# Seeing Topological Charges by Naked Eyes

Matheus S. M. de Sousa  Department of Physics, PUC-Rio, 22451-900 Rio de
Janeiro, Brazil  Antonio L. Cruz  Department of Physics, PUC-Rio, 22451-900
Rio de Janeiro, Brazil  Wei Chen  Department of Physics, PUC-Rio, 22451-900
Rio de Janeiro, Brazil

###### Abstract

The opacity of graphene is known to be approximately given by the fine
structure constant $\alpha$ times $\pi$. We point out the fact that the
opacity is roughly independent of the frequency of the light can be attributed
to the topological charge of the Dirac point. As a result, one can literally
see the topological charge by naked eyes from the opacity of graphene, and
moreover it implies that the fine structure constant is topologically
protected. A similar analysis applied to 3D topological insulators suggests
that their opacity in the infrared region is also given by $\pi\alpha$, which
may be seen by naked eyes through an infrared lens, and ideally is independent
of the thickness of the material since it is contributed solely from the
topological surface states. For 3D Dirac or Weyl semimetals, the optical
absorption power is linear to the frequency in the infrared region, with a
linearity given by the fine structure constant and the topological charge of
Weyl points.

Introduction.- A fundamentally important issue in the research of topological
materials is how the topological order can be unambiguously detected in
experiments. Conventionally, the most straightforward way is to detect the
effects related to the metallic edge or surface states, such as the quantized
Hall conductanceKlitzing  _et al._ (1980); König  _et al._ (2007) or zero bias
conductanceMourik  _et al._ (2012), which only occur in topologically
nontrivial phase and is directly determined by the bulk topological invariant.
On the other hand, to our knowledge, there has yet been a bulk material
property in the macroscopic scale that can be felt by human perception and be
attributed to the topological order. This is in sharp contrast to the
materials that possess Landau order parameters. For instance, magnetic order
can be simply perceived from the force that repels or attracts two bar
magnets, and superconductivity can be understood as the mechanism behind the
magnetic levitation.

In this Letter, we point out that the topological charge of various materials
containing gapless Dirac cones can actually be seen by naked eyes, either
directly or through an infrared lens, owing to the optical absorption power
that is proportional to the fine structure constant $\alpha=e^{2}/4\pi\hbar
c\varepsilon_{0}\approx 1/137$ times a factor given by the topological charge
${\cal C}$. This feature originates from a correspondence between quantum
geometry and topological ordervon Gersdorff and Chen (2021), and implies
$\alpha$ can be extracted in a topologically protected manner from the
frequency- or polarization-dependence of optical absorption in different
materials. We will use several prototype topological materials including
graphene, 3D topological insulators (TIs), and 3D Dirac and Weyl semimetals to
explicitly demonstrate how $\alpha$ and ${\cal C}$ appear in the optical
absorption power, and then address how various factors in real materials, such
as hexagonal warping, spin-orbit coupling, and many different sources of
scattering can alter this simple picture, which nevertheless still serves as a
pedagogical example to perceive the topological charge in the macroscopic
scale.

Relating optical absorption power to quantum geometry.- Our survey starts by
considering the quantum geometry of valance band statesMatsuura and Ryu
(2010); von Gersdorff and Chen (2021); Chen and von Gersdorff (2022). We will
reserve the index $n$ for valence bands, $m$ for conduction bands, $\ell$ for
all the bands, and their energies at momentum ${\bf k}$ are denoted by
$\varepsilon_{\ell}^{\bf k}$. For a gaped topological material with $N_{-}$
valence bands, the fully antisymmetric valence band Bloch state at momentum
${\bf k}$ is $|u^{\rm val}({\bf
k})\rangle=\epsilon^{n_{1}n_{2}...n_{N-}}|n_{1}\rangle|n_{2}\rangle...|n_{N_{-}}\rangle/\sqrt{N_{-}!}$.
The quantum metric of this state is defined fromProvost and Vallee (1980)

$\displaystyle|\langle u^{\rm val}({\bf k})|u^{\rm val}({\bf k+\delta
k})\rangle|=1-\frac{1}{2}g_{\mu\nu}({\bf k})\delta k^{\mu}\delta k^{\nu},$ (1)

whose diagonal component $g_{\mu\mu}$ can be calculated from each band byvon
Gersdorff and Chen (2021)

$\displaystyle g_{\mu\mu}({\bf k})=\langle\partial_{\mu}u^{\rm
val}|\partial_{\mu}u^{\rm val}\rangle-\langle\partial_{\mu}u^{\rm val}|u^{\rm
val}\rangle\langle u^{\rm val}|\partial_{\mu}u^{\rm val}\rangle$
$\displaystyle=\sum_{nm}\langle\partial_{\mu}n|m\rangle\langle
m|\partial_{\mu}n\rangle.$ (2)

To relate the quantum metric to optical responses, we introduce a quantum
metric spectral functionChen and von Gersdorff (2022)

$\displaystyle g_{\mu\mu}({\bf
k},\omega)=\sum_{\ell<\ell^{\prime}}\langle\partial_{\mu}\ell|\ell^{\prime}\rangle\langle\ell^{\prime}|\partial_{\mu}\ell\rangle$
$\displaystyle\times\left[f(\varepsilon_{\ell}^{\bf
k})-f(\varepsilon_{\ell^{\prime}}^{\bf
k})\right]\delta(\omega+\frac{\varepsilon_{\ell}^{\bf
k}}{\hbar}-\frac{\varepsilon_{\ell^{\prime}}^{\bf k}}{\hbar})$
$\displaystyle=\frac{V_{D}}{\pi e^{2}\hbar\omega}\sigma_{\mu\mu}({\bf
k},\omega),$ (3)

which at zero temperature frequency-integrates to the quantum metric
$\lim_{T\rightarrow 0}\int_{0}^{\infty}d\omega\,g_{\mu\mu}({\bf
k},\omega)=g_{\mu\mu}({\bf k})$, where $V_{D}$ is the volume of the
$D$-dimensional unit cell, and $\sigma_{\mu\mu}({\bf k},\omega)$ is the finite
temperature longitudinal optical conductivity at momentum ${\bf k}$de Souza
_et al._ (2023).

The optical conductivity measured in real space is given by the momentum
integration

$\displaystyle\sigma_{\mu\mu}(\omega)=V_{D}\int\frac{d^{D}{\bf
k}}{(2\pi\hbar)^{D}}\,\sigma_{\mu\mu}({\bf k},\omega)$
$\displaystyle=\frac{\pi e^{2}}{\hbar^{D-1}}\,\omega\int\frac{d^{D}{\bf
k}}{(2\pi)^{D}}\,g_{\mu\mu}({\bf k},\omega)\equiv\frac{\pi
e^{2}}{\hbar^{D-1}}\,\omega\,{\cal G}_{\mu\mu}(\omega),$ (4)

which defines what we call the fidelity number spectral function ${\cal
G}_{\mu\mu}(\omega)$de Souza  _et al._ (2023). Furthermore, applying an
oscillating electric field polarized in $\mu$ direction
$E_{\mu}(\omega,t)=E_{0}\cos\omega t$ to the system induces a current that
oscillates accordingly
$j_{\mu}(\omega,t)=\sigma_{\mu\mu}(\omega)E_{0}\cos\omega t$, where $E_{0}$ is
the strength of the field. Thus the optical absorption power per unit cell at
frequency $\omega$ isde Souza  _et al._ (2023)

$\displaystyle W_{a}^{\mu}(\omega)=\langle
j_{\mu}(\omega,t)E_{\mu}(\omega,t)\rangle_{t}=\frac{1}{2}\sigma_{\mu\mu}(\omega)E_{0}^{2}$
$\displaystyle=\frac{\pi e^{2}}{2\hbar^{D-1}}\,E_{0}^{2}\omega\,{\cal
G}_{\mu\mu}(\omega),$ (5)

where the time average gives $\langle\cos^{2}\omega t\rangle_{t}=1/2$. The
main point of the present work is how $W_{a}^{\mu}(\omega)$ is related to the
topological charge ${\cal C}$ and fine structure constant $\alpha$ in a
topologically protected manner, as we elaborate below for several different
topological materials.

Opacity of pristine graphene.- The low energy band structure of graphene can
be described by the tight-binding model on a honeycomb lattice with nearest-
neighbor hopping $H=\sum_{\langle
ij\rangle\sigma}t\,c_{i\sigma}^{{\dagger}}c_{j\sigma}$ with $t\approx 2.8$eV,
and we denote the distance between neighboring carbon atoms by
$a=0.142$nmCastro Neto  _et al._ (2009). For each spin species, the low energy
Homiltonian in the momentum space may be obtained by an expansion around the
two Dirac points ${\bf K}$ and ${\bf K}^{\prime}$, yielding the linear Dirac
Hamiltonian Bernevig and Hughes (2013)

$\displaystyle H_{0}^{{\bf K},{\bf K^{\prime}}}({\bf k})=v_{F}\left(\pm
k_{y}\sigma_{x}-k_{x}\sigma_{y}\right),$ (6)

where $v_{F}=3ta/2\hbar$ is the Fermi velocity. This linearized model well
describes the linear band structure up to energy $\sim 1$eV, and hence is a
suitable model for the optical absorption in the visible light range. To
proceed, we introduce the spin-valley index $\gamma=\left\\{{\bf
K}\uparrow,{\bf K}\downarrow,{\bf K}^{\prime}\uparrow,{\bf
K}^{\prime}\downarrow\right\\}$ and denote the valence and conduction band
states by $|n^{\gamma}\rangle$ and $|m^{\gamma}\rangle$, whose eigenenergies
$\varepsilon_{n}^{\bf k}=-v_{F}k$ and $\varepsilon_{m}^{\bf k}=v_{F}k$ do not
depend on $\gamma$. The key point in our formalism is the relation between the
topological charge ${\cal C}=1/2$ and the azimuthal quantum metric
$g_{\phi\phi}^{\gamma}=1/4$ of any flavor $\gamma$Sup ; von Gersdorff and Chen
(2021)

$\displaystyle-\oint\frac{d\phi}{2\pi}\langle n^{\bf
K\uparrow}|i\partial_{\phi}|n^{\bf
K\uparrow}\rangle=\oint\frac{d\phi}{2\pi}\langle n^{\bf
K^{\prime}\uparrow}|i\partial_{\phi}|n^{\bf K^{\prime}\uparrow}\rangle$
$\displaystyle=1/2\equiv{\cal C}=|\langle
n^{\gamma}|i\partial_{\phi}|n^{\gamma}\rangle|=\sqrt{g_{\phi\phi}^{\gamma}}.$
(7)

Putting this result into Eq. (3) and (4), one obtains a fidelity number
spactral function that contains a prefactor of ${\cal C}^{2}$

$\displaystyle{\cal G}_{\mu\mu}(\omega)=\frac{{\cal
C}^{2}}{\pi\omega}\left[f\left(-\frac{\hbar\omega}{2}\right)-f\left(\frac{\hbar\omega}{2}\right)\right].$
(8)

In addition, from Eq. (3), the optical conductivity contributed by the flavor
$\gamma$ is $\sigma_{\mu\mu}^{\gamma}({\bf k},\omega)=\pi
e^{2}\hbar\omega\,g_{\mu\mu}^{\gamma}({\bf k},\omega)/A_{cell}$, where
$A_{cell}=3\sqrt{3}a^{2}/2$ is the unit cell area of the hexagonal lattice,
and hence the conductivity measured in real space is
$\sigma_{\mu\mu}(\omega)=\sum_{\gamma}\int\frac{d^{2}{\bf
k}}{\hbar^{2}A_{BZ}}\,\sigma_{\mu\mu}^{\gamma}({\bf k},\omega)=\frac{\pi
e^{2}}{\hbar}\,\omega\,{\cal G}_{\mu\mu}(\omega)$, which links the optical
conductivity of grapheneFalkovsky and Varlamov (2007); Falkovsky (2008) to the
spectral function ${\cal G}_{\mu\mu}(\omega)$, where
$A_{BZ}=8\pi^{2}/3\sqrt{3}a^{2}$ is the area of the BZ.

For an incident light with electric field $E_{0}$, the incident power per unit
cell of area $A_{cell}$ is $W_{i}=c\varepsilon_{0}E_{0}^{2}/2$. Using Eq. (5),
the opacity at polarization $\mu$ and frequency $\omega$ is

$\displaystyle{\cal
O}(\omega)=\frac{W_{a}^{\mu}(\omega)}{W_{i}}=4\pi^{2}\alpha\omega\,{\cal
G}_{\mu\mu}(\omega)$ $\displaystyle=\pi\alpha\times 4{\cal
C}^{2}\left[f\left(-\frac{\hbar\omega}{2}\right)-f\left(\frac{\hbar\omega}{2}\right)\right],$
(9)

which does not depend on the polarization $\mu$. In the zero temperature
limit, the Fermi factor drops out, so one obtains

$\displaystyle\lim_{T\rightarrow 0}{\cal O}(\omega)=\pi\alpha\times 4{\cal
C}^{2}=\pi\alpha\approx 2.3\%.$ (10)

This is the seminal result of the $2.3\%$ opacity (or transmittance) of
grapheneNair  _et al._ (2008); Stauber  _et al._ (2008); Bruna and Borini
(2009); Weber  _et al._ (2010), and our analysis brings in several new aspects
to this result: (1) The independence of frequency $\omega$ is due to the
topological charge ${\cal C}=1/2$, or equivalently the azimuthal quantum
metric $g_{\phi\phi}^{\gamma}$ according to Eq. (25), that is independent of
the circle of radius $k=\hbar\omega/2v_{F}$ at which the conduction band
electrons are excited by the light. In addition, $g_{\phi\phi}^{\gamma}$ is
also independent of the angle $\phi$, hence the opacity does not depend on the
polarization $\mu$ either. Since the visible light produced by common light
sources usually contains a wide range of polarization and frequency, the
reason that graphene always shows $2.3\%$ opacity is due to the topological
protection of the Dirac cone, and it implies that one can literally see the
topological charge ${\cal C}$ by naked eyes through the opacity. (2) One may
use the plateau of frequency or polarization dependence of opacity to extract
$\alpha$. This procedure echoes the extraction of von Klitzing constant
$h/e^{2}$ from the Hall plateaux as a function of magnetic field in the
quantum Hall effect (QHE)Klitzing  _et al._ (1980), which is also
topologically protectedThouless  _et al._ (1982) (note that the magnetic field
can also be converted into the cyclotron frequency). This topological
protection implies that the measured $\alpha$ should be independent of many
details of the system, i.e., ideally any 2D materials that have a Dirac code
should have the same $\pi\alpha$ opacity, which may describe a variety of 2D
materials such as graphynesMalko  _et al._ (2012a, b), B2SZhao  _et al._
(2018), siliceneEzawa (2013); Houssa  _et al._ (2015); Sadeddine  _et al._
(2017), germaneneAcun  _et al._ (2015), etcWehling  _et al._ (2014);
Balendhran  _et al._ (2015); Wang  _et al._ (2015).

Despite this appealing connection between the opacity and topological charge,
the above simple results suffer a great challenge from many realistic factors
in 2D materials containing Dirac cones, as we formulate in the supplementary
material and discuss belowSup . The first is the realistic band structure
beyond the simple description of Eq. (6), which contains complications such as
hexagonal warping and next-nearest-neighbor hopping. Investigating these
effects by means of a tight-binding model leads us to conclude that the next-
nearest-neighbor hopping actually has no effect on either the opacity or the
topological charge. However, the tight-binding band structure does feature an
opacity that increases with frequency until the transition between van Hove
singularities $\hbar\omega\approx 2t$, which makes the opacity in the visible
light range to be $2.46\%<{\cal O}(\omega)<2.71\%$, as has already been
investigated theoretically and observed experimentallyNair  _et al._ (2008);
Stauber  _et al._ (2008); Kuzmenko  _et al._ (2008); Bruna and Borini (2009);
Weber  _et al._ (2010); Nair  _et al._ (2010). Nevertheless, since human eyes
can hardly distinguish such a small deviation, the roughly $\approx 2.3\%$
opacity observed by human eyes can still serve as a pedagogical example to
perceive the topological charge in graphene in the macroscopic scale. On the
other hand, for 2D materials whose transition between van Hove singularities
falls in the visible light range, such as silicene simulated by $t=1.6$eVEzawa
(2015) and shown in Fig. 1 (a), then the opacity will strongly depend on the
color of the light. This result indicates that Dirac cone materials with
larger hopping $t$, i.e., the linearity of Dirac cone extends beyond visible
light range, are more ideal to visualize the topologically protected constant
opacity. Nevertheless, because the opacity in the infrared region should still
be $\pi\alpha$ for materials with a small $t$, the constant opacity may still
be perceived by human eyes through an infrared lens. Finally, the finite
temperature data in Fig. 1 (a) indicate that the thermal broadening in Eq. (9)
only reduces the opacity in the low frequency region $\hbar\omega\apprle
k_{B}T$, which is far below and hence has negligible effect on the visible
light rangeStauber  _et al._ (2008); Kuzmenko  _et al._ (2008).

Secondly, the realistic band structure also influences the polarization
dependence of the opacitySup , yielding a $15\%$ to $30\%$ variation with the
polarization angle $\phi_{E}$ in the visible light range, as shown in Fig. 1
(b). This variation is mainly contributed from the optical Hall conductance
$\sigma_{xy}(\omega)$ originated from $g_{xy}$Sup , and hence one can actually
estimate the magnitude of $\sigma_{xy}(\omega)$ by simply rotating a graphene
sheet against a polarized light and seeing how its opacity varies. However,
this also indicates that it is hard to associate the opacity against a
polarized light to the topological charge since it is not a constant of
$\phi_{E}$, and suggesting that only unpolarized light that averages over
$\phi_{E}$ can serve this purpose.

Figure 1: (a) The opacity of graphene ${\cal O}(\omega)/\pi\alpha$ at $T=$
300K, 1000K (a rather unrealistic temperature just to demonstrate the effect),
and silicene at $T=0$ as a function of frequency of the unpolarized light. (b)
The dependence of the opacity of graphene on the polarization angle $\phi_{E}$
at $T=0$. (c) The effect of RSOC on the opacity of graphene at $T=0$. (d) The
effect of impurities on the opacity of linear Dirac model, with impurity
density $n_{imp}$ and strength $V$. The inset shows schematically how the
energy shift caused by the real part of self-energy blocks the optical
transition.

Thirdly, for 2D materials that are not perfectly flat but have some buckling
structure, the breaking of inversion symmetry can induce a Rashba spin-orbit
coupling (RSOC). By incorporating the RSOC of strength $\lambda_{R}$ into the
honeycomb latticede Sousa  _et al._ (2021), we obtain an opacity that shows an
anomaly at frequency $\omega\sim 3\lambda_{R}$ owing to the splitting of the
bands, as shown in Fig. 1 (c). Thus if the RSOC of some material happens to
fall in the visible light region $1.8$eV$\apprle 3\lambda_{R}\apprle 3.1$eV,
then its opacity may strongly depend on the color of the light. However, such
a strong RSOC seems rather unlikely, since most of known 2D materials have an
RSOC of the order of 10meVLiu  _et al._ (2011), and hence its influence on the
visible light range is negligible.

Finally, concerning various sources of scattering, graphene under the
influence of electron-electron interaction has been investigated
intensivelyHerbut  _et al._ (2008); Elias  _et al._ (2011), which only yields
a very small correction to the opacitySheehy and Schmalian (2009). In
contrast, we consider the effect of random point-like impuritiesAndo  _et al._
(2002); Sup . The results shown in Fig. 1 (d) indicate an opacity that is
strongly suppressed in the low frequency regime $\hbar\omega\apprle
2n_{imp}V$, which is caused by the real part of the self-energy that acts like
a chemical potential, rendering the valence band states with energy less than
$\varepsilon_{n}^{\bf k}<n_{imp}V$ empty. As a result, the optical absorption
is strongly suppressed and hence the material actually becomes more
transparent at $\hbar\omega\apprle 2n_{imp}V$, as indicated by the inset of
Fig. 1 (d). This suppression is in qualitative agreement with the experiment
in fluorinated graphene, where a reduced low frequency opacity is observed,
and the reduction region can be extended to the visible light range by
increasing fluorine concentrationNair  _et al._ (2010). In fact, changing the
chemical potential should also cause such a suppression, as has been observed
experimentallyLi  _et al._ (2008). Thus to see the topological charge from the
opacity in the visible light range, a clean and unbiased graphene is needed.
Finally, we remark that the opacity is frequency-independent only if the
topological material is gapless, as we demonstrate in the supplementary
material using 2D Chern insulator as a counterexampleSup ; de Souza  _et al._
(2023).

Opacity of 3D TIs.- Interestingly, although 3D TIs like Bi2Se3 and Sb2Te3 have
a bulk gapZhang  _et al._ (2009); Liu  _et al._ (2010), we predict that their
opacity in the infrared region is also $\pi\alpha$ just like graphene owing to
the gapless topological surface states. The bulk of 3D TI usually have a
direct band gap $2M\sim 0.5$eV, so in the topologically trivial phase it
should have no optical absorption and hence ideally be transparent in the
infrared region. However, in the topologically nontrivial phase, the TI has
surface states in the top and bottom surfaces that the light passes through,
each has two spin species, and hence one may label them by the four spin-
surface flavors $\gamma=({\rm top}\uparrow,{\rm top}\downarrow,{\rm
bottom}\uparrow,{\rm bottom}\downarrow)$. Each $\gamma$ is described by one of
the Hamiltonians in Eq. (6) with a proper assignment of $k_{x}$ and
$k_{y}$Zhang  _et al._ (2009); Liu  _et al._ (2010); Qi and Zhang (2011),
yielding the same opacity $\pi\alpha$ as graphene, and is also independent of
the frequency and polarization in the infrared region owing to the topological
charge of the surface state, equivalently the bulk topological invariant due
to the bulk-edge correspondence. Moreover, the opacity should be independent
of the thickness of the TI provided the material is thicker than the decay
length of surface states (same as the correlation length $\hbar v_{F}/M\sim$
nmChen  _et al._ (2017); Chen and Schnyder (2019)), and is the same for all 3D
TIs with a direct band gap, which is ready to be verified by naked eyes
through an infrared lens. Our result also implies that the opacity changes
when the 3D TI undergoes a topological phase transition, which may help to
detect the transition by optical means. Finally, we remark that the
transmittance of Bi2Se3 grown on a relatively transparent fluorophlogopite
mica has been measured, which however exhibits a rapid Fabry–Pérot oscillation
with frequencyPeng  _et al._ (2012), indicating that a free standing single-
crystal 3D TI thin film may be necessary to obtain the constant $\pi\alpha$
opacity.

Optical absorption of 3D Dirac and Weyl semimetals.- We proceed to consider
the low energy sector of type-I 3D Weyl semimetals, such as TaAs and TaPWeng
_et al._ (2015); Huang  _et al._ (2015), which contain pairs of Weyl pointsYan
and Felser (2017); Armitage  _et al._ (2018); Lv  _et al._ (2021). Their low
energy eigenstates can be labeled by the valley index
$\gamma=\left\\{1,2...N_{W}\right\\}$, where $N_{W}$ is the number of Weyl
points. The Hamiltonian in the infrared region for each $\gamma$ is well
described by the Dirac model

$\displaystyle H^{\gamma}({\bf
k})=\pm\left(vk_{x}\sigma_{x}+vk_{y}\sigma_{y}+vk_{z}\sigma_{z}\right)=\pm{\bf
d}\cdot{\bm{\sigma}},\;\;\;$ (11)

and we assume a cubic lattice of unit cell volume $V_{D}=a^{3}$ for
simplicity. As detailed in the supplementary materialSup , the absorption
power summing over the three crystalline directions and then divided by the
incident power per unit cell volume $W_{i}=c\varepsilon_{0}E_{0}^{2}/2a$
yields

$\displaystyle\sum_{\mu=x,y,z}\frac{W_{a}^{\mu}(\omega)}{W_{i}}$
$\displaystyle=\alpha|{\cal
C}|\left(\frac{N_{W}a}{2v}\right)\omega\left[f\left(-\frac{\hbar\omega}{2}\right)-f\left(\frac{\hbar\omega}{2}\right)\right].$
(12)

whose zero temperature limit is linear in frequency, as has been pointed out
theoreticallyHosur  _et al._ (2012); Bácsi and Virosztek (2013); Timusk  _et
al._ (2013); Ashby and Carbotte (2014) and experimentally observedXu  _et al._
(2016). Our result further suggests that the optical absorption power summing
over three crystalline directions is directly proportional to the module of
the topological charge $|{\cal C}|$. In addition, since it is proportional to
the module, even if two Weyl nodes of opposite chirality merge together to
form a Dirac semimetalYoung  _et al._ (2012); Mañes (2012), the absorption
power is still that described by Eq. (12), whose linearity in frequency has
been observed experimentallyChen  _et al._ (2015). Physically, this means that
Dirac and Weyl semimetals appear darker under higher frequency light in the
infrared region, which should be detectable by human eyes through an infrared
lens, with a darkness given by the fine structure constant $\alpha$, module of
the topological charge $|{\cal C}|$, lattice constant $a$, Fermi velocity $v$,
and the number of Weyl points $N_{W}$. Finally, we remark that 3D topological
semimetals have surface states too, but since the bulk already absorbs light,
the contribution from the surface states should be negligible in the bulk
limit.

Conclusions.- In summary, we clarify that the approximate frequency-
independence of the opacity of disorder-free and unbiased graphene can be
attributed to the topological charge of Dirac points. In other words, the
roughly $\pi\alpha\approx 2.3\%$ opacity of graphene against any sources of
unpolarized visible light is a manifestation of topological charge. The same
analysis applied to 3D TIs suggests that their opacity in the infrared region
ideally is also the bulk topological invariant times $\pi\alpha$ independent
of frequency, polarization, and thickness of the material. In contrast, for 3D
Weyl and Dirac semimetals, the fine structure constant and topological charge
also determine the linear dependence of optical absorption power on frequency.
All these features suggest that the fine structure constant extracted from the
opacity or optical absorption power of these materials is topologically
protected, offering a very accessible way to estimate the constant in a
topologically protected manner.

##  Appendix A Detail of linear Dirac model calculation

For the linear Dirac model in Eq. (6) of the main text, the eigenstates of
either spin, say spin up, do not depend on the module $k$ but only on the
polar angle $\phi$ of the momentum, and are given explicitly by

$\displaystyle|n^{\bf
K\uparrow}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
ie^{i\phi}\end{array}\right),\;\;\;|m^{\bf
K\uparrow}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
-ie^{i\phi}\end{array}\right),$ (17) $\displaystyle|n^{\bf
K^{\prime}\uparrow}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
ie^{-i\phi}\end{array}\right),\;\;\;|m^{\bf
K^{\prime}\uparrow}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\
-ie^{-i\phi}\end{array}\right),\;\;\;$ (22)

and the expression is the same for spin $\downarrow$. The topological charge
${\cal C}$ per spin at each of the two Dirac points ${\bf K}$ and ${\bf
K^{\prime}}$ is given by integrating the valence band Berry connection along a
closed loop of radius $k$ circulating the Dirac points

$\displaystyle\oint\frac{d\phi}{2\pi}\langle n^{\bf
K\uparrow}|i\partial_{\phi}|n^{\bf
K\uparrow}\rangle=-\oint\frac{d\phi}{2\pi}\langle n^{\bf
K^{\prime}\uparrow}|i\partial_{\phi}|n^{\bf K^{\prime}\uparrow}\rangle$
$\displaystyle=-1/2\equiv-{\cal C},$ (23)

which has opposite signs at the two Dirac points.

We now elaborate the relation between this topological charge and the quantum
metric, and how this relation eventually manifests in the opacity of graphene.
The quantum metric for the valence band state of spin-valley flavor $\gamma$
is defined from the overlap $|\langle n^{\gamma}({\bf k})|n^{\gamma}({\bf
k+\delta k})\rangle|=1-g_{\mu\nu}^{\gamma}\delta k^{\mu}\delta k^{\nu}/2$. In
particular, because $|n^{\gamma}\rangle$ in Eq. (22) does not depend on the
module $k$, the only nonzero component in the polar coordinates
$\left\\{\mu,\nu\right\\}=\left\\{k,\phi\right\\}$ is the azimuthal component

$\displaystyle g_{\phi\phi}^{\gamma}=|\langle
m^{\gamma}|i\partial_{\phi}|n^{\gamma}\rangle|^{2}=|\langle
n^{\gamma}|i\partial_{\phi}|n^{\gamma}\rangle|^{2}={\cal
C}^{2}=\frac{1}{4},\;\;\;$ (24)

which is the same for all the four flavors of $\gamma$. Comparing with Eq.
(23), and use the fact that $\langle
n^{\gamma}|i\partial_{\phi}|n^{\gamma}\rangle$ is a constant, one sees that
the topological charge is actually given by the azimuthal quantum metric

$\displaystyle{\cal C}=|\langle
n^{\gamma}|i\partial_{\phi}|n^{\gamma}\rangle|=\sqrt{g_{\phi\phi}^{\gamma}}.$
(25)

This relation, which has been called the metric-curvature correspondence, is
the key to identify the opacity of graphene with the topological charge, as we
shall see below.

The quantum metric in the Cartesian coordinates
$\left\\{\mu,\nu\right\\}=\left\\{x,y\right\\}$ can be obtained from Eq. (24)
by utilizing $\partial_{x}=\cos\phi\,\partial_{k}-(\sin\phi/k)\partial_{\phi}$
and $\partial_{y}=\sin\phi\,\partial_{k}+(\cos\phi/k)\partial_{\phi}$.
Moreover, because in the angular integration $\int_{0}^{2\pi}d\phi$ in the
calculation of optical absorption below, the choice of $\phi=0$ is arbitrary,
so for any polarization $\mu=\left\\{x,y\right\\}$ we can simply set

$\displaystyle
g_{\mu\mu}^{\gamma}=\frac{\sin^{2}\phi}{k^{2}}g_{\phi\phi}^{\gamma}=\frac{\sin^{2}\phi}{k^{2}}\,{\cal
C}^{2}.$ (26)

It follows that the quantum metric spectral function in Eq. (3) of the main
text for the flavor $\gamma$ is

$\displaystyle g_{\mu\mu}^{\gamma}({\bf
k},\omega)=g_{\mu\mu}^{\gamma}\left[f(\varepsilon_{n}^{\bf
k})-f(\varepsilon_{m}^{\bf k})\right]\delta(\omega+\frac{\varepsilon_{n}^{\bf
k}}{\hbar}-\frac{\varepsilon_{m}^{\bf k}}{\hbar}).\;\;\;$ (27)

After a momentum integration and summation over $\gamma$, one obtains the
fidelity number spactral function in Eq. (8) of the main text.

Figure 2: (a) The definition of coordinates and various vectors on the
honeycomb lattice. (b) The band structure simulated by $t=1$,
$t^{\prime}=0.036$, and $\lambda_{R}=0.2$.

##  Appendix B Effect of next-nearest-neighbor hopping and Rashba spin-orbit
coupling

The effect of next-nearest-neighbor hopping and Rashba spin-orbit coupling
(RSOC) may be investigated by incorporating them into the tight-binding model.
For this purpose, we investigate the following model defined on a honeycomb
lattice

$\displaystyle H$ $\displaystyle=$ $\displaystyle-t\sum_{\langle
ij\rangle,\sigma}c_{i\sigma}^{{\dagger}}c_{j\sigma}+t^{\prime}\sum_{\langle\langle
ij\rangle\rangle,\sigma}c_{i\sigma}^{{\dagger}}c_{j\sigma}$ (28)
$\displaystyle+$ $\displaystyle i\lambda_{R}\sum_{\langle
ij\rangle,\alpha,\beta}c_{i\alpha}^{{\dagger}}\left({\bm{\sigma}}_{\alpha\beta}\times{\bf
d}_{ij}\right)^{z}c_{j\beta}.$

Here $c^{{\dagger}}_{i\sigma}(c_{i\sigma})$ creates (annihilates) an electron
of spin $\sigma$ on the lattice site $i$, and $\langle ij\rangle$ and
$\langle\langle ij\rangle\rangle$ indicate nearest-neighbor and next-nearest-
neighbor lattice sites with the corresponding hopping $-t$ and $t$. The
$\lambda_{R}$ is the RSOC coupling constant,
${\bm{\sigma}}=(\sigma^{x},\sigma^{y},\sigma^{z})$ are the spin Pauli
matrices, ${\bf d}_{ij}$ is the vector connecting the site $i$ to $j$.
Defining the basis
$\psi=\left(A\uparrow,B\uparrow,A\downarrow,B\downarrow\right)$ and the
Fourier transformation $c_{Ii\sigma}=\sum_{\bf k}e^{i{\bf k\cdot
r}_{i}}c_{I{\bf k}\sigma}$, where $I=\left\\{A,B\right\\}$ denotes the two
sublattices and ${\bf r}_{i}$ the unit cell position, the Hamiltonian
$H=\sum_{\bf kIJ\alpha\beta}c_{I{\bf k}\alpha}^{{\dagger}}H_{I\alpha
J\beta}({\bf k})c_{J{\bf k}\beta}$ is described by the matrix

$\displaystyle H_{I\alpha J\beta}({\bf
k})=\left(\begin{array}[]{cccc}t^{\prime}Z^{\prime}&tZ^{\ast}&0&\lambda_{R}Y^{\ast}\\\
tZ&t^{\prime}Z^{\prime}&\lambda_{R}X^{\ast}&0\\\
0&\lambda_{R}X&t^{\prime}Z^{\prime}&tZ^{\ast}\\\
\lambda_{R}Y&0&tZ&t^{\prime}Z^{\prime}\end{array}\right),\;\;\;\;$ (33)
$\displaystyle Z\equiv e_{1}+e_{2}+e_{3},$ $\displaystyle Z^{\prime}\equiv
e^{\prime}_{1}+e^{\prime}_{2}+e^{\prime}_{3},$ $\displaystyle
X\equiv\frac{-1-i\sqrt{3}}{2}e_{1}^{\ast}+\frac{-1+i\sqrt{3}}{2}e_{2}^{\ast}+e_{3}^{\ast},$
$\displaystyle
Y\equiv\frac{1+i\sqrt{3}}{2}e_{1}+\frac{1-i\sqrt{3}}{2}e_{2}-e_{3},$ (34)

where the nearest neighbor ${\bm{\delta}}_{a}$ and next nearest neighbor
vectors ${\bm{\delta}}^{\prime}_{a}$ and the corresponding phase factors are

$\displaystyle{\bm{\delta}}_{1}=\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right),\;\;\;{\bm{\delta}}_{2}=\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right),\;\;\;{\bm{\delta}}_{3}=\left(-1,0\right),$
$\displaystyle{\bm{\delta}}^{\prime}_{1}=\left(-\frac{3}{2},-\frac{\sqrt{3}}{2}\right),\;\;\;{\bm{\delta}}^{\prime}_{2}=\left(\frac{3}{2},-\frac{\sqrt{3}}{2}\right),\;\;\;{\bm{\delta}}^{\prime}_{3}=\left(0,\sqrt{3}\right),$
$\displaystyle e_{a}=e^{i{\bf
k}\cdot{\bm{\delta}}_{a}},\;\;\;e^{\prime}_{a}=2\cos{\bf
k}\cdot{\bm{\delta}}^{\prime}_{a},$ (35)

These vectors and the resulting band structure using $t=2.8$eV$\equiv 1$ as
the energy unit, together with $t^{\prime}=0.1$eV$\equiv 0.036$ and
$\lambda_{R}=0.56$eV$\equiv 0.2$ (a rather large RSOC just to demonstrate the
splitting of bands), are shown in Fig. 2. From the Hamiltonian matrix in Eq.
(34), one can calculate the velocity operators by ${\hat{j}}_{\mu}({\bf
k})=e\,\partial_{\mu}H({\bf k})$, which may then be used to calculate the
optical conductivity and subsequently the quantum metric spectral function.

From the form of the Hamiltonian in Eq. (34), one can immediately conclude
that the next-nearest-neighbor hopping $t^{\prime}$ does not affect the
quantum metric and opacity of graphene, simply because it enters the diagonal
element of the Hamiltonian in the form of an identity matrix
$t^{\prime}Z^{\prime}\times I_{4\times 4}$. This means that $t^{\prime}$ does
give a $k$-dependent deformation to the dispersion without modifying the
eigenstates, and the deformation is the same for all the bands at ${\bf k}$
and hence it does not affect the $\delta(\omega+\varepsilon_{\ell}^{\bf
k}/\hbar-\varepsilon_{\ell^{\prime}}^{\bf k}/\hbar)$ condition in the quantum
metric spectral function either. As a result, the optical conductivity and
opacity are not affected by $t^{\prime}$.

##  Appendix C Polarization dependence of opacity

To calculate the dependence of the direction of polarization
${\hat{\bm{\mu}}}$ of the light, we consider ${\hat{\bm{\mu}}}$ to be pointing
at the polar angle $\phi_{E}$ on the $xy$-plane of graphene. The electric
field and the corresponding current operator in this situation may be
decomposed into

$\displaystyle{\bf E}=E_{0}\left(\cos\phi_{E}{\hat{\bf
x}}+\sin\phi_{E}{\hat{\bf y}}\right),$
$\displaystyle{\hat{j}}_{E}=\cos\phi_{E}{\hat{j}}_{x}+\sin\phi_{E}{\hat{j}}_{y}.$
(36)

The usual linear response theory requires to calculate the correlator
$\langle\left[{\hat{j}}_{E},{\hat{j}}_{E}\right]\rangle$, which yields an
optical conductivity (suppressing $({\bf k},\omega)$)

$\displaystyle\sigma_{E}$ $\displaystyle=$
$\displaystyle\cos^{2}\phi_{E}\sigma_{xx}+\sin^{2}\phi_{E}\sigma_{yy}$ (37)
$\displaystyle+\sin\phi_{E}\cos\phi_{E}\left(\sigma_{xy}+\sigma_{yx}\right),$

which is equivalent to calculating the quantum metric (note that
$g_{xy}=g_{yx}$)

$\displaystyle g_{E}$ $\displaystyle=$
$\displaystyle\cos^{2}\phi_{E}g_{xx}+\sin^{2}\phi_{E}g_{yy}$ (38)
$\displaystyle+2\sin\phi_{E}\cos\phi_{E}g_{xy}.$

The opacity will then carry the same angular dependence

$\displaystyle{\cal O}_{E}(\omega,\phi_{E})$ $\displaystyle=$
$\displaystyle\cos^{2}\phi_{E}{\cal O}_{xx}(\omega)+\sin^{2}\phi_{E}{\cal
O}_{yy}(\omega)$ (39) $\displaystyle+2\sin\phi_{E}\cos\phi_{E}{\cal
O}_{xy}(\omega),$

where ${\cal O}_{\mu\nu}(\omega)$ corresponds to the contribution coming from
$g_{\mu\nu}$. Note that if a light source is not polarized, then averaging
over the angle yields an opacity ${\overline{\cal O}}(\omega)=\left[{\cal
O}_{xx}(\omega)+{\cal O}_{yy}(\omega)\right]/2$, meaning that the contribution
from the off-diagonal element of the quantum metric $g_{xy}$ drops out.
However, for a polarized light, because ${\cal O}_{xx}(\omega)\approx{\cal
O}_{yy}(\omega)$ (for instance, $({\cal O}_{xx}(\omega),{\cal
O}_{yy}(\omega),{\cal O}_{xy}(\omega))=(1.175,1.182,0.277)\times\pi\alpha$ for
blue light $\hbar\omega=3.1$eV), the variation of ${\cal
O}_{E}(\omega,\phi_{E})$ in Eq. (39) as a function of the polarization angle
$\phi_{E}$ mainly comes from the $2\sin\phi_{E}\cos\phi_{E}{\cal
O}_{xy}(\omega)$ term contributed from the optical Hall conductance
$\sigma_{xy}(\omega)$, which is originated from the off-diagonal element of
the quantum metric $g_{xy}$.

##  Appendix D Effect of impurities

We proceed to utilize a many-body formalism to investigate the opacity of
graphene under the influence of impurities. To give an analytical result at
low energy, we adopt the spinless linear Dirac model of the basis $(c_{A{\bf
k}},c_{B{\bf k}})$ near the ${\bf K}$ point as described by Eq. (6) of the
main text, and consider the potential scattering

$\displaystyle{\hat{V}}=V\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right),$ (42)

where $V$ is the strength of impurity potential. We will focus on only one
spin species near the ${\bf K}$ valley, since all the four spin-valley flavors
give the same result. The corresponding vertex for the
$\ell=\left\\{n,m\right\\}$ band is

$\displaystyle V_{\bf kk^{\prime}}=\langle\ell{\bf k}|{\hat{V}}|\ell{\bf
k^{\prime}}\rangle=\langle n\phi|{\hat{V}}|n\phi^{\prime}\rangle$
$\displaystyle=\frac{V}{2}\left[1+e^{i(\phi^{\prime}-\phi)}\right],$ (43)

yielding the $T$-matrix for the $\ell$ band

$\displaystyle T_{\bf
kk^{\prime}}^{\ell}(\omega)=\frac{V}{2}\left[1+e^{i(\phi^{\prime}-\phi)}\right]\sum_{s=0}^{\infty}\left(\frac{V}{2}\int\frac{k_{1}dk_{1}}{2\pi/a^{2}}G_{\ell}({\bf
k_{1}},\omega)\right)^{s}$
$\displaystyle=\frac{\frac{V}{2}\left[1+e^{i(\phi^{\prime}-\phi)}\right]}{1-\frac{V}{2}\int\frac{k_{1}dk_{1}}{2\pi/a^{2}}G_{\ell}({\bf
k_{1}},\omega)}.$ (44)

So we are lead to the integration of the retarded Green’s function over the
module of the momentum, assuming that it has a cut-off $0<k_{1}<\pi/a$

$\displaystyle\int_{0}^{\pi/a}\frac{k_{1}dk_{1}}{2\pi/a^{2}}G_{\ell}({\bf
k_{1}},\omega)$
$\displaystyle=\int_{0}^{\pi/a}\frac{k_{1}dk_{1}}{2\pi/a^{2}}\left[\frac{1}{\omega\pm
vk}-i\pi\delta(\omega\pm vk)\right],$ (45)

where $+$ is for the valence band $\ell=n$ and $-$ is for the conduction band
$\ell=m$, which leads to the result

$\displaystyle T_{\bf
kk^{\prime}}^{n}(\omega)=\frac{\frac{V}{2}\left[1+e^{i(\phi-\phi^{\prime})}\right]}{1-\frac{Va^{2}}{4\pi
v^{2}}\left[\frac{\pi v}{a}-\omega\ln\left|\frac{\pi
v/a+\omega}{\omega}\right|\right]-i\frac{Va^{2}\omega}{4v^{2}}\theta(0>\omega>-\pi
v/a)},$ $\displaystyle T_{\bf
kk^{\prime}}^{m}(\omega)=\frac{\frac{V}{2}\left[1+e^{i(\phi-\phi^{\prime})}\right]}{1-\frac{Va^{2}}{4\pi
v^{2}}\left[-\frac{\pi v}{a}-\omega\ln\left|\frac{\pi
v/a-\omega}{\omega}\right|\right]+i\frac{Va^{2}\omega}{4v^{2}}\theta(0<\omega<\pi
v/a)},$ (46)

where the $\theta$-function ensures the range of the frequency $\omega$. The
self-energy at momentum ${\bf k}$ is simply the $T$-matrix at ${\bf
k=k^{\prime}}$ multiplied by the density of the impurity $\Sigma_{\ell}({\bf
k},\omega)=n_{imp}T_{\bf kk}^{\ell}(\omega)$, which can be used to calculate
the spectral function by

$\displaystyle A_{\ell}({\bf k},\omega)=-\frac{1}{\pi}{\rm Im}\,G_{\ell}({\bf
k},\omega)$ $\displaystyle=-\frac{1}{\pi}\frac{{\rm Im}\Sigma_{\ell}({\bf
k},\omega)}{(\omega\pm vk-{\rm Re}\Sigma_{\ell}({\bf k},\omega))^{2}+{\rm
Im}\Sigma_{\ell}({\bf k},\omega)^{2}}.$ (47)

After the spectral function is obtained, we further use the full Green’s
function approximation to obtain the dressed quantum metric spectral function
$\tilde{g}_{\mu\nu}^{\gamma}({\bf k},\omega)$ for the spin-valley flavor
$\gamma$ by

$\displaystyle\tilde{g}_{\mu\nu}^{\gamma}({\bf k},\omega)$ $\displaystyle=$
$\displaystyle\hbar
g_{\mu\nu}^{\gamma}\int_{-\infty}^{\infty}d\varepsilon\,A_{n}({\bf
k},\varepsilon)A_{m}({\bf k},\varepsilon+\hbar\omega)$ (48)
$\displaystyle\times\left[f(\varepsilon)-f(\varepsilon+\hbar\omega)\right]$

where $g_{\mu\nu}^{\gamma}$ is the unperturbed quantum metric. This dressed
spectral function can then be used to calculate the opacity using the
formalism presented in the main text. The spectral function $A_{\ell}({\bf
k},\omega)$ shown in Fig. 3 exhibits a broadening of quasilarticle peak that
gradually diminishes as $k\rightarrow 0$ owing to the imaginary part of self-
energy, and a positive energy shift relative to the Dirac point of the
pristine graphene due to the real part roughly given by the density of
impurities times their strength $n_{imp}V$. This positive energy shift
strongly suppresses the optical conductivity in the low frequency regime
$\hbar\omega\apprle 2n_{imp}V$, as argued in the main text.

Figure 3: Single-particle spectral function for the valence band $A_{n}({\bf
k},\omega)$ and the conduction band $A_{m}({\bf k},\omega)$ of disordered
graphene, where we fix the impurity density at $n_{imp}=10\%$ and plot the
results at impurity potential (a) $V=0.28$eV and (b) $V=4.2$eV. The dashed
line labels the Dirac point of the pristine graphene. The result suggests a
shift of energy spectrum towards positive energy caused by the impurities.

##  Appendix E Opacity of a Chern insulator

We remark that the constant opacity $\pi\alpha$ seems to manifest only if the
unperturbed topological material is gapless. As a counter example, consider a
2D Chern insulator described by adding a mass term $M\sigma_{z}$ into the
linear Dirac model in Eq. (6) of the main text, which gaps out the Dirac cone.
The resulting fidelity number spectral function is

$\displaystyle{\cal G}_{\mu\mu}(\omega)$ $\displaystyle=$
$\displaystyle\left[\frac{1}{16\pi\omega}+\frac{M^{2}}{4\pi\hbar^{2}\omega^{3}}\right]$
(49)
$\displaystyle\times\left[f\left(-\frac{\hbar\omega}{2}\right)-f\left(\frac{\hbar\omega}{2}\right)\right]_{\omega\geq
2|M|/\hbar}$

which yields an opacity that depends on the band gap $M$ and is not a constant
of frequency. Since the opacity depends on the color of the light and is not
directly proportional to the Chern number, it is hard to argue that one can
see the bulk topological invariant by naked eyes in this case.

##  Appendix F Detail of the optical absorption power of Weyl semimetal

Starting from Eq. (11) of the main text, the topological charge of each Weyl
point in a Weyl semimetal is calculated by the integration of Berry curvature
over a spherical surface of any radius that encloses the Weyl point, whose
sign depends on the chirality

$\displaystyle{\cal C}=\pm\frac{1}{4\pi}\int d\phi\int
d\theta\frac{1}{d^{3}}\varepsilon^{ijk}d_{i}\partial_{\theta}d_{j}\partial_{\phi}d_{k}$
$\displaystyle=\pm\frac{1}{4\pi}\int d\phi\int d\theta\sin\theta=\pm 1,$ (50)

and we denote the integrand by $J_{\bf
k}=\varepsilon^{ijk}d_{i}\partial_{\theta}d_{j}\partial_{\phi}d_{k}/d^{3}=\sin\theta$.
On the other hand, the quantum metric of each spin-valley flavor $\gamma$ on
the spherical surface is

$\displaystyle
g_{\theta\theta}^{\gamma}=\frac{1}{4},\;\;\;g_{\phi\phi}^{\gamma}=\frac{1}{4}\sin^{2}\theta,\;\;\;g_{\theta\phi}^{\gamma}=g_{\phi\theta}^{\gamma}=0,$
(51)

which satisfies the metric-curvature correspondence

$\displaystyle\sqrt{\det g^{\gamma}}=\frac{1}{4}|J_{\bf k}|.$ (52)

In the Cartesian coordinates, the metric is given by

$\displaystyle
g_{\mu\mu}^{\gamma}=\frac{1}{4k^{4}}\left(k^{2}-k_{\mu}^{2}\right),\;\;\;g_{\mu\nu}^{\gamma}|_{\mu\neq\nu}=-\frac{k_{\mu}k_{\nu}}{4k^{4}}.$
(53)

and hence the trace of the quantum metric at ${\bf k}$ for each spin-valley
flavor $g_{xx}^{\gamma}+g_{yy}^{\gamma}+g_{zz}^{\gamma}=1/2k^{2}$ only depends
on the module $k$ but not the direction ${\hat{\bf k}}$ of the momentum. As a
result, using the definition of $g_{\mu\mu}^{\gamma}({\bf k},\omega)$ and
${\cal G}_{\mu\mu}(\omega)$ in the main text with $D=3$, the conductivity in
real space summing over three crystalline directions is

$\displaystyle\sum_{\mu=x,y,z}\sigma_{\mu\mu}(\omega)=\frac{N_{W}e^{2}\omega}{4\pi\hbar^{2}}\left[\frac{1}{4\pi}\int
d\phi\int d\theta\sin\theta\right]$ $\displaystyle\times\int
dk\left[f\left(-vk\right)-f\left(vk\right)\right]\frac{\hbar}{2v}\delta(k-\frac{\hbar\omega}{2v})$
$\displaystyle=\frac{N_{W}e^{2}\omega|{\cal C}|}{8\pi\hbar
v}\left[f\left(-\frac{\hbar\omega}{2}\right)-f\left(\frac{\hbar\omega}{2}\right)\right],$
(54)

since the bracket $\left[...\right]$ in the second line is precisely the
topological charge $|{\cal C}|$ in Eq. (50). This averaged conductivity then
yields the optical absorption power summing over three directions described by
Eq. (12) of the main text.

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# Humans in Humans Out:
On GPT Converging Toward Common Sense in both Success and Failure

Philipp Koralus Faculty of Philosophy
University of OxfordInstitute for Ethics in AI
University of OxfordSt. Catherine’s College, Oxford
<EMAIL_ADDRESS>Department of Computer Science
University of OxfordSt. Catherine’s College, Oxford Vincent Wang-Maścianica
Department of Computer Science
University of OxfordSt. Catherine’s College, Oxford
<EMAIL_ADDRESS>
###### Abstract

Increase in computational scale and fine-tuning has seen a dramatic
improvement in the quality of outputs of large language models (LLMs) like GPT
[20, 4]. Given that both GPT-3 and GPT-4 were trained on large quantities of
human-generated text, we might ask to what extent their outputs reflect
patterns of human thinking, both for correct and incorrect cases. The Erotetic
Theory of Reason (ETR) provides a symbolic generative model of both human
success and failure in thinking, across propositional [13], quantified [19],
and probabilistic reasoning [18], as well as decision-making [17]. We
presented GPT-3, GPT-3.5, and GPT-4 with 61 central inference and judgment
problems from a recent book-length presentation of ETR [14], consisting of
experimentally verified data-points on human judgment and extrapolated data-
points predicted by ETR, with correct inference patterns as well as fallacies
and framing effects (the ETR61 benchmark). ETR61 includes classics like
Wason’s card task, illusory inferences, the decoy effect, and opportunity-cost
neglect, among others. GPT-3 showed evidence of ETR-predicted outputs for 59%
of these examples, rising to 77% in GPT-3.5 and 75% in GPT-4. Remarkably, the
production of human-like fallacious judgments increased from 18% in GPT-3 to
33% in GPT-3.5 and 34% in GPT-4. This suggests that larger and more advanced
LLMs may develop a tendency toward more human-like mistakes, as relevant
thought patterns are inherent in human-produced training data. According to
ETR, the same fundamental patterns are involved both in successful and
unsuccessful ordinary reasoning, so that the ”bad” cases could paradoxically
be learned from the ”good” cases. We further present preliminary evidence that
ETR-inspired prompt engineering could reduce instances of these mistakes.

##  1 Introduction

Large language models (LLMs) like GPT-4 have become increasingly important due
to their wide-ranging capabilities and are already having a transformative
impact in many applications. LLMs are designed to generate human-like text, as
well as other types of outputs, as reasonable responses to prompts. While
exact figures for GPT-4 have not been released, GPT-3 was trained on hundreds
of billions of words from diverse sources, including websites, books, and
articles. Besides very large data-sets of human-generated text, GPT-3.5 and
GPT-4 both also benefited from a human-guided fine-tuning training regime
[25]. Given that these systems have been trained on human input rather than
formal knowledge representations and their outputs are determined by
extrapolation of patterns found in those inputs (essentially a form of next-
word prediction), rather than by formal deductions, we may ask whether these
systems approximate human outputs in _failure_ , as well as in success. To
give structure to this investigation, we use an empirically motivated model of
human reason, the erotetic theory.

###  1.1 Success and failure in human reason through the lens of the erotetic
theory

Human reason has a two-faced nature. On the one hand, it has an unparalleled
range of successes to its credit, notably science and modern civilisation. On
the other hand, humans suffer from a variety of systematic failures of reason.
Many such experimentally documented failures have been widely popularised in
the literature [9, 2]. According to the erotetic (or “question-based”) theory
of reason (ETR), the two-faced nature of human reason is the result of one set
of basic principles. We treat incoming information, as well as choice
scenarios, as raising questions and we aim to resolve these questions as
directly as possible. In many situations, the dynamics of resolving questions
as directly as possible yields correct judgments, but it can also lead us
astray. Luckily, the processes proposed by ETR are guaranteed to produce
correct judgments, by the relevant standards of classical validity,
probabilistic coherence, and rational choice theory, provided that the agent
is in _erotetic equilibrium_ , e.g. in a state from which the judgment could
be reached regardless of what further questions might be raised.
Unfortunately, such equilibrium is often too costly to achieve. A key tenet of
the theory is that common-sense judgment is following the same core
principles, regardless of whether it is succeeding or leading us astray; the
only difference is whether we are robust with respect to further questions. We
will illustrate the basic idea with a simple example.

  * (1)

You have a hand of several cards. There is at least an ace and a queen in the
hand or at least a king and a jack. There is an ace in the hand. What if
anything follows?

A majority of participants in a variety of studies of this type of pattern
freely conclude that it follows that there is a queen in the hand [22, 16]. A
moment’s reflection shows that this is not the case: the premises are
compatible with a situation in which there is an ace, no queen, but a king and
a jack. The erotetic theory diagnoses the human tendency toward this fallacy
as follows: we take the premise on board as if it was a question, ‘‘Am I in an
ace and queen situation, or in a king and jack situation?’’ We then treat the
next premise as a maximally strong answer, ‘‘You’re in an ace situation!’’
Your ace situation has a queen in it as well, so you conclude that there is a
queen.

ETR holds that the same question/answer dynamics are behind ordinary judgment
more broadly, including cases in which we draw correct inferences. For
example, consider _modus ponens_. From ‘‘If there is an ace in the hand, then
there is king in the hand’’ and ‘‘There is an ace in the hand’’, virtually
everybody readily infers ‘‘There is a king in the hand’’. ETR also treats this
as a case of the same question/answer dynamic we have already considered: we
take the conditional premise on board as a question, ‘‘Am I in an ace and king
situation, or am I in a non-ace situation?’’ We then take the second premise
as, ‘‘You are in an ace situation!’’, and, again, treat it as a maximally
strong answer, yielding the conclusion that there is a king. In other words,
the same basic dynamics involved in correct reasoning are also involved in
fallacious reasoning. The key difference, according to ETR, is that fallacious
judgments are not in equilibrium with respect to further questions that might
be asked. Returning to the first question, suppose I have taken on board: ‘‘Am
I in an ace and queen situation, or in a king and jack situation?’’ If I now
further ask myself, e.g. ‘‘What about further situations with respect to the
question of whether there is an ace?’’, I end up with, ‘‘Am I in an ace and a
queen situation, a king, jack, and ace situation, or a king, jack, and no ace
situation?’’ If I now take on board ‘‘You are in an ace situation!’’ as a
maximally strong answer as before, I no longer get the conclusion that there
is a king. The fallacious inference was not an erotetic equilibrium inference.
However, if we gave the _modus ponens_ inference the same treatment of asking
further questions, we would still arrive at the conclusion that there is a
king. This is because correct inferences are erotetic equilibrium inferences;
they are robust to further questions.

The patterns of judgment errors that can be explained erotetically are not
limited to logical inference. Similar patterns can be identified in reasoning
about probability. For example, it is well-known that humans have a tendency
in certain cases to rank the probability of a conjunction as higher than the
probability of individual conjuncts, violating the axioms of probability [21].
If we are told that someone is a math genius as well as an athletic
outdoorswoman, we might naïvely judge it to be more likely for this person to
be a computer scientist and active in the climbing community, than for this
person to be active in the climbing community. The ETR diagnosis is that
‘‘You’re in a math genius and athletic outdoorswoman situation!’’ more
strongly answers in favour of the conjunction (provided no further questions
are asked).

To consider another example from the domain of decision-making, it is known
that we suffer from irrational opportunity-cost neglect [7]. If we are told
that we have saved some money for fun in a hypothetical scenario, there is a
greater chance we will say ‘‘Buy!’’ to ‘‘Buy an entertaining video or don’t
buy an entertaining video?’’ than if we are asked ‘‘Buy an entertaining video
or save your money for other purchases?’’ The ETR diagnosis is that ‘‘Fun!’’
more strongly answers in favour of buying the video than in favour of not
buying the video – unless you remind yourself that not buying the video is a
situation that likely has other potential fun purchases down the line, and
make that part of what your decision-question considers.

The question/answer dynamic proposed by ETR is represented set-theoretically,
treating questions fundamentally as sets of alternative states, with the
dynamics of inference and judgment treated as update rules on sets. ETR to-
date covers propositional reasoning [13, 15], first-order quantified
relational reasoning [19], reasoning with uncertainty [18] and decision-making
[17]. ETR is symbolic, in the sense that its outputs are determined by sets of
states and the atoms that occur within those states. ETR is a generative
theory, in the sense that it produces concrete inferences from a given
sequence of inputs. In other words, we can ask it what if anything follows
from a sequence of premises, or what the decision is for a given decision-
question and a given a set of priorities. The theory has been motivated by a
large number of data-points from the empirical literature on human reasoning
and judgment, and has also been corroborated by experiments on ETR-specific
predictions. For example, answering a question is an inherently ordered
process, since you cannot meaningfully treat something as an answer without
having first taken on board a question. ETR correctly predicts that the extent
to which people fall for our opening example (1) of a fallacious inference is
lessened if the order of premises is reversed [16]. After all, if we are first
given ‘‘You are in an ace situation!’’ and then ‘‘Am I in an ace and queen
situation, or in a king and jack situation?’’ it is harder to treat the former
as an answer to the latter.

###  1.2 Predicting failure patterns

According to ETR, the same dynamics of reason underly common sense thinking in
both success and failure. This would lead us to expect that if we were to
train a system to approximate the dynamics of common sense thinking based on
human-produced text, the familiar fallacious judgment patterns should emerge
as well. On the ETR view, this would be the case even if the text used as a
training basis vastly over-represents cases of _correct_ common-sense
judgment. This is relevant since we expect that GPT would have been trained
somewhat judiciously to approximate high-quality output and to solve problems
with the hope of objective correctness. Yet, on the ETR view, it would not be
a a surprise if a better approximation of _good_ common-sense reasoning also
yields a better approximation of _bad_ common-sense reasoning. The ETR view of
how the successes and failures of reason are linked contrasts with views that
might take mistakes to be the result of special purpose heuristics. In
addition, insofar as judgment errors are made as a result of approximating
patterns of common-sense judgment as envisaged by ETR, we would further expect
that prompts to take on board broader questions should reduce the occurrence
of fallacious judgments.

To investigate these hypotheses, we studied the outputs of GPT with key
examples from a recent book-length treatment of ETR as inputs. We then
examined whether ETR-inspired prompt engineering can reduce the incidence of
certain fallacious judgments that the GPT systems produced.

##  2 Study 1: Reasoning patterns in GPT with the ETR61 benchmark

We logged the outputs of GPT-3, GPT-3.5, and GPT-4 for a set of 61 inference
and decision-making problems for which ETR makes predictions about common-
sense responses. We will refer to this set as the ETR61 benchmark. ETR61 was
created by compiling the main examples throughout a book-length presentation
of ETR [14], including some novel content variations. The examples covered a
variety of types of problems, across logical and probabilistic inference, as
well as decision-making. Many examples were variants of cases that have been
shown experimentally to elicit incorrect or incoherent responses in humans.
Some additional examples are reproduced below.

  * (2)

Some blue cards are textured. All square cards are blue. What if anything
follows?

Human participants tend to fallaciously conclude that some square cards are
textured [10].

  * (3)

There are several cards on the table, which have a letter on one side and a
number on the other side. One card shows and E, one card shows a C, one card
shows a 4, and one card shows a 5. Which cards do you have to turn over to
determine if the following statement is true? If a card has an E on one side
then it has a four on the other side.

In this type of problem the majority of human participants omit to turn over
the ”5” card, even though it is necessary to do so to determine whether the
conditional is falsified [23].

  * (4)

[Abbreviated] Linda is thirty-one years old. She majored in philosophy. As a
student, she was deeply concerned with issues of discrimination and social
justice. Please rank order by probability (highest to lowest) the following:
Linda is a bank teller. Linda is a bank teller and is active in the feminist
moment.

Human participants have a tendency to rank the probability of the conjunction
higher than the probability of the conjuncts [21], violating the axioms of
probability theory.

  * (5)

Which of the following subscription would you be most likely to purchase. 1.
Economist.com subscription - US $59.00. One-year subscription to
Economist.com. Includes online access to all articles from The Economist since
1997. [2. Print subscription - US $ 125.00. One-year subscription to the print
edition of The Economist.] 3. Print & web subscription - US $125.00. One-year
subscription to the print edition of The Economist and online access to all
articles from The Economist since 1997.

Human participants have been reported as being more likely to pick option 3 if
the dominated option 2 (which nobody chose) is included in the menu [2],
violating the axioms of rational choice theory.

In constructing the ETR61 benchmark, no systematic commitment was made to
replicate the vignettes from human experiments exactly (though most of them
were exact replications). We changed the wording and parameters of various
problems to overcome the resistance of GPT to make recommendations, and to
deal with the possibility of some problems having been strongly present in the
training data. We only varied examples in ways that left the ETR predictions
on those problems unchanged.

###  2.1 Methodology

The GPT-3 model used was accessed at https://platform.openai.com/playground,
with parameters text-davinci-003 at temperature 0.7, max-output 256 tokens,
top P = 1, 0 frequency and presence penalities, best-of-1. We refer to ChatGPT
as GPT-3.5 and ChatGPT with GPT-4 backend as just GPT-4. Both GPT-3.5 and
GPT-4 models used were accessed at https://chat.openai.com/chat in March 2023,
using the default 3.5 and 4 models.

For each version of GPT, we used ETR61 as prompts under two conditions:
production and query. Each question in each condition was entered into a
refreshed context-free instance of the model. In the production condition,
questions were asked in the form of premises ended by ‘‘What, if anything,
follows?". Answers that that did not mention the main ETR-predicted conclusion
but included statements that imply it were further prompted with a topical
query. For GPT-3, which had a relatively limited max-output length, submission
was repeated whenever answers were cut midway.

In the query condition, questions were asked in the form of premises ended by
‘‘Does it follow that X?" in place of ‘‘What, if anything follows?" where $X$
was the ETR-predicted response. No follow-up questions were applied in the
query condition. For both cases, appropriate semantic variants of ‘‘What, if
anything follows?" or ‘‘Does it follow that X?" were assigned for questions
involving probability rankings and decision-making.

For every question in both conditions, correctness was recorded and the
transcript logged. Correctness was judged by the following criteria. A correct
declaration of the answer anywhere was marked correct regardless of the
ensuing justification. This gave GPT-3 an inflated correctness rate, as it
often produced nonsense justifications. Mindful of poor native numerical
ability in language models [8] (and workarounds [24]), for questions
concerning probability, correctness was awarded for either correct answers or
correct procedure _e.g._ correct application of Bayes’ rule. For rankings of
subjective probability, correctness was awarded for probabilistic consistency
_e.g._ $\mathbf{P}(A)\geq\mathbf{P}(A\&B)$ anywhere is inconsistent. For
decision questions, correctness was awarded for consistent decision-making
across materially equivalent framings of the choice scenarios.

###  2.2 Results

We adopt the convention of statistical significance at $p\leq 0.05$. All
significance tests are Wilcoxon signed-rank as implemented in Mathematica
version 13.0.1.0. by the SignedRankTest method. Since it is known [3] that
different computer packages for rank-sum statistics may yield different
judgements of significance at the $p=0.05$ level, we report fair significance
($0.01\leq p\leq 0.05$) in orange and strong significance ($p\leq 0.01$) in
red. We introduce each test condition with a gloss, we report the percentage
rate for each test across the GPTs, and we provide the statistical
significance of the change in rates; we tabulate the change in rate from GPT-3
to GPT-3.5 as “v3-v3.5”, GPT-3.5 to GPT-4 as “v3.5-v4”, and GPT-3 to GPT-4 as
“v3-v4”.

We found that GPT-3.5 produced fewer correct answers than either GPT-3 or
GPT-4. In Table 1 we observe a dip-then-rise in the rate of production of
correct answers across generations, and a steady-then-rise in the endorsement
of correct answers. We observe a steady-then-sharp-rise in correctness and
consistency, taken to be both the production and endorsement of correct
answers.

_How correct are GPTs as productive reasoners in ETR61?_
---
Correct answer produced | Percentage | GPT-3 | GPT-3.5 | GPT-4
56% | 48% | 64%
Cross-generational statistical significance | v3-v3.5 | v3.5-v4 | v3-v4
0.3248 | 0.0346 | 0.3752
_How often do GPTs endorse correct conclusions in ETR61?_
Correct answer endorsed | Percentage | GPT-3 | GPT-3.5 | GPT-4
48% | 52% | 72%
Cross-generational statistical significance | v3-v3.5 | v3.5-v4 | v3-v4
0.6701 | 0.0150 | 0.0045
_How often is GPT consistently correct in ETR61?_
Correct production and endorsement | Percentage | GPT-3 | GPT-3.5 | GPT-4
30% | 30% | 59%
Cross-generational statistical significance | v3-v3.5 | v3.5-v4 | v3-v4
1 | 0.0004 | 0.0017
Table 1: Cross-generational tests for correctness in reasoning across GPTs.

_More advanced models exhibit more common sense._ In Table 2 we observe upward
trends in rates of answers produced and answers endorsed that are predicted by
ETR as common-sense judgments. There is a rise-then-steady pattern when we
consider any instance of common-sense reasoning in either production or
endorsement. While these trends are not statistically significant stepwise
between generations, there is strong evidence that GPT-4 is overall more
common-sense congruent than GPT-3.

_How often do GPTs produce ETR-predicted common-sense answers?_
---
Production predicted by ETR | Percentage | GPT-3 | GPT-3.5 | GPT-4
49% | 61% | 72%
Cross-generational statistical significance | v3-v3.5 | v3.5-v4 | v3-v4
0.1134 | 0.1316 | 0.0030
_How often do GPTs endorse ETR-predicted common-sense answers?_
Endorsement predicted by ETR | Percentage | GPT-3 | GPT-3.5 | GPT-4
36% | 49% | 54%
Cross-generational statistical significance | v3-v3.5 | v3.5-v4 | v3-v4
0.1060 | 0.5255 | 0.0354
_How often do GPTs exhibit ETR-predicted common-sense responses?_
Either above predicted by ETR | Percentage | GPT-3 | GPT-3.5 | GPT-4
59% | 77% | 75%
Cross-generational statistical significance | v3-v3.5 | v3.5-v4 | v3-v4
0.0083 | 0.8213 | 0.0268
Table 2: Cross-generational tests for common-sense reasoning across GPTs.

_More advanced models are more prone to ETR-predicted fallacies._ We consider
fallacies to be ”mistakes of common sense” that correspond to incorrect
answers that are predicted by the default reasoning procedure in ETR. We note
in Table 3 that production and endorsement of fallacies rises from GPT-3 to
GPT-3.5, and either holds steady or drops slightly from GPT-3.5 to GPT-4.
Remarkably, in every case, the more advanced GPT-4 is more fallacy-prone than
GPT-3.

_How often do GPTs produce ETR-predicted human-like errors?_
---
Production fallacious | Percentage | GPT-3 | GPT-3.5 | GPT-4
18% | 33% | 34%
Cross-generational statistical significance | v3-v3.5 | v3.5-v4 | v3-v4
0.0140 | 0.8121 | 0.0135
_How often do GPTs endorse ETR-predicted human-like errors?_
Fallacy endorsed | Percentage | GPT-3 | GPT-3.5 | GPT-4
18% | 23% | 20%
Cross-generational statistical significance | v3-v3.5 | v3.5-v4 | v3-v4
0.4281 | 0.5941 | 0.8213
_How often do GPTs fall prey to ETR-predicted human-like errors?_
Fallacy produced or endorsed | Percentage | GPT-3 | GPT-3.5 | GPT-4
26% | 43% | 36%
Cross-generational statistical significance | v3-v3.5 | v3.5-v4 | v3-v4
0.0135 | 0.2669 | 0.1647
Table 3: Cross-generational tests for fallacious reasoning across GPTs.

_GPT-4 produces significantly more fallacies than it endorses._ We performed
several intra-generational comparisons, reported in Table 4. First, we know on
reasoning problems, human participants often produce different results
depending on whether they are asked to produce a conclusion or whether they
are asked to whether a given conclusion follows [10]. We observe that later
generations appeared to produce more fallacies than they endorsed, and we
report that GPT-4 produces significantly more fallacies than it endorses.

Do GPTs produce answers other than what they endorse?
---
ETR predicts production vs. ETR predicts endorsement | GPT-3 | GPT-3.5 | GPT-4
0.6552 | 0.3248 | 0.1169
Do GPTs differ in production vs. verification performance?
Production correct vs. endorsement correct | GPT-3 | GPT-3.5 | GPT-4
0.3429 | 0.5588 | 0.1450
Is there a difference in rate of production vs. endorsement of fallacies?
Fallacy produced vs. fallacy endorsed | GPT-3 | GPT-3.5 | GPT-4
1 | 0.2610 | 0.0134
Table 4: Intra-generational tests across GPTs.

###  2.3 Discussion

We observe that while we see improvements in correct reasoning with later
generations of GPT, we also see an increase in the susceptibility to
systematic fallacies. Remarkably, GPT-4 did not produce fewer ETR-predicted
fallacious conclusions than GPT-3.5; while GPT-3.5 in fact produced more of
these fallacious conclusions than GPT-3. This is would be expected if a
mixture of more training (including reinforcement learning from human
feedback) and increased model-size increase the fidelity with which human
common-sense reasoning patterns (as conceived of by ETR) are approximated.
Unfortunately, exact details of the model differences have not been made
public, making it difficult to theorise more specifically about the mechanisms
responsible for the observed differences in model behaviour.

Some of the relevant outputs produced to GPT 4 were in fact reminiscent of the
dynamics of reasoning described by ETR. Though these observations have to be
interpreted with caution, they are potentially suggestive.

  * _Prompt:_ Either Jane is kneeling by the fire and she is looking at the TV or else Mark is standing at the window and he is peering into the garden. Jane is kneeling by the fire. What if anything follows?

  * _GPT4:_ Since Jane is kneeling by the fire, it follows that the first part of the statement is true. [$\dots$] So, what follows is that Jane is kneeling by the fire and looking at the TV.

This is reminiscent of the pattern described in section 1.1, according to
which entire disjuncts are treated as alternatives in a question, and the
subsequent premise is used as a maximally strong answer, effectively selecting
the alternative with the greatest overlap, instead of engaging with the logic
of disjunction. We also found some evidence of the ETR-predicted order effect
discussed in section 1.1: GPT4 produced the relevant fallacious conclusion for
"There is an ace and a queen, or else a king and a ten. There is a king." but
not for "There is a king. There is an ace and a queen, or else a king and a
ten."

##  3 Study 2: Prompt engineering to mitigate fallacies

As in the case of human judgment, the presence of systematic mistakes raises
the question of whether we can mitigate these mistakes by following a better
reasoning strategy. Adding global instructions like “reason step-by-step” can
improve the accuracy of LLM outputs [11]. According to ETR, mistakes are due
to not having taken on board sufficiently broad questions before treating
further information as an answer. We examined whether a simplistic version of
this idea could improve GPT performance.

###  3.1 Method

For each of GPT-3, 3.5, and 4, we ran augmented prompts based on a subset of
ETR61, following a similar procedure that described previously. For each
version of GPT, we only used those original prompts that had produced
incorrect judgments that were predicted by the basic step in the ETR default
reasoning procedure. In other words, we only examined those cases that yielded
mistakes and where those mistakes were most directly generated by the
question/answer dynamic at the core of ETR.

To construct the augmented prompts we used the following templates:

(Control)

Reason step-by-step for the following problem. [Original prompt inserted here]

(ETR)

Answer the following question according to this procedure: First, list the
premises. Second, turn each premise into a question to make a new list of
questions; treat questions as possible alternatives. Third, reason step-by-
step using both lists, keeping track of alternatives. [Original prompt
inserted here]

###  3.2 Results

We found that both the control prompt template and the ETR prompt template
reduced the incidence of incorrect judgment outputs. In the case of GPT3.5,
the ETR prompt was better than the control prompt at a level that was
statistically significant.

Are fallacies blocked by erotetic prompting?
---
Model | GPT-3 | GPT-3.5 | GPT-4
Fallacies blocked by control prompt | 4/8 | 4/17 | 8/19
Fallacies blocked by ETR prompt | 4/8 | 14/17 | 7/19
Stat. sig. | 1 | 0.0045 | 0.7656
Table 5: Prompting GPT-3.5 to reason erotetically was significantly better at
blocking fallacies compared to step-by-step prompting. On other models both
prompts had comparable effect on blocking fallacies.

##  4 Conclusion and future work

The results of Study 1 suggest that GPT-3.5 and GPT-4 outputs are
significantly more similar to common-sense judgment than GPT-3 outputs.
Remarkably, this also applies to patterns of fallacious judgments. In fact,
the _production_ of fallacious judgments from ETR61, including deductive
reasoning, judgments about probability, and decision-making, increases from
GPT-3 to GPT-4. On the view of human reasoning proposed by ETR, this is not
surprising. ETR holds that both successful and unsuccessful uses of common
sense are based on the same principles, so that a better approximation of the
dynamics of common-sense, as provided by more training and larger models,
would also generate more common-sense fallacies, even if the LLMs are
primarily learning from success cases. Study 2 suggests that some of the
insights from ETR could inform strategies for prompt engineering to lessen the
incidence of common-sense fallacies in LLMs like GPT.

Study 1 bears on the question of how future improvements of LLMs relate to
data scale. Given that the later generation LLMs examined were in fact more
prone to human-like fallacies, even though they involved more processing on
larger data sets, we speculate that the relevant problems are not easily
remedied by more generic training on human-generated text, which will have the
relevant thought patterns ingrained in it. However, ETR suggests a possible
solution. With an implementation of ETR, it would be possible to generate
synthetic datasets of arbitrary size and complexity of fallacy-prone problems
and their correct solutions that could be used to train against them.
Similarly, an implementation of ETR, could be used to construct a dynamic
generative benchmark to test LLM performance. This may help further pave the
way toward LLMs as _cleaned-up_ common-sense approximators.

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