\section{ARPES of non-polar (110) surface}



\subsection{A. Yb divalency and wide EDC}


\begin{figure*}[t]
\includegraphics[width=13.5 cm]{sfig1eps.pdf}
\caption
(a) Angle-integrated 120 eV spectra of YbB$_{6}$\ (110) surface
demonstrating the pure divalency of Yb,
and ruling out the calculations in Fig. \ref{sdft1}(b,c).
(Inset) Magnified view of the overlapping B-$sp$ and Yb$^{3+}$ regions
showing the weaker B-$sp$ cross section for $s$-polarization
and the background subtraction for quantitative valence analysis.
Also shown (dashed line) for comparison is the signature spectral line shape
for Yb$^{3+}$ $f^{13} \rightarrow f^{12}$ multiplet peaks
acquired from an aged (001) cleaved surface of YbB$_{6}$.
(b) Low-energy electron diffraction of the prepared (110) facet
showing 1 $\times$ 1 surface order.
(c) Wide-energy 120 eV X-point spectrum of the YbB$_{6}$\ (110) surface
showing the divalent Yb $4f_{7/2}$ peak at 1.05 eV
and overplotted DFT bands along X-M.
}
\label{valence}
\end{figure*}


The deep binding energy (BE) of the bulk Yb 4\emph{f} levels
and lack of Yb$^{2+}$ spectral weight near the Fermi level $E_\textrm{F}$\
is strong experimental evidence for pure divalency in YbB$_{6}$.
However, another important test for mixed-valency is the observation
and quantification of the relative amount of Yb$^{3+}$ spectral weight
in valence band photoemission measurements.
Previous XPS and UPS measurements \cite{sm-Kakizaki93,sm-Nanba93}
observed distinct Yb$^{3+}$ contributions to Yb 4\emph{p}
core-level and valence band spectra,
but the origin was ascribed to surface contributions.
Similarly, a weak ferromagnetic moment corresponding to 2\% bulk trivalency
was initially reported in YbB$_{6}$\ \cite{sm-Gavilano03}.
But later measurements showed stoichiometric YbB$_{6}$\ to be diamagnetic,
with surface ferromagnetism or paramagnetism appearing only
for non-stoichiometric samples \cite{sm-Kim07}.


In Fig. \ref{valence}(a), we demonstrate the pure divalency of Yb
from a wide valence band spectrum for a prepared YbB$_{6}$\ (110)
surface measured at $h\nu$ = 120 eV.
The spectrum exhibits Yb$^{2+}$ $4f_{7/2}$ and $4f_{5/2}$ peaks
that dominate over the very weak 4 - 12 eV BE range
where Yb$^{3+}$ spectral weight would be found.
Overlapping in this energy range are B-\emph{sp} valence band states
whose relative cross section is low at this photon energy
and which can be further suppressed using \emph{s}-polarization
of the incident photons, as shown in the Fig. \ref{valence}(a) inset.

The line shape of the valence band structure,
shown in the expanded view of this 4 $\sim$ 12 eV range,
exhibits no evidence for any Yb$^{3+}$ spectral weight
whose distinct $f^{13} \rightarrow f^{12}$ final state multiplet peak structure
is illustrated in the inset of Fig. \ref{valence}(a) (dashed line).
This example of a Yb$^{3+}$ spectrum comes from a cleaved (001) surface
that initially looks like the pure divalent (110) surface spectra,
but then develops within only 4 hours the appearance of this Yb$^{3+}$ signature,
indicating the trivalent conversion of surface Yb atoms due to residual gas adsorption.
In contrast, the (110) surface with coplanar Yb and  B$_{6}$\  termination
and 1 $\times$ 1 surface order,  shown in Fig. \ref{valence}(b),
exhibits no spectral changes after greater than 12 hours of measurement.
Even if one integrates the B-\emph{sp} spectral weight
of the \emph{s}-polarization spectrum,
after employing the background subtraction shown in Fig. \ref{valence}(a),
and assigns it to Yb$^{3+}$,
one obtains an upper bound contribution of less than 1\%.
This tight experimental upper bound on the pure divalency of Yb in YbB$_{6}$\
immediately rules out both of the theoretical predictions of $f$-$d$ \cite{sm-Weng14}
and $p$-$d$ \cite{sm-Neupane15,sm-Chang15} band inversion scenarios,
both of which predict mixed Yb valency.


A 12 eV wide angle-resolved spectrum measured at the 120 eV bulk X-point,
shown in Fig. \ref{valence}(c), also illustrates the wide B-\emph{sp} band dispersions
in the 4 - 12 eV range, and the absence of any $k$-independent Yb$^{3+}$ peaks.
The spectrum additionally illustrates the 1 eV and 2.3 eV binding energies
of the Yb $4f_{7/2}$ and Yb $4f_{5/2}$ states, respectively.
The Yb 4$f$ peaks are observed to be uniform across the prepared (110) sample surface,
in contrast to the reported energy shifts between cleaves,
and spatial inhomogeneity of the cleaved (001) surface \cite{sm-Xia14,sm-Xu14,sm-Frantzeskakis14}.
The boron hole band that forms the valence band maximum
is observed to disperse from below the Yb $4f_{5/2}$ state
and significantly hybridize with the \emph{f}-states
as it passes through towards $E_\textrm{F}$.
A small band width expansion of 1.05 of the DFT bands
was required to best match experiment.




\subsection{B. X-point location}


\begin{figure}[b]
\includegraphics[width=8.5 cm]{sfig2eps.pdf}
\caption
(a) Normal-emission photon-dependent map with the sample $(1\bar{1}0)$ direction
aligned to vertical spectrometer slit.
The identity of the $\Gamma$-point defines the inner potential ($V_{0}$ = 14 eV).
The vertical line indicates the off-normal plane of the photon-dependent maps
in Fig. 2(a) of the main text.
(b) Angle-dependent map at 120 eV, showing the $k$-location of the X-point spectrum
used for the band gap analysis Fig. 1 in the main text.
}
\label{xpoint}
\end{figure}


A suitable bulk X-point location for the band gap determination
was chosen based on photon-dependent and angular-dependent mapping of the momentum-space.
First a normal-emission photon-dependent $k_{x}-k_{z}$ map
was acquired with the sample $(1\bar{1}0)$ axis
aligned vertically with the spectrometer slit.
An energy cut at $-1.1$ eV, between the two 4\emph{f} states,
highlights a strong feature at 61 eV,
as shown in Fig. \ref{xpoint}(a) that aligns to the bulk Brillouin zone (BZ)
using an inner potential parameter of 14 eV.

Then the sample is rotated in azimuth to align the sample (100) axis
to the vertical spectrometer slit and an off-normal photon-dependent map
was acquired at fixed $k_{x} = -0.53 {\AA}^{-1}$
as indicated in Fig. \ref{xpoint}(a).
The $k_{y}-k_{z}$ map at fixed $k_{x}$,
presented in Fig. 2(a) of the main text,
with an energy cut 0.3 eV below $E_\textrm{F}$\
probing the valence band hole dispersion,
shows two strong hole-band features
at $k_{y}$ = 0 for $h\nu$ = 76 eV and 120 eV
as well as at four weaker intensity second BZ X-points
at $k_{y} = \pm1.5 {\AA}^{-1}$.

$h\nu$ = 120 eV was chosen for the $k_{x}-k_{y}$ angular-dependent map
in Fig. \ref{xpoint}(b) with the same sample orientation.
Similar to the photon-dependent map, strong valence band hole intensities
are observed at three different X-points at $k_{y}$ = 0 and also additionally
at four more X-points in the second BZ at $k_{y} = \pm1.5 {\AA}^{-1}$.
The bulk X-point at $k_{x} = -0.53 {\AA}^{-1}$ was used
for Fig. 1(e) in the main text
and was checked for consistency at other bulk X-points.


\subsection{C. K-dose}


\begin{figure}[b]
\includegraphics[width=8.5 cm]{sfig3eps.pdf}
\caption
(a) X-point spectrum at $h\nu$ = 76 eV of the YbB$_{6}$\ (110) surface before K-dosing
showing $E_\textrm{F}$\ pinned to the bottom of the Yb 5$d$ conduction band.
(b) K-dosing EDC image of the valence and conduction band energy shifts.
(c) X-point spectrum after K-dosing allowing
conduction band dispersion analysis for band gap determination.
}
\label{K-dose}
\end{figure}


To further characterize the conduction band minimum energy,
surface K-dosing was used to create a small $n$-type band bending
until the electron pocket was sufficiently deep
to observe its non-parabolic dispersion.
For this experiment, an X-point spectrum was obtained at 76 eV instead of 120 eV
due to the strong $d$-electron ($p$-hole) band intensity
for $s$ ($p$) polarization, allowing a
good quantitative analysis of the non-parabolic conduction (valence)
band dispersion and semiconductor gap evaluation
(see subsection D).

Fig. \ref{K-dose}(a) shows the X-point $p$-polarization spectrum before K-dosing.
Fig. \ref{K-dose}(b) shows a time/dosing dependent
energy distribution curve (EDC) cut at X
illustrating the energy shifts of the valence and conduction bands
as well as the Yb 4\emph{f} state.
The final EDC spectrum, shown in Fig. \ref{K-dose}(c),
is a sum of $p$- and $s$-polarizations
due to the weakening of the \emph{p}-hole band with K-dosing.
Because a rigid shift of both valence and conduction bands
is observed in Fig. \ref{K-dose}(b)
with no discernible surface band gap narrowing,
we can claim that this K-dosed analysis is reflective of a bulk gap
of magnitude 0.32 eV, based on non-parabolic dispersion visual fits to
the polarization-dependent data that best highlight either
the hole valence band or electron conduction band dispersions \cite{sm-Denlinger15}.



\subsection{D. Non-parabolic Dispersion}

A two-band $k\cdot p$ model for non-parabolic valence and conduction bands
in the presence of small band gaps \cite{sm-Kane57} was used to assist
in the determination of the valence band maximum and conduction band minimum energies.
The two band $k\cdot p$ correction to the free electron parabolic dispersion
can be written simply as \cite{sm-Lopez93}:
\begin{equation}
\varepsilon(k) = \frac{E_{g}}{2}\Bigg(\sqrt{1+\frac{4}{E_{g}}
	\cdot\frac{\hbar^{2}k^{2}}{2m^{*}}}-1\Bigg),
\label{kp}
\end{equation}
where $E_{g}$ and $m^{*}$ are the energy gap parameter
and the electron effective mass, respectively.

The energy gap dependence of Eq. (\ref{kp}) exhibits
the well-known progression towards a purely linear V-shaped dispersion at zero gap,
corresponding to a Dirac semimetal.
The valence band dispersion in Fig. 1(d) of the main text exhibits
linearity away from a distinctly rounded band maximum
and is nicely fit using the nominal experimental gap value of 0.3 eV
and an effective mass of $m^{*}$ = 0.12.
This dispersion shape for intermediate small gap values
contrasts sharply with topological Dirac cone surface states
in Bi$_{2}$X$_{3}$ (X = Se, Te), where the dispersion is mostly linear
near the Dirac point and deviates further away ({\it e.g.}, hexagonal warping).

The conduction band dispersion, revealed after K-dosing in Fig. \ref{K-dose}(c),
is similarly fit to $E_{g}$ = 0.3 eV and $m^{*}$ = 0.1.


\section{Computational details}

The band calculations were performed by using
the full-potential linearized augmented plane-wave (FLAPW)
band method, as implemented in the WIEN2K package \cite{sm-wien2k}.
For the DFT calculations, the PBE (Perdew-Burke-Ernzerhof) exchange-correlation
functional was used in the GGA (generalized-gradient approximation).
The spin-orbit coupling (SOC) was included in a way of the second variational method.
In the GGA + SOC + $U$ method, a correlation energy of $U=7$ eV was
chosen to obtain the correct experimental value of the
Yb 4\emph{f} BE of $\approx$ 1 eV,
which is consistent with the previous calculations \cite{sm-Jun07,sm-Kunes04}.
The mBJ potential corresponds to an orbital-independent
semilocal exchange potential
that mimics the behavior of orbital-dependent potentials
and has been shown to provide band gap corrections
in good agreement with the improved many-body
but more computation-demanding GW calculation \cite{sm-TB09,sm-Singh10}.
A single pass open-core pseudopotential GW calculation
was also performed using VASP \cite{sm-vasp},
which effectively removes the Yb 4\emph{f} levels from the calculation.
This is reasonable in view of the deep 4\emph{f} BE in YbB$_{6}$.


In the GGA + $U$ ($U$: on-site Coulomb interaction of Yb $4f$ electrons) calculations,
the scheme of a fully-localized limit was used for the 4\emph{f} electrons.
The BZ integration was done with a 17 $\times$ 17 $\times$ 17 $k$-mesh, and,
in the FLAPW calculations, muffin-tin radii
$R_{MT}$'s of Yb and B were chosen to be 2.50 and 1.59 a.u.,
respectively, and the plane-wave cutoff was $R_{MT}K_{max}$ = 7.

For the GW calculation, a single pass open-core pseudopotential
G$_{0}$W$_{0}$ calculation was performed using VASP
with a 4 $\times$ 4 $\times$ 4 $k$-mesh,
400 unoccupied bands and 80 mesh size in the omega frequency.


YbB$_{6}$ crystallizes in the CsCl-type structure, like SmB$_{6}$.
The ambient pressure calculations were performed
using the experimental YbB$_{6}$ lattice parameter of \emph{a} = 4.1792 {\AA}
and internal crystal parameter $x_{B}$ = 0.202
that specifies the relative inter-octahedron B-B bond length \cite{sm-Jun07}.
For pressure-dependent calculations,
a linear relation between the lattice constant and applied pressure up to 30 GPa
was established using $x$-ray diffraction data in Ref. \cite{sm-Zhou15},
before starting the $x_{B}$ relaxation.
To obtain the structure information under 50 GPa,
where the lattice constant was not accessible in Ref. \cite{sm-Zhou15},
the fully relaxed calculation was performed.
All electronic structure calculations under pressure
were done using the mBJ + SOC + $U$ (7 eV) scheme.
We have checked the pressure dependent $U$ value
by a constrained LDA calculation \cite{sm-Madsen05}
and found that $U$ $\approx$ 7 eV is obtained over the whole pressure range,
which could be interpreted as showing the strong localization of the Yb $4f$ orbital.


\begin{figure}[t]
\includegraphics[width=8.5 cm]{sfig4eps.pdf}
\caption
DFT-Wien2k band structures of YbB$_{6}$.
(a) GGA + SOC + $U$ (7 eV) calculation yields a semimetallic
$p$-$d$ overlap with anticrossing gaps.
(b) GGA + SOC + $U$ (4 eV) calculation yields SOC enhanced
gapping of overlapping $p$-$d$ states,
similar to the VASP calculation by Neupane \emph{et al.} \cite{sm-Neupane15}.
(c) GGA + SOC + $U$ (0 eV) calculation yields $f$-$d$ band inversion,
similar to the LDA + Gutzwiller calculation by Weng \emph{et al.} \cite{sm-Weng14}.
}
\label{sdft1}
\end{figure}




\section{DFT + SOC + $U$ band structures for Y\MakeLowercase{b}B$_{6}$\ with varying $U$}



\begin{figure*}[t]
\includegraphics[width=15.5 cm]{sfig5eps.pdf}
\caption
(a) The GGA + SOC + $U$ calculations using Wien2k,
showing the persistence of the \emph{p}-\emph{d} semimetal overlap
up to unphysically high values of $U$.
Red, blue, and green color symbols represent
Yb 4\emph{f}, Yb 5\emph{d}, and B 2\emph{p} characters, respectively.
(b) Summary comparison of the \emph{p}-\emph{d} band overlap
and the binding energy of Yb 4\emph{f} band ($E_{4f}$)
between our Wien2k and the VASP calculations by Chang \emph{et al.}
\cite{sm-Neupane15,sm-Chang15}.
The SOC-induced \emph{p}-\emph{d} band gap by Chang \emph{et al.} (SO gap) are
also plotted.
}
\label{sdft2}
\end{figure*}


Two recent calculations predict YbB$_{6}$
to be a topological insulator (TI) based on $f$-$d$ band inversion \cite{sm-Weng14}
or $p$-$d$ inversion \cite{sm-Neupane15,sm-Chang15}.
The first was an LDA + Gutzwiller calculation with $U$ = 6 eV
that erroneously predicted a topological $f$-$d$ inverted band structure
very similar to SmB$_{6}$\ \cite{sm-Lu13},
with a highly incorrect 4$f$ BE of $\approx$ 0.1 eV.
The LDA + Gutzwiller result \cite{sm-Weng14} is empirically very similar
to a $U$ = 0 DFT result in Fig. \ref{sdft1}(c) in terms of the 4$f$ BE
and the $f$-$d$ band inversion at $E_\textrm{F}$.


A second proposed topological band structure for YbB$_{6}$,
based on DFT + $U$ pseudopotential VASP calculations
using small $U$ values of 4 eV \cite{sm-Neupane15} or 5 eV \cite{sm-Chang15},
obtains a $p$-$d$ band overlap
but with a small insulating gap throughout the bulk BZ,
similar to that in Fig. \ref{sdft1}(b) \cite{sm-Replica}.
The effect of the smaller $U$ values is a smaller Yb 4$f$ BE,
0.3 (0.5) eV for $U$ = 4 (5) eV, whose closer proximity to
the $p$-$d$ overlap region at $E_\textrm{F}$\
results in significant hybridization of the Yb 4$f$ and B 2$p$ states.
This allows the Yb 4$f$ SOC to open a larger $p$-$d$ gap
that otherwise is intrinsically tiny
due to the smallness of the 5$d$ SOC.
The hybridization of Yb 4$f$ states into the $p$-$d$ overlap region
also results in a mixed-valent Yb 4$f$-occupation evaluated
to be $n_{f}$ $\approx$ 13.8,
which does not agree with the experimental divalent state of Yb.
Chang \emph{et al.} \cite{sm-Chang15} also reported that
the SOC-induced gapping of the $p$-$d$ overlap at $U$ = 4 $\sim$ 5 eV
evolves into a topologically trivial $p$-$d$ gap above $U$ = 6.5 eV.
This result is in contradiction to the earlier literature result \cite{sm-Jun07}
and our Wien2k result shown in Fig. \ref{sdft1}(a).
As shown in Fig. \ref{sdft2}, the DFT + $U$ calculations
for increasing $U$ yield only a small gradual decrease
in the $p$-$d$ overlap energy
without opening up a semiconducting $p$-$d$ gap
even for unphysically large $U$ values as high as 15 eV.
This is consistent with a decreasing influence of the Yb 4$f$ states
on the $p$-$d$ states as they are pushed farther away from $E_\textrm{F}$.
Namely, large $U$ is not the formative parameter for the semiconductor gap.





One might consider that the $U$ $>$ 6.5 eV VASP prediction
of a topologically trivial semiconductor
is in agreement with the Wien2k mBJ + SOC + $U$ semiconductor result in Fig. 3(b)
of main text.
However this is entirely fortuitous because the VASP calculation does not contain
the band gap renormalization treatment of the mBJ or GW calculations in Fig. 3(b).
Chang \emph{et al.} \cite{sm-Chang15} also presented a GW calculation for YbB$_{6}$\
but for $U$ = 0 with the goal of demonstrating that GW alone
cannot produced the $p$-$d$ band inversion.
In contrast, the VASP GW calculation in Fig. 3(b) is an open-core calculation,
which simulates a large effective $U$ that removes the Yb 4$f$ states far from $E_\textrm{F}$.
Thus the GW calculation in Fig. 3(b) treats
the $p$ and $d$ correlations that renormalize the $p$-$d$ overlap
into a small semiconductor gap.


\begin{figure}[b]
\includegraphics[width=8.5 cm]{sfig6eps.pdf}
\caption
DFT-VASP open-core slab calculations for YbB$_{6}$\ (001) surface.
For the purpose of getting a trivial \emph{p}-\emph{d} gap without the mBJ scheme,
a lattice constant of YbB$_{6}$\ is enlarged to be $a = 4.5 {\AA}$.
(a) B$_{6}$\ termination.
(b) Yb termination.
Both of terminations give topologically trivial surface states.
The surface states (red lines) below $E_\textrm{F}$\ are mostly of B-$p$ character,
while those above $E_\textrm{F}$\ in (b) have mixed B $p$ and Yb $d$ characters.
}
\label{slab}
\end{figure}




We have also checked DFT + $U$ electronic structures using VASP.
The crystal structure was fully relaxed using the GGA + SOC scheme
($a$ = 4.13 {\AA}, $x_{B}$ = 0.201).
The resultant crystal structure was quite similar to the experimental structure.
Band structures of YbB$_{6}$\ were calculated with various $U$ values
in both relaxed and experimental structures,
and it was found that the band structures for the two cases are almost the same.
We have found that our VASP results are quite consistent with our Wien2k results.
There is a sensitivity of the $p$-$d$ overlap to
the internal structural parameter $x_{B}$ \cite{sm-Lee05}.
One could speculate that differences in structural input could be
a possible source of the discrepancy between our Wien2k (and VASP)
and the published VASP calculations \cite{sm-Neupane15,sm-Chang15},
but no structural information (lattice constant or $x_{B}$)
was provided in Refs. \cite{sm-Neupane15,sm-Chang15}.



Figure \ref{sdft2}(b) shows the comparison between the VASP results
by Chang \emph{et al.} \cite{sm-Chang15} vs. our Wien2k results,
which highlights the discrepancy for $U$ $>$ 6 eV.
We can note discrepancies between two in
(i) the binding energy of Yb 4$f$ band ($E_{4f}$)
and (ii) $p$-$d$ overlap.
The VASP results of the latter by Chang \emph{et al.}
look merely shifted Wien2k results.


In summary, we note  that the incorrect $f$-$d$ overlap \cite{sm-Weng14},
and $p$-$d$ overlap \cite{sm-Neupane15,sm-Chang15} calculations
did not cite the earlier Wien2k calculation of Ref. \cite{sm-Jun07},
and hence failed to compare and discuss their discrepancies with the published literature.
We also note that the complicated intermixing of all three $p$, $d$ and $f$ states
near $E_\textrm{F}$\ with interplay of 4$f$ SOC and mixed valency discussed in the main text
for the high-pressure phase of YbB$_{6}$\  is the same physics at the heart of
the erroneous claim of a TI $p$-$d$ overlap gap at ambient pressure \cite{sm-Neupane15, sm-Chang15},
albeit errantly realized from a too small value of $U$.


A remaining open question about the ambient pressure phase
is the quantitative disagreement between the experimental 0.3 eV and theoretical 0.1 eV gaps,
which would be worthwhile to be investigated theoretically.
Further theoretical investigation of the high pressure phase
is warranted to address whether any low energy scale dynamical correlations
({\it e.g.} Kondo effect) can emerge from the small mixed valency,
or whether any non-trivial topological $p$-$d$ parity inversion effects
can be realized while in a partially gapped semimetallic state.





\section{DFT slab calculation for Y\MakeLowercase{b}B$_{6}$\ (001) surface}


In order to examine the surface states,
we have also carried out DFT open-core slab calculations for the YbB$_{6}$\ (001) surface
using VASP.
We have used slab structures composed of 9 Yb and 8 B$_{6}$\ layers for Yb termination
and 8 Yb and 9 B$_{6}$\ layers for B$_{6}$\ termination,
and for both terminations, $\approx$ 30 {\AA} vacuum layers were considered.
We have first relaxed the slab structures for both terminations,
and then self-consistent calculations were performed
with a 10 $\times$ 10 $\times$ 1 $k$-mesh.
For the purpose of getting a topologically trivial $p$-$d$ bulk gap in the
DFT + SOC + $U$ (7 eV) without employing mBJ,
we employed an increased lattice constant of $a = 4.5 {\AA}$
in the slab calculations.

As shown in Fig. \ref{slab}, surface states (red lines) appear
for both B$_{6}$\ and Yb terminations.
These surface states, however, are just topologically trivial surface states,
which are often observed in the hexaboride systems \cite{sm-Zhu13,sm-Monnier04}.
The Dirac-cone-like surface states observed in ARPES are not obtained.
In fact, such Dirac cones are expected to be quantum well states
coming from the band bendings in the polar (001) surfaces with different
charge terminations \cite{sm-Frantzeskakis14,sm-Denlinger15}.


The DFT open-core slab calculations for the non-polar (110) surface
also produce similar surface states, one deep surface band of B $p$ character
and two surface bands near $E_\textrm{F}$\ of mixed B $p$ and Yb $d$ characters
that arise from broken bond states from $two$ surface B atoms  \cite{sm-CJKang15}.
All of those surface states are topologically trivial.



\section{Link between Y\MakeLowercase{b}B$_{6}$\ and S\MakeLowercase{m}B$_{6}$\ }


There are strong differences in issues of the two materials YbB$_{6}$\ and SmB$_{6}$\,
{\it i.e.}, pure-divalency and $p$-$d$ overlap/semiconductor gap issues in YbB$_{6}$\
and mixed valency and $f$-$d$ hybridization gap issues in SmB$_{6}$.
Also note that while YbB$_{6}$\ (001) ARPES shows ``false" in-gap states
that are actually band-bending quantum well states,
the YbB$_{6}$\ (110) ARPES show no states within the 0.3 eV semiconductor gap,
thereby disproving the TI scenario for YbB$_{6}$.

In contrast for SmB$_{6}$\ the 20 meV hybridization gap exhibits similar
ARPES in-gap states for both the polar (001)
and non-polar (110) surfaces \cite{sm-Denlinger16}.
The existence of the X-point in-gap states for SmB$_{6}$\
is insensitive to the different polar surface terminations,
and similar in-gap states are observed
for the non-polar (110) surface too \cite{sm-Denlinger16}.
Hence the polar nature alone cannot explain the in-gap states,
although polar effects are still being discussed for the (001) surface,
{\it e.g.}, in the context of 2D-like $d$-states outside the gap \cite{sm-Hlawenka15}.

We have also checked the mBJ scheme for SmB$_{6}$.
The reason why we used the mBJ scheme for YbB$_{6}$\ is to consider
the correlation effects of delocalized Yb 5$d$ and B 2$p$ electrons near $E_\textrm{F}$.
While it is very crucial for YbB$_{6}$ that the mBJ correction
changes the $p$-$d$ overlap into a $p$-$d$ trivial gap,
for SmB$_{6}$, the Sm 4$f$ bands are dominating near $E_\textrm{F}$,
and so the mBJ method does not play a crucial role.
In fact, we have confirmed for SmB$_{6}$ that the DFT and mBJ results are essentially
the same \cite{sm-CJKang15}.



For SmB$_{6}$, dynamical mean-field theory (DMFT) is essential
to address the dynamic energy correlations
that result in strong low energy scale renormalization of the $f$-states
near $E_\textrm{F}$\ as well as the mixed valency.
In contrast, for YbB$_{6}$, with the large binding energy of the 4$f$ states and pure divalency,
one would not expect any low energy scale dynamical correlations
to emerge from DMFT calculations.
Indeed we have checked and confirmed for YbB$_{6}$ that DFT + DMFT gives
essentially identical results as DFT + $U$,
{\it i.e.}, a proper 4$f$ binding energy and an uncorrected $p$-$d$ overlap \cite{sm-CJKang15}.
The latter is because the correlation effects of delocalized $p$ and $d$ electrons are
not treated within DMFT.





