# AUTOLYCUS: Exploiting Explainable AI (XAI) for Model Extraction Attacks
against Decision Tree Models

Abdullah Caglar Oksuz  Case Western Reserve University
<EMAIL_ADDRESS>Anisa Halimi  IBM Research
<EMAIL_ADDRESS>Erman Ayday  Case Western Reserve University
<EMAIL_ADDRESS>
(January 2023)

###### Abstract

Model extraction attack is one of the most prominent adversarial techniques to
target machine learning models along with membership inference attack and
model inversion attack. On the other hand, Explainable Artificial Intelligence
(XAI) is a set of techniques and procedures to explain the decision making
process behind AI. XAI is a great tool to understand the reasoning behind AI
models but the data provided for such revelation creates security and privacy
vulnerabilities. In this poster, we propose AUTOLYCUS, a model extraction
attack that exploits the explanations provided by LIME to infer the decision
boundaries of decision tree models and create extracted surrogate models that
behave similar to a target model.

##  I Introduction

Model inversion [1] and model extraction (stealing) attacks [2] are advanced
state-of-the-art adversarial techniques that compromise the security and
privacy of machine learning (ML) models. These attacks can be utilized to
extract sensitive information from the models such as training data and hyper-
parameters. Increasing number of ML-as-a-service systems which provide ML
models trained with confidential data and these models’ publicly accessible
query interfaces are investigated in the literature for growing security and
privacy concerns.

Investigation of the interpretability of black-box AI models gave rise to a
new research topic Explainable Artificial Intelligence (XAI) [3]. XAI aims to
describe the inner workings of AI models -that are challenging to be
understood by humans- such that the model accuracy, fairness, transparency,
and outcomes can be characterized [4]. One such method is local interpretable
model agnostic explanations (LIME) [5] that identifies the most important
features (that influences the decision-making of AI) of a given sample by
generating locally surrogate dataset around that sample.

Capabilities of adversarial ML attacks can be enhanced by combining the
additional vulnerabilities XAI produces. Hence, we propose a novel model
extraction attack called AUTOLYCUS that employs model explanations from LIME
to generate smarter samples with less queries than the state-of-the-art
attacks for revealing the decision boundaries of decision tree models and
extract their functionality to locally trained surrogate models. AUTOLYCUS
also can draw on information from the samples obtained by other resources.
These resources may include previously sampled queries and other datasets. We
demonstrate the effectiveness of our method in different datasets by using
accuracy, $R_{\textit{test}}$ and # of queries metrics.

##  II Related Work

There exist known privacy attacks to machine learning models, such as
membership inference attacks to determine whether a target data sample was
part of the training dataset or not, attribute inference attacks to infer
sensitive information based on the observed ones, model inversion attacks [1]
to reconstruct data samples from a target ML model, and model extraction
attacks [2] to extract model parameters/hyper-parameters from a target ML
model. With the introductions of model explanations [3] such as LIME, SHAP,
LRP etc., it was shown that some of the aforementioned attacks (e.g.,
membership inference attacks and model inversion attacks) can be improved by
exploiting explanations. In this work, we investigate how to exploit the
explanations provided by LIME [5] to reconstruct a target decision tree model.

##  III Methodology

C[1]¿p#1 L[1]¿p#1 In most ML-as-a-services, non-adversary users such as Alice
can send a query $Q(s_{i})$ per sample $s_{i}$ to a model $M$ which is stored
on cloud or locally. In order to reflect under what circumstances can
AUTOLYCUS be utilized, we set some preliminaries.

For simplicity, we assume that the architecture of $M$ and the general
constraints of features in samples are known by Alice. Unknown values for
features in $s_{i}$ are not permitted. Hence, missing features in $s_{i}$
should be filled with averages or permitted random values, for $M$ to respond.
$M$ responds to $Q(s_{i})$ by providing only a prediction label and
explanations using features such that the reasoning behind the particular
prediction is explained.

An adversary user Bob can send queries $Q(s_{i})$ to the model $M$ just like
Alice does. However, Bob exploits explanations to conduct a model extraction
attack on $M$ using targeted queries. Targeted queries are the new samples
$s_{i}^{1},s_{i}^{2},...$ to be queried, guided by the explanations. These new
samples differ from $s_{i}$ in only single feature such that the decision
boundaries of $M$ in prediction are extracted. Recursive procedure of sending
queries and creating new samples from the responses of those queries is
referred as TRAV-A shortened for traversal algorithm in this paper. TRAV-A can
be initialized by a single sample filled with valid random values. But for
efficiency, having at least a single sample for each prediction label (or node
in decision trees) is recommended. Bob can also utilize multiple samples from
other resources as auxiliary information for less queries. TRAV-A explores and
generates new queries until it collects a pre-determined amount of samples
from each prediction class. Bob uses these samples as a surrogate dataset to
train a new model $M^{{}^{\prime}}$ with the same architecture as $M$.

1:function TRAV-A($D_{\textit{smp}}$, Exp, $M_{\textit{trg}}$, $c_{n}$,
$f_{n}$, $L_{b}$, $U_{b}$)

2: samples, v_samples, preds $\leftarrow
D_{\textit{smp}},D_{\textit{smp}}[\textbf{0}],\left\\{\right\\}$

3: n_visits $\leftarrow\\{0,0,...0\\}$ $\triangleright$ ($c_{n}$ times)

4: query $\leftarrow 0$

5: while samples $\neq\emptyset\textbf{ and}$ ANY(nvisits) $\leq L_{b}$ do

6: query++

7: curr $\leftarrow$ samples.POP()

8: pred $\leftarrow$ model.PREDICT(curr)

9: if n_visits[pred] $<U_{b}$ then

10: n_visits[pred]++

11: preds.ADD(pred)

12: v_samples.ADD(curr)

13: e $\leftarrow$ PARSE_EXP(curr, Exp, $M_{\textit{trg}}$, $c_{n}$, $f_{n}$)

14: cand_samples $\leftarrow$ GENERATE_SMP(e)

15: for i in cand_samples do

16: if i $\notin$ samples, v_samples then

17: samples.PUSH(i)

18: return samples, preds, query

##  IV Experiments and Discussion

###  IV-A Datasets & Models & Metrics

For the experiments, we have used Iris, Crop and German Credit (numeric)
datasets. The details about the datasets and the decision tree models are
provided in Table I.

Table I: Dataset and Model Properties  Dataset # Samples # Features # Classes
Depth Leaves Iris 150 4 3 2 3 Crop 2200 7 22 10 26 German Credit 1000 24 2 11
80

We measured the success of our extraction using; model accuracy,
$R_{\textit{test}}$ [2] (classification similarity against a neutral dataset)
and the number of queries as our metrics.

###  IV-B Results

We evaluated our approach on the datasets mentioned in Section IV-A.
Experiments on the Iris dataset yielded comparable accuracy to the target
model even if the surrogate dataset that trains the surrogate model is very
minimal. That is due to Iris dataset’s low tree depth, small node size and
small number of features. Experiments on Crop and German Credit datasets were
more informative to test the effectiveness of our method. For Crop Dataset,
the target model is trained with 800 samples. The model has 97.5% accuracy and
a comparable model with 90% $R_{\textit{test}}$ is achieved with only 50% of
the data. German credit dataset consists of both continuous and categorical
variables and extracted surrogate samples followed the constraints the
datasets have, making them realistic for utilization.

Figure 1: Impact of surrogate dataset size to the accuracy of the model
resulting from model extraction attack

In Figure 1, the target model is a decision tree classifier trained with 400
samples from the German Credit dataset and its accuracy is 68.5%. Bob has
access to 50 samples from the German Credit dataset, which consists of 2% of
the target’s model training dataset. When Bob sends queries of 20% target’s
data size, with an average of only 209 queries, the surrogate model’s accuracy
raised from 58% to 63% and $R_{\textit{test}}=70\%$ is achieved. In [2] 100%
reconstruction in German Credit dataset was achieved with 1722 queries.
However, AUTOLYCUS partially reconstructs 70% of the model with 8 times less
queries. Hence, it can be utilized as a budget option with softer requirements
against platforms that provide AI explanations.

##  V Conclusion

In this poster, we demonstrated a new model extraction attack AUTOLYCUS
against decision-tree models. AUTOLYCUS exploits XAI to enhance the
capabilities of state-of-the-art methods by softening the requirements needed
for reconstruction like decreasing the number of queries and utilizing
existing samples as auxiliary information.

## References

  * [1] M. Fredrikson, S. Jha, and T. Ristenpart, “Model inversion attacks that exploit confidence information and basic countermeasures,” in _Proceedings of the 22nd ACM SIGSAC Conference on Computer and Communications Security_ , ser. CCS ’15. New York, NY, USA: Association for Computing Machinery, 2015, p. 1322–1333. [Online]. Available: https://doi.org/10.1145/2810103.2813677
  * [2] F. Tramèr, F. Zhang, A. Juels, M. K. Reiter, and T. Ristenpart, “Stealing machine learning models via prediction apis,” in _Proceedings of the 25th USENIX Conference on Security Symposium_ , ser. SEC’16. USA: USENIX Association, 2016, p. 601–618.
  * [3] A. Barredo Arrieta, N. Díaz-Rodríguez, J. Del Ser, A. Bennetot, S. Tabik, A. Barbado, S. Garcia, S. Gil-Lopez, D. Molina, R. Benjamins, R. Chatila, and F. Herrera, “Explainable artificial intelligence (xai): Concepts, taxonomies, opportunities and challenges toward responsible ai,” _Information Fusion_ , vol. 58, pp. 82–115, 2020. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S1566253519308103
  * [4] “Explainable artificial intelligence (xai).” [Online]. Available: https://www.ibm.com/watson/explainable-ai
  * [5] M. T. Ribeiro, S. Singh, and C. Guestrin, “”why should i trust you?”: Explaining the predictions of any classifier,” in _Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_ , ser. KDD ’16. New York, NY, USA: Association for Computing Machinery, 2016, p. 1135–1144. [Online]. Available: https://doi.org/10.1145/2939672.2939778


Present address: ]Google Quantum AI, Mountain View, CA, USA

# Absence of edge reconstruction for quantum Hall edge channels in graphene
devices

Alexis Coissard  Adolfo G. Grushin  Univ. Grenoble Alpes, CNRS, Grenoble INP,
Institut Néel, 38000 Grenoble, France  Cécile Repellin  Univ. Grenoble Alpes,
CNRS, LPMMC, 38000 Grenoble, France  Louis Veyrat  Univ. Grenoble Alpes, CNRS,
Grenoble INP, Institut Néel, 38000 Grenoble, France  Kenji Watanabe  Research
Center for Functional Materials, National Institute for Materials Science, 1-1
Namiki, Tsukuba 305-0044, Japan  Takashi Taniguchi  International Center for
Materials Nanoarchitectonics, National Institute for Materials Science, 1-1
Namiki, Tsukuba 305-0044, Japan  Frédéric Gay  Hervé Courtois  Hermann Sellier
Benjamin Sacépé<EMAIL_ADDRESS>[  Univ. Grenoble Alpes, CNRS,
Grenoble INP, Institut Néel, 38000 Grenoble, France

###### Abstract

Electronic edge states in topological insulators have become a major paradigm
in physics. The oldest and primary example is that of quantum Hall (QH) edge
channels that propagate along the periphery of two-dimensional electron gases
(2DEGs) under perpendicular magnetic field. Yet, despite 40 years of intensive
studies using a variety of transport and scanning probe techniques, imaging
the real-space structure of QH edge channels has proven difficult, mainly due
to the buried nature of most 2DEGs in semiconductors. Here, we show that QH
edge states in graphene are confined to a few magnetic lengths at the crystal
edges by performing scanning tunneling spectroscopy up to the edge of a
graphene flake on hexagonal boron nitride. These findings indicate that QH
edge states are defined by boundary conditions of vanishing electronic
wavefunctions at the crystal edges, resulting in ideal one-dimensional chiral
channels, free of electrostatic reconstruction. We further evidence a uniform
charge carrier density at the edges, contrasting with conjectures on the
existence of non-topological upstream modes. The absence of electrostatic
reconstruction of quantum Hall edge states has profound implications for the
universality of electron and heat transport experiments in graphene-based
systems and other 2D crystalline materials.

### Introduction

In 1982, two years after the discovery of the quantum Hall effect [1], B.
Halperin predicted the existence of edge states carrying the electron flow
along sample periphery [2]. These edge states, which form unidirectional
(chiral) ballistic conduction channels, have been pivotal in understanding
most of the transport properties of the QH effect [3, 4]. They have served as
an extraordinarily versatile platform for a multitude of quantum coherent
experiments [5], culminating recently in the evidence of fractional statistics
in the fractional QH effect [6] and the possibility of anyon braiding through
interferometry [7].

The existence of edge states was initially inferred as a consequence of the
boundary conditions imposed by the physical edges on the electron
wavefunctions [2]. The energy of the electron states that are condensed into
Landau levels increases upon approaching the edge due to the hard-wall
boundary conditions, opening new conduction channels –the QH edge channels–
spatially located at their intersection with the Fermi level [2] (see Fig.
1B-C). Inclusion of a smooth electrostatic confining potential, which is
experimentally used to define edges in 2DEGs buried in semiconductor
heterostructures, enriches the picture with the concept of edge reconstruction
[8]. There, the Coulomb interaction energy dominates the confining potential,
leading to a transformation of the edge states into a series of wide
compressible channels separated by incompressible strips. In the opposite case
of a sharp potential, the Coulomb interaction is not relevant and the single
particle picture is valid. Edge reconstruction mechanisms have further proven
to be of paramount importance in the fractional QH regime where additional co-
and/or counter-propagative or even neutral modes [9, 10, 11] can emerge and
complexify charge and heat transport [12, 13, 14].

Figure 1: Tunneling spectroscopy of quantum Hall edge states. A, B, Schematics
of the experiment. A PtIr tip is glued at the extremity of one prong of a
piezoelectric tuning fork to enable imaging both in STM (by regulating the
tunneling current $I_{\text{t}}$) and in AFM (by regulating the frequency
shift of the tuning fork). Graphene lies atop an insulating hBN flake and is
contacted by a Cr/Pt/Au electrode to apply the sample bias $V_{\text{b}}$. A
back-gate voltage $V_{\text{g}}$ applied to the Si/SiO2 substrate enables to
tune the Fermi level $E_{F}$ in graphene. Graphene edges are first located by
AFM under perpendicular magnetic field, $B$. The tip is then moved from the
graphene bulk to the edge in STM to perform tunneling spectroscopy of QH edge
channels. C, Landau level spectrum [15, 16, 17] as a function of energy $E$
(normalized to the first cyclotron gap $\varepsilon_{0}$) and position. The
Landau levels disperse at an armchair edge on the scale of the magnetic length
$l_{B}$. Their intersect with the Fermi level defines the QH edge channels. D,
E, Topographic image (E) and its $z$ profile averaged on the $y$ direction (D)
of the graphene edge obtained in STM. We consider that the tip apex is located
above the graphene edge at the maximum of the $z$ profile. F, Atomic
resolution of the graphene honeycomb lattice measured in STM a few nanometers
away from the edge. The vertical blue line indicates the crystal edge
orientation deduced from (E). G, Kekulé-bond order imaged in charge-neutral
graphene [18] at $V_{\text{g}}$=-5 $V$ at a distance of $20\>$nm from the
edge.

Non-reconstructed edge states can substantially clarify QH edge transport with
virtually ideal one-dimensional edge states [19] and new regimes of intra- and
inter-channel interactions. Contrary to semiconductor heterostructures, two-
dimensional crystalline materials like graphene, for which physical edges are
crystal edges, may be archetypical systems hosting such edge states. For
graphene and its massless, linear band structure, QH edge states without
confining electrostatic potential are expected to be the exact eigenstates of
the Dirac equations derived with vanishing boundary conditions at the armchair
or zigzag edge [15, 16, 17]. Akin to Halperin’s original prediction [2], these
solutions for edges states are maximally confined to a few magnetic lengths
$l_{B}=\sqrt{\hbar/eB}$ ($\hbar$ is the reduced Planck constant, $e$ the
electron charge and $B$ the magnetic field) from the crystal edge, leaving no
room for edge reconstruction.

Here, we unveil the real-space structure of the quantum Hall edge states of
graphene lying on an insulating hexagonal boron nitride flake (hBN) and
evidence the absence of edge reconstruction, by performing scanning tunneling
spectroscopy up to the graphene crystal edge, under strong perpendicular
magnetic field. We achieved this by overcoming the long-standing experimental
challenge [20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] of approaching a
scanning tunneling tip to the edge without crashing it on the insulating
substrate that borders the graphene flake, by means of a prior localization of
the graphene edge by atomic force microscopy (AFM). We purposely used a home-
made hybrid scanning microscope [18] capable of operating alternatively in AFM
and scanning tunneling microscopy (STM) mode, thanks to a PtIr STM tip glued
onto a piezoelectric tuning fork acting as a force sensor [31, 32] for AFM
(see Fig. 1A). Our sample schematized in Fig. 1A and B consists of a graphene
monolayer deposited on a hBN flake sitting on a Si/SiO2 substrate that serves
as a back-gate electrode (see Methods). The graphene flake is contacted by a
Cr/Pt/Au tri-layer that allows to apply a voltage bias $V_{\text{b}}$ and
collect a tunnel current $I_{\text{t}}$ via the STM tip. All experiments
presented here are performed at a temperature of 4.2 K and a perpendicular
magnetic field of 14 T.

### Results

Quantum Hall edge states spectroscopy

Figure 1E displays a STM topographic image taken in constant current mode to
the graphene edge, initially coarsely located by AFM (see Fig. S1). The height
profile of this image (Fig. 1D) shows a large flat area, and a slight bump on
the left part of the scan. This bump results from the tip-graphene interaction
lifting up the graphene edge when the tip is right above it [33]. This bump
allows us to locate the edge of the graphene crystal with an accuracy of a few
nanometers (see SI). To the left of the bump, the tip dips towards the hBN
substrate, on which a tip crash is avoided by a height limit of the STM
controller. Atomic scale imaging of the honeycomb lattice shown in Fig. 1F
gives insight into the graphene lattice termination. The edge orientation in
Fig. 1E, which is reported in Fig. 1F with the blue line, indicates an
armchair termination.

Figure 2: Sharp quantum Hall edge states. A, Evolution of the tunneling
conductance d$I_{\text{t}}$/d$V_{\text{b}}$ as a function of the distance from
graphene edge measured at charge neutrality ($V_{\text{g}}=-5.4$ V). The half-
filled zeroth Landau level is split into two sublevels LL${}_{0^{+}}$ and
LL${}_{0^{-}}$ due to quantum Hall ferromagnetism [18]. B, Schematics of the
tunneling into QH edge states. Due to the spatial extent
$R_{\text{c}}^{N}=l_{B}(2|N|+1)^{1/2}$ of the LLN wave functions, the
tunneling electrons probe at one point contributions from all states up to
distances of about $R_{\text{c}}^{N}$ (red and orange gaussians in the middle
panel for LLN and LLN+1, respectively). The resulting density of states
features sharp Landau level peaks in the bulk, i.e. at distance $d_{1}$ from
the edge, and a smooth profile close to the edge, at a distance $d_{2}\sim
l_{B}$, due to the energy broadening of Landau levels [15]. Additionally, when
approaching the edge, the tip starts to probe the edge states of the lower
Landau levels, pushed at higher energies by the presence of the physical edge,
and overlapping with the highly-degenerate bulk states. The resulting peaks in
the density of states thus exhibit a spectral weight redistribution toward
higher energies, which leads to a suppression of the Landau level peak height
in the tunneling conductance (bottom panel, in solid blue each individual $N$
and $N+1$ Landau level peak, and in dashed blue the overall density of
states). C, Spatial maps of the tunneling conductance
d$I_{\text{t}}$/d$V_{\text{b}}$ at the energies of the Landau levels. D,
Individual spectra taken from (A) at different distances from the edge
indicated by the color-coded arrows in (A).

The central result of this work is shown in Figure 2, which presents the
evolution of the Landau levels upon approaching the immediate proximity of the
graphene edge in the region shown in Fig. 1E, under a magnetic field of 14 T.
We first study charge-neutral graphene by tuning the density with the back-
gate voltage set at $V_{\text{g}}=-5.4$ V. Tunneling spectroscopy of Landau
levels [34, 35, 36, 37] results in a series of peaks in the tunneling
conductance $G(V_{\text{b}})=\text{d}I_{\text{t}}/\text{d}V_{\text{b}}$ that
is proportional to the local density of states. We show in Fig. 2A the
tunneling conductance $G(d_{\text{edge}},V_{\text{b}})$ as a function of tip
distance perpendicular to the graphene edge $d_{\text{edge}}$, and bias
voltage $V_{\text{b}}$. Far from the edge, Landau levels are readily
identified as bright conductance peaks that we label LLN, where $N$ is the
Landau level index. These conductance peaks are conspicuously stable upon
approaching the edge on the left of the figure. Within 40 nm from the edge, we
observe a suppression of the Landau level peak heights (see individual spectra
in Fig. 2D) starting at distances that depend on the Landau level (the higher
the Landau index, the further from the edge). Figure 2C shows spatial maps of
the tunneling conductance at the voltage bias of the Landau level peaks. For
each Landau level peak, darker areas corresponding to Landau level peak
suppression appear further and further from the edge as the Landau level index
increases.

These findings contrast with the expectation for a smooth confining potential
at the edges, for which the Landau level spectrum would have continuously
shifted in energy, following the confining potential as the edge is
approached. Since the tunneling conductance probes states on the scale of the
electron wavefunction, that is, the cyclotron radius
$R_{\text{c}}^{N}=l_{B}(2|N|+1)^{1/2}$ for Landau level index $N$, the
suppression of the Landau level peaks, here, reflects a spreading of the
spectral weight to higher energy due to an abrupt edge state dispersion at the
physical edge, on a very short scale of the order of the magnetic length (see
Fig. 2B). This suppression of the tunneling density of states of the Landau
levels, which has been observed on graphene on a conductive graphite substrate
[28], is therefore direct evidence of QH edge states sharply confined at the
edges. Ultimately, on the last few nanometers from the edge, the Landau level
peaks disappear completely, and the redistribution of Landau level spectral
weight yields a V-shape like tunneling density of states (see Fig. 2D).

Figure 3: Theoretical tunneling density of states. A, Schematic of the
simulated edge-geometry. We considered a Kekulé-bond order [38, 39, 18] as
broken-symmetry state [40], with lattice vectors that triple the unit cell
compared to pristine graphene. B, Corresponding local density of states (LDOS)
as a function of the distance from the armchair graphene edge,
$d_{\mathrm{edge}}$, normalized by $l_{B}$, for charge-neutral graphene. The
Kekulé-bond order splits the zeroth Landau level into two sublevels
LL${}_{0^{+}}$ and LL${}_{0^{-}}$ with an energy gap chosen to match the
experimentally measured value of $50$ m$e$V [18]. The white dashed lines are
the numerically computed Landau levels of graphene nano-ribbon with armchair
termination. Due to position momentum locking, the Landau Levels disperse as
they approach the physical edge as sketched in Fig. 1B. C, Individual spectra
taken from (B) at different distances from the edge. Solid lines show cuts at
different $d_{\mathrm{edge}}$, while dashed lines show spectral asymmetry
emerging from a single disorder realization of an on-site disorder potential
with strength $W/t=0.3$ (see Methods).

In this measurement we have set the Fermi level at charge neutrality, that is,
at Landau level filling factor $\nu=0$, which leads to a splitting of the
zeroth Landau level (see split peaks labeled LL${}_{0^{+}}$ and LL${}_{0^{-}}$
in Fig. 2A) with the opening of an interaction-induced gap at $V_{\text{b}}=0$
V (see Ref. [18]). This splitting signals the broken-symmetry state [40] at
charge neutrality with the Kekulé-bond order [38, 39, 18]. Interestingly, we
identified the Kekulé-bond order at 20 nm of the edge in Fig. 1G, indicating
that this broken-symmetry state, which develops in the bulk, is robust even in
the very proximity of the edge [41].

To substantiate our finding we performed numerical simulations of the local
density of states of a charge neutral graphene graphene ribbon with an
armchair edge under perpendicular magnetic field (see. Fig. 3A) [15, 16, 17].
We computed the Landau levels of the lattice Hamiltonian of nearest-neighbor
hopping energy $t$. We assumed a Kekulé-bond order with a gap at half-filling
of the zeroth Landau level of $50$ m$e$V, as measured experimentally [18]. The
eigenstates for a ribbon with periodic boundary conditions along $\hat{y}$ are
shown in Fig. 3B as white dashed lines. The Landau level eigenstates disperse
as their average $x$ position, locked to their $k_{y}$ momentum, approaches
the physical edge of graphene [42], as schematized in Fig. 1B. Fig. 3B shows
the clean local density of states, which integrates the eigenstates weighted
by the amplitude of the wavefunctions, as a function of the distance to the
edge normalized by $l_{B}$, $d_{\mathrm{edge}}/l_{B}$, averaged over each unit
cell (see Methods). The range of $d_{\mathrm{edge}}/l_{B}$ coincides with the
range of displacement in Fig. 2A, allowing direct comparison with the
experimental data. The resulting Landau level peaks are suppressed at higher
values of $d_{\mathrm{edge}}$ the higher their Landau level index, and on the
same spatial scale as observed experimentally in Fig. 2A. This reduction of
spectral weight is more visible in Fig. 3C where we plot spectra for different
$d_{\mathrm{edge}}$ (solid lines) including a single realization of on-site
disorder (dashed lines). The latter breaks the particle-hole symmetry of the
spectrum, and thus may contribute to the asymmetries observed in the zeroth
landau level peaks.

On the charge accumulation on the edges

The question of charge carrier homogeneity is critical for graphene transport.
A body of work has shown anomalous asymmetry in some transport properties
supplemented by scanning probe investigations [43, 44, 45], which points to a
charge carrier accumulation at the graphene edges. Its origin may be either
electrostatic stray field of the back-gate electrode [46] or chemical doping
due to edge treatments (etching) or dangling bonds. In the QH effect, such an
accumulation could open up additional counter-propagative edge channels and
produce dissipation [46, 44, 45].

Figure 4: Charge density inhomogeneity on pristine edges. A, Spatial evolution
of the tunneling conductance up to the edge at filling factor $\nu=2$ for
$V_{\text{g}}=4$ V. B-D, Tunneling conductance gate maps as a function of
sample bias $V_{\text{b}}$ and gate voltage $V_{\text{g}}$. At a distance
$d_{\text{edge}}=36\>\text{nm}=5.3\>l_{B}$ from the edge, in (B), we observe
the staircase pattern of Landau levels of the graphene bulk. At closer
distances from the edge, the Landau level peaks in the staircase pattern
starts to blur (C) and mostly vanishes in (D). In the three panels, the
opening of the $\nu=0$ gap as a function of $V_{\text{g}}$ is indicated by
white dashed lines, and the back-gate voltage of the charge-neutrality point
$V_{\text{g}}^{\text{CNP}}$ is identified by the maximum of the gap. The blue
dashed lines indicate the spectra at $V_{\text{g}}=4$ V, which coincide with
the back-gate voltage of (A). The faint diagonal lines that translate into
faint horizontal lines above the LL0 peak in (A) result from residual charging
effects in the tunneling process. Their downward dispersion near the edge in
(A) is consistent with the charge carrier variation in (E). E, Evolution of
$V_{\text{g}}^{\text{CNP}}$ and charge carrier density $n_{0}$ determined from
tunneling conductance gate maps as a function of the distance from the edge
$d_{\text{edge}}$. The position of the charge carrier density shift coincides
with the upward shift of the LL0 in (A). Error bars correspond to the range of
gate voltage where the $\nu=0$ gap opens in the gate maps.

In tunneling experiments, a charge inhomogeneity on the edge would result in
an energy shift of the Landau level spectrum as a whole due to a local change
of the Landau level filling factor. Our measurements in Fig. 2 provides a
first insight on this issue with a remarkable stability of the Landau level
peaks in energy that indicates that a possible charge accumulation is not
large enough to depin the chemical potential from the zeroth Landau level
[18]. In particular, it is lower than the value $\delta n=6.8\times 10^{11}$
cm-2 required to fill the zeroth Landau level and reach $\nu=2$ at 14 T, which
would produce a visible energy shift of the Landau level spectrum that we do
not observe.

To enhance the sensitivity of the spectroscopy to possible charge
inhomogeneities, we performed similar measurements at filling factor $\nu=2$
($V_{\text{g}}=4$ V), when the Fermi level is pinned by localized states in
the cyclotron gap separating LL0 from LL1. There, due to the little density of
localized states as compared to the highly degenerate Landau levels, a small
variation of charge density would result in a substantial shift of the Landau
levels in the tunneling spectra. Figure 4 displays the spatial evolution of
the tunneling conductance up to the edge, at $\nu=2$ and 14 T. As in Fig. 2,
the Landau level peaks (LL0, LL-1 and LL-2) stay at the same energy over the
scan and vanish at about 20 nm from the edge, clearly indicating the absence
of charge accumulation. We further performed systematic gate-tuned tunneling
spectroscopy maps at various locations, from 500 nm to 5 nm from the edge (see
SI). Figure 4B-D displays three of these maps taken close to the edge. We
observe in Fig. 4B the usual staircase pattern of the Landau level peaks due
to the successive pinning of the Fermi energy in the Landau levels [47, 48,
18], which allows us to precisely identify the back-gate voltage of the charge
neutrality point $V_{\text{g}}^{\text{CNP}}$. As shown in Fig. 4E that
displays $V_{\text{g}}^{\text{CNP}}$ as a function of the distance from the
edge, there is no charge accumulation from 500 nm to 20 nm to the edge, and
only within 20 nm of the edge we measure a variation $\delta n=(-1.5\pm
1.1)\times 10^{11}\>\text{cm}^{-2}$.

Interestingly, such a charge density variation near the edge at 14 T yields a
little variation $\delta\nu=0.4$ of local filling factor, which would have no
consequence on the QH edge transport properties. Extrapolating at lower field,
however, $\delta\nu=2$ would be reached at a magnetic field of 3 T, thus
potentially affecting edge transport with additional modes. Yet, the very
small spatial scale of this charge accumulation cannot explain recent scanning
probes experiments evidencing indirect, sometimes out-of-equilibrium responses
within hundreds of nanometers from the edge [44, 45]. We conjecture that this
charge accumulation in our particular case is related to the tip-graphene
interaction when the tip reaches and lift up the graphene edge (see
Supplementary Section III).

### Discussion

The issue of charge accumulation on the edge and the ensuing emergence of
upstream modes [44, 45] were put forth as an alternative interpretation [49]
for the signature of helical edge transport in charge-neutral graphene [50,
51]. Although we cannot exclude that the stray field of the back gate
electrode may accumulate charges at high back-gate voltages, that is, away
from charge neutrality point, and over a long distance [46], our results show
that this accumulation is absent at low back-gate voltage, thus invalidating
the doubts raised [49] on the existence of the quantum Hall topological
insulator phase in charge neutral-graphene [50, 51]. Still, it may be
interesting to revisit non-local transport in non-linear regime [49] in view
of the exact spatial structure of the QH edge states in graphene.

Regarding edge reconstruction, a wealth of fractional and integer quantum Hall
states exhibit complex sequences of reconstructed edge channels, including
additional integer and/or fractional as well as neutral modes [9, 10, 11].
Whereas the smooth electrostatic potential in GaAs and other semiconductors
reconstructs edge states into wide compressible stripes of the order of $\sim
100$ nm (see Ref. [29]), the graphene QH edge states confined on a very short
length scale, at few magnetic lengths on the physical edge, pose new
constraints and limits for such a reconstruction, opening the investigation of
universal transport and thermal properties [19]. Moreover, in such a strongly
confined configuration, an enhancement of inter-edge-states interactions can
be expected, which makes the picture of independent chiral channels irrelevant
in this case, thus impacting charge and heat equilibration [52, 53, 54]. This
should impact QH interferometry [55] in graphene systems [56, 57] and other
coherent experiments [5], for which the independence, exact positions and
nature of edge modes are crucial parameters to address anyon physics as well
as other interaction-driven phenomena, such as charging effects [58], spin-
charge separation [59] or electron pairing [60].

Note: A very recent work (https://arxiv.org/abs/2210.01831) reports a
complementary tunneling spectroscopy study of electrostatically-defined QH
edge states at a $pn$ junction.

### methods

#### Sample fabrication

The graphene/hBN heterostructure was assembled from exfoliated flakes with the
van der Waals pick-up technique using a polypropylene carbonate (PPC) polymer
[61]. The stack with graphene on top of the hBN flake was deposited using the
method described in Ref. [62] on a highly p-doped Si substrate with a $285$ nm
thick SiO2 layer. Electron-beam lithography using a PMMA resist was used to
pattern a guiding markerfield on the whole $5\times 5$ mm2 substrate to drive
the STM tip toward the device and to locate the graphene edge. Cr/Pt/Au
electrodes contacting the graphene flake were also patterned by electron-beam
lithography and metalized by e-gun evaporation. The sample was thermally
annealed at $350\,^{\circ}$C in vacuum under an halogen lamp to remove resist
residues and clean graphene, before being mounted into the STM where it was
heated in situ during the cooling to $4.2\;$K.

#### Measurements

Experiments were performed with a home-made hybrid scanning tunneling
microscope (STM) and atomic force microscope (AFM) operating at a temperature
of $4.2\;$K in magnetic fields up to $14\;$T. The sensor consists of a hand-
cut PtIr tip glued on the free prong of a tuning fork, the other prong being
glued on a Macor substrate. Once mounted inside the STM, the tip is roughly
aligned over the sample at room temperature. The AFM mode was used first for
coarse navigation at $4.2\;$K on the sample surface to align the tip onto
graphene and then for locating coarsely the graphene edge, see SI. The STM
imaging in constant-height mode of the edge, done subsequently, yields a fine
identification. Scanning tunneling spectroscopy (STS) was performed using a
lock-in amplifier technique with a modulation frequency of $263\;$Hz and rms
modulation voltage between $1-5\;$mV depending on the spectral range of
interest. Current Imaging Tunneling Spectroscopy (CITS) measurements were
acquired by starting far from the edge, with a grid whose slow $x$-axis is
perpendicular to the edge direction (as imaged by STM) and the $y$-axis is
parallel to the edge with a size of a few tens of nanometers. A safety
condition is added to the tip vertical $z$-position controller to prevent the
crashing into the hBN flake beyond the graphene edge : if the $z$-position
reaches a threshold (typically $3\>$nm below the $z$-position of the tip
estimated close to the edge), the tip is withdrawn and the CITS ends. Imaging
of the Kekulé-bond order was carried out in STM constant-height mode after
tuning the graphene to charge neutrality with the back gate, at a bias voltage
corresponding to the energy of the LL${}_{0^{+}}$ peak (see Ref. [18] for
details).

#### Theoretical simulations

To compute the local density of states shown in Fig. 3 we use the simulation
software Kwant [63]. First we create a honeycomb lattice in a square system of
size $L_{x}\times L_{y}=130\times 130$, in units of graphene’s lattice
constant $a$. The unit-cell for the Kekulé order is tripled compared to
pristine graphene, and is defined by the reciprocal vectors
$\mathbf{a}_{1}=a(3\sqrt{3}/2,3/2)$ and $\mathbf{a}_{2}=a(-3\sqrt{3}/2,3/2)$,
see Fig. 3A. To calculate the local density of states $\rho(E,x)$ at a given
energy $E$ and Kekulé unit cell $x$, we average over the six sites weighted by
the corresponding wave-function,
$\rho(E,x)=\sum_{\alpha}|\psi_{\alpha}(x)|^{2}\delta(E-E_{\alpha})$, where
$\alpha$ runs over the six unit cell sites. We compute the local density of
states spectra, shown in Fig. 3B as a colormap, using the kernel polynomial
method [64] with a target energy resolution of $\Delta E/t=0.005$, and a
magnetic field of $\phi/\phi_{0}=0.005$ in units of the magnetic flux
$\phi_{0}=h/e$. The dashed line spectra of Fig. 3B maps are obtained for a
finite nano-ribbon of width $L_{y}$, with an arm-chair edge parallel to the
$\hat{y}$ direction, as in Ref. [42]. We allow the edge to be mis-aligned with
the Kekulé lattice vectors, as observed experimentally in Fig. 1F and G.
Lastly, the solid lines in Fig. 3C show cuts of the local density of states
spectra show in Fig. 3B. The dashed lines are calculated adding a single
disorder realization obtained by adding a random on-site potential
$V_{\mathrm{dis}}$ at each site to the clean local density of states spectra
described above. The disorder strength at each site is drawn from a uniform
distribution in the interval $[-W,W]$ with $W/t=0.3$.

### Data availability

All data needed to evaluate the conclusions in the paper are present in the
paper and/or the Supplementary Materials.

### Acknowledgments

We thank C. Déprez, B. Halperin, M. Feigelman, M. Goerbig, M. Guerra, D.
Perconte, H. Vignaud, and W. Yang for valuable discussions. We thank F.
Blondelle, D. Dufeu, Ph. Gandit, D. Grand, G. Kapoujyan, D. Lepoittevin, J.-F.
Motte and P. Plaindoux for technical support in setting up of the experimental
system. Samples were prepared at the Nanofab facility of the Néel Institute.
This work has received funding from the European Union’s Horizon 2020 research
and innovation program ERC grants QUEST No. 637815 and SUPERGRAPH No. 866365,
and the Marie Sklodowska-Curie grant QUESTech No. 766025. A. G. G.
acknowledges financial support by the ANR under the grant ANR-18-CE30-0001-01
(TOPODRIVE).

### Author contributions

AC fabricated the sample and performed the measurements. AC, AGG, CR, HS and
BS analysed the data. AGG and CR conducted the theoretical analysis. LV
assembled the STM microscope. KW and TT supplied the hBN crystals. FG provided
technical support on the experiment. BS conceived and supervised the project,
designed the experimental set-up, and wrote the paper with inputs from all co-
authors.

### Competing Interests

The authors declare that they have no competing financial interests.

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##  Supplementary Information

###  I Sample details and AFM mapping

The sample AC04 studied is this work is a heterostructure made of a graphene
sheet atop a hexagonal boron nitride (hBN) flake, assembled by van der Waals
stacking, and then deposited on a p++Si/SiO2 substrate to enable back gating
of the charge carrier density in graphene. The voltage bias $V_{\text{b}}$ is
applied using a Cr/Pt/Au contact patterned by e-beam lithography and covering
partially the graphene sheet, leaving a large fraction of the perimeter
accessible by the tip for imaging and tunneling spectroscopy of the edge
states, see Fig. S1a and b. The graphene bulk properties of this sample have
been presented in Ref. [18].

Figure S1: Sample AC04 and AFM mapping. a, The graphene/hBN heterostructure is
deposited on a Si++/SiO2 substrate that serves as a back-gate electrode to
tune the charge carrier density. Graphene is biased with a Cr/Pt/Au contact
patterned by e-beam lithography on one of its edge, leaving others accessible
for the tunneling spectroscopy of QH edge states. b, Optical image of the
device. The graphene and hBN flakes are outlined by white and blue dashed
lines, respectively. c, AFM mapping of the graphene sheet at $T=4.2\>$K and
$B=14\>$T, with three high-resolution images of the edges. The gold contact
used to bias graphene is visible on the right of the images. The edge studied
in this work is indicated by the white arrow. d, Evolution of the quality
factor $Q$ of the tuning fork with magnetic field.

The STM tip is brought atop the graphene sheet by AFM imaging of the coding
markerfield patterned on the whole chip surface. This guiding process is done
after about ten AFM images. An AFM mapping of graphene and its boundary with
the underlying hBN performed at $B=14\>$T is shown in Fig. S1c. High-
resolution AFM images of some edges are placed in overlay. These images reveal
that the vacuum annealing employed to clean the graphene left some resist
residues that have migrated toward the edges, forming bright spots in-between
which edges are clean. In this work we focus on the edge indicated by the
white arrow, which is also the direction of the Current Imaging Tunneling
Spectroscopy (CITS) measurement grids performed from the bulk of graphene to
the edge. Note that the tuning fork we used here still displays a relatively
high quality factor in magnetic field, with $Q\sim 4000$ at 14 T (Fig. S1d).

###  II Localization of graphene edges on hBN

Figure S2: Locating graphene armchair edge. $16\times 7\>$nm2 STM image in
constant height mode of a graphene edge. On the left side the tunneling
current vanishes, indicating the tip is atop hBN. Inset : $5\times 5\>$nm2 STM
image of the honeycomb lattice a few nanometers away from the edge. When
reporting the edge direction (blue dashed line) on the honeycomb lattice, it
coincides with an armchair edge orientation.

We show in Fig. S2 a STM image at $B=14\>$T of the edge of graphene indicated
by the white arrow in Fig. S1c, which provides a very accurate identification
of the edge position. It is obtained in constant height mode: before STM
imaging, we approach the STM tip in tunneling contact with the graphene in
order to measure the setpoint tunneling current (typically $1\>$nA) and next
switch off the $Z$-regulation for imaging. This mode allows a safe imaging of
graphene edge since the tip would not crash down on the insulating hBN, but
would rather simply measure zero tunneling current as seen on the left part of
the STM image in Fig. S2. However, on the very edge of the graphene flake, the
honeycomb lattice is not resolved due to the instability of the tunneling
current at this location. As a result, the meaningful information in Fig. S2
is the vanishing of the tunneling current when the tip reaches the hBN, which
constitutes a clear identification of the edge location with nanometer-scale
precision. Using this image of the edge, we can estimate its direction as
indicated by the dashed blue line in Fig. S2. We report this line on the
honeycomb lattice of the inset taken a few nanometers away and identify the
armchair orientation for this edge.

We believe that the instability of the tunneling current measured on the very
edge of the graphene in Fig. S2 stems from the local lifting of the graphene
sheet edge from the hBN flake, each time the STM tip scans over it, due to
electrostatic interactions with the tip.

###  III Tip-induced lifting of the graphene edge and definition of the edge
position

We discuss here another way to locate the edge by means of a CITS grid
spectroscopy measurement of the spatial dispersion of the Landau level (LL)
spectrum toward the boundary. The grid spectroscopy is set to start far away
in graphene bulk and to finish a few nanometers beyond the edge, previously
located with STM images. Moreover, the slow $x$-axis direction of the grid is
chosen to be perpendicular to the edge. A safety condition is added to the
$Z$-controller to prevent the tip from crashing into hBN : if the $Z$-position
of the tip goes below a threshold (typically $3\>$nm below the $Z$-position of
the tip estimated close to the edge), the tip is withdrawn and the CITS ends.

Figure S3: CITS to the graphene edge: locating the edge. a,b,
$z(d_{\text{edge}},y)$ topographic map (a) and $z(d_{\text{edge}})$ profile
(b) obtained from a CITS toward the armchair edge and beyond. We distinguish
three regimes : the flat horizontal profile (red area) where the electrons
tunnel in bulk graphene, see c, the sharp increase of $z$ (blue region) where
the electrons tunnel from the tip apex very close to the edge in a situation
where the sheet is strongly lifted, see d, and the decrease of $z$ when the
tip apex is moved outside the graphene sheet, with a residual tunneling
between the edge of graphene and other atoms on the side of the tip, see e,
preventing the tip from crashing into hBN. The edge position is taken at the
position of the maximum of the topographic profile.

We show in Fig. S3a the topographic map $z(d_{\text{edge}},y)$ obtained from a
CITS toward the graphene armchair edge identified in Fig. S2.
$d_{\text{edge}}$ is the distance from the armchair edge, while $y$ is the
lateral coordinate parallel to the edge. The topographic map features a clean
and flat bulk graphene on a $80\times 10\>$nm2 area next to the edge. When the
tip is situated a few nanometers away from the edge, the
$z(d_{\text{edge}},y)$ map reveals inhomogeneous bright spots. Though one can
first think about residues, the small height of these spots, around
$1-3\>\angstrom$, rules out this hypothesis.

We rather attribute these large spots to the lifting of the edge of the
graphene sheet, as illustrated in Fig. S3d. The attractive van der Waals force
of the tip was shown [33] to lift locally a graphene sheet lying on a SiO2
substrate on a typical height of $1\>\angstrom$. Although we do not observe
such lifting in Fig. S3a in bulk graphene (either because the deformation
follows the tip such that we eventually observe an overall flat background, or
because the deformation of the graphene sheet on hBN is more difficult, since
the adhesion interactions between both materials are more important than
between graphene and SiO2), we can assume that the graphene flake is more
easily deformed at the edge by the force of the tip, and therefore the lifting
is larger there than in the bulk.

The lifting of the edge is well visible in the height profile of Fig. S3b,
obtained by averaging the topographic map along the $y$ direction (parallel to
the edge). The $z$ profile features a flat region corresponding to bulk
graphene (with variations of less than $1\>\angstrom$), and a hump of
$3\>\angstrom$ height at the edge. After that, the tip quickly moves down by
several nanometers until it meets the safety condition of the $Z$-controller,
which stops the CITS. We attribute this lowering of the $z$ position to the
fact that the tip apex has gone beyond the edge of graphene, but tunneling
remains possible with some other higher atoms of the tip close to the apex,
see Fig. S3d. This makes the measurement of a tunneling current possible even
when the apex itself is lying on hBN, yet this current is highly unstable.

From this model we assume the position of the edge of graphene (i.e. the tip
apex is atop the edge) is given by the maximum of the hump in the $z$ profile,
and from this origin we compute $d_{\text{edge}}$ the distance from the edge,
which we use in the main text and the following figures.

###  IV Additional tunneling conductance maps at the edge

We show in this section two additional tunneling conductance maps acquired
along the same armchair edge, but a few tens of nanometers away from the map
shown in Fig. 2 of the main text. The back-gate voltage is fixed at
$V_{\text{g}}=-5\>$V, corresponding to filling factor $\nu=0$.

Figure S4: Landau level spectroscopy toward armchair edge (1). a, Topographic
profile $z(d_{\text{edge}})$ and map $z(d_{\text{edge}},y)$ near the graphene
edge ($l_{B}=6.85\>$nm at $B=14\>$T). b, Tunneling conductance as a function
of the distance to the edge and sample bias, toward the edge. c, Tunneling
conductance spatial maps at the different LL energies. The panel at
$V_{\text{b}}=-9\>$mV shows the spatial map of the $\nu=0$ gap.

In Figs. S4 and S5, the panels (a) show the topographic map
$z(d_{\text{edge}},y)$ and the profile $z(d_{\text{edge}})$ obtained by
averaging the map on the lateral $y$-dimension. Bulk graphene appears flat and
clean, with a corrugation of at most $1\>\angstrom$ on a distance of 100 and
$300\>$nm, respectively. When approaching the edge on the left,
$z(d_{\text{edge}})$ increases by around $2-3\>\angstrom$ due to the tip-
induced lifting of the graphene sheet edge. In Fig. S4 the CITS grid
spectroscopy did not go beyond the edge: the edge position is rather roughly
estimated using the STM image in Fig. 1d from the main text. The same goes for
Fig. S5.

Panels (b) show the tunneling conductance toward the armchair edge as a
function of the distance to the edge and the sample bias. The same qualitative
observations as that of the main text can be made for the two edges: the
Landau level peaks do not disperse when approaching the edges but vanish. The
splitting of the LL0 is well visible and the gap stays open down to the edge
where it even gets more pronounced.

Panel (c) in Fig. S4 shows the tunneling conductance as a function of the
distance to the edge and the $y$-direction parallel to the edge, at different
bulk Landau level energies $E_{N}$.

Figure S5: Landau level spectroscopy toward armchair edge (2). a, Topographic
profile $z(d_{\text{edge}})$ near the graphene edge ($l_{B}=6.85\>$nm at
$B=14\>$T). b, Tunneling conductance as a function of the distance to the edge
and voltage bias. Figure S6: a, Peak energy for LLN as a function of the
distance from the edge, extracted from the LDOS map in Fig. S4. The dashed
lines show the fitted energies using the parameter $v_{\text{F}}^{*}$ obtained
in (c). b, Peak height for the same LLN (same color code). c, Evolution of the
effective Fermi velocity $v_{\text{F}}^{*}$ toward the edge obtained from the
fit of LL-1, LL1, LL2, LL3 and LL4 positions (when visible). d, Carrier
density $n$ computed from the Dirac point position $E_{\text{D}}$ obtained
from the same fit. $n$ presents a residual bulk value of $n_{0}\approx 7\times
10^{9}\>$cm-2 and vanishes from $40\>$nm from the edge.

We now consider in more details the tunneling conductance map shown in Fig.
S4b. We plot in Fig. S6a the evolution of the positions in energy $E_{N}$ of
the visible LLN peaks and in Fig. S6b the variation of their height as a
function of $d_{\text{edge}}$. The amplitude of the peaks decreases as we
approach the edge until peaks merge into a V-shape background at the edge
where they are no longer visible. In particular, LL4 vanishes at $9\,l_{B}$
from the edge, LL3 at $3\,l_{B}$ whereas LL2 and LL±1 disappear at $l_{B}$.
The amplitude of LL1 also vanishes way faster than the other LLN of higher
index $N$. In addition to the peak vanishing at the edge, we can also notice
in Fig. S4b and S6a a weak dispersion toward higher energy of the LLN peaks
close to the edge (on a length of around $\sim 6\,l_{B}$ from the edge), see
Ref. [28].

Furthermore, we can fit the positions $E_{N}$ of LLN≠0 at each
$d_{\text{edge}}$ (for every visible LL at this point) with respect to
equation $E_{N}=E_{\text{D}}+\text{sign}(N)v_{\text{F}}^{*}\sqrt{2\hbar
e|N|B}$ to extract an effective Fermi velocity $v_{\text{F}}^{*}$ and an
estimate of the Dirac point position $E_{\text{D}}$ as a function of
$d_{\text{edge}}$. These results are shown in Figs. S6c for $v_{\text{F}}^{*}$
and S6d for $E_{\text{D}}$, which is converted into charge carrier density $n$
using $n=-\text{sign}(E_{\text{D}})\frac{1}{\pi}[E_{\text{D}}/\hbar
v_{\text{F}}^{*}]^{2}$. The bulk value $v^{*}_{\text{F,bulk}}=1.42\times
10^{6}\>$m.s-1 is consistent with a renormalization of the Fermi velocity due
to the enhancement of electron-electron interactions at charge neutrality [65,
47, 48], as characterized in a previous work [18] for the same sample AC04.
Below $7\,l_{B}$ the effective Fermi velocity starts to increase toward the
armchair edge due to the dispersion of the LL peaks, reaching
$v^{*}_{\text{F,edge}}=1.6\times 10^{6}\>$m.s-1 at $l_{B}$ from the edge. As
for the carrier density, we obtain a residual value $n_{0}\approx 7\times
10^{9}\>$cm-2 in bulk graphene (in agreement with a back-gate voltage tuned at
$\nu=0$). Below $60\>$nm the density is seen to decrease and eventually
vanishes at $40\>\text{nm}=5\,l_{B}$ from the edge. A similar decrease of the
density with respect to its bulk value has also been observed around $l_{B}$
from graphene edge on graphite [28]. Finally, we use the
$v_{\text{F}}^{*}(d_{\text{edge}})$ and $E_{\text{D}}(d_{\text{edge}})$
parameters to plot in Fig. S6a the fitted energies of each LL (black dashed
lines). We notice a good agreement with the experimental points, especially
for the dispersing parts.

###  V Tunneling conductance gate maps at different distances from the edge

We show in this section additional tunneling conductance gate maps (Fig. S7)
used to plot the evolution of the charge-neutrality point
$V_{\text{g}}^{\text{CNP}}$ as a function of the distance from the edge in
Fig. 3c of the main text. $V_{\text{g}}^{\text{CNP}}$ is estimated at the
middle of the $\nu=0$ gap opening when LL0 pins the Fermi level at zero bias.
This gap due to exchange interaction is indeed expected to be maximal at
charge neutrality (i.e. half filling).

In each panel we indicate the opening of the $\nu=0$ gap by yellow dashed
lines, observed as :

  * -

either as a typical gap opening between both LL${}_{0^{\pm}}$ such as in panel
(i,j),

  * -

either as a kink toward negative energies in the LL${}_{0^{-}}$ peak around
zero sample bias, when the LL${}_{0^{+}}$ is not visible, such as in panels
(b-h),

  * -

either as a kink in the other LL peaks or charging peaks when not easily
visible for LL0, such as in panel (a).

Note that the opening of the $\nu=0$ gap at zero sample bias also induces a
shift in energy of other LL peaks (and also of charging peaks), which enables
unambiguous identification of the charge-neutrality point. Still these shifts
in energy do not occur strictly at constant gate voltage due to tip-induced
gating, see for instance the red dashed line in Fig. S7c.

Figure S7: Tunneling conductance gate maps at decreasing distances from the
edge. For every panel, we identify the $\nu=0$ gap that opens in charge-
neutral graphene as a kink in the LL0 peak when it pins the Fermi level at
zero sample bias. The range of gate voltage where the gap opens is indicated
by yellow dashed lines. The charge-neutrality point
$V_{\text{g}}^{\text{CNP}}$ is then assumed to be in the middle of this range,
where the $\nu=0$ gap is maximal, see white arrow. In panels (a-h), only the
LL0- peak is well visible, the LL${}_{0^{+}}$ peak is hindered. In panel (a),
the kink at charge-neutrality is hardly visible for LL0, we rather identify it
in the charging peaks below it.


# Proposed theorems for lifts of the extended almost complex structures on the
complex manifold

Mohammad Nazrul Islam Khan
Department of Computer Engineering, College of Computer
Qassim University, Buraydah, Saudi Arabia
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

It is well known that the tensor field $J$ of type (1,1) on the manifold $M$
is an almost complex structure if $J^{2}=-I,I$ is an identity tensor field and
the manifold $M$ is called the complex manifold. Let ${}^{k}M$ be the $k$
order extended complex manifold of the manifold $M$. A tensor field $J_{k}$ on
${}^{k}M$ is called extended almost complex structure if $(J_{k})^{2}=-I$. The
present paper aims to study the higher order complete and vertical lifts of
the extended almost complex structures on an extended complex manifold
${}^{k}M$. The proposed theorems on the Nijenhuis tensor of an extended almost
complex structure $J_{k}$ on the extended complex manifold ${}^{k}M$ are
proved. Also, a tensor field $\tilde{J_{k}}$ of type (1,1) is introduced and
shows that it is an extended almost complex structure. Finally, the Lie
derivative concerning higher-order lifts is studied and basic results on the
almost analytic complex vector concerning an extended almost complex structure
on ${}^{k}M$ are investigated.

2020 Mathematics Subject Classification: 28A51, 53C15, 32Q60.

Keywords: Complex manifold, Complete lift, Vertical lift, Nijenhuis tensor,
Extended almost complex structure.

##  1 Introduction

The lifting theory has precious vicinity in the differentiable geometry due to
the fact it is feasible to generalize to the geometric structures on any
manifold to extensions through lift function. The geometric structures; for
example, an almost complex structure, an almost product structure, on the base
manifold admit lifts, namely, the complete and vertical lifts, to the
canonical extended manifold. Tekkoyun et al [3] studied the geometric
structures of an extended complex manifold ${}^{k}M$ of $k$-th order of the
complex manifold $M$ and established higher-order vertical and complete lifts
of functions, vector fields, and 1-forms on $M$ to ${}^{k}M$. Tekkoyun and
Civelek [1, 2] investigated higher order complete, vertical and horizontal
lifts of the complex structures on the complex manifold $M$ to the extended
complex manifold ${}^{k}M$. Das and the author [13] have studied almost
$r$-contact structures on the tangent bundle using the complete and vertical
lifts. The geonmtric structures such as almost complex, metallic structure,
almost Hermitian structure, etc. on the manifold to the tangent bundle are
studied by earlier investigators [5, 6, 7, 8, 9, 12, 14, 15].

The major focus of this work can be summarized as follows:

  * •

The proposed theorem proves that the higher order complete lift $F^{c^{k}}$ of
an almost complex structure $F$ of type (1,1) is an extended almost complex
structure on ${}^{k}M$.

  * •

The Nijenhuis tensor of an extended almost complex structure $F^{c^{k}}$ on
${}^{k}M$ is determined.

  * •

A tensor field $\tilde{J_{k}}$ of type (1,1) on ${}^{k}M$ is introduced and
shows that it is an extended almost complex structure on ${}^{k}M$.

  * •

The basic results on the almost analytic complex vector concerning an extended
almost complex structure on ${}^{k}M$ are investigated.

The paper is organized as follows: Section 2 describes a brief account of an
extended almost complex manifold, higher-order lifts of complex function,
vector field, 1-form, and tensor field of type (1,1). In Section 3, higher-
order lifts of the Nijenhuis tensor and integrability conditions are
calculated. In Section 4, a tensor field $\tilde{J_{k}}$ of type (1,1) is
introduced on the extended complex manifold ${}^{k}M$ and shows that it is an
extended almost complex structure on ${}^{k}M$. Finally, the Lie derivative
concerning higher-order lifts is studied and some basic results on the almost
analytic complex vector concerning an extended almost complex structure on
${}^{k}M$ are investigated.

##  2 Preliminaries

Let $M$ be $2m$-real dimensional manifold and ${}^{k}M$ its $k$-th order
extended manifold. A tensor field $J_{k}$ of type (1,1) on ${}^{k}M$ is called
an extended almost complex structure on ${}^{k}M$ if $J_{k}$ is endomorphism
of the tangent space $Tp(^{k}M)$ such that $(J_{k})^{2}=-I$ at every point $p$
of ${}^{k}M$. An extended manifold ${}^{k}M$ with an extended almost complex
structure $J_{k}$ is called an extended almost complex manifold [3].

Suppose $(x^{ri},y^{ri})$ be a system of real coordinate defined at any point
$p$ of ${}^{k}M$ covered by neighborhood ${}^{k}U$ and
$\\{\frac{\partial}{\partial x^{ri}},\frac{\partial}{\partial y^{ri}}\\}$ be
the natural base over the real field $R$ of the tangent space $Tp(^{k}M)$ of
${}^{k}M$. The manifold ${}^{k}M$ is called an extended complex manifold if
${}^{k}M$ is covered by neighborhood ${}^{k}U$ and a local coordinate system
$(x^{ri},y^{ri})$ defined in the neighborhood ${}^{k}U$ such that

$\displaystyle J_{k}(\frac{\partial}{\partial x^{ri}})$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial y^{ri}},$ (2.1) $\displaystyle
J_{k}(\frac{\partial}{\partial y^{ri}})$ $\displaystyle=$
$\displaystyle-\frac{\partial}{\partial x^{ri}}.$ (2.2)

If $k=0,J_{0}$ is called an almost complex structure and a manifold
${}^{0}M=M$ with an almost complex structure $J_{0}$ is said to be the complex
manifold.

Let $z^{ri}=x^{ri}+iy^{ri},i=\sqrt{-1}$ be an extended complex local
coordinate system on a neighborhood ${}^{k}U$ of any point $p$ of ${}^{k}M$.
Then $\frac{\partial}{\partial z^{ri}}=\frac{1}{2}\\{\frac{\partial}{\partial
x^{ri}}-i\frac{\partial}{\partial
y^{ri}}\\},\frac{\partial}{\partial\bar{z}^{ri}}=\frac{1}{2}\\{\frac{\partial}{\partial
x^{ri}}+i\frac{\partial}{\partial y^{ri}}\\}$ and the endomorphism $J_{k}$ is
given by

$J_{k}(\frac{\partial}{\partial z^{ri}})=i\frac{\partial}{\partial
z^{ri}},J_{k}(\frac{\partial}{\partial\bar{z}^{ri}})=-i\frac{\partial}{\partial\bar{z}^{ri}}.$
(2.3)

Let $M$ be any complex manifold and ${}^{k}M$ its $k$-th order extension. If
$f$ is a function on $M$, then the functions $f^{v^{k}}$ and $f^{c^{k}}$
denote the vertical and complete lifts of the function $f$ on ${}^{k}M$,
respectively and given by [3, 4]

$f^{v^{k}}=f\circ\tau M\circ\tau^{2}M\circ...\circ\tau^{k-1}M,$ (2.4)

and

$f^{c^{k}}=\dot{z}^{ri}(\frac{\partial f^{c^{k}}}{\partial
z^{ri}})^{v}+\dot{\bar{z}}^{ri}(\frac{\partial
f^{c^{k}}}{\partial\bar{z}^{ri}})^{v},$ (2.5)

where $\tau^{k-1}M\rightarrow^{k}M^{k-1}M$ is a canonical projection.

The extended properties for the vertical and complete lifts of the complex
functions are given as follows:

$\displaystyle i)~{}~{}(f+g)^{v^{r}}$ $\displaystyle=$ $\displaystyle
f^{v^{r}}+g^{v^{r}},(f.g)^{v^{r}}=f^{v^{r}}.g^{v^{r}}$ $\displaystyle
ii)~{}~{}(f+g)^{c^{r}}$ $\displaystyle=$ $\displaystyle
f^{c^{r}}+g^{c^{r}},(f.g)^{c^{r}}=\sum^{r}_{j=0}C^{r}_{j}f^{c^{r-j}v^{j}}.g^{c^{j}v^{r-j}},$
(2.6)

where $f$ and $g$ are the complex functions and $C^{r}_{j}$ is the
combination.

Let $X$ be a complex vector field with the local expression
$X=Z^{0i}\frac{\partial}{\partial
z^{ki}}+\bar{Z}^{0i}\frac{\partial}{\partial\bar{z}^{ki}}$. Then the local
expression of the vertical and complete lifts of $X$ to ${}^{k}M$ are,
respectively, given by

$X^{v^{k}}=(Z^{0i})^{v^{k}}\frac{\partial}{\partial
z^{ki}}+(\bar{Z}^{0i})^{v^{k}}\frac{\partial}{\partial\bar{z}^{ki}}.$ (2.7)

and

$X^{c^{k}}=C^{r}_{j}(Z^{0i})^{v^{k-r}c^{r}}\frac{\partial}{\partial
z^{ri}}+C^{r}_{j}(\bar{Z}^{0i})^{v^{k-r}c^{r}}\frac{\partial}{\partial\bar{z}^{ri}}.$
(2.8)

The extended properties for the vertical and complete lifts of the complex
vector fields are given as follows:

$\displaystyle i)~{}~{}(X+Y)^{v^{r}}$ $\displaystyle=$ $\displaystyle
X^{v^{r}}+Y^{v^{r}},(X+Y)^{c^{r}}=X^{c^{r}}+Y^{c^{r}},$ $\displaystyle
ii)~{}~{}X^{v^{k}}(f^{c^{k}})$ $\displaystyle=$
$\displaystyle(Xf)^{v^{k}},~{}~{}~{}X^{c^{k}}(f^{c^{k}})=(Xf)^{c^{k}},$
$\displaystyle iii)~{}~{}~{}~{}(fX)^{v^{r}}$ $\displaystyle=$ $\displaystyle
f^{v^{r}}.X^{v^{r}},(fX)^{c^{r}}=\sum^{r}_{j=0}C^{r}_{j}f^{c^{r-j}v^{j}}.X^{c^{j}v^{r-j}},$
(2.9) $\displaystyle iv)~{}~{}X^{v^{r}}(f^{v^{r}})$ $\displaystyle=$
$\displaystyle
0,~{}~{}X^{c^{r}}(f^{c^{r}})=(Xf)^{c^{r}},~{}X^{c^{r}}(f^{v^{r}})=X^{v^{r}}(f^{c^{r}})=(Xf)^{v^{r}},$
$\displaystyle v)~{}~{}[X^{v^{r}},Y^{v^{r}}]$ $\displaystyle=$ $\displaystyle
0,[X^{c^{r}},Y^{c^{r}}]=[X,Y]^{c^{r}},[X^{v^{r}},Y^{c^{r}}]=[X^{c^{r}},Y^{v^{r}}]=[X,Y]^{v^{r}},$

where $X,Y$ are the complex vector fields and $f$ is the complex function.

Let $\alpha$ be a complex 1-form with the local expression
$\alpha=\alpha_{0i}dz^{0i}+\bar{\alpha}_{0i}\bar{dz}^{0i}$. Then the local
expression of the vertical and complete lifts of $\alpha$ to ${}^{k}M$ are,
respectively, given by

$\alpha^{v^{k}}=(\alpha_{0i})^{v^{k}}dz^{0i}+(\bar{\alpha}_{0i})^{v^{k}}\bar{dz}^{0i},$
(2.10)

and

$\alpha^{c^{k}}=(\alpha_{0i})^{c^{k-r}v^{r}}dz^{ri}+(\bar{\alpha}_{0i})^{c^{k-r}v^{r}}d\bar{z}^{ri}.$
(2.11)

The extended properties for the vertical and the complete lifts of complex
1-forms are given as follows [3]:

$\displaystyle i)(\alpha+\lambda)^{v^{r}}$ $\displaystyle=$
$\displaystyle\alpha^{v^{r}}+\lambda^{v^{r}},(\alpha+\lambda)^{c^{r}}=\alpha^{c^{r}}+\lambda^{c^{r}},$
$\displaystyle ii)~{}~{}~{}(f\alpha)^{v^{r}}$ $\displaystyle=$ $\displaystyle
f^{v^{r}}\alpha^{v^{r}},(f\alpha)^{c^{r}}=\sum^{r}_{j=0}C^{r}_{j}f^{c^{r-j}v^{j}}\alpha^{c^{r}v^{r-j}}.$
(2.12)

Let $M$ be any complex manifold and ${}^{k}M$ its $k$-th order extension. If
$F$ be a tensor field of type (1,1). Then

$\displaystyle\alpha^{c^{k}}(F^{c^{k}})$ $\displaystyle=$
$\displaystyle(\alpha F)^{c^{k}},~{}~{}\alpha^{v^{k}}(F^{v^{k}})=(\alpha
F)^{v^{k}},$ $\displaystyle F^{c^{k}}(X^{c^{k}})$ $\displaystyle=$
$\displaystyle(FX)^{c^{k}},~{}~{}F^{v^{k}}(X^{c^{k}})=(FX)^{v^{k}},$ (2.13)

where $X$ and $\alpha$ are a vector field and a 1-form respectively.

##  3 Some calculations on the Nijenhuis tensor on the extended complex
manifold

In this section, the basic properties of the tensor field of type (1,1) on an
extended complex manifold ${}^{k}M$ are studied. The proposed theorem proves
that the higher order complete lift $F^{c^{k}}$ of a tensor field $F$ of type
(1,1) is an extended almost complex structure to an extended complex manifold
${}^{k}M$ if $F$ is an almost complex structure on $M$. The Nijenhuis tensor
of an extended almost complex structure $F^{c^{k}}$ on the extended complex
manifold ${}^{k}M$ is determined.

Let $F$ and $G$ be the complex tensor fields of type (1,1) and consider them
as fields of linear endomorphisms of tangent spaces of complex manifold $M$.
Let the eteration of the endomorphisms $F$ and $G$ be denoted by $FG$ i.e.
$(FG)X=F(GX)$, where $X$ being an arbitrary complex vector field [18].

###### Theorem 3.1

Let $M$ be any complex manifold and ${}^{k}M$ its $k$-th order extension. If
$F$ and $G$ be the complex tensor fields of type (1,1). Then

$\displaystyle(a)~{}~{}~{}(FG)^{c^{k}}=F^{{c^{k}}G^{c^{k}}},$
$\displaystyle(b)~{}~{}~{}~{}~{}I^{c^{k}}=I,$ (3.1)

where $I$ is an identity tensor field.

Proof. (i) Let $M$ be any complex manifold and let $F$ and $G$ be complex
tensor fields of type (1,1). Then

$\displaystyle(FG)^{c^{k}}X^{c^{k}}$ $\displaystyle=$
$\displaystyle((FG)X)^{c^{k}}=(F(GX))^{c^{k}}=F^{c^{k}}(GX)^{c^{k}}$
$\displaystyle=$ $\displaystyle
F^{c^{k}}G^{c^{k}}X^{c^{k}}=(F^{c^{k}}G^{c^{k}})X^{c^{k}}$
$\displaystyle\Rightarrow(FG)^{c^{k}}$ $\displaystyle=$ $\displaystyle
F^{c^{k}}G^{c^{k}},$ (3.2)

for an arbitrary complex vector field $X$.

Second part of the theorem is proved by using similar devices.

Hence the theorem is proved.

Definition 1 Let $M$ be any complex manifold and ${}^{k}M$ its $k$-th order
extension. If the tensor field $F$ of type (1,1) on $M$ satisfies the equation

$F^{2}+I=0,$ (3.3)

where $I$ is an identity tensor field. Then $F$ is called an almost complex
structure on $M$ [17].

###### Theorem 3.2

Let $F$ be a tensor field of type (1,1) on $M$. Then $F^{c^{k}}$ defines an
extended almost complex structure to ${}^{k}M$ if and only if $F$ defines an
almost complex structure on $M$.

Proof. Let $F$ be an almost complex structure on $M$, then

$F^{2}+I=0.$

Taking the complete lifts on both sides the above equation to ${}^{k}M$, then

$\displaystyle(F^{2}+I)^{c^{k}}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle(F^{2})^{c^{k}}+I$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle(F^{2})^{c^{k}}$ $\displaystyle=$
$\displaystyle-I\Rightarrow(F^{c^{k}})^{2}=-I.$

Therefore, $F^{c^{k}}$ defines an extended almost complex structure to
${}^{k}M$.

Hence the theorem is proved.

The Nijenhuis tensor $N_{F}$ of $F$ is a tensor field of type (1,2) defined by
[11, 18]

$N_{F}(X,Y)=[FX,FY]+F^{2}[X,Y]-F[FX,Y]-F[X,FY],$ (3.4)

where $X$ and $Y$ are the complex vector fields in $M$.

###### Theorem 3.3

Let $M$ be any complex manifold and ${}^{k}M$ its $k$-th order extension. Then

$(N_{F})^{c^{k}}=N_{F^{c^{k}}},$

where $X,Y$ are arbitrary complex vector fields.

Proof. Taking lifts of both sides of equation (3.4) and using equation (2),
then

$\displaystyle(N_{F})^{c^{k}}$ $\displaystyle=$
$\displaystyle([FX,FY]+F^{2}[X,Y]-F[FX,Y]-F[X,FY])^{c^{k}},$ $\displaystyle=$
$\displaystyle[F^{c^{k}}X^{c^{k}},F^{c^{k}}Y^{c^{k}}]+(F^{c^{k}})^{2}[X^{c^{k}},Y^{c^{k}}]-F^{c^{k}}[F^{c^{k}}X^{c^{k}},Y^{c^{k}}]-F^{c^{k}}[X^{c^{k}},F^{c^{k}}Y^{c^{k}}],$
$\displaystyle=$ $\displaystyle N_{F^{c^{k}}}.$

Hence the theorem is proved.

###### Theorem 3.4

Let $F$ be a tensor field of type (1,1) on $M$. Then an extended almost
complex structure $F^{c^{k}}$ is integrable to ${}^{k}M$ if and only if an
almost complex structure $F$ is integrable on $M$.

Proof. Let $N_{F^{c^{k}}}$ and $N_{F}$ be the Nijenhuis tensors of $F^{c^{k}}$
and $F$, respectively. Then

$N_{F^{c^{k}}}=(N_{F})^{c^{k}},$ (3.5)

since $F$ is integrable, then

$N_{F}=0.$

From equation (3.5), the obtained equation is

$N_{F^{c^{k}}}=0.$

Thus, $F^{c^{k}}$ is integrable to ${}^{k}M$ if and only if $F$ is integrable
on $M$.

Hence the theorem is proved.

##  4 Extended almost complex structures

In this section, a tensor field $\tilde{J_{k}}$ of type (1,1) on ${}^{k}M$ is
introduced and shows that it is an extended almost complex structure on
${}^{k}M$.

Suppose that there is given, on a complex manifold $M$, a tensor field $F$ of
type (1,1), a vector field $U$ and a 1-form $\omega$ satisfying

$\displaystyle F^{2}$ $\displaystyle=$
$\displaystyle-I+U\otimes\omega,~{}~{}~{}FU=0,$ $\displaystyle\omega\circ F$
$\displaystyle=$ $\displaystyle 0,~{}~{}~{}\omega(U)=0.$ (4.1)

The structure $(F,U,\omega)$ is said to be an almost contact structure on $M$
[18].

###### Theorem 4.1

Let $M$ be any complex manifold and ${}^{k}M$ its $k$-th order extension
endowed with an almost contact structure $(F,U,\omega)$, then

$\tilde{J_{k}}=F^{c^{k}}+U^{v^{k}}\otimes\omega^{v^{k}}-U^{c^{k}}\otimes\omega^{c^{k}}$
(4.2)

is an extended almost complex structure on ${}^{k}M$.

Proof. Taking complete lifts on both sides of equation (4) to ${}^{k}M$, then
[10]

$\displaystyle(F^{c^{k}})^{2}$ $\displaystyle=$
$\displaystyle-I+U^{v^{k}}\otimes\omega^{c^{k}}+U^{c^{k}}\otimes\omega^{v^{k}},$
$\displaystyle F^{c^{k}}U^{v^{k}}$ $\displaystyle=$ $\displaystyle
0,~{}~{}F^{c^{k}}U^{c^{k}}=0,~{}~{}\omega^{v^{k}}\circ
F^{c^{k}}=0,~{}~{}\omega^{c^{k}}\circ F^{c^{k}}=0,$ (4.3)
$\displaystyle\omega^{v^{k}}(U^{v^{k}})$ $\displaystyle=$ $\displaystyle
0,~{}~{}\omega^{c^{k}}(U^{c^{k}})=0,~{}~{}\omega^{c^{k}}(U^{v^{k}})=1,~{}~{}\omega^{v^{k}}(U^{v^{k}})=0.$

Let us introduce a tensor field $\tilde{J_{k}}$ of type (1,1) on ${}^{k}M$ as

$\tilde{J_{k}}=F^{c^{k}}+U^{v^{k}}\otimes\omega^{v^{k}}-U^{c^{k}}\otimes\omega^{c^{k}}.$
(4.4)

Using equations (4) and (4.4), the obtained equation is

$\tilde{J_{k}}^{2}=-I.$

This shows that $\tilde{J_{k}}$ is an extended almost complex structure on
${}^{k}M$.

###### Corollary 4.1

Let $M$ be a complex manifold and $\tilde{J_{k}}$ defined in equation (4.4) is
an extended almost complex structure on ${}^{k}M$, then

$\displaystyle\tilde{J_{k}}X^{v^{k}}$ $\displaystyle=$
$\displaystyle(FX)^{v^{k}}-(\omega(X))^{v^{k}}U^{c^{k}},$
$\displaystyle\tilde{J_{k}}X^{c^{k}}$ $\displaystyle=$
$\displaystyle(FX)^{c^{k}}+(\omega(X))^{v^{k}}U^{v^{k}}-(\omega(X))^{v^{k}}U^{c^{k}},$
(4.5)

where $X$ is a complex vector field and $\omega$ is a 1-form.

###### Corollary 4.2

Let $M$ be a complex manifold and $\tilde{J_{k}}$ is an extended almost
complex structure on ${}^{k}M$ such that $\omega(X)=0$, then

$\displaystyle\tilde{J_{k}}X^{v^{k}}$ $\displaystyle=$
$\displaystyle(FX)^{v^{k}},~{}~{}~{}\tilde{J_{k}}X^{c^{k}}=(FX)^{c^{k}},$
$\displaystyle\tilde{J_{k}}U^{v^{k}}$ $\displaystyle=$
$\displaystyle-U^{c^{k}},~{}~{}~{}\tilde{J_{k}}U^{c^{k}}=-U^{v^{k}},$ (4.6)

where $X$ is a complex vector field and $\omega$ is a 1-form.

##  5 Lie derivative with respect to higher order lifts on the extended
complex manifold

In this section, a study is done on the Lie derivative with respect to a
complex vector field on ${}^{k}M$. The basic results on the almost analytic
complex vector concerning an extended almost complex structure on ${}^{k}M$
are found.

Let $M$ be any complex manifold and $X$ be a complex vector field. The
differential transformation $\pounds_{X}$ is called Lie derivative with
respect to $X$ if

$\pounds_{X}f=Xf,~{}~{}~{}\pounds_{X}Y=[X,Y],$ (5.1)

where $f$ is complex function on $M$.

The Lie derivative $\pounds_{X}F$ of a tensor field $F$ of type (1, 1) on $M$
with respect to a complex vector field $X$ is defined by [18]

$(\pounds_{X}F)Y=[X,FY]-F[X,Y],$ (5.2)

where $[,]$ is Lie bracket.

###### Theorem 5.1

Let $M$ be the complex manifold and ${}^{k}M$ its $k$-th order extension. The
Lie derivation $\pounds_{X}$ with respect to a complex vector field $X$, then

(a)
$\displaystyle~{}~{}~{}~{}\pounds_{X^{v^{r}}}f^{v^{r}}=0,~{}~{}\pounds_{X^{v^{r}}}f^{c^{r}}=(\pounds_{X}f)^{v^{r}},$
(b)
$\displaystyle~{}~{}~{}~{}\pounds_{X^{c^{r}}}f^{v^{r}}=(\pounds_{X}f)^{v^{r}},~{}~{}\pounds_{X^{c^{r}}}f^{c^{r}}=(\pounds_{X}f)^{c^{r}},$
(5.3) (c)
$\displaystyle~{}~{}~{}~{}\pounds_{X^{v^{r}}}Y^{v^{r}}=0,~{}~{}\pounds_{X^{v^{r}}}Y^{C}=(\pounds_{X}Y)^{v^{r}},$
(d)
$\displaystyle~{}~{}~{}~{}\pounds_{X^{c^{r}}}Y^{v^{r}}=(\pounds_{X}Y)^{v^{r}},~{}~{}\pounds_{X^{c^{r}}}Y^{c^{r}}=(\pounds_{X}Y)^{c^{r}},$

where $f$ is a complex function in $M$.

Proof. The proof follows in an obvious manner on using equation (2).

###### Theorem 5.2

Let $M$ be the complex manifold and ${}^{k}M$ its $k$-th order extension. Let
an extended almost complex structure $\tilde{J}_{k}$ defined by equation (4.4)
and $\omega(Y)=0$ on ${}^{k}M$. The Lie derivation $\pounds_{X}$ with respect
to a complex vector field $X$, then

(a) $\displaystyle~{}~{}~{}~{}(\pounds_{X^{v^{r}}}\tilde{J_{k}})Y^{v^{r}}=0,$
(b)
$\displaystyle~{}~{}~{}~{}(\pounds_{X^{v^{r}}}\tilde{J_{k}})Y^{c^{r}}=\left((\pounds_{X}F)Y\right)^{v^{r}}-((\pounds_{X}\omega)Y)^{v^{r}}U^{c^{r}},$
(c)
$\displaystyle~{}~{}~{}~{}(\pounds_{X^{c^{r}}}\tilde{J_{k}})Y^{v^{r}}=\left((\pounds_{X}F)Y\right)^{v^{r}}-((\pounds_{X}\omega)Y)^{v^{r}}U^{c^{r},}$
(5.4) (d)
$\displaystyle~{}~{}~{}~{}(\pounds_{X^{c^{r}}}\tilde{J_{k}})Y^{c^{r}}=\left((\pounds_{X}F)Y\right)^{c^{r}}+((\pounds_{X}\omega)(Y))^{v^{r}}U^{v^{r}}-((\pounds_{X}\omega)(Y))^{c^{r}}U^{c^{r}},$

where $F$ is a tensor field of type (1,1).

Proof. Let $M$ be any complex manifold and $\omega(Y)=0$. Then

$\displaystyle\mbox{(5.5a)}~{}~{}(\pounds_{X^{v^{r}}}\tilde{J_{k}})Y^{v^{r}}$
$\displaystyle=$
$\displaystyle\pounds_{X^{v^{r}}}\left[F^{c^{k}}+U^{v^{k}}\otimes\omega^{v^{k}}-U^{c^{k}}\otimes\omega^{c^{k}}\right]Y^{v^{k}},$
$\displaystyle-$
$\displaystyle\left[F^{c^{k}}+U^{v^{k}}\otimes\omega^{v^{k}}-U^{c^{k}}\otimes\omega^{c^{k}}\right]\pounds_{X^{v^{r}}}Y^{v^{k}}$
$\displaystyle=$
$\displaystyle\pounds_{X^{v^{r}}}F^{c^{k}}Y^{v^{k}}+\pounds_{X^{v^{r}}}(U^{v^{k}}\otimes\omega^{v^{k}})Y^{v^{k}}-\pounds_{X^{v^{r}}}(U^{c^{k}}\otimes\omega^{c^{k}})Y^{v^{k}},$
$\displaystyle=$
$\displaystyle\pounds_{X^{v^{r}}}(FY)^{v^{k}}+\pounds_{X^{v^{r}}}(\otimes\omega(Y))^{v^{k}}U^{v^{k}}-\pounds_{X^{v^{r}}}(\otimes\omega(Y))^{v^{k}}U^{c^{k}},$
$\displaystyle=$ $\displaystyle
0,~{}~{}~{}as~{}\omega(Y)=0,~{}as\pounds_{X^{v^{r}}}Y^{v^{k}}=0.$
$\displaystyle\mbox{(5.5b)}~{}~{}(\pounds_{X^{v^{r}}}\tilde{J_{k}})Y^{c^{r}}$
$\displaystyle=$
$\displaystyle\pounds_{X^{v^{r}}}\left[F^{c^{k}}+U^{v^{k}}\otimes\omega^{v^{k}}-U^{c^{k}}\otimes\omega^{c^{k}}\right]Y^{c^{k}}$
$\displaystyle-$
$\displaystyle\left[F^{c^{k}}+U^{v^{k}}\otimes\omega^{v^{k}}-U^{c^{k}}\otimes\omega^{c^{k}}\right]\pounds_{X^{v^{r}}}Y^{c^{k}},$
$\displaystyle=$
$\displaystyle\pounds_{X^{v^{r}}}F^{c^{k}}Y^{c^{k}}+\pounds_{X^{v^{r}}}(\otimes\omega^{v^{k}}(Y)^{c^{k}})U^{v^{k}}-\pounds_{X^{v^{r}}}(\otimes\omega^{c^{k}}(Y)^{c^{k}})U^{c^{k}},$
$\displaystyle-$ $\displaystyle
F^{c^{k}}\pounds_{X^{v^{r}}}Y^{c^{k}}-\omega^{v^{k}}(\pounds_{X^{v^{r}}}Y^{c^{k}})U^{v^{k}}+\omega^{c^{k}}(\pounds_{X^{v^{r}}}Y^{c^{k}})U^{c^{k}},$
$\displaystyle=$
$\displaystyle((\pounds_{X}F)Y)^{v^{r}}-((\pounds_{X}\omega)Y)^{v^{k}}U^{c}.$

The other formulas the theorem are proved by using similar devices.

###### Corollary 5.1

If we put $Y=U$ , an extended almost complex structure $\tilde{J}_{k}$ defined
by equation (4.4) and $\omega(Y)=0$, we have

(a)
$\displaystyle~{}~{}~{}~{}(\pounds_{X^{v^{r}}}\tilde{J_{k}})U^{v^{r}}=-(\pounds_{X}U)^{v^{r}},$
(b)
$\displaystyle~{}~{}~{}~{}(\pounds_{X^{v^{r}}}\tilde{J_{k}})U^{c^{r}}=((\pounds_{X}F)U)^{v^{r}}-((\pounds_{X}\omega)(U))^{v^{r}}U^{c^{r}},$
(c)
$\displaystyle~{}~{}~{}~{}(\pounds_{X^{c^{r}}}\tilde{J_{k}})U^{v^{r}}=((\pounds_{X}F)U)^{c^{r}}-[X,U]^{c^{r}}-((\pounds_{X}\omega)(U))^{c^{r}}U^{c^{r}},$
(5.5) (d)
$\displaystyle~{}~{}~{}~{}(\pounds_{X^{c^{r}}}\tilde{J_{k}})U^{c^{r}}=((\pounds_{X}F)U)^{c^{r}}+[X,U]^{v^{r}}+((\pounds_{X}\omega)(U))^{v^{r}}U^{v^{r}}-((\pounds_{X}\omega)(U))^{c^{r}}U^{c^{r}},$

where $F$ is a tensor field of type (1,1).

###### Theorem 5.3

Let $M$ be any complex manifold and ${}^{k}M$ its $k$-th order extension
endowed with an almost contact structure $(F,U,\omega)$. Then the vertical
lift $X^{v^{r}}$ of a complex vector field $X$ given in $M$ is an almost
analytic with respect to the extended almost complex structure $\tilde{J_{k}}$
defined by equation (4.4) on ${}^{k}M$ if and only if the conditions

$\pounds_{X}F=0,~{}~{}~{}~{}\pounds_{X}U=0,~{}~{}~{}~{}\pounds_{X}\omega=0$

are satisfied on $M$.

Proof. As a consequence of equations (5.2a), (5.2b), (5.1a) and (5.1b), it
follows that the condition

$\pounds_{X^{v^{r}}}J_{k}=0$

is equivalent to the conditions

$\pounds_{X}F=0,~{}~{}~{}~{}\pounds_{X}U=0,~{}~{}~{}~{}\pounds_{X}\omega=0.$

###### Theorem 5.4

Let $M$ be any complex manifold and ${}^{k}M$ its $k$-th order extension
endowed with an almost contact structure $(F,U,\omega)$. Then the vertical
lift $X^{c^{r}}$ of a complex vector field $X$ given in $M$ is almost analytic
with respect to the an extended almost complex structure $\tilde{J_{k}}$
defined by equation (4.4) on ${}^{k}M$ if and only if the conditions

$\pounds_{X}F=0,~{}~{}~{}~{}\pounds_{X}U=CU,~{}~{}~{}~{}\pounds_{X}\omega=-C\omega,$

$C$ being a non-zero constant, are satisfied on $M$.

Proof. As a consequence of equations (5.2c), (5.2d), (5.1c) and (5.1d), it
follows that the condition

$\pounds_{X^{c^{r}}}J_{k}=0$

is equivalent to the conditions

$\pounds_{X}F=0,~{}~{}~{}~{}\pounds_{X}U=CU,~{}~{}~{}~{}\pounds_{X}\omega=-C\omega.$

## Conflict of Interests

The authors declare that there is no conflict of interests.

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