# Exposing nontrivial flat bands and superconducting pairing in infinite-layer
nickelates

Ruiqi Zhang  Department of Physics and Engineering Physics, Tulane University,
New Orleans, LA 70118, USA  Cheng-Yi Huang  Department of Physics,
Northeastern University, Boston, MA 02115, USA  Mehdi Kargarian  Department of
Physics, Sharif University of Technology, Tehran, Iran  Rahul Verma
Department of Condensed Matter Physics and Materials Science, Tata Institute
of Fundamental Research, Mumbai 400005, India  Robert S. Markiewicz
Department of Physics, Northeastern University, Boston, MA 02115, USA  Arun
Bansil  Department of Physics, Northeastern University, Boston, MA 02115, USA
Jianwei Sun<EMAIL_ADDRESS>Department of Physics and Engineering Physics,
Tulane University, New Orleans, LA 70118, USA  Bahadur Singh
<EMAIL_ADDRESS>Department of Condensed Matter Physics and Materials
Science, Tata Institute of Fundamental Research, Mumbai 400005, India

###### Abstract

Flat bands coupled with magnetism and topological orders near or at the Fermi
level are well known to drive exotic correlation physics and unconventional
superconductivity. Here, based on first-principles modeling combined with an
in-depth symmetry analysis, we reveal the presence of topological flat bands
involving low-energy Ni-$3d_{z^{2}}$ states in the recently discovered
superconductor LaNiO2. Our analysis demonstrates that LaNiO2 is an Axion
insulator with $\mathbb{Z}_{4}=2$ and that it supports topological crystalline
insulating states protected by the glide mirror symmetries. The topological
flat bands in LaNiO2 are also shown to host odd-parity superconductivity. Our
study indicates that the nickelates would provide an interesting materials
platform for exploring the interplay of flat bands, topological states, and
superconductivity.

Introduction. Flat bands in quantum materials provide a robust basis for
investigating many-body physics, including non-Fermi liquids and
unconventional superconductivity interwoven with nontrivial electronic states
Regnault  _et al._ (2022); Ma  _et al._ (2020); Yin  _et al._ (2022); Tang
_et al._ (2011); Sun  _et al._ (2011); Cao  _et al._ (2018). Suppression of
electronic kinetic energy in a flat band produces electron localization in
real space and the emergence of highly degenerate electronic states. Although
models and materials have been proposed Tang  _et al._ (2011); Sun  _et al._
(2011), the experimental realization of flat bands has been limited mainly to
moiré lattice materials such as twisted bilayer graphene (TBLG) and kagome
materials Cao  _et al._ (2018); Yin  _et al._ (2022). The mechanism in moiré
materials involves the flattening of the Dirac cone dispersion through spatial
variations of the interlayer coupling that restructures the underlying lattice
and yields a smaller Brillouin zone Bistritzer and MacDonald (2011). The moiré
materials although exhibit nontrivial states coexisting with superconducting,
magnetic, and other correlated states, but controlling their properties
requires complex tuning of structural parameters. For this reason,
stoichiometric crystals with flat bands are more appealing for exploring the
physics of high electron density coupled with nontrivial topology Regnault
_et al._ (2022); Ma  _et al._ (2020); Yin  _et al._ (2022), and the kagome
materials are drawing intense current interest Yin  _et al._ (2022); Chen  _et
al._ (2021); Kang  _et al._ (2022); Jiang  _et al._ (2023); Singh (2023); Wang
_et al._ (2023); Kang  _et al._ (2020); Singh (2023). However, the flat bands
in the kagome materials usually lie away from the Fermi level Yin  _et al._
(2022), and therefore, it is not clear how these bands participate in driving
superconductivity. It is important therefore to find superconducting materials
that support low-energy flat bands that coexist with other correlated states
Regnault  _et al._ (2022); Ma  _et al._ (2020); Yin  _et al._ (2022).

The cuprates have been shown recently to host high-order van Hove
singularities (VHSs) and flat bands which could drive correlated phases such
as superconductivity and supermetals Markiewicz  _et al._ (2023). Here we
characterize the topological flat bands and superconducting pairing symmetries
of the recently discovered infinite-layer (IL) nickelate superconductors that
have emerged as a promising analog of the cuprates Li  _et al._ (2019); Osada
_et al._ (2020); Zeng  _et al._ (2022); Osada  _et al._ (2021a); Sawatzky
(2019); Norman (2020); Pickett (2021). Extensive experimental and theoretical
efforts have been devoted to finding similarities and differences between the
infinite-layer (IL) nickelates and the cuprates Hepting  _et al._ (2020); Liu
_et al._ (2020); Choi  _et al._ (2020a); Jiang  _et al._ (2019); Lechermann
(2020); Leonov  _et al._ (2020); Lee and Pickett (2004); Li  _et al._ (2020);
Sakakibara  _et al._ (2020); Botana and Norman (2020); Karp  _et al._ (2020);
Goodge  _et al._ (2021); Jiang  _et al._ (2020); Zhang  _et al._ (2021, 2022);
Lane  _et al._ (2023). For instance, competing charge orders Rossi  _et al._
(2022); Krieger  _et al._ (2022); Tam  _et al._ (2022), superconductivity
Rossi  _et al._ (2022); Osada  _et al._ (2021b), and magnetism Fowlie  _et
al._ (2022); Lu  _et al._ (2021) has been reported recently in the $undoped$
IL-nickelates Li  _et al._ (2019); Ortiz  _et al._ (2021); Lu  _et al._
(2021). These findings sharply contrast with the $undoped$ cuprates, which
host a well-defined AFM ground state. The pairing symmetry in IL-nickelates,
however, remains unclear Sakakibara  _et al._ (2020); Gu  _et al._ (2020);
Kitatani  _et al._ (2020); Chow  _et al._ (2022). Although the participation
of Ni 3$d_{z^{2}}$-like flat bands in superconducting pairing is indicated
Kreisel  _et al._ (2022), their topological character and the associated
pairing symmetries have not been identified Gao  _et al._ (2020).

In this Letter, we unveil the presence of topological flat bands in the parent
nickelate superconductor LaNiO2 and discuss the associated pairing symmetries.
LaNiO2 has been shown to support various low-energy competing magnetic phases
with $C$-type antiferromagnetic ($C-$AFM) ground state Zhang  _et al._ (2021,
2022). The $C-$AFM state supports flat bands primarily composed of Ni
$3d_{z^{2}}$ states with VHSs at the Fermi level, which would enhance many-
body interactions and drive exotic correlated phenomena Zhang  _et al._
(2021); Choi  _et al._ (2020b). We delineate the topological character of
LaNiO2 and its nontrivial topological state. Our in-depth symmetry analysis of
the bulk bands and surface states shows that the $C-$AFM phase realizes an
axion insulator state with $\mathbb{Z}_{4}=2$ and a glide-mirror-symmetry
protected topological crystalline insulating (TCI) state. We also characterize
the superconducting pairing symmetries associated with the bulk flat bands at
the Fermi level and discuss the possibility of an odd-parity pairing in
LaNiO2. Our study indicates that LaNiO2 is a materials promising platform for
exploring the interplay of topological flat bands and unconventional
superconductivity.

Methods. All calculations were performed by using the projector-augmented wave
method Kresse and Joubert (1999) as implemented in the Vienna ab initio
simulation package (VASP) Kresse and Hafner (1993); Kresse and Furthmüller
(1996). A high-energy cutoff of 520 eV was used to truncate the plane-wave
basis set. The exchange-correlation effects were treated using the strongly
constrained and appropriately normed (SCAN) density functional without any
adjustable parameters Sun  _et al._ (2015). The crystal structures and ionic
positions were fully optimized using a force convergence criterion of 0.01
eV/Å for each atom along with a total energy tolerance of 10-6 eV. Notably,
substantial recent studies have shown the efficacy of SCAN for accurately
modeling ground states and electronic structure of strongly correlated systems
such as the cuprates Zhang  _et al._ (2020a); Furness  _et al._ (2018); Lane
_et al._ (2018) and nickelates Zhang  _et al._ (2021, 2022), binary $3d$
oxides Zhang  _et al._ (2020b), and especially the f-electron system SmB6
Zhang  _et al._ (2022) due to the reduction of self-interaction error in
comparison with other widely used density functionals Zhang  _et al._ (2020b).
We adopted a 16 $\times$ 16 $\times$ 16 $\Gamma$-centered $k$ mesh to sample
the primitive bulk Brillouin zone (BZ) and total energy calculations. Spin-
orbit coupling (SOC) effects were included self-consistently. Topological
properties were calculated by generating a material-specific tight-binding
model Hamiltonian using the VASP2WANNIER90 interface Mostofi  _et al._ (2014).
La $f$ and $d$, Ni $d$, and O $p$ states were included for constructing the
Wannier functions. The surface spectral weight of semi-infinite slabs was
calculated using an iterative Green’s function method Guinea  _et al._ (1983);
Sancho  _et al._ (1984).

Crystal structure and electronic properties. The parent nonmagnetic LaNiO2
crystallizes in the tetragonal Bravais lattice with space group $P4/mmm$
($\\#123,D_{4h}$). The La, Ni, and O atoms occupy Wyckoff positions
$1d:~{}\\{\frac{1}{2},\frac{1}{2},\frac{1}{2}\\}$, $1a:~{}\\{0,0,0\\}$, and
$2f:~{}\\{\frac{1}{2},0,0\\}$, respectively, in the lattice. The magnetic unit
cell of $C-$AFM phase is constructed considering a
$\sqrt{2}\times\sqrt{2}\times 1$ supercell of nonmagnetic LaNiO2 (see Fig.
1(a)). In the $C-$AFM phase, the magnetic moments of Ni atoms are aligned
parallel to the tetragonal $c$ axis as illustrated in Fig. 1(b). Particularly,
the magnetic interaction of Ni atoms is antiferromagnetic in the plane whereas
it is ferromagnetic in the out-of-plane direction. The calculated magnetic
moment of Ni is $\sim 1~{}\mu_{B}$, which is consistent with previous DFT+$U$
Choi  _et al._ (2020b) and DMFT calculations Leonov  _et al._ (2020). Note
that despite the absence of long-range orders in LaNiO2 even at low
temperatures, experiments reported the presence of local magnetic moments on
Ni atoms and strong AFM interactions with competing charge orders in
nickelates Fowlie  _et al._ (2022); Lu  _et al._ (2021). In this context, our
recent study has discovered a variety of low-energy competing magnetic phases
of LaNiO2 with $C-$AFM phase as the lowest energy state Zhang  _et al._ (2021,
2022). We thus focus on the symmetry properties and topological properties of
the $C-$AFM phase. It retains the $D_{4h}$ point-group symmetry of the parent
phase when the SOC effects are included. It contains 16 symmetry operations
including an inversion $\mathcal{I}$ symmetry and glide-mirror
$\mathcal{G}_{v}=\\{\mathcal{M}_{v}|\frac{1}{2}\frac{1}{2}0\\}$ symmetries
highligted in Fig. 1(b). Figure 1(c) shows the first BZ of the $C-$AFM phase
with various high-symmetry points.

Figure 1:  (a) A $2\times 2$ representation of parent nonmagnetic structure of
LaNiO2 with the $C-$AFM unit cell marked in red color. Green, blue, and yellow
balls represent La, Ni, and O atoms, respectively. (b) The perspective view of
the $C-$AFM phase of LaNiO2, where $\mathcal{M}_{110}$ and
$\mathcal{M}_{1\bar{1}0}$ planes are highlighted. The green arrows denote
magnetic moment directions. (c) The first Brillouin zone (BZ) of $C-$AFM with
relevant high-symmetry points. The $\mathcal{M}_{110}$ and
$\mathcal{M}_{1\bar{1}0}$ planes are also shown. (d)-(e) Calculated bulk band
structure of the $C-$AFM phase of LaNiO2 (d) without and (e) with spin-orbit
coupling. The orbital decompositions to various states are identified.

The bulk band structure of the $C-$AFM phase without SOC is shown in Fig.
1(d). A pronounced antiferromagnetic splitting of $\sim$2eV in Ni
$3d_{x^{2}-y^{2}}$ bands is evident in accord with earlier DFT+$U$ Choi  _et
al._ (2020b) and experimental studies Tam  _et al._ (2022). More importantly,
we observe the presence of a nearly flat band pinned precisely at the Fermi
level on the $k_{z}=\pi/c$ plane. This flat band intersects with another
itinerant band to form nodal band crossings along the $Z-R$ and $A-Z$ high-
symmetry directions. The detailed orbital analysis shows that the flat band is
predominantly composed of Ni 3$d_{z^{2}}$ orbitals and crosses with O 2$p$ and
Ni 3$d_{xz/yz}$ derived itinerant band to form a nodal band crossing. Upon
considering the SOC effects in Fig. 1(e), a minuscule gap opens up these
crossing points, forming a continuous local band gap between the valence and
conduction bands.

Flat band and the nontrivial topological state. To characterize the flat band
and nontrivial state, we present the close-up of the band structure without
and with SOC in Figs. 2(a) and 2(b), respectively. Interestingly, a scan of
the nodal crossings in the full BZ unfolds a nearly dispersionless nodal line
on the $k_{z}=\pi/c$ plane without SOC as illustrated in the inset of Fig.
2(a). The inclusion of SOC drives an inverted insulating gap of $\leq 5$ meV
between band-1 and band-2 in Fig. 2(b). For clarity, we denote the two
intersecting bands at the Fermi level as band-1 and band-2 in Fig. 2(b).
Band-1 primarily arises due to strong hybridization between the Ni 3$d$ and O
2$p$ orbitals (Fig. 2(f)), whereas band-2 exhibits Ni 3$d_{z^{2}}$ and La 5$d$
states as shown in Fig. 2(g). These findings not only infer a nontrivial
character of the $C-$AFM phase but also underscore the necessity of a
multiband model to accurately describe the low-energy physics in nickelates.
We present a three-dimensional ($E-k_{x}-k_{y}$) rendition of the band
structure near the Fermi level in Fig. 2(d). It highlights the nearly
isotropic nature of the flat bands at the $k_{z}=\pi/c$ plane. Such a flat
band drives high-order Van Hove singularities (VHSs) in the density of states
(Fig. 2(e)) similar to $high-T_{c}$ cuprates superconductors Markiewicz  _et
al._ (2023). The presence of flat bands and VHSs at the Fermi level can
introduce competition among various lattice-distorted and correlated
electronic states in nickelates as found in experiments. To determine the
origin of the flat bands, we plot the charge density of band-1 and band-2 in
Figs. 2(f)-2(g), and 2(h)-2(i). We find a power-law decay of Ni wavefunctions
on the $k_{z}=\pi/c$ plane, which results in highly restricted in-plane
hopping between Ni spins. In contrast, the out-of-plane Ni wavefunctions are
less localized, giving rise to bands with notable dispersion. This scenario
aligns with Hund’s coupling rules, signifying that in-plane hopping terms are
significantly suppressed due to the conservation of spin while hopping along
the $c$-axis is free from such a constraint.

Figure 2: Closeup of the bulk band structure of $C-$AFM phase (a) without and
(b) with spin-orbit coupling. Band-1 and band-2 are marked in (b) to
facilitate the discussion (see text for details). (c) The evolution of the
mirror winding number on the $k_{x}=k_{y}$ plane. The non-zero mirror Chen
number shows the TCI phase in the $C-$AFM phase of LaNiO2. (d) $E-k_{x}-k_{y}$
plot resolving the crossings band structure on the $k_{z}=\pi/c$ plane. (e)
Calculated density of states (DOS). A large peaked DOS at the Fermi level
highlights VHSs due the the flat band energy. (f)-(g) The charge density
associated with band-1 and band-2 represented in the crystal lattice. (h)-(i)
The charge density associated with band-1 and band-2 represented in the
$k_{z}=\pi/c$ plane. A power decay of the Bloch wavefunctions associated with
Ni atoms is revealed.

The presence of an inverted band gap between band-1 and band-2 in the $C-$AFM
phase of LaNiO2 hints towards a topological nontrivial state. To reveal the
nontrivial band character, we compute the parity-based higher-order
$\mathbb{Z}_{4}$ invariant which is given as

$\mathbb{Z}_{4}=\sum_{i=1}^{8}\sum_{n=1}^{n_{occ}}\frac{1+\xi_{n}(\Lambda_{i})}{2}~{}\mathrm{mod~{}4}$
(1)

where $\xi_{n}$ is the parity eigenvalue ($\xi_{n}=\pm 1$) of $n^{th}$ band at
time-reversal invariant momentum point $\Lambda_{i}$. The $\mathbb{Z}_{4}$ is
well defined even for a magnetic material preserving inversion symmetry. A
centrosymmetric insulator with $\mathbb{Z}_{4}=2$ indicates an axion insulator
state with a quantized topological magnetoelectric effect with axion coupling
$\theta=\pi$ Xu  _et al._ (2019). Based on the parity eigenvalues of occupied
states, we obtained $\mathbb{Z}_{4}$ of 2, which demonstrates that the $C-$AFM
phase of LaNiO2 is an antiferromagnetic axion insulator. It can support the
half-quantized quantum Hall effect if the associated surface states are
gapped.

Since the system respects
$\mathcal{G}_{v_{1}}=\\{\mathcal{M}_{v_{1}}|\frac{1}{2}\frac{1}{2}0\\}$, we
also evaluated the mirror Chern number on the $k_{x}=k_{y}$ plane. The
evolution of the Wannier charge centers (WCCs) associated with two mirror
eigen-sectors is plotted in Fig. 2(c), which reveals the nontrivial binding of
WCCs and a mirror Chern number of -1. The nontrivial mirror Chern number
guarantees the existence of nontrivial Dirac cone states on the surface
respecting glide-mirror
$\mathcal{G}_{v_{1}}=\\{\mathcal{M}_{v_{1}}|\frac{1}{2}\frac{1}{2}0\\}$
symmetry. We present the (001) surface band spectrum for Ni-O and La
terminated surfaces in Fig. 3. In Figs. 3(a)-3(b), we illustrate the two
possible (001) surface terminations that expose the Ni-O and La atomic planes
in the surface layer, respectively. The surface electronic structures
associated with the Ni-O and La terminated surfaces are shown in Figs.
3(c)-(e) and 3(f)-3(h), respectively. The clear surface Dirac cone states are
seen crossings at the $\bar{\Gamma}$ point on both the terminations. However,
these surface states show distinct energy-momentum dispersion on different
terminated surfaces. Specifically, the surface states on the Ni-O surface show
a nearly flat energy dispersion with a bandwidth of $\sim$0.1eV (Figs.
3(c)-3(d)). The surface states overlap with the projected flat bulk bands for
the Ni-O surface at the Fermi level in full surface BZ as revealed in the
Fermi band contours in Fig. 3(e). In contrast, the surface states on the La
terminated surface have a larger bandwidth with a higher group velocity (Figs.
3(f)-3(g)). The associated Fermi band contour in Fig. 3(h) is separated from
bulk bands in momentum space. The reason for such a difference in energy
dispersion on two terminated surfaces can be attributed to a suppressed
hopping strength on Ni-O termination due to anti-parallel magnetic moments
between Ni atoms while there is no such restriction for non-magnetic La
termination on the surface.

We turn now to discuss the symmetry protection of the observed surface Dirac
fermions. The glide mirror plane
$\mathcal{G}_{v_{1}}:(x,y,z)\rightarrow\left(y+\frac{1}{2},x+\frac{1}{2},z\right)$
reflects the states in momentum space at $(k_{x},k_{y},k_{z})$ to
$(k_{y},k_{x},k_{z})$. The states on the $(k_{x},k_{x},k_{z})$ plane can be
labeled by glide eigenvalues. To obtain these glide eigenvalues, the relation
$\hat{\mathcal{G}}_{v_{1}}^{2}=\\{-E|110\\}$ holds for a spin-full system with
action on the Bloch states as
$\hat{\mathcal{G}}_{v_{1}}^{2}|\mathbf{k}\rangle=-e^{-i(k_{x}+k_{y})}|\mathbf{k}\rangle$.
Therefore, the glide eigenvalues on the $(k_{x},k_{x},k_{z})$ plane read as
$g_{\pm}=\pm ie^{-ik_{x}}$. In particular, on the line $k_{x}=k_{y}=0$, the
glide eigenvalues become pure imaginary, $g_{\pm}=\pm i$, and the signs are
swapped on the line $k_{x}=k_{y}=\pi$, i.e., $g_{\pm}=\mp i$. There could be
thus unavoided band crossing on the surface preserving the glide symmetry
($\bar{\Gamma}-\bar{M}$ line on (001) surface) as found at $\bar{\Gamma}$
point in Fig. 3.

Similarly, another vertical glide mirror plane
$\mathcal{G}_{v_{2}}:(x,y,z)\rightarrow\left(x+\frac{1}{2},-y+\frac{1}{2},z\right)$
dictates that the states at $(k_{x},k_{y},k_{z})$ reflects to
$(k_{x},-k_{y},k_{z})$. The eigenstates on $k_{y}=\\{0,~{}\pi\\}$ planes can
be thus labeled by $\mathcal{G}_{v_{2}}$ glide eigenvalues. Using the
relations
$\hat{\mathcal{G}}_{v_{2}}^{2}=\\{-E|110\\},~{}~{}~{}\hat{\mathcal{G}}_{v_{2}}^{2}|\mathbf{k}\rangle=-e^{-i(k_{x}+k_{y})}|\mathbf{k}\rangle$,
we obtain $g_{\pm}(k_{x},k_{y}=0)=\pm
ie^{-ik_{x}/2},~{}~{}~{}~{}g_{\pm}(k_{x},k_{y}=\pi)=\pm e^{-ik_{x}/2}$ on the
$k_{y}=\\{0,~{}\pi\\}$ planes. On the glide symmetric plane $k_{y}=0$ and for
the subspace restricted to the $k_{z}$ axis (i.e., $k_{x}=0$), the glide
eigenvalues become $g_{\pm}=\pm i$. This dictates the band degeneracy at
$\bar{\Gamma}$ and thus, the surface states can be gapless along
$\bar{\Gamma}-\bar{X}$ as well. These symmetry properties indicate that the
surface Dirac fermions are protected by glide mirror symmetry, and thus, the
$C-$AFM phase of LaNiO2 realizes a symmetry-protected topological crystalline
insulator state.

Figure 3: Lattice structure of a semi-infinite slab of the $C-$AFM phase of
LaNiO2 with (a) NiO and (b) La terminations. (c)-(e) The calculated (001)
surface band structure for NiO surface termination (c) with a closeup of
surface bands in (d) and associated Fermi band counter in (e). The dark blue
color marks the surface states. The horizontal dashed line marks the Fermi
level. (f)-(h) Same as (c)-(e) but for La surface termination.

Possible nontrivial superconducting states. The preceding analysis
demonstrates that LaNiO2 is a topological crystalline insulator with
nontrivial surface Dirac Fermion states. These states could become
superconducting through the bulk proximity effect. We now analyze the
possibility of nontrivial superconducting pairing in the bulk states of
LaNiO2. Under the glide operation, the symmetric kinetic bulk Hamiltonian
transforms as
$\hat{\mathcal{G}}(\mathbf{k})H(\mathbf{k})\hat{\mathcal{G}}(\mathbf{k})^{\dagger}=H(\mathcal{M}_{v}\mathbf{k})$,
where
$\hat{\mathcal{G}}(\mathbf{k})~{}(\equiv\\{\mathcal{M}_{v_{1}/v_{2}}|\frac{1}{2}\frac{1}{2}0\\})$
represents the glide mirror operator in momentum space acting on spin,
orbital, and sublattice degrees of freedom. The superconducting model with gap
function $\Delta(\mathbf{k})$ is described by the BdG Hamiltonian,

$H_{\mathrm{BdG}}(\mathbf{k})=\begin{pmatrix}H(\mathbf{k})&\Delta(\mathbf{k})\\\
\Delta^{\dagger}(\mathbf{k})&-H^{\mathrm{T}}(-\mathbf{k})\end{pmatrix}.$ (2)

The bulk symmetries imply that
$\hat{\mathcal{G}}_{\mathrm{BdG}}(\mathbf{k})H_{\mathrm{BdG}}(\mathbf{k})\hat{\mathcal{G}}_{\mathrm{BdG}}(\mathbf{k})^{\dagger}=H_{\mathrm{BdG}}(\mathcal{M}_{v}\mathbf{k})$.
Here the symmetry operations are extended to the particle-hole space as
$\hat{\mathcal{G}}_{\mathrm{BdG}}(\mathbf{k})=\hat{\mathcal{G}}(\mathbf{k})\oplus[\hat{\mathcal{G}}^{\pm}(-\mathbf{k})]^{*}$.
To satisfy this symmetry constraint, the gap function should transform as
$\hat{\mathcal{G}}(\mathbf{k})\Delta(\mathbf{k})\hat{\mathcal{G}}^{\mathrm{T}}(-\mathbf{k})=\pm\Delta(\mathcal{M}_{v}\mathbf{k})$.
We define
$\hat{\mathcal{G}}^{\pm}(-\mathbf{k})=\pm\hat{\mathcal{G}}(-\mathbf{k})$
depending on the even (+) or odd (-) transformation of the gap function under
the symmetry operation. For odd-symmetric gap functions, the lattice symmetry
operation and particle-hole transformation,
$\hat{\mathcal{C}}H_{\mathrm{BdG}}(\mathbf{k})\hat{\mathcal{C}}^{-1}=-H_{\mathrm{BdG}}(-\mathbf{k})$
should anticommute i.e. $\\{\hat{\mathcal{C}},\hat{\mathcal{G}}\\}=0$. These
symmetry constraints restrict the Hilbert space of Bloch states to one-
dimensional lines $k_{x}=k_{y}=0$ and $k_{x}=k_{y}=\pi$, which corresponds to
$\Gamma-Z$ and $M-A$ symmetry directions in the bulk BZ, respectively. The
Bloch states on these symmetry lines thus acquire pure imaginary glide
eigenvalues $g_{\pm}=\pm i$:
$\mathcal{H}=\bigoplus_{k_{z}}\left(\mathcal{H}_{k_{z}}^{+i}\oplus\mathcal{H}_{k_{z}}^{-i}\right)$.

Let us consider superconducting states with odd-symmetric gap functions. Each
eigenvalue sector is preserved under the action of particle-hole symmetry
operation due to the anticommutation relation
$\\{\hat{\mathcal{C}},\hat{\mathcal{G}}\\}=0$, i.e., a glide eigenstate with
definite eigenvalue is preserved upon the action of $\hat{\mathcal{C}}$ Nogaki
_et al._ (2021). Therefore, each sector is a one-dimensional superconductor
that breaks the time-reversal symmetry. It belongs to the $D$ class of AZ
classification. The superconducting states in this class are characterized by
$\mathbb{Z}_{2}$ invariant defined as

$\nu^{\pm}=\frac{1}{2\pi}\oint A^{\pm}(k)dk,$ (3)

where
$A^{\pm}(k)=i\sum_{n}\langle\psi^{\pm}_{n}(k)|\partial_{k}\psi^{\pm}_{n}(k)\rangle$
is the sum of the Berry connection of occupied BdG states with a definite
glide eigenvalue, and the integration is over the loops $Z-\Gamma-Z$ or
$A-M-A$. For an odd-parity superconductor, the topological invariant is
$\mathbb{Z}_{2}$ nontrivial and it is shown that the invariant can be deduced
from the structure of the Fermi surface of the normal state. The invariant is
related to the band structure as Fu and Berg (2010); Sato (2010):

$(-1)^{\nu^{\pm}}=\prod_{n}\mathrm{sgn}\varepsilon^{\pm}_{n}(\Lambda_{i})\mathrm{sgn}\varepsilon^{\pm}_{n}(\Lambda_{j}),$
(4)

where $\varepsilon^{\pm}_{n}(\Lambda_{i})$ is the eigenstate of the normal
Hamiltonian $H(\mathbf{k})$ at $\Lambda_{i}=\Gamma(M)$ and $\Lambda_{j}=Z(A)$.
From (4), the invariant is defined modulo 2, and therefore, it can be written
as
$(-1)^{\nu^{\pm}}=(-1)^{p_{0}}~{}~{}~{}\Rightarrow~{}~{}~{}\nu^{\pm}=p_{0}~{}~{}\mathrm{mod}~{}~{}2,$
where $p_{0}$ is the number of times the energy bands with a given glide
eigenvalue cross the Fermi level between $\Lambda_{i}$ and $\Lambda_{j}$.
Using the band structure shown in Figs. 1(e) and 2(b), we give the sign of
energy bands crossing the Fermi level in Table 1. These results show that
$\mathbb{Z}_{2}$ invariant, $\nu^{\pm}=+1$, is nontrivial for an odd-parity
superconductor. This implies that the surface perpendicular to the $z$
direction should support zero-energy modes at $\bar{\Gamma}$.

As a possible odd-parity pairing candidate, the superconducting state could be
a pair-density wave (PDW) with $B_{2u}$ representation. This irreducible
representation allows for $d$-wave superconductor, $k_{x}^{2}-k_{y}^{2}$,
within the layers while the sign of the superconducting gap alternates between
the layers. A similar PDW state has been proposed as topological crystalline
superconductivity in heavy-fermion CeRh2As2 Nogaki  _et al._ (2021).

Table 1: Sign of the mirror eigenvalues of energy bands crossing Fermi level at $\Lambda$ in Fig. 1(e). $\Lambda$ | $\Gamma$ | X | M | Z | R | A
---|---|---|---|---|---|---
$\mathrm{sgn}$[$\varepsilon(\Lambda)$] | + | + | + | - | - | +

The symmetry of the magnetic state also allows a $\mathbb{Z}_{2}$
classification of possible gapped superconducting states in the layered
nickelates. The magnetic state breaks both the time-reversal and translational
symmetry. Nevertheless, a combination of these operations is preserved in the
antiferromagnetic state Mong  _et al._ (2010). We denote the time-reversal
operation by $\Theta$, and the half-cell translation by $T_{1/2}\equiv
T_{\mathbf{a}/2+\mathbf{b}/2}$. The combined operation, $\tilde{\Theta}=\Theta
T_{1/2}$, is the symmetry of the magnetic state. For simplicity, let us define
new lattice vectors $\mathbf{a}_{1}=\mathbf{a}+\mathbf{b}$,
$\mathbf{a}_{2}=\mathbf{a}-\mathbf{b}$ and $\mathbf{a}_{3}=\mathbf{c}$. The
corresponding components of wave vector reads as
$k_{1}=\mathbf{k}\cdot\mathbf{a}_{1}$, $k_{2}=\mathbf{k}\cdot\mathbf{a}_{2}$
and $k_{3}=\mathbf{k}\cdot\mathbf{a}_{3}$. In momentum space,
$T_{1/2}^{2}=e^{ik_{1}},~{}\tilde{\Theta}^{2}=-e^{ik_{1}}$, and
$\tilde{\Theta}H_{\mathrm{BdG}}(k_{1},k_{2},k_{3})\tilde{\Theta}^{-1}=H_{\mathrm{BdG}}(-k_{1},-k_{2},-k_{3})$
with $\Theta^{2}=-1$ for spin-$1/2$ systems. On the plane $k_{1}=0$
($k_{x}+k_{y}=0$) or another $C_{4}$-symmetric equivalent plane, i.e., the
plane $k_{2}=0$ for half translation along $\mathbf{a}_{2}$, we have
$\tilde{\Theta}^{2}=-1$. Thus, in analogous to time-reversal invariant
insulators, the Hilbert space of Bloch states with gapped spectrum restricted
to the plane $(k_{2},k_{3})$ can be classified by $\mathbb{Z}_{2}$ invariant
$\nu=1$ ($\nu=0$) corresponds to topological (trivial) ground state. Note that
there is no such classification on the $k_{1}=\pi$ plane since
$\tilde{\Theta}^{2}=+1$. Therefore, the symmetry of the magnetic state allows
for a $\mathbb{Z}_{2}$ classification of a fully gapped superconductor on the
$(k_{2},k_{3})$ plane. However, whether the fully gapped superconductor on
this plane is topologically nontrivial depends on the details of the band
structure of $H_{\mathrm{BdG}}$, which constitutes an interesting problem for
future studies.

Conclusion.– It is widely accepted in the cuprates that the competing
inhomogeneous phases and VHSs play a key role in producing their
superconducting state. This, however, is not widely discussed in the recently
discovered nickelate superconductors despite their similarities with the
cuprates. Our in-depth first-principles calculations and symmetry analysis
reveal the existence of a flat band in the undoped nickelate superconductor
LaNiO2. This flat band mainly involves Ni 3$d_{z^{2}}$ states and it crosses a
hybrid state composed of Ni 3$d_{xz/yz}$ and O 2$p$ orbitals to generate a
band inversion in the bulk spectrum. We show that the $C-$AFM phase of LaNiO2
realizes an axion insulator state with a higher-order invariant
$\mathbb{Z}_{4}=2$ and mirror-symmetry-protected topological crystalline
insulator state with nontrivial surface states. We also show that the
superconducting state in LaNiO2 possesses a nontrivial topological character.
Our study not only highlights the similarities between the $undoped$
nickelates and the $doped$ cuprate superconductors but it also shows that the
nickelates would provide an interesting materials platform for exploring the
interplay of topological states and nontrivial superconductivity.

## Acknowledgements

R.Z. and J.S. acknowledge the support of the U.S. Office of Naval Research
(ONR) Grant No. N00014-22-1-2673 and it benefited from NSF’s Advanced
Cyberinfrastructure Coordination Ecosystem, and the National Energy Research
Scientific Computing Center. The work at Northeastern University was supported
by the National Science Foundation through NSF-ExpandQISE award #2329067 and
it benefited from Northeastern University’s Advanced Scientific Computation
Center and the Discovery Cluster. The work at TIFR Mumbai was supported by the
Department of Atomic Energy of the Government of India under project number
12-R&D-TFR-5.10-0100 and benefitted from the computational resources of TIFR
Mumbai.

R. Z., C.-Y. H., and M. K. contributed equally to this work.

## References

  * Regnault  _et al._ (2022) Nicolas Regnault, Yuanfeng Xu, Ming-Rui Li, Da-Shuai Ma, Milena Jovanovic, Ali Yazdani, Stuart S. P. Parkin, Claudia Felser, Leslie M. Schoop, N. Phuan Ong, Robert J. Cava, Luis Elcoro, Zhi-Da Song, and B. Andrei Bernevig, “Catalogue of flat-band stoichiometric materials,” Nature 603, 824–828 (2022).
  * Ma  _et al._ (2020) Da-Shuai Ma, Yuanfeng Xu, Christie S. Chiu, Nicolas Regnault, Andrew A. Houck, Zhida Song, and B. Andrei Bernevig, “Spin-orbit-induced topological flat bands in line and split graphs of bipartite lattices,” Physical Review Letters 125, 266403 (2020).
  * Yin  _et al._ (2022) Jia-Xin Yin, Biao Lian, and M. Zahid Hasan, “Topological kagome magnets and superconductors,” Nature 612, 647–657 (2022).
  * Tang  _et al._ (2011) Evelyn Tang, Jia-Wei Mei, and Xiao-Gang Wen, “High-temperature fractional quantum hall states,” Phys. Rev. Lett. 106, 236802 (2011).
  * Sun  _et al._ (2011) Kai Sun, Zhengcheng Gu, Hosho Katsura, and S. Das Sarma, “Nearly flatbands with nontrivial topology,” Phys. Rev. Lett. 106, 236803 (2011).
  * Cao  _et al._ (2018) Yuan Cao, Valla Fatemi, Shiang Fang, Kenji Watanabe, Takashi Taniguchi, Efthimios Kaxiras, and Pablo Jarillo-Herrero, “Unconventional superconductivity in magic-angle graphene superlattices,” Nature 556, 43–50 (2018).
  * Bistritzer and MacDonald (2011) Rafi Bistritzer and Allan H. MacDonald, “Moiré bands in twisted double-layer graphene,” Proceedings of the National Academy of Sciences 108, 12233–12237 (2011).
  * Chen  _et al._ (2021) Hui Chen, Haitao Yang, Bin Hu, Zhen Zhao, Jie Yuan, Yuqing Xing, Guojian Qian, Zihao Huang, Geng Li, Yuhan Ye, Sheng Ma, Shunli Ni, Hua Zhang, Qiangwei Yin, Chunsheng Gong, Zhijun Tu, Hechang Lei, Hengxin Tan, Sen Zhou, Chengmin Shen, Xiaoli Dong, Binghai Yan, Ziqiang Wang, and Hong-Jun Gao, “Roton pair density wave in a strong-coupling kagome superconductor,” Nature 599, 222–228 (2021).
  * Kang  _et al._ (2022) Mingu Kang, Shiang Fang, Jonggyu Yoo, Brenden R. Ortiz, Yuzki M. Oey, Jonghyeok Choi, Sae Hee Ryu, Jimin Kim, Chris Jozwiak, Aaron Bostwick, Eli Rotenberg, Efthimios Kaxiras, Joseph G. Checkelsky, Stephen D. Wilson, Jae-Hoon Park, and Riccardo Comin, “Charge order landscape and competition with superconductivity in kagome metals,” Nature Materials 22, 186–193 (2022).
  * Jiang  _et al._ (2023) Zhicheng Jiang, Zhengtai Liu, Haiyang Ma, Wei Xia, Zhonghao Liu, Jishan Liu, Soohyun Cho, Yichen Yang, Jianyang Ding, Jiayu Liu, Zhe Huang, Yuxi Qiao, Jiajia Shen, Wenchuan Jing, Xiangqi Liu, Jianpeng Liu, Yanfeng Guo, and Dawei Shen, “Flat bands, non-trivial band topology and rotation symmetry breaking in layered kagome-lattice rbti3bi5,” Nature Communications 14, 4892 (2023).
  * Singh (2023) Bahadur Singh, “Rotation rearranges electrons,” Nature Physics (2023), 10.1038/s41567-023-02237-7.
  * Wang  _et al._ (2023) Yaojia Wang, Heng Wu, Gregory T. McCandless, Julia Y. Chan, and Mazhar N. Ali, “Quantum states and intertwining phases in kagome materials,” Nature Reviews Physics (2023), 10.1038/s42254-023-00635-7.
  * Kang  _et al._ (2020) Mingu Kang, Shiang Fang, Linda Ye, Hoi Chun Po, Jonathan Denlinger, Chris Jozwiak, Aaron Bostwick, Eli Rotenberg, Efthimios Kaxiras, Joseph G. Checkelsky, and Riccardo Comin, “Topological flat bands in frustrated kagome lattice CoSn,” Nature Communications 11, 1–9 (2020).
  * Markiewicz  _et al._ (2023) Robert S. Markiewicz, Bahadur Singh, Christopher Lane, and Arun Bansil, “Investigating the cuprates as a platform for high-order van hove singularities and flat-band physics,” Communications Physics 6, 292 (2023).
  * Li  _et al._ (2019) Danfeng Li, Kyuho Lee, Bai Yang Wang, Motoki Osada, Samuel Crossley, Hye Ryoung Lee, Yi Cui, Yasuyuki Hikita, and Harold Y. Hwang, “Superconductivity in an infinite-layer nickelate,” Nature 572, 624–627 (2019).
  * Osada  _et al._ (2020) Motoki Osada, Bai Yang Wang, Berit H. Goodge, Kyuho Lee, Hyeok Yoon, Keita Sakuma, Danfeng Li, Masashi Miura, Lena F. Kourkoutis, and Harold Y. Hwang, “A Superconducting Praseodymium Nickelate with Infinite Layer Structure,” Nano Letters 20, 5735–5740 (2020).
  * Zeng  _et al._ (2022) Shengwei Zeng, Changjian Li, Lin Er Chow, Yu Cao, Zhaoting Zhang, Chi Sin Tang, Xinmao Yin, Zhi Shiuh Lim, Junxiong Hu, Ping Yang, and Ariando Ariando, “Superconductivity in infinite-layer nickelate la1-xcaxnio2 thin films,” Science Advances 8, 9927 (2022).
  * Osada  _et al._ (2021a) Motoki Osada, Bai Yang Wang, Berit H. Goodge, Shannon P. Harvey, Kyuho Lee, Danfeng Li, Lena F. Kourkoutis, and Harold Y. Hwang, “Nickelate Superconductivity without Rare‐Earth Magnetism: (La,Sr)NiO2,” Advanced Materials 33 (2021a), 10.1002/adma.202104083.
  * Sawatzky (2019) George A. Sawatzky, “Superconductivity seen in a non-magnetic nickel oxide,” Nature 572, 592–593 (2019).
  * Norman (2020) Michael R. Norman, “Entering the Nickel Age of Superconductivity,” Physics 13, 85 (2020).
  * Pickett (2021) Warren E. Pickett, “The dawn of the nickel age of superconductivity,” Nature Reviews Physics 3, 7–8 (2021).
  * Hepting  _et al._ (2020) M. Hepting, D. Li, C. J. Jia, H. Lu, E. Paris, Y. Tseng, X. Feng, M. Osada, E. Been, Y. Hikita, Y.-D. Chuang, Z. Hussain, K. J. Zhou, A. Nag, M. Garcia-Fernandez, M. Rossi, H. Y. Huang, D. J. Huang, Z. X. Shen, T. Schmitt, H. Y. Hwang, B. Moritz, J. Zaanen, T. P. Devereaux, and W. S. Lee, “Electronic structure of the parent compound of superconducting infinite-layer nickelates,” Nature Materials 19, 381–385 (2020).
  * Liu  _et al._ (2020) Zhao Liu, Zhi Ren, Wei Zhu, Zhengfei Wang, and Jinlong Yang, “Electronic and magnetic structure of infinite-layer NdNiO2: trace of antiferromagnetic metal,” npj Quantum Materials 5, 31 (2020).
  * Choi  _et al._ (2020a) Mi Young Choi, Kwan Woo Lee, and Warren E. Pickett, “Role of 4f states in infinite-layer NdNiO2,” Physical Review B 101, 20503 (2020a).
  * Jiang  _et al._ (2019) Peiheng Jiang, Liang Si, Zhaoliang Liao, and Zhicheng Zhong, “Electronic structure of rare-earth infinite-layer $r\mathrm{Ni}{\mathrm{o}}_{2}(r=\mathrm{La},\mathrm{Nd})$,” Phys. Rev. B 100, 201106 (2019).
  * Lechermann (2020) Frank Lechermann, “Late transition metal oxides with infinite-layer structure: Nickelates versus cuprates,” Physical Review B 101, 081110 (2020), 1911.11521 .
  * Leonov  _et al._ (2020) I. Leonov, S. L. Skornyakov, and S. Y. Savrasov, “Lifshitz transition and frustration of magnetic moments in infinite-layer ${\mathrm{ndnio}}_{2}$ upon hole doping,” Phys. Rev. B 101, 241108 (2020).
  * Lee and Pickett (2004) K.-W. Lee and W. E. Pickett, “Infinite-layer $\mathrm{La}\mathrm{Ni}{\mathrm{o}}_{2}$: ${\mathrm{ni}}^{1+}$ is not ${\mathrm{cu}}^{2+}$,” Phys. Rev. B 70, 165109 (2004).
  * Li  _et al._ (2020) Danfeng Li, Bai Yang Wang, Kyuho Lee, Shannon P. Harvey, Motoki Osada, Berit H. Goodge, Lena F. Kourkoutis, and Harold Y. Hwang, “Superconducting dome in ${\mathrm{nd}}_{1-x}{\mathrm{sr}}_{x}{\mathrm{nio}}_{2}$ infinite layer films,” Phys. Rev. Lett. 125, 027001 (2020).
  * Sakakibara  _et al._ (2020) Hirofumi Sakakibara, Hidetomo Usui, Katsuhiro Suzuki, Takao Kotani, Hideo Aoki, and Kazuhiko Kuroki, “Model construction and a possibility of cupratelike pairing in a new ${d}^{9}$ nickelate superconductor $(\mathrm{Nd},\mathrm{Sr}){\mathrm{nio}}_{2}$,” Phys. Rev. Lett. 125, 077003 (2020).
  * Botana and Norman (2020) A. S. Botana and M. R. Norman, “Similarities and differences between ${\mathrm{lanio}}_{2}$ and ${\mathrm{cacuo}}_{2}$ and implications for superconductivity,” Phys. Rev. X 10, 011024 (2020).
  * Karp  _et al._ (2020) Jonathan Karp, Antia S. Botana, Michael R. Norman, Hyowon Park, Manuel Zingl, and Andrew Millis, “Many-body electronic structure of ${\mathrm{ndnio}}_{2}$ and ${\mathrm{cacuo}}_{2}$,” Phys. Rev. X 10, 021061 (2020).
  * Goodge  _et al._ (2021) Berit H. Goodge, Danfeng Li, Kyuho Lee, Motoki Osada, Bai Yang Wang, George A. Sawatzky, Harold Y. Hwang, and Lena F. Kourkoutis, “Doping evolution of the Mott–Hubbard landscape in infinite-layer nickelates,” Proceedings of the National Academy of Sciences 118, e2007683118 (2021).
  * Jiang  _et al._ (2020) Mi Jiang, Mona Berciu, and George A. Sawatzky, “Critical Nature of the Ni Spin State in Doped ${\mathrm{NdNiO}}_{2}$,” Phys. Rev. Lett. 124, 207004 (2020).
  * Zhang  _et al._ (2021) Ruiqi Zhang, Christopher Lane, Bahadur Singh, Johannes Nokelainen, Bernardo Barbiellini, Robert S. Markiewicz, Arun Bansil, and Jianwei Sun, “Magnetic and f-electron effects in LaNiO2 and NdNiO2 nickelates with cuprate-like 3$d_{x^{2}-y^{2}}$ band,” Communications Physics 4, 118 (2021).
  * Zhang  _et al._ (2022) Ruiqi Zhang, Christopher Lane, Johannes Nokelainen, Bahadur Singh, Bernardo Barbiellini, Robert S. Markiewicz, Arun Bansil, and Jianwei Sun, “Peierls distortion driven multi-orbital origin of charge density waves in the undoped infinite-layer nickelate,” arXiv e-prints (2022), 10.48550/arXiv.2207.00184.
  * Lane  _et al._ (2023) Christopher Lane, Ruiqi Zhang, Bernardo Barbiellini, Robert S. Markiewicz, Arun Bansil, Jianwei Sun, and Jian-Xin Zhu, “Competing incommensurate spin fluctuations and magnetic excitations in infinite-layer nickelate superconductors,” Communications Physics 6, 90 (2023).
  * Rossi  _et al._ (2022) Matteo Rossi, Motoki Osada, Jaewon Choi, Stefano Agrestini, Daniel Jost, Yonghun Lee, Haiyu Lu, Bai Yang Wang, Kyuho Lee, Abhishek Nag, Yi-De Chuang, Cheng-Tai Kuo, Sang-Jun Lee, Brian Moritz, Thomas P. Devereaux, Zhi-Xun Shen, Jun-Sik Lee, Ke-Jin Zhou, Harold Y. Hwang, and Wei-Sheng Lee, “A broken translational symmetry state in an infinite-layer nickelate,” Nat. Phys. 18, 869–873 (2022).
  * Krieger  _et al._ (2022) G. Krieger, L. Martinelli, S. Zeng, L. E. Chow, K. Kummer, R. Arpaia, M. Moretti Sala, N. B. Brookes, A. Ariando, N. Viart, M. Salluzzo, G. Ghiringhelli, and D. Preziosi, “Charge and spin order dichotomy in ${\mathrm{ndnio}}_{2}$ driven by the capping layer,” Phys. Rev. Lett. 129, 027002 (2022).
  * Tam  _et al._ (2022) Charles C. Tam, Jaewon Choi, Xiang Ding, Stefano Agrestini, Abhishek Nag, Mei Wu, Bing Huang, Huiqian Luo, Peng Gao, Mirian García-Fernández, Liang Qiao, and Ke-Jin Zhou, “Charge density waves in infinite-layer NdNiO2 nickelates,” Nat. Mater. 21, 1116–1120 (2022), 2112.04440 .
  * Osada  _et al._ (2021b) Motoki Osada, Bai Yang Wang, Berit H. Goodge, Shannon P. Harvey, Kyuho Lee, Danfeng Li, Lena F. Kourkoutis, and Harold Y. Hwang, “Nickelate superconductivity without rare-earth magnetism: (la,sr)nio2,” Advanced Materials 33, 2104083 (2021b).
  * Fowlie  _et al._ (2022) Jennifer Fowlie, Marios Hadjimichael, Maria M. Martins, Danfeng Li, Motoki Osada, Bai Yang Wang, Kyuho Lee, Yonghun Lee, Zaher Salman, Thomas Prokscha, Jean-Marc Triscone, Harold Y. Hwang, and Andreas Suter, “Intrinsic magnetism in superconducting infinite-layer nickelates,” Nat. Phys. 18, 1043–1047 (2022).
  * Lu  _et al._ (2021) H. Lu, M. Rossi, A. Nag, M. Osada, D. F. Li, K. Lee, B. Y. Wang, M. Garcia-Fernandez, S. Agrestini, Z. X. Shen, E. M. Been, B. Moritz, T. P. Devereaux, J. Zaanen, H. Y. Hwang, Ke-Jin Zhou, and W. S. Lee, “Magnetic excitations in infinite-layer nickelates,” Science 373, 213–216 (2021).
  * Ortiz  _et al._ (2021) R. A. Ortiz, P. Puphal, M. Klett, F. Hotz, R. K. Kremer, H. Trepka, M. Hemmida, H. A. Krug von Nidda, M. Isobe, R. Khasanov, H. Luetkens, P. Hansmann, B. Keimer, T. Schäfer, and M. Hepting, “Magnetic correlations in infinite-layer nickelates: an experimental and theoretical multi-method study,” arXiv e-prints , arXiv:2111.13668 (2021), arXiv:2111.13668 [cond-mat.str-el] .
  * Gu  _et al._ (2020) Yuhao Gu, Sichen Zhu, Xiaoxuan Wang, Jiangping Hu, and Hanghui Chen, “A substantial hybridization between correlated Ni-d orbital and itinerant electrons in infinite-layer nickelates,” Communications Physics 3, 84 (2020).
  * Kitatani  _et al._ (2020) Motoharu Kitatani, Liang Si, Oleg Janson, Ryotaro Arita, Zhicheng Zhong, and Karsten Held, “Nickelate superconductors—a renaissance of the one-band Hubbard model,” npj Quantum Materials 5, 59 (2020).
  * Chow  _et al._ (2022) L. E. Chow, S. Kunniniyil Sudheesh, Z. Y. Luo, P. Nandi, T. Heil, J. Deuschle, S. W. Zeng, Z. T. Zhang, S. Prakash, X. M. Du, Z. S. Lim, Peter A. van Aken, Elbert E. M. Chia, and A. Ariando, “Pairing symmetry in infinite-layer nickelate superconductor,” arXiv e-prints , arXiv:2201.10038 (2022), arXiv:2201.10038 [cond-mat.supr-con] .
  * Kreisel  _et al._ (2022) Andreas Kreisel, Brian M. Andersen, Astrid T. Rømer, Ilya M. Eremin, and Frank Lechermann, “Superconducting instabilities in strongly correlated infinite-layer nickelates,” Phys. Rev. Lett. 129, 077002 (2022).
  * Gao  _et al._ (2020) Jiacheng Gao, Shiyu Peng, Zhijun Wang, Chen Fang, and Hongming Weng, “Electronic structures and topological properties in nickelates Lnn+1NinO2n+2,” National Science Review 8 (2020).
  * Choi  _et al._ (2020b) Mi-Young Choi, Warren E. Pickett, and Kwan-Woo Lee, “Fluctuation-frustrated flat band instabilities in ${\mathrm{ndnio}}_{2}$,” Phys. Rev. Research 2, 033445 (2020b).
  * Kresse and Joubert (1999) G. Kresse and D. Joubert, “From ultrasoft pseudopotentials to the projector augmented-wave method,” Physical Review B 59, 1758–1775 (1999).
  * Kresse and Hafner (1993) G. Kresse and J. Hafner, “Ab initio molecular dynamics for open-shell transition metals,” Physical Review B 48, 13115–13118 (1993).
  * Kresse and Furthmüller (1996) G Kresse and J. Furthmüller, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,” Physical Review B 54, 11169–11186 (1996).
  * Sun  _et al._ (2015) Jianwei Sun, Adrienn Ruzsinszky, and Johnp Perdew, “Strongly Constrained and Appropriately Normed Semilocal Density Functional,” Physical Review Letters 115, 036402 (2015).
  * Zhang  _et al._ (2020a) Yubo Zhang, Christopher Lane, James W. Furness, Bernardo Barbiellini, John P. Perdew, Robert S. Markiewicz, Arun Bansil, and Jianwei Sun, “Competing stripe and magnetic phases in the cuprates from first principles,” Proceedings of the National Academy of Sciences 117, 68–72 (2020a).
  * Furness  _et al._ (2018) James W. Furness, Yubo Zhang, Christopher Lane, Ioana Gianina Buda, Bernardo Barbiellini, Robert S. Markiewicz, Arun Bansil, and Jianwei Sun, “An accurate first-principles treatment of doping-dependent electronic structure of high-temperature cuprate superconductors,” Communications Physics 1, 11 (2018).
  * Lane  _et al._ (2018) Christopher Lane, James W. Furness, Ioana Gianina Buda, Yubo Zhang, Robert S. Markiewicz, Bernardo Barbiellini, Jianwei Sun, and Arun Bansil, “Antiferromagnetic ground state of La2CuO4 : A parameter-free $ab-initio$ description,” Physical Review B 98, 125140 (2018).
  * Zhang  _et al._ (2020b) Yubo Zhang, James Furness, Ruiqi Zhang, Zhi Wang, Alex Zunger, and Jianwei Sun, “Symmetry-breaking polymorphous descriptions for correlated materials without interelectronic u,” Phys. Rev. B 102, 045112 (2020b).
  * Zhang  _et al._ (2022) Ruiqi Zhang, Bahadur Singh, Christopher Lane, Jamin Kidd, Yubo Zhang, Bernardo Barbiellini, Robert S. Markiewicz, Arun Bansil, and Jianwei Sun, “Critical role of magnetic moments in heavy-fermion materials: Revisiting ${\mathrm{smb}}_{6}$,” Phys. Rev. B 105, 195134 (2022).
  * Mostofi  _et al._ (2014) Arash A. Mostofi, Jonathan R. Yates, Giovanni Pizzi, Young-Su Lee, Ivo Souza, David Vanderbilt, and Nicola Marzari, “An updated version of wannier90: A tool for obtaining maximally-localised wannier functions,” Computer Physics Communications 185, 2309–2310 (2014).
  * Guinea  _et al._ (1983) F. Guinea, C. Tejedor, F. Flores, and E. Louis, “Effective two-dimensional hamiltonian at surfaces,” Phys. Rev. B 28, 4397–4402 (1983).
  * Sancho  _et al._ (1984) MP Lopez Sancho, JM Lopez Sancho, and Jessy Rubio, “Quick iterative scheme for the calculation of transfer matrices: application to mo (100),” J. Phys. F: Met. Phys 14, 1205 (1984).
  * Xu  _et al._ (2019) Yuanfeng Xu, Zhida Song, Zhijun Wang, Hongming Weng, and Xi Dai, “Higher-order topology of the axion insulator ${\mathrm{euin}}_{2}{\mathrm{as}}_{2}$,” Phys. Rev. Lett. 122, 256402 (2019).
  * Nogaki  _et al._ (2021) Kosuke Nogaki, Akito Daido, Jun Ishizuka, and Youichi Yanase, “Topological crystalline superconductivity in locally noncentrosymmetric ${\mathrm{cerh}}_{2}{\mathrm{as}}_{2}$,” Phys. Rev. Res. 3, L032071 (2021).
  * Fu and Berg (2010) Liang Fu and Erez Berg, “Odd-parity topological superconductors: Theory and application to ${\mathrm{cu}}_{x}{\mathrm{bi}}_{2}{\mathrm{se}}_{3}$,” Phys. Rev. Lett. 105, 097001 (2010).
  * Sato (2010) Masatoshi Sato, “Topological odd-parity superconductors,” Phys. Rev. B 81, 220504 (2010).
  * Mong  _et al._ (2010) Roger S. K. Mong, Andrew M. Essin, and Joel E. Moore, “Antiferromagnetic topological insulators,” Phys. Rev. B 81, 245209 (2010).


11institutetext: Newcastle University, Newcastle upon Tyne NE1 7RU, United
Kingdom

# Local Morphological Measures Confirm that Folding within Small Partitions of
the Human Cortex Follows Universal Scaling Law

Karoline Leiberg 11 Christoforos Papasavvas 11 Yujiang Wang 11

###### Abstract

The universal scaling law of cortical morphology describes cortical folding as
the covariance of average grey matter thickness, pial surface area, and
exposed surface area. It applies for mammalian species, humans, and across
lobes, however it remains to be shown that local cortical folding obeys the
same rules. Here, we develop a method to obtain morphological measures for
small regions across the cortex and correct surface areas by curvature to
account for differences in patch size, resulting in a map of local morphology.
It enables a near-pointwise analysis of morphological variables and their
regional changes due to processes such as healthy ageing. We confirm
empirically that the theorised covariance of morphological measures still
holds at this level of local partition sizes as predicted, justifying the use
of independent variables derived from the scaling law to identify regional
differences in folding, subject-specific abnormalities, and local effects of
ageing.

##  1 Introduction

Cortical morphology is a useful imaging-based biomarker for a range of
applications, including aging and disease. Measures that describe the shape
and folding of a brain, such as gyrification and cortical thickness, are being
used both over the entire cortex and locally to identify differences in
individuals and cohorts of subjects [1, 2, 3, 4]. Such measures are often
studied separately, without taking their interaction into account.

Only recently, a universal scaling law describing the interaction of cortical
morphology measures has been proposed [5]. This scaling law captures the
folding of the cortex as the covariance between the measure of average
cortical thickness $T$, total pial surface area $A_{t}$ and exposed surface
area $A_{e}$ in the equation

$A_{t}\sqrt{T}=kA_{e}^{1.25},$ (1)

where $k$ is a constant. Empirically, $k$ varies slightly between age groups,
and has been interpreted as the tension/pressure applied to the cortical
surface [8]. This relation has been shown to hold for different mammalian
species [5], in human hemispheres [8], and within human cortices across the
lobes [9]. It says that the three variables are interdependent. For example,
if two brains of a similar age have the same total area, but one is thicker,
it will also have more exposed area - and hence less gyrification. The scaling
law can be visualised by the analogy of crumpling a piece of paper into a
ball: the thickness of the paper, its total area and the force used to crush
it determine the exposed surface area of the ball [5].

The scaling law additionally allows us to derive three linearly independent,
interpretable morphological variables as linear combinations of $\log T^{2}$,
$\log A_{t}$ and $\log A_{e}$ [7]:

$K=\log A_{t}+\frac{1}{4}\log T^{2}-\frac{5}{4}\log A_{e},$ (2) $I=\log
A_{t}+\log T^{2}+\log A_{e},$ (3) $S=\frac{3}{2}\log A_{t}-\frac{9}{4}\log
T^{2}+\frac{3}{4}\log A_{e}.$ (4)

$K$ is derived directly from Eq. (1) by taking logarithms and arranging for
$\log k$. It is a measure of tension/pressure acting on the cortex, or, in the
analogy, pressure applied to the paper ball. $I$ describes the overall size of
the brain or paper ball. Changes in $I$ correspond to an isometric scaling of
the cortex. Lastly, $S$ is the inner product of $K$ and $I$ that contains
information about shape; it can be thought of as the “folding technique” of
the paper ball. Using $K$, $I$, and $S$ avoids covarying variables, whilst
keeping them interpretable. Otherwise, analysing cortical thinning without
accounting for surface area changes could miss signs of atrophy that can be
discovered when accounting for the covariance [7].

However, one distinct challenge remains in regionalising these independent
measures of cortical morphology, $K$, $I$, and $S$. It is clear that the
scaling law cannot hold for arbitrarily small regions. The goal of this paper
is to show that the scaling law still holds for areas smaller than lobes,
defined independently of them. We first develop a method to find the raw
variables $T$, $A_{t}$, and $A_{e}$ in small patches across the cortex, and
correct the surface areas so they are independent of the size chosen for
patches. This allows us to compare the folding within the regions to that of
other partitions of the cortex or even the full hemisphere [9].

If local cortical folding follows the scaling law, it justifies the use of
$K$, $I$, and $S$ for further morphological analyses. For example, we will
show that they can be used to get a better understanding of regional age-
related cortical atrophy.

##  2 Methods

###  2.1 MRI data and processing

To asses the rules of local cortical folding in a cohort of healthy subjects
in a large age range, we used the NKI Rockland Sample data
(http://fcon˙1000.projects.nitrc.org/indi/pro/nki.html [6]).

The MRIs were preprocessed with the FreeSurfer 6.0 recon-all pipeline to
obtain the fine, triangular mesh representing the grey matter surface, as well
as the grey matter thickness on each point of the mesh. We then ran the local
gyrification index processing stream to obtain the outer smooth (exposed) pial
surface.

Out of the 929 subjects for whom data was available, 95 subjects were rejected
due to inadequate image quality, motion artifacts or parts of their grey
matter missing. We performed visual quality inspections on a random sample of
50 subjects and applied manual corrections where needed. We deemed overall
image quality and FreeSurfer processing to be adequate. The remaining sample
consists of 834 subjects, 508 female and 326 male, in an age range from 6 to
85 years. A full list of subjects used can be found on github
(https://github.com/KarolineLeiberg/folding˙pointwise).

###  2.2 Extraction of local morphological measures

The following describes the method in which morphological measures were
computed for each point on the pial surface. The full code is available on
github: https://github.com/KarolineLeiberg/folding˙pointwise.

The pial surface and smooth pial surface are reduced to 5% and 10% of their
original resolution respectively, using the downsampling function of the
MATLAB iso2mesh toolbox. This retains the main features of cortical folds, but
reduces the number of vertices in the pial to about 7000 per hemisphere.

The thickness map is converted from the original pial surface to the
downsampled pial by assigning the value of the nearest pial vertex for each
downsampled pial point. The pial thickness is subsequently transformed from
being a pointwise measure to a measure for each face in the mesh
(https://github.com/cnnp-lab/CorticalFoldingAnalysisTools).

Then, for each point $p$ on the downsampled pial surface, a local patch around
$p$ is defined as all vertices within a radius of 25mm, measured as Euclidean
distance, that are directly connected to $p$ through neighbouring points which
are also in the patch - this is to avoid e.g. disconnected patches across two
gyri that are very close to each other, but the connecting sulcus is not
included. A visualisation of the process can be found in Figure 1 a-c. A
corresponding smooth patch is defined by first finding the nearest downsampled
pial point for each downsampled smooth pial point, and then including smooth
pial points if their neighbouring pial point is in the patch. (Fig. 1 d).

Figure 1: Defining a patch on the pial around a point. a-c Extracts of the
process of adding neighbouring points to the patch. c The final pial surface
patch around this point. d The corresponding patch on the smooth pial.

The total surface area $A_{t}^{p}$ for the patch around $p$ is then computed
as the sum of all triangular faces of the downsampled pial which are contained
in $p$’s patch. Similarly, the average cortical thickness $T^{p}$ is the
average thickness across those faces. The exposed surface area $A_{e}^{p}$ is
computed as the sum of face areas within the patch on the downsampled smooth
pial.

A convex hull is fitted over $p$’s downsampled smooth pial patch, and the
integrated Gaussian Curvature of the patch around $p$ is approximated as the
sum of Gaussian curvatures of all points on the convex hull which do not lie
on the edge of the smooth patch (https://github.com/cnnp-
lab/CorticalFoldingAnalysisTools).

Having computed the morphological measures for regions around each point on
the downsampled pial surface, the data is converted back to the original, full
pial, again using values of the nearest neighbour. This is done so the data
can later be projected to the FreeSurfer average subject to compare across
subjects.

###  2.3 Surface area reconstruction

As shown in Wang et al. [9], the size of regions into which a cortex is
partitioned fully determines its surface areas and thus its location in the
space of $A_{t}$, $A_{e}$ and $T$, making it impossible to compare patches to
each other and infer whether they align on the plane predicted by the scaling
law. Each patch’s surface areas are thus reconstructed to what they would be
if the patch was a full hemisphere of the same gyrification index, using the
proportion of curvature it contains, approximated via its convex hull. This
correction is done using the formulas [9]

$A_{t}^{\prime p}=A_{t}^{p}*\frac{4\pi}{I_{G}^{p}},$ (5) $A_{e}^{\prime
p}=A_{e}^{p}*\frac{4\pi}{I_{G}^{p}},$ (6)

where $A_{t}^{p}$ and $A_{e}^{p}$ are the total and exposed surface areas of
patch $p$ before correction, and $A_{t}^{\prime p}$ and $A_{e}^{\prime p}$ are
the values after correction using the integrated Gaussian curvature
$I_{G}^{p}$ over $p$. This correction of a patch’s surface areas by its
integrated Gaussian curvature preserves its average cortical thickness,
gyrification index, and the average Gaussian curvatures of its total and
exposed areas [9].

Points that lie on particularly flat parts of the cortex or have very small
surface areas tend to have convex hulls with integrated curvatures close to
zero. In such cases, the correction of the surface areas leads to an
overcorrection, inflating the surface area to unreasonably large values. For
simplicity, we will exclude these points from our analysis, choosing not to
investigate if they also obey the scaling law at this stage. All points within
10mm to the left and right of the midline of the brain and all points with a
curvature below 0.16 are excluded. We will discuss later on how future work
may be able to investigate these more challenging parts of the cortex, where
the Gaussian curvature of the convex hull is not a good representation of the
proportion of the patch of interest.

###  2.4 Fitting the scaling law within subjects

We assess if local folding follows the scaling law by regressing within each
subject over all patches in the two-dimensional projection of the scaling law,
where

$X=\log A_{e}^{\prime p},$ (7) $Y=\log(A_{t}^{\prime p}\sqrt{T^{p}}).$ (8)

Here, $A_{t}^{\prime p}$ and $A_{e}^{\prime p}$ are the surface areas after
correcting by curvature, $T^{p}$ is the observed thickness. The slope of the
regression is a subject-specific estimation of the exponent of $A_{e}$ in Eq.
(1). We verify if each subject’s local folding follows the scaling law by
seeing how close the slope is to 1.25.

###  2.5 Local age effects

To quantify the effect ageing has on morphology in the raw measures and in the
independent variables, we used the FreeSurfer function mri_surf2surf to
convert all subjects’ morphology maps (i.e. pial surfaces with morphological
measures computed at each point) to the same surface space, where they can be
compared and analysed as a group. We compute the independent variables $K$,
$I$, and $S$ on each point of the pial surface. We then fit linear regression
models pointwise across all subjects, including both sex and age as
covariates. Points represented in fewer than 10 subjects of either sex were
excluded to ensure the regression was representative of the population. We
then use the coefficient of the ageing covariate at each point and for each
variable as an indicator of the local effect of age-related atrophy on that
measure.

##  3 Results

###  3.1 Surface area reconstruction by Gaussian curvature breaks down in
insula and on midline

We apply Gaussian curvature corrections (Eq. (5),(6)) to all points to
reconstruct surface areas of patches to what they would be for a full
hemisphere, but find that some points have approximately zero integrated
Gaussian curvature (Fig. 2a shows the convex hull over one example patch).
When we look across all points of the cortex, we observe a distinct subset of
points that display zero curvature (Fig. 2b). These points are usually located
in the insula or particularly deep sulci, where the patch has a small, overall
convex, exposed surface, or on the midline, where the exposed surface might be
large, but very flat. The midline additionally contains points with very large
curvature, where the convex hull over a patch can contain an angle of 90° or
more.

Correcting points by extremely small curvatures leads to overcorrections (Fig.
2c): Whilst most points naturally align in the $X$-$Y$-plane (Eq. (7),(8))
with a slope of 1.25 after the surface areas are reconstructed, points with
curvatures near zero are being overcorrected to the top-right of the plot. For
simplicity, we therefore exclude these points from our further analysis,
acknowledging that Gaussian curvature of the convex hull is not a good
representation of the proportion of those patches. This affects around 20% of
all points on the pial surface.

Figure 2: Gaussian curvature and its effect on reconstructing surface areas in
one example subject. a Example of patch with zero curvature. b Distribution of
curvature across patches. c Sample of 150 patches plotted in the X-Y-plane,
both before (round marker) and after (asterisk) surface area corrections.
Yellow lines with slope 1 indicate the shift of points caused by the curvature
corrections. Green points are included in further analysis, purple points are
excluded due to their low curvature or position relative to the midline. The
black line is a regression line through the green points after correction.

###  3.2 Local folding follows scaling law

To verify whether the scaling law still applies in small patches of the
cortex, meaning if the measures of $T$, $A_{t}$, and $A_{e}$ covary locally as
predicted, we fit the scaling law within subjects, estimating the slope
between points in $X$ and $Y$ as described in Eq. (7) and (8). We find that
the slope for each subject, i.e. the subject-specific estimates of the
coefficient of $A_{e}$ in Eq. (1), are distributed around a mean of 1.23 (Fig.
3), only slightly lower than the observed coefficient of 1.25 for hemispheres
and lobes. This shows that the scaling law still holds at this level of patch
sizes, meaning that when looking at small, local regions of the cortex with
spherical radii of 25mm, the average thickness, total area and exposed area
covary predictably according to Eq. (1). Note that the subjects follow the
scaling law without any age or sex corrections, because age differences only
affect the scaling law intercept ($K$), but not the slope [8].

Figure 3: Distribution of slopes observed in subjects when fitting a
regression through points across the cortex in the $X$-$Y$-plane. The black
line marks the median of observed slopes.

###  3.3 $K$, $I$ and $S$ have additional value when observing ageing effects

As we have seen, the folding follows the scaling law even locally, which
implies that $K$, $I$, and $S$ are theoretically independent. We convert to
these variables to see if they add insight to the differences in local
morphology.

When we look at the effect of healthy ageing on local morphology, we see a
decrease in the raw variables $A_{t}$, $A_{e}$, and $T$ at varying rates in
most areas of the brain (Fig. 4 a-c). A decrease in surface areas can be
interpreted as a flattening of the cortex, since less surface area within a
constant radius indicates less folding. The independent variables show an
isometric shrinking of most areas, with an overall loss of tension (Fig. 4
d-e).

In the raw variables, we do not see systematic differences in how the areas
around the upper and lower precentral gyrus are affected by atrophy. We do
however see such a difference in the shape term $S$ (Fig. 4f): $S$ decreases
in the upper precentral gyrus with ageing, but the lower part is relatively
unaffected. This is one example of the added value from switching to
independent variables; we gain information otherwise hidden in the covariance
of $A_{t}$, $A_{e}$, and $T$.

Figure 4: Local effects of healthy ageing in the left and right hemisphere.
a-c Raw variables $T$, $A_{t}$, and $A_{e}$ (logged). d-f Independent
variables $K$, $I$, and $S$.

##  4 Discussion

On a big data set covering a large age range, we have demonstrated that the
local folding of the brain follows the universal scaling law, meaning the grey
matter thickness, total surface area and exposed surface area in small regions
covary according to the same rule as the whole cortex. This result extends
previous work done in this area, which had shown that morphology in full
hemispheres and lobes adheres to the scaling law, by confirming empirically
that the minimum size of regions for which the folding rule applies has a
radius of under 25mm.

Our method works well for most of the cortex. However, we had to exclude
around 20% of points on the pial surface from our analysis, predominantly
located on the midline and in the insula. For these points, we founds that the
Gaussian curvature was not an adequate representation of the proportion of the
patch. For future iterations of the method, we plan to improve on this. An
alternative way to approximate patch curvature would be to compute the
integrated curvature of the patch’s smooth pial surface, rather than its
convex hull. However, this surface may already contain too much local
information of folding for our purpose. Another idea is to inflate the pial
surface to a sphere and use the proportion of surface area or curvature of the
patch’s representation on the sphere for corrections.

Another improvement of the method could be achieved by making the radius
defining the patch around each point on the pial adaptive depending on the
thickness at that point. This flexible way of patch definitions uses the
smallest radius required to find enough exposed surface area at each point,
making the morphological map as close to pointwise as possible. We expect the
ideal patch radii to be around 20-30mm for human brains (spanning at least one
sulcus/gyrus), which is why we chose a value in this range for the fixed-size
method.

The local applicability of the scaling law allows us to use it in terms of
independent components derived from it, to further analyse local morphology.
In our results we have shown local differences in how the cortex is affected
by the process of ageing in the independent variables, which were not clear
from the raw variables alone. Our method could have other practical
applications, such as finding local abnormalities in patient groups compared
to a control cohort, indicating regional effects of dysfunction. It might also
be used to detect subject-specific abnormalities, indicating areas of the
brain that fold atypically.

####  4.0.1 Acknowledgements.

KL was supported by the Centre for Doctoral Training in Cloud Computing for
Big Data (EP/L015358/1). We thank the members of CNNP lab (www.cnnp‐lab.com)
and Bruno Mota for discussions of the method and results.

## References

  * [1] Chaudhary, S., Kumaran, S.S., Goyal, V., Kaloiya, G.S., Kalaivani, M., Jagannathan, N.R., Sagar, R., Mehta, N., Srivastava, A.K.: Cortical thickness and gyrification index measuring cognition in Parkinson’s disease. International Journal of Neuroscience pp. 1–10 (may 2020)
  * [2] Frangou, S., Modabbernia, A., Williams, S.C.R., Fuentes-claramonte, P., Glahn, D.C.: Cortical thickness across the lifespan : Data from 17 , 075 healthy individuals aged 3 – 90 years. Human Brain Mapping (November 2020), 1–21 (2021)
  * [3] Galovic, M., Van Dooren, V.Q., Postma, T.S., Vos, S.B., Caciagli, L., Borzì, G., Cueva Rosillo, J., Vuong, K.A., De Tisi, J., Nachev, P., Duncan, J.S., Koepp, M.J.: Progressive Cortical Thinning in Patients with Focal Epilepsy. JAMA Neurology 76(10), 1230–1239 (oct 2019)
  * [4] Libero, L.E., Schaer, M., Li, D.D., Amaral, D.G., Nordahl, C.W.: A Longitudinal Study of Local Gyrification Index in Young Boys With Autism Spectrum Disorder. Cerebral Cortex 29(6), 2575–2587 (jun 2019)
  * [5] Mota, B., Herculano-Houzel, S.: Cortical folding scales universally with surface area and thickness, not number of neurons. Science 349(6243), 74–77 (jul 2015)
  * [6] Nooner, K.B., Colcombe, S.J., Tobe, R.H., Mennes, M., Benedict, M.M., Moreno, A.L., Panek, L.J., Brown, S., Zavitz, S.T., Li, Q., Sikka, S., Gutman, D., Bangaru, S., Schlachter, R.T., Kamiel, S.M., Anwar, A.R., Hinz, C.M., Kaplan, M.S., Rachlin, A.B., Adelsberg, S., Cheung, B., Khanuja, R., Yan, C., Craddock, C.C., Calhoun, V., Courtney, W., King, M., Wood, D., Cox, C.L., Kelly, A.M., Di Martino, A., Petkova, E., Reiss, P.T., Duan, N., Thomsen, D., Biswal, B., Coffey, B., Hoptman, M.J., Javitt, D.C., Pomara, N., Sidtis, J.J., Koplewicz, H.S., Castellanos, F.X., Leventhal, B.L., Milham, M.P.: The NKI-Rockland sample: A model for accelerating the pace of discovery science in psychiatry. Frontiers in Neuroscience 6(OCT) (2012)
  * [7] Wang, Y., Leiberg, K., Ludwig, T., Little, B., Necus, J.H., Winston, G., Vos, S.B., de Tisi, J., Duncan, J.S., Taylor, P.N., Mota, B.: Independent components of human brain morphology. NeuroImage 226, 117546 (feb 2021)
  * [8] Wang, Y., Necus, J., Kaiser, M., Mota, B.: Universality in human cortical folding in health and disease. Proceedings of the National Academy of Sciences of the United States of America 113(45), 12820–12825 (nov 2016)
  * [9] Wang, Y., Necus, J., Rodriguez, L.P., Taylor, P.N., Mota, B.: Human cortical folding across regions within individual brains follows universal scaling law. Communications Biology 2(1), 1–8 (dec 2019)