††thanks: These authors contributed equally to the work.††thanks: These
authors contributed equally to the work.

# Emergent $\mathcal{D}_{8}^{(1)}$ spectrum and topological soliton excitation
in CoNb2O6

Ning Xi  Department of Physics and Beijing Key Laboratory of Opto-electronic
Functional Materials and Micro-nano Devices, Renmin University of China,
Beijing 100872, China  Xiao Wang  Tsung-Dao Lee Institute, Shanghai Jiao Tong
University, Shanghai, 201210, China  Yunjing Gao  Tsung-Dao Lee Institute,
Shanghai Jiao Tong University, Shanghai, 201210, China  Yunfeng Jiang  School
of Phyiscs and Shing-Tung Yau Center, Southeast University, Nanjing 210096,
China  Rong Yu<EMAIL_ADDRESS>Department of Physics and Beijing Key
Laboratory of Opto-electronic Functional Materials and Micro-nano Devices,
Renmin University of China, Beijing 100872, China  Tsung-Dao Lee Institute,
Shanghai Jiao Tong University, Shanghai, 201210, China  Key Laboratory of
Quantum State Construction and Manipulation (Ministry of Education), Renmin
University of China, Beijing, 100872, China  Jianda Wu<EMAIL_ADDRESS>Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai, 201210,
China  School of Physics & Astronomy, Shanghai Jiao Tong University, Shanghai,
200240, China  Shanghai Branch, Hefei National Laboratory, Shanghai 201315,
China

###### Abstract

Quantum integrability emerging near a quantum critical point (QCP) is
manifested by exotic excitation spectrum that is organised by the associated
algebraic structure. A well known example is the emergent $E_{8}$
integrability near the QCP of a transverse field Ising chain (TFIC), which was
long predicted theoretically and initially proposed to be realised in the
quasi-one-dimensional (q1D) quantum magnet CoNb2O6. However, later
measurements on the spin excitation spectrum of this material revealed a
series of satellite peaks that cannot be described by the $E_{8}$ Lie algebra.
Motivated by these experimental progresses, we hereby revisit the spin
excitations of CoNb2O6 by combining numerical calculation and analytical
analysis. We show that, as effects of strong interchain fluctuations, the
spectrum of the system near the 1D QCP is characterised by the
$\mathcal{D}_{8}^{(1)}$ Lie algebra with robust topological soliton
excitation. We further show that the $\mathcal{D}_{8}^{(1)}$ spectrum can be
realised in a broad class of interacting quantum systems. Our results advance
the exploration of integrability and manipulation of topological excitations
in quantum critical systems.

##  I Introduction

Enhanced quantum fluctuations near QCPs can give rise to rich emergent
phenomena [1, 2, 3, 4], including enhanced symmetry, deconfined fractional
excitations, and emergent integrability. As a prototypical model for studying
quantum criticality, the TFIC, which can be realised in a number of q1D
quantum magnets, continues to be a research hotspot [1, 2, 5, 6, 7, 8, 9, 10,
11, 12, 13]. In q1D magnets, the interchain couplings, though weak, are
relevant perturbation that causes 3D magnetic ordering of the system. As a
consequence, the genuine 1D QCP is usually hidden inside the 3D ordered phase
as illustrated in Fig. 1(a). This makes the critical behaviour even more
complex and intriguing [13, 14]. The significance of the 1D quantum
criticality in a TFIC is manifested by its spin excitation spectrum, and in
the 3D magnetic ordered phase, the interchain interaction can be treated as an
effective weak longitudinal field $\tilde{h}$ that confines the quasiparticles
at critical into gapped bound states [8, 15] with quantum $E_{8}$
integrability, whose spectrum and scattering matrix are organised by the
$E_{8}$ Lie algebra [16, 2].

The above scenario for the emergent $E_{8}$ integrability was long predicted
theoretically [16] and the $E_{8}$ spectrum was initially proposed to be
realised in the q1D Ising magnet CoNb2O6 [2]. Recently, the full $E_{8}$
spectrum and well defined dispersive $E_{8}$ quasiparticles have been observed
in another q1D Ising magnet, BaCo2V2O8, under transverse magnetic field [8,
17, 18, 15]. As for CoNb2O6, however, results based on recent measurements are
still controversial.

CoNb2O6 is a renowned q1D Ising magnet with zigzag ferromagnetic (FM) chains
along the crystalline $c$ axis forming a frustrated isosceles triangular
lattice in the $a$-$b$ plane as depicted in Fig. 1(c) and (d). The FM
intrachain coupling $J$ is much stronger than the antiferromagnetic (AFM)
interchain ones, $J_{i}$ and $J_{i}^{\prime}$ [19, 20, 21, 22]. Experiments
suggest that a 1D QCP in the TFIC universality class at $H_{c}^{\rm{1D}}\simeq
5.0-5.3$ T [2, 11, 12, 13, 22, 23, 10] is hidden in the 3D AFM ordered phase
not far from the 3D QCP (at $H_{c}^{\rm{3D}}\simeq 5.5$ T [2]). The seminal
inelastic neutron scattering (INS) measurement [2] provides a first evidence
of $E_{8}$ spectrum: The energy ratio of two lowest peaks identified is close
to the golden ratio, which is the exact mass ratio of the two lightest $E_{8}$
particles, when the system is tuned approaching $H_{c}^{\rm{1D}}$. Recent THz
spectroscopy measurements with much higher energy resolution, however,
revealed numerous additional satellite excitation modes that surpass the
$E_{8}$ description [23, 24]. Besides, recent INS results indicate that the
low-energy spectrum is influenced by the domain wall (DW) interaction
associated with the reduced lattice symmetry of CoNb2O6 [9]. These results
raise crucial questions about the adequacy of the Ising universality and the
emergent $E_{8}$ spectrum.

In this article, we reexamine the spin excitations of CoNb2O6 near
$H_{c}^{\rm{1D}}$ by performing iTEBD calculation and field theoretical
analysis. We demonstrate that the low-energy physics in the vicinity of
$H_{c}^{\rm{1D}}$ is controlled by the TFIC universality and the DW
interaction is irrelevant. By suitably incorporating interchain spin
fluctuations enhanced by spin frustration and the proximity to a 3D QCP, we
find a remarkable result that the spectrum of CoNb2O6 is not consistent with
the $E_{8}$ algebra but can be described by the $\mathcal{D}_{8}^{(1)}$ Lie
algebra associated with the Ising${}_{h}^{2}$ integrable model. The satellite
peaks incompatible with the $E_{8}$ algebra in the THz measurements are re-
identified as the $\mathcal{D}_{8}^{(1)}$ soliton and/or breather excitations.
The topological single soliton excitation, usually forbidden in other systems,
is found to be robust over a finite field range. We propose that the
$\mathcal{D}_{8}^{(1)}$ spectrum can be realised in a broad class of q1D
quantum magnets. These results expand the realm of emergent phenomena in
quantum magnetic systems.

Figure 1:  (a): Sketched phase diagram for CoNb2O6 under a transverse field
($H{\kern 5.59998pt/\kern-8.00003pt/\kern 5.59998pt}b$), which consists of a
3D AFM phase at low temperatures and a prominent 1D quantum critical regime
above the 3D ordering temperature. The (putative) 1D QCP at $H_{c}^{\rm{1D}}$
is hidden inside the 3D ordered phase close to the 3D QCP. The interchain
interaction confines continuous critical excitations into gapped bound states
that are characterised by the emergent $\mathcal{D}_{8}^{(1)}$ Lie algebra.
(b): Sketched phase diagram showing crossover between two integrabilities by
tuning $J_{i}/\tilde{h}$ in the minimal model. (c) and (d): Illustration of
the crystal structure of CoNb2O6 in the $a$-$b$ plane [in (c)] and in the
$b$-$c$ plane [in (d)], showing Co2+ ions (purple) inside the edge-sharing O2-
octahedra (orange) and translucent coordinate Nb-O octahedra. The Co2+ ions
construct an isosceles triangular lattice with AFM interchain couplings
$J_{i}$ and $J_{i}^{\prime}$ in the $a$-$b$ plane, and form a zigzag chain
with FM intrachain coupling $J$ alone the $c$ axis.

##  II A minimal model for CoNb2O6

We consider the following Hamiltonian for CoNb2O6,

$\mathcal{H}=\mathcal{H}_{\text{chain}}+\mathcal{H}_{\text{ic}},$ (1)

where $\mathcal{H}_{\text{chain}}$ and $\mathcal{H}_{\text{ic}}$ include the
intra- and inter-chain interactions, respectively. The intrachain Hamiltonian
proposed to precisely describe the magnetic properties of CoNb2O6 reads as [9]

$\displaystyle\mathcal{H}_{\text{chain}}$
$\displaystyle=J\sum_{j}\left[-S^{z}_{j}S^{z}_{j+1}-\varepsilon(S^{x}_{j}S^{x}_{j+1}+S^{y}_{j}S^{y}_{j+1})\right.$
(2)
$\displaystyle\left.+\lambda_{af}S^{z}_{j}S^{z}_{j+2}+(-1)^{j}\lambda_{dw}(S^{z}_{j}S^{y}_{j+1}+S^{y}_{j}S^{z}_{j+1})\right],$
$\displaystyle-g\mu_{0}H\sum_{j}S^{y}_{j}$

where the first two terms form a 1D XXZ model in the FM Ising limit, the third
term refers to an AFM interaction between the next-nearest neighbouring
(n.n.n.) spins, the fourth term is the so called DW interaction associated
with the two DW bound state excitations, and the last term comes from the
applied transverse magnetic field (along the crystalline $b$ axis). The AFM
and DW terms originate from the zigzag geometry of the chain. Following Ref.
9, we take $J=2.7607$ meV, $\varepsilon=0.239$, $\lambda_{af}=0.1507$,
$\lambda_{dw}=0.1647$, and $g=3.100$, which were shown to accurately describe
the low-energy excitations of CoNb2O6. We note that more refined parameters
for high-energy and strong-field excitations were recently proposed [22, 25].

It is believed that the dominant interchain interaction is also of Ising-type,
so we consider the following Hamiltonian

$\mathcal{H}_{\text{ic}}=\sum_{j,\langle\alpha,\beta\rangle}J_{i}S^{z}_{j,\alpha}S^{z}_{j,\beta},$
(3)

where chain labels $\alpha$, $\beta$ run over n.n. chains. Given the weak
coupling $J_{i}$, $\mathcal{H}_{\text{chain}}$ is usually treated at a chain
mean-field level when the system is inside the 3D ordered phase,
$\mathcal{H}_{\text{ic}}\approx-\tilde{h}\sum_{j}S^{z}_{j,\alpha}$, where
$\tilde{h}=-J_{i}\sum_{\beta}\langle S^{z}_{j,\beta}\rangle$ is an effective
longitudinal field acting on a single chain from its neighbours.

However, the frustrated interchain alignment causes cancellation of the
effective fields from neighbouring chains and the interchain fluctuations,
which are further enhanced by the proximity to the 3D QCP, must be treated in
a more proper way. We then adopt a cluster mean-field approximation. Taking
into account the two-sublattice nature of the 3D AFM order, we pick up a
minimal unit consisting of two n.n. chains. The interchain couplings between
these two chains are treated exactly, while the couplings to other chains are
considered at the chain mean-field level as described above. The Hamiltonian
of the minimal model then reads as

$\mathcal{H}_{\text{min}}=\sum_{m=1,2}\mathcal{H}_{\text{chain}}^{(m)}+J_{i}\sum_{j=1}^{N}S^{z}_{j,1}S^{z}_{j,2}-\tilde{h}\sum_{j=1}^{N}\sum_{m=1,2}S^{z}_{j,m},$
(4)

where $m$ is the chain index, and $\tilde{h}$ refers to the effective
longitudinal field introduced by the chain mean-field approximation. Note that
the minimal model recovers to two decoupled Ising chains when $J_{i}=0$, and
an $E_{8}$ integrability is expected in this case. We will show in the
following that the system crosses over to a novel quantum integrable class
exhibiting the $\mathcal{D}_{8}^{(1)}$ mass spectrum in the
$J_{i}/\tilde{h}\rightarrow\infty$ limit [Fig. 1(b)].

##  III Emergent integrabilities of the minimal model

By comparing the experimental and calculated spectra and computing the
critical exponents in the minimal model, we first show that the non-Ising
terms, including the DM interaction, are irrelevant to the TFIC universality
of the 1D QCP [see Supplemental Materials (SM) [26]]. Then we analyse the
emergent integrabilities of the minimal model in Eq. (4). Without $\tilde{h}$,
each chain at $H_{c}^{\rm{1D}}$ is described by a $(1+1)$D conformal field
theory (CFT) with central charge $c=1/2$. Either a small $\tilde{h}$ field or
a small coupling $J_{i}$ can push the ladder system away from criticality into
a field-induced or intrinsic ordered phase, respectively. The former one has
the system emerged the $E_{8}$ integrability emerges [27], while the later one
drives the system approaching the Ising${}_{h}^{2}$ integrability. In the
$\tilde{h}=0$ limit, the low-energy physics is described by the following
Ising${}_{h}^{2}$ field theory

$\mathcal{A}_{\text{field}}=\sum_{\alpha=1,2}\mathcal{A}_{c=1/2}^{(\alpha)}+\lambda\int{\rm
d}x~{}\sigma^{(1)}(x)\sigma^{(2)}(x),$ (5)

where $\mathcal{A}_{c=1/2}^{(\alpha)}$ denotes the $c=1/2$ CFT action for the
critical chain $\alpha$, $\sigma^{(\alpha)}(x)$ and $\lambda$ refer to the
correspondences of $S^{z}_{j,\alpha}$ and interchain coupling in the
continuous limit, respectively. In the scaling limit [$a\;({\text{lattice
spacing}}),\lambda\rightarrow 0$ with finite $\lambda/a$], the theory exhibits
emergent Ising${}^{2}_{h}$ integrability characterised by the
$\mathcal{D}_{8}^{(1)}$ Lie algebra [28]. In the most general case where both
$\tilde{h}$ and $J_{i}$ are present, the mass spectrum of the minimal model in
Eq. (4) crosses over from $E_{8}$$\times$$E_{8}$ to $\mathcal{D}_{8}^{(1)}$,
as illustrated in Fig. 1(d).

As for CoNb2O6, the ordered moment at $H_{c}^{\rm{1D}}$ should be tiny since
$H_{c}^{\rm{1D}}$ is very close to $H_{c}^{\rm{3D}}$. This, together with the
frustrated alignment of chains, suppresses the effective field $\tilde{h}$. We
therefore expect $J_{i}\gg\tilde{h}$ in CoNb2O6, so that its excitation
spectrum near $H_{c}^{\rm{1D}}$ is characterised by the
$\mathcal{D}_{8}^{(1)}$ algebra. Note that this is different from the case of
BaCo2V2O8, whose 1D QCP is located deep inside the 3D ordered phase [8].
There, the chain mean-field theory is a good approximation such that the
excitation spectrum is well described by the $E_{8}$ algebra.

Figure 2: (a): Calculated zone-centre spectral functions at $H_{c}^{\rm{1D}}$
of an Ising ladder with $J=1$ meV, $J_{i}=0.36~{}J$ and $\tilde{h}=0$; (b):
same as (a) but for the minimal model with $\lambda_{i}=0.1~{}J$ and
$\tilde{h}=0$; (c): THz absorption spectrum of CoNb2O6 at $H_{c}=5$ T and
$T=0.25$ K, adapted from Ref. 23. (d): Calculated zone-centre spectral
functions at $H_{c}^{\rm{1D}}$ of the single-chain model in Eq. (2) with the
effective longitudinal field $\tilde{h}=0.034~{}J$, exhibiting the $E_{8}$
structure. In each panel, the vertical dashed lines at peak positions
correspond to the masses of quasiparticles or bound states of multi-particles
in the particular $\mathcal{D}_{8}^{(1)}$ [in (a)-(c)] or $E_{8}$ [in (d)]
model. Spectra at $k=\pi/2$ are also shown as light-coloured lines in panels
(b) and (d) to demonstrate the zone-folding effect. Blue and red arrows refer
to peaks associated with detailed microscopic Hamiltonian (see text). Note
that many multiparticle modes are located in the shaded regimes, which give
rise to multiple peak or plateau like spectrum.

##  IV Emergent $\mathcal{D}_{8}^{(1)}$ spectrum of CoNb2O6

The integrable Ising${}_{h}^{2}$ model in Eq. (5) possesses excitations
associated with the $\mathcal{D}_{8}^{(1)}$ algebra, which is characterised by
a total of 8 types of particles, including one soliton ($S_{+1}$) and one
antisoliton ($S_{-1}$), each with mass $m_{\text{s}}$, as well as 6 breathers
$B_{n}$ with masses $m_{n}=2m_{\text{s}}\sin(n\pi/14)$, $(n=1,\dots,6)$ [28],
which are referred to $\mathcal{D}_{8}^{(1)}$ particles in the following. We
expect that peaks corresponding to these quasiparticles appear in the
excitation spectrum of the minimal model at $\tilde{h}=0$ within the energy-
momentum range where the Ising universality dominates. To check this, we
numerically calculate the excitation spectrum at $k=0$ of an Ising ladder with
$J_{i}=0.36J$ at $H_{c}^{\rm{1D}}$. As shown in Fig. 2(a), we identify a
series of peaks with energies precisely corresponding to masses of
$\mathcal{D}_{8}^{(1)}$ particles. Besides the single particle peaks, we also
identify edges of several multiparticle excitation continua. Interestingly,
the most prominent peak in the spectrum at about $0.74$ meV originates from
the unusual topological single (anti)soliton. Also note that we cannot resolve
any spectral signature corresponding to single $B_{1}$, $B_{3}$, or $B_{5}$
breather. This verifies a recent theoretical prediction that these odd-parity
breathers cannot be excited from the ground state because of symmetry
restriction [29]. The agreement between the numerical and analytical results
indicates that the $\mathcal{D}_{8}^{(1)}$ physics is robust even for a
sizeable interchain coupling.

We then compare our result with the spectrum obtained in a recent high-
resolution THz measurement [23] for CoNb2O6 near $H_{c}^{\rm{1D}}$ (in Fig.
2(c)). Surprisingly, most peaks in the experiment can be assigned to single or
multiple $\mathcal{D}_{8}^{(1)}$ particles, all the way up to about $2$ meV.
To understand the two peaks (labelled by arrows) not captured by the Ising
ladder model, we further calculate the spectrum of the minimal model with the
non-Ising interactions and at $\tilde{h}=0$. As shown in Fig. 2(b), peaks
consistent with the experiment emerge at arrow positions. We find these peaks
are indeed associated with details of the microscopic model, instead of the
$\mathcal{D}_{8}^{(1)}$ algebraic structure. In fact, the peak labelled by the
red arrow appears at both $k=0$ and $k=\pi/2$, indicating it is a zone-folding
peak caused by the n.n.n. AFM interaction $J_{\rm{AF}}$. The peak at the blue
arrow, however, is attributed to the confining effect from the DW interaction,
which gives rise to additional bound states out of the $2m_{1}$ continuum of
the $\mathcal{D}_{8}^{(1)}$ spectrum (see Fig. 2(a)).

We further calculate the spectrum of a single chain under an effective field
$\tilde{h}$, which is characterised by the $E_{8}$ algebra. As shown in Fig.
2, many features (satellite peaks) of the experimental spectrum are missing in
the $E_{8}$ model but can be well captured by the $\mathcal{D}_{8}^{(1)}$ one.
In Fig. 3(a), we plot the field dependence of several characteristic energies
extracted from peaks in experimental spectra presented in Refs. 23 and 24. The
energy ratios best fit to the $\mathcal{D}_{8}^{(1)}$ mass spectrum at about
$5$ T. (Putative) $H_{c}^{\rm{1D}}$ determined in this way is very close to
the value ($\sim 5.1-5.3$ T) determined from a previous NMR measurement [13].
Note that by assuming the $E_{8}$ structure, the same data gave
$H_{c}^{\rm{1D}}\simeq 4.75$ T [23] which deviated much to the NMR value. All
these results unambiguously justify the emergent $\mathcal{D}_{8}^{(1)}$ mass
spectrum at low energies in CoNb2O6.

As illustrated in Fig. 2, Fig. 3(a), and Ref. 23, with the transverse field
increasing near $H_{c}^{\rm{1D}}$, one signature of the spectrum is the
splitting of $E_{8}$-like peaks where the $\mathcal{D}_{8}^{(1)}$ particles
arise. Note that the mass ratio of $(m_{1}+m_{s})/m_{2}\approx 1.665$ in
$\mathcal{D}_{8}^{(1)}$ and that of $m_{2}/m_{1}\approx 1.618$ in $E_{8}$ are
very close. Taking into account the field dependence of peak positions, it is
inadequate to assert any emergent integrability from calculating mass ratios
of few low-energy peaks. An observation of a full spectrum is crucial.

From Fig. 3(a), we find that the mass ratio $m_{s}/m_{2}$ varies little over a
finite field range near $H_{c}^{\rm{1D}}$, whereas the mass ratios of other
$\mathcal{D}_{8}^{(1)}$ particles show much stronger field dependence. This is
because the single (anti)soliton, corresponding to a single domain wall, is a
topological excitation that is robust against local perturbations. A single
soliton is usually forbidden. Here it is inherent to the
$\mathcal{D}_{8}^{(1)}$ algebra and stabilised as confined by the interchain
coupling.

Figure 3:  (a) Field dependence of several peak positions in the measured THz
spectra extracted from Refs. 23 and 24. We take the lowest-energy peak (i.e.
the $m_{2}$ of the $\mathcal{D}_{8}^{(1)}$ at QCP) as the energy unit. The
low-energy peaks fit to the $\mathcal{D}_{8}^{(1)}$ algebraic structure best
at about $H_{c}^{\rm{1D}}\simeq 5.0$ T. (b) The calculated spectrum of CoNb2O6
at $H_{c}^{\rm{1D}}$ in the entire BZ. The purple dotted lines are analytical
predictions of several $\mathcal{D}_{8}^{(1)}$ quasiparticle dispersions.

##  V Discussions and conclusions

In Fig. 3(b) we show the calculated spectrum of the minimal model in the full
BZ using the same parameters as in Fig. 2(b). Clear dispersive bands
corresponding to well defined $\mathcal{D}_{8}^{(1)}$ quasiparticles show up
in low energies. The bands span about 30% of the BZ and follow the predicted
relativistic dispersion of $\mathcal{D}_{8}^{(1)}$ particle
$\omega=\sqrt{m_{n}^{2}c^{4}+k^{2}c^{2}}$, where $m_{n}$ refers to the mass of
the $n$-th quasiparticle, and $c$ is a characteristic velocity. Moreover, the
masses follow scaling relation $m_{n}\sim J_{i}^{4/7}$ [Fig. 3(c)] as
predicted from the Ising${}_{h}^{2}$ integrable theory (SM [26]). This is to
be contrast to the $E_{8}$ model in which $m_{n}\sim J_{i}^{8/15}$.
Interestingly, the dispersion and scaling relation apply to both the
topological (anti)soliton excitation and the odd-parity breathers (which are
invisible in the excitation spectrum) [28, 29]. The unusual topological
properties and symmetric restrictions make these particles potentially useful
resources in quantum information. All these features deserve to be explored by
spectral measurements such as INS and NMR.

Even though we do not perform fine tuning of model parameters, our minimal
model describes the spectrum of CoNb2O6 quantitatively well. This suggests
that our model, being a cluster mean-field approximation, captures the correct
emergent low-energy physics of the system. We can understand this as follows:
Fixing the transverse field to $H_{c}^{\rm{1D}}$, the interchain coupling is
definitely relevant when going from decoupled chains to a ladder. It drives
the system from critical to inside the ordered phase and causes emergent
$\mathcal{D}_{8}^{(1)}$ mass spectrum. We can further extend the cluster by
including more chains, until it covers the full 3D system, where the
approximation becomes exact. During this procedure, the system becomes non-
integrable. But we expect that the interchain coupling only causes less
relevant perturbation to the $\mathcal{D}_{8}^{(1)}$ mass spectrum. This is
indeed verified by our numerical calculation on 4 coupled Ising chains (Fig.
S5 of the SM [26]). Interestingly, we find that the spin frustration plays a
crucial role in stabilising the $\mathcal{D}_{8}^{(1)}$ spectrum. In the
frustrated case, the spectrum resembles the $\mathcal{D}_{8}^{(1)}$ of two
decoupled ladders, whereas the spectrum turns to $E_{8}$ in the unfrustrated
case.

Note that although we show the robustness of the $\mathcal{D}_{8}^{(1)}$ mass
spectrum for CoNb2O6, the experimental spectral intensity may crucially depend
on both the microscopic details of interactions and experimental setup, and is
generically non-universal. It is worth further noting that the emergent
$\mathcal{D}_{8}^{(1)}$ mass spectrum is not limited to CoNb2O6, but can be
applied to a large class of quantum magnets. It obviously shows up in spectra
of weakly coupled quantum Ising ladders with rung interaction much weaker than
that along the ladder direction. Moreover, we expect our argument for CoNb2O6
can be well applied to other weakly coupled Ising chains when
$H_{c}^{\rm{1D}}$ is close to $H_{c}^{\rm{3D}}$. It would also be interesting
to experimentally test the crossover from $E_{8}$ to $\mathcal{D}_{8}^{(1)}$
physics by tuning the distance of $H_{c}^{\rm{1D}}$ to $H_{c}^{\rm{3D}}$ via
either rotating the field direction or applying a pressure. Last but not
least, the physics should also be realised in specifically designed Rydberg
atom systems.

In conclusion, we show that the spectrum of CoNb2O6 near its 1D QCP is
described by the emergent $\mathcal{D}_{8}^{(1)}$ algebra. We show the
emergent $\mathcal{D}_{8}^{(1)}$ mass spectrum contains breather and exotic
single-soliton excitations. Our results advance the study on quantum
integrability and topological excitations in quantum magnets.

##  VI Acknowledgments

We thank Linhao Li for helpful discussions. This work is supported by the
National Key R&D Program of China (Grant No. 2023YFA1406500), the National
Natural Science Foundation of China (Grant Nos. 12334008, 12274288, and
12174441), the Innovation Program for Quantum Science and Technology Grant No.
2021ZD0301900, and Natural Science Foundation of Shanghai with Grant No.
20ZR1428400.

## References

  * Sachdev [2011] S. Sachdev,  _Quantum Phase Transitions_ (Cambridge University Press, Cambridge, England, 2011) pp. 1–521.
  * Coldea  _et al._ [2010a] R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Science 327, 177 (2010a).
  * Senthil  _et al._ [2004] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, and M. P. A. Fisher, Science 303, 1490 (2004), https://www.science.org/doi/pdf/10.1126/science.1091806 .
  * Cui  _et al._ [2023] Y. Cui, L. Liu, H. Lin, K.-H. Wu, W. Hong, X. Liu, C. Li, Z. Hu, N. Xi, S. Li, R. Yu, A. W. Sandvik, and W. Yu, Science 380, 1179 (2023).
  * Cui  _et al._ [2019] Y. Cui, H. Zou, N. Xi, Z. He, Y. X. Yang, L. Shu, G. H. Zhang, Z. Hu, T. Chen, R. Yu, J. Wu, and W. Yu, Phys. Rev. Lett. 123, 067203 (2019).
  * Wang  _et al._ [2018] Z. Wang, T. Lorenz, D. I. Gorbunov, P. T. Cong, Y. Kohama, S. Niesen, O. Breunig, J. Engelmayer, A. Herman, J. Wu, K. Kindo, J. Wosnitza, S. Zherlitsyn, and A. Loidl, Phys. Rev. Lett. 120, 207205 (2018).
  * Steinberg  _et al._ [2019] J. Steinberg, N. P. Armitage, F. H. L. Essler, and S. Sachdev, Phys. Rev. B 99, 035156 (2019).
  * Zou  _et al._ [2021] H. Zou, Y. Cui, X. Wang, Z. Zhang, J. Yang, G. Xu, A. Okutani, M. Hagiwara, M. Matsuda, G. Wang, G. Mussardo, K. Hódsági, M. Kormos, Z. He, S. Kimura, R. Yu, W. Yu, J. Ma, and J. Wu, Phys. Rev. Lett. 127, 077201 (2021).
  * Fava  _et al._ [2020] M. Fava, R. Coldea, and S. A. Parameswaran, Proc. Natl. Acad. Sci. U.S.A. 117, 25219 (2020).
  * Morris  _et al._ [2021] C. M. Morris _et al._ , Nat. Phys. 17, 832 (2021).
  * Liang  _et al._ [2015] T. Liang, S. M. Koohpayeh, J. W. Krizan, T. M. McQueen, R. J. Cava, and N. P. Ong, Nature Communications 6, 7611 (2015).
  * Xu  _et al._ [2022] Y. Xu, L. S. Wang, Y. Y. Huang, J. M. Ni, C. C. Zhao, Y. F. Dai, B. Y. Pan, X. C. Hong, P. Chauhan, S. M. Koohpayeh, N. P. Armitage, and S. Y. Li, Phys. Rev. X 12, 021020 (2022).
  * Kinross  _et al._ [2014] A. W. Kinross, M. Fu, T. J. Munsie, H. A. Dabkowska, G. M. Luke, S. Sachdev, and T. Imai, Phys. Rev. X 4, 031008 (2014).
  * Zhang  _et al._ [2020a] Z. Zhang, K. Amelin, X. Wang, H. Zou, J. Yang, U. Nagel, T. Rõõm, T. Dey, A. A. Nugroho, T. Lorenz, J. Wu, and Z. Wang, Phys. Rev. B 101, 220411 (2020a).
  * Wang  _et al._ [2023] X. Wang, K. Puzniak, K. Schmalzl, C. Balz, M. Matsuda, A. Okutani, M. Hagiwara, J. Ma, J. Wu, and L. Bella, arXiv:2308.00249 (2023).
  * Zamolodchikov [1989a] A. Zamolodchikov, in  _Integrable Sys Quantum Field Theory_ (Academic Press, San Diego, 1989) pp. 641–674.
  * Zhang  _et al._ [2020b] Z. Zhang, K. Amelin, X. Wang, H. Zou, J. Yang, U. Nagel, T. Rõõm, T. Dey, A. A. Nugroho, T. Lorenz, J. Wu, and Z. Wang, Phys. Rev. B 101, 220411 (2020b).
  * Wang  _et al._ [2021] X. Wang, H. Zou, K. Hódsági, M. Kormos, G. Takács, and J. Wu, Phys. Rev. B 103, 235117 (2021).
  * Kobayashi  _et al._ [1999] S. Kobayashi, S. Mitsuda, M. Ishikawa, K. Miyatani, and K. Kohn, Phys. Rev. B 60, 3331 (1999).
  * Kobayashi  _et al._ [2000] S. Kobayashi, S. Mitsuda, and K. Prokes, Phys. Rev. B 63, 024415 (2000).
  * Kobayashi  _et al._ [2004] S. Kobayashi, H. Okano, T. Jogetsu, J. Miyamoto, and S. Mitsuda, Phys. Rev. B 69, 144430 (2004).
  * Woodland  _et al._ [2023a] L. Woodland, D. Macdougal, I. M. Cabrera, J. D. Thompson, D. Prabhakaran, R. I. Bewley, and R. Coldea, Phys. Rev. B 108, 184416 (2023a).
  * Amelin  _et al._ [2020] K. Amelin, J. Engelmayer, J. Viirok, U. Nagel, T. Rõ om, T. Lorenz, and Z. Wang, Phys. Rev. B 102, 104431 (2020).
  * Amelin  _et al._ [2022] K. Amelin, J. Viirok, U. Nagel, T. R. o. om, J. Engelmayer, T. Dey, A. A. Nugroho, T. Lorenz, and Z. Wang, Journal of Physics A: Mathematical and Theoretical 55, 484005 (2022).
  * Woodland  _et al._ [2023b] L. Woodland, I. Lovas, M. Telling, D. Prabhakaran, L. Balents, and R. Coldea, Phys. Rev. B 108, 184417 (2023b).
  * [26] “See Supplemental Information [http://link…] for details about the iTEBD method, comparison of experimental and calculated spectra, the Ising universality at the 1D QCP of the single-chain model, scaling of the quasiparticle masses with $J_{i}$ with the $\mathcal{D}_{8}^{(1)}$ and $E_{8}$ integrabilities, and calculated spectra of 4 weakly coupled Ising chains, which also includes Refs. 24, 10.” .
  * Zamolodchikov [1989b] A. B. Zamolodchikov, Int. J. Mod. Phys. A 04, 4235 (1989b).
  * LeClair  _et al._ [1998a] A. LeClair, A. Ludwig, and G. Mussardo, Nuclear Physics B 512, 523 (1998a).
  * Gao  _et al._ [2024] Y. Gao, X. Wang, N. Xi, Y. Jiang, R. Yu, and J. Wu, arXiv:2402.11229 (2024).
  * Coldea  _et al._ [2010b] R. Coldea, D. A. Tennant, E. M. Wheeler, E. Wawrzynska, D. Prabhakaran, M. Telling, K. Habicht, P. Smeibidl, and K. Kiefer, Science 327, 177 (2010b).
  * LeClair  _et al._ [1998b] A. LeClair, A. Ludwig, and G. Mussardo, Nucl. Phys. B 512, 523 (1998b).
  * Delfino and Mussardo [1995] G. Delfino and G. Mussardo, Nucl. Phys. B 455, 724 (1995).

##  Appendix A Details of the iTEBD Simulations

In this section we provide a concise overview of the process of calculating
the dynamical structure factors (DSFs) in the infinite matrix product
representation. Initially, we compute the zero-temperature space-time
correlation $\left\langle\hat{O}(0,0)\hat{O}(r,t)\right\rangle$ by employing
the iTEBD method. The calculation of the space-time correlation of unitary
operators is a routine operation within the iTEBD framework. This space-time
correlation can be expressed as

$\displaystyle\left\langle\hat{O}(0,0)\hat{O}(r,t)\right\rangle$
$\displaystyle=\left\langle\psi_{G}\right|\hat{O}(0)e^{-it\hat{H}}\hat{O}(r)e^{it\hat{H}}\left|\psi_{G}\right\rangle$
(6)
$\displaystyle=\left\langle\psi_{L}\right|e^{-it\hat{H}}\hat{O}(r)e^{it\hat{H}}\left|\psi_{G}\right\rangle.$

In our study the local observable is a local spin operator
$S^{\alpha}\;(\alpha=x,\;y,\;z)$ which is both Hermitian and unitary. The
unitarity of the local spin operator guarantees the process
$\left\langle\psi_{L}\right|=\left\langle\psi_{G}\right|\hat{O}(0)$ will not
alter the canonical form of the ground state $\left\langle\psi_{G}\right|$.

Subsequently, standard operations of real-time evolution can be applied to the
matrix product states $\left\langle\psi_{G}\right|\hat{O}(0)$ and
$\left|\psi_{G}\right\rangle$. Then the dynamical structure factor of
$\hat{O}$ can be determined by performing a Fourier transformation on
$\left\langle\hat{O}(0,0)\hat{O}(r,t)\right\rangle$.

In practice, a sixth-order Suzuki-Trotter decomposition is utilised to
minimise time-step errors. The values of the time step $\tau$ and the number
of steps $N$ are optimised during the calculation. A larger $\tau$ results in
a larger Trotter error and a narrower range of energy. As $N$ increases,
truncation errors also increase, and the memory cost grows quadratically.
However, taking a large number of steps can improve the energy resolution. To
balance computing resources and acceptable errors, we choose a truncation
dimension $D=160$ for the computation of a ladder, and $D=80$ for the
computation of a single chain. We also set $\tau=0.6J^{-1}$ and $N=2000$ as
the maximum number of steps. Note that with the sixth-order decomposition,
this parameter set already generates more precise results than taking
$\tau=0.04J^{-1}$ in a second-order decomposition.

##  Appendix B Comparison of experimental and calculated spectra

Figure 4: (a) The measured THz spectra of CoNb2O6 at 2.5 K with transverse
field ranging from 0 to 12 T in 0.5 T steps, from bottom to top. Data adapted
from Ref. 10. (b) The calculated spectra at $k=0$ of the single-chain model
with a longitudinal field $\tilde{h}\simeq 0.01J$ under transverse field
ranging from 0 to 6 T in 0.2 T steps, from bottom to top. (c) The calculated
spectra at the zone centre of the transverse field Ising ladder with
$J_{i}=0.1J$ under transverse fields ranging from $0.42~{}J$ to $0.59~{}J$ in
$0.01~{}J$ steps, from bottom to top.

In this section, we present calculated zero-momentum spectra of the minimal
model described by the Hamiltonian in Eq. (4) of the main text in Fig. 4(b)
and (c). For comparison, we also show the experimental spectra, adapted from
Ref. 10, in Fig. 4(a).

We first show in Fig. 4(b) the spectra of a single chain under several
transverse field values. Here the 3D ordering effect is treated by a
longitudinal field $\tilde{h}$. This corresponds to setting $J_{i}=0$ in the
minimal model of Eq. (4). We can identify several characteristic peaks whose
excitation energies increase with increasing the transverse field. Their
energy positions and field dependences are in accord with the measured ones,
indicating that the single chain Hamiltonian in Eq. (2) of the main text
already captures main features of the spectrum of CoNb2O6. However, we observe
that several low-energy peaks in the experimental spectra split when
increasing the field approaching the 1D QCP, $H_{c}^{\rm{1D}}$. This splitting
is missed by the single chain model. To understand this splitting, we computed
the low-energy spectra of an Ising ladder under several transverse fields near
$H_{c}^{\rm{1D}}$, and the results are shown in Fig. 4(c). The spectra exhibit
several different features to those of a single chain. First, the high-energy
modes whose energy increase with increasing field disappear, implying that
these modes are associated with the DW and other non-Ising terms of the
microscopic Hamiltonian (see the next section for further discussions). Then,
we focus on the low-energy excitation peaks. Away from $H_{c}^{\rm{1D}}$, they
look similar to those of a single chain under an effective longitudinal field.
With $H_{c}^{\rm{1D}}$ approaching, the low-energy peaks in the spectrum split
and evolve with the field. Their energies and spectral weights redistribute
and ultimately develop a structure following the $\mathcal{D}_{8}^{(1)}$
algebra at $H_{c}^{\rm{1D}}$, which include soliton and breather excitations,
as addressed in the main text. Upon further increasing the field, the single
soliton peak persists at finite energies, while the peaks corresponding to
breathers shift to lower energies and decay gradually. The behaviour of these
low-energy peaks is highly similar to that observed in the experiment shown in
Fig. 4(a). Similar behaviours of the spectra have been recently observed in
several other THz measurements in CoNb2O6 [24, 10, 23].

It is worth noting that the critical field of the single chain in the
calculation is slightly lower than the experimental value. This could be
associated with either the absence of the interchain coupling in the
calculation or calibration of magnetic field in experiments. Nevertheless,
this slight mismatch does not affect the overall field dependent behaviours of
the spectra, especially for those gapped modes associated with the domain wall
(DW) and other non-Ising interactions.

##  Appendix C Ising universality and Effects of the DW interaction

Figure 5: (a) Plot of the transverse field evolution of several characteristic
energies extracted from peaks of spectral functions at $k=0$. Red points
represent the lowest excitation mode of the Ising-type, which becomes critical
at $H_{c}^{\rm{1D}}$. Deep-coloured points are calculated from the single
chain model [Eq. (2) in the main text] without an effective longitudinal field
$\tilde{h}$, and light-coloured ones are with $\tilde{h}$. Purple points
represent modes associated the DW and other non-Ising interactions of the
model. Dashed lines are guides to the eye with a factor of 2 difference in
their slopes, in accord with the $(1+1)$D Ising universality. (b) The
calculated dynamical structure factors of the single chain model at
$H_{c}^{\rm{1D}}$ in the entire BZ. The Ising criticality is characterised by
the linear dispersive mode guided by the blue dashed lines.

As described in the main text, the microscopic Hamiltonian along the chain in
Eq. (2) contains several non-Ising terms. Here we show that these terms,
especially the DW interaction, do not affect the Ising universality at
$H_{c}^{\rm{1D}}$ (referring to the 1D QCP of a single chain, instead of the
ladder). We have calculated the spectral functions at $k=0$ of a single chain
under different transverse fields, described by Eq. (2) [see Fig. 4(b)]. From
peaks of the spectral functions, we can identify several characteristic
energies. Their field evolution is shown in Fig. 5(a). In a TFIC, the
excitation gap decreases with increasing field in the ordered state. But the
gap of the lowest energy mode initially increases with the field, as an effect
of the DW interaction. Keep increasing the field, the lowest energy mode
decreases after an avoid level crossing with another higher energy mode at
about $H\simeq 2$ T. Approaching $H_{c}^{\rm{1D}}$, the lowest excitation gaps
for both $H>H_{c}^{\rm{1D}}$ and $H<H_{c}^{\rm{1D}}$ decrease linearly, and
the gap ratio at the same distance to $H_{c}^{\rm{1D}}$ between the two sides
is $2$. These features imply the TFIC universality of the QCP is unaffected by
the DW and other non-Ising interactions.

We then provide more evidences for the TFIC universality by examining the
critical behaviours of magnetisation and entanglement entropy in the vicinity
of $H_{c}^{\rm{1D}}$. As shown in Fig. 6, the order parameter, the
magnetisation $M$, scales with the transverse field as
$M\sim(H_{c}^{\rm{1D}}-H)^{1/8}$. Moreover, the entanglement entropy scales
with the length of the chain segment as $E\sim\frac{c}{3}\ln L$ with the
central charge $c=1/2$. These results further confirm that the single-chain
model belongs to the (1+1)D Ising universality class near $H_{c}^{\rm{1D}}$.

Figure 6: Results obtained via the iTEBD calculation on the single chain
Hamiltonian in Eq. (2) of the main text. (a) The average of magnetisation $M$
versus the transverse field $H$. (b) Same data as in (a) but are plotted
against $H_{c}^{\rm{1D}}-H$ on a log-log scale with the critical field
determined to be $H_{c}^{\rm{1D}}=0.29822$. The blue line is fit to the
scaling function $M\propto(H_{c}^{\rm{1D}}-H)^{\beta}$ with the order
parameter exponent $\beta=1/8$. (c) Entanglement entropy versus the length $L$
of the chain segment in the semi-log scale. The fitted slope value agrees with
a central charge of $c=1/2$.

In fact, by including the interactions in Eq. (2) term by term and compare the
corresponding spectra, we can identify the modes associated with each term. We
find that the modes whose energies increase with increasing transverse field
as shown in Fig. 4(b) and Fig. 5(a) are associated with the DW and other non-
Ising terms of the Hamiltonian. For $H\sim H_{c}^{\rm{1D}}$, the DW related
modes become dominant in the spectrum above about $2$ meV, as shown in Fig.
5(a). To see this more clearly, we show the spectrum at the QCP in the entire
Brillouin zone (BZ) in Fig. 5(b). At low energies, there is a linear
dispersive mode enveloping a continuum, aligning with the prediction of the
Ising model. Around $2$ meV, the spectrum exhibits significant folding and
flattening (compared to the typical bandwidth of the Ising model $\sim
2J\simeq 5.4$ meV) due to the DW interaction. These features are consistent
with recent THz and INS measurements [10, 24, 22, 30].

##  Appendix D Scaling of masses of the $\mathcal{D}_{8}^{(1)}$ and $E_{8}$
quasiparticles

Figure 7: Scaling behaviours of several low-energy excitation modes in the
spectrum at $k=0$ with the interchain coupling $J_{i}$ at $H_{c}^{\rm{1D}}$ in
an Ising ladder ($\tilde{h}=0$) and a single chain ($\tilde{h}\propto J_{i}$),
respectively. The extracted scaling exponents, $4/7$ in the ladder model and
$8/15$ in the single chain, are consistent with the predicted values from the
Ising${}_{h}^{2}$ and $E_{8}$ integrable models.

According to the Ising${}^{2}_{h}$ quantum integrable field theory [31], a
standard scaling applies to masses of the $\mathcal{D}_{8}^{(1)}$ particles

$m_{n}\sim\lambda^{1/(2-d)},$ (7)

where $\lambda$ is the effective interchain coupling, and $d=1/4$ is the
scaling dimension of the interchain interaction. This leads to
$m_{n}\sim\lambda^{4/7}\sim J_{i}^{4/7}$ for all $\mathcal{D}_{8}^{(1)}$
particles including both soliton, anti-soliton, and breathers.

A similar scaling relation,

$m_{n}\sim\tilde{h}^{1/(2-d)},$ (8)

holds for $E_{8}$ particles with $d=1/8$ [32], where according to the chain
mean-field approximation, the effective field $\tilde{h}\propto J_{i}$.
Therefore, we expect that $m_{n}\sim J_{i}^{8/15}$. In Fig. 7, as well as Fig.
3(c) of the main text, we show the $J_{i}$ dependence of energies of several
low-energy excitations at zone centre in the transverse field Ising ladder,
which correspond to $m_{2}$, $m_{s}$, $m_{1}+m_{s}$, and $m_{4}$,
respectively. All these modes show the same scaling behaviour consistent with
the theoretical prediction. This scaling property provides further evidence in
support of the $\mathcal{D}_{8}^{(1)}$ mass spectrum, and can be
experimentally detected. We also show in Fig. 7 the scaling behaviour of
several $E_{8}$ quasiparticles for comparison. As expected, they fall into a
different scaling, with the power-law exponent $8/15$. This difference in the
scaling behaviour can be used to determine the exact nature of the emergent
integrability.

##  Appendix E Spectra of 4 weakly coupled Ising chains

Figure 8: (a)-(c) Calculated spectra at $k=0$ and $H_{c}^{\rm{1D}}$ of 4
weakly coupled Ising chains [see inset of panel (a)] with (a) $J_{i}=0$ and
$J_{i}^{\prime}=0.1~{}J$, (b) $J_{i}=0.05~{}J$ and $J_{i}^{\prime}=0.1~{}J$,
and (c) $J_{i}=0.1~{}J$ and $J_{i}^{\prime}=0$. In each panel, the vertical
dashed lines at peak positions correspond to the masses of quasiparticles or
edges of multiparticle bound states. The spectra in (a) and (b) are well
described by the $\mathcal{D}_{8}^{(1)}$ algebra, whereas the one in (c) is
consistent with the $E_{8}$ model. (d)-(f) are corresponding spectra in the
entire BZ. The inset of panel (a) shows a sketch of how the 4 Ising chains are
aligned. $J_{i}$ and $J_{i}^{\prime}$ refer to the nearest- and next-nearest-
neighbour interchain Ising couplings, respectively.

In the main text, we have shown, via a cluster mean-field approximation, that
the spectrum of CoNb2O6 at $H_{c}^{1D}$ is well described by the
$\mathcal{D}_{8}^{(1)}$ algebra. As for the validity of this approximation, we
argue that we can extend the cluster by coupling more chains. Although the
system becomes non-integrable, the interchain fluctuations only cause less
relevant perturbation to the $\mathcal{D}_{8}^{(1)}$ spectrum. To see this is
a valid argument, we hereby present calculated spectra of 4 coupled Ising
chains in Fig. 8. The spectrum in panel (a) is for two decoupled ladders. We
see it is identical to the one of an Ising ladder, which is precisely
described by the $\mathcal{D}_{8}^{(1)}$ algebra. In panel (b), we consider
the case where the 4-chain system is fully frustrated with interchain
couplings $J^{\prime}_{i}=2J_{i}$. In this case, one can easily check that the
effective longitudinal field $\tilde{h}$ applied on each chain is exactly
zero. This system is non-integrable. However, as shown in Fig. 8(b), the peaks
still resemble the $\mathcal{D}_{8}^{(1)}$ mass spectrum up to about $E\sim
J$. In panel (c), we consider the unfrustrated case. Interestingly, the
spectrum resembles that of the $E_{8}$ model instead of
$\mathcal{D}_{8}^{(1)}$.

In the comparison we observe that the frustration plays a crucial role for the
$\mathcal{D}_{8}^{(1)}$ spectrum. The frustrated alignment of chains causes
cancellation of effective longitudinal field $\tilde{h}$. Once $\tilde{h}$ is
suppressed, the interchain fluctuations only add a perturbation to the
$\mathcal{D}_{8}^{(1)}$ spectrum, as expected. However, in the unfrustrated
case, the longitudinal field $\tilde{h}$ dominates and drives the system to
the $E_{8}$ integrability. As for CoNb2O6, the chains are indeed aligned in a
frustrated way, and together with the proximity to the 3D QCP, the
longitudinal field $\tilde{h}$ is substantially suppressed. In this case, as
we illustrated in the 4-chain case, the interchain fluctuations only add a
weak perturbation to the $\mathcal{D}_{8}^{(1)}$ mass spectrum at low energy.


# Simultaneous Multi-Pivot Neural Machine Translation

Raj Dabre1 Aizhan Imankulova2 Masahiro Kaneko3 Abhisek Chakrabarty4
National Institute of Information and Communications Technology1,4
CogSmart2 Tokyo Institute of Technology3
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Parallel corpora are indispensable for training neural machine translation
(NMT) models, and parallel corpora for most language pairs do not exist or are
scarce. In such cases, pivot language NMT can be helpful where a pivot
language is used such that there exist parallel corpora between the source and
pivot and pivot and target languages. Naturally, the quality of pivot language
translation is more inferior to what could be achieved with a direct parallel
corpus of a reasonable size for that pair. In a real-time simultaneous
translation setting, the quality of pivot language translation deteriorates
even further given that the model has to output translations the moment a few
source words become available. To solve this issue, we propose multi-pivot
translation and apply it to a simultaneous translation setting involving pivot
languages. Our approach involves simultaneously translating a source language
into multiple pivots, which are then simultaneously translated together into
the target language by leveraging multi-source NMT. Our experiments in a low-
resource setting using the N-way parallel UN corpus for Arabic to English NMT
via French and Spanish as pivots reveals that in a simultaneous pivot NMT
setting, using two pivot languages can lead to an improvement of up to 5.8
BLEU.

##  1 Introduction

Neural machine translation (NMT) Bahdanau et al. (2015) is an end-to-end
approach that is known to give the state of the art results for a variety of
language pairs. Unfortunately, most language pairs have little to no parallel
corpora, which are indispensable for NMT. In such situations, unsupervised,
zero-shot, and zero-resource NMT approaches have been developed, which have
enabled translation between such pairs. In particular, the most
straightforward and most effective approach is the pivot language NMT approach
which involves using a pivot language such that the source language sentence
is translated into the pivot language, which is then translated into the
target language. Pivot language translation quality is often poorer than the
direct translation quality, even if a small amount of direct parallel corpus
exists. Naturally, it is expected that pivot translation quality further
degrades in a real-time translation setting where simultaneous NMT (SNMT) Gu
et al. (2019); Ma et al. (2019); Zheng et al. (2019b) approaches are used.

SNMT approaches are those that perform a translation with partial source
language sentences, which are fed to the model a word/token at a time. The
most straightforward simultaneous NMT approach is the wait-k approach Ma et
al. (2019) where the model starts generating translations, a word at a time
when $K$ source words/tokens are available. Note that simultaneous NMT
approaches already exhibit more inferior translation quality as compared to
their non-simultaneous counterparts Gu et al. (2019); Ma et al. (2019); Zhang
et al. (2020). Naturally, simultaneous pivot language NMT, where both the
source to pivot and pivot to target language models are simultaneous NMT
models, is expected to suffer from a drastically reduced translation quality.
Fortunately, nothing prevents us from using multiple pivot languages.

Figure 1: Our multi-pivot NMT architecture. When each component model is an
SNMT model we obtain a simultaneous multi-pivot NMT (SMPNMT) model.

Previous work in statistical machine translation (SMT) Utiyama and Isahara
(2007); Bertoldi et al. (2008) has shown that using multiple pivot languages
can significantly improve translation quality as opposed to using a single
pivot language Dabre et al. (2015). In SMT, phrase-table triangulation can be
done for each pivot language leading to a new phrase table following which all
phrase tables are combined using an interpolation approach. In the case of
NMT, however, we can resort to multi-source neural machine translation (MSNMT)
Dabre et al. (2017); Zoph and Knight (2016) which can be helpful. MSNMT is a
variation of NMT which involves translating a sentence written in two
different languages into a target language. The premise of MSNMT is that each
source language has phenomena that are hard to translate. Thus by providing
multiple sources, the decoder is able to better disambiguate the hard-to-
translate phenomena in the individual sources leading to an improvement in
translation quality. In the pivot language NMT setup, if the pivot to target
model is replaced with an MSNMT model, then we can achieve multi-pivot NMT
(MPNMT). Finally, when the source to pivot and multi-pivot to target models
are simultaneous NMT models, we arrive at simultaneous multi-pivot NMT
(SMPNMT), which is the main idea in this paper. Our experiments on Arabic to
English translation via French and Spanish as pivot languages show that multi-
pivot NMT significantly outperforms single-pivot NMT in a non-SNMT setting and
in an SNMT setting, SMPNMT outperforms single-pivot SNMT by over 5.8 BLEU.
This paper is intended to be a proof of concept that multi-pivot NMT works and
is useful. We leave experimentation on a variety of other language pairs in
realistic settings for the future.

##  2 Related Work

The ideas in this paper revolve around pivot language NMT, SNMT and MSNMT.

Pivot language translation helps improve the accuracy of low/zero-resource
translation. In the case of SMT, pivot translation was done using chaining, or
phrase table triangulation Wu and Wang (2007); Utiyama and Isahara (2007).
Dabre et al. (2015) showed that in SMT, leveraging multiple pivot languages
can significantly improve the translation quality which was one of the
motivations for our work. In NMT, although zero-shot translation Johnson et
al. (2017) is an option, chaining is much simpler to use and gives reasonable
results Firat et al. (2016), and hence we rely on it in this paper. Cheng et
al. (2017) first proposed the pivot based neural machine translation by
jointly training source to pivot and pivot to target models, which improves
the effectiveness of chaining. Currey and Heafield (2019) proposed an
alternative way of using a monolingual pivot-language data for zero-resource
NMT via back-translation Sennrich et al. (2016).

SNMT work are mainly divided into two groups based on their policies Zheng et
al. (2019b): SNMT with fixed policy Dalvi et al. (2018); Ma et al. (2019);
Zheng et al. (2019b) and SNMT with adaptive policy Cho and Esipova (2016); Gu
et al. (2017); Zheng et al. (2019a, c, 2020). Models with adaptive policy
dynamically decide whether the model should READ or WRITE tokens with some
waiting criteria. On the other hand, models with the fixed policy have much
simpler architecture and lower latency compared to more complicated models
with the adaptive policy. As a study utilizing additional information for
SNMT, it has been shown that the image information related to the translated
sentence contributes to performance improvement Imankulova et al. (2020);
Caglayan et al. (2020).

Zoph and Knight (2016) first proposed MSNMT using multiple encoders for each
source language and a single decoder for the target language. Dabre et al.
(2017), on the other hand, showed that simply concatenating multiple source
sentences is sufficient, which helps avoid the need for modification of the
NMT architecture. However, this prevents the special handling of individual
languages and thus, we rely on the multi-encoder approach. Other multi-source
NMT papers include using mixture-of-experts Garmash and Monz (2016),
incorporating linearized parses of the source data Currey and Heafield (2018),
and addressing missing data in multi-source MT Nishimura et al. (2018). One
major limitation of MSNMT is that N-way parallel corpora are required, and
although Nishimura et al. (2018) partially address this issue, it is important
to note that MSNMT can be achieved without any N-way corpora Firat et al.
(2016). Our idea currently requires N-way corpora simply to establish its
efficacy, but we will relax this constraint in the future.

##  3 Background

We briefly introduce some background concepts related to pivot NMT, SNMT and
MSNMT.

###  3.1 Pivot NMT

The most straightforward approach for pivot-language NMT involves chaining two
NMT models, one for source to pivot language translation and one for a pivot
to target language translation. There are other sophisticated approaches such
as joint training both models, but our objective is to enhance the quality of
the chaining approach via multiple pivots.

###  3.2 SNMT

The most straightforward approach for SNMT is the wait-k approach Ma et al.
(2019) with a fixed policy. As tokens are fed to the encoder one at a time, we
have to rely on a unidirectional encoder that cannot attend to future tokens.
Once the encoder has been fed $K$ tokens, the decoder starts generating a
token at a time. This means that at a given time, the $i^{th/st}$ decoding
step can only see $K+i-1$ encoder token representations. Once the whole input
sentence is available, wait-k behaves like regular NMT except with a
unidirectional encoder.

###  3.3 MSNMT

A standard encoder-decoder NMT model can be converted to an MSNMT model by
having $N$ encoders, one for each source language. Consider the case of 2
source languages with 2 encoders. Consider source sentences $s_{1}$ and
$s_{2}$ with lengths $l_{1}$ and $l_{2}$. Suppose that the encoder
representations are $E_{s_{1}}$ and $E_{s_{2}}$ with dimensions $(l_{1},h)$
and $(l_{1},h)$ respectively where $h$ is the hidden size. For each time step
in the decoder, we first compute attentions for each source sentence
independently as $A_{s_{1}}$ and $A_{s_{2}}$ with hidden sizes $h$.
$A_{s_{1}}$ and $A_{s_{2}}$ are then passed through a gating mechanism to
determine their relative importance and generate the final attention $A$.
$A=w*A_{s_{1}}+(1-w)*A_{s_{2}}$ where $w={\rm
sigmoid}(W[A_{s_{1}}:A_{s_{1}}]+b)$. $W$ is a weight matrix of dimensions
$(2h,h)$ and $b$ is a bias of dimension $h$. $:$ represents concatenation.

An MSNMT model typically requires N-way corpora for training which we use in
this paper, but this need not be the case as shown by Firat et al. (2016). An
MSNMT model can be converted into its wait-k SNMT equivalent by enforcing the
unidirectional constraint on each encoder. The decoder generates a token at a
time once $K$ tokens have been fed to both encoders. In reality, each encoder
can receive tokens at different rates, but for simplicity, we assume that they
receive tokens at the same time.

##  4 MPNMT and SMPNMT

Figure 1 gives an overview of our approach. Combining the SNMT with MSNMT lead
us to multi-pivot NMT (PNMT). MPNMT involves translating the source into N
pivot languages via bilingual or multilingual NMT models and then translating
these N pivot languages into the target language via an MSNMT model. The
bilingual or multilingual NMT models are trained using parallel corpora, and
the MSNMT models are trained using N-way corpora (although this can be
achieved without bilingual corpora too Firat et al. (2016)). Replacing the
source to pivot and pivot to target models in the MPNMT framework with their
(wait-k) SNMT equivalents leads us to simultaneous MPNMT (SMPNMT).

##  5 Experimental Settings

###  5.1 Datasets and preprocessing

We use the UN dataset and focus on Arabic to English translation using French
and Spanish as pivot languages for our experiments. The UN corpus is N-way
parallel for the training, development and test set splits. As our purpose is
not to establish state-of-the-art results but to validate the efficacy of our
multi-pivot method, we consider a sample of 200,000 training sentences that
are also N-way parallel. This also helps simulate a low-resource setting where
multi-pivot translation should be extremely beneficial. We do not perform any
pre-processing on our own and let our implementation handle it via its
internal mechanisms.

###  5.2 Implementation, Training and Evaluation details

For our experiments, we modify the implementation of the transformer model in
the tensor2tensor toolkit111https://github.com/tensorflow/tensor2tensor
(v1.15.4). The tensor2tensor toolkit has internal tokenization and sub-word
segmentation mechanisms, which we use with a subword vocabulary of size
16,000. The subword vocabularies for the encoder and decoder are separate. In
the case of multiple pivot languages (multi-source models) the encoder is
shared222Simply put, in the French+Spanish multi-source (pivot to target
model) model, the subword vocabulary is shared between French and Spanish but
the English target vocabulary is separate. between the pivot languages which
helps cut down the number of parameters. We use hyperparameter configurations
corresponding to the “transformer_base” model. We save and evaluate our models
on the development set every 1000 batches with BLEU Papineni et al. (2002) as
the evaluation metric. We train our models till the BLEU score does not grow
for 10 consecutive evaluations. We average the last 10 saved checkpoints and
then decode the model. As we are working in a simultaneous translation
setting, using greedy search makes sense as tokens should be output one at a
time 333It’s possible to consider a sophisticated beam search method, but that
is beyond the scope of this paper..

###  5.3 Models to Compare

We train and evaluate full sentence and wait-k models for k=1,2,4,6 and 8 for
the following models.

  * •

Direct Baselines: Direct Arabic to English.

  * •

Pivot Baselines: Arabic to pivot to English using a single pivot of French or
Spanish.

  * •

Our proposed models: Arabic to French+Spanish to English using both French and
Spanish as pivots.

All source to pivot models are unidirectional bilingual models, although it is
possible to train a single one-to-many model to translate from a source to
multiple pivots. When the source and pivot models are both wait-k, there are a
total of 25 combinations of wait-k for source to pivot and pivot to target.
The effective wait-k is the sum of the wait-k’s of individual models.

##  6 Results

| Type |  | Pivot
---
Language
BLEU
1 | Direct | N/A | 34.0
2 | Pivot | French | 27.5
3 | Pivot | Spanish | 25.1
4 |  | Pivot (Ours)
---
| French+Spanish
---
29.9

Table 1: BLEU scores for baseline Arabic to English models. Results are shown
for (1) direct Arabic to English, (2) Arabic to English via French as a pivot,
(3) Arabic to English via Spanish as a pivot and (4) Arabic to English via
both French and Spanish as a pivot.

###  6.1 Full Sentence Results

We first show the results for full-sentence direct, single-pivot and multi-
pivot NMT models in Table 1. It is clear that when compared to directly
translating into the target language (row 1), translating through a pivot
language (rows 2 and 3) dramatically hurts the translation quality. The
translation quality drops by 6.5 and 8.9 BLEU for French and Spanish as
pivots, respectively. However, when both pivot languages are used together,
the translation quality shows a significant improvement of 2.4 BLEU compared
to the best translation score obtained using French as the pivot. In other
words, the multi-pivot model is only 4.1 BLEU below the direct model as
opposed to 6.5 to 8.9 BLEU below the direct model when single pivots were
used. This result is important because the pivot language sentence is
inherently noisy and translating into the target language makes it even
noisier. However, combining two noisy translations via multi-source NMT can
significantly enhance the translation quality. Note that the direct
translation model score is the upper limit of translation quality if a
parallel corpus is available. Hence, it acts as a reference upper limit score
that we wish to approach via multi-pivot translation.

###  6.2 Simultaneous NMT Results

Table 2 contains results for simultaneous models for Arabic to English
translation directly and via pivots.

####  6.2.1 SNMT vs Full Sentence Models

First, when comparing the direct wait-k results against direct full-sentence
results, it can be seen that as the value of k increases, the gap between
full-sentence and SNMT drops rapidly, and when k is 8, the gap drops to 2.9
BLEU. Depending on the cost-benefit requirement, we can increase or decrease
the value of k to yield the desired translation quality. Next, comparing
wait-k pivot models to their non wait-k equivalents, we can observe that the
wait-k equivalents give significantly more unsatisfactory results. Although
this is expected, it must be noted that the drop in translation quality is
even more severe (up to 6.2 BLEU (21.3 vs 27.5 with French as a pivot))
compared to the drop for direct translation models (only 2.9 BLEU). However,
this is where using multiple pivots comes to the rescue. Compared to the
multi-pivot full-sentence model, the wait-k equivalent is only 2.8 BLEU points
weaker. Thus it is clear that by employing multiple pivots, we can reduce the
BLEU drop from 6.2 BLEU (using one pivot) to 2.8 BLEU (using two pivots). We
suppose that using additional pivots may help close this gap even further and
will verify it in the future.

k |  | 1 | 2 | 4 | 6 | 8
---|---|---|---|---|---|---
Direct |  | 19.1 | 22.5 | 26.3 | 30.0 | 31.1
Pivot |  | P2T-k
French | S2P-k | 1 | 2 | 4 | 6 | 8
1 | 7.3 | 9.3 | 11.9 | 14.1 | 15.4
2 | 8.4 | 10.6 | 13.7 | 16.1 | 17.3
4 | 9.6 | 12.3 | 16.1 | 18.7 | 19.8
6 | 9.9 | 12.8 | 16.8 | 19.6 | 21.0
8 | 10.1 | 12.9 | 16.8 | 19.7 | 21.3
Spanish | 1 | 5.8 | 8.2 | 11.0 | 13.0 | 14.6
2 | 6.6 | 9.3 | 12.3 | 14.7 | 16.3
4 | 7.3 | 10.4 | 13.9 | 16.4 | 18.0
6 | 7.7 | 11.0 | 14.9 | 17.1 | 19.1
8 | 7.6 | 11.1 | 15.1 | 17.9 | 20.0
1 | 10.4 | 12.4 | 15.1 | 16.7 | 18.3
French+ | 2 | 11.7 | 14.1 | 17.3 | 18.7 | 20.0
Spanish | 4 | 13.9 | 16.6 | 20.1 | 21.6 | 23.1
(Ours) | 6 | 14.4 | 17.6 | 21.3 | 23.0 | 25.2
| 8 | 14.4 | 18.1 | 21.9 | 23.8 | 27.1

Table 2: BLEU scores for wait-k Arabic to English models. The values of k are
1, 2, 4, 6 and 8. In the case of pivot models, two wait-k models are involved
leading to 25 combinations. S2P and P2T are short for source to pivot and
pivot to target. Results are shown for (1) Direct Arabic to English, (2)
Arabic to English via French as a pivot, (3) Arabic to English via Spanish as
a pivot and (4) Arabic to English via both French and Spanish as a pivot.

####  6.2.2 SNMT pivot vs SNMT direct models

The best multi-pivot result (27.1 BLEU) with an effective wait-k of 16 is 4
BLEU points below the best direct translation result (31.1 BLEU) with a wait-k
of 8. Although this drop in translation quality is undesirable, consider the
best single pivot results using French (21.3 BLEU) or Spanish (20.0 BLEU),
which are 9.8 and 11.1 BLEU below the direct. Compared to the single pivot
models, multi pivot models manage to recover most of the loss in translation
quality. This shows that multiple pivots are more useful in an SNMT setting as
compared to a full sentence setting. Once again, the reader should note that
direct model scores are upper limits when parallel corpora exist, which may
not always be the case.

####  6.2.3 Comparing S2P And P2T Models

When source and pivot models are both wait-k, the effective wait-k is the sum
of the individual wait-k’s. Consider an effective wait-k of 8, which can be
obtained by individual wait-k combinations of (2, 6), (4, 4) and (6, 2) for
source to pivot (S2P) and pivot to target (P2T). When French is used as the
pivot, the corresponding scores are 16.1, 16.1 and 12.8 BLEU. In the case of
Spanish, the scores are 14.7, 13.9 and 11.0. Finally, when both pivot
languages are used, the scores are 18.7, 20.1 and 17.6. All these scores show
that a longer wait-k for the P2T NMT model is more important than a longer
wait-k for the S2P NMT model. This makes sense as the P2T model is responsible
for the final translation and hence needs to have as much context as possible
for translation. This might also mean that the P2T model is the bottleneck,
and multi-pivot NMT addresses a part of this bottleneck. In the future, we
will investigate how to reduce the need for long wait-k’s for P2T models.

##  7 Conclusion

This paper proposed multi-pivot NMT and extended it to a simultaneous NMT
setting where the source to pivot and pivot to target models are wait-k
simultaneous NMT models. We showed that in a full sentence setting, multiple
pivots can lead to an improvement of up to 2.4 BLEU and in an SNMT setting,
multiple pivots can lead to an improvement of up to 5.8 BLEU. This shows that
multiple pivots are even more critical in real-time translation settings. The
work presented in this paper, due to its preliminary nature, relies on N-way
corpora for multi-source (pivots in our work) translation, however previous
work on multi-source translation Firat et al. (2016) has shown that we do not
need N-way corpora at all. Our future work will focus on obviating the need
for N-way corpora as well as pushing the limits of translation quality via a
larger number of pivot languages.

## References

  * Bahdanau et al. (2015) Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. 2015.  Neural machine translation by jointly learning to align and translate.  In _3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings_.
  * Bertoldi et al. (2008) N. Bertoldi, M. Barbaiani, M. Federico, and R. Cattoni. 2008.  Phrase-based statistical machine translation with pivot languages.  In _Proceedings of the International Workshop on Spoken Language Translation_ , pages 143–149, Huawaii.
  * Caglayan et al. (2020) Ozan Caglayan, Julia Ive, Veneta Haralampieva, Pranava Madhyastha, Loïc Barrault, and Lucia Specia. 2020.  Simultaneous machine translation with visual context.  In _EMNLP_ , pages 2350–2361. Association for Computational Linguistics.
  * Cheng et al. (2017) Yong Cheng, Qian Yang, Yang Liu, Maosong Sun, and Wei Xu. 2017.  Joint training for pivot-based neural machine translation.  In _Proceedings of the 26th International Joint Conference on Artificial Intelligence_ , pages 3974–3980.
  * Cho and Esipova (2016) Kyunghyun Cho and Masha Esipova. 2016.  Can neural machine translation do simultaneous translation?  _arXiv preprint arXiv:1606.02012_.
  * Currey and Heafield (2018) Anna Currey and Kenneth Heafield. 2018.  Multi-source syntactic neural machine translation.  In _Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing_ , pages 2961–2966. Association for Computational Linguistics.
  * Currey and Heafield (2019) Anna Currey and Kenneth Heafield. 2019.  Zero-resource neural machine translation with monolingual pivot data.  In _Proceedings of the 3rd Workshop on Neural Generation and Translation_ , pages 99–107, Hong Kong. Association for Computational Linguistics.
  * Dabre et al. (2017) Raj Dabre, Fabien Cromieres, and Sadao Kurohashi. 2017.  Enabling multi-source neural machine translation by concatenating source sentences in multiple languages.  In _Proceedings of MT Summit XVI, vol.1: Research Track_ , pages 96–106, Nagoya, Japan.
  * Dabre et al. (2015) Raj Dabre, Fabien Cromieres, Sadao Kurohashi, and Pushpak Bhattacharyya. 2015.  Leveraging small multilingual corpora for SMT using many pivot languages.  In _Proceedings of the 2015 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 1192–1202, Denver, Colorado. Association for Computational Linguistics.
  * Dalvi et al. (2018) Fahim Dalvi, Nadir Durrani, Hassan Sajjad, and Stephan Vogel. 2018.  Incremental decoding and training methods for simultaneous translation in neural machine translation.  In _Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 2 (Short Papers)_ , pages 493–499. Association for Computational Linguistics.
  * Firat et al. (2016) Orhan Firat, Baskaran Sankaran, Yaser Al-Onaizan, Fatos T. Yarman Vural, and Kyunghyun Cho. 2016.  Zero-resource translation with multi-lingual neural machine translation.  In _Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing_ , pages 268–277, Austin, Texas. Association for Computational Linguistics.
  * Garmash and Monz (2016) Ekaterina Garmash and Christof Monz. 2016.  Ensemble learning for multi-source neural machine translation.  In _Proceedings of COLING 2016, the 26th International Conference on Computational Linguistics: Technical Papers_ , pages 1409–1418, Osaka, Japan. The COLING 2016 Organizing Committee.
  * Gu et al. (2017) Jiatao Gu, Graham Neubig, Kyunghyun Cho, and Victor OK Li. 2017.  Learning to translate in real-time with neural machine translation.  In _Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics (Volume 1, Long Papers)_ , pages 1053–1062.
  * Gu et al. (2019) Jiatao Gu, Yong Wang, Kyunghyun Cho, and Victor O.K. Li. 2019.  Improved zero-shot neural machine translation via ignoring spurious correlations.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 1258–1268, Florence, Italy. Association for Computational Linguistics.
  * Imankulova et al. (2020) Aizhan Imankulova, Masahiro Kaneko, Tosho Hirasawa, and Mamoru Komachi. 2020.  Towards multimodal simultaneous neural machine translation.  In _WMT_ , pages 594–603. Association for Computational Linguistics.
  * Johnson et al. (2017) Melvin Johnson, Mike Schuster, Quoc V. Le, Maxim Krikun, Yonghui Wu, Zhifeng Chen, Nikhil Thorat, Fernanda Viégas, Martin Wattenberg, Greg Corrado, Macduff Hughes, and Jeffrey Dean. 2017.  Google’s multilingual neural machine translation system: Enabling zero-shot translation.  _Transactions of the Association for Computational Linguistics_ , 5:339–351.
  * Ma et al. (2019) Mingbo Ma, Liang Huang, Hao Xiong, Renjie Zheng, Kaibo Liu, Baigong Zheng, Chuanqiang Zhang, Zhongjun He, Hairong Liu, Xing Li, Hua Wu, and Haifeng Wang. 2019.  STACL: Simultaneous translation with implicit anticipation and controllable latency using prefix-to-prefix framework.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 3025–3036.
  * Nishimura et al. (2018) Yuta Nishimura, Katsuhito Sudoh, Graham Neubig, and Satoshi Nakamura. 2018.  Multi-source neural machine translation with missing data.  In _Proceedings of the 2nd Workshop on Neural Machine Translation and Generation_ , pages 92–99. Association for Computational Linguistics.
  * Papineni et al. (2002) Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. 2002.  BLEU: a method for automatic evaluation of machine translation.  In _Proceedings of the 40th Annual Meeting on Association for Computational Linguistics_ , pages 311–318.
  * Sennrich et al. (2016) Rico Sennrich, Barry Haddow, and Alexandra Birch. 2016.  Improving neural machine translation models with monolingual data.  In _Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 86–96, Berlin, Germany. Association for Computational Linguistics.
  * Utiyama and Isahara (2007) Masao Utiyama and Hitoshi Isahara. 2007.  A comparison of pivot methods for phrase-based statistical machine translation.  In _Human Language Technologies 2007: The Conference of the North American Chapter of the Association for Computational Linguistics; Proceedings of the Main Conference_ , pages 484–491, Rochester, New York. Association for Computational Linguistics.
  * Wu and Wang (2007) Hua Wu and Haifeng Wang. 2007.  Pivot language approach for phrase-based statistical machine translation.  _Machine Translation_ , 21(3):165–181.
  * Zhang et al. (2020) Biao Zhang, Philip Williams, Ivan Titov, and Rico Sennrich. 2020.  Improving massively multilingual neural machine translation and zero-shot translation.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 1628–1639, Online. Association for Computational Linguistics.
  * Zheng et al. (2019a) Baigong Zheng, Renjie Zheng, Mingbo Ma, and Liang Huang. 2019a.  Simpler and faster learning of adaptive policies for simultaneous translation.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 1349–1354.
  * Zheng et al. (2019b) Baigong Zheng, Renjie Zheng, Mingbo Ma, and Liang Huang. 2019b.  Simultaneous translation with flexible policy via restricted imitation learning.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 5816–5822.
  * Zheng et al. (2019c) Renjie Zheng, Mingbo Ma, Baigong Zheng, and Liang Huang. 2019c.  Speculative beam search for simultaneous translation.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 1395–1402.
  * Zheng et al. (2020) Renjie Zheng, Mingbo Ma, Baigong Zheng, Kaibo Liu, and Liang Huang. 2020.  Opportunistic decoding with timely correction for simultaneous translation.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 437–442.
  * Zoph and Knight (2016) Barret Zoph and Kevin Knight. 2016.  Multi-source neural translation.  In _Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 30–34, San Diego, California. Association for Computational Linguistics.