ChiralTopo-SI

# Topological phase transition in chiral graphene nanoribbons:
from edge bands to end states

Jingcheng Li∗ CIC nanoGUNE-BRTA, 20018 Donostia-San Sebastián, Spain  Sofia
Sanz∗ Donostia International Physics Center (DIPC), 20018 Donostia-San
Sebastián, Spain  Nestor Merino-Díez∗ CIC nanoGUNE-BRTA, 20018 Donostia-San
Sebastián, Spain  Donostia International Physics Center (DIPC), 20018
Donostia-San Sebastián, Spain  Centro de Física de Materiales CSIC-UPV/EHU,
20018 Donostia-San Sebastián, Spain  Manuel Vilas-Varela∗ Centro Singular de
Investigación en Química Biolóxica e Materiais Moleculares (CiQUS), and
Departamento de Química Orgánica, Universidade de Santiago de Compostela,
Spain  Aran Garcia-Lekue  Donostia International Physics Center (DIPC), 20018
Donostia-San Sebastián, Spain  Ikerbasque, Basque Foundation for Science,
48013 Bilbao, Spain  Martina Corso  Centro de Física de Materiales CSIC-
UPV/EHU, 20018 Donostia-San Sebastián, Spain  Dimas G. de Oteyza △ Donostia
International Physics Center (DIPC), 20018 Donostia-San Sebastián, Spain
Ikerbasque, Basque Foundation for Science, 48013 Bilbao, Spain  Thomas
Frederiksen△ Donostia International Physics Center (DIPC), 20018 Donostia-San
Sebastián, Spain  Ikerbasque, Basque Foundation for Science, 48013 Bilbao,
Spain  Diego Peña△ Centro Singular de Investigación en Química Biolóxica e
Materiais Moleculares (CiQUS), and Departamento de Química Orgánica,
Universidade de Santiago de Compostela, Spain  Jose Ignacio Pascual△ CIC
nanoGUNE-BRTA, 20018 Donostia-San Sebastián, Spain  Ikerbasque, Basque
Foundation for Science, 48013 Bilbao, Spain

###### Abstract

Precise control over the size and shape of graphene nanostructures allows
engineering spin-polarized edge and topological states, representing a novel
source of non-conventional $\pi$-magnetism with promising applications in
quantum spintronics. A prerequisite for their emergence is the existence of
robust gapped phases, which are difficult to find in extended graphene
systems: only armchair graphene nanoribbons (GNRs) show a band gap that,
however, closes for any other GNR orientation. Here we show that semi-metallic
chiral GNRs (chGNRs) narrowed down to nanometer widths undergoes a topological
phase transition, becoming first topological insulators, and transforming then
into trivial band insulators for the narrowest chGNRs. We fabricated
atomically precise chGNRs of different chirality and size by on surface
synthesis using predesigned molecular precursors. Combining scanning
tunnelling microscopy (STM) measurements and theory simulations, we follow the
evolution of topological properties and bulk band gap depending on the width,
length, and chirality of chGNRs. The first emerging gapped phases are
topological, protected by a chiral interaction pattern between edges. For
narrower ribbons, the symmetry of the interaction pattern changes, and the
topological gap closes and re-opens again as a trivial band insulator. Our
findings represent a new platform for producing topologically protected spin
states and demonstrates the potential of connecting chiral edge and defect
structure with band engineering.

$*$ These authors contributed equally to this work.

Band topological classification of materials has been successfully applied to
predict and explain the emergence of exotic states of matter such as Quantum
Spin Hall (QSH) edge states in topological insulators Hasan and Kane (2010);
Qi and Zhang (2011); Kane and Mele (2005) or topological superconductivity
Sato and Ando (2017). The potential of this classification relies on the
protection of the topological order by a symmetry, that can undergo a
topological phase transition when the symmetry is changed. Symmetry Protected
Topological (SPT) phase transitions were observed in artificial semiconducting
systems such as two-dimensional quantum wells of HgTeKönig  _et al._ (2007)
and one-dimensional organic polymers Cirera  _et al._ (2020). The key element
is the existence of two gapped SPT phases, the traditional (trivial) band
insulator and the non-trivial topological insulating state, separated by a
metallic state.

In spite of being a semimetal, graphene has the potential to build up SPT
phases by opening an energy gap around the Fermi level and endowing the
lattice with an additional chiral symmetric interaction Haldane (1988). For
example, one-dimensional SPT phases were engineered inside the band-gap of
armchair GNRs by modelling a one-dimensional Su-Schrieffer-Heeger (SSH) chain
Su  _et al._ (1979) with edge moieties containing localized in-gap states Cao
_et al._ (2017); Gröning  _et al._ (2018); Rizzo  _et al._ (2018); Sun  _et
al._ (2020); Rizzo  _et al._ (2020). In zigzag GNRs (ZGNRs), however, the
existence of zero-energy edge bands Nakada  _et al._ (1996); Fujita  _et al._
(1996) brings the system into a metallic state that prevents the appearance of
gaped topological phases. As proposed by Kane and Mele Kane and Mele (2005),
the presence of spin-orbit interaction can open a gap in bulk graphene and
turn the zero-energy modes into QSH edge states. However, the expected gap
induced by spin orbit interaction in graphene is very small, and this effect
could only be present at very low temperatures Sichau  _et al._ (2019).

Here, we demonstrate that sizeable topological insulating phases emerge in
narrow chiral GNRs driven by the interaction between the opposing edges. The
term chiral GNRs (chGNRs) refers to the large set of ribbons extending along
low-symmetry crystallographic directions ($n,m$) of graphene. The genuine
zero-energy edge bands of ZGNRs persist in chGNRs via the accumulation of
states at zero energy over zigzag sites, including their spin polarization in
the presence of Coulomb electron-electron interactions Yazyev  _et al._
(2011); Tao  _et al._ (2011). However, chiral ribbons are more sensitive to a
reduced width than ZGNRs, allowing to easily produce gapped GNRs with
inherited chirality from the edge reconstruction.

Figure 1: Edge states in zigzag and chiral GNRs:. a Edge structure of wide
zigzag GNRs, and chiral GNRs with chiral vectors (3,1) and (2,3).
Superimposed, tight-binding (TB) wave-function amplitude and spin polarization
of their zero energy modes. The blue-shaded area indicates the rectangular
graphene building block used for constructing a family of chiral GNRs, with
edge structure alternating z zigzag and $a$-1/2 armchair segments. For the
both chiral ribbons $z=3$, while $a=1$ and 2, respectively (details in the
text and in Supplementary Note LABEL:SN:STRUCT). b Band structure of 40-carbon
wide ZGNR, and (3,1) and (2,3) chiral GNRs from TB simulations. The slight
dispersion of the flat bands is due to the inclusion of second-nearest
neighbour interactions in our model LABEL:SN:TBWF. c Band gap values of (3,1)
and (2,3) chiral GNRs (purple diamond and blue squared symbols), as a function
of the width (grey circles indicate the zero gap of ZGNRs). The value of the
computed topological invariant $\mathbb{Z}_{2}$ is indicated. A topological
transition occurs for ribbons at the width $w=6$.

Prediction of a SPT phase transition: We consider a family of chGNRs
customized from a basic rectangular aromatic block of length z (number of
zigzag unit cells) and width $w$ (number of carbon atoms across), blue-
shadowed in Fig. 1a. Chiral GNRs along any chiral direction ($n,m$) can be
obtained simply by repeating and shifting these blocks along their armchair
edges an amount of $a-1/2$ armchair unit cells, and connecting them with C-C
bonds. The edges of the resulting chGNR alternate z zigzag and a-1/2 armchair
sites to accommodate its orientation to the chiral vector, such that
$(n,m)=(z+1-a,2a-1)$ (see Supplementary Note LABEL:SN:STRUCT). This edge
reconstruction promotes the localization of zero-energy edge states at the $z$
zigzag segments, while the perpendicular armchair spacers act as potential
barriers between them Santos  _et al._ (2013). For example, our tight-binding
(TB) simulations in Fig. 1a,b for wide chGNRs (theory methods in Supplementary
Part LABEL:SN:TBwide) show the presence of zero-energy bands localized at the
zigzag edge segments, reminiscent of the edge bands in the zigzag edges of
graphene, which decay towards the center of the ribbon. Therefore, these
ribbons lie in a metallic phase, with electron mobility that can be described
as hopping between zigzag segments along the chGNR edge.

However, this metallic phase vanishes in narrow ribbons due to hybridization
of bands at opposing edges. Interestingly, the emergent gapped phase
corresponds to a SPT insulating phase characterized by an invariant
$\mathbb{Z}_{2}=1$, as obtained from the computed Zak phase $\gamma_{z}=\pi$
Zak (1989) of the occupied bands (using a rectangular unit cell enclosing the
blue block in Fig. 1a). The non-trivial topological class of this phase turns
out to be a global property of narrow chGNRs of this family, protected by the
symmetry of a chiral hybridization pattern between edges.

Our simulations also find that the topological phase vanishes upon further
reducing the width of the ribbon. The gap closes and reopens again as a
trivial band insulator, characterized by $\mathbb{Z}_{2}=0$. In agreement with
the properties of SPT phases, this new trivial state is connected with a
symmetry change in the interaction pattern between the edges. Figure 1c
exemplifies the transition for ribbons with ($n=3$,$m=1$) and (2,3) chiral
vectors, by plotting their evolution of the energy gap with the width $w$.
First, a sizable energy gap opens up as the ribbons are narrowed down, which
corresponds to a one-dimensional topological insulating phase, and at a
critical width of $w\sim 6$, the gap closes, and reopens, now in a trivial
phase.

Figure 2: Synthetic strategy to produce chGNRs  combining solution and on-
surface synthesis. a-d, solution synthesis protocols for producing molecular
precursors 1, 2, 3, for the synthesis of 3,1,$w$-chGNRs with different widths
($w=4$, 6 and 8), and precursor 4 for 3,2,8-chGNRs. Synthesis details are
described in the text and in the Methods section. e-h, chemical structures of
chGNRs after Ullmann coupling and CDH of the four molecular precursors in a-d,
respectively. Red(blue) carbon bonds highlight the zigzag(armchair) segments
of the chGNRs. Green circles highlight the carbon atoms across the GNRs, which
measure the width of the GNRs. The GNRs are named following the sequence of
$z$ zigzag sites, a brominated site (see LABEL:SN:STRUCT), and $w$ width, as
$z,a,w$-chGNRs. i-l, STM overview images of the formed chGNRs on a Au(111)
surface after Ullmann coupling and CDH of the corresponding molecular
precursors 1-4 (sample bias $V_{s}=1$ V, scale bar as labeled in l).

Fabrication of chGNRs: To demonstrate the predicted topological phase
transition described above, we fabricated several members of this chGNR family
with different chiral vectors and widths (Fig. 2) though a combination of
customized organic precursors and on-surface synthesis over a gold (111)
surface Talirz  _et al._ (2016); Clair and de Oteyza (2019). Our strategy
started with the synthesized poli-[n’]anthracene precursor molecules 1, 2, 3,
and 4 shown in Fig. 2a-d for the synthesis of chGNRs with $z=3$ edge
structure. Their customized structures composed of an increasing number of
anthacene units and Br functionalization sites a, were designed for obtaining
(3,1) chGNRs with different widths (using 1, 2, 3) and chiral angle (e.g.
(2,3) chGNR, using 4) through a sequence of OSS steps. For a fixed $z$, the
Br-substitution site, labelled a in LABEL:SN:STRUCT, determines the ($n,m$)
chiral vector by steering the Ullmann-like connection between molecular
precursors with a shift of $a-1/2$ armchair unit lengths (see Fig. 1a). The
number of anthracene units [n’] determines the width $w=2$[n’] of the ribbon.
Hence, in the following we label the ribbons of this family as z,a,w-chGNR
(LABEL:SN:STRUCT).

The GNR precursors were prepared in solution, as shown in Fig. 2a-d. Compound
1, which is formed by the linking of two anthracenes and constitutes the
molecular precursor of 3,1,4-chGNR, was obtained by Zn-promoted reductive
coupling of bromoanthrone 5, followed by dehydration De Oteyza  _et al._
(2016). The trisanthracene 2, precursor of 3,1,6-chGNRs, was obtained in one
pot from dibromoanthraquinone 6, by addition of two equivalents of
9-anthracenyl lithium (7) followed by reduction with a mixture of HI and
H3PO2. Similarly, the tetrakisanthracene 3, precursor of 3,1,8-chGNRs, was
synthesized by reaction of compound 8 with organolithium 7, followed by a
reduction step. Finally, compound 4, precursor of 3,2,8-chGNRs, was prepared
from derivative 9 following a similar procedure (see Supplementary Part
LABEL:SN:synthesis for further details). In the OSS step, each precursor was
independently sublimated onto a clean Au(111) surface held at room temperature
and step-wisely annealed to $T_{1}$ to induce their Ullmann-like
polymerization. Subsequently, a further annealing to $T_{2}$ activated the
cyclodehydrogenation (CDH) of the polymers into the targeted chiral graphene
nanoribbons with chiral vectors (3,1) and (2,3) and with different widths
(Fig. 2(e-h)). STM images of the resulting structures (Fig. 2i-l show the
characteristic straight, and planar shape of the ribbons, confirming their
successful synthesis.

Figure 3: Emergence of edge bands and gaps in 3,1,w-chGNRs. a-c, Bond-resolved
constant height $dI/dV$ images of 3,1,4-, 3,1,6-, and 3,1,8-chGNRs,
respectively ($V=2$ mV). The images were acquired with a CO-terminated tip.
All the images share the same scale bar as labeled in c. The chemical
structure of the basic aromatic block is superimposed on the images. d,e,
$dI/dV$ spectra taken at the locations as noted on the images. The spectra are
shifted for clarity. f, Band diagrams in black, blue and red are the TB
simulated band structure of 3,1,4-, 3,1,6- and 3,1,8-chGNRs, respectively.
Inset shows a zoom of the bands around zero energy. The wavefunction at
$\Gamma$ ($k=0$) and X ($k=\pi/a$) of VB of 3,1,4, and 3,1,8-chGNRs are shown
in the left and right panels in the figure, with their inversion symmetry
indicated.

Emergence of edge bands in wide 3,1,w-chGNRs: We compare first the effect of
increasing the width on the electronic structure of 3,1,w-chGNRs. Bond-
resolved STM images shown in Fig. 3a-c (obtained by measuring constant height
current maps at $V=2$ mV using a CO-terminated tip Gross  _et al._ (2009);
Kichin  _et al._ (2011)) reproduce the hexagonal ring patterns of the
different ribbons, in agreement with the chemical structures in Fig. 2g-i.
However, the wider 3,1,6- and 3,1,8-chGNRs show, on top of the ring structure,
a characteristic current increase over the edges, which is absent in the
3,1,4-chGNR images. These brighter edges unveil a larger density of states
(DOS) around the Fermi energy, this being an experimental evidence for the
emergence of edge bands in the wider ribbons. This is further corroborated by
comparing differential conductance spectra ($dI/dV$) on the different ribbons,
as shown in Fig. 3d. The spectral plots 2 and 3, measured at the edges of
3,1,6- and 3,1,8-chGNRs, respectively, show a pronounced increase of $dI/dV$
signal around zero bias, that is absent over the central part of the ribbons
(spectral plot 4). In contrast, a wide band gap of $\sim$ 0.7 eV with no DOS
enhancement around the Fermi level is found all over the 3,1,4-chGNRs (plot 1
in Fig. 3d) Merino-Díez  _et al._ (2018).

The emergence of edge bands close to zero-energy in 3,1,6- and 3,1,8-chGNRs is
reproduced by our TB (Fig. 3f) and density functional theory simulations
(LABEL:SN:DFTband) of the band structure of infinitely long 3,1,$w$-chGNRs.
The relatively large band gap ($E_{g}$=0.26 eV) of the 3,1,4-chGNR closes
abruptly for the wider ribbons, whose valence and conduction bands (VB and CB)
apparently merge at zero energy and flatten, being these the edge-localized
states resolved in the experiments. However, as we show in the inset of Fig.
3f, the frontier bands of 3,1,6- and 3,1,8-chGNRs do not overlap at zero, but
remain gapped. Contrary to a monotonous gap closing, the theoretical energy
gap is very small for 3,1,6-chGNRs ($\sim$8 meV), and opens again for the
wider 3,1,8-chGNRs ($\sim$29 meV). Only for $w\geq 12$ the gap closes
definitively (see Fig. 1c).

The origin of the mini-gap reopening from $w=6$ to $w=8$ is connected with a
gap inversion due to a change in valence band’s topology with the width. This
can be deduced by comparing maps of the wave function amplitude and phase
distribution at $k=0$ and at $k=\pi/a$, shown in Fig. 3f. The VB of
3,1,4-chGNRs (LABEL:SN:TBWF) maintains an odd inversion symmetry with respect
to the center of the unit cell in all $k$-space, whereas for 3,1,8-chGNRs it
changes parity (from even, at $k=0$, to odd at $k=\pi/a$), revealing a band
inversion at $k=0$. As a consequence, the wave function acquires a net phase
as it disperses along the Brillouin space. To connect these differences in
parity with topological classes, we computed the Zak phase $\gamma_{z}$ Zak
(1989) of the occupied band structure for every ribbon and obtained their
$\mathbb{Z}_{2}$ invariant, as described in LABEL:SN:ZAK. The 3,1,4-chGNRs
accumulate a global Zak phase of $\gamma_{z}=0$ and, hence, are in a trivial
topological phase, i.e. $\mathbb{Z}_{2}$=0. The intermediate case,
3,1,6-chGNRs, has a very small gap that changes topological phase depending on
details of the simulation (LABEL:SN:TBWF) and thus, we consider here as the
transition metallic case. For the wider ribbon, 3,1,8-chGNR, we obtain
$\gamma_{z}=\pi$ in accordance with a non-trivial SPT phase
($\mathbb{Z}_{2}=1$), thus accounting for the gap reopening found in the
simulations.

Figure 4: SSH model and simulations of the band topology of chGNRs:  a,
Modified Su-Shrieffer-Heeger model to explain the SPT phase transition of
chGNRs with the width. b, Evolution of the energy gap value and sign with the
chGNR width, obtained by fitting the mSSH model in a to frontier bands from
3NN TB simulations. (Left ordinate) comparison of intra- ($t_{i}^{0}$) and
inter-edge ($t_{i}^{-1}+t_{i}^{+1}$) hopping elements obtained from the fit,
indicating that the topological transition occurs when both are balanced. c
Color plots of the energy gap value and sign for the family of $(z,a,w)$
chGNRs, showing that the topological transition is a global property of this
family of chiral ribbons. Marked squares indicate the ribbons studied in this
work. The red/blue color scale refers to the positive/negative sign of
$E_{g}$, defined as $(-1)^{\mathbb{Z}_{2}}$, where the $\mathbb{Z}_{2}$
invariant is obtained from the Zak phase (LABEL:SN:ZAK).

Su-Shrieffer-Heeger model prediction of a SPT phase transition: The presence
of band gaps in narrow chGNRs and the SPT phase transition can be explained
using the modified Su-Shrieffer-Heeger (mSSH) model Su  _et al._ (1979)
depicted in Figure 4 and in LABEL:SN:mSSH. We can describe the 3,1,$w$-chGNR
as a chain of singly-occupied states localized at zigzag edge sites Golor  _et
al._ (2014), with hopping matrix elements $t$ along the edge, and width-
dependent hopping terms $t_{i}$, $t_{i}^{{}^{\prime}}$, and
$t_{i}^{{}^{\prime\prime}}$ between states at opposing edges. For very wide
ribbons only the edge hopping term $t$ is relevant, and the ribbon’s edges
enclose metallic one dimensional bands, as pictured in Fig. 1b. To simulate
the emergence of gapped SPT phases during chGNR narrowing, we fitted their VB
and CB obtained from 3NN TB simulations using the mSSH model (LABEL:SN:mSSH),
and obtained that a gap opens when the three elements representing the hoping
between opposing edges becomes sizable (Fig. 4b). Initially, interactions
between the diagonal neighbours $t_{i}^{{}^{\prime}}$ and
$t_{i}^{{}^{\prime\prime}}$ (i.e. inter-cell hopping between edges) dominate
over confronted zigzag elements (intra-cell hopping, $t_{i}$). This chiral
interaction pattern causes a gapped phase with negative sign, defined from the
${\mathbb{Z}_{2}}$ invariant as $(-1)^{\mathbb{Z}_{2}}$, and explains the non-
trivial band topology of the 3,1,8-chGNR. However, the interaction pattern
reverses for narrower ribbons, and the intra-cell hopping element $t_{i}$
dominates over the others, leading now to a gapped phase with positive sign
($\mathbb{Z}_{2}=0$) (Fig. 4b), with a SPT phase transition close to the $w=6$
case.

Inspired by the mSSH model, we performed TB simulations for the z,a,w-chGNR
family. We computed the band structure and the total Zak phase $\gamma_{z}$ of
occupied bands for the set $0<z\leq 10$, $a<6$, and $w\leq 12$, comprising
ribbons with chiral angle from 4.5∘ to 80∘ (LABEL:SN:STRUCT). The resulting
band gap values $E_{g}$ and sign $(-1)^{Z_{2}}$ are represented in Fig. 4c.
The results show that the SPT phase transition found for the 3,1,$w$-chGNRs is
a global property of the $z,a,w$-chGNR family. All chiral ribbons show a
similar trend with the width: the gap-less phase of wide ribbons, with edge
states as in Fig. 1a, transforms first into a one-dimensional topological
insulating phase and then into a trivial phase below a critical width.

SPT boundary states: To experimentally confirm the existence of different
gapped SPT phases in this family of chGNRs, we analyze the origin of the
persisting low-bias substructure in $dI/dV$ spectra appearing over the zero-
bias peaks along the edge (shown in Fig. 3e). It is expected that a non-
trivial bulk-boundary correspondence in topological chGNRs of finite length
leads to in-gap states localized at the GNR termini and associated to SPT
boundary states. Correspondingly, our TB simulations for finite ribbons
reproduce sharp zero-energy states distributed around the ends of a
3,1,8-chGNR (Fig. 5a,b), which are absent in the narrower ribbons, with
opposite topological class LABEL:SN:PTendSlength. The experimental $dI/dV$
maps measured at low sample bias, like in Fig. 5c, confirm the presence of
these boundary states in 3,1,8-chGNRs, appearing as a peculiar signal
enhancements over the edge’s termini, with symmetry and extension similar to
the simulated LDOS in Fig. 5a,b. Additionally, $dI/dV$ spectra over these
brighter regions show a sharp peak centered at $2$ meV (Fig. 5d), and slowly
decaying towards the interior of the GNR edge (LABEL:318sts). In the middle of
the ribbon, the VB and CB onsets appear as two peaks at $\sim\pm 10$ meV,
delimiting a band gap of barely $20\pm 4$ meV.

Figure 5: SPT boundary states of 3,1,8- and 3,2,8-chGNRs: a,f, Zero-energy
LDOS distribution of finite 3,1,8-chGNRs and 3,2,8-chGNRs of 9 precursor units
obtained with TB simulations and, b,g, their corresponding Density of States
(DOS) over the center (bulk) and the termini surface showing the localization
of zero-energy boundary states. c,h, Constant height $dI/dV$ maps ($V=0$ mV)
of 3,1,8-chGNRs and 3,2,8-chGNRs with 9 and 10 precursor units respectively
and, d,i, $dI/dV$ spectra taken over the termini and edge’s center (as
indicated with colored crosses). e,j, Comparison of $dI/dV$ spectra taken on
the termini and on center of short ribbons confirm the survival of topological
states in ribbons as small as 4 and 2 precursor units for the 3,1,8- and
3,2,8-chGNR, respectively. Insets show constant height $dI/dV$ maps (V=0 mV).
Spectra in d,e,j are shifted for clarity. Blue dashed lines in d,e,i,j
indicate the VB and CB onset features.

The small band gap $E_{g}$ of these chGNRs accounts for the slow decay of the
end states inside the ribbon (see LABEL:SN:PTendSlength). In short ribbons,
end states from opposing termini may overlap and open an hybridization gap
$E_{\Delta}$ Wang  _et al._ (2016) that can hinder the observation of end
states when $E_{\Delta}>E_{g}$. In spite of the large spatial extension of the
end states in 3,1,8-chGNRs, and the very small gap $E_{g}$, we found that SPT
boundary states survive for ribbons with only four precursor units (PUs)
length, while vanish completely in chGNRs with three PUs or less, whose
spectra is fully gapped (Fig. 5e). The survival of end states in short ribbons
is confirmed by our TB simulations, which also find the opening of an
unusually small hybridization gap (LABEL:SN:PTendSlength). The weak
interaction between end states is due to their peculiar distribution in the
chiral backbone, where each end state lays at opposing ribbon’s edges (as
expected from the mSSH model), thus reducing their overlap in short ribbons.

Contrasting with the topological character of 3,1,8-chGNR, end states are
absent in narrower ribbons, proving their trivial insulating phase and
confirming the existence of a width controlled topological phase transition in
this family of chGNRs. For 3,1,6-chGNRs, low bias peaks are spaced by tens of
meVs, with a central one pinned close above the Fermi level (Fig. 3e). As we
show in LABEL:SN:316sts, these peaks correspond to valence and conducting
bands, discretized in quantum-well states due to the finite length of the
ribbon Carbonell-Sanromà  _et al._ (2017); Wang  _et al._ (2017); Merino-Díez
_et al._ (2018). The peak pinned above the Fermi level coincides with the VB
onset, partially depopulated in response to the large electron affinity of the
Au(111) substrate Gröning  _et al._ (2018); Kimouche  _et al._ (2015); Merino-
Díez  _et al._ (2017). The first peak above the VB corresponds to the CB
onset, and no sub-gap features neither signal at the chGNR ends is observed,
in agreement with their trivial semiconducting character.

Topological insulating phase of 3,2,8-chGNR: The simulations from Fig. 4c also
illustrate that the size of the chGNR band gaps increase with the chiral
angle, with $E_{g}$ varying from just a few tens of meV for lower-angle
ribbons, to almost one electron-volt for some orientations. This property
allows engineering robust topological chGNRs with wider gaps than for the
3,1,8-chGNR. For example, we note that the theoretical gap of the 3,2,8-chGNR
shown in Fig. 4 amounts to 199 meV and is inverted (i.e. $\mathbb{Z}_{2}=1$).
Correspondingly, the simulations for finite ribbons of this kind reveal zero-
energy topological modes at the termini (Fig. 5f,g).

To confirm this topological insulating state, we studied 3,2,8-chGNRs
fabricated using the modified precursor 4. As depicted in Fig. 2, the modified
halogen substitution of this precursor (at $a=2$ sites) steers the formation
of 3,2,8-chGNRs on a Au(111) surface at elevated temperatures. These ribbons
are oriented along a (2,3) vector of the graphene lattice, and alternate three
zigzag with one and a half armchair sites along the edges. The bond resolved
STM current image in Fig.5h confirms the successful OSS of 3,2,8-chGNRs by
revealing its characteristic ring structure over the bulk part of the ribbon.
However, the STM image also reproduces over the edges a characteristic signal
enhancement the resembles the SPT boundary states in the TB LDOS maps of Fig.
5f. Furthermore, $dI/dV$ spectra measured over the chGNR ends shows sharp
peaks pinned at zero bias, while over the bulk region of the ribbon a bare
$\sim$ 300 meV band gap is found (Fig.5i). These end states are then readily
identified as SPT boundary states, as predicted by our TB simulations, thus
confirming the $\mathbb{Z}_{2}=1$ topological class also in this case. As for
the 3,1,8-chGNRs, these end states are pinned slightly above $E_{F}$, being
partially depopulated. However, due to the larger band gap of this ribbon,
these end states are more localized at the terminations
(LABEL:SN:PTendSlength), and, hence, they are readily detected in even shorter
ribbons, with only two precursors units (Fig. 5j).

The wider band gap and the larger degree of localization augment the potential
of ribbons with this chirality to become spin-polarized Lawrence  _et al._
(2020); Friedrich  _et al._ (2020) in the charge undoped state and in the
presence of electron-electron interactions. The generalized behaviour
described here for this family of chiral GNRs represents a novel route to
manufacture graphene ribbons with metallic edge bands and to transform them
into topological states in graphene platforms. It further shows that endowing
graphene with a chiral interaction is an effective method to induce gapped
phases. We envision that this method could be extended not only to other
chiral geometries in one-dimensional nanoribbons Jiang and Louie (2021), but
also to two-dimensional porous graphene networks Moreno  _et al._ (2018), or
Moiré 2D systems, in which combination of flat-bands with chiral symmetries
might lead to novel SPT phases.

###### Acknowledgements.

We gratefully acknowledge financial support from the Agencia Estatal de
Investigación (AEI) from projects No MAT2016-78293, PID2019-107338RB and
FIS2017-83780-P, and from the Maria de Maeztu Units of Excellence Programme
MDM-2016-0618, the Xunta de Galicia (Centro singular de investigación de
Galicia, accreditation 2016–2019, ED431G/09), the University of the Basque
Country (Grant IT1246-19), the Basque Departamento de Educación through the
PhD scholarship no. PRE_2019_2_0218 (S.S.), and the European Regional
Development Fund. We also acknowledge funding from the European Union (EU)
H2020 program, through the ERC (grant agreement No. 635919) and FET Open
project SPRING (grant agreement No. 863098). Methods and Supplementary Notes
and Figures can be downloaded from https://www.springfetopen.eu/publications/
Correspondence and requests for materials should be addressed to: D.G.O.
(d$\\_$g$\\_$oteyza@ehu.eus), T.F. (thomas$\\_$frederiksen@ehu.eus), D.P.
(diego.pena@usc.es), or J.I.P<EMAIL_ADDRESS>
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# Discrete analog of the Jacobi set for vector fields††thanks: This work was
supported by the Ministry of Education and Science of the Republic of
Kazakhstan (program 0115PK03029) and Russian Foundation for Basic Research
(grant 15-01-01671a).

A.N. Adilkhanov111 National Laboratory “Astana”, Nazarbayev University, 53,
Kabanbay Batyr Ave., Astana, 010000, Republic of Kazakhstan;
<EMAIL_ADDRESS>A.V. Pavlov222 North-Eastern Federal University, 677000
Yakutsk, Russia<EMAIL_ADDRESS>and I.A. Taimanov333 Sobolev Institute of
Mathematics, 630090 Novosibirsk, and Novosibirsk State University, 630090
Novosibirsk, Russia<EMAIL_ADDRESS>
###### Abstract

The Jacobi set is a useful descriptor of mutual behavior of functions defined
on a common domain. We introduce the piecewise linear Jacobi set for general
vector fields on simplicial complexes. This definition generalizes the
definition of the Jacobi set for gradients of functions introduced by
Edelsbrunner and Harer.

_2010 Mathematics Subject Classification: 68U05, 55C99, 57R25._

_Keywords:_ Jacobi set, vector fields, simplicial complex.

##  1 Introduction

In this article we give a construction of a piecewise linear analog of the
Jacobi set for vector fields. This set serves as a descriptor of the relation
between vector fields defined on a common domain.

For the gradient fields of Morse functions
$f_{1},\dots,f_{k}:D\to{\mathbb{R}}$, where $D$ is a domain in
${\mathbb{R}}^{N}$, or more generally an $N$-dimensional manifold, the Jacobi
set is the subset of $D$ formed by all points at which the gradients of these
functions are linearly dependent. This set can be used for extracting useful
information about the mutual behavior of multiple functions [1]. As Jacobi
sets for a pair of functions on the plane it appears for different reasons in
[2] (see also [3]), and in general form it was introduced in [4].

For applications, it is helpful to have a discrete analog of the Jacobi set,
and such an analog for functions defined on triangulated complexes was
introduced in [4]. In the same article, the problem of extending the proposed
methods to general vector fields was posed. We demonstrate how to do that on
the example of pairs of vector fields on the plane.

##  2 The piecewise linear Jacobi set

We recall the main definitions and results from [4].

A _Morse function_ on a compact manifold $M$ is a function $f:M\to\mathbb{R}$
that has only a finite number of critical points, where the matrix of second
derivatives is nondegenerate and the function values are distinct from each
other. For two Morse functions $f,g:M\to\mathbb{R}$ defined on a compact
manifold $M$, their _Jacobi set_ is defined as the set $J(f,g)$ where their
gradients are linearly dependent. Equivalently, $J(f,g)$ can be described as
the set of all critical points of functions $h_{\lambda}=f+\lambda g$ and
$e_{\lambda}=\lambda f+g$ for all $\lambda\in\mathbb{R}$. For two generic
Morse functions $f$, $g$ having no common critical points, $J(f,g)$ is a
1-submanifold of $M$.

In the discrete case of two functions $f$, $g$ defined on the vertices of a
triangulation $K$ of a $d$-dimensional manifold, we can extend $f$ and $g$ to
PL-functions on the entire complex $K$ and view each as a limit of a series of
smooth functions. Motivated by this viewpoint, the discrete Jacobi set
${\mathbb{J}(f,g)}$ is introduced [4] as a 1-dimensional subcomplex of $K$
consisting of edges $uv$, with multiplicity, along which, in the limit, the
critical points of $h_{\lambda}$ and $e_{\lambda}$ travel as $\lambda$ varies.

To state the precise definition, we need some notation. Let $K$ be a
simplicial complex. The _star_ of its simplex $\sigma$ is the set of all
simplices containing $\sigma$, and the _link_ $\mathop{\mathrm{Lk}}\sigma$
consists of all simplices in the closure of the star of $\sigma$ that are
disjoint from $\sigma$. Note that $\mathop{\mathrm{Lk}}\sigma$ is itself a
complex. Let $h$ be a real-valued function on the vertices of a simplicial
complex $K$. For a simplex $\sigma\in K$, define the lower link
$\mathop{\underline{\mathrm{Lk}}}\sigma$ in $K$ with respect to $h$ to be the
portion of $\mathop{\mathrm{Lk}}v$ that bounds the set of all simplices in the
star of $v$ that have $v$ as the vertex with the maximal value of $h$.

Consider an edge $uv\in K$. We disregard the edges with $g(u)=g(v)$. Denote by
$\lambda^{*}=\lambda^{*}(u,v)$ the value of $\lambda$ that equalizes the
values of the linear combination $f+\lambda g$ at both ends of the edge:
$f(u)+\lambda^{*}g(u)=f(v)+\lambda^{*}g(v)$. Denote this linear combination by
$h$: $h=f+\lambda^{*}g$. The link of the vertex $u$ is a triangulation of a
$(d-1)$-sphere containing $v$. The multiplicity of the edge $uv$ is defined as
the sum of reduced Betti numbers $\tilde{\beta}_{i}$ of the lower link
$\mathop{\underline{\mathrm{Lk}}}uv$ with respect to $h$. The _piecewise
linear Jacobi set_ of two functions $f$, $g$ on $K$ is defined as the one-
dimensional subcomplex $\mathbb{J}(f,g)$ of $K$ consisting of all edges having
nonzero multiplicity, together with their endpoints.

|
---|---
(a) | (b)
Figure 1: The link of an edge in a triangulation of (_a_) 2-manifold (_b_)
3-manifold

We now review the general definition for the special cases of 2- and
3-dimensional simplicial complexes.

In two dimensions, the star of an edge $uv$ consists of two 2-simplices
neighboring along $uv$, and $\mathop{\mathrm{Lk}}uv$ is just two vertices $a$,
$b$ opposite $uv$ in these 2-simplices (fig. 1,  _a_). Thus, the edge $uv$
belongs to $\mathbb{J}(f,g)$ if and only if the values $h(a)$, $h(b)$ are
either both greater or both smaller than the value of $h(u)=h(v)$, where
$h=f+\lambda^{*}g$, and $\lambda^{*}=(f(u)-f(v))/(g(v)-g(u))$. This condition
can be rewritten in terms of function differences in adjacent vertices. For
any function $f:K\to{\mathbb{R}}$ and edge $xy$, denote $df(xy)=f(y)-f(x)$.
Then we can write

$\begin{array}[]{rcl}uv\in\mathbb{J}(f,g)&\Leftrightarrow&dh(ua)\text{ and
}dh(ub)\text{ have the same sign,}\\\ &&\text{where }h=f+\lambda^{*}g\text{~{}
and~{}}\lambda^{*}=df(uv)/df(vu).\end{array}$ (1)

This condition is actually symmetric in $u$ and $v$, since
$dh(vx)=dh(ux)-dh(uv)$ for any vertex $x$, and $dh(uv)=0$.

In a triangulation of a three-dimensional manifold, the link
$\mathop{\mathrm{Lk}}uv$ of an edge $uv$ is a triangulation of a circle. The
multiplicity of an edge $uv$ in the Jacobi set $\mathbb{J}(f,g)$ is equal to
the sum of the reduced Betti numbers of the lower link
$\mathop{\mathrm{\underline{Lk}}}{uv}$ with respect to $h$:

$\tilde{\beta}_{-1}+\tilde{\beta}_{0}+\tilde{\beta}_{1},$

where $\tilde{\beta}_{-1}$ is 1 if the lower link is empty, and 0 otherwise;
$\tilde{\beta}_{0}$ is one less than the number of connected components in
$\mathop{\mathrm{Lk}}\underline{uv}$ if this number is positive, and 0
otherwise; $\tilde{\beta}_{1}$ is 1 if the lower link is the entire circle,
and 0 otherwise.

In fig. 1,  _b_ , the link of $uv$ is shown in bold lines. This link is a
triangulation of a circle. Denote its consecutive vertices by $a_{0}$,
$a_{1}$, …, $a_{k-1}$, and put $a_{k}=a_{0}$. As previously, $dh(xy)$ stands
for the difference $h(y)-h(x)$. Count the number of times the difference
$dh(ua_{i})$ changes from negative to positive along the circle:

$\beta_{0}=\\#\\{i:~{}dh(ua_{i})<0\text{~{} and ~{}}dh(ua_{i+1})>0,~{}0\leq
i<k\\}.$

Then the multiplicity of the edge $uv$ in $\mathbb{J}(f,g)$ is
$|\beta_{0}-1|$. In particular, if the lower link of the edge $uv$ with
respect to $h$ consists of just a single component that is not the entire
circle, the edge does not belong to the Jacobi set.

Figure 2: Zigzags in a simplicial Jacobi set (red) for functions
$f=((x-1)^{2}+y^{2})((x+1)^{2}+y^{2})$ (dotted level lines) and
$g=(x-1)^{2}+(y-1)^{2}$ (dashed level lines). The triangulation is obtained
from a square grid with step $\frac{1}{6}$. The continuous black line is the
smooth Jacobi set.

The functions $h_{\lambda}(v)=f+\lambda g$ are linear in $\lambda$ for any
given $v$. Because of that, any $v\in\mathop{\mathrm{Lk}}u$ changes its status
as inside/outside the lower link of $u$ with respect to $h_{\lambda}$ exactly
once as $\lambda$ grows from $-\infty$ to $+\infty$, namely at
$\lambda=\lambda^{*}(u,v)=df(uv)/dg(vu)$. So in dimension 2, the number of
connected components $\beta_{0}$ in the lower link of $u$ is either the same
at both extremes, or $0$ for one of them and $1$ at the other. Obviously, for
an edge $uv\notin\mathbb{J}$, passing $\lambda^{*}(u,v)$ does not change
$\beta_{0}$. For $uv\in\mathbb{J}$, passing $\lambda^{*}$ either changes
$\beta_{0}$ by one or does not change $\beta_{0}=1$ if on either side of
$\lambda^{*}$ the lower link is all of the link of $u$. Counting the parity of
$\beta_{0}$, we see that the number of edges $uv\in\mathbb{J}$ for a fixed
vertex $u$, i.e. the degree of $u$ in $\mathbb{J}$, must be even. A similar
argument, after unfolding each multiple critical point into multiple simple
critical points, holds in any dimension:

Even-degree lemma [4]. The degree of any vertex $u$ in $\mathbb{J}$ is even.

Although the even-degree lemma guarantees that the discrete Jacobi set can be
represented as a continuous polyline, it may contain spurious cycles and
zigzags, becoming incovienently large. For example, if the simplicial complex
in question is a fine enough regular triangulation of a plane, the discrete
Jacobi set may appear to fill entire 2-dimensional regions on the plane (fig.
2). A variety of simplification techniques exist for the smooth as well as
discrete versions of the Jacobi set ([6], [7]).

##  3 The piecewise linear Jacobi set for vector fields

The main idea behind our definition is as follows. The gradient

$df=\left(\frac{\partial f}{\partial x^{1}},\dots,\frac{\partial f}{\partial
x^{N}}\right)$

of a function $f:D\to{\mathbb{R}}$ is in fact a $1$-form which is a linear
form on vector fields. Indeed, its value for a vector field $X$ is the
derivative of $f$ in the direction of $X$:

$D_{X}f=X^{i}\frac{\partial f}{\partial x^{i}},$

where we assume the summation over the repeated index. To obtain the gradient
vector field we have to raise the index by using some non-degenerate quadratic
form $g^{ik}$ (usually the inverse of the metric tensor $g_{ik}$):

$(\nabla f)^{i}=g^{ik}\frac{\partial f}{\partial x^{k}},$

where again we assume the summation over the repeated index $k$. The Euclidean
metric is given by the tensor

$g_{ik}=g^{ik}=\begin{cases}1&\mbox{for $i=k$}\\\
0&\mbox{otherwise},\end{cases}$

the gradient of the function and the gradient vector field look the same, but
in general coordinates their numerical expressions are different. We refer for
detailed discussion, for instance, to [5].

Since the lowering of the index (the convolution)

$X^{i}\to Y_{k}=g_{ik}X^{i}$

maps linearly dependent vector fields into linearly dependent $1$-forms, it is
enough to define the Jacobi sets for $1$-forms.

For a triangulated complex $K$, $1$-forms $Y$ are linear functions on oriented
$1$-chains, i.e., on oriented edges:

$Y(uv)\in{\mathbb{R}}\ \ \ \mbox{where $uv$ is an oriented edge in $K$}.$

We interpret the Edelsbrunner–Harer definition of the Jacobi set of two
gradient vector fields as the definition of the Jacobi set of two $1$-forms
that are coboundaries of linear functions on the vertices of $K$:

$Y(uv)=df(uv)=f(v)-f(u).$

For a triangulation of a smooth manifold $K$ and a smooth function
$f:K\to{\mathbb{R}}$ the discretization of its gradient (covector) field is
exactly given by the formula above where $f$ is evaluated in the vertices of
the triangulation.

Given a smooth $1$-form $\omega$ on a triangulated manifold, we have to
construct a $1$-form on oriented edges. The most natural way is to consider an
edge as an oriented path and take an integral of $\omega$ over the path:

$Y(uv)=\int_{u}^{v}\omega.$

For a smooth gradient field $df$ in Euclidean space we get

$Y(uv)=\int^{u}_{v}df=f(v)-f(u).$

A non-gradient vector field corresponds to a non-closed 1-form. Circular
integrals of such a form may not vanish, so generally, it is not true that
$Y(uw)=Y(uv)+Y(vw)$.

Let $K$ be a simplicial complex that is a triangulation of a $d$-dimensional
manifold, and $F$, $G$ be discrete 1-forms given by their values on all
oriented edges $uv$ of $K$:

$F(uv)=-F(vu),\quad G(uv)=-G(vu).$

Denote by $H_{\lambda}$ the linear combination $F+\lambda G$. For each edge
$uv$ with $G(uv)\neq 0$, as previously, denote by $\lambda^{*}$ the value of
the coefficient that makes this linear combination vanish along $uv$:

$\lambda^{*}=\frac{F(uv)}{G(vu)},\quad H_{\lambda^{*}}(uv)=0.$

For a vertex $w$ in $\mathop{\mathrm{Lk}}uv$ define the average of the values
of the form $H_{\lambda^{*}}$ on the edges connecting $u$ and $v$ to $w$:

$h(w)=\frac{1}{2}(H_{\lambda^{*}}(uw)+H_{\lambda^{*}}(vw)).$

Multiplicity of an edge $uv$ is defined as the sum of the reduced Betti
numbers of the lower link of $uv$ with respect to $h$, and we define the
_Jacobi set $\mathbb{J}(F,G)$ of two discrete 1-forms $F$ and $G$_ as the one-
dimensional subcomplex of $K$ consisting of all edges having nonzero
multiplicity, together with their endpoints.

Figure 3: The edge test for 1-forms on a 2-dimensional complex

In two dimensions ($d=2)$, this definition means that the Jacobi set of $F$
and $G$ consists of all edges $uv$ for which the average of the values of
$H_{\lambda^{*}}=F+\lambda^{*}G$ along $ua$ and $va$ has the same sign as the
average of its values along $ub$ and $vb$, where $a$ and $b$ are the two
points of the link of $uv$ (fig. 3).

$uv\in\mathbb{J}(F,G)\Leftrightarrow(H_{\lambda^{*}}(ua)+H_{\lambda^{*}}(va))(H_{\lambda^{*}}(ub)+H_{\lambda^{*}}(vb))>0.$
(2)

Note that, as was the case for the condition (1), this condition is also
symmetric in $u$, $v$. It is symmetric in $F$ and $G$ as well when all values
of the forms $F$, $G$ on the edges are nonzero.

However, the even degree lemma no longer holds for nongradient 1-forms. This
is illustrated below for the approximation of the Jacobi set for two smooth
1-forms on the plane (fig. 4). The smooth Jacobi set is the set of points
where the forms are linearly dependent, and is shown with continuous green
lines, while the piecewise linear Jacobi set for the triangulation of a square
grid with step size $h=0.1$ is shown in red.

Figure 4: Continuous Jacobi set (green line) and its piecewise linear
approximation for the forms $F(x,y)=(y+1)dx+2(x+1)dy$ and
$G(x,y)=(2x-3y)dx+(2x+3y)dy$, grid step size $h=0.1$

Still, as can be seen in table 1, with the refinement of the grid the
approximation converges to the smooth Jacobi set.

In applications, a vector field $X$ is usually given by its coordinates on a
plane grid. A reasonable approximation for the integrals of the corresponding
1-form $Y$ on the edges are scalar products of the mean value of the vector
field on the edge with the edge vector itself:

$Y(uv)=\frac{1}{2}(X_{v}+X_{u},uv).$

We have also tested our definition for three different regular triangulations
on the plane, shown in fig. 5: the diagonal grid $T_{1}$ (invariant with
respect to rotations by $\pi$), crossed $T_{2}$ (invariant with respect to
rotations by $\pi/4$), and hexagonal $T_{3}$ (invariant with respect to
rotations by $\pi/6$).

---
(_a_) T1 (_b_) T2 (_c_) T3
Figure 5: Plane triangulations

Results of these calculations are shown in fig. 6. As was the case for the
Jacobi sets of Morse functions, the approximations differ, with no clear
winner.

For better connectivity of the produced approximation, the edge test (2) can
be modified to include cases where the absolute value of at least one of the
factors is smaller than some threshold value $\varepsilon$:

$\begin{array}[]{r@{}l}\mathbb{J}(F,G)=\\{&uv\in
K\mid(H_{\lambda^{*}}(ua)+H_{\lambda^{*}}(va))(H_{\lambda^{*}}(ub)+H_{\lambda^{*}}(vb))>0.\\\
&\text{or \>}|H_{\lambda^{*}}(ua)+H_{\lambda^{*}}(va)|<\varepsilon\;\text{\>
or \>}|H_{\lambda^{*}}(ub)+H_{\lambda^{*}}(vb)|<\varepsilon\\}\end{array}$ (3)

This will improve the connectivity at the cost of thickening the Jacobi set.

In fig. 7, we show the smooth Jacobi set, and in table 1 illustrate the
dependence of approximation, using the T1 triangulation scheme, on the grid
step size $h$ and the threshold $\varepsilon$ in (3) for the forms

$F(x,y)=y(x^{2}+y^{2}+1)dx-x(x^{2}+y^{2}-1)dy,~{}~{}G(x,y)=(2x-3y-6)dx+(2x-3y)dy.$

As in the case of the Jacobi set for functions, numerically approximated
Jacobi set for vector fields may turn out to be very complicated. Sometimes,
it might be an indication of a strong similarity between the vector fields, as
in fig. 8. However, it would be interesting to develop methods for its
simplification similar to those proposed in [6], [7].

$\varepsilon$ | $h=0.1$ | $h=0.05$ | $h=0.01$
---|---|---|---
$0.0$ |  |  |
$0.00005$ |  |  |
$0.0001$ |  |  |
$0.001$ |  |  |
$0.005$ |  |  |
Table 1: Approximations for the Jacobi set in fig. 7 for various $h$ and
$\varepsilon$

Figure 6: Piecewise linear Jacobi set of $F(x,y)=(y+1)dx+2(x+1)dy$ and
$G(x,y)=(2x-3y)dx+(2x+3y)dy$ for various triangulations on the plane

Figure 7: The fields $f$, $g$ and the smooth Jacobi set $J(F,G)$ for the
corresponding forms
$F(x,y)=y(x^{2}+y^{2}+1)dx-x(x^{2}+y^{2}-1)dy$, $G(x,y)=(2x-3y-6)dx+(2x-3y)dy$

Figure 8: The discrete Jacobi set for the nongradient wind speed vector field
and the gradient of temperature over a part of Asia Pacific. GFS forecast data
for 1 November 2018 used [8].

## References

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