for itself,
potentially then leading us to reassess what we see as moral? For instance,
the aggregation of societal preferences has the potential to produce a morally
better system than that of any individual alone, and may even enable us to
then identify general principles that underlie our own decision-making [7].

##  2 EMOTION, SENTIENCE AND MORALITY

It has been debated, especially following Jeremy Bentham’s and John Stuart
Mill’s formulation of ‘the greater good for the greatest number’, whether the
stoic characteristic of machines limits their ability to carry out moral
decision-making because of their incapacity to care. The dual-process theory
of moral judgment, which proposes the idea of dichotomous thinking, with a
fast, instinctive and emotional process driving us alongside a slower and more
logical, reason-guided process [8], has led to the question: Does this
emotional capacity of humans lead us to moral subjectivity? Or does it allow
us to decide between what is wrong or right? Thinking of humans as having two
different systems may play an important role in developing AI machines that
have both reason and the capacity to care. ‘Cognitive emotion’ could be built
into machines, whereby one allows for weights to be added to, for example,
‘human life’, thereby producing somewhat similar caring systems for human life
[8].

However, the innate stoicism of machines may be advantageous in reducing the
moral biases we have as humans, one of the limiting factors as to why it is
hard to come to a consensus [8]. The idea of ‘embodiment’ and whether
artificial moral decision-making should be extended with emotional and sensory
components is interesting. However, various issues come to light when
extensive sensor systems are involved. For example, these are more vulnerable
to attack, and people may become worried about privacy issues.

Whether or not a sentient agent is achievable remains to be explored. The
symbiosis between technology and humans will become ever more important, and
similar to a jury determining someone’s innocence, or a group of doctors
discussing how to proceed with the most complex medical problems, there are
simply certain decisions that require several inputs before coming to a
conclusion [3]. It is therefore imperative that any system taking on
responsibilities that would otherwise be discussed by a group of people should
have a notification state. This state would ensure that a board of human
overseers intervene and is involved in the final decision when conflicts
arise.

##  3 PROPOSING A HYBRID APPROACH

Figure 1: A hierarchical structure could be developed for building ethical AI
by combining virtue ethics, deontology and utilitarianism.

How do we, as humans, learn our morals and values? Why do they differ so much
globally as we have seen with the Moral Machine experiment? Children grow to
imitate and follow what they have been bought up to see. It is the complex
interaction between nature and nurture that ultimately shapes our values, as
is becoming more notable in the field of behavioral epigenetics.

With the advent of apprenticeship learning, i.e. inverse reinforcement
learning, it would be reasonable to try and develop similar artificial systems
that would learn by viewing the behaviors of humans [9]. As opposed to
reinforcement learning which uses rewards and punishments to learn certain
behaviors, inverse reinforcement learning would involve an AI system observing
human behaviors and figuring out the goal of the behavior itself. This paves a
way for coding the complexity of human ethical values without having to
explicitly code every rule for every possible scenario. However, this
highlights the need once again for stringent thought into the data AI should
learn from. A child growing up with abusive parents would possibly learn that
it is right to behave in such a way. An excellent example of how this type of
exploitation could affect AI is Tay bot, which learned racial remarks from
Twitter trolls [12].

Regarding the data an AI should learn from, we could look to the actions of
the moral exemplars [3], and thus to the agent-based ethics approach proposed
by Aristotle: virtue ethics. This emphasizes an agent’s moral character in
their actions, rather than emphasizing duties (deontology) or consequences
(utilitarianism). Aristotle himself stresses that virtue is gained through
habit, thus a virtuous person consistently embodies the right actions because
they have built the right character.

For instance, in a similar manner to how we have courts and ordered decision-
making processes in place to judge one’s character with a rigorous review
panel, it may be possible to have an equivalent scrutiny panel for AI. We
could use this as part of a hierarchical structure for building moral AI, in
which we could combine choice aspects of different moral paradigms, such as
virtue ethics, deontology and utilitarianism, an example of which we provide
in (Figure 1).

Furthermore, tying back to one of the first points regarding the differences
in moral ideologies across different cultures, it is important to consider the
principles that can be taken from other philosophers, for example, Confucian
or Buddhist virtues. The four pillars of Confucian ethics (Yen, love and
empathy; Yi, righteousness; Li, veneration and comity; Zhi, wisdom) could also
be considered when choosing virtuous people for data collection as the
‘teachers’, or moral exemplars, of the AI.

Hagendorff (2022) performed a reductionist clustering approach to meta-studies
on AI ethics guidelines to distill four fundamental AI virtues: Justice,
honesty, responsibility and care [13]. The author suggests that these four
virtues are behind the majority of practical AI ethical principles discussed
in the literature today, such as fairness, transparency, accountability and
social cohesion. Notably, prudence and fortitude were identified as second-
order virtues that aim to counteract factors, such as innate bias and peer
influence. This provides an important starting point for addressing how we
might define and identify moral exemplars for developing ethical AI.

As shown in Figure 1, following the selection of ‘teachers’, or moral
exemplars, a refinement process will take place: data will be chosen through
the selection of actions that comply with very specific rules imposed that are
decided upon a priori, as constituting duties in a deontological fashion. This
could be followed by a selection of valid actions which exhibit consequences
desired according to utilitarianism or other frameworks, such as being desired
by the majority as suggested by Noothigattu et al. (2018) [11].

It is necessary to highlight an important hypothetical scenario described by
Wallach and Allen, regarding AI-based trading systems that lead to overly high
oil prices [8]. This would lead to other automated systems moving to coal as
an alternative source of energy to reduce costs – a massive influx in coal
demand and the need for full-time production could lead to an explosion and
power outage. As one can see, the channels of multiple automated decision
systems in interaction and under the influence of one another could lead to
catastrophic human and economic consequences. It is with that, that there has
to be a higher-order framework in place to assess all potential interactions
between AI systems, for example, the AI influencing the trade of oil with an
AI deciding which power source to use.

This ties into the self-awareness embodiment of AI and awareness of other
systems as well as humans. Hence, the cognitive simulation theory seems
another way to help avoid such scenarios where AI could be programmed through
internal simulations of actions with predictions of their consequences,
instead of logic statements [14]. In this paper, a multi-layer architecture
involving a controller that generates several actions to be performed to reach
a certain goal is proposed. These actions are then sent to an ethical layer
that simulates each action using the current state of the environment, the
human involved and the robot. Internal states of the human and robot are
retrieved and used in conjunction to evaluate the best action to take.

With this, it is important to determine the tests and simulations that must be
carried out even before AI systems are given higher moral authority. Devin
Gonier (2018) introduced the notion of ’tricking’ AI models based on the
Hawthorne effect - the observer affects the outcome of an experiment [15].
Based on this, if we were to constantly observe a model, the outcome should be
in our favor. We could have two scenarios. The computer thinks it moves from a
simulation to the real world, but in reality, it stays in simulation mode.
Conversely, the model could think it’s in a simulation but actually in the
real world. Ultimately, keeping the model in a ’doubt’ mode enables control.

Could we then have parallel systems running that would each take a vote on the
best action? If the AI working on trading systems could predict that high oil
prices would result in decisions to move away from oil by other logically-
programmed machines, it would know to avoid such a scenario. Such a framework
could be combined with the inverse reinforcement learning approach where the
AI learns from observing actions from exemplars, and figuring out the goals
and internal states of impacted agents. In doing so, an extensive learned base
could be created to enable internal simulations of actions and accurate
predictions of consequences without the need for explicit rules. Such a
process would take time and experience to learn, hence the introduction of
regulatory sandboxes, which will be discussed in the next section, are a key
way to ensure a controlled application of ethical AI.

##  4 AI GOVERNANCE PRINCIPLES

Ultimately, the development of moral machines heavily relies on stringent
rules in place for deploying such systems. At the very least, it should be
proven that accountability has been covered, the system is transparent for
inspection, cannot be easily manipulated (i.e. strong security measures have
been employed to reduce the chance of hackers intercepting) and that the AI
has a high level of predictability that can match the majority of humans.
Regarding opacity and accountability, it may be possible to have parallel
functioning systems, the black box - referring to complex models that are not
highly interpretable to humans - and it’s more transparent shadow that aims to
conduce step-by-step thought processes. Prospector is one such example,
whereby researchers have developed an interactive visual representation of
predictive models that aims to help us understand how and why data points are
predicted, and the predictive value of different features [16]. Notably, the
authors of this paper stress that we need to move away from solely evaluating
performance metrics, such as accuracy.

Research is being done to ensure AI is more transparent, for example,
gradient-weighted class activation mapping is one method that attempts to
explain convoluted neural networks used in histology or magnetic resonance
imaging labeling. Maps are created to demonstrate the most relevant areas of
images used in the classification. Such techniques enable histopathologists or
radiologists to gain some insight into the decision-making process. Whilst
this provides information on how a model made a decision, current explanatory
methods typically do not justify whether a decision was appropriate or not.
Notably, Ghassemi, Oakden-Rayner and Beam discussed the idea that humans
unknowingly use heat maps to understand whether the areas identified are
justifiable based on intuition, and thus, bias can be introduced. Humans do
tend to over-trust computers, and explainability methods could reduce the
ability to identify errors and/or unreasonably increase our confidence in
algorithmic decisions [17].

Indeed, detailed logs must be made available by the AI to help understand any
decision and to further guide its development. As suggested by Mitchell et al.
(2019), documentation detailing the performance of models in different
contexts (e.g. age, race, sex) should be provided [18]. Increasing numbers of
publications relating to frameworks for dataset generation and AI development
are becoming available and a compilation of these academic works must be
acknowledged in any regulatory documents going forward.

Incorporating these techniques and tools into the AI development process can
help us work towards creating AI systems that reflect the world we want to
live in. By fostering transparency, accountability, and fairness, we can
ensure that AI systems adhere to the principles of virtue ethics in the age of
AI.

Various governments, including the UK, US and Japan, have leaned towards the
use of ‘regulatory sandboxes’, defined as a testbed for innovation in a
controlled real-life environment prior to launch. The 2023 UK policy paper, ”A
pro-innovation approach to AI regulation”, highlights the use of sandboxes to
encourage innovation and provide an adaptable response to regulating AI [19].
This contrasts with proposals for EU legislation (the EU AI Act or the
Liability Directive) which take a different approach with more comprehensive
regulations, including the General Data Protection Regulation (GDPR) that
applies to AI development and data sharing across EU member states [20]. GDPR
emphasizes data subject rights, data minimization, and transparency. The UK’s
Data Protection Act 2018 incorporates the GDPR into UK law, ensuring similar
data protection standards for AI development and data sharing. Nevertheless,
Spain has piloted the first regulatory sandbox on AI in the EU to aid in
generating future guidelines [20]. Many countries have also adopted a sector-
specific approach to regulations, impacting AI development and data sharing in
various industries, such as finance, energy and healthcare.

As technology constantly evolves, guidelines will also have to evolve with it.
There is no set of rules that can foresee all future problems that arise. This
is also why it is important to have interdisciplinary research involving
engineers, philosophers, neuroscientists, psychologists and policymakers to
ensure the best possible outcomes. It may be that regulations require
companies to involve a multidisciplinary team during development. University
education may also need to introduce varied modules (i.e. ML courses include a
philosophy and ethics module, and medical degrees include ML models) to
prepare future generations for such collaborative tasks. Regulatory workshops
should be made more regular and accessible to inform and guide companies
during the development process. Either way, there is still much yet to debate
regarding human ethics before machines can become the moral entities we want
them to become. If we are to take the bottom-up approach to train AI, we must
ensure the data is a reflection of the world we want to live in.

Encouraging open discussions among individuals and communities with different
views will be important in decision-making. This could be done by creating
national and international forums, where society can share experiences and
develop recommendations for AI governance. Such forums can provide a platform
for dialogue, consensus-building, and the exchange of best practices. By
fostering a culture of dialogue and debate, society can collectively engage
with moral dilemmas to understand and overcome the nuances and complexities.
Additionally, individuals can further develop the ability to critically
analyze moral dilemmas through the promotion of moral education and critical
thinking from an early age, enabling them to understand different viewpoints
and appreciate the diversity of moral values across cultures and traditions.
We must also ensure that decision-making processes are inclusive and represent
the diversity of society. This will help to avoid the domination of a single
moral perspective and to embrace multiple moral views.

##  5 TOWARDS MORAL AI

In conclusion, we have looked at the matter of building moral machines from
different angles, such as those of technical approaches, culture, emotion,
sentience, and governance. Our main contribution is the proposal of a hybrid
approach using a combination of top-down and bottom-up and a further
combination of different moral paradigms to get around the different obstacles
and limitations discussed. We have also discussed some of the practical issues
and techniques relevant to implementing our approach.

Ultimately, determining what constitutes good morals is an ongoing and
collaborative effort that requires input from various stakeholders and an
openness to learn, grow, and adapt. By engaging in these processes, society
can work towards establishing shared moral values that contribute to the
greater good. It is in doing so, that we may be able to start to answer some
of the questions we, as humans, hold regarding morality and what it is to act
with moral grounding. This collaborative approach can help create a more
responsible, inclusive, and beneficial AI landscape for all.

##  6 ACKNOWLEDGMENTS

Reneira Seeamber was supported by the UKRI CDT in AI for Healthcare
http://ai4health.io (Grant No. EP/S023283/1). For the purpose of open access,
the authors have applied a Creative Commons Attribution (CC BY) license to the
Accepted Manuscript.

## REFERENCES

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