# Symmetry-enforced nodal lines in the band structures of vacancy-engineered
graphene

Matheus S. M. de Sousa  Department of Physics, PUC-Rio, 22451-900 Rio de
Janeiro, Brazil  Fujun Liu  Nanophotonics and Biophotonics Key Laboratory of
Jilin Province, School of Physics, Changchun University of Science and
Technology, Changchun, 130022, P.R.China  Mariana Malard  Faculdade UnB
Planaltina, Universidade de Brasília, Brasília-DF, Brazil  Fanyao Qu
Instituto de Física, Universidade de Brasília, Brasília-DF, Brazil  Wei Chen
Department of Physics, PUC-Rio, 22451-900 Rio de Janeiro, Brazil

###### Abstract

We elaborate that single-layer graphene with periodic vacancies can have a
band structure containing nodal lines or nodal loops, opening the possibility
of graphene-based electronic or spintronic devices with novel functionalities.
The principle is that by removing carbon atoms such that the lattice becomes
nonsymmorphic, every two sublattices in the unit cell will map to each other
under glide plane operation. This mapping yields degenerate eigenvalues for
the glide plane operation, which guarantees that the energy bands must stick
together pairwise at a boundary of the Brillouin zone. Moving away from the
Brillouin zone boundary causes the symmetry-enforced nodal lines to split,
resulting in accidental nodal lines caused by the crossings of the split
bands. Moreover, the density of states at the Fermi level may be dramatically
enhanced if the nodal lines crosses the Fermi level. The nodal lines occur a
variety of vacancy configurations even in the presence of Rashba spin-orbit
coupling. Finally, our theory also explains the nodal loops surrounding the
entire Brillouin zone of a chevron-type nanoporous graphene fabricated in a
recent experiment.

##  I Introduction

A nodal-line semimetal (NLSM) is a novel phase of matter, characterized by
band crossings along lines, loops, or even circles in the Brillouin zone
(BZ)Fang2015 . Both theoretical and experimental results have demonstrated
that NLSMs possess various interesting properties, such as chiral anomaly
Zyuzin2012 , extremely large magnetoresistance Gao2017 , photo-induced
anomalous Hall effects Liu2018 , high thermal conductivity, giant intrinsic
charge mobility, non-Abelian statistics and superconductivity Alicea2011 ,
which have strongly motivated the research on NLSMsFeng2017 ; Liu2018 ;
Nie2020 ; Zunge2018 . Concerning the mechanisms for the formation of these
nodal lines and nodal loops, they can be either accidental or symmetry-
enforcedZuo2019 ; Schoop2016 ; Fang2015 ; YangSY2018 . The former are related
to various spatial or nonspatial symmetries, such as the Dirac nodal lines
protected by reflection, space-time inversion, or rotation symmetry, and can
be adiabatically destroyed by tuning parameters of the material, such as spin-
orbit coupling (SOC). Symmetry enforced NLSM phases Schoop2016 ; Fang2015 , on
the other hand, emerge in crystals with nonsymmorphic symmetries, which
warrants global stability to the associated nodal lines.

A recent work proves a remarkable principle to engineer nodal lines in two-
dimensional (2D) materials: By periodically removing atoms such that the
lattice becomes nonsymmorphic, the material becomes a robust NLSMLiu21 ;
Liu21_2 . As a case study, it was shown that the band structures of a number
of vacancy-engineered borophenes display nodal lines originated from a
nonsymmorphic glide-plane symmetry. In this paper, we demonstrate that vacancy
engineering can in fact be applied to another important 2D material, namely
the single-layer graphene. As the first member of the 2D family, graphene
exhibits a variety of extraordinary properties with huge impact to applied
research, including unprecedented high strength and flexibility, ultralow
weight, ultrahigh carrier nobilities, high optical transparency and high
thermal conductivity Castro-2009 . However, these physical properties rely on
the linear Dirac cones at low energy which are rather frail to SOC, especially
the Rashba SOC, which may hinder its application in SOC-based devices. In
particular, Rashba SOC is known to be tunable by a gate voltage in
graphene/transition-metal dichalcogenide heterostructuresYang16 ; Wang16 ;
Yang17 ; Offidani17 ; Safeer19 ; Ghiasi19 ; Benitez20 , which may be used to
engineer a variety of spintronic effects, such as the recently discovered edge
current, edge spin current, and bias-voltage free spin torque in geometrically
confined graphenedeSousa21 . A vacancy-engineered graphene that supports nodal
lines even in the presence of Rashba SOC could thus open up new possible
functionalities. Moreover, we find that the density of states (DOS) near the
Fermi level is dramatically enlarged if the nodal lines passes the Fermi
level, which is expected to strongly impact the electronic and magnetic
properties of the proposed structures. This connects with studies of twisted
bilayer graphene which hosts superconductivity when tuned to special “magic
angles” at which isolated and relatively flat bands appear Cao-2018 .

Besides nodal lines, here we demonstrate that vacancy-engineering also allows
to generate nodal loops surrounding the entire boundary of the BZ.
Interestingly, the scheme additionally gives rise to accidental nodal lines
and loops inside the BZ that are robust to Rashba SOC. Regarding the
feasibility of our proposal, various experimental techniques such as self-
aligned anisotropic etchingShi11 , copolymer lithographyBai10 , nano-network
maskingJung14 , nanosphere lithographyWang13 , and nitrogenationMahmood-2018
have been employed to fabricate graphene with vacancies, often called graphene
nanomesh or holey graphene. In particular, we will elaborate that the nodal
loops in fact have already been realized in a recent experiment that
fabricates a nonsymmorphic chevron-type nanoporous grapheneJacobse20 ,
although this feature seems to be overlooked. Additionally, to further
illustrate the generality of our scheme, we show that a vacancy-engineered
square lattice also supports nodal loops. Thus we anticipate that the proposed
vacancy engineering principle may be further exploited to design Rashba SOC-
active nodal-line spintronic or electronic devices in a wide variety of 2D
materials.

Naturally nonsymmorphic materials (i.e. that contain glide-plane or screw-axis
symmetries without lattice engineering) are expected to possess symmetry-
enforced band degeneraciesYoung15 ; Yamakage16 ; Zhao16 ; Wieder18 . Three-
dimensional realizations of such materials have been predicted in hexagonal
compoundsZhang18 . Our aim is to put forward a simple and practical method to
create 2D nonsymmorphic materials which contains multiple nodal-lines or
nodal-loops. We argue theoretically and illustrate numerically that it can
done simply by periodically removing atoms from common monoatomic and
nonmagnetic sheets, using graphene as a concrete example.

##  II NLSM phases in vacancy-engineered graphene

###  II.1 Vacancy-engineered graphene with a single glide plane

The lattices engineered from graphene are denoted by CN, where $N$ is the
number of sublattices in the rectangular unit cell.We firstly consider a
lattice that belongs to the wallpaper group $p2mg$Aroyo2016 , which has a
glide plane going along ${\hat{\bf y}}$, and a reflection plane along
${\hat{\bf x}}$, as shown in Fig. 1 (a) for a C10 configuration. In this case,
we demonstrate that every two of the $N$ bands must stick together and form
$N/2$ symmetry-enforced nodal lines at the BZ boundary $k_{y}=\pm\pi$. In
contrast to the previous works about vacancy-engineered nodal lines that are
based on analyzing how the pairwise degenerate eigenvalues of the
nonsymmorphic symmetry operator constrain the band structureLiu21 ; Liu21_2 ,
in this work we present a new formalism based on a general feature of these
nonsymmorphic vacancy configurations that enforces the nodal lines
irrespective of the detail of the Hamiltonian. The general feature is that
every two sublattices form a pair that map to each other under glide-plane
operation, which we call a glide pair. Denoting the position of the unit cell
to be $(x,y)$ and each sublattice to be $(A,B,C...)$, there are two kinds of
glide pairs that we call type-I and type-II. The electron annihilation
operators $c_{x,y}^{\rm sub}$ for a type-I/II glide pair $(A,B)/(C,D)$
transform under glide-plane operation as

$\displaystyle{\rm Type\;I}:\;c_{-x,-y}^{A}\rightarrow
c_{x,-y}^{B},\;\;\;c_{-x,-y}^{B}\rightarrow c_{x,-y+1}^{A},$
$\displaystyle{\rm Type\;II}:\;c_{-x,-y}^{C}\rightarrow
c_{x-1,-y}^{D},\;\;\;c_{-x,-y}^{D}\rightarrow c_{x-1,-y+1}^{C}.\;\;\;$ (1)

The above transformations convey that the glide plane reflects the
$x$-coordinate of a type-I pair, whereas in a type-II pair the glide plane
reflects the $x$-coordinate and then translates to a neighboring unit cell
along $x$. This translation is due to a shift between the center of the unit
cell and the glide plane along the $x$-direction (c.f. Fig. 1 (a)). In both
type-I and type-II pairs, the reflection of the $x$-coordinate is followed by
a translation along $y$ which lands at the neighboring unit cell for the
$B\rightarrow A$ and $D\rightarrow C$ transformations. The presence of the two
orthogonal translations means that the glide plane is, simultaneously, a
nonsymmorphic and an off-centered symmetryLiu21 ; Malard2018 .

With the transformations in Eq. (1) and arrangement of the basis functions
according to the $N_{1}$ type-I and $N_{2}$ type-II glide pairs, the $N\times
N$ glide-plane operator $G({\bf k})$ of the whole system is block diagonal,
with $N_{1}+N_{2}=N/2$ blocks of $2\times 2$ matrices

$\displaystyle G({\bf k})=N_{1}\,g_{1}({\bf k})\oplus N_{2}\,g_{2}({\bf
k})\Updownarrow_{k_{x}},$ $\displaystyle g_{1}({\bf
k})=\left(\begin{array}[]{cc}&e^{-ik_{y}}\\\
1&\end{array}\right),\;\;\;g_{2}({\bf
k})=\left(\begin{array}[]{cc}&e^{-ik_{x}-ik_{y}}\\\
e^{-ik_{x}}&\end{array}\right),$ (6) (7)

where $\Updownarrow_{k_{x}}$ takes $k_{x}$ to $-k_{x}$. The lattice described
by the Bloch Hamiltonian $H({\bf k})$ has glide-plane symmetry if
$\left[H({\bf k}),G({\bf k})\right]=0$. It then follows that there are
$\left\\{N_{1},N_{1},N_{2},N_{2}\right\\}$-fold degenerate glide plane
symmetry eigenvalues
$\left\\{+e^{-ik_{y}/2},-e^{-ik_{y}/2},+e^{-ik_{x}-ik_{y}/2},-e^{-ik_{x}-ik_{y}/2}\right\\}$.
The simultaneous eigenstates of $G({\bf k})$ and $H({\bf k})$ satisfy

$\displaystyle G({\bf k})|\psi_{n1\pm}({\bf k})\rangle=\pm
e^{-ik_{y}/2}\Updownarrow_{k_{x}}|\psi_{n1\pm}({\bf k})\rangle,$
$\displaystyle G({\bf k})|\psi_{n2\pm}({\bf k})\rangle=\pm
e^{-ik_{x}-ik_{y}/2}\Updownarrow_{k_{x}}|\psi_{n2\pm}({\bf k})\rangle,$
$\displaystyle H({\bf k})|\psi_{nI\pm}({\bf
k})\rangle=E_{nI\pm}(k_{x},k_{y})|\psi_{nI\pm}({\bf k})\rangle.\;\;\;\;\;\;$
(8)

where $n=\left\\{1,2...\right\\}$ is the band index, $I=\left\\{1,2\right\\}$
stems from the two types of glide pairs. In Appendix A, we elaborate that
combining the appropriate transformation properties of the eigenstate with the
fact that $k_{y}=0$ and $k_{y}=2\pi$ are the same point, we arrive at a
condition for the eigenenergies

$\displaystyle E_{nI-}(k_{x},2\pi)=E_{nI+}(k_{x},0),$ $\displaystyle
E_{nI+}(k_{x},2\pi)=E_{nI-}(k_{x},0),$ (9)

meaning that at a given $\left\\{k_{x},n,I\right\\}$, the two bands with
energy $E_{nI+}(k_{x},k_{y})$ and $E_{nI-}(k_{x},k_{y})$ swap and hence must
cross each other somewhere in $0\leq k_{y}\leq 2\pi$. Applying the same
argument to the BZ boundary $k_{y}=\pm\pi$ further dictates that the band
crossing must occur at the BZ boundary

$\displaystyle E_{nI+}(k_{x},\pi)=E_{nI-}(k_{x},\pi).$ (10)

Thus every two bands with parameters $\left\\{nI+\right\\}$ and
$\left\\{nI-\right\\}$ at any $k_{x}$ have to stick together at $k_{y}=\pi$,
yielding $N/2$ nodal-lines there.

Figure 1: Lattice of C10 (a) before and (b) after structural optimization.
There are 10 C atoms in the unit cell defined by lattice vectors $\vec{a}$ and
$\vec{b}$. The lattices have two symmetries: symmorphic reflection plane $R$
perpendicular to $\vec{b}$ and nonsymmorphic glide plane $G$ perpendicular to
$\vec{a}$. (c) Band structure of C10 along the high-symmetry line
$\Gamma$-X-V-$\Gamma$-Y-V, with the Fermi level set at zero energy. The black
arrow and circles indicate the symmetry-enforced nodal lines and the circles
indicate accidental nodal lines, respectively. The corresponding density of
states is shown to the right of the band structure. (d) Contour plot of the
two kinds of nodal-lines by projecting them on the $k_{x}$-$k_{y}$ plane, with
the color bar indicating the inter-band energy gap. The symmetry-enforced
nodal line is at the $k_{y}=\pm\pi$ boundary of the Brillouin zone, while the
accidental one is inside.

The validity of the above analysis is further supported by our first
principles calculations for the band structure of various nonsymmorphic
configurations in the $p2mg$ group. The band structures are obtained using the
QUANTUM ESPRESSO package Giannozzi2009 . The kinetic energies cutoff for wave
function (ecutwfc) and for charge density (ecutrho) are set to 500 and 45 Ry,
respectively. Perdew, Burke and Ernzerhof (PBE) form of the generalized
gradient approximation (GGA) is adopted for the exchange-correlation energy
JPPerdew . Numerical integrations in the BZ are evaluated with the Monkhorst-
Pack mesh of $10\times 10\times 1$. All structures are relaxed until the total
energy converges to within $10^{-4}$ eV during the self-consistent loop, with
forces converged to 0.1 eV/nm, while employing the Methfessele-Paxton method
with a smearing of 0.2 eV width. With the optimized geometry of the graphene
structure and the corresponding self-consistent ground state computed with
Quantum ESPRESSO, we use Wannier90 Mostofi2014  to map the ground-state wave
functions onto a maximally localized Wannier function basis, and employ an
adaptive $k$-mesh strategy to extract the matrices to build the real-space
tight-binding model in the basis of the $s$, $p_{x}$, $p_{y}$ and $p_{z}$
orbitals of C atoms. We find that although the vacancy-engineered lattice
distorts after the geometry optimization process as shown in Fig. 1(b) for
C10, the relaxed lattice still belongs to the same $p2mg$ group and satisfies
all the requirements in our argument.

Figure 1 (c) displays the band structure of C10 within a 2.0 eV window
centered at the Fermi energy, obtained by density functional theory (DFT)
calculation. Along the Y-V direction which corresponds to $k_{y}=\pi$, every
two pairs of bands stick together to form a nodal line, as predicted. We note
that, in the absence of spin-orbit coupling, all bands are completely spin-
degenerate throughout the Brillouin zone (BZ). Hence, the nodal lines formed
along the Y-V BZ edge are four-fold degenerate, with a two-fold degeneracy
from spin and a two-fold degeneracy enforced by the glide-plane symmetry. In
addition, from the DOS shown in Figure 1 (c), we see that the DOS no longer
vanishes linearly near the chemical potential as in pristine
grapheneCastro-2009 , but has a finite value due to the more complicated band
structure. Figure 1(d) shows the contour plot on the $k_{x}$-$k_{y}$ plane of
the glide-plane-enforced nodal lines indicated by the arrow in Fig. 1(c), as
well as of accidental nodal lines indicated by the black circles. The latter
arise from the accidental crossing of the splitting bands as they disperse
from the symmetry enforced nodal lines. Accidental nodal lines have also been
observed in holey grapheneChen-2018 , but the mechanism therein is unrelated
to crystalline symmetry, and therefore the resulting nodal lines are unstable
against SOC. In contrast, the accidental nodal lines shown in the interior of
the BZ in Fig. 1(d) arise from splitting the symmetry enforced ones, and hence
are robust against SOC and any glide-plane symmetry-preserving perturbations.

Figure 2: Lattice of C44 (a) before and (b) after structural optimization.
There are 44 C atoms in the unit cell defined by lattice vectors $\vec{a}$ and
$\vec{b}$. The lattices have two orthogonal glide-planes $G_{x}$ and $G_{y}$.
(c) Band structure of C44 along the path $\Gamma$-X-V-$\Gamma$-Y-V, where the
black arrows and circles indicate the symmetry-enforced and accidental nodal
loops, respectively. The corresponding density of states is shown to the right
of the band structure. (d) Contour plot of the two kinds of nodal loops by
projecting them on the $k_{x}$-$k_{y}$ plane, with the color bar indicating
the inter-band energy gap. The symmetry-enforced nodal loop surrounds the
Brillouin zone boundary, while the accidental one is inside.

###  II.2 Vacancy-engineered graphene with two orthogonal glide planes

We proceed to discuss vacancy-engineered graphene that contain two orthogonal
glide planes going along ${\hat{\bf x}}$ and ${\hat{\bf y}}$ directions, with
the corresponding glide plane operators $G_{x}$ and $G_{y}$. Figure 2(a) shows
the lattice structure of C44 which belongs to the wallpaper group $p2gg$. Even
after the lattice relaxation, C44 still hosts two orthogonal glide planes, as
seen in Fig. 2(b). Because the Hamiltonian commutes with both glide-plane
operators, $\left[H({\bf k}),G_{x,y}({\bf k})\right]=0$, we are able to label
the common eigenstates by the quantum numbers
$\left\\{n,\alpha,\beta\right\\}$, where $n$ is the band index,
$\alpha=I_{x}\pm$ labels the eigenvalues of $G_{x}$, and $\beta=I_{y}\pm$
labels the eigenvalues of $G_{y}$. Through generalizing the argument for one
glide plane to two orthogonal glide planes, as detailed in Appendix B, we
arrive at the condition

$\displaystyle E_{n\alpha I_{y}-}(k_{x},\pi)=E_{n\alpha I_{y}+}(k_{x},\pi),$
$\displaystyle
E_{nI_{x}-\beta}(\pi,k_{y})=E_{nI_{x}+\beta}(\pi,k_{y}),\;\;\;\;$ (11)

where $E_{n\alpha\beta}(k_{x},k_{y})$ are the eigenenergies. Thus every two
bands are forced to stick together all around the BZ boundary, forming $N/2$
symmetry enforced nodal loops.

The above assertion is verified in the DFT band structure of C44 in Fig. 2 (c)
which clearly displays band crossings for every two pairs of spin-degenerate
bands along the lines X-V and Y-V, yielding the outer four-fold degenerate
nodal loop shown in the contour plot of Fig. 2 (d), and an enlarged DOS near
the Fermi level. The band structure also exhibits accidental nodal loops
inside the BZ, which are caused by crossings of bands as they split from the
symmetry-enforced nodal-loops at the BZ boundary.

###  II.3 Effect of intrinsic and Rashba SOC on nodal lines and nodal loops

In this section, we use C44 to elaborate that in the presence of Rashba SOC,
even though the spin degeneracy of the band structure is lifted, the glide
plane symmetry still forces every two spin-split bands to stick together at
the BZ boundary, ensuring the existence of nodal lines and nodal loops, which
are now two-fold degenerate. To demonstrate this effect, we consider the
nearest-neighbor tight-binding model of graphene with Rashba SOC described by
the Hamiltonian

$\displaystyle H$ $\displaystyle=$ $\displaystyle t\sum_{\langle
ij\rangle,\sigma}c_{i\sigma}^{{\dagger}}c_{j\sigma}+i\lambda_{SOC}\sum_{\langle
ij\rangle,\alpha,\beta}c_{i\alpha}^{{\dagger}}\left({\boldsymbol{\sigma}}_{\alpha\beta}\times{\bf
d}_{ij}\right)^{z}c_{j\beta}$ (12) $\displaystyle+U\sum_{i\in
vac,\sigma}c_{i,\sigma}^{{\dagger}}c_{i\sigma}.\;\;\;$

Here $c_{i\sigma}$ is the electron annihilation operator of spin $\sigma$ on
the lattice site $i$, $t$ is the hopping amplitude between nearest neighbor
lattice sites $\langle ij\rangle$, $\lambda_{SOC}$ is the coupling constant of
Rashba SOC caused by breaking the inversion symmetry in the out-of-plane
direction ${\hat{\bf z}}$,
${\boldsymbol{\sigma}}=(\sigma^{x},\sigma^{y},\sigma^{z})$ are the Pauli
matrices, ${\bf d}_{ij}$ is the vector connecting site $i$ to site $j$. A very
large on-site potential $U\sim 100t$ is applied on the vacancy sites $i\in
vac$ to conveniently create the desired vacancy configuration.

Figure 3: (a) The band structure of the C44 configuration in Fig. 2 without
Rashba SOC simulated by a nearest-neighbor tight-binding model. The nodal
loops in the BZ boundary $X-V$ and $Y-V$ are four-fold degenerate. (b) The
band structure of C44 in the presence of Rashba SOC, which shows that despite
the spin-splitting of the bands inside the BZ, every two bands still stick
together at the BZ boundary to form nodal loops. The nodal loops are two-fold
degenerate in this case due to the lifting of the spin degeneracy.

We will use this tight-binding model to examine the effect of Rashba SOC on
the C44 configuration in Fig. 2, which has two orthogonal glide planes. In the
pristine C44 without Rashba SOC, the two orthogonal glide planes cause every
two bands to stick together at the BZ boundary, and in addition there is spin
degeneracy, so the nodal loops in Fig. 2 (a) are in fact four-fold degenerate.
In contrast, Fig. 3 (b) shows that at a finite Rashba SOC, the spin degeneracy
is lifted everywhere inside the BZ as expected. Nevertheless, every two spin-
split bands still merge together to form a two-fold degenerate nodal loop at
the BZ boundary. In short, the Rashba SOC splits the spin degeneracy of the
nodal loops and hence changes their degeneracy from four-fold to two-fold, but
the glide-plane symmetry still ensures the existence of nodal loops at the BZ
boundary.

Figure 4: (a) A vacancy configuration containing two orthogonal glide planes
$G_{x}$ and $G_{y}$ engineered from a square lattice, and (b) its tight-
binding band structure along high-symmetry lines. The black arrows and circles
indicate the symmetry-enforced and accidental nodal loops, respectively.

###  II.4 Vacancy-engineered NLSM from a square lattice

In this section, we elaborate that our proposal is in fact a general principle
not limited to graphene, but also applicable to other 2D materials with
different lattice structures. Figure 4 (a) shows a crystal structure vacancy-
engineered from a square lattice, which belongs to wallpaper group $p4gm$ that
contains two orthogonal glide planes. The band structure obtained by tight-
binding model with nearest-neighbor hopping $H=\sum_{\langle
ij\rangle}t\,c_{i}^{{\dagger}}c_{j}$ contains nodal-loops at BZ boundary just
like C44, as shown in Figure 4 (b). This result indicates that our vacancy
engineering principle can be generically applied to any 2D lattices regardless
the structural and chemical details of the host system, and it is based solely
on crystalline symmetries.

###  II.5 Experimental realization in nanoporous graphene

Concerning the experimental realization of our proposal, a particularly
promising route is the so-called bottom-up type of approach to nanoporous
graphene. In this type of approach, one starts from small clusters of some
precursor molecules, and choose a suitable chemical environment such that they
self-assemble into lattice structures with periodic vacancies. This technique
has been applied to grow graphene with periodic nanometer size poresMoreno18 ;
Jacobse20 . In fact, the experimental vacancy configuration realized in Ref.
Jacobse20, , called chevron-type nanoporous graphene (C-NPG), is
nonsymmorphic. The lattice belongs to wallpaper group $p2gg$ that contains
glide planes in two orthogonal crystalline directions, similar to the C44
example discussed in Sec. II.2. Therefore, the C-NPG should contain nodal
loops surrounding the entire BZ edge according to our theory, as have also
been confirmed by DFT calculations (see Fig. 3 E of Ref. Jacobse20, ),
although this feature has not been emphasized. However, this C-NPG contains a
band gap $\sim$ eV at the Fermi level, and all the nodal loops form outside
the band gap. Thus we anticipate that some experimental efforts is needed to
search for other nanoporous configurations that contain nodal lines or loops
crossing the Fermi level, such that the DOS may be enhanced instead of
reduced.

##  III Conclusions

In summary, we elaborate that vacancy-engineered nonsymmorphic graphene
exhibits band structures with multiple symmetry enforced nodal lines or nodal
loops at the BZ boundary. This mechanism is based on the formation of glide
pairs of the sublattices, which manifests regardless the original material is
semimetallic, like graphene, or metallic, like a square lattice. In addition,
accidental nodal lines and nodal loops can also occur inside the BZ. In fact,
such a nonsymmorphic vacancy configurations have been realized experimentally
in a nanoporous grapheneJacobse20 , and the existence of nodal loops in this
configuration has been confirmed by DFT calculations, although it has not been
emphasized. Our mechanism thus opens a new direction to explore vacancy-
triggered NLSMs which can coexist with other material properties like Rashba
SOC, and is even compatible with other types of vacancy-engineered band
structures such as flat bandsdeSousa21_flatband , hence may be exploited to
fabricate novel NLSM-based electronic or spintronic devices. Moreover, shall
the nodal lines cross the Fermi level, the finite DOS at the Fermi level is
expected to dramatically alter thermal, electric and magnetic properties of
the material compared to those in pristine graphene. We anticipate that this
nonsymmorphic vacancy-engineering principle can be widely applied to change
the band structure of a great variety of 2D materials, with the accompanying
change of physical properties that awaits to be explored.

###### Acknowledgements.

We thank exclusively D. Kochan for the discussion about SOC. W.C. is supported
by the productivity in research fellowship from CNPq.

##  Appendix A Nodal lines enforced by a single glide plane

We now give a detailed formalism for the nodal lines and nodal loops enforced
by nonsymmorphic symmetry of the vacancy engineered lattices, starting from
the nodal lines in the situation that the lattice contains only one glide
plane. We will consider the spinless situation for simplicity, but the
argument can be easily generalized to include spin. First we elaborate why the
eigenvalues of $G({\bf k})$ in Eq. (8) contain the momentum-swapping operation
$\Updownarrow_{k_{x}}$ using a simple example. Consider the minimal situation
of $N=2$ sublattice as an example, in which $G({\bf
k})=g(k_{y})\Updownarrow_{k_{x}}$, and we intend to diagonalize it to obtain
the eigenvalues $\lambda_{\pm}({\bf k})$

$\displaystyle g({\bf k})|\phi_{\pm}({\bf k})\rangle=\lambda_{\pm}({\bf
k})|\phi_{\pm}({\bf k})\rangle.$ (13)

Denoting the eigenstate by

$\displaystyle|\phi_{\pm}({\bf
k})\rangle=\left(\begin{array}[]{c}u_{\pm}(k_{x},k_{y})\\\
v_{\pm}(k_{x},k_{y})\end{array}\right),$ (16)

the eigenvalue problem leads to

$\displaystyle g({\bf k})|\phi_{\pm}({\bf
k})\rangle=\left(\begin{array}[]{cc}&e^{-ik_{y}}\\\
1&\end{array}\right)\Updownarrow_{k_{x}}\left(\begin{array}[]{c}u_{\pm}(k_{x},k_{y})\\\
v_{\pm}(k_{x},k_{y})\end{array}\right)$ (21)
$\displaystyle=\left(\begin{array}[]{cc}&e^{-ik_{y}}\\\
1&\end{array}\right)\left(\begin{array}[]{c}u_{\pm}(-k_{x},k_{y})\\\
v_{\pm}(-k_{x},k_{y})\end{array}\right)$ (26)
$\displaystyle=\lambda_{\pm}(k_{x},k_{y})\left(\begin{array}[]{c}u_{\pm}(k_{x},k_{y})\\\
v_{\pm}(k_{x},k_{y})\end{array}\right),$ (29)

we see that this equation in general cannot be solved, because there needs not
be a relation between $u_{\pm}(-k_{x},k_{y})$ and $u_{\pm}(k_{x},k_{y})$, or
between $v_{\pm}(-k_{x},k_{y})$ and $v_{\pm}(k_{x},k_{y})$. Thus the correct
way to diagonalize it is to maintain the $\Updownarrow_{k_{x}}$ in the
eigenvalues

$\displaystyle g({\bf k})|\phi_{\pm}({\bf k})\rangle=\lambda_{\pm}({\bf
k})\Updownarrow_{k_{x}}|\phi_{\pm}({\bf k})\rangle,$ (30)

such that the diagonalization leads to

$\displaystyle g({\bf k})|\phi_{\pm}({\bf
k})\rangle=\left(\begin{array}[]{cc}&e^{-ik_{y}}\\\
1&\end{array}\right)\Updownarrow_{k_{x}}\left(\begin{array}[]{c}u_{\pm}(k_{x},k_{y})\\\
v_{\pm}(k_{x},k_{y})\end{array}\right)$ (35)
$\displaystyle=\left(\begin{array}[]{cc}&e^{-ik_{y}}\\\
1&\end{array}\right)\left(\begin{array}[]{c}u_{\pm}(-k_{x},k_{y})\\\
v_{\pm}(-k_{x},k_{y})\end{array}\right)$ (40)
$\displaystyle=\lambda_{\pm}(k_{x},k_{y})\Updownarrow_{k_{x}}\left(\begin{array}[]{c}u_{\pm}(k_{x},k_{y})\\\
v_{\pm}(k_{x},k_{y})\end{array}\right)$ (43)
$\displaystyle=\lambda_{\pm}(k_{x},k_{y})\left(\begin{array}[]{c}u_{\pm}(-k_{x},k_{y})\\\
v_{\pm}(-k_{x},k_{y})\end{array}\right),$ (46)

and hence one can solve for the coefficients $u_{\pm}(-k_{x},k_{y})$ and
$v_{\pm}(-k_{x},k_{y})$ with eigenvalues $\lambda_{\pm}=\pm e^{-ik_{y}/2}$.
This argument can be arbitrarily generalize to unit cells that contain more
glide pairs.

We then consider the fact that, at a fixed $k_{x}$, the Hamiltonian at
$k_{y}=0$ and $k_{y}=2\pi$ is the same, $H(k_{x},0)=H(k_{x},2\pi)$, and so is
the glide-plane operator, $G(k_{x},0)=G(k_{x},2\pi)$. At a fixed $k_{x}$ and
band index $n$, the symmetry eigenvalues at $k_{y}=0$ and $k_{y}=2\pi$ are,

$\displaystyle
G(k_{x},0)|\psi_{n1\pm}(k_{x},0)\rangle=\pm\Updownarrow_{k_{x}}|\psi_{n1\pm}(k_{x},0)\rangle,$
$\displaystyle
G(k_{x},2\pi)|\psi_{n1\pm}(k_{x},2\pi)\rangle=\mp\Updownarrow_{k_{x}}|\psi_{n1\pm}(k_{x},2\pi)\rangle,$
$\displaystyle G(k_{x},0)|\psi_{n2\pm}(k_{x},0)\rangle=\pm
e^{-ik_{x}}\Updownarrow_{k_{x}}|\psi_{n2\pm}(k_{x},0)\rangle,$ $\displaystyle
G(k_{x},2\pi)|\psi_{n2\pm}(k_{x},2\pi)\rangle=\mp
e^{-ik_{x}}\Updownarrow_{k_{x}}|\psi_{n2\pm}(k_{x},2\pi)\rangle,$

Combining this with $G(k_{x},0)=G(k_{x},2\pi)$ implies that one must be able
to find a gauge in which

$\displaystyle|\psi_{nI+}(k_{x},0)\rangle=|\psi_{nI-}(k_{x},2\pi)\rangle,$
$\displaystyle|\psi_{nI-}(k_{x},0)\rangle=|\psi_{nI+}(k_{x},2\pi)\rangle.$
(48)

It then follows that the eigenenergies satisfy

$\displaystyle
H(k_{x},0)|\psi_{nI+}(k_{x},0)\rangle=E_{nI+}(k_{x},0)|\psi_{nI+}(k_{x},0)\rangle$
$\displaystyle=H(k_{x},2\pi)|\psi_{nI+}(k_{x},2\pi)\rangle=E_{nI-}(k_{x},2\pi)|\psi_{nI-}(k_{x},2\pi)\rangle$
$\displaystyle=E_{nI-}(k_{x},2\pi)|\psi_{nI+}(k_{x},0)\rangle,$ (49)

since $H(k_{x},0)=H(k_{x},2\pi)$. This and a similar argument leads to

$\displaystyle E_{nI-}(k_{x},2\pi)=E_{nI+}(k_{x},0),$ $\displaystyle
E_{nI+}(k_{x},2\pi)=E_{nI-}(k_{x},0).$ (50)

Thus at given $k_{x}$, $n$, and $I$ the two bands $E_{nI+}(k_{x},k_{y})$ and
$E_{nI-}(k_{x},k_{y})$ must cross each other somewhere in $0\leq k_{y}\leq
2\pi$.

We can apply the same argument to the BZ boundary $k_{y}=\pm\pi$, which has
symmetry eigenvalues

$\displaystyle G(k_{x},\pi)|\psi_{n1\pm}(k_{x},\pi)\rangle=\mp
i\Updownarrow_{k_{x}}|\psi_{n1\pm}(k_{x},\pi)\rangle,$ $\displaystyle
G(k_{x},-\pi)|\psi_{n1\pm}(k_{x},-\pi)\rangle=\pm
i\Updownarrow_{k_{x}}|\psi_{n1\pm}(k_{x},-\pi)\rangle,$

and similarly for the eigenstate with index $I=2$. Because
$G(k_{x},\pi)=G(k_{x},-\pi)$, there exists a gauge in which the eigenstates
satisfy

$\displaystyle|\psi_{nI+}(k_{x},\pi)\rangle=|\psi_{nI-}(k_{x},-\pi)\rangle,$
$\displaystyle|\psi_{nI-}(k_{x},\pi)\rangle=|\psi_{nI+}(k_{x},-\pi)\rangle.$
(52)

Using $H(k_{x},\pi)=H(k_{x},-\pi)$, the same procedure in Eq. (49) leads to

$\displaystyle E_{nI+}(k_{x},\pi)=E_{nI-}(k_{x},-\pi)=E_{nI-}(k_{x},\pi),$
(53)

where in the last equality we have used the fact that $k_{y}=\pi$ and
$k_{y}=-\pi$ are the same point on the boundary of a rectangular BZ, thus
completing the proof to Eq. (10).

To be more concrete about the notion of glide pairs, using the numbering of
sublattices and the glide vector in Fig. 5 (a), the glide pairs defined with
respect to the glide plane $G$ for the C10 configuration in Fig. 1 are

$\displaystyle{\rm Type-I}:\;$ $\displaystyle(1,6),\;\;\;(3,8),\;\;\;(5,10),$
$\displaystyle{\rm Type-II}:\;$ $\displaystyle(2,7),\;\;\;(4,9).$ (54)

For each of the 3 type-I pairs, mapping from left to right under $G$ remains
in the same unit cell, but mapping from right to left under $G$ moves to the
next unit cell in $+{\hat{\bf y}}$ direction; For each of the 2 type-II pairs,
mapping from left to right moves to the next unit cell in $-{\hat{\bf x}}$
direction while from right to left moves to the next unit cell along
$-{\hat{\bf x}}+{\hat{\bf y}}$. As result, the glide plane operator is that in
Eq. (7) with $N_{1}=3$ and $N_{2}=2$, and so follows the discussion in this
section.

Figure 5: The numbering of sublattices for (a) the C10 example in Fig. 1 and
(b) the C44 example in Fig. 2.

##  Appendix B Nodal-loops enforced by two orthogonal glide planes

We proceed to discuss vacancy engineered graphene that belong to the wallpaper
groups that have two orthogonal glide planes denoted by $G_{x}$ and $G_{y}$.
In these wallpaper groups, a specific sublattice $A$ is mapped to another one
$B$ under $G_{x}$, but it is mapped to a different one $C$ under $G_{y}$. In
other words, the glide pair arrangements are different for $G_{x}$ and
$G_{y}$. Thus if we arrange the basis according to the glide pairs of $G_{y}$,
then $G_{y}$ will take the block-diagonal form of Eq. (7), but $G_{x}$ will
not be block-diagonal in this basis because it has a different glide pair
assignment. Nevertheless, $G_{x}$ will have
$\left\\{N_{1x},N_{1x},N_{2x},N_{2x}\right\\}$ degenerate eigenvalues
according to the numbers of type I and type II glide pairs defined for this
glide plane, and $G_{y}$ will have
$\left\\{N_{1y},N_{1y},N_{2y},N_{2y}\right\\}$ degenerate eigenvalues
regardless how the basis is arranged. Because the Hamiltonian commutes with
both of them, $\left[H({\bf k}),G_{x}({\bf k})\right]=0$ and $\left[H({\bf
k}),G_{y}({\bf k})\right]=0$, one must be able to label the eigenstates by the
quantum numbers $\left\\{n,\alpha,\beta\right\\}$, where $n$ is the band
index, $\alpha=I_{x}\pm$ labels the eigenvalues of $G_{x}$, and
$\beta=I_{y}\pm$ labels the eigenvalues of $G_{y}$. The eigenstates satisfy

$\displaystyle G_{x}({\bf k})|\psi_{n1\pm\beta}({\bf k})\rangle=\pm
e^{-ik_{x}/2}\Updownarrow_{k_{y}}|\psi_{n1\pm\beta}({\bf k})\rangle,$
$\displaystyle G_{x}({\bf k})|\psi_{n2\pm\beta}({\bf k})\rangle=\pm
e^{-ik_{x}/2-ik_{y}}\Updownarrow_{k_{y}}|\psi_{n2\pm\beta}({\bf k})\rangle,$
$\displaystyle G_{y}({\bf k})|\psi_{n\alpha 1\pm}({\bf k})\rangle=\pm
e^{-ik_{y}/2}\Updownarrow_{k_{x}}|\psi_{n\alpha 1\pm}({\bf k})\rangle,$
$\displaystyle G_{y}({\bf k})|\psi_{n\alpha 2\pm}({\bf k})\rangle=\pm
e^{-ik_{x}-ik_{y}/2}\Updownarrow_{k_{x}}|\psi_{n\alpha 2\pm}({\bf k})\rangle,$
$\displaystyle H({\bf k})|\psi_{n\alpha\beta}({\bf
k})\rangle=E_{n\alpha\beta}({\bf k})|\psi_{n\alpha\beta}({\bf k})\rangle.$
(55)

Following the same argument for the $p2mg$ group in the previous section, we
obtain

$\displaystyle E_{n\alpha I_{y}\mp}(k_{x},2\pi)=E_{n\alpha
I_{y}\pm}(k_{x},0),$ $\displaystyle
E_{nI_{x}\mp\beta}(2\pi,k_{y})=E_{nI_{x}\pm\beta}(0,k_{y}),$ (56)

implying a band crossing in the range $0\leq k_{x}\leq 2\pi$ at any fixed
$k_{y}$, and another band crossing in the range $0\leq k_{y}\leq 2\pi$ at any
fixed $k_{x}$. The argument applied to the BZ boundary also leads to

$\displaystyle E_{n\alpha I_{y}-}(k_{x},\pi)=E_{n\alpha
I_{y}+}(k_{x},-\pi)=E_{n\alpha I_{y}+}(k_{x},\pi),$ $\displaystyle
E_{nI_{x}-\beta}(\pi,k_{y})=E_{nI_{x}+\beta}(-\pi,k_{y})=E_{nI_{x}+\beta}(\pi,k_{y}).\;\;\;\;$
(57)

Thus every two bands are forced to stick together at the BZ boundary, forming
$N/2$ symmetry enforced nodal loops surrounding the BZ boundary.

For the C44 example in Fig. 2, using the numbering of sublattices in Fig. 5
(b), the glide pairs defined with respect to the glide plane $G_{x}$ are

$\displaystyle{\rm Type-I}:\;$
$\displaystyle(1,10),\;\;\;(2,11),\;\;\;(3,12),\;\;\;(7,4),$
$\displaystyle(8,5),\;\;\;(9,6),$ $\displaystyle{\rm Type-II}:\;$
$\displaystyle(13,42),\;\;\;(14,43),\;\;\;(15,44),\;\;\;(40,16),$
$\displaystyle(41,17),\;\;\;(18,38),\;\;\;(19,39),\;\;\;(35,20),$
$\displaystyle(36,21),\;\;\;(37,22),\;\;\;(23,32),\;\;\;(24,33),$
$\displaystyle(25,34),\;\;\;(29,26),\;\;\;(30,27),\;\;\;(31,28).$

For the 6 type-I pair, mapping from left to right under $G_{x}$ remains in the
same unit cell, but from right to left moves to the next unit cell in
${\hat{\bf x}}$ direction; For the 14 type-II pairs, mapping from left to
right moves to the next unit cell in the $-{\hat{\bf y}}$ direction, whereas
mapping from right to left moves to the next unit cell in the ${\hat{\bf
x}}-{\hat{\bf y}}$ direction. As a result, the glide plane operator is that
defined in Eq. (7) with $N_{1x}=6$ and $N_{2x}=14$ and swapping
$\left\\{k_{x},k_{y}\right\\}\rightarrow\left\\{k_{y},k_{x}\right\\}$. On the
other hand, the glide pair assignment is different for the glide plane
$G_{y}$, which are

$\displaystyle{\rm Type-I}:\;$
$\displaystyle(1,26),\;\;\;(2,25),\;\;\;(3,24),\;\;\;(4,23),$
$\displaystyle(7,32),\;\;\;(8,31),\;\;\;(9,30),\;\;\;(10,29),$
$\displaystyle(13,38),\;\;\;(14,37),\;\;\;(15,36),\;\;\;(16,35),$
$\displaystyle(18,42),\;\;\;(19,41),\;\;\;(20,40),$ $\displaystyle{\rm Type-
II}:\;$ $\displaystyle(5,28),\;\;\;(6,27),\;\;\;(11,34),\;\;\;(12,33),$ (59)
$\displaystyle(17,39),\;\;\;(21,44),\;\;\;(22,43),$

where the mapping follows that described after Eq. (54), yielding the glide
plane operator $G_{y}$ given by Eq. (7) with $N_{1y}=15$ and $N_{2y}=7$, and
so follows the discussion in this section.

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# Entropy of irregular points
for some dynamical systems

Katrin Gelfert  Instituto de Matemática Universidade Federal do Rio de
Janeiro, Av. Athos da Silveira Ramos 149, Cidade Universitária - Ilha do
Fundão, Rio de Janeiro 21945-909, Brazil<EMAIL_ADDRESS>,  Maria José
Pacifico  Instituto de Matemática Universidade Federal do Rio de Janeiro, Av.
Athos da Silveira Ramos 149, Cidade Universitária - Ilha do Fundão, Rio de
Janeiro 21945-909, Brazil<EMAIL_ADDRESS>and  Diego Sanhueza  Instituto
de Matemática Universidade Federal do Rio de Janeiro, Av. Athos da Silveira
Ramos 149, Cidade Universitária - Ilha do Fundão, Rio de Janeiro 21945-909,
Brazil<EMAIL_ADDRESS>
###### Abstract.

We derive sufficient conditions for a dynamical systems to have a set of
irregular points with full topological entropy. Such conditions are verified
for some nonuniformly hyperbolic systems such as positive entropy surface
diffeomorphisms and rational functions on the Riemann sphere.

###### Key words and phrases:

irregular points, topological entropy, specification property, Lyapunov
exponents

###### 2000 Mathematics Subject Classification:

Primary: 37D25, 37C45, 28D99, 37F10

This research has been supported [in part] by CAPES – Finance Code 001 and by
CAPES- and CNPq-grants. MJP was partially supported by FAPERJ

##  1\. Introduction

Given a continuous map $f\colon X\to X$ on a compact metric space $X$ and a
continuous observable $\varphi\colon X\to\mathbb{R}$, its Birkhoff average
along the orbit

$\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^{j}(x))$

(for $x\in X$ for which this limit exists) play an important role because of
their intimate relation with convergence in the weak$\ast$ topology. Recall
that the set of $\varphi$-_irregular points_ for which the Birkhoff averages
do not converge,

$\hat{X}(f,\varphi):=\Big{\\{}x\in
X\colon\lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^{j}(x))\text{
does not exist}\Big{\\}},$

is universally null. Ruelle [19] coined the term of “points with historical
behaviour” because they trace the history of the system, whereas points whose
Birkhoff’s sum converge only warn average behaviour. Although
$\hat{X}(f,\varphi)$ is not detected by any invariant measure, it can be
“large” from another point of view such as, for example, fractal dimension,
entropy, or general topology. This type of question is typical in multifractal
analysis. Irregular points form an essential part of the _multifractal
decomposition_ of $X$ (relative to $\varphi$),

$X=\bigcup_{\alpha\in\mathbb{R}\cup\\{\infty\\}}X_{\alpha}(f,\varphi)\,\dot{\cup}\,\hat{X}(f,\varphi),$

where $X_{\alpha}(f,\varphi)$ denotes the set of _$\varphi$ -regular points_
with average $\alpha$,

$X_{\alpha}(f,\varphi):=\Big{\\{}x\in
X\colon\displaystyle\lim_{n\to\infty}\frac{1}{n}\displaystyle\sum_{j=0}^{n-1}\varphi(f^{j}(x))=\alpha\Big{\\}}.$

Denote also the _set of irregular points_ of $f$ by

$\hat{X}(f):=\bigcup\Big{\\{}\hat{X}(f,\varphi)\colon\varphi\colon
X\to\mathbb{R}\text{ continuous}\Big{\\}}.$

Our focus here will be on entropy. Although any of the above defined sets is
$f$-invariant, in general it is noncompact and we rely on the concept of
topological entropy introduced in [4]. We denote by $h(f,A)$ the _topological
entropy_ of $f$ on $A\subset X$. The following estimates hold true in general

$0\leq h(f,\hat{X}(f,\varphi))\leq h(f,\hat{X}(f))\leq h(f,X).$

Each inequality can be strict. Recall, for instance, the example of a minimal
dynamical system (hence satisfying $\hat{X}(f)=\emptyset$) with positive
entropy in [11]. To our best knowledge, there is no nontrivial (that is, say,
topologically transitive) example for which $0<h(f,\hat{X}(f))<h(f,X)$.

Previous approaches to analyze of the regular and the irregular part of the
spectrum commonly require “orbit-gluing properties”. For example, [17, 8]
consider the case of a subshift of finite type. In [21], just assuming the
specification property of $f$, there is stated a _restricted variational
principle_

$h(f,X_{\alpha}(f,\varphi))=\sup\Big{\\{}h_{\mu}(f)\colon\mu\
f\mbox{-invariant,}\int{\varphi}\,d\mu=\alpha\Big{\\}}.$

On the other hand, “chaotic dynamics” quite commonly gives rise to irregular
sets which are dense and have full entropy. For a full shift of two symbols
$\sigma\colon\Sigma_{2}\to\Sigma_{2}$, by [17, Lemma 6] it holds
$h(\sigma,\hat{\Sigma}_{2}(\sigma))=h(\sigma,\Sigma_{2})$. If $f$ satisfies
the specification property, it is a consequence of [20, Theorem 4] (see also
[5, Theorem 3]) that $\hat{X}(f)$ is nonempty and of [7] that it has full
entropy.

The specification property roughly says that given any number of arbitrarily
long orbit segments, there exists an orbit which stays $\varepsilon$-close to
each segment and between segments there are only a bounded number of
iterations whose number only depends on $\varepsilon$ (we refer to [20, 22]
for the full definition). For example, for a basic set of an Axiom A
diffeomorphism, the existence of a Markov partition enables a symbolic
description of orbits and shadowing permits to “connect” orbit segments which
were symbolically coded previously. As such partitions can be chosen with
arbitrarily small diameter, this guarantees the existence of “arbitrarily
specified orbits”.

In general, specification, or any of it’s weaker versions, does not hold. We
will illustrate that it is also often not required in this strong sense to
deduce “maximal historic behavior”. Here the key observation is that such
strong hypotheses are not required globally, but only for invariant subsystems
whose entropies are sufficiently large and that entropy in some sense is “a
local quantifier”.

We start by considering the set of irregular points and not _a priori_ fixing
any observable, assuming hyperbolicity.

######  Theorem A.

For any Axiom A $C^{1}$-diffeomorphism $f\colon X\to X$ on a $n$-dimensional
closed manifold $X$, $n\geq 2$, it holds

$h(f,\hat{X}(f))=h(f,X).$

In a “nonuniformly hyperbolic context”, the proof of the next result takes
into consideration approximations in the weak$\ast$ topology and in entropy of
positive-entropy ergodic measures by horseshoes.

######  Theorem B.

For any $C^{1+\alpha}$-diffeomorphism $f\colon X\to X$ on a closed surface $X$
it holds

$h(f,\hat{X}(f))=h(f,X).$

A version of Theorem B was obtained independently in [2, Corollary 2.4].

To guarantee that $h(f,\hat{X}(f,\varphi))$ is positive (or even large), some
hypotheses on the observable are absolutely necessary. Indeed, if $\varphi$ is
cohomologous to a constant, then $\hat{X}(f,\varphi)$ is empty (and hence has
zero entropy). If $\hat{X}(f,\varphi)$ is nonempty, for a topologically mixing
subshift of finite type (or any topologically conjugate system), by [1] this
set carries full topological entropy. This result was generalized to maps
satisfying the almost specification property ([22], see also [16] for the
context of flows) or the orbit gluing property (see [14] and further
references therein). Moreover, by [6], assuming the asymptotic average
shadowing property (AASP for short) and a certain condition on the measure
center, the set $\hat{X}(f,\varphi)$ is either residual or empty. It is not
known if AASP in general implies that $\hat{X}(f,\varphi)$, if nonempty, has
full entropy.

The following result provides a sufficient condition on $\varphi$ to “detect”
historic behavior and forces $h(f,\hat{X}(f,\varphi))$ to be large (or even
full). Let us denote by $\mathfrak{M}(f)$ the set of $f$-invariant Borel
probability measures.

######  Theorem C.

Let $f\colon X\to X$ and $\varphi\colon X\to\mathbb{R}$ be continuous
functions on a compact metric space $X$ such that there is a sequence
$(\Gamma_{n})_{n}$ of compact $f$-invariant subsets of $X$ and numbers
$\ell_{n}\in\mathbb{N}$ such that $f^{\ell_{n}}|_{\Gamma_{n}}$ has the almost
specification property and

(1)
$\inf_{\mu\in\mathfrak{M}(f|_{\Gamma_{n}})}\int\varphi\,d\mu<\sup_{\mu\in\mathfrak{M}(f|_{\Gamma_{n}})}\int\varphi\,d\mu.$

Then, it holds

$\limsup_{n\to\infty}h(f,\Gamma_{n})\leq h(f,\hat{X}(f,\varphi))\leq
h(f,\hat{X}(f))=h(f,X).$

######  Remark 1.1.

In many cases (such as, for example, under the hypotheses of Theorem D below),
the sets $\Gamma_{n}$ to verify the hypotheses of Theorem C can be chosen to
be nested, that is, $\Gamma_{n}\subseteq\Gamma_{n+1}$, and so it suffices to
show that (1) holds for some $n$ (and hence for all $n^{\prime}\geq n$).

_A priori_ , fixing some observable $\varphi\colon X\rightarrow\mathbb{R}$,
the value $h(f,\hat{X}(f,\varphi))$ may be much smaller than $h(f,X)$. To
exemplify this, recall that a $C^{1}$ diffeomorphism $f$ is Axiom A if its
nonwandering set $\Omega(f)$ is hyperbolic and the periodic points of $f$ are
dense in $\Omega(f)$. Now, given a $C^{1}$ Axiom A diffeomorphism $f$ with
nontrivial spectral decomposition
$\Omega(f)=\Omega_{1}\cup\cdots\cup\Omega_{\ell}$ such that
$h(f,\Omega_{i})<h(f,\Omega_{j})$ for some index pair $i\neq j$, then for
$\varphi\colon M\to\mathbb{R}$ such that $\varphi|_{\Omega_{i}}$ is not
cohomologous to a constant and $\varphi|_{\Omega_{j}}\equiv 0$ it holds

$\hat{X}(f,\varphi)\neq\emptyset\quad\mbox{and}\quad
h(f,\hat{X}(f,\varphi))<\max_{j\neq i}h(f,\Omega_{j})=h(f,X).$

Finally, we invoke Theorem C in a setting which can also be considered
“nonuniformly hyperbolic”. Note that this setting, in particular in the case
where $f$ is a rational function has a critical point inside its Julia set, is
in general very far from being hyperbolic or satisfying specification. A
classical example is the Ulam-von Neumann map $z\mapsto z^{2}-2$ with Julia
set being the closed interval $[-2,2]$ and critical point $z=0$.

######  Theorem D.

Suppose that $f$ is a rational function of degree $d\geq 2$ on the Riemann
sphere and $J(f)$ its Julia set. Then

$h(f,\hat{J}(f))=h(f,J(f))=\log d.$

Moreover, there are no critical points in $J(f)$, then for
$\varphi:=\log|f^{\prime}|$ it holds

$\hat{J}(f,\varphi)\neq\emptyset\quad\text{ and }\quad
h(f,\hat{J}(f,\varphi))=h(f,\hat{J}(f))=h(f,J(f))$

if, and only if, $f$ is not of the form $f(z)=z^{\pm d}$.

In Theorem D, for the second claim we assume that there are no $f$-critical
points in $J(f)$ in order to guarantee that $\varphi$ is continuous. An
analogous statement in the general case, including critical points and
extending the concept of $\varphi$-irregular points to general observables,
can be derived following, for example, techniques in [10].

In Section 2 we recall some preliminaries. In Section 3 we prove Theorems A,
B, and C. In Section 4 we study rational maps and prove Theorem D.

##  2\. Preliminaries

Note that $\hat{X}(f,\varphi)$ is empty whenever $\varphi$ is “essentially
constant”. More precisely, consider the space $C(X)$ of continuous observables
$\varphi\colon X\to\mathbb{R}$, equipped with the usual sup-norm
$\lVert\varphi\rVert:=\sup\lvert\varphi\rvert$. Two functions $\varphi,\psi\in
C(X)$ are _cohomologous_ (with respect to $f$) if there exists $u\in C(X)$
such that $\psi=\varphi+u-u\circ f$. Any function which is cohomologous to a
constant function is a _coboundary_. The following facts are immediate.

######  Proposition 2.1.

For every $\varphi\in C(X)$, $c_{1},c_{2}\in\mathbb{R}$, $c_{1}\neq 0$, and
$k\in\mathbb{N}$,
$\hat{X}(f,\varphi)=\hat{X}(f,c_{1}\varphi+c_{2})=\hat{X}(f,\varphi\circ{f^{k}})$.
If $\varphi,\psi\in C(X)$ are cohomologous, then
$\hat{X}(f,\varphi)=\hat{X}(f,\psi)$. For every coboundary $\varphi$, it holds
$\hat{X}(f,\varphi)=\emptyset$.

######  Remark 2.2.

For $\varphi\in C(X)$ the following facts are equivalent:

  * 1)

$\inf_{\mu\in\mathfrak{M}(f)}\int\varphi\,d\mu<\sup_{\mu\in\mathfrak{M}(f)}\int\varphi\,d\mu$,

  * 2)

$\varphi$ is not in the closure of the subset of coboundaries,

  * 3)

$\frac{1}{n}\sum_{j=0}^{n-1}\varphi\circ f^{j}$ does not converge pointwise to
a constant.

See, for example [22, Lemma 2.1], where further properties are stated. It is
clear that if $\hat{X}(f,\varphi)\neq\emptyset$, then property 3) (and hence
any other) holds true. Assuming the almost specification property, by [22,
Theorem 4.1] property 1) (and hence any other) implies
$\hat{X}(f,\varphi)\neq\emptyset$.

In general, $f$ can have a complicated dynamics and it can be more convenient
to analyze certain subsystems. To this end, we state the following lemma which
is straightforward to show.

######  Lemma 2.3.

For each $\ell\in\mathbb{N}$ and $\varphi\in C(X)$, it holds

$\hat{X}(f,\varphi)=\bigcup_{j=0}^{\ell-1}\hat{X}(f^{\ell},\varphi\circ
f^{j}).$

In particular, $\hat{X}({f^{\ell}},\varphi)\subseteq\hat{X}(f,\varphi)$ and
$\hat{X}(f)=\hat{X}(f^{\ell})$.

Observe that $\hat{X}({f^{\ell}},\varphi)$ can be a proper subset of
$\hat{X}(f,\varphi)$.

##  3\. Proof of Theorems A, B, and C

###### Proof of Theorem C.

For each $n\geq 1$, [22, Theorem 4.1] implies

$h(f^{\ell_{n}},\Gamma_{n})=h\big{(}f^{\ell_{n}}|_{\Gamma_{n}},\hat{X}(f^{\ell_{n}}|_{\Gamma_{n}},\varphi|_{\Gamma_{n}})\big{)}$

and hence Lemma 2.3 together with
$\hat{X}(f|_{\Gamma_{n}},\varphi|_{\Gamma_{n}})\subseteq\hat{X}(f,\varphi)$
and monotonicity of entropy gives

$h(f,\Gamma_{n})=h\big{(}f|_{\Gamma_{n}},\hat{X}(f^{\ell_{n}}|_{\Gamma_{n}},\varphi|_{\Gamma_{n}})\big{)}\leq
h\big{(}f,\hat{X}(f,\varphi)\big{)}\leq h(f,\hat{X}(f))\leq h(f,X).$

Letting $n\to\infty$, this implies the claim. ∎

###### Proof of Theorem B.

If $h(f,X)=0$, the result is immediate.

If $h(f,X)>0$, by the variational principle for entropy, there is a sequence
of ergodic measures $(\mu_{n})_{n\geq 1}$ such that $h_{\mu_{n}}(f)\to
h(f,X)$. By Ruelle’s inequality, any $\mu_{n}$ with positive entropy is
hyperbolic. Let us hence consider $\mu$ hyperbolic ergodic with entropy
arbitrarily close to $h(f,X)$. By Katok’s horseshoe construction [12, Theorem
S.5.9], for every $\varepsilon>0$, there exists a basic set $\Gamma\subset M$
and $m\in\mathbb{N}$ such that $f^{m}|_{\Gamma}$ is topologically mixing and
hence has the specification property, and
$h_{\mu}(f)-\varepsilon<h(f,\Gamma)$. Hence,

$\begin{split}h(f,\Gamma)&=\frac{1}{m}h(f^{m},\Gamma)=\frac{1}{m}h(f^{m},\hat{X}(f^{m}|_{\Gamma}))\\\
\text{\tiny{by Lemma
\ref{lem:Irregularesfk}}}\quad&\leq\frac{1}{m}h(f^{m},\hat{X}(f|_{\Gamma}))\\\
&=h(f|_{\Gamma},\hat{X}(f|_{\Gamma}))\leq h(f,\hat{X}(f)).\end{split}$

Choosing $\mu$ with entropy arbitrarily close to $h(f,X)$ implies the claim. ∎

In higher dimensions there may not exist any hyperbolic ergodic measure and
hence the strategy of the proof of Theorem B does not work. In an _a priori_
hyperbolic context, we invoke [22, Theorem 4.1] (or [1]) directly.

###### Proof of Theorem A.

By hypothesis, we can invoke the spectral decomposition theorem. Hence, the
set of nonwandering points splits as

$\Omega(f)=\Omega_{1}\cup\ldots\cup\Omega_{m},$

where for every $i=1,\ldots,m$ there is $\ell_{i}\in\mathbb{N}$ such that
$\Omega_{i}=X_{1,i}\cup\ldots\cup X_{\ell_{i},i}$ such that
$f^{\ell_{i}}|_{X_{k,i}}$ is topologically mixing, for each $1\leq
k\leq\ell_{i}$, and hence has the specification property. As

$h(f,X)=\max_{i}h(f,\Omega_{i})=\max_{i}\frac{1}{\ell_{i}}h(f^{\ell_{i}},X_{k,i})\leq
h(f,\hat{X}(f)),$

this proves the claim. ∎

##  4\. Irregular points in Julia sets and proof of Theorem D

In this section, let $f\colon\overline{\mathbb{C}}\to\overline{\mathbb{C}}$ be
a rational function of degree $d:=\deg(f)\geq 2$ and consider its Julia set
$J=J(f)$. We estimate the entropy of the set of $\varphi$-irregular points for
the _geometric potential_ $\varphi(z):=\log\,\lvert f^{\prime}\rvert$. This
observable, along with its scaled version $t\varphi$ for $t\in\mathbb{R}$,
plays an important role in the thermodynamics formalism (see [18, 23] and
references therein). Its Birkhoff averages are simply the Lyapunov exponents.
If $f$ has no critical points in $J$ then $\varphi$ is continuous on $J$.

Recall that $J$ is nonempty, compact, and coincides with the closure of the
set of repelling periodic points. Moreover, $f|_{J}$ is topologically exact
and its entropy equals $h(f,J)=\log d$ (see, for example, [3, 15]).

A compact $f$-invariant set $R\subset J$ is a _uniformly expanding repeller_
if $f|_{R}$ is _hyperbolic_ , that is, there is $n\in\mathbb{N}$ so that

$\inf\\{|(f^{n})^{\prime}(z)|\colon z\in R\\}>1,$

topologically transitive, and isolated, that is, there exists an open
neighborhood $U\supset R$ such that $f^{n}(z)\in U$ for all $n\geq 0$ implies
$z\in R$. Note that if a uniformly expanding repeller $R$ is such that
$f^{\ell}|_{R}$ is topologically mixing then $f^{\ell}|_{R}$ has the
specification property.

A rational function $f$ is _hyperbolic_ if $f|_{J}$ is hyperbolic. Every
rational function without critical points in $J$ is either hyperbolic or
admits a _parabolic point_ , that is, a periodic point $z=f^{p}(z)\in J$ such
that $(f^{p}(z))^{\prime}(z)$ is a root of unity (see [23, Theorem 3.2]).
Given an ergodic Borel probability measure
$\mu\in\mathfrak{M}_{\mathrm{erg}}(f)$ consider its Lyapunov exponent

$\mathcal{X}(\mu):=\int\log\,\lvert f^{\prime}\rvert\,d\mu=\int\varphi\,d\mu.$

Let
$\alpha^{-}:=\inf\\{\mathcal{X}(\nu)\colon\nu\in\mathfrak{M}_{\mathrm{erg}}(f)\\}$
and
$\alpha^{+}:=\sup\\{\mathcal{X}(\nu)\colon\nu\in\mathfrak{M}_{\mathrm{erg}}(f)\\}$.
Let $\mu_{0}$ be the (unique, hence ergodic) measure of maximal entropy.
Recall

$0<\log d\leq\mathcal{X}(\mu_{0})\leq\alpha^{+}.$

The following preliminary result can be proved in several different ways, for
example building upon nowadays available description of the spectrum of
Lyapunov exponents and approximation techniques (see, for example, [18,
Chapter 11] and references therein). We provide a direct proof.

######  Lemma 4.1.

If $f|_{J}$ has a parabolic point, then there is a hyperbolic ergodic measure
$\nu$ supported on a periodic orbit satisfying
$0<\mathcal{X}(\nu)<\mathcal{X}(\mu_{0})$.

###### Proof.

First note that, as $f|_{J}$ is topologically exact, for every $\delta>0$
there is $N(\delta)\in\mathbb{N}$ such that for every $x\in J$ and $N\geq
N(\delta)$ it holds $f^{N}(B(x,\delta))\supset J$.

By contradiction, suppose that all hyperbolic periodic points have the same
exponent $\alpha:=\mathcal{X}(\mu_{0})$.

Given $\varepsilon\in(0,\alpha/4)$, topological exactness together with our
assumptions implies that there is $\delta_{1}>0$ such that for every
hyperbolic periodic point $w=f^{p}(w)$ the set $U=B(w,\delta_{1})$ satisfies
$f^{p}(U)\supset U$ and

(2) $\frac{|(f^{p})^{\prime}(x)|}{|(f^{p})^{\prime}(y)|}\leq e^{p\varepsilon}$

for all $x,y\in U$. Let $N_{1}=N(\delta_{1})$.

Let $\operatorname{\rm Crit}(f)$ be the set of all the critical points of
$f\colon\overline{\mathbb{C}}\to\overline{\mathbb{C}}$. Fix a parabolic point
$z=f^{q}(z)$ and let $\delta_{0}>0$ so that
$\cup_{k=0}^{q-1}f^{k}(\operatorname{\rm Crit}(f))$ and $B(z,\delta_{0})$ are
disjoint. Hence, for any $\delta\in(0,\delta_{0})$ and any preimage of
$V=B(z,\delta)$ by $f^{q}$ is univalent; denote by
$\operatorname{Comp}_{z}f^{-q}(V)$ its connected component containing $z$. By
the Koebe Distortion Theorem, there is $C(\delta)>1$ satisfying $C(\delta)\to
1$ as $\delta\to 0$ so that for all $x,y\in
V^{\prime}:=\operatorname{Comp}_{z}f^{-q}(V)$ it holds

(3) $\frac{|(f^{q})^{\prime}(x)|}{|(f^{q})^{\prime}(y)|}\leq C(\delta).$

Fix some $\delta\in(0,\delta_{0})$ and let $N_{2}=N(\delta)$.

Recall that there are repelling periodic points of arbitrarily large period.
We choose a periodic point $w=f^{p}(w)$ with period $p$ large enough, whose
choice is specified below. Let

$W^{\prime}:=\operatorname{Comp}_{z}f^{-(N_{2}+q)}(U)\subseteq V^{\prime}.$

By the above, we have

$W^{\prime}\subset f^{N_{1}}(U)\subset f^{N_{1}+p}(U)$

and thus there exists a periodic point $x=f^{k}(x)\in W^{\prime}$ with period
$k=q+N_{1}+p+N_{2}$. By the distortion estimates (2) and (3), it holds

$|(f^{k})^{\prime}(x)|\leq 1\cdot C(\delta)\cdot|(f^{p})^{\prime}(w)|\cdot
e^{p\varepsilon}\cdot D^{N_{1}+N_{2}},\quad\text{ where }\quad
D:=\max_{J}|f^{\prime}|,$

together with the analogous lower bound. In particular, $x$ is hyperbolic and

$\begin{split}\frac{1}{k}\log|(f^{k})^{\prime}(x)|&\leq\frac{1}{q+p+N_{1}+N_{2}}(\log{C(\delta)}+(N_{1}+N_{2})\log{D})+\\\
&\phantom{\leq}+\frac{p}{q+p+N_{1}+N_{2}}\alpha+\frac{p}{q+p+N_{1}+N_{2}}\varepsilon.\end{split}$

Now choosing $w$ with period $p$ large enough, we obtain that
$\log|(f^{k})^{\prime}(x)|/k<\alpha/2$, contradiction. ∎

###### Proof of Theorem D.

We start by proving the first claim about the entropy of the set of irregular
points. By Ruelle’s inequality, the ergodic measure of maximal entropy,
$\mu_{0}$, is hyperbolic and satisfies $\int\log\,\lvert
f\rvert\,d\mu_{0}\geq\log d>0$. Hence, by [18, Theorem 11.6.1], there exist a
sequence $(\Gamma_{n})_{n}$ of uniformly expanding repellers satisfying
$h(f,\Gamma_{n})\to h(f,J)$ as $n\to\infty$ and numbers $m_{n}\in\mathbb{N}$
such that $f^{m_{n}}|_{\Gamma_{n}}$ mixing. In particular,
$f^{m_{n}}|_{\Gamma_{n}}$ has the specification property. As $\Gamma_{n}$ has
positive entropy, $\mathfrak{M}(f,\Gamma_{n})$ is not a singleton and, in
particular, there exists $\phi_{n}\in C(J)$ distinguishing measures in
$\mathfrak{M}(f,\Gamma_{n})$ and hence satisfying (1). Invoking [9, Lemma 2],
in fact $(\Gamma_{n})_{n}$ can be chosen to be increasing, that is,
$\Gamma_{n}\subset\Gamma_{n+1}\subset\ldots$. Thus, there exists some common
$\phi\in C(J)$ satisfying (1) for every $n\in\mathbb{N}$. Thus, by Theorem C,
we obtain

$h(f,\hat{J}(f,\phi))=h(f,\hat{J}(f))=h(f,J)=\log d.$

In the second part of this proof, we assume that $f$ has no critical points in
$J$ and hence $\varphi=\log\,\lvert f^{\prime}\rvert$ is continuous on $J$.

If $f|_{J}$ is hyperbolic, then it has the shadowing property and topological
exactness implies that $f|_{J}$ is topologically mixing. Hence, $f|_{J}$
satisfies the specification property (see, for example, [13, Lemma 9]) and
thus $h(f,\hat{J}(f,\varphi))=h(f,J)$ follows from [22, Theorem 4.1].

Otherwise, if $f|_{J}$ has a parabolic point, by Lemma 4.1, there is a
$f$-invariant measure $\nu$ supported on a periodic hyperbolic orbit
$\Gamma_{\nu}$ satisfying

$0<\int\log|f^{\prime}|\,d\nu=\mathcal{X}(\nu)<\mathcal{X}(\mu_{0})=\int\log|f^{\prime}|\,d\mu_{0}.$

By [18, Theorem 11.6.1], there exist mixing uniformly expanding Cantor
repellers $\Gamma_{n}\subseteq J$ satisfying

$h(f,\Gamma_{n})\geq h(f,J)-\frac{1}{n}$

so that any invariant measure $\mu_{n}$ supported in $\Gamma_{n}$ satisfies

$\int\log|f^{\prime}|\,d\nu<\int\log|f^{\prime}|\,d\mu_{0}-\frac{1}{n}\leq\int\log|f^{\prime}|\,d\mu_{n}.$

In particular, for $n$ large, $\Gamma_{\nu}$ and $\Gamma_{\mu_{n}}$ are
disjoint. By [9, Lemma 2], there exist transitive uniformly expanding Cantor
repellers $\Lambda_{n}\supseteq\Gamma_{n}\cup\Gamma_{\nu}$. In particular,
there is $\ell_{n}\in\mathbb{N}$ such that $f^{\ell_{n}}|_{\Lambda_{n}}$
satisfies the specification property, $\nu,\mu_{n}$ are supported in
$\Lambda_{n}$, and

$h(f,\Lambda_{n})\geq h(f,J)-\frac{1}{n}.$

Hence, by Theorem C it follows $h(f,\hat{J}(f,\varphi))=h(f,J)$.

It remains to analyze the case when the spectrum is “trivial”.

######  Claim 1.

If $f$ has no critical points in $J$, then the following facts are equivalent:

  * (1)

$\hat{J}(f,\varphi)=\emptyset$,

  * (2)

$\alpha^{-}=\alpha^{+}$,

  * (3)

$f$ is (conjugate to) $z\mapsto z^{\pm d}$.

###### Proof.

Check that if $f|_{J}$ is hyperbolic then $\alpha^{-}=\alpha^{+}$ if and only
if $\varphi$ is a coboundary and, in particular by Remark 2.2, (2) and (1) are
equivalent. Recall that if $f|_{J}$ is hyperbolic and $\varphi$ a coboundary
then by [24, Corollary in Section 7 and Proposition 8], $f$ is conjugate to
$z^{\pm d}$. Clearly, if $f$ is conjugated to $z\mapsto z^{\pm d}$, then
$\varphi\equiv\log d$ and $\alpha^{-}=\alpha^{+}$, hence (3) implies (2). This
proves equivalence of (1), (2), and (3) for $f$ hyperbolic.

Moreover, observe that none of the three cases occurs if $f$ has any parabolic
point. Indeed, if $\lambda$ is the $f$-ergodic measure supported on a
parabolic periodic orbit then (2) does not hold as

$\alpha^{-}=\mathcal{X}(\lambda)=0<\log d=\mathcal{X}(\mu_{0})\leq\alpha^{+}.$

Clearly, (3) does not hold either. Finally, the above second part of the proof
of the theorem implies that in this case $\hat{J}(f,\varphi)\neq\emptyset$. ∎

This proves the theorem. ∎

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