# Stacking-induced Chern insulator

Marwa Mannaï1 Jean-Noël Fuchs2<EMAIL_ADDRESS>Frédéric Piéchon3
<EMAIL_ADDRESS>Sonia Haddad1,4
<EMAIL_ADDRESS>1Laboratoire de Physique de la Matière Condensée,
Faculté des Sciences de Tunis, Université Tunis El Manar, Campus Universitaire
1060 Tunis, Tunisia
2Sorbonne Université, CNRS, Laboratoire de Physique Théorique de la Matière
Condensée, LPTMC, 75005 Paris, France
3Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405,
Orsay, France
4 Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba
277-8581, Japan

(August 27, 2024)

###### Abstract

Graphene can be turned into a semimetal with broken time-reversal symmetry by
adding a valley-dependent pseudo-scalar potential that shifts the Dirac point
energies in opposite directions, as in the modified Haldane model. We consider
a bilayer obtained by stacking two time-reversed copies of the modified
Haldane model, where conduction and valence bands cross to give rise to a
nodal line in each valleys. In the AB stacking, the interlayer hopping lifts
the degeneracy of the nodal lines and induces a band repulsion, leading
surprisingly to a chiral insulator with a Chern number $C=\pm 2$. As a
consequence, a pair of chiral edge states appears at the boundaries of a
ribbon bilayer geometry. In contrast, the AA stacking does not show nontrivial
topological phases. We discuss possible experimental implementations of our
results.

Introduction. –  A Chern insulator Hasan-Rev ; Qi-Rev ; Bansil  is a two-
dimensional (2D) topological insulator with broken time reversal symmetry
(TRS), where chiral edge states emerge in a ribbon geometry with
counterpropagating directions. These edge states are the hallmark of the bulk
topological properties described by a topological invariant, the Chern number
$C$, indicating the number of the chiral edge channels. A Chern insulator
exhibits quantum anomalous Hall effect (QAHE) AQHE-rev ; Yang ; Khan ; Ni ;
Esslinger ; Kee ; acoustic ; Serlin1 ; Serlin2 ; resistance ; tuneC
introduced by Haldane in his seminal paper Haldane .

The Haldane model Haldane  (HM) describes a honeycomb lattice with complex
hopping integrals between next nearest-neighbors (NNN), creating staggered
magnetic fluxes, which break TRS. The NNN hopping terms are characterized by a
complex phase $\Phi$ which has the same sign in the two sublattices of the
honeycomb lattice.

A modified Haldane model (mHM) has been proposed in Ref. Franz,  where TRS is
broken by a valley-dependent pseudo-scalar potential which shifts, oppositely,
the energies of the Dirac points in the two valleys. The potential is
generated by the sign flip of the complex phase $\Phi$ in one of the honeycomb
sublattices. The system turns into a semi-metal with a Fermi surface
consisting, at half filling, of a hole pocket, in a valley, and an equal-sized
electron pocket in the opposite valley (Fig. 1(a)).

Figure 1: (a) Fermi surface of the monolayer mHM at half filling. A valley-
dependent scalar potential shifts, oppositely, the energies of the Dirac
cones. The Fermi surface consists into an electron pocket (blue line) in one
valley and a hole pocket (red line) in the opposite valley. (b) and (c) Nodal
lines (red and blue lines) of the mHM on uncoupled bilayers where the
intralayer pseudo-scalar potential have opposite signs corresponding (b) to
$\Phi_{1}=-\Phi_{2}$, and (c) to phases of opposite signs and unequal
magnitudes ($\Phi_{1}\Phi_{2}<0$), where $\Phi_{l}$, ($l=1,2$) is the
intralayer complex phase of the NNN hopping integrals.

Imagine, now, a bilayer structure of the HM where the layers, indexed by
$l=1,2$, host Haldane phases labeled by the Chern numbers $C_{l}$. A naive
intuition tells us that the total Chern number is $C=C_{1}+C_{2}$, as reported
in Ref. Dutta,  for the AA bilayer footnote HM AA-AB . In particular, a
vanishing Chern number $C=0$ is expected if the layers have opposite
chiralities resulting from oppositely broken TRS Dutta ; footnote HM AA-AB .
This expectation is at the heart of the Kane and Mele Kane  idea, where the
layer index is replaced by the spin projection and the two opposite TRS copies
of the spin polorized HM give rise to a vanishing ($\mathbb{Z}$) Chern number.
The latter leaves room to another ($\mathbb{Z}_{2}$) TRS protected topological
invariant corresponding to a quantized spin Hall insulator.

In this Letter, we consider two stacked layers of the mHM, where the semi-
metallic layers break TRS in opposite ways (Fig. 1) and have undefined Chern
number $C_{l}=\emptyset$ ($l=1,2$). The resulting bilayer structure shows
counter-intuitive behaviors. Depending on the precise stacking order (AA, AB
or BA), we find that the resulting system may be gapless (AA) with
$C=\emptyset$ or topologically gapped with a Chern number $C\neq C_{1}+C_{2}$
but $C=\pm 2$ (AB/BA). In other words, and in contrast with the HM bilayer,
the stacking order in the mHM bilayer is a key parameter controlling the gap
opening and the emergence of chirality. Understanding the origin of the
topological gapped phases and the corresponding chirality is the main
objective of the present letter.

Bilayer modified Haldane model. –  We start with the HM on a honeycomb lattice
with a unit cell containing two different types of atoms denoted A and B. The
corresponding spinless fermionic Hamiltonian is Haldane

$\displaystyle H_{H}=t\sum_{\langle
i,j\rangle}c_{i}^{\dagger}c_{j}+t_{2}\sum_{\langle\langle
i,j\rangle\rangle}e^{-i\Phi_{ij}}c_{i}^{\dagger}c_{j}+\sum_{i}M_{i}c_{i}^{\dagger}c_{i},$
(1)

where $t$ ($t_{2}$) is the hopping integral to first (second) nearest
neighbors, $c_{i}$ annihilate a spinless fermion on atom ($i$),
$\Phi_{ij}=\nu_{ij}\Phi$ is the complex phase of NNN hopping integrals and
$\nu_{ij}=\pm 1$ according to the pattern given in Fig. 2 (b). The last term
describes the Semenoff masses where $M_{i}=M$ ($-M$) for A (B) atoms.

Figure 2: (a) Modified Haldane model on AB stacked bilayer. The monolayer unit
cell atoms are denoted by $A_{l}$ and $B_{l}$, where $l=1,2$ is the layer
index. $\boldsymbol{\delta}_{i}$ ($i=1,2,3$) are the vectors connecting
nearest neighboring (NN) atoms, $\left(\mathbf{a}_{1},\mathbf{a}_{2}\right)$
is the lattice basis and $\Phi$ is the phase of the complex NNN hoppings. The
solid (dashed) arrows indicate the pattern of the complex NNN hopping terms in
the sublattice $A$ ($B$). The blue and orange lines correspond, respectively,
to layer $1$ and $2$. (b) NNN hopping in the Haldane model and in (c) the
modified Haldane model.

In the sublattice pseudo-spin basis, the HM Bloch Hamiltonian can be written
as

$\displaystyle
H_{H}(\mathbf{k})=a^{0}_{\mathbf{k}}\sigma_{0}+b_{\mathbf{k}}\sigma_{x}+c_{\mathbf{k}}\sigma_{y}+\left(a_{\mathbf{k}}+M\right)\sigma_{z},$
(2)

where $\boldsymbol{\sigma}$ and $\sigma_{0}$ are, respectively, the sublattice
Pauli and $2\times 2$ identity matrices,
$b_{\mathbf{k}}=\Re\left(f_{\mathbf{k}}\right)$,
$c_{\mathbf{k}}=-\Im\left(f_{\mathbf{k}}\right)$,
$f_{\mathbf{k}}=t\sum^{3}_{i=1}e^{i\mathbf{k}\cdot\boldsymbol{\delta}_{i}}$,
$a^{0}_{\mathbf{k}}=2t_{2}\cos\Phi\sum^{3}_{i=1}\cos\left(\mathbf{k}\cdot\mathbf{a}_{i}\right)$
and
$a_{\mathbf{k}}=-2t_{2}\sin\Phi\sum^{3}_{i=1}\sin\left(\mathbf{k}\cdot\mathbf{a}_{i}\right)$
is the Haldane mass. The vectors $\boldsymbol{\delta}_{i}$ ($i=1,2,3$) connect
an atom to its first neighbors and ($\mathbf{a}_{1},\mathbf{a}_{2}$) is the
Bravais lattice basis given by (see Fig. 2 (a)):
$\mathbf{a}_{1}=\sqrt{3}a\mathbf{e}_{x},\,\mathbf{a}_{2}=-\frac{\sqrt{3}}{2}a\mathbf{e}_{x}+a\frac{3}{2}\mathbf{e}_{y}$,
where $a$ is the distance between nearest neighbors. We also define
$\mathbf{a}_{3}=-\left(\mathbf{a}_{1}+\mathbf{a}_{2}\right)$.

The modified Haldane model can be derived from the HM (Eq. 2) by flipping the
sign of the phase $\Phi$ on one sublattice, as shown in Fig. 2 (c). The
corresponding Hamiltonian mHM symm1  can be deduced from Eq. 2 by changing
$a_{\mathbf{k}}\sigma_{z}$ by $a_{\mathbf{k}}\sigma_{0}$

$\displaystyle
H_{mH}(\mathbf{k})=\left(a_{\mathbf{k}}+a^{0}_{\mathbf{k}}\right)\sigma_{0}+b_{\mathbf{k}}\sigma_{x}+c_{\mathbf{k}}\sigma_{y}+M\sigma_{z},$
(3)

which describes a semi-metal (due to band overlap) for $|M|\leq M_{c}\equiv
3\sqrt{3}t_{2}\sin\Phi$, and a trivial insulator otherwise. In the gapless
phase, antichiral edge states emerge at the boundaries of a ribbon structure
Franz . In contrast with the chiral edge states, these edge channels propagate
in the same direction and are counterbalanced by bulk modes Franz .

Since $a_{\mathbf{k}}=-a_{-\mathbf{k}}$ for $\Phi\not\equiv 0\mod{[\pi]}$, the
HM and the mHM break TRS as
$\mathcal{T}^{\dagger}H_{\alpha}(\mathbf{k})\mathcal{T}\neq
H_{\alpha}(-\mathbf{k})$, where $\alpha=HM,mMH$, $\mathcal{T}=K$ and $K$
denotes complex conjugation. Charge conjugation $\mathcal{C}=\sigma_{z}K$ and
the sublattice chiral symmetries $\mathcal{S}=\sigma_{z}$ are also broken as
$\mathcal{C}^{\dagger}H_{\alpha}(\mathbf{k})\mathcal{C}\neq-
H_{\alpha}(-\mathbf{k})$ and
$\mathcal{S}^{\dagger}H_{\alpha}(\mathbf{k})\mathcal{S}\neq-
H_{\alpha}(\mathbf{k})$. The HM belongs to the A class of topological
insulators AZ  characterized by a $\mathbb{Z}$ invariant (Chern number) HM
symm  while the mHM is a semi-metal or a trivial insulator as stated before.

As the AA-stacked mHM bilayer, discussed in section III of the Supplemental
Material supp , does not show nontrivial topological behavior, we consider the
AB-stacked bilayer and denote by $\Phi_{l}$ ($l=1,2$) the complex phase in
layer ($l$). The mHM in Bernal bilayer, is described by the following Bloch
Hamiltonian written, in the basis of the four orbitals forming the unit cell
($A_{1},B_{1},A_{2},B_{2}$), (Fig. 2 (a)), as

$\displaystyle
H_{B}(\mathbf{k})=\begin{pmatrix}A_{1,\mathbf{k}}+M_{1}&f_{\mathbf{k}}&0&2t_{\perp}\\\
f^{\ast}_{\mathbf{k}}&A_{1,\mathbf{k}}-M_{1}&0&0\\\
0&0&A_{2,\mathbf{k}}+M_{2}&f_{\mathbf{k}}\\\
2t_{\perp}&0&f^{\ast}_{\mathbf{k}}&A_{2,\mathbf{k}}-M_{2}\end{pmatrix},$ (4)

where we only considered the interlayer coupling $2t_{\perp}$ between dimer
sites $\left(A_{1},B_{2}\right)$ mHM BA . Here $M_{l}$ ($l=1,2$) is the layer
Semenoff mass, and $A_{l,\mathbf{k}}=a_{l,\mathbf{k}}+a^{0}_{l,\mathbf{k}}$,
where

$\displaystyle a_{l,\mathbf{k}}$ $\displaystyle=$
$\displaystyle-2t_{2}\sin\Phi_{l}\sum_{i}\sin\left(\mathbf{k}\cdot\mathbf{a}_{i}\right),$
$\displaystyle a^{0}_{l,\mathbf{k}}$ $\displaystyle=$ $\displaystyle
2t_{2}\cos\Phi_{l}\sum_{i}\cos\left(\mathbf{k}\cdot\mathbf{a}_{i}\right).$ (5)

Introducing the layer pseudospin $\boldsymbol{\tau}$ Pauli matrices and the
corresponding identity matrix $\tau_{0}$, $H_{B}(\mathbf{k})$ (Eq. 4) reduces
to

$\displaystyle H_{B}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\left(b_{\mathbf{k}}\sigma_{x}+c_{\mathbf{k}}\sigma_{y}\right)\tau_{0}+2t_{\perp}\left(\sigma_{+}\tau_{+}+\sigma_{-}\tau_{-}\right)$
(6) $\displaystyle+$
$\displaystyle\frac{1}{2}\left(A_{1}+A_{2}\right)\sigma_{0}\tau_{0}+\frac{1}{2}\left(A_{1}-A_{2}\right)\sigma_{0}\tau_{z}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left(M_{1}+M_{2}\right)\sigma_{z}\tau_{0}+\frac{1}{2}\left(M_{1}-M_{2}\right)\sigma_{z}\tau_{z},$

where $\sigma_{\pm}=\frac{1}{2}\left(\sigma_{x}\pm i\sigma_{y}\right)$ and
$\tau_{\pm}=\frac{1}{2}\left(\tau_{x}\pm i\tau_{y}\right)$.

This Hamiltonian has some similarity with the Hamiltonian given by Eq. 1 in
Ref. SOC, , where the authors have studied a modified Kane and Mele Kane
model of a graphene layer on a substrate. The Hamiltonian of Ref. SOC,
belongs to the class AII AZ ; AZ2  characterized by a $\mathbb{Z}_{2}$
invariant, whereas the Hamiltonian given by Eq. 6 belongs to class A AZ ; AZ2
labeled by a $\mathbb{Z}$ invariant. Indeed, since
$a_{l,-\mathbf{k}}=-a_{l,\mathbf{k}}$ (Eq. 5), the system breaks TRS,
$\mathcal{T}=K$, the charge conjugation represented by
$\mathcal{C}=\sigma_{z}\tau_{0}K$, with $\mathcal{C}^{2}=\mathds{1}$ and the
chirality $\mathcal{S}=\tau_{0}\sigma_{z}$ footnote sym .

To discuss the topological class of the system, one needs to analyze the
presence of gaps in the energy spectrum of the Hamiltonian $H_{B}(\mathbf{k})$
(Eq. 6) and, in particular, for vanishing Semenoff masses ($M_{l}=0,\,l=1,2$)
where both layers are semi-metals.

We consider, for simplicity, the case where
$\Phi_{1}=-\Phi_{2}=\pm\frac{\pi}{2}$ to drop the energy shift term
$a^{0}_{l,\mathbf{k}}$ which does not affect the band topology. In this case,
the energy spectrum of $H_{B}(\mathbf{k})$ (Eq. 6) shows a particle-hole
symmetry footnote Eg  and is given by

$\displaystyle
E_{\alpha_{1},\alpha_{2}}(\mathbf{k})=\alpha_{1}\sqrt{A_{\mathbf{k}}+\alpha_{2}B_{\mathbf{k}}},$
(7)

where $\alpha_{i}=\pm 1$ and

$\displaystyle
A_{\mathbf{k}}=a^{2}_{\mathbf{k}}+|f_{\mathbf{k}}|^{2}+2t_{\perp}^{2},B_{\mathbf{k}}=2\sqrt{|f_{\mathbf{k}}|^{2}\left(a^{2}_{\mathbf{k}}+t_{\perp}^{2}\right)+t_{\perp}^{4}}.$
(8)

The eigenenergies given by Eq. 7 obey the inequalities
$E_{-,+}(\mathbf{k})\leq E_{-,-}(\mathbf{k})\leq E_{+,-}(\mathbf{k})\leq
E_{+,+}(\mathbf{k})$ and the gap separating the energy bands around the zero-
energy is
$\Delta=\min_{\mathbf{k}}\left(\Delta_{\mathbf{k}}\right)=2\sqrt{A_{\mathbf{k}}-B_{\mathbf{k}}}$.
The gap closing condition is

$\displaystyle\left(|f_{\mathbf{k}}|^{2}-a^{2}_{\mathbf{k}}\right)^{2}=-4a^{2}_{\mathbf{k}}t_{\perp}^{2},$
(9)

which can be satisfied in two cases: (i) for $a_{\mathbf{k}}=0$ ($t_{2}=0$),
which corresponds to an AB graphene bilayer with only nearest neighboring (NN)
hopping terms and a quadratic contact point between the bands at zero energy
McCann ; (ii) for uncoupled layers ($t_{\perp}=0$) where Eq. 9 reduces to
$|a_{\mathbf{k}}|=|f_{\mathbf{k}}|$, which defines two non-intersecting closed
loops in the Brillouin zone (BZ) and results into two valley closed Fermi
lines, originating from two overlapping bands (Fig. 1). The gap (Eq. 9) is
therefore finite as soon as $t_{\perp}$ and $t_{2}$ are non-zero footnote . It
comes out that the mHM in AB stacked bilayer, with opposite complex phases
($\Phi_{1}=-\Phi_{2}$), is gapped by the interlayer hopping process which
hybridizes the conduction and the valence bands, forming nodal lines in each
valley (Fig. 1), and opens a gap due to band repulsion .

At this point, a question arises: what are the possible values of the Chern
number $C$ of the gapped phases resulting from the instability of the Fermi
lines? Numerically, we find that $C=\pm 2$ when the two phases
$\Phi_{l}\in\left[-\pi,\pi\right]$ ($l=1,2$) are of opposite signs
($\Phi_{1}\Phi_{2}<0$) and that the system is gapless when they are of the
same sign.

In order to understand these findings, we proceed in two steps. We first
derive an effective $2\times 2$ Hamiltonian in the limit of large interlayer
coupling ($t_{2}\ll t_{\perp}$) at which it is simple to get an analytical
expression of the Chern number of the highest filled energy band, denoted
$E_{-,-}(\mathbf{k})$ (Eq. 7) around half-filling. Restricting the analytical
calculations to this limit is fully justified in the case of vanishing
Semenoff masses since the topology of the system is unchanged when the ratio
$t_{2}/t_{\perp}$ crosses from $t_{2}/t_{\perp}\ll 1$ to the opposite limit
$t_{2}/t_{\perp}\gg 1$. The reason is that the gap does not close as soon as
$a_{\mathbf{k}}$ or/and $t_{\perp}$ are turned on (Eq. 9), which prevents any
topological phase transition.

In a second step, we calculate the energy-spectrum of the mHM on AB-stacked
nanoribbons to bring out the signature of the chiral edge states corresponding
to the bulk Chern insulating phase.

Effective two-band model. –  In the limit of large interlayer coupling
($t_{2}\ll t_{\perp}$), the energy bands corresponding to the dimer
$(A_{1},B_{2})$, coupled by $t_{\perp}$, are pushed away from zero-energy. The
lowest energy bands around half-filling can then be described by an effective
$2\times 2$ model written in the subspace of the uncoupled orbitals $A_{2}$
and $B_{1}$.

To derive the low energy Hamiltonian, we use the Löwdin partitioning method
Lowdin ; McCann . The effective $2\times 2$ Hamiltonian reduces, as we show in
section II of the supplemental material supp , to

$\displaystyle
H_{\text{eff}}({\mathbf{k}})=-2t_{\perp}\frac{f_{\mathbf{k}}^{2}}{X^{2}}\sigma_{+}-2t_{\perp}\frac{f_{\mathbf{k}}^{\ast
2}}{X^{2}}\sigma_{-}+\left[\frac{1}{2}\left(M_{1}+M_{2}\right)-a_{\mathbf{k}}\left(1-\frac{|f_{\mathbf{k}}|^{2}}{X^{2}}\right)\right]\sigma_{z}+\frac{1}{2}\left(M_{2}-M_{1}\right)\sigma_{0}$
(10)

where the $\boldsymbol{\sigma}$ Pauli matrices are now written in the
$(A_{2},B_{1})$ basis and
$X^{2}=(a_{\mathbf{k}}+M_{1})(a_{\mathbf{k}}+M_{2})+4t_{\perp}^{2}$.

For $M_{1}=M_{2}=0$, the eigenenergies are
$E_{\text{eff},\pm}(\mathbf{k})=\pm\sqrt{a^{2}_{\mathbf{k}}\left(1-\frac{|f_{\mathbf{k}}|^{2}}{X^{2}}\right)^{2}+4t^{2}_{\perp}\frac{|f_{\mathbf{k}}|^{4}}{X^{4}}}$.
$E_{\text{eff},-}$ is equal, to the leading order in
$\frac{|f_{\mathbf{k}}|}{t_{\perp}}$, to $E_{--}(\mathbf{k})$ (Eqs. 7, 8).

In order to characterize the topology of the Hamiltonian given by Eq. (10), we
consider the limit $M_{l},\,t_{2}\ll t_{\perp}$ ($l=1,2$) footnote limit  and
expand $H_{\text{eff}}$ around the Dirac points $\pm\mathbf{K}$, so that Eq.
(10) becomes

$\displaystyle
H_{\text{eff}}(\mathbf{q})=-\frac{\hbar^{2}v^{2}}{t_{\perp}}\left(q^{2}_{x}-q^{2}_{y}\right)\sigma_{x}+2\tau_{z}\frac{\hbar^{2}v^{2}}{t_{\perp}}q_{x}q_{y}\sigma_{y}+\left[\frac{1}{2}\left(M_{1}+M_{2}\right)\mp
3\sqrt{3}t_{2}\tau_{z}\right]\sigma_{z}+\frac{1}{2}\left(M_{2}-M_{1}\right)\sigma_{0},$
(11)

where $\mathbf{q}=\mathbf{k}-\tau\mathbf{K}$ and $\tau$ is the valley index.
The two first terms in Eq. (11) describe the low energy Hamiltonian of a
Bernal bilayer graphene McCann  while the $\sigma_{z}$ term contains both a
Haldane $a_{\mathbf{k}}\sim 3\sqrt{3}t_{2}\tau$ and a Semenoff
$\frac{1}{2}\left(M_{1}+M_{2}\right)$ mass terms.

The Chern number associated to the lowest band of this two-band Hamiltonian is
Sticlet

$\displaystyle C=\sum_{\tau}\frac{1}{2}\chi\operatorname{sgn}(m_{\tau}),$ (12)

where $m_{\tau}=\frac{1}{2}\left(M_{1}+M_{2}\right)\mp 3\sqrt{3}t_{2}\tau$ is
the total mass and $\chi=-2\tau$ is the chirality of this quadratic band
contact point McCann .

For vanishing Semenoff masses ($M_{l}=0$) where topological phase transitions
are prohibited, $\operatorname{sgn}(m_{\tau})=\mp\tau$, which gives $C=\pm 2$.
This result is in agreement with our numerical calculations of the Chern
number $C$ of the four energy bands $E_{\alpha_{1},\alpha_{2}}(\mathbf{k})$
(Eqs. 7, 8): $C_{-,+}=0$, $C_{-,-}=\pm 2$, $C_{+,-}=\mp 2$, and $C_{+,+}=0$
giving rise, for the occupied bands, to a total Chern number $C=2\,(-2)$ for
$\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}(-\frac{\pi}{2})$.

The chiral insulating phases occur in the case where the complex phase
$\Phi_{1}$ and $\Phi_{2}$ have opposite signs and the Chern number of the
lowest energy band is defined as far as the system is gapped footnote V supp .

Zigzag and armchair ribbons. –  Figure 3 shows the band structure of AB
stacked nanoribbons with zigzag boundaries described by the mHM with complex
phases $\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}$. For a non vanishing interlayer
coupling $t_{\perp}$, a gap opens at half filling and two pairs of chiral edge
states emerge.

$\begin{array}[]{cc}\includegraphics[width=108.405pt]{mHM-AB-
ZZa.eps}&\includegraphics[width=108.405pt]{mHM-AB-ZZb.eps}\\\
\includegraphics[width=119.24615pt]{mHM-AB-
ZZc.eps}&\includegraphics[width=97.56383pt]{mHM-AB-ZZ-proba.eps}\end{array}$

Figure 3: Band structure of an AB bilayer mHM on zigzag nanoribbons of a width
$W=60$ atoms. Calculations are done for $t_{2}=0.1t$,
$\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}$, and $M_{1}=M_{2}=0$. The interlayer
hopping is (a) $t_{\perp}=0$, (b) $t_{\perp}=0.5t$ and (c) $t_{\perp}=0.8t$.
(d) Probability distributions of the edge states denoted by A, B, C and D in
(c) located at the energy indicated by the dashed line.

The corresponding probabilities are depicted in Fig. 3 (d) indicating that the
two edge modes, appearing at the left side of the ribbon, have the same
velocity and the right boundary supports also two edge modes having the same
direction of propagation, but counterpropagating with the left side channels.
This feature confirms that the coupled semi-metal ribbons turn into a
topological Chern insulator with a Chern number $C=2$. By flipping the signs
of the complex phases $\Phi_{1}=-\Phi_{2}=-\frac{\pi}{2}$, the direction of
propagation of the chiral edge states at each boundary is reversed, which
results into a Chern number $C=-2$.

In figure 4, we represent the band structure of the mHM bilayer ribbons with
armchair boundaries in the case of $\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}$ and for
vanishing Semenoff masses. Under the interlayer coupling, the system becomes a
$C=2$ Chern insulator.

$\begin{array}[]{cc}\includegraphics[width=108.405pt]{mHM-AB-
AC.eps}\includegraphics[width=108.405pt]{mHM-AB-AC-proba.eps}\end{array}$

Figure 4: (a) Band structure of an AB bilayer mHM on armchair nanoribbons of a
width $W=70$ atoms and for $t_{2}=0.1t$, $t_{\perp}=0.8t$, $M_{1}=M_{2}=0$ and
$\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}$. (b) Corresponding probability
distributions of the edge states denoted by A, B, C and D in (a).

The analytical expression of the Chern number (Eq. 12) is derived in the limit
$M_{l},\,t_{2}\ll t_{\perp}$. To go beyond this limit, we perform numerical
calculations for different values of $M_{l}$, $t_{2}$ and $t_{\perp}$ supp .

For uncoupled layers, the closed Fermi lines survive if the layers remain
semi-metallic, which is the case for $|M_{l}|\leq M_{lc}\equiv
3\sqrt{3}t_{2}|\sin\Phi_{l}|$, ($l=1,2$), where the critical mass $M_{lc}$
marks the transition between the semi-metallic phase ($M_{l}\leq M_{lc}$) to
the gapped phase ($M_{l}>M_{lc}$) of the monolayer mHM. By turning on the
interlayer coupling, our numerical calculations show that, for realistic
hopping integrals ($t_{\perp}\sim t_{2}$ the system is gapped and becomes a
Chern insulator. The corresponding Chern number is $C=\pm 2$ regarding the
presence of two channels of chiral edge states at each boundary of a bilayer
strip. If the Semenoff mass overcomes the Haldane mass $|M_{lc}|<|M_{l}|$, the
monolayers are trivial insulators and the interlayer hopping $t_{\perp}$
brings the mHM bilayer to a trivial gapped phase with a vanishing Chern number
$C=0$, regardless of the stacking order (AA, AB or BA) as we showed in section
IV of the supplemental material supp .

In the case of the AA bilayer mHM, we show in section III of the supplemental
material supp , that under the interlayer hopping, the system turns into a
semi-metal (trivial insulator) in the absence (presence) of Semenoff mass
terms, regardless of the nature of the ribbon’s boundaries.

Discussion. –  We now explain how the chirality emerges due to the stacking
order of the layers. This feature can be understood from the schematic
representation of the interlayer hopping illustrated in Fig. 5, where we
consider the case of opposite complex phases $\Phi_{1}=-\Phi_{2}$. Imagine a
situation of finite $t_{2}$ and of slowly turning on $t_{\perp}$ in order to
see the emergence of a Chern (resp. trivial) insulator in the case of AB
(resp. AA) stacking. In AB-stacked bilayers, the interlayer hopping couples
the sublattice $A_{1}$ and $B_{2}$ where the fluxes flow in the same direction
(see Fig. 5(a)). Consequently, the dimer chirality dominates and gives rise to
a pair of chiral edge states at each layer end. If the interlayer hopping
concerns the $B_{1}$ and the $A_{2}$ atoms (BA-stacking), the flux flows are
flipped in comparison with the case where the dimer is
$\left(A_{1},B_{2}\right)$ and the system gains an opposite chirality.

$\begin{array}[]{c}\includegraphics[width=151.76633pt]{mHM-AB-interp.eps}\\\
\includegraphics[width=151.76633pt]{mHM-AA-interp.eps}\end{array}$

Figure 5: Schematic interpretation of the stacking-dependent chirality of the
mHM bilayers in the case of opposite complex phases $\Phi_{1}=-\Phi_{2}$. (a)
In the AB stacking, the interplane hopping $t_{\perp}$ couples the fluxes of
$A_{1}$ and $B_{2}$ sublattices flowing in the same direction, giving rise to
a dominant clockwise chirality and a pair of chiral edge states in each layer.
(b) In the AA stacked layers, the $A_{1}$ ($B_{1}$) and the $A_{2}$ ($B_{2}$)
sublattices have opposite fluxes with no dominant chirality.

However, in the AA stacked bilayer mHM, the $A_{1}$ and the $A_{2}$
sublattices have opposite fluxes which cannot result in a dominant chirality
(see Fig. 5 (b)), and the system, if gapped, cannot support chiral edge
states. In Ref. supp, , we discuss, in sections I and III, the topology of the
HM and the mHM in AA stacked bilayers.

Conclusion. –  We discussed the topological properties of the modified Haldane
model Franz  in a bilayer of honeycomb lattice where the TRS is broken,
oppositely, in the uncoupled semi-metallic layers. We found that, in the AB
stacked bilayer, the interlayer hopping drives the system into a topological
insulator, with a Chern number $C=\pm 2$. The smoking gun of this insulating
phase is the systematic emergence of two channels of chiral edge states at the
boundaries of the AB bilayer ribbons, regardless of the boundary nature
(zigzag/armchair) of the ribbons bearded ; bearded-Ref . However, the modified
Haldane model in the AA stacked bilayer is found to be a semi-metal or a
trivial insulator depending on the value of the Semenoff masses. Our results
are schematically summarized in Fig. 6, where we also give the behavior of the
HM in AA and AB stacked bilayers. This stacking-induced Chern insulator could
be readily realized in bilayers of electrical circuits Private-elec  and
photonic crystals Private-pho  by coupling two time reversal copies of the mHM
which was already implemented in a microwave-scale gyromagnetic photonic
crystal Zhou  and in electrical circuits Yang21 . An experimental
implementation with real material can be achieved in the hexagonal
$\text{WSe}_{2}$, where the Dirac cone shift is due to spin orbit coupling
Franz . We propose that a Chern insulator with $C=\pm 2$ can be hosted by a
Bernal bilayer of $\text{WSe}_{2}$ where one layer is hole-doped while the
other is electron-doped. The present work could be extended to multilayer
structures of semi-metals with broken TRS, opening the way to tunable Chern
insulators, as realized with heterostructures of topological insulators Ctunue
. We also expect a twist-induced Chern insulator Guy  in a moiré superlattice
of twisted honeycomb bilayers Herrero , where AA stacked domains form a
triangular lattice alternating with AB and BA regions. A gapless (or a gapped
trivial insulating) state, emerging in the AA domains, may coexist with Chern
insulating phases in the AB and the BA regions with, respectively, a Chern
number $C=\pm 2$ and $C=\mp 2$ Guy .

Figure 6: Summary of the stacking-dependent properties of the Haldane (HM) and
the modified Haldane (mHM) models in bilayer honeycomb lattices. $\Phi_{l}$
($l=1,2$) is the complex phase of the NNN hopping term in layer $l$, $C_{l}$
is the corresponding Chern number while $C$ is the total Chern number of the
lowest occupied bands at half-filling. $C=\emptyset$ means an undefined Chern
number. The yellow cells indicate the case showing unexpected topological
behavior (i.e. $C\neq C_{1}+C_{2}$), which has been studied in the main text.
The other cases are discussed in the Supplemental Material supp .

Acknowledgments. –  This work was supported by the Tunisian Ministry of Higher
Education and Scientific Research. S. H. acknowledges the LPS in Orsay for
financial support and kind hospitality. We thank A. Meszaros for fruitful
discussions and we acknowledge B. Zhang, Y. Chong and Y. Yang for discussions
about the experimental implementation of our results.

## References

  * (1) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * (2) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
  * (3) A. Bansil, H. Lin, and T. Das, Rev. Mod. Phys. 88, 021004 (2016).
  * (4) K. He, Y. Wang and Q.-K. Xue, Annu. Rev. Condens. Matter Phys. 9, 329 (2018).
  * (5) Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, Phys. Rev. Lett. 114, 114301 (2015).
  * (6) A. Khanikaev, R. Fleury, S. Mousavi, and A. Alu, Nat. Commun. 6, 8260 (2015).
  * (7) X. Ni, C. He, X.-C. Sun, X.-P. Liu, M.-H. Lu, L. Feng and Y.-F. Chen, New J. Phys. 17 053016 (2015).
  * (8) G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, T. Uehlinger, D. Greif and T. Esslinger, Nature 515, 237 (2014).
  * (9) H.-S. Kim and H-Y Kee, npj Quantum Materials, 2, 20 (2017).
  * (10) Y. Ding, Y. Peng, Y. Zhu, X. Fan, J. Yang, B. Liang, X. Zhu, X. Wan, and J. Cheng, Phys. Rev. Lett. 122, 014302 (2019).
  * (11) M. Serlin, C. L. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu, K. Watanabe, T. Taniguchi, L. Balents, and A. F. Young, Science 367, 900 (2020).
  * (12) C. L. Tschirhart, M. Serlin, H. Polshyn, A. Shragai, Z. Xia, J. Zhu, Y. Zhang, K. Watanabe, T. Taniguchi, M. E. Huber, and A. F. Young, Science 372, 1323 (2021).
  * (13) Y. Okazaki, T. Oe, M., Kawamura, R. Yoshimi, S. Nakamura, S. Takada, M. Mogi, K. S. Takahashi, A. Tsukazaki, M. Kawasaki, Y. Tokura and N.-H. Kaneko et al., Nat. Phys. 18, 25 (2022).
  * (14) Z. Li, Y. Han, and Z. Qiao, Phys. Rev. Lett. 129, 036801 (2022).
  * (15) F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).
  * (16) E. Colomés and M. Franz, Phys. Rev. Lett. 120, 086603 (2018).
  * (17) S. Bhattacharjee, S.Bandyopadhyay, D. Sen, and A. Dutta, Phys. Rev. B 103, 224304 (2021).
  * (18) In section II of the supplemental material supp , we show that $C=C_{1}+C_{2}$ regardless of the layer stacking order (AA/AB).
  * (19) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005); ibid, 146802 (2005).
  * (20) The system breaks TRS and parity $\mathcal{P}$ even for $M=0$ and it is invariant under $\mathcal{TP}$ transformation for $M=0$. For $\Phi\equiv\frac{\pi}{2}\mod{[\pi]}$ and $M=0$, $H_{mH}(\mathbf{k})$ the Hamiltonian given by Eq. 3 conserves charge-conjugation symmetry as the HM.
  * (21) A. Altland, and M. R. Zirnbauer, Phys. Rev. B 55, 1142 (1997).
  * (22) For $\Phi\equiv\frac{\pi}{2}\mod{[\pi]}$ and $M=0$, the HM switches to the D class AZ ; AZ2  where the charge-conjugation symmetry is conserved. Moreover, in the absence of Semenoff mass, the HM conserves parity $\mathcal{P}=\sigma_{x}$: $\mathcal{P}^{\dagger}H_{H}(\mathbf{k})\mathcal{P}=H_{H}(-\mathbf{k})$ but is not invariant under $\mathcal{TP}$ transformation.
  * (23) For details, see the Supplemental Material.
  * (24) In section V of the supplemental material supp , we present a generic Hamiltonian $H_{B}(\theta,\mathbf{k})$ interpolating continuously between AB $(\theta=0)$ and BA $(\theta=\frac{\pi}{2})$ stackings, without going through the AA stacking.
  * (25) T. Frank, P. Högl, M. Gmitra, D. Kochan, and J. Fabian, Phys. Rev. Lett. 120, 156402 (2018).
  * (26) C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016).
  * (27) In the case of opposite complex phases $\Phi_{1}=-\Phi_{2}$ and masses $M_{1}=-M_{2}$, the system is invariant under inversion $\mathcal{P}=\sigma_{x}\tau_{x}$. Moreover, in the absence of Semenoff masses and for $\Phi_{1}=-\Phi_{2}=\pm\frac{\pi}{2}$, the energy-shift terms ($a^{0}_{l,\mathbf{k}}=0$ (Eq. 5)) vanish, and the system becomes invariant under charge conjugation, which results in a $\mathcal{CP}=\tau_{x}\sigma_{y}K$ symmetry.
  * (28) In the presence of the Semenoff masses, the energy-spectrum has no more the particle-hole symmetry since the charge-conjugation symmetry is broken. However, it keeps the inversion symmetry for $M_{1}=-M_{2}$.
  * (29) E. McCann and V. I. Fal’ko, Phys. Rev. Lett. 96, 086805 (2006).
  * (30) Taking into account the Semenoff masses, a simple analytical expression of the energy spectrum can be derived for $\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}$ and $M_{1}=M_{2}=M$, where the charge-conjugation and the parity symmetries are broken. The $A_{\mathbf{k}}$ and $B_{\mathbf{k}}$ terms given by Eq. (8) become: $A_{\mathbf{k}}=a^{2}_{\mathbf{k}}+|f_{\mathbf{k}}|^{2}+2t_{\perp}^{2}+M^{2}$ and $B_{\mathbf{k}}=2\sqrt{|f_{\mathbf{k}}|^{2}\left(a^{2}_{\mathbf{k}}+t_{\perp}^{2}\right)+t_{\perp}^{4}+a^{2}_{\mathbf{k}}M^{2}+2a_{\mathbf{k}}t_{\perp}^{2}M}$.
  * (31) P.‐O. Löwdin, J. Chem. Phys. 19, 1396 (1951).
  * (32) In this limit, it is possible to derive, around the Dirac points, a simple form of the Hamiltonian given by Eq. 10.
  * (33) D. Sticlet, F. Piéchon, J.-N. Fuchs, P. Kalugin, and P. Simon, Phys. Rev. B 85, 165456 (2012).
  * (34) For details see section V of the supplemental material supp .
  * (35) Stacking-induced Chern insulator is expected to occur in mHM on AB bilayer ribbons with bearded edges regarding the presence of zero energy states as in the zigzag ribbons. However, we did not focus on this type of ribbons since they are less stable compared to the zigzag and the armchair ones bearded-Ref .
  * (36) M. Kohmoto and Y. Hasegawa, Phys. Rev. B 76, 205402 (2007).
  * (37) Y. Chong and Y. Yang, private communication.
  * (38) B. Zhang, private communication
  * (39) P. Zhou, G.-G. Liu, Y. Yang, Y.-H. Hu, S. Ma, H. Xue, Q. Wang, L. Deng, and B. Zhang Phys. Rev. Lett. 125, 263603 (2020).
  * (40) Y. Yang, D. Zhu, Z. Hang, and Y. Chong, Sci. China Phys. Mech. Astron. 64, 257011 (2021).
  * (41) Y.-F. Zhao, R. Zhang, R. Mei, L. -J. Zhou, H. Yi, Y.-Q. Zhang, J. Yu, R. Xiao, K. Wang, N. Samarth, M. H. W. Chan, C.-X. Liu, and C.-Z. Chang, Nature 588, 419 (2020).
  * (42) M. Mannaï, G. Trambly de laissardière, S. Haddad, J.-N. Fuchs and F. Piéchon, in preparation.
  * (43) Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E. Kaxiras, and P. Jarillo-Herrero, Nature, 556, 43 (2018).

## Supplemental material for stacking-induced Chern insulator

##  I I. Haldane model in bilayer honeycomb lattice

We consider a HM bilayer with AB and AA stackings where the layers are assumed
to have complex NNN phases $\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}$, to drop the
global energy shift ($a^{0}_{\mathbf{k}}=0$ (Eq. 5)) which does not affect the
topology of the system. In the basis of the four orbitals forming the unit
cell ($A_{1},B_{1},A_{2},B_{2}$) the corresponding Hamiltonians can be written
as

$\displaystyle H_{\text{AA-
HM}}(\mathbf{k})=\begin{pmatrix}a_{\mathbf{k}}+M_{1}&f_{\mathbf{k}}&2t_{\perp}&0\\\
f^{\ast}_{\mathbf{k}}&-a_{\mathbf{k}}-M_{1}&0&2t_{\perp}\\\
2t_{\perp}&0&-a_{\mathbf{k}}+M_{2}&f_{\mathbf{k}}\\\
0&2t_{\perp}&f^{\ast}_{\mathbf{k}}&a_{\mathbf{k}-M2}.\end{pmatrix}$ (S.1)

$\displaystyle H_{\text{AB-
HM}}(\mathbf{k})=\begin{pmatrix}a_{\mathbf{k}}+M_{1}&f_{\mathbf{k}}&0&2t_{\perp}\\\
f^{\ast}_{\mathbf{k}}&-a_{\mathbf{k}}-M_{1}&0&0\\\
0&0&-a_{\mathbf{k}}+M_{2}&f_{\mathbf{k}}\\\
2t_{\perp}&0&f^{\ast}_{\mathbf{k}}&a_{\mathbf{k}-M2}.\end{pmatrix}$ (S.2)

These Hamiltonians can be expressed, using the layer and the sublattice
pseudospin matrices $\boldsymbol{\sigma}$ and $\boldsymbol{\tau}$, as

$\displaystyle H_{\text{AA-HM}}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\left(b_{\mathbf{k}}\sigma_{x}+c_{\mathbf{k}}\sigma_{y}\right)\tau_{0}+2t_{\perp}\sigma
0\tau_{x}+a_{\mathbf{k}}\sigma_{z}\tau_{z}+\frac{1}{2}\left(M_{1}+M_{2}\right)\sigma_{z}\tau_{0}+\frac{1}{2}\left(M_{1}-M_{2}\right)\sigma_{z}\tau_{z},$
(S.3) $\displaystyle H_{\text{AB-HM}}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\left(b_{\mathbf{k}}\sigma_{x}+c_{\mathbf{k}}\sigma_{y}\right)\tau_{0}+t_{\perp}\left(\sigma_{x}\tau_{x}-\sigma_{y}\tau_{y}\right)+a_{\mathbf{k}}\sigma_{z}\tau_{z}+\frac{1}{2}\left(M_{1}+M_{2}\right)\sigma_{z}\tau_{0}+\frac{1}{2}\left(M_{1}-M_{2}\right)\sigma_{z}\tau_{z}.$
(S.4)

where $a_{\mathbf{k}}$ is given by Eq. 5 in the main text.

$H_{\text{AA-HM}}$ ($H_{\text{AB-HM}}$) breaks TRS $\mathcal{T}=K$, the charge
conjugation, represented by $\mathcal{C}=\sigma_{z}\tau_{z}K$
($\mathcal{C}=\sigma_{z}\tau_{0}K$) with $\mathcal{C}^{2}=\mathds{1}$, and the
chirality $\mathcal{S}=\tau_{z}\sigma_{z}$ ($\mathcal{S}=\tau_{0}\sigma_{z}$).

In the following, we will show based on numerical band structure calculations
on bilayer ribbons, that coupling two HM with opposite chiralities
($C_{1}=-C_{2}$), resulting from oppositely broken TRS ($\Phi_{1}=-\Phi_{2}$),
gives rise, as expected, to a trivial Chern insulator with $C=C_{1}+C_{2}=0$.
We will discuss the stacking order, the nature of the ribbon edges (zigzag or
armchair) and the effect of the intralayer Semenoff masses $M_{l}$, where
$l=1,2$ is the layer index. The case of AA stacking was discussed in Ref.
Dutta,  for a fixed value of the mass term $M_{l}$.

Figure S.1 shows the band structure of the AB bilayer HM on zigzag ribbons for
$\Phi_{1}=\Phi_{2}=\frac{\pi}{2}$, $M_{1}=M_{2}=0$ and at different values of
the interlayer hopping $t_{\perp}$ . Starting from uncoupled ($t_{\perp}=0$)
chiral layers, with equal Chern number $C_{1}=C_{2}=\pm 1$, the system turns,
under the interlayer coupling, into a Chern insulator with a Chern number
$C=\pm 2$ characterized by a pair of chiral edge states propagating at the
boundaries of each layer as shown in Fig. 6 of the main text.

$\begin{array}[]{ccc}\includegraphics[width=65.04366pt]{HM-AB-
ZZ-a.eps}\includegraphics[width=65.04366pt]{HM-AB-
ZZ-b.eps}\includegraphics[width=65.04366pt]{HM-AB-ZZ-c.eps}\end{array}$

Figure S.1: Tight binding calculations of the electronic band structure of an
AB bilayer HM on zigzag nanoribbons of a width $W=60$ atoms. The interlayer
hopping is (a) $t_{\perp}=0$, (b) $t_{\perp}=0.5t$ and (c) $t_{\perp}=0.8t$.
Calculations are done for $\Phi_{1}=\Phi_{2}=\frac{\pi}{2}$, $M_{1}=M_{2}=0$
and $t_{2}=0.1t$, where $t$ is the NN hopping integral.

The $C=\pm 2$ Chern insulating phase occurs as far as the Semenoff mass
$|M_{l}|<|M_{lc}|$, where

$\displaystyle M_{lc}=3\sqrt{3}t_{2}\sin\Phi_{l}.$ (S.5)

$M_{lc}$ is the critical mass at which the transition from a topological phase
($C_{l}=\pm 1$) to a trivial gapped phase ($C_{l}=0$), takes place in the
monolayer HM Haldane .

This feature is depicted in Fig. S.2, showing that the pair of chiral edge
states appears at the boundaries of the bilayer ribbons only if both layers
have the same chirality Haldane .

$\begin{array}[]{ccc}\includegraphics[width=65.04366pt]{HM_AB_mass_a.eps}\includegraphics[width=65.04366pt]{HM_AB_mass_b.eps}&\includegraphics[width=65.04366pt]{HM_AB_mass_c.eps}\\\
\includegraphics[width=65.04366pt]{HM_AB_mass_d.eps}\includegraphics[width=65.04366pt]{HM_AB_mass_e.eps}&\includegraphics[width=65.04366pt]{HM_AB_mass_f.eps}&\end{array}$

Figure S.2: Electronic band structure of an AB bilayer HM on zigzag
nanoribbons of a width $W=60$ atoms. Calculations are done for $t_{2}=0.1t$,
$\Phi_{1}=\Phi_{2}=\frac{\pi}{2}$, $t_{\perp}=0.5t$ and for (a)
$M_{1}=M_{2}=0$, (b) $M_{1}=M_{2}=\sqrt{3}t_{2}$, (c)
$M_{1}=-M_{2}=\sqrt{3}t_{2}$, (d) $M_{1}=0,M_{2}=3\sqrt{3}t_{2}$, (e)
$M_{1}=0,M_{2}=5\sqrt{3}t_{2}$, (f)
$M_{1}=5\sqrt{3}t_{2},M_{2}=5\sqrt{3}t_{2}$.

This behavior is independent of the nature (zigzag or armchair) of the ribbon
boundaries as shown in Fig. S.3.

$\begin{array}[]{cc}\includegraphics[width=65.04366pt]{HM-AB-
ZZ-b.eps}\includegraphics[width=65.04366pt]{HM_AC_AB_b.eps}\end{array}$

Figure S.3: Electronic band structure of an AB bilayer HM on (a) zigzag and
(b) armchair nanoribbons of a width $W=60$ atoms. Calculations are done for
$t_{2}=0.1t$, $t_{\perp}=0.5t$, $\Phi_{1}=\Phi_{2}=\frac{\pi}{2}$ and
$M_{1}=M_{2}=0$.

Regardless of the stacking type (AB or AA), the bilayer HM is Dutta : $(i)$ a
trivial insulator, if the layers have opposite Chern numbers $C_{1}=-C_{2}$,
$(ii)$ a topological chiral insulator with $C=\pm 2$, if the layers have the
same chirality ($C_{1}=C_{2}$), $(iii)$ and a Chern insulator with $C=\pm 1$
if one layer has a non-vanishing Chern number $C_{1}=\pm 1$ and the other
layer is a trivial insulator $C_{2}=0$, as depicted in Fig. S.4 showing the
band structure of an AA bilayer HM on zigzag ribbons.

$\begin{array}[]{cc}\includegraphics[width=65.04366pt]{HM-
AA-a.eps}\includegraphics[width=65.04366pt]{HM-AA-b.eps}\\\
\includegraphics[width=65.04366pt]{HM-
AA-c.eps}\includegraphics[width=65.04366pt]{HM-AA-d.eps}\end{array}$

Figure S.4: Electronic band structure of an AA Bilayer HM on zigzag
nanoribbons of a width $W=60$ atoms for $t_{\perp}=0.5t$, (a)
$\Phi_{1}=\Phi_{2}=\frac{\pi}{2}$, $M_{1}=M_{2}=0$, (b)
$\Phi_{1}=\Phi_{2}=-\frac{\pi}{2}$, $M_{1}=M_{2}=0$, (c)
$\Phi_{1}=\Phi_{2}=\frac{\pi}{2}$, $M_{1}=0$, $M_{2}=5\sqrt{3}t_{2}$ and (d)
$\Phi_{1}=\frac{\pi}{2},\Phi_{2}=0$, $M_{1}=\sqrt{3}t_{2}$, $M_{2}=0$.
Calculations are done for $t_{2}=0.1t$ in (a), (b) and (d) and $t_{2}=0.2t$ in
(c).

##  II  II. Modified Haldane model in AB stacked layers

To derive the low energy Hamiltonian given by Eq. 10 in the main text, we use
the Löwdin partitioning method Lowdin ; McCann  in the case of bilayer
graphene. For simplicity, we consider the case
$\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}$ to remove the energy-shift terms
$a^{0}_{l,\mathbf{k}}$ (Eq. 5). We rewrite the full Hamiltonian (Eq. 4), in
the basis ($A_{2},B_{1},A_{1},B_{2}$) as

$\displaystyle
H_{B}(\mathbf{k})=\begin{pmatrix}H_{\alpha\alpha}&H_{\alpha\beta}\\\
H_{\beta\alpha}&H_{\beta\beta}\end{pmatrix},$ (S.6)

where

$\displaystyle H_{\alpha\alpha}=\begin{pmatrix}-a_{\mathbf{k}}+M_{2}&0\\\
0&a_{\mathbf{k}}-M1\end{pmatrix},\,H_{\alpha\beta}=H_{\beta\alpha}\begin{pmatrix}0&f_{\mathbf{k}}\\\
f^{\ast}_{\mathbf{k}}&0\end{pmatrix},\,H_{\beta\beta}=\begin{pmatrix}a_{\mathbf{k}}+M_{1}&t_{\perp}\\\
t_{\perp}&-a_{\mathbf{k}}-M_{2}\end{pmatrix}.$ (S.7)

The corresponding effective Hamiltonian is McCann

$\displaystyle
H_{\text{eff}}({\mathbf{k}},E)=H_{\alpha\alpha}+H_{\alpha\beta}\left(E-H_{\beta\beta}\right)^{-1}H_{\beta\alpha},$
(S.8)

which reduces in the limit $M_{l}\sim t_{2}\ll t_{\perp}$ ($l=1,2$), and for
$E\sim 0$ to

$\displaystyle H_{\text{eff}}({\mathbf{k}},E=0)\equiv
H_{\text{eff}}({\mathbf{k}})\sim
H_{\alpha\alpha}-\frac{1}{X^{2}}H_{\alpha\beta}H_{\beta\beta}H_{\beta\alpha},$
(S.9)

where
$X^{2}=\left(a_{\mathbf{k}}+M_{1}\right)\left(a_{\mathbf{k}}+M_{2}\right)+4t^{2}_{\perp}$.

Assuming $M_{l}\frac{|f_{\mathbf{k}}|^{2}}{X^{2}}\ll M_{l^{\prime}}\sim
t_{2}\,(l,l^{\prime}=1,2)$, the corresponding effective Hamiltonian gives rise
to Eq. 10 of the main text.

##  III III. Modified Haldane model in AA stacked layers

$\begin{array}[]{cc}\includegraphics[width=65.04366pt]{mHM-AA-
ZZ-a.eps}\includegraphics[width=65.04366pt]{mHM-AA-AC-b.eps}\\\
\includegraphics[width=65.04366pt]{mHM-AA-
ZZ-c.eps}\includegraphics[width=65.04366pt]{mHM-AA-ZZ-d.eps}\end{array}$

Figure S.5: Band structure of the mHM on AA nanoribbons of a width of $W=60$
atoms with (a,c,d) zigzag and (b) armchair boundaries. Calculations are done
for $t_{2}=0.1t$, $t_{\perp}=0.5t$, (a) and (b)
$\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}$, $M_{1}=M_{2}=0$, while in (c)
$\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}$, $M_{1}=-M_{2}=\sqrt{3}t_{2}$ and (d)
$\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}$, $M_{1}=0$, $M_{2}=\sqrt{3}t_{2}$.

Figure S.5 shows the band structure of the mHM in AA stacked ribbons with
zigzag and armchair boundaries in the case of opposite complex phases
$\Phi_{1}=-\Phi_{2}$. In the absence of the Semenoff masses ($M_{1}=M_{2}=0$),
the system remains gapless under the interlayer coupling. However, it turns to
a trivial insulator if the layers have Semenoff mass terms.

Therefore, in the absence of the Semenoff masses, the Fermi surface (Fig. 1)
of the mHM in AA stacked bilayer is, contrary to the AB stacking, stable
against the interlayer hopping which cannot induce a gap opening.

To understand the Fermi surface stability, we start by writing the
corresponding Hamiltonian in the basis of the four orbitals forming the unit
cell ($A_{1},B_{1},A_{2},B_{2}$) and we consider, for simplicity, the case of
opposite complex NNN phases $\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}$ to have a
vanishing global energy shift ($a^{0}_{\mathbf{k}}=0$ (Eq. 5))

$\displaystyle H_{AA-
mHM}(\mathbf{k})=\begin{pmatrix}a_{\mathbf{k}}&f_{\mathbf{k}}&2t_{\perp}&0\\\
f^{\ast}_{\mathbf{k}}&a_{\mathbf{k}}&0&2t_{\perp}\\\
2t_{\perp}&0&-a_{\mathbf{k}}&f_{\mathbf{k}}\\\
0&2t_{\perp}&f^{\ast}_{\mathbf{k}}&-a_{\mathbf{k}}\end{pmatrix}.$ (S.10)

This Hamiltonian can be written, using the layer and the sublattice pseudospin
matrices $\boldsymbol{\sigma}$ and $\boldsymbol{\tau}$, as

$\displaystyle H_{\text{AA-mHM}}(\mathbf{k})$ $\displaystyle=$
$\displaystyle\left(b_{\mathbf{k}}\sigma_{x}+c_{\mathbf{k}}\sigma_{y}\right)\tau_{0}+2t_{\perp}\sigma_{0}\tau_{x}+a_{\mathbf{k}}\sigma_{0}\tau_{z},$
(S.11)

where $a_{\mathbf{k}}$ is given by Eq. 5 in the main text.

The Hamiltonian of Eq. S.11 breaks TRS, $\mathcal{T}=K\tau_{x}$, the charge
conjugation, represented by $\mathcal{C}=\sigma_{z}\tau_{z}K$ with
$\mathcal{C}^{2}=\mathds{1}$, and the chirality
$\mathcal{S}=\tau_{z}\sigma_{z}$.

The gap separating the two bands, $E_{-,-}(\mathbf{k})$ and
$E_{+,-}(\mathbf{k})$, around the zero energy is
$\Delta=\mathrm{min}_{\mathbf{k}}\left(\Delta_{\mathbf{k}}\right)$, where

$\displaystyle\Delta_{\mathbf{k}}=2\sqrt{A_{\mathbf{k}}-B_{\mathbf{k}}},\;A_{\mathbf{k}}=a^{2}_{\mathbf{k}}+|f_{\mathbf{k}}|^{2}+4t^{2}_{\perp},\;B_{\mathbf{k}}=2|f_{\mathbf{k}}|\sqrt{a^{2}_{\mathbf{k}}+4t^{2}_{\perp}}$
(S.12)

$\Delta_{\mathbf{k}}=0$ leads to

$\displaystyle|f_{\mathbf{k}}|^{2}=a^{2}_{\mathbf{k}}+4t^{2}_{\perp},$ (S.13)

which defines a closed Fermi line.

For $a_{\mathbf{k}}=0$, Eq. S.13 corresponds to the Fermi line of the AA
graphene bilayer in the absence of NNN hopping terms.

For $t_{\perp}=0$, Eq. S.13 describes the mHM in AA bilayer with a particle-
hole Fermi line obeying to $|f_{\mathbf{k}}|=|a_{\mathbf{k}}|$.

By turning on $t_{\perp}$, this Fermi line is, simply, shifted but cannot be
gapped (Eq. S.13). The mHM on AA bilayer remains, then, metallic for vanishing
Semenoff masses.

##  IV IV. Modified Haldane model in AB stacked layers: effect of Semenoff
masses

The effect of Semenoff mass on the mHM in monolayer graphene nanoribbon is
represented in Fig. S.6 showing that the mass term lifts the degeneracy of the
antichiral edge modes which survive as far as $M<M_{c}\equiv
3\sqrt{3}t_{2}\sin\Phi$.

$\begin{array}[]{cccc}\includegraphics[width=43.36133pt]{mHM-ML-
mass-a.eps}\includegraphics[width=43.36133pt]{mHM-ML-
mass-b.eps}\includegraphics[width=43.36133pt]{mHM-ML-
mass-c.eps}\includegraphics[width=43.36133pt]{mHM-ML-mass-d.eps}\end{array}$

Figure S.6: Band structure of the mHM in monolayer graphene nanoribbon with
zigzag boundaries and a width $W=60$ atoms. Calculations are done for
$t_{2}=0.1t$, $\Phi=\frac{\pi}{2}$, (a) $M=0$, (b) $M=0.1t_{2}$, (c)
$M=M_{c}\equiv 3\sqrt{3}t_{2}$, and (d) $M_{2}=4\sqrt{3}t_{2}$.

In Fig. S.7, we plot the band structure of the mHM on AB bilayer honeycomb
lattices with different choices of intralayer Semenoff masses.

$\begin{array}[]{cccc}\includegraphics[width=43.36133pt]{mHM-supp-
mass-a.eps}\includegraphics[width=43.36133pt]{mHM-supp-
mass-b.eps}\includegraphics[width=43.36133pt]{mHM-supp-
mass-c.eps}\includegraphics[width=43.36133pt]{mHM-supp-mass-d.eps}\\\
\includegraphics[width=43.36133pt]{mHM-supp-
mass-e.eps}\includegraphics[width=43.36133pt]{mHM-supp-
mass-f.eps}\includegraphics[width=43.36133pt]{mHM-supp-
mass-g.eps}\includegraphics[width=43.36133pt]{mHM-supp-mass-h.eps}\end{array}$

Figure S.7: Band structure of the mHM in AB stacked bilayer zigzag ribbons of
a width $W=60$ atoms with $t_{2}=0.1t$, $t_{\perp}=0.5t$,
$\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}$, (a) $M_{1}=M_{2}=0$, (b) $M_{1}=0$,
$M_{2}=t_{2}$, (c) $M_{1}=0$, $M_{2}=3\sqrt{3}t_{2}$ (d) $M_{1}=0$,
$M_{2}=6\sqrt{3}t_{2}$, (e) $M_{1}=M_{2}=\sqrt{3}t_{2}$, (f)
$M_{1}=-M_{2}=\sqrt{3}t_{2}$, (g) $M_{1}=M_{2}=3\sqrt{3}t_{2}$, (h)
$M_{1}=M_{2}=4\sqrt{3}t_{2}$.

Figure S.7 shows that, in AB bilayer mHM, the chiral edge states emerge as far
as $M_{l}<M_{lc}$ (Eq. S.5), for which the nodal lines, originating from the
overlap of the two layer bands, can occur, as discussed in the main text 1.

##  V  V. Modified Haldane model in AB stacked layers: effect of the NNN
hopping complex phase

It is noteworthy that the induced Chern insulator in the AB bilayer mHM occurs
as far as $\Phi_{1}$ and $\Phi_{2}$ have opposite signs and not only in the
case where $\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}$, which we considered to obtain
simple analytical expressions. This feature is shown in Fig. S.8.

$\begin{array}[]{c}\includegraphics[width=65.04366pt]{mHM-
phi-a.eps}\includegraphics[width=65.04366pt]{mHM-
phi-b.eps}\includegraphics[width=65.04366pt]{mHM-phi-c.eps}\end{array}$

Figure S.8: Band structure of the mHM in AB stacked bilayer ribbons of a width
$W=60$ atoms, for $t_{2}=0.1t$, $t_{\perp}=0.5t$, $M_{1}=M_{2}=0$ and (a)
$\Phi_{1}=-\Phi_{2}=\frac{\pi}{3}$, $\Phi_{1}=\frac{\pi}{3}$,
$\Phi_{2}=-\frac{\pi}{4}$ and (c) $\Phi_{1}=0,\Phi_{2}=-\frac{\pi}{2}$.

##  VI VI. Modified Haldane model in bilayer graphene: from AB to BA stacking

We consider a generic Hamiltonian that, continuously, interpolates between AB
$(\theta=0)$ and BA $(\theta=\pi/2)$ stacking of two single graphene layers
described by the mHM. The interlayer hopping terms, for a given value of
$\theta$ between these limits, does not represent physical couplings. We take,
for simplicity, vanishing Semenoff masses and inplane complex phases
$\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}$. The Bloch Hamiltonian written in the basis
($A_{1},B_{1},A_{2},B_{2}$) is

$\displaystyle
H_{B}(\mathbf{k},\theta)=\begin{pmatrix}a_{\mathbf{k}}&f_{\mathbf{k}}&0&2t_{\perp}c_{\theta}\\\
f^{\ast}_{\mathbf{k}}&a_{\mathbf{k}}&2t_{\perp}s_{\theta}&0\\\
0&2t_{\perp}s_{\theta}&-a_{\mathbf{k}}&f_{\mathbf{k}}\\\
2t_{\perp}c_{\theta}&0&f^{\ast}_{\mathbf{k}}&-a_{\mathbf{k}}\end{pmatrix}$
(S.14)

where
$f_{\mathbf{k}}=t\sum_{i=1}^{3}e^{i\mathbf{k}\cdot\boldsymbol{\delta}_{i}}$
and
$a_{\mathbf{k}}=-2t_{2}\sum_{i=1}^{3}\sin\left(\mathbf{k}\cdot\mathbf{a}_{i}\right)$.

The AB (BA) stacking corresponds to $c_{\theta}\equiv\cos\theta=1$ and
$s_{\theta}\equiv\sin\theta=0$ ($c_{\theta}=0$, $s_{\theta}=1$).

By varying $\theta$, the system can be interpolated between the two stacking
configurations, without going through the AA stacking. Using the intralayer
and interlayer pseudo-spin matrices $\boldsymbol{\sigma}$ and
$\boldsymbol{\tau}$, the Bloch Hamiltonian of Eq. S.14 becomes

$\displaystyle
H_{B}(\mathbf{k},\theta)=\left(b_{\mathbf{k}}\sigma_{x}+c_{\mathbf{k}}\sigma_{y}\right)\tau_{0}+2t_{\perp}\left[\left(c_{\theta}\sigma_{+}+s_{\theta}\sigma_{-}\right)\tau_{+}+2t_{\perp}\left(c_{\theta}\sigma_{-}+s_{\theta}\sigma_{+}\right)\tau_{-}\right],$
(S.15)

where $b_{\mathbf{k}}$, $c_{\mathbf{k}}$, $\sigma_{\pm}$ and $\tau_{\pm}$ are
given in the main text.

The Hamiltonian of Eq. S.15 breaks TRS, chirality but is invariant under
charge-conjugation and inversion since $M_{l}=0$ and
$\Phi_{1}=-\Phi_{2}=\frac{\pi}{2}$. The corresponding energy spectrum is
expressed, as in Eq. 7 of the main text with

$\displaystyle A_{\mathbf{k}}$ $\displaystyle=$ $\displaystyle
a^{2}_{\mathbf{k}}+|f_{\mathbf{k}}|^{2}+2t_{\perp}^{2},$ $\displaystyle
B_{\mathbf{k}}$ $\displaystyle=$ $\displaystyle
2\sqrt{|f_{\mathbf{k}}|^{2}\left(a^{2}_{\mathbf{k}}+t_{\perp}^{2}\right)+c^{2}_{2\theta}t_{\perp}^{4}+s_{2\theta}t_{\perp}^{2}\left(b^{2}_{\mathbf{k}}-c^{2}_{\mathbf{k}}\right)}$

The gap separating the lowest energy band around zero energy is
$\Delta=\min_{\mathbf{k}}\left(\Delta_{\mathbf{k}}\right)=2\sqrt{A_{\mathbf{k}}-B_{\mathbf{k}}}$.

$\Delta_{\mathbf{k}}$ closes, for uncoupled layers ($t_{\perp}=0$), for
$A_{\mathbf{k}}=B_{\mathbf{k}}$ which defines, as we have seen in the main
text, two non-intersecting closed Fermi lines. For a non vanishing interlayer
hopping $t_{\perp}$, $\Delta_{\mathbf{k}}$ can be expressed as

$\displaystyle\Delta_{\mathbf{k}}=2\frac{\sqrt{A^{2}_{\mathbf{k}}-B^{2}_{\mathbf{k}}}}{A_{\mathbf{k}}+B_{\mathbf{k}}}\sim\frac{2t_{\perp}}{|f_{\mathbf{k}}|}\sqrt{\left(1-s_{2\theta}\right)b^{2}_{\mathbf{k}}+\left(\lambda^{2}_{\mathbf{k}}-c^{2}_{2\theta}\right)t_{\perp}^{2}},$
(S.17)

where we introduced the parameter $\lambda_{\mathbf{k}}$ defined as

$\displaystyle
a^{2}_{\mathbf{k}}=|f_{\mathbf{k}}|^{2}+\left(2\lambda_{\mathbf{k}}-1\right)t_{\perp}^{2}$
(S.18)

In Eq. S.17, the numerator is given by its exact expression, whereas the
denominator has been approximated to the zeroth order in $t_{\perp}$.

According to Eq. S.17, $\Delta_{\mathbf{k}}$ cannot vanish for
$\theta\neq\pi/4,3\pi/4$.

For $\theta=\pi/4$, $\Delta_{\mathbf{k}}=0$ if $\lambda_{\mathbf{k}}=0$, which
give rise, according to Eq. S.18, to a closed loop defined by
$a^{2}_{\mathbf{k}}+t_{\perp}^{2}=|f_{\mathbf{k}}|^{2}$.

For $\theta=3\pi/4$, the closing of the gap $\Delta_{\mathbf{k}}$ requiers
$\lambda_{\mathbf{k}}=0$ and $b_{\mathbf{k}}=0$.

The critical values $\theta=\pi/4,3\pi/4$ correspond to the semimetallic phase
marking the transition from the topological Chern insulator phase $C=2$,
occurring for $0\leq\theta<\pi/4$ to the Chern insulator phase $C=-2$,
associated to $3\pi/4<\theta\leq\pi$. At this topological phase transition,
the gap closes at the four Dirac points of the bilayer system where, right
after the transition, the signs of the gap flip, inducing a variation $\Delta
C=-4$ of the Chern number.

The flipping of the Chern number sign, at the crossing form the AB to BA
stacking, could be understood from Fig. 5. Such crossing can be regarded as a
sign change of the complex phases: since the inplane sublattices have opposite
fluxes, the AB stacking corresponds to the dimer ($A_{1}$,$B_{2}$) for which
the complex phases are $\Phi_{1}=-\Phi_{2}=\pi/2$ while the BA stacking is
ascribed to the dimer ($B_{1}$,$A_{2}$) with $\Phi_{1}=-\Phi_{2}=-\pi/2$ (Fig.
5 of the main text).


# Online Multi-Target Tracking Using Recurrent Neural Networks

Anton Milan1, S. Hamid Rezatofighi1, Anthony Dick1, Ian Reid1 Konrad
Schindler2
1School of Computer Science, The University of Adelaide
2Photogrammetry and Remote Sensing, ETH Zürich
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present a novel approach to online multi-target tracking based on recurrent
neural networks (RNNs). Tracking multiple objects in real-world scenes
involves many challenges, including _a)_ an a-priori unknown and time-varying
number of targets, _b)_ a continuous state estimation of all present targets,
and _c)_ a discrete combinatorial problem of data association. Most previous
methods involve complex models that require tedious tuning of parameters.
Here, we propose for the first time, an end-to-end learning approach for
online multi-target tracking. Existing deep learning methods are not designed
for the above challenges and cannot be trivially applied to the task. Our
solution addresses all of the above points in a principled way. Experiments on
both synthetic and real data show promising results obtained at $\approx$300
Hz on a standard CPU, and pave the way towards future research in this
direction.

## Introduction

Tracking multiple targets in unconstrained environments is extremely
challenging. Even after several decades of research, it is still far from
reaching the accuracy of human labelling. (_cf_.  _MOTChallenge_ (?)). The
task itself constitutes locating all targets of interest in a video sequence
and maintaining their identity over time. One of the obvious questions that
arises immediately is how to model the vast variety of data present in
arbitrary videos that may include different view points or camera motion,
various lighting conditions or levels of occlusion, a varying number of
targets, _etc_. Tracking-by-detection has emerged as one of the most
successful strategies to tackle this challenge. Here, all “unused” data that
is available in a video sequence is discarded and reduced to just a few single
measurements per frame, typically by running an object detector. The task is
then to associate each measurement to a corresponding target, _i.e_. to
address the problem of data association. Moreover, due to clutter and an
unknown number of targets, the option to discard a measurement as a false
alarm and a strategy to initiate new targets as well as terminate exiting ones
must be addressed.

Figure 1: A schematic illustration of our architecture. We use RNNs for
temporal prediction and update as well as track management. The combinatorial
problem of data association is solved via LSTMs for each frame.

With the recent rise of deep learning, there has been surprisingly little work
related to multi-target tracking. We presume that this is due to several
reasons. First, when dealing with a large number of parameters, deep models
require huge amounts of training data, which is not yet available in the case
of multi-target tracking. Second, both the data and the desired solution can
be quite variable. One is faced with both discrete and continuous variables,
unknown cardinality for input and output, and variable lengths of video
sequences. One interesting exception in this direction is the recent work of
Ondrúška and Posner (?) that introduces deep recurrent neural networks to the
task of state estimation. Although this work shows promising results, it only
demonstrates its efficacy on simulated data with near-perfect sensor
measurements, a known number of targets, and smooth, linear motion. Their
follow-up work introduces real-world measurements and multi-class scenarios
(?), however, in both cases, tracking is formulated as estimating the world
occupancy, without explicit data association.

With this paper, we make an important step towards end-to-end model learning
for online tracking of multiple targets in realistic scenarios. Our main
contributions are as follows:

  1. 1.

Inspired by the well-studied Bayesian filtering idea, we present a recurrent
neural network capable of performing all multi-target tracking tasks including
prediction, data association, state update as well as initiation and
termination of targets within a unified network structure (Fig. 1). One of the
main advantages of this approach is that it is completely model-free, _i.e_.
it does not require any prior knowledge about target dynamics, clutter
distributions, _etc_. It can therefore capture linear (_cf_. Kalman filter),
non-linear (_cf_. particle filter), and higher-order dependencies.

  2. 2.

We further show, that a model for the challenging combinatorial problem of
data association including birth and death of targets can be learned entirely
from data. This time-varying cardinality component demonstrates that it is
possible to utilise RNNs not only to predict sequences with fixed-sized input
and output vectors, but in fact to infer unordered sets with unknown
cardinality.

  3. 3.

We present a way to generate arbitrary amounts of training data by sampling
from a generative model.

  4. 4.

Qualitative and quantitative results on simulated and real data show
encouraging results, confirming the potential of this approach. We firmly
believe that it will inspire other researchers to extend the presented ideas
and to further advance the performance.

## Related Work

##### Multi-object tracking.

A multitude of sophisticated models have been developed in the past to capture
the complexity of the problem at hand. Early works include the multiple
hypothesis tracker (MHT) (?) and joint probabilistic data association (JPDA)
(?). Both were developed in the realm of radar and sonar tracking but were
considered too slow for computer vision applications for a long time. With the
advances in computational power, they have found their way back and have
recently been re-introduced in conjunction with novel appearance models (?),
or suitable approximation methods (?). Recently, a large amount of work
focused on simplified models that could be solved to (near) global optimality
(?; ?; ?; ?). Here, the problem is cast as a linear program and solved via
relaxation, shortest-path, or min-cost algorithms. Conversely, more complex
cost functions have been considered in (?; ?), but without any theoretical
bounds on optimality. The optimization techniques range from quadratic boolean
programming, over customised alpha-expansion to greedy constraint propagation.
More recently, graph multi-cut formulations (?) have also been employed.

##### Deep learning.

Early ideas of biologically inspired learning systems date back many decades
(?). Later, convolutional neural networks (also known as CNNs) and the back
propagation algorithm were developed and mainly applied to hand-written digit
recognition (?). However, despite their effectiveness on certain tasks, they
could hardly compete with other well-established approaches. This was mainly
due to their major limitation of requiring huge amounts of training data in
order not to overfit the high number of parameters. With faster multi-
processor hardware and with a sudden increase in labelled data, CNNs have
become increasingly popular, initiated by a recent breakthrough on the task of
image classification (?). CNNs achieve state-of-the-art results in many
applications (?; ?) but are restrictive in their output format. Conversely,
_recurrent_ neural networks (RNNs) (?) include a loop between the input and
the output. This not only enables to simulate a memory effect, but also allows
for mapping input sequences to arbitrary output sequences, as long as the
sequence alignment and the input and output dimensions are known in advance.

Our work is inspired by the recent success of recurrent neural nets (RNNs) and
their application to language modeling (?). However, it is not straightforward
to apply the same strategies to the problem of multi-target tracking for
numerous reasons. First, the state space is multi-dimensional. Instead of
predicting one character or one word, at each time step the state of all
targets should be considered at once. Second, the state consists of both
continuous and discrete variables. The former represents the actual location
(and possibly further properties such as velocities) of targets, while a
discrete representation is required to resolve data association. Further
indicator variables may also be used to infer certain target states like the
track state, the occlusion level, _etc_. Third, the desired number of outputs
(_e.g_. targets) varies over time. In this paper, we introduce a method for
addressing all these issues and demonstrate how RNNs can be used for end-to-
end learning of multi-target tracking systems.

## Background

### Recurrent Neural Networks

Broadly speaking, RNNs work in a sequential manner, where a prediction is made
at each time step, given the previous state and possibly an additional input.
The core of an RNN is its hidden state $h\in\mathbb{R}^{n}$ of size $n$ that
acts as the main control mechanism for predicting the output, one step at a
time. In general, RNNs may have $L$ layers. We will denote $h_{t}^{l}$ as the
hidden state at time $t$ on layer $l$. $h^{0}$ can be thought of as the input
layer, holding the input vector, while $h^{L}$ holds the final embedded
representation used to produce the desired output $y_{t}$. The hidden state
for a particular layer $l$ and time $t$ is computed as $h^{l}_{t}=\tanh
W^{l}\left(h^{l-1}_{t},h^{l}_{t-1}\right)^{\top},$ where $W$ is a matrix of
learnable parameters.

The RNN as described above performs well on the task of motion prediction and
state update. However, we found that it cannot properly handle the
combinatorial task of data association. To that end, we consider the long
short-term memory (LSTM) recurrence (?). Next to the hidden state, the LSTM
unit also keeps an embedded representation of the state $c$ that acts as a
memory. A gated mechanism controls how much of the previous state should be
“forgotten” or replaced by the new input (see Fig. 2, right, for an
illustration). More formally, the hidden representations are computed as
$h^{l}_{t}=o\odot\tanh\left(c^{l}_{t}\right)$ and $c^{l}_{t}=f\odot
c^{l}_{t-1}+i\odot g,$ where $\odot$ represents element-wise multiplication.
The input, output and forget gates are all vectors of size $n$ and model the
memory update in a binary fashion using a sigmoid function:

$i,o,f=\sigma\left[W^{l}\left(h^{l-1}_{t},h^{l}_{t-1}\right)^{\top}\right],$
(1)

with a separate weight matrix $W^{l}$ for each gate.

### Bayesian Filtering

In Bayseian filtering, the goal is to estimate the true state $x{}$ from noisy
measurements $z{}$. Under the Markov assumption, the state distribution at
time $t$ given all past measurements is estimated recursively as

$p(x_{t}|z_{1:t})\propto p(z_{t}|x_{t})\int
p(x_{t}|x_{t-1})p(x_{t-1}|z_{1:t-1})dx_{t-1},$ (2)

where $p(z_{t}|x_{t})$ is the last observation likelihood and
$p(x_{t}|x_{t-1})$ the state transition probability. Typically, Eq. (2) is
evaluated in two steps: a _prediction_ step that evaluates the state dynamics,
and an _update_ step that corrects the belief about the state based on the
current measurements. Two of the most widely used techniques for solving the
above equation are Kalman filter (?) and particle filter (?). The former
performs exact state estimation under linear and Gaussian assumptions for the
state and measurements models, while the latter approximates arbitrary
distributions using sequential importance sampling.

$h_{t}$$x_{t}$$z_{t+1}$$\mathcal{A}_{t+1}$$\mathcal{E}_{t}$$\mathcal{E}_{t+1}$$\mathcal{E}^{*}_{t+1}$$x_{t+1}$$x^{*}_{t+1}$$h_{t+1}$

$h_{i}$$\mathcal{C}_{t+1}$$h_{i+1}$$c_{i}$$\mathcal{A}^{i}_{t+1}$$c_{i+1}$

Figure 2: Left: An RNN-based architecture for state prediction, state update,
and target existence probability estimation. Right: An LSTM-based model for
data association.

When dealing with multiple targets, one is faced with two additional
challenges. 1) Before the state update can be performed, it is crucial to
determine which measurements are associated with which targets. A number of
algorithms have been proposed to address this problem of data association
including simple greedy techniques, and sophisticated probabilistic approaches
like JPDA (see (?) for an overview). 2) To allow for a time-varying number of
targets, it is necessary to provide a mechanism to spawn new targets that
enter the scene, and remove existing ones that disappear indefinitely. Like
data association, this task is non-trivial, since each unassigned measurement
can potentially be either the start of a new trajectory or a false alarm.
Conversely, a missing measurement for a certain target could mean that the
target has disappeared, or that the detector has failed. To address this
challenge, online tracking approaches typically base their decisions about
births and deaths of tracks on heuristics that consider the number of
consecutive measurement errors.

## Our Approach

We will now describe our approach to cast the classical Bayesian state
estimation, data association as well as track initiation and termination tasks
as a recurrent neural net, allowing for full end-to-end learning of the model.

### Preliminaries and Notation

We begin by defining $x_{t}\in\mathbb{R}^{N\cdot D}$ as the vector containing
the states for all targets at one time instance. In our setting, the targets
are represented by their bounding box coordinates $(x,y,w,h)$, such that
$D=4$. Note that it is conceptually straightforward to extend the state to an
arbitrary dimension, _e.g_. to incorporate velocity, acceleration or
appearance model. $N$ is the number of interacting targets that are
represented (or tracked) simultaneously in one particular frame and
$x_{t}^{i}$ refers to the state of the $i^{\text{th}}$ target. $N$ is what we
call the network’s _order_ and captures the spatial dependencies between
targets. Here, we consider a special case with $N=1$ where all targets are
assumed to move independently. In other words, the same RNN is used for each
target. Similar to the state vector above, $z_{t}\in\mathbb{R}^{M\cdot D}$ is
the vector of all measurements in one frame, where $M$ is maximum number of
detections per frame.

The assignment probability matrix $\mathcal{A}\in[0,1]^{N\times(M+1)}$
represents for each target (row) the distribution of assigning individual
measurements to that target, _i.e_. $\mathcal{A}_{ij}\\!\equiv\\!p(i\text{
assigned to }j)$ and $\forall i:\sum_{j}\mathcal{A}_{ij}=1$. Note that an
extra column in $\mathcal{A}$ is needed to incorporate the case that a
measurement is missing. Finally, $\mathcal{E}\in[0,1]^{N}$ is an indicator
vector that represents the existence probability of a target and is necessary
to deal with an unknown and time-varying number of targets. We will use
($\sim$) to explicitly denote the ground truth variables.

### Multi-Target Tracking with RNNs

As motivated above, we decompose the problem at hand into two major blocks:
state prediction and update as well as track management on one side, and data
association on the other. This strategy has several advantages. First, one can
isolate and debug individual components effectively. Second, the framework
becomes modular, making it easy to replace each module or to add new ones.
Third, it enables one to (pre)train every block separately, which not only
significantly speeds up the learning process but turns out to be necessary in
practice to enable convergence. We will now describe both building blocks in
detail.

### Target Motion

Let us first turn to state prediction and update. We rely on a temporal RNN
depicted in Fig. 2 (left) to learn the temporal dynamic model of targets as
well as an indicator to determine births and deaths of targets (see next
section). At time $t$, the RNN outputs four values111We omit the RNN’s hidden
state $h_{t}$ at this point in order to reduce notation clutter.  for the next
time step: A vector $x^{*}_{t+1}\in\mathbb{R}^{N\cdot D}$ of predicted states
for all targets, a vector $x_{t+1}\in\mathbb{R}^{N\cdot D}$ of all updated
states, a vector $\mathcal{E}_{t+1}\in(0,1)^{N}$ of probabilities indicating
for each target how likely it is a real trajectory, and
$\mathcal{E}^{*}_{t+1}$, which is the absolute difference to
$\mathcal{E}_{t}$. This decision is computed based on the current state
$x_{t}$ and existence probabilities $\mathcal{E}_{t}$ as well as the
measurements $z_{t+1}$ and data association $\mathcal{A}_{t+1}$ in the
following frame. This building block has three primary objectives:

  1. 1.

Prediction: Learn a complex dynamic model for predicting target motion in the
absence of measurements.

  2. 2.

Update: Learn to correct the state distribution, given target-to-measurement
assignments.

  3. 3.

Birth / death: Learn to identify track initiation and termination based on the
state, the measurements and the data association.

The prediction $x^{*}_{t+1}$ for the next frame depends solely on the current
state $x_{t}$ and the network’s hidden state $h_{t}$. Once the data
association $\mathcal{A}_{t+1}$ for the following frame is available, the
state is updated according to assignment probabilities. To that end, all
measurements and the predicted state are concatenated to form
$\hat{x}=[z_{t+1};x^{*}_{t+1}]$ weighted by the assignment probabilities
$\mathcal{A}_{t+1}$. This is performed for all state dimensions. At the same
time, the track existence probability $\mathcal{E}_{t+1}$ for the following
frame is computed.

#### Loss.

A loss or objective is required by any machine learning algorithm to compute
the goodness-of-fit of the model, _i.e_. how close the prediction corresponds
to the true solution. It is typically a continuous function, chosen such that
minimising the loss maximises the performance of the given task. In our case,
we are therefore interested in a loss that correlates with the tracking
performance. This poses at least two challenges. First, measuring the
performance of multi-target tracking is far from trivial (?) and moreover
highly dependent on the particular application. For example, in vehicle
assistance systems it is absolutely crucial to maintain the highest precision
and recall to avoid accidents and to maintain robustness to false positives.
On the other hand, in sports analysis it becomes more important to avoid ID
switches between different players. One of the most widely accepted metrics is
the multi-object tracking accuracy (MOTA) (?) that combines the three error
types mentioned above and gives a reasonable assessment of the overall
performance. Ideally, one would train an algorithm directly on the desired
performance measure. This, however, poses a second challenge. The MOTA
computation involves a complex algorithm with non-differentiable zero-gradient
components, that cannot easily be incorporated into an analytical loss
function. Hence, we propose the following loss that satisfies our needs:

$\begin{split}\mathcal{L}(x^{*},x,\mathcal{E},\widetilde{x},\widetilde{\mathcal{E}})=&\underbrace{\frac{\lambda}{ND}\sum\|x^{*}-\widetilde{x}\|^{2}}_{\text{prediction}}+\\\
&\underbrace{\frac{\kappa}{ND}\|x-\widetilde{x}\|^{2}}_{\text{update}}+\underbrace{\nu\mathcal{L}_{\mathcal{E}}+\xi\mathcal{E}^{*},}_{\text{birth/death
+ reg.}}\end{split}$ (3)

where $x^{*},x$, and $\mathcal{E}$ are the predicted values, and
$\widetilde{x}$ and $\widetilde{\mathcal{E}}$ are the true values,
respectively. Note that we omit the time index here for better readability. In
practice the loss for one training sample is averaged over all frames in the
sequence.

The loss consists of four components. Let us first concentrate on the first
two, assuming for now that the number of targets is fixed. Intuitively, we aim
to learn a network that predicts trajectories that are close to the ground
truth tracks. This should hold for both, predicting the target’s motion in the
absence of any measurements, as well as correcting the track in light of new
measurements. To that end, we minimise the mean squared error (MSE) between
state predictions and state update and the ground truth.

### Initiation and Termination

Tracking multiple targets in real-world situations is complicated by the fact
that targets can appear and disappear in the area of interest. This aspect
must not be ignored but is difficult to model within the fixed-sized vector
paradigm in traditional neural network architectures. We propose to capture
the time-varying number of targets by an additional variable
$\mathcal{E}\in(0,1)^{N}$ that mimics the probability that a target exists
($\mathcal{E}=1$) or not ($\mathcal{E}=0$) at one particular time instance. At
test time, we then simply discard all targets for which $\mathcal{E}$ is below
a threshold ($0.6$ in our experiments).

#### Loss.

Figure 3: The effect of the pairwise smoothness prior on the existence
probability. See text for details.

The last two terms of the loss in Eq. (3) guide the learning to predict the
existence of each target at any given time. This is necessary to allow for
target initiation and termination. Here, we employ the widely used binary
cross entropy (BCE) loss

$\mathcal{L}_{\mathcal{E}}(\mathcal{E},\widetilde{\mathcal{E}})=\widetilde{\mathcal{E}}\log\mathcal{E}+(1-\widetilde{\mathcal{E}})\log(1-\mathcal{E})$
(4)

that approximates the probability of the existence for each target. Note that
the true values $\widetilde{\mathcal{E}}$ here correspond to a box function
over time (_cf_. Fig. 3, left). When using the BCE loss alone, the RNN learns
to make rather hard decisions, which results in track termination at each
frame when a measurement is missing. To remedy this, we propose to add a
smoothness prior $\mathcal{E}^{*}$ that essentially minimises the absolute
difference between two consecutive values for $\mathcal{E}$.

### Data Association with LSTMs

Arguably, the data association, _i.e_. the task to uniquely classify the
corresponding measurement for each target, is the most challenging component
of tracking multiple targets. Greedy solutions are efficient, but do not yield
good results in general, especially in crowded scenes with clutter and
occlusions. Approaches like JPDA are on the other side of the spectrum. They
consider _all_ possible assignment hypotheses jointly, which results in an NP-
hard combinatorial problem. Hence, in practice, efficient approximations must
be used.

In this section, we describe an LSTM-based architecture that is able to learn
to solve this task entirely from training data. This is somewhat surprising
for multiple reasons. First, joint data association is in general a highly
complex, discrete combinatorial problem. Second, most solutions in the output
space are merely permutations of each other w.r.t. the input features.
Finally, any possible assignment should meet the one-to-one constraint to
prevent the same measurement to be assigned to multiple targets. We believe
that the LSTM’s non-linear transformations and its strong memory component are
the main driving force that allows for all these challenges to be learned
effectively.

Method | Rcll$\uparrow$ |  Prcn$\uparrow$ | MT$\uparrow$ |  ML$\downarrow$ |  FP$\downarrow$ |  FN$\downarrow$ |  IDs$\downarrow$ |  FM$\downarrow$ | MOTA$\uparrow$ | MOTP$\uparrow$
---|---|---|---|---|---|---|---|---|---|---
Kalman-HA | 28.5 | 79.0 | 32 | 334 | 3,031 | 28,520 | 685 | 837 | 19.2 | 69.9
Kalman-HA2* | 28.3 | 83.4 | 39 | 354 | 2,245 | 28,626 | 105 | 342 | 22.4 | 69.4
JPDAm* | 30.6 | 81.7 | 38 | 348 | 2,728 | 27,707 | 109 | 380 | 23.5 | 69.0
RNN_HA | 37.8 | 75.2 | 50 | 267 | 4,984 | 24,832 | 518 | 963 | 24.0 | 68.7
RNN_LSTM | 37.1 | 73.5 | 50 | 260 | 5,327 | 25,094 | 572 | 983 | 22.3 | 69.0
Table 1: Tracking results on the MOTChallenge training dataset. *Denotes
offline post-processing.

To support this claim, we demonstrate the capability of LSTM-based data
association on the example of replicating the linear assignment problem. Our
model is illustrated in Figures 1 and 2 (right). The main idea is to exploit
the LSTM’s temporal step-by-step functionality to predict the assignment for
each target one target at a time. The input at each step $i$, next to the
hidden state $h_{i}$ and the cell state $c_{i}$, is the _entire_ feature
vector. For our purpose, we use the pairwise-distance matrix
$\mathcal{C}\in\mathbb{R}^{N\times M}$, where
$\mathcal{C}_{ij}=\|x^{i}-z^{j}\|_{2}$ is the Euclidean distance between the
predicted state of target $i$ and measurement $j$. Note that it is straight-
forward to extend the feature vector to incorporate appearance or any other
similarity information. The output that we are interested in is then a vector
of probabilities $\mathcal{A}^{i}$ for one target and all available
measurements, obtained by applying a softmax layer with normalisation to the
predicted values. Here, $\mathcal{A}^{i}$ denotes the $i^{\text{th}}$ row of
$\mathcal{A}$.

Figure 4: Results of our tracking method on a 20-frame long synthetic sequence
with clutter. Top: Ground truth ($x$-coordinate vs. time). Middle: Our
reconstructed trajectories. Bottom: The existence probability $\mathcal{E}$
for each target. Note the delayed initiation and termination, _e.g_. for the
top-most track (yellow) in the middle. This an inherent limitation of any
purely online approach that cannot be avoided.

#### Loss.

To measure the misassignment cost, we employ the common negative log-
likelihood loss

$\mathcal{L}(\mathcal{A}^{i},\tilde{a})=-\log(\mathcal{A}_{i\tilde{a}}),$ (5)

where $\tilde{a}$ is the correct assignment and $\mathcal{A}_{ij}$ is the
target $i$ to measurement $j$ assignment probability, as described earlier.

### Training Data

It is well known that deep architectures require vast amounts of training data
to avoid overfitting the model. Huge labelled datasets like ImageNET (?) or
Microsoft COCO (?) have enabled deep learning methods to unfold their
potential on tasks like image classification or pixel labelling.
Unfortunately, mainly due to the very tedious and time-consuming task of video
annotation, only very limited amount of labelled data for pedestrian tracking
is publicly available today. We therefore resort to synthetic generation by
sampling from a simple generative trajectory model learned from real data. To
that end, we first learn a trajectory model from each training sequence. For
simplicity, we only estimate the mean and the variance of two features: the
start location $x_{1}$ and the average velocity $\bar{v}$ from all annotated
trajectories in that sequence. For each training sample we then generate up to
$N$ tracks by sampling from a normal distribution with the learned parameters.
Note that this simplistic approach enables easy generation of realistic data,
but does not accomodate any observations.

### Implementation Details

Figure 5: Influence of two exemplar hyper-parameters on the overall
performance on the MOTChallenge benchmark, measured by MOTA. The optimal
parameter is marked with a red circle. Note that this graph shows the
performance of our prediction/update RNN block for only one target ($N=1$),
which explains the relatively low MOTA.

We implemented our framework in Lua and Torch7. Both our entire code base as
well as pre-trained models are publicly
available.222https://bitbucket.org/amilan/rnntracking Finding correct hyper-
parameters for deep architectures still remains a non-trivial task (?). In
this section we will point out some of the most important parameters and
implementation details. We follow some of the best practices found in the
literature (?; ?), such as setting the initial weights for the forget gates
higher ($1$ in our case), and also employ a standard grid search to find the
best setting for the present task.

##### Network size.

The RNN for state estimation and track management is trained with one layer
and 300 hidden units. The data association is a more complex task, requiring
more representation power. To that end, the LSTM module employed to learn the
data association consists of two layers and 500 hidden units.

##### Optimisation.

We use the RMSprop (?) to minimise the loss. The learning rate is set
initially to $0.0003$ and is decreased by $5\%$ every $20\,000$ iterations. We
set the maximum number of iterations to $200\,000$, which is enough to reach
convergence. The training of both modules takes approximately 30 hours on a
CPU. With a more accurate implementation and the use of GPUs we believe that
training can be sped up significantly.

##### Data.

The RNN is trained with approximately $100$K $20$-frame long sequences. The
data is divided into mini-batches of $10$ samples per batch and normalised to
the range $[-0.5,0.5]$, w.r.t. the image dimensions. We experimented with the
more popular zero-mean and unit-variance data normalisation but found that the
fixed one based on the image size yields superior performance.

## Experiments

Method | MOTA$\uparrow$ | MOTP$\uparrow$ | MT%$\uparrow$ | ML%$\downarrow$ | FP$\downarrow$ |  FN$\downarrow$ |  IDs$\downarrow$ |  FM$\downarrow$ |  FPS$\uparrow$
---|---|---|---|---|---|---|---|---|---
MDP (Xiang et al. 2015) | 30.3% | 71.3% | 13.0 | 38.4 | 9,717 | 32,422 | 680 | 1,500 | 1.1
SCEA (?) | 29.1% | 71.7% | 8.9 | 47.3 | 6,060 | 36,912 | 604 | 1,182 | 6.8
JPDAm* (?) | 23.8% | 68.2% | 5.0 | 58.1 | 6,373 | 40,084 | 365 | 869 | 32.6
TC_ODAL (?) | 15.1% | 70.5% | 3.2 | 55.8 | 12,970 | 38,538 | 637 | 1,716 | 1.7
RNN_LSTM (ours) | 19.0% | 71.0% | 5.5 | 45.6 | 11,578 | 36,706 | 1,490 | 2,081 | 165.2
Table 2: Tracking results on the MOTChallenge test dataset. *Denotes an
offline (or delayed) method.

Figure 6: Our RNN tracking results on the MOTChallenge sequence ADL-Rundle-3.
Frames 104, 149, 203, and 235 are shown. The colour of each bounding box
indicates the person identity.

To demonstrate the functionality of our approach, we first perform experiments
on simulated data. Fig. 4 shows an example of the tracking results on
synthetic data. Here, five targets with random birth and death times are
generated in a rather cluttered environment. The initiation / termination
indicators are illustrated in the bottom row.

We further test our approach on real-world data, using the MOTChallenge 2015
benchmark (?). This pedestrian tracking dataset is a collection of 22 video
sequences (11/11 for training and testing, respectively), with a relatively
high variation in target motion, camera motion, viewing angle and person
density. The evaluation is performed on a server using unpublished ground
truth. Next to precision and recall, we show the number of mostly tracked
($>\\!\\!80\%$ recovered) and mostly lost ($<\\!\\!20\%$ recovered)
trajectories (?), the number of false positive (FP), false negative (FN)
targets, identity swaps (IDs) and track fragmentations (FM). MOTA and MOTP are
the widely used CLEAR metrics (?) and summarise the tracking accuracy and
precision, respectively. Arrows next to each metric indicate weather higher
($\uparrow$) or lower ($\downarrow$) values are better.

##### Baseline comparison.

We first compare the proposed approach to three baselines. The results on the
training set are reported in Tab. 1. The first baseline (Kalman-HA) employs a
combination of a Kalman filter with bipartite matching solved via the
Hungarian algorithm. Tracks are initiated at each unassigned measurement and
terminated as soon as a measurement is missed. This baseline is the only one
that fully fulfils the online state estimation without any heuristics, time
delay or post-processing. The second baseline (Kalman-HA2) uses the same
tracking and data association approach, but employs a set of heuristics to
remove false tracks in an additional post-processing step. Finally, JPDAm is
the full joint probabilistic data association approach, recently proposed in
(?), including post-processing. We show the results of two variants of our
method. One with learned motion model and Hungarian data association, and one
in which both components were learned from data using RNNs and LSTMs. Both
networks were trained separately. Our learned model performs favourably
compared to the purely online solution (Kalman-HA) and is even able to keep up
with similar approaches but without any heuristics or delayed output. We
believe that the results can be improved further by learning a more
sophisticated data association technique, such as JPDA, as proposed by Milan
_et al_. (?), or by introducing a slight time delay to increase robustness.

##### Benchmark results.

Next, we show our results on the benchmark test set in Tab. 2 next to three
_online_ methods. The current leaderboard lists over 70 different trackers,
with the top ones reaching over $50\%$ MOTA. Even though the evaluation is
performed by the benchmark organisers, there are still considerable
differences between various submissions, that are worth pointing out. First,
all top-ranked trackers use their own set of detections. While a better
detector typically improves the tracking result, the direct comparison of the
tracking method becomes rather meaningless. Therefore, we prefer to use the
provided detections to guarantee a fair setting. Second, most methods perform
so-called _offline_ tracking, _i.e_. the solution is inferred either using the
entire video sequence, or by peeking a few frames into the future, thus
returning the tracking solution with a certain time delay. This is in contrast
to our method, which aims to strictly compute and fix the solution with each
incoming frame, before moving to the next one. Finally, it is important to
note that many current methods use target appearance or other image features
like optic flow (?) to improve the data association. Our method does not
utilise any visual features and solely relies on geometric locations provided
by the detector. We acknowledge the usefulness of such features for pedestrian
tracking, but these are often not available in other application, such as
_e.g_. cell or animal tracking. We therefore refrain from including them at
this point.

Overall, our approach does not quite reach the top accuracy in pedestrian
online tracking (?), but is two orders of magnitude faster. Fig. 6 shows some
example frames from the test set.

## Discussion and Future Work

We presented an approach to address the challenging problem of data
association and trajectory estimation within a neural network setting. To the
best of our knowledge, this is the first approach that employs recurrent
neural networks to address online multi-target tracking. We showed that an
RNN-based approach can be utilised to learn complex motion models in realistic
environments. The second, somewhat surprising finding is that an LSTM network
is able to learn one-to-one assignment, which is a non-trivial task for such
an architecture. We firmly believe that, by incorporating appearance and by
learning a more robust association strategy, the results can be improved
significantly.

##### Acknowledgments.

This work was supported by ARC Linkage Project LP130100154, ARC Laureate
Fellowship FL130100102 and the ARC Centre of Excellence for Robotic Vision
CE140100016.

## References

  * [Bae and Yoon 2014] Bae, S.-H., and Yoon, K.-J.  2014\.  Robust online multi-object tracking based on tracklet confidence and online discriminative appearance learning.  In CVPR.
  * [Bar-Shalom and Fortmann 1988] Bar-Shalom, Y., and Fortmann, T. E.  1988\.  Tracking and Data Association.  Academic Press.
  * [Berclaz et al. 2011] Berclaz, J.; Fleuret, F.; Türetken, E.; and Fua, P.  2011\.  Multiple object tracking using k-shortest paths optimization.  IEEE T. Pattern Anal. Mach. Intell. 33(9):1806–1819.
  * [Bernardin and Stiefelhagen 2008] Bernardin, K., and Stiefelhagen, R.  2008\.  Evaluating multiple object tracking performance: The CLEAR MOT metrics.  Image and Video Processing 2008(1):1–10.
  * [Butt and Collins 2013] Butt, A. A., and Collins, R. T.  2013\.  Multi-target tracking by Lagrangian relaxation to min-cost network flow.  In CVPR.
  * [Choi 2015] Choi, W.  2015\.  Near-online multi-target tracking with aggregated local flow descriptor.  In ICCV.
  * [Doucet, Godsill, and Andrieu 2000] Doucet, A.; Godsill, S.; and Andrieu, C.  2000\.  On sequential monte carlo sampling methods for bayesian filtering.  Statistics and Computing 10(3):197–208.
  * [Eigen and Fergus 2015] Eigen, D., and Fergus, R.  2015\.  Predicting depth, surface normals and semantic labels with a common multi-scale convolutional architecture.  In ICCV.
  * [Fortmann, Bar-Shalom, and Scheffe 1980] Fortmann, T. E.; Bar-Shalom, Y.; and Scheffe, M.  1980\.  Multi-target tracking using joint probabilistic data association.  In IEEE Conference on Decision and Control.
  * [Goller and Küchler 1996] Goller, C., and Küchler, A.  1996\.  Learning task-dependent distributed representations by backpropagation through structure.  In ICNN, 347–352.  IEEE.
  * [Greff et al. 2015] Greff, K.; Srivastava, R. K.; Koutník, J.; Steunebrink, B. R.; and Schmidhuber, J.  2015\.  LSTM: A search space odyssey.  arXiv:1503.04069 [cs].  arXiv: 1503.04069.
  * [Hochreiter and Schmidhuber 1997] Hochreiter, S., and Schmidhuber, J.  1997\.  Long short-term memory.  Neural Comput. 9(8):1735–1780.
  * [Hong Yoon et al. 2016] Hong Yoon, J.; Lee, C.-R.; Yang, M.-H.; and Yoon, K.-J.  2016\.  Online multi-object tracking via structural constraint event aggregation.  In CVPR.
  * [Ivakhnenko and Lapa 1966] Ivakhnenko, A. G., and Lapa, Valentin, G.  1966\.  Cybernetic predicting devices.
  * [Jiang, Fels, and Little 2007] Jiang, H.; Fels, S.; and Little, J. J.  2007\.  A linear programming approach for multiple object tracking.  In CVPR.
  * [Kalman 1960] Kalman, R. E.  1960\.  A new approach to linear filtering and prediction problems.  Transactions of the ASME–Journal of Basic Engineering 82(Series D):35–45.
  * [Karpathy, Johnson, and Li 2015] Karpathy, A.; Johnson, J.; and Li, F.-F.  2015\.  Visualizing and understanding recurrent networks.  arXiv:1506.02078.
  * [Kim et al. 2015] Kim, C.; Li, F.; Ciptadi, A.; and Rehg, J. M.  2015\.  Multiple hypothesis tracking revisited: Blending in modern appearance model.  In ICCV.
  * [Krizhevsky, Sutskever, and Hinton 2012] Krizhevsky, A.; Sutskever, I.; and Hinton, G. E.  2012\.  Imagenet classification with deep convolutional neural networks.  In NIPS, 1097–1105.
  * [Leal-Taixé et al. 2015] Leal-Taixé, L.; Milan, A.; Reid, I.; Roth, S.; and Schindler, K.  2015\.  MOTChallenge 2015: Towards a benchmark for multi-target tracking.  arXiv:1504.01942 [cs].
  * [LeCun et al. 1998] LeCun, Y.; Bottou, L.; Bengio, Y.; and Haffner, P.  1998\.  Gradient-based learning applied to document recognition.  Proceedings of the IEEE 86(11):2278–2324.
  * [Leibe, Schindler, and Van Gool 2007] Leibe, B.; Schindler, K.; and Van Gool, L.  2007\.  Coupled detection and trajectory estimation for multi-object tracking.  In ICCV.
  * [Li, Huang, and Nevatia 2009] Li, Y.; Huang, C.; and Nevatia, R.  2009\.  Learning to associate: Hybridboosted multi-target tracker for crowded scene.  In CVPR.
  * [Lin et al. 2014] Lin, T.-Y.; Maire, M.; Belongie, S.; Bourdev, L.; Girshick, R.; Hays, J.; Perona, P.; Ramanan, D.; Zitnick, C. L.; and Dollár, P.  2014\.  Microsoft COCO: Common objects in context.  arXiv:1405.0312 [cs].
  * [Milan et al. 2017] Milan, A.; Rezatofighi, S. H.; Garg, R.; Dick, A.; and Reid, I.  2017\.  Data-driven approximations to NP-hard problems.  In AAAI.
  * [Milan, Roth, and Schindler 2014] Milan, A.; Roth, S.; and Schindler, K.  2014\.  Continuous energy minimization for multitarget tracking.  IEEE T. Pattern Anal. Mach. Intell. 36(1):58–72.
  * [Milan, Schindler, and Roth 2013] Milan, A.; Schindler, K.; and Roth, S.  2013\.  Detection- and trajectory-level exclusion in multiple object tracking.  In CVPR.
  * [Ondruska and Posner 2016] Ondruska, P., and Posner, I.  2016\.  Deep tracking: Seeing beyond seeing using recurrent neural networks.  In AAAI.
  * [Ondruska et al. 2016] Ondruska, P.; Dequaire, J.; Zeng Wang, D.; and Posner, I.  2016\.  End-to-end tracking and semantic segmentation using recurrent neural networks.  In RSS Workshop on Limits and Potentials of Deep Learning in Robotics.
  * [Reid 1979] Reid, D. B.  1979\.  An algorithm for tracking multiple targets.  IEEE Transactions on Automatic Control 24(6):843–854.
  * [Rezatofighi et al. 2015] Rezatofighi, H. S.; Milan, A.; Zhang, Z.; Shi, Q.; Dick, A.; and Reid, I.  2015\.  Joint probabilistic data association revisited.  In ICCV.
  * [Russakovsky et al. 2014] Russakovsky, O.; Deng, J.; Su, H.; Krause, J.; Satheesh, S.; Ma, S.; Huang, Z.; Karpathy, A.; Khosla, A.; Bernstein, M.; Berg, A. C.; and Fei-Fei, L.  2014\.  ImageNet Large Scale Visual Recognition Challenge.
  * [Tang et al. 2016] Tang, S.; Andres, B.; Andriluka, M.; and Schiele, B.  2016\.  Multi-person tracking by multicuts and deep matching.  In ECCV Workshop on Benchmarking Multi-Target Tracking.
  * [Tieleman and Hinton 2012] Tieleman, T., and Hinton, G.  2012\.  Rmsprop: Divide the gradient by a running average of its recent magnitude.  Coursera: Neural networks for machine learning.
  * [Vinyals et al. 2015] Vinyals, O.; Toshev, A.; Bengio, S.; and Erhan, D.  2015\.  Show and tell: A neural image caption generator.  In CVPR.
  * [Wang et al. 2012] Wang, T.; Wu, D.; Coates, A.; and Ng, A.  2012\.  End-to-end text recognition with convolutional neural networks.  In 2012 21st International Conference on Pattern Recognition (ICPR), 3304–3308.
  * [Xiang, Alahi, and Savarese 2015] Xiang, Y.; Alahi, A.; and Savarese, S.  2015\.  Learning to track: Online multi-object tracking by decision making.  In ICCV, 4705–4713.
  * [Zhang, Li, and Nevatia 2008] Zhang, L.; Li, Y.; and Nevatia, R.  2008\.  Global data association for multi-object tracking using network flows.  In CVPR.