#  Topological phase transitions generated by order from quantum disorder

Fadi Sun1 and Jinwu Ye1,2 1 The School of Science, Great Bay University,
Dongguan, Guangdong, 523000, China
2 Department of Physics and Astronomy, Mississippi State University, MS 39762,
USA

###### Abstract

The order from quantum disorder (OFQD) phenomenon was first discovered in
quantum spin systems in geometric frustrated lattice. Similar phenomenon was
also discovered in interacting bosonic systems or quantum spin systems with
spin-orbit coupling in a bipartite lattice. Here we show that the OFQD also
leads to a topological phase transition. We demonstrate this new connection in
the experimentally realized weakly interacting Quantum Anomalous Hall system
of spinor bosons in an optical lattice. There are two classes of topological
phenomena: the first class is a perturbative one smoothly connected to the
non-interacting limit. The second one is a non-perturbative one which has no
analog in the non-interacting limit. Their experimental detections are also
discussed.

1\. Introduction. — Searching for new topological phases and topological phase
transitions in various materials or artificial systems are fascinating
frontiers in modern condensed matter physics kane ; zhang . The quantum
anomalous Hall (QAH) is the simplest topological phase with non-vanishing
Chern number QAHthe . The non-interacting fermionic QAH has been
experimentally realized in Cr doped Bi(Sb)2Te3 thin films QAHthe ; QAHexp . On
the other hand, the weakly interacting bosonic analogy of QAH model has also
been successfully realized via spinor bosons 87Rb 2dsocbec . It is crucial to
study the topological properties of bosonic QAH model, and find possible deep
connections between the fermionic QAH model and the bosonic QAH model.

It is well-known that the fermionic QAH model has two bands carrying opposite
Chern numbers. In the non-interacting limit, when $|h/t|<4$, the corresponding
Chern number of the lower and upper band are $C_{1}=+\mathrm{sgn}(h)$ and
$C_{2}=-\mathrm{sgn}(h)$, respectively; when $|h/t|>4$, the corresponding
Chern number of the lower and upper bands is $C_{1}=C_{2}=0$. However, the
topological properties of weakly interacting bosonic quantum anomalous Hall
model maybe much involved. Due to its bosonic nature, an interaction must be
considered at the very beginning. Unlike the fermionic QAH model, the weakly
interacting bosonic QAH model is always in some spin-orbital superfluid phases
with the spontaneous U(1) symmetry breaking. To the quadratic level, there are
always Bogliubov quasi-particle excitations above these exotic superfluid
phases. If neglecting the cubic or quartic interactions between them, which is
justified near any quantum phase transitions (QPT), one may ask the question
on what are the topology of these Bogliubov quasi-particle bands and possible
TPTs. In this Letter, we map out the Chern number of the bosonic Bogliubov
band at the quadratic level.

In Ref.NOFQD , we work out quantum phases and quantum phase transitions in
NOFQD . It is the order from quantum disorder (OFQD) phenomena which leads to
the quantum ground states at $h=0$. It is the nearly order from quantum
disorder (NOFQD) phenomena which leads to the quantum phase transition at
$h=h_{c}$. In this work, we focus on the topological aspects of the same
system. So the results to be achieved are complementary to those achieved in
NOFQD . We find that there are two kinds of topological phases and TPTs. The
first can be considered as the remnants from the non-interacting fermion
limit, so it reduce to this limit smoothly as the interaction gets very small.
It can also be called perturbative regime. The second has no analog or
counterpart in the non-interacting fermion limit, so it is a completely new
feature due to the interaction. We will show that it is completely induced by
the non-perturbative OFQD phenomenon at $h=0$ discovered in NOFQD . It may
also be called perturbative regime.

In addition to the OFQD at $h=0$ and the QPT at $h=h_{c}$ induced by the NOFQD
discovered in NOFQD , we find two critical fields $h_{2}>h_{1}>h_{c}>0$. At
the upper critical field $h=h_{2}$, there is a conic band touching at the
$R=(\pi,\pi)$ point of the Brillouin zone (BZ), so the lower band $C_{1}$ and
upper band $C_{2}=-C_{1}$ Chern number changes from $(C_{1},C_{2})=(+1,-1)$
below $h_{2}$ to $(C_{1},C_{2})=(0,0)$ above $h_{2}$; At the lower critical
field $h_{1}<h_{2}$, there is a conic band touching at the $X=(\pi,0)$ and
$Y=(0,\pi)$ point of the BZ, so $(C_{1},C_{2})=(-1,+1)$ changes below $h_{1}$
to $(C_{1},C_{2})=(+1,-1)$ above $h_{1}$. Both $h_{1}$ and $h_{2}$ only
depends on $t$ and $U$, but independent of the SOC $t_{s}$, and approaches to
the corresponding non-interacting value $h_{20}=4t$ and $h_{10}=0$
respectively. We conclude the topology in the regime $|h|>|h_{1}|$ belongs to
the first perturbative class.

We show that $h=h_{1}>h_{c}$ which is due to the NOFQD NOFQD . When $h<h_{c}$,
the ground state becomes a XY-CAFM SF phase which breaks $[C_{4}\times
C_{4}]_{D}\to 1$ and leads to 4 bosonic Bogliubov bands in the reduced BZ. We
find there is always a gap between the second and the third band and calculate
the combined Chern numbers of the two lower bands and that of the two upper
bands $(C_{1+2},C_{3+4})=(-1,+1)$. So there is no change on the topological
band structure across the QPT at $h=h_{c}$. However, the change comes from
$h=0$ where the OFQD leads to two Dirac points at in-commensurate momenta.
Their positions not only depend on $t,U$, but also the SOC $t_{s}$ and related
by the remaining two Mirror symmetries. As the three parameters change, the
two Dirac points approach to each other, then collide at a degenerate momentum
$\Lambda=(\pi/2,\pi/2)$, then bounce off along the normal direction. Any
$h\neq 0$ opens the gap on the two Dirac points, so there is a TPT from $h<0$
with the combined Chern number $(C_{1+2},C_{3+4})=(1,-1)$ to $h>0$ with
$(C_{1+2},C_{3+4})=(-1,1)$, which is completely induced by the OFQD at $h=0$.
We construct effective action to study the TPT and find it always contains an
exotic Doppler shift term. The topology in the regime $|h|<|h_{1}|$ belongs to
the second non-perturbative class. We also critically comment on the common
conceptual mistakes made in the previous theoretical or experimental
literatures to attempt to evaluate edge modes within the bulk gaps associated
with the bosonic bulk Chern numbers. We also elucidate the physical meanings
of the combined Chern numbers. Finally, we discuss the experimental detection
of these topological phenomena, especially the ones induced by the OFQD near
$h=0$.

2\. The Hamiltonian, quantum phases and quantum phase transitions (QPT). — The
recently experimentally realized two-component spinor Bose-Hubbard Hamiltonian
with a spin-orbit coupling 2dsocbec  can be written as

$\displaystyle\mathcal{H}=-\\!\sum_{i}[a_{i}^{\dagger}(t\sigma_{z}\\!-\\!it_{\text{s}}\sigma_{x})a_{i+x}\\!+\\!a_{i}^{\dagger}(t\sigma_{z}\\!-\\!it_{\text{s}}\sigma_{y})a_{i+y}\\!+\\!h.c.]$
$\displaystyle\qquad-h\sum_{i}(n_{i\uparrow}-n_{i\downarrow})+U\sum_{i}n^{2}_{i}-\mu\sum_{i}n_{i}\>.$
(1)

where $\sigma=\uparrow$ or $\downarrow$ denotes the 87Rb atoms in the state
$|1,m_{F}=0\rangle$ or $|1,m_{F}=-1\rangle$, respectively. Since the
experiment can only achieve relatively weak spin-orbit coupling, our
discussion will focus on regime $|t_{s}|<2|t|$.

The quantum phase diagram of Eq.(1) was studied in Ref.NOFQD . Especially, we
found a new phenomenon we named Nearly order from quantum disorder (NOFQD)
which captures the delicate competition between the effective potential
generated by the order from quantum disorder (OFQD) and the Zeeman field. It
is this competition which splits a putative first order quantum phase
transition (QPT) at $h=0$ into two second order QPTs at $h=\pm h_{c}$. Here we
briefly summarize the main results: when $h>h_{c}$, it is in the Z-FM
superfluid with the condensate wavefunction $\Psi\propto\chi_{\uparrow}$; when
$h<h_{c}$, it is the XY-CAFM superfluid with the condensate wavefunction
$\Psi\propto\cos(\theta/2)\chi_{\uparrow}+e^{i(\mathbf{Q}\cdot\mathbf{r}+\phi)}\sin(\theta/2)\chi_{\downarrow}$,
where $\mathbf{Q}=(\pi,\pi)$ is the orbital ordering wave-vector,
$\theta=\arccos(h/h_{c})$, and $\phi=\pm\pi/4,\pm 3\pi/4$. The $h_{c}$ was
estimated as $h_{c}\sim n_{0}U^{2}t_{s}^{2}/t^{3}$.

However, Ref.NOFQD  fucus only on the quantum phases and QPTs, ignored the
topological features of the Bogoliubov bands, also did not touch what are the
connections of these quantum phases and QPTs to the topological nature of the
QAH Hamiltonian. This work will address these outstanding open problems.
Without loss of generality, we can set $h,t,t_{\text{s}}>0$ and then discuss
the topology of the Bogoliubov excitation bands at $h>h_{c}$ and $h<h_{c}$
separately.

3\. Evaluating the Chern number of the bosonic quasi-particle band from the
lattice theory. — To exam the topology of the Bogoliubov excitation bands, we
need not only the eigenvalues but also the eigenvectors, thus one need also to
pay attention to the Bogoliubov transformations. After replacing the bosonic
operator by its average plus a quantum fluctuation, we obtain a quadratic
Bogoliubov Hamiltonian via expanding the Hamiltonian to the second order in
the quantum fluctuations. Due to the spontaneous U(1) symmetry breaking, the
quadratic Hamiltonian in $\mathbf{k}$-space, $H_{\mathbf{k}}$, is a $2N\times
2N$ matrix,

$\displaystyle\mathcal{H}=E_{0}+\frac{1}{2}\sum_{\mathbf{k}}(\psi_{\mathbf{k}}^{\dagger},\psi_{-\mathbf{k}})H_{\mathbf{k}}\begin{pmatrix}\psi_{\mathbf{k}}\\\
\psi_{-\mathbf{k}}^{\dagger}\\\ \end{pmatrix}\>,$ (2)

where
$\psi_{\mathbf{k}}^{\dagger}\\!=\\!(\psi_{1,\mathbf{k}}^{\dagger},\cdots,\psi_{N,\mathbf{k}}^{\dagger})$.
The $1,\cdots,N$ indices count the spin degrees of freedom and also the
momentum resulting from the spontaneous translational symmetry breaking i9n
the ground state. As shown in NOFQD , when $h>h_{c}$, the ground state is the
Z-FM phase, we have $N=2$ and
$\psi_{\mathbf{k}}^{\dagger}=(\psi_{\mathbf{k}\uparrow}^{\dagger},\psi_{\mathbf{k}\downarrow}^{\dagger})$;
when $h<h_{c}$, the ground state is the XY-AFM phase, we have $N=4$ and
$\psi_{\mathbf{k}}^{\dagger}=(\psi_{\mathbf{k}\uparrow}^{\dagger},\psi_{\mathbf{k}\downarrow}^{\dagger},\psi_{\mathbf{k}+\mathbf{Q}\uparrow}^{\dagger},\psi_{\mathbf{k}+\mathbf{Q}\downarrow}^{\dagger})$.
where $\vec{Q}=(\pi,\pi)$.

Diagonalizing Eq.(2) by a $2N\times 2N$ Bogoliubov transformation matrix
$T_{\mathbf{k}}$, so that
$T_{\mathbf{k}}^{\dagger}H_{\mathbf{k}}T_{\mathbf{k}}=\mathrm{diag}(\omega_{1,\mathbf{k}},\cdots,\omega_{N,\mathbf{k}},\omega_{1,-\mathbf{k}},\cdots,\omega_{N,-\mathbf{k}})$
where $\mathrm{diag}(\cdots)$ means a diagonal matrix with diagonal elements
$\cdots$, and $(\begin{smallmatrix}\psi_{\mathbf{k}}\\\
\psi_{-\mathbf{k}}^{\dagger}\\\
\end{smallmatrix})=T_{\mathbf{k}}(\begin{smallmatrix}\alpha_{\mathbf{k}}\\\
\alpha_{-\mathbf{k}}^{\dagger}\\\ \end{smallmatrix})$, we obtain

$\displaystyle\mathcal{H}=E_{0}+\sum_{n,\mathbf{k}}\omega_{n,\mathbf{k}}(\alpha_{n,\mathbf{k}}^{\dagger}\alpha_{n,\mathbf{k}}+1/2),$
(3)

where $\mathbf{k}$ sums over the corresponding Brillouin Zone (BZ),
$\omega_{n,\pm\mathbf{k}}>0$, $n=1,\cdots,N$ are the $N$ Bogoliubov excitation
bands.

In contrast to the fermionic cases, to keep the bosonic commutation relations,
the $T_{\mathbf{k}}$ is required to be a para-unitary matrix instead of a
unitary one, which means
$T_{\mathbf{k}}^{\dagger}\tau_{3}T_{\mathbf{k}}=T_{\mathbf{k}}\tau_{3}T_{\mathbf{k}}^{\dagger}=\tau_{3}$
and $\tau_{3}=\sigma_{z}\otimes I_{N\times N}$. The Berry curvature of the
$n$-th Bogoliubov excitation band can be calculated via
$F_{n}(\mathbf{k})=i[\epsilon_{\mu\nu}\tau_{3}\partial_{\mu}T_{\mathbf{k}}^{\dagger}\tau_{3}\partial_{\nu}T_{\mathbf{k}}]_{nn}$,
where $\mu,\nu=k_{x},k_{y}$. The Chern number of the $n$-th Bogoliubov
excitation band can be evaluated via a integral

$\displaystyle
C_{n}=\frac{1}{2\pi}\int_{\text{BZ}}d^{2}\mathbf{k}\>F_{n}(\mathbf{k})\>,$ (4)

where the integral is over the corresponding BZ.

4\. Evaluating the Chern number of the bosonic quasi-particle band from the
continuum theory. — Since the band closing is the signature of a topological
phase transition (TPT), the change of band Chern number may also be computed
in terms of a continuum theory.

Near the band touching point, the typical effective Hamiltonian can be
expressed in terms of the polar coordinate $(k,\xi)$ with $k_{x}=k\cos\xi$ and
$k_{y}=k\sin\xi$,

$\displaystyle
H_{\mathbf{k}}=v_{x}k^{n}\cos(n\xi)\sigma_{x}+v_{y}k^{n}\sin(n\xi)\sigma_{y}-\Delta\sigma_{z}$
(5)

where $|v_{x}|=|v_{y}|=v$ and $n=1,2,3,\cdots$ stands for the order ( or
charge ) of the touching point.

The corresponding dispersion relation is
$\epsilon(\mathbf{k})=\pm\sqrt{\Delta^{2}+v^{2}k^{2n}}$, and
$\epsilon(\mathbf{k})\sim k^{n}$ as $\Delta\to 0$ approaching the TPT. The
Berry curvature of the lower branch is $F_{-}(\mathbf{k})=-v_{x}v_{y}\Delta
k^{-2+2n}n^{2}/|\epsilon(k)|^{3}$, thus its Chern number is

$\displaystyle C_{-}=\frac{1}{2\pi}\int
d^{2}\mathbf{k}F_{-}(\mathbf{k})=-\frac{n}{2}\mathrm{sgn}(v_{x}v_{y}\Delta)\>.$
(6)

where the $sgn$ indicates the chirality of the Dirac boson. Therefore the band
touching through the changing sign of $\Delta$ from positive to negative
across the TPT leads to a change of Chern number $\Delta
C_{=}-n\,\mathrm{sgn}(v_{x}v_{y})$. When there are multiple band touching
points, the total change of Chern number needs to sum up all the contributions
from all the band touching points.

In the superfluid phases, due to the BEC at $k=0$, the Bogoliubov
transformation matrix $T_{k}$ diverges at $k=0$, the Chern number of the
lowest band $C_{1}$ may not be well-defined, but those of the higher bands
$C_{n>1}$ are usually well-defined, and quantized to be integers. Ueda2015 .
In this Letter, we are still able to calculate $C_{1}$ by excluding the
momentum $k$ where the BEC resides. The results are summarized in Fig. 1. When
comparing the non-interacting fermionic QAH model with the interacting bosonic
QAH model, we find that the bosonic interaction leads to new and additional
patterns of topological bands. In the following sections, we present the
details of calculations leading to Fig.1.

Figure 1:  The band Chern numbers of (a) The non-interacting fermionic QAH,
and (b) The weak-interacting bosonic QAH. The topological non-trivial regions
($C_{i}\neq 0$) in (b) is slightly larger than (a). The regime $|h|>h_{1}$
region is smoothly connected to the non-interacting limit in (a), so can be
called perturbative region. While the entire $|h|<h_{1}$ region is induced by
the non-perturabative OFQD phenomenon at $h=0$. It has no non-interacting
analog, so may be called the non-perturabative region which shrinks to zero as
$U$ goes $0$.

5\. The topology of the Bogoliubov band when $h>h_{c}$.— In the $h>h_{c}$
case, the ground state is the Z-FM phase, there are $N=2$ energy bands, the
Bogoliubov transformation matrix $T_{k}$ is a $4\times 4$ matrix. The
corresponding band structure is plotted in Fig.2 where

$\displaystyle h_{2}$ $\displaystyle=\sqrt{4t(4t+n_{0}U)}$ $\displaystyle
h_{1}$ $\displaystyle=\sqrt{2t(2t+n_{0}U)}-2t$ (7)

When $h=h_{2}$, the two bands conically touch at $k_{R}=(\pi,\pi)$; when
$h=h_{1}$, they conically touch at $k_{X}=(\pi,0)$ and $k_{Y}=(0,\pi)$. A
direct evaluation of Eq.(4) on the lattice shows that: when $h>h_{2}$, the
lower band and the upper band Chern number are $C_{1}=C_{2}=0$ respectively;
when $h_{1}<h<h_{2}$, $(C_{1},C_{2})=(1,-1)$; when $h<h_{1}$,
$(C_{1},C_{2})=(-1,1)$. Below, we construct continuum theories to confirm the
change of the band Chern number.

At $h=h_{2}$, one needs to expand the Hamiltonian around $k_{R}=(\pi,\pi)$,
the $T_{k}$ at $k_{R}$ tells the eigenmodes are
$\alpha_{1,R}=(1/\sqrt{8})[2(4+n_{0}U/t)^{-1/4}+(4+n_{0}U/t)^{1/4}]\psi_{R\uparrow}-(1/\sqrt{8})[2(4+n_{0}U/t)^{-1/4}-(4+n_{0}U/t)^{1/4}]\psi_{R\uparrow}^{\dagger}$
and $\alpha_{2,R}=\psi_{R\downarrow}$, with the eigen-energy
$\omega_{h2}\\!=\\!2\sqrt{\smash[b]{4t(4t\\!+\\!n_{0}U)}}=2h_{2}$ When $h$
deviates slightly from $h_{2}$, defining $k=k_{R}+q$ and projecting the
original Hamiltonian onto these eigenmodes lead to the effective Hamiltonian:

$\displaystyle
H_{R}=\omega_{h2}-vq_{x}\sigma_{x}-vq_{y}\sigma_{y}-\Delta\sigma_{z}$ (8)

where $omega_{h2}\\!=2h_{2},\Delta=2(h-h_{2})$,
$v\\!=\\!t_{\text{s}}\sqrt{\smash[b]{2[1\\!+\\!(8t\\!+\\!n_{0}U)/\omega_{h2}\\!]}}$.
Note that the constant term $omega_{h2}$ shows it is an excited energy which
has a direct experimental consequence CIT .

Then the dispersion takes the form

$\displaystyle\omega_{1,2}(q)=\omega_{h2}\mp\sqrt{\Delta^{2}+v^{2}(q_{x}^{2}+q_{y}^{2})}\>.$
(9)

Thus the change of band Chern number is $\Delta C_{1}=-{\rm sgn}(v^{2})=-1$.
This analysis is consistent with the numerical result $C_{1}=0$ at $h>h_{2}$
and $C_{1}=+1$ at $h<h_{2}$, therefore $0-(+1)=-1$.

At $h=h_{1}$, one needs to expand the Hamiltonian around $k_{X}=(\pi,0)$ and
$k_{Y}=(0,\pi)$, the $T_{k}$ at $k_{X}$ and $k_{Y}$ tells the eigenmodes are
$\alpha_{1,X(Y)}=(1/\sqrt{8})[2(2+n_{0}U/t)^{-1/4}+(2+n_{0}U/t)^{1/4}]\psi_{X(Y)\uparrow}-(1/\sqrt{8})[2(2+n_{0}U/t)^{-1/4}-(2+n_{0}U/t)^{1/4}]\psi_{X(Y)\uparrow}^{\dagger}$
and $\alpha_{2,X(Y)}=\psi_{X(Y)\downarrow}$, with the eigen-energy
$\omega_{h1}\\!=\\!2\sqrt{\smash[b]{2t(2t\\!+\\!n_{0}U)}}=2h_{1}+4t$. When $h$
deviates slightly from $h_{1}$, defining $k=k_{X(Y)}+q$ and projecting the
original Hamiltonian onto these eigenmodes lead to the effective Hamiltonian

$\displaystyle H_{X}$
$\displaystyle=\omega_{h1}-vq_{x}\sigma_{x}+vq_{y}\sigma_{y}-\Delta\sigma_{z}$
$\displaystyle H_{Y}$
$\displaystyle=\omega_{h1}+vq_{x}\sigma_{x}-vq_{y}\sigma_{y}-\Delta\sigma_{z}$
(10)

where $\omega_{h1}\\!=2h_{1}+4t,\Delta=2(h-h_{1})$,
$v\\!=\\!t_{\text{s}}\sqrt{\smash[b]{2[1\\!+\\!(4t\\!+\\!n_{0}U)/\omega_{h1}\\!]}}$.
Again, the constant term $omega_{h2}$ shows it is an excited energy which has
a direct experimental consequence CIT .

Then the dispersion of $H_{X}$ or $H_{Y}$ takes the same form

$\displaystyle\omega_{1,2}(q)=\omega_{h1}\mp\sqrt{\Delta^{2}+v^{2}(q_{x}^{2}+q_{y}^{2})}\>.$
(11)

Thus the change of band Chern number is $\Delta C_{1}=2{\rm sgn}(v^{2})=2$.
This analysis is consistent with the numerical result $C_{1}=+1$ at $h>h_{1}$
and $C_{1}=-1$ at $h<h_{1}$, therefore $+1-(-1)=2$.

Since $\lim_{U\to 0}h_{2}=4t$ and $\lim_{U\to 0}h_{1}=0$, this result suggests
that the topology of the Bogoliubov excitation bands at $h>h_{c}$ is smoothly
connecting to that of the non-interacting limit. Besides, the fact
$h_{2}=\sqrt{4t(4t+n_{0}U)}>4t$ at any $U>0$ suggests that the region
displaying a non-trivial topology ( with a non-zero Chern number) is enlarged
with an increasing $U>0$.

One can also determine the relation between $h_{1}$ and $h_{c}$ when $U/t$ is
small. When $U/t$ is small, $h_{1}=\sqrt{2t(2t+n_{0}U)}-2t\sim n_{0}U/2$,
while $h_{c}\sim n_{0}U^{2}t_{s}^{2}/t^{3}\ll h_{1}$. Thus in the weak
coupling limit $U/t\ll 1$, there is always a window for $h_{c}<h<h_{1}$ as
shown in Fig.1b.

Figure 2: The perturbative topological band structure of Bogoliubov excitation
bands at $h>h_{c}$ and near (a) $h_{2}$ and (b) $h_{1}$. It is smoothly
connected to the non-interacting limit. The parameters are $t=1$, $t_{s}=1/2$,
$n_{0}U=1$, thus $h_{2}\approx 4.47$ and $h_{1}\approx 0.48$. When $h=h_{2}$,
there is a Dirac conical band touch at the R point $k_{R}=(\pi,\pi)$ of the
Brillouin zone (BZ). When $h=h_{1}$, there is a Dirac conical band touch at
the X point $k_{X}=(\pi,0)$ of the BZ. The $[C_{4}\times C_{4}]_{\text{diag}}$
symmetry tells there is also a Dirac conical band touch at the Y point
$k_{Y}=(0,\pi)$ of the BZ.

6\. The topology of the Bogoliubov band when $0<h<h_{c}$ and the absence of
the TPT across the QPT at $h=h_{c}$.— In the $h<h_{c}$ case, this corresponds
to Eq.(9) and Eq.(10) in Ref.NOFQD . Because of spontaneous translational
symmetry breaking, we have $N=4$ energy bands and $T_{k}$ is an $8\times 8$
matrix. From the NOFQD analysis in Ref.NOFQD , we know the $h=0$ ground-state
requires $\theta=\pi/2$, and $0<h<h_{c}$ ground-state requires
$\theta_{h}=\arccos(h/h_{c})$. We substitute $\theta=\theta_{h}$ back into the
Eq.(2), then calculate the band Chern numbers via Eq.(4). In principle, there
should be an additional term due to nearly order-from-quantum disorder (NOFQD)
correction NOFQD , but we expect this correction does not change the topology
of the Bogoliubov bands. From the band structure shown in Fig.3, we find that
the two lower bands touch, the two upper bands also touch, and there is always
a band gap between the two groups when decreasing $\theta_{h}$ from $\pi/2$ to
$0$ (or equivalently, increasing $h$ from $0$ for $h_{c}$). Note that two
lower/upper bands touch at the X point $k_{X}=(\pi,0)$, which is a momentum
far away from $k=0$, thus the NOFQD effects ignored so far should not change
the band touching behaviors. Due to the band touching, we need to consider the
combination of the Chern numbers $C_{1+2}$ and $C_{3+4}$ when $h<h_{c}$. Our
numerical evaluation of the integral Eq.(4) gives $(C_{1+2},C_{3+4})=(-1,+1)$.
Recall the result of $h$ just above $h_{c}$ achieved in the last subsection
also gives $(C_{1},C_{2})=(-1,+1)$, which is the same pattern as
$(C_{1+2},C_{3+4})=(-1,+1)$. So we conclude that there is not any change in
the topology across the quantum phase transition at $h=h_{c}$.

Figure 3: The Non-perturbative band structure of Bogoliubov excitation bands
at (a) $h>h_{c}$, (b) $h=h_{c}$, (c) $h=h_{c}/\sqrt{2}$, (d) $h=0$. The
parameters are $t=1$, $t_{s}=1/2$, $n_{0}U=1$. It has no analog in the non-
interacting limit. When $h>0$, there is always a band gap between two upper
bands and two lower bands. (a) is just a re-plot of the bottom subfigure of
Fig.2(b) in the reduced Brillouin zone (BZ), so the 2 bands in the BZ changes
to 4 bands in the reduced BZ. As shown in Sec.C in the SM, two lower (upper)
bands also touch along the whole line from $(\pi,0)$ to $(0,-\pi)$ As shown in
(a) and (b), when $h\geq h_{c}$, they also touch along the whole line from
$\Lambda$-point to the X-point along the boundary of the RBZ. $\omega_{1}$ is
the linear gapless Goldstone mode due to the $U(1)$ symmetry breaking. Note
that due to the dropping of the NOFQD effects NOFQD , the roton mode
$\omega_{1}$ is still a spurious quadratic gapless mode at $k=0$.
Incorporating them back will open a small roton gap at $k=0$, but will not
change the Chern numbers of topology of the band. When $h=0$ in (d), there is
a Dirac conical band touch at $K_{1}=(\pi-k_{0},k_{0})$ of Brillouin zone; and
the mirror symmetry $M_{1}$ with respect to $k_{x}=k_{y}$ tells there is also
a Dirac conical band touch at $K_{2}=(k_{0},\pi-k_{0})$.

7\. The TPT at $h=0$ and the absence of any QPT at $h=0$. — As shown in Fig.2,
the topology changes at $h=h_{1}$ with the conic band crossings at
$k_{X}=(\pi,0)$ and $k_{Y}=(0,\pi)$, also at $h=h_{2}$ with the conic band
crossing at $k_{R}=(\pi,\pi)$. The origin of these TPTs can be traced back and
mapped to the corresponding non-interacting limit of fermions.

In this section, we show that there is also a conic band crossing at $h=0$ in
Fig.3(d) at some in-commensurate momentum $(k_{0x},k_{0y})$ between the
$\Lambda$-point and X-point, which is not any high symmetry points. As shown
in Fig.S2, there are two such Dirac points in the RBZ. The positions of the
two Dirac points are in-commensurate and also depend on $t,t_{s}$ and $U$, so
it comes from the interaction induced OFQD NOFQD . In contrast to the TPTs at
$h=h_{1}$ and $h=h_{2}$, the TPT at $h=0$ does not have any non-interacting
analog. sl This is also the cental result achieved in this manuscript .

At $h=0$, the ground state wavefunction is the XY-AFM with $\theta=0$ and
$\phi=-\pi/4$. Solving the band-touching condition
$\omega_{2}(k)=\omega_{3}(k)$ leads to the following three cases: (I) when
$n_{0}U/t<8t_{s}/(\sqrt{2}t-t_{s})$, two solutions are $K_{1}=(\pi-
k_{0},k_{0})$ and $K_{2}=(k_{0},\pi-k_{0})$, with
$k_{0}=\arcsin[\sqrt{2}tn_{0}U/(t_{s}(8t+n_{0}U))]<\pi/2$; (II) when
$n_{0}U/t>8t_{s}/(\sqrt{2}t-t_{s})$, two solutions are
$P_{1,2}=\pm(p_{0},p_{0})$, with
$p_{0}=\arcsin[(4t/(8t+n_{0}U))\sqrt{4t(4t+n_{0}U)/(4t^{2}-2t_{s}^{2})}]<\pi/2$;
(III) when $n_{0}U/t=8t_{s}/(\sqrt{2}t-t_{s})$, the two solutions merge into
just one at $\Lambda=(\pi/2,\pi/2)$. Intuitively, as shown in Fig.S2, when
keeping $h=0$, increasing $n_{0}U/t$ from $0^{+}$ to
$8t_{s}/(\sqrt{2}t-t_{s})$, the $\omega_{2,3}$ conically touch at $K_{1,2}$;
then at $n_{0}U/t=8t_{s}/(\sqrt{2}t-t_{s})$, $K_{1}$ and $K_{2}$ collide at
$\Lambda$-point; further increasing $n_{0}U/t$, they bounce off along the two
opposite directions along the perpendicular direction footnote , so
$\omega_{2,3}$ conic touches at $P_{1,2}$.

Around $h=0$, defining $k=K_{1,2}+q$ or $k=P_{1,2}+q$, we can similarly expand
the Hamiltonian around the two minima of $\omega_{2,3}$ to find the effective
Hamiltonian:

$\displaystyle
H_{1}=\omega_{0}+c_{\parallel}q_{\parallel}+v_{\parallel}q_{\parallel}\sigma_{x}+v_{\perp}q_{\perp}\sigma_{y}\\!-\Delta\sigma_{z}\>,$
$\displaystyle
H_{2}=\omega_{0}-c_{\parallel}q_{\parallel}-v_{\parallel}q_{\parallel}\sigma_{x}-v_{\perp}q_{\perp}\sigma_{y}\\!-\Delta\sigma_{z}\>,$
(12)

where $\omega_{0}=8t(4t+n_{0}U)/(8t+n_{0}U)$,
$q_{\parallel}=(q_{x}-q_{y})/\sqrt{2},q_{\perp}=(q_{x}+q_{y})/\sqrt{2}$, and
$\Delta=hn_{0}U\sqrt{t/(4t+n_{0}U)}$ for regime (I),
$q_{\parallel}=(q_{x}+q_{y})/\sqrt{2},q_{\perp}=(q_{x}-q_{y})/\sqrt{2}$, and
$\Delta=4htt_{s}/\sqrt{2t^{2}-t_{s}^{2}}$ for regime (II). The effective
velocities $v_{\parallel},v_{\perp},c_{\parallel}>0$ are proportional to
$\cos(k_{0})$ or $\cos(p_{0})$ for regime (I) and regime (II) respectively and
are listed in the appendix B.

The dispersion of $H_{i},i=1,2$ takes the form

$\displaystyle\omega_{2,3}(q)=\omega_{0}-(-1)^{i}c_{\parallel}q_{\parallel}\mp\sqrt{\Delta^{2}+v_{\parallel}^{2}q_{\parallel}^{2}+v_{\perp}^{2}q_{\perp}^{2}}\>.$
(13)

It is necessary to stress that there exists a Doppler shift term, the $c$
term, in $H_{1,2}$, which is the salient feature unique to the OFQD induced
TPT, not shared by any non-interacting counter-part. Of course, it does not
affect the value of band Chern numbers.

From the effective Hamiltonian Eq.(12), Eq.(6) tells the change of band Chern
number is $\Delta C=-2{\rm sgn}(v_{\parallel}v_{\perp})=-2$ when $\Delta$
changes from positive to negative. This result is consistent with the
numerical evaluation via Eq.(4), which gives $C_{1+2}=-1$ at $h>0$ and
$C_{1+2}=+1$ at $h<0$, therefore $-1-(+1)=-2$.

At $h=0$, in the colliding point $n_{0}U/t=8t_{s}/(\sqrt{2}t-t_{s})$, the
$K_{1}$ and $K_{2}$ merge at $\Lambda=(\pi/2,\pi/2)$-point. Expanding around
the $\Lambda$-point leads to the dispersion:

$\displaystyle\omega_{2,3}(q)=\omega_{0}(q)\mp\sqrt{u_{1}^{2}q^{4}_{\perp}+u_{2}q^{4}_{\parallel}+wq^{2}_{\parallel}q^{2}_{\perp}},$
(14)

where $u_{1}=t^{2}/(\sqrt{2}t_{s})$,
$u_{2}=t^{2}t_{s}/[\sqrt{2}(2t^{2}-t_{s}^{2})]$,
$w=t^{2}(t^{2}-t_{s}^{2})/(2t^{2}-t_{s}^{2})$, and
$\omega_{0}(q)=4t+2\sqrt{2}t_{s}+DS$ where one can see the Doppler shift term
$DS=[t_{s}/\sqrt{2}][-q^{2}_{\parallel}+q^{2}_{\perp}t_{s}^{2}/(2t^{2}-t_{s}^{2})]$
also gets to the quadratic order.

Note that $4u_{1}^{2}u_{2}^{2}>w^{2}$ is ensured by the condition
$0<t_{s}<\sqrt{2}t$, so the square root is always positive define. Since
$w$-term won’t lead to a gap close, one can ignore $w$-term and rescale $q$ to
make $\omega_{2,3}(q)$ isotropic. The $\omega_{2,3}(q)\sim q^{2}$ is
consistent with the fact that $v_{\parallel}$, $v_{\perp}$, $c_{\parallel}$ in
Eq.(12) vanish as $k_{0}$ or $p_{0}$ approaching $\pi/2$. When $h$ deviates
from $0$, the effective Hamiltonian belongs to $n=2$ case of Eq.(5). Thus,
although there is only one band touching point with the monopole charge $n=2$,
the change of Chern number is still $-2$.

In short, regardless of the value of $n_{0}U$, the change of the Chern number
from $h=0^{+}$ to $h=0^{-}$ is always $-2$. So the scattering process in Fig.
S1, is not a TPT.

8\. Dramatic differences between the non-interacting fermionic band structure
and the bosonic Bogoliubov band structure at the quadratic level.— It is
important to stress some crucial differences between the fermionic band
structure and the bosonic Bogliubov band structure at the quadratic level and
also pointed out the common mistakes made in all the previous literatures to
calculate the edge modes associated with the bosonic Bogliubov band structure:

In the fermionic or bosonic QAH problem, the Time reversal symmetry is broken,
the Chern number is not protected by any symmetry, its definition involves no
symmetry requirement. The former is really non-interacting, while the bosonic
Bogoliubov band is non-interacting at the quadratic level only where one
dropped the cubic and quartic interactions and all the higher order
interactions. All these interactions are not important except near the QPT at
$h=h_{c}$. In fact, as shown by the RG analysis, near the QPT, the two quartic
interaction in Eq.M32 are marginally irrelevant. But both are relevant inside
the XY-AFM phase and in fact, leads to the symmetry breaking inside the phase.

For the fermionic problem, there are always edge states associated with the
topological phases. This is justified, because a single fermion operator never
condenses. One tempts to also calculate the edge modes within the band gaps in
Fig.2 and Fig.3 associated with the bosonic Bogoliubov band. Unfortunately,
this kind of calculation has mathematical meanings, but makes no physical
sense: this is because Eq.M3 is based on the BEC at $k=0$ in the Z-FM phase,
Eq.M9 with the parametrization Eq. M11 are based on the BEC at both $k=0$ and
$Q=(\pi,\pi)$ in the XY-AFM phase. Unfortunately, they are ill-defined in a
strip geometry. Ref.Ueda2015  and many other previous works, just copied the
same method from its fermionic counterpart to evaluate edge modes associated
with the bosonic Bogoliubov band: They simply transfer the quadratic
Bogoliubov band in momentum space to real space, then solve the edge modes in
the strip geometry by putting a periodic boundary condition along one
direction, then the open boundary condition along the other. Unfortunately,
this approach is well-planned mathematically, but is not self-consistent
physically, because it ignored the root of BEC which leads to the quadratic
bands in the first place. As stressed above, the BEC root is ill-defined in
the strip geometry. So it makes no physical sense to talk about the edge
modes. The claims made in some experimental works to measure the edge modes
have no physical grounds.

9\. Experimental detections.— So far, the experiment2dsocbec  has detected the
non-interacting fermionic Chern numbers Fig.1a using the highly excited
bosonic spinors, so it just mimic the single particle properties of the
Hamiltonian Eq.1 using a spinor boson with SOC. Its main purpose is to
demonstrate a possible realization of the bosonic Hamiltonian Eq.1 using cold
atoms. The common drawbacks of most cold atom experiments is just to
demonstrate a possibility to simulate a Hamiltonian which has been well
studied in materials and claim the ability to tune the parameters. But in
reality, it is rare to go beyond just a simulation to demonstrate new many-
body phenomenon or topological phenomena due to many-body interactions..

Here we showed that there are two kinds of topological band structures (1) The
one in the regime $|h|>|h_{1}|$ is smoothly connected to the non-interacting
fermionic one. So it can also be called perturbative regime. Even so, the two
critical fields $h_{1,2}$ listed in Eq.7 still depends on the interaction
$n_{0}U$. All the parameters in the effective Hamiltonian Eq.8 and 10 also
depend on the interaction $n_{0}U$. (2) The one in the regime $-h_{1}<h<h_{1}$
has no analog in the non-interacting counter-part, so it is completely due to
the many-body interaction. More specifically, due to the non-perturbative
OFQD. So it can also be called non-perturbative regime. It is also an
experimentally easily accessible regime in the cold atom systems.

It is extremely important to push the already 6 year old experiment2dsocbec
beyond the single-particle picture: to detect this novel purely interaction
induced topological phenomena. In contrast to the detections suggested in
NOFQD  which are all equilibrium properties in the ground states and low
energy excitations, here are the excited states near the high energy
$\omega_{h2},\omega_{h1}$ and $\omega_{0}$ in Eq.8,10 and 12 near the high
momentum $R=(\pi,\pi)$ or $X=(\pi,0),Y=(0,\pi)$ and tunable in-commensurate
momenta $K_{1,2}$ or $P_{1,2}$ respectively CIT . We suggest to also measure
the scattering process of the two bosonic Dirac points at $h=0$ displayed in
Fig.S1 by the Bragg spectroscopy becbragg ; bragg1 ; bragg2  or the momentum
resolved interband transitions mrit  adapted to the excited states.

10\. Conclusions.— For any interacting Hamiltonian with a non-trivial topology
in the non-interacting limit such as the QAH one Eq.1, there are always two
aspects quantum and topology. The former focus on the ground states and
quantum phase transitions which may only depend on the low energy excitations
around the band minima in the weak interaction limit. OFQD is needed to even
determine the ground state. NOFQD is needed to determine the QPT at $h=h_{c}$.
The latter focus on the global structure of the bands, therefore also the
excited states to capture the global topology CIT . However, the Bogliubov
quasi-particle band picture breaks down near the QPT near $h=h_{c}$ where a RG
analysis is needed to capture all the physics well beyond the quadratic band
picture.

However, as classified in NOFQD , there are two kinds of OFQD, the first
response trivially to a deformation, no NOFQD phenomenon emerging from the
OFQD. The second response highly non-trivially to a deformation and lead to
NOFQD phenomenon. It would be interesting to see if there are also two classes
here, the first leads to a TPT, as is the case presented in this work, the
second does not lead to any TPT. The two criterions coincide in the present
case. It is also worthy to see if the two criterions coincide in other cases.

The OFQD and the topological invariants such as Chern number seem are two very
different concepts. The first concept is a completely many-body quantum
phenomenon. While the latter is mainly a non-interacting topological
phenomenon. Here we show that there are deep connections between the two in
the context of the experimentally realized weakly interacting Quantum
Anomalous Hall system of spinor bosons in an optical lattice. We expect this
new effects could also be realized in frustrated quantum spin systems which
leads to topological phase transitions of magnons.

Acknowledgements

J. Ye thanks Prof. Gang Tian for the hospitality during his visit at the
School of Science of Great Bay University.

## References

  * (1) M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  * (2) X. L. Qi and S. C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  * (3) Rui Yu, Wei Zhang, Hai-Jun Zhang, Shou-Cheng Zhang, Xi Dai1, Zhong Fang, Quantized Anomalous Hall Effect in Magnetic Topological Insulators, Science 329, 61 (2010).
  * (4) Cui-Zu Chang, Jinsong Zhang, Xiao Feng, Jie Shen, Zuocheng Zhang, Minghua Guo1, Kang Li, Yunbo Ou, Pang Wei, Li-Li Wang, Zhong-Qing Ji, Yang Feng1, Shuaihua Ji, Xi Chen, Jinfeng Jia1, Xi Dai2, Zhong Fang2, Shou-Cheng Zhang3, Ke He2,?, Yayu Wang1,?, Li Lu2, Xu-Cun Ma2, Qi-Kun Xue1, Experimental Observation of the Quantum Anomalous Hall Effect in a Magnetic Topological Insulator, Science 340, 167 (2013).
  * (5) Zhan Wu, Long Zhang, Wei Sun, Xiao-Tian Xu, Bao-Zong Wang, Si-Cong Ji, Youjin Deng, Shuai Chen, Xiong-Jun Liu, Jian-Wei Pan, Realization of two-Dimensional spin-orbit coupling for Bose-Einstein condensates, Science 354, 83 (2016).
  * (6) Fadi Sun and Jinwu Ye, Nearly order from quantum disorder phenomena, arXiv:1903.11134, substantially revised version.
  * (7) Shunsuke Furukawa1 and Masahito Ueda, Excitation band topology and edge matter waves in Bose-Einstein condensates in optical lattices, New J. Phys. 17 (2015) 115014.
  * (8) Si-Cong Ji, Long Zhang, Xiao-Tian Xu, Zhan Wu, Youjin Deng, Shuai Chen, and Jian-Wei Pan, Softening of Roton and Phonon Modes in Bose-Einstein Condensate with Spin-Orbit Coupling, Phys. Rev. Lett. 114, 105301 (2015).
  * (9) Jinwu Ye, J.M. Zhang, W.M. Liu, K.Y. Zhang, Yan Li, W.P. Zhang, Light scattering detection of various quantum phases of ultracold atoms in optical lattices, Phys. Rev. A 83, 051604 (R) (2011).
  * (10) Jinwu Ye, K.Y. Zhang, Yan Li, Yan Chen and W.P. Zhang, Optical Bragg, atom Bragg and cavity QED detections of quantum phases and excitation spectra of ultracold atoms in bipartite and frustrated optical lattices, Ann. Phys. 328 (2013), 103-138.
  * (11) Tarruell, L., Greif, D.,Uehlinger, T., Jotzu, G.Esslinger, T. Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice. Nature 483, 302 (2012).
  * (12) Our calculations are only valid in the regime $t_{s}/t<\sqrt{2}$ and $n_{0}U/t$ below the SF-MI transition. If we fix $t_{s}$ to be some typical value between 0 and $\sqrt{2}t$, i.e. $t=1$ and $t_{s}=1/2$, then $8t_{s}/(\sqrt{2}t-t_{s})\approx 4.38$, while the typical $(U/t)_{c}$ in the Bose Hubbard model at $n=1$ for the SF-MI is about $16$. Thus both (I) and (II) are reachable in teh SF phase.
  * (13) Fadi Sun, Yu Yi-Xiang, Jinwu Ye and WuMing Liu, A new universal ratio in random matrix theory and Chaotic to integral transition in type-I and type-II hybrid Sachdev-Ye-Kitaev models, Fadi Sun, Yu Yi-Xiang, Jinwu Ye, and W. M. Liu, Phys. Rev. B 104, 235133 (2021). Here, the Chaotic to integral transition (CIT) also happens in the excited bulk states which is not a QPT which involves only ground state and low energy excitations.

Supplemental Materials: Topological phase transitions induced by order from
quantum disorder

In this Supplemental Materials we provide some details of the band structures
which are needed to construct the effective Hamiltonian near the various TPTs.

##  I  A. Symmetry analysis and the scattering process of the two bosonic
Dirac points at $h=0$

The high symmetry points in the 1st Brillouin zone are defined as

$\displaystyle\Gamma=(0,0),~{}\Lambda=(\pi/2,\pi/2),~{}R=(\pi,\pi),~{}X=(\pi,0),~{}Y=(0,\pi).$
(S1)

Two additional special $k$-points are defined as the band minima ( maxima ) of
$\omega_{3}(k)$ ( $\omega_{2}(k)$ ) when $|h|<h_{c}$, they are functions of
spin-orbit coupling $t_{s}$ and the interaction $U$.

At $h=0$, if the interaction strength $U$ is sufficiently small
$n_{0}U/t<8t_{s}/(\sqrt{2}t-t_{s})$, the $\omega_{3}(k)$ has two minima at
$k=K_{1}=(\pi-k_{0},k_{0})$ and $k=K_{2}=(k_{0},\pi-k_{0})$, where

$\displaystyle
k_{0}=\arcsin\Big{[}\frac{\sqrt{2}tn_{0}U}{t_{s}(8t+n_{0}U)}\Big{]}\>.$ (S2)

When $n_{0}U$ goes to $0$, the $k_{0}$ will become $0$, thus $K_{1}$ becomes
$X$-point and $K_{2}$ becomes $Y$-point.

At $h=0$, if the interaction strength $U$ is relatively large
$n_{0}U/t>8t_{s}/(\sqrt{2}t-t_{s})$, the $\omega_{3}(k)$ has two minima at
$k=P_{1}=(+p_{0},+p_{0})$ and $k=P_{2}=(-p_{0},-p_{0})$, where

$\displaystyle
p_{0}=\arcsin\Big{[}\frac{4t}{8t+n_{0}U}\sqrt{\frac{4t(4t+n_{0}U)}{4t^{2}-2t_{s}^{2}}}\Big{]}$
(S3)

At $h=0$, if $n_{0}U/t=8t_{s}/(\sqrt{2}t-t_{s})$, the $\omega_{3}(k)$ has only
one minima at $k=\Lambda=(\pi/2,\pi/2)$ with the monopole charge $n=2$.

In summary, pictorially as illustrated in Fig.S1, at $h=0$, increasing
$n_{0}U$ from $0^{+}$ to the value greater than $8tt_{s}/(\sqrt{2}t-t_{s})$,
the two minima of $\omega_{3}(k)$ approach each other along the line
$k_{0x}+k_{0y}=\pi$, then collide at $\Lambda$-point, then the two minima
bounce off along the opposite direction along the perpendicular direction
$k_{0x}=k_{0y}$.

The point group symmetry of the square lattice is $C_{4v}$. In the presence of
spin-orbit coupling, the $C_{4v}$ symmetry becomes $[C_{4}\times
C_{4}]_{\text{diag}}$ symmetry, which is a joint 4-fold rotation of spin and
orbit. Besides, there are also two mirror symmetries $M_{1,2}$, which also
inherit from the $C_{4v}$ point group symmetry. The mirror symmetry $M_{1}$ is
a composition of time reversal $i\to-i,\vec{\sigma}\to-\vec{\sigma}$, the
orbit reflection $k_{x}\leftrightarrow k_{y}$, and the spin rotation
$(\sigma_{x},\sigma_{y},\sigma_{z})\to(\sigma_{y},\sigma_{x},-\sigma_{z})$.
The mirror symmetry $M_{2}$ is a composition of the orbit reflection
$\vec{k}\to-\vec{k}$, and the spin rotation
$(\sigma_{x},\sigma_{y},\sigma_{z})\to(-\sigma_{x},-\sigma_{y},\sigma_{z})$.

When $h>h_{c}$, the ground state is the Z-FM superfluid phase where both
$[C_{4}\times C_{4}]_{\text{diag}}$ and $M_{1,2}$ symmetries are unbroken.

When $h<h_{c}$, it is the XY-CAFM superfluid where the $[C_{4}\times
C_{4}]_{\text{diag}}\to 1$ is broken while $M_{1,2}$ remain unbroken. It is
the $M_{1}$ symmetry which guarantees the two minima $K_{1}=(k_{0},\pi-k_{0})$
and $K_{2}=(\pi-k_{0},k_{0})$ are related by $k_{x}\leftrightarrow k_{y}$. It
is the $M_{2}$ symmetry which guarantees the two minima
$P_{1}=(+p_{0},+p_{0})$ and $P_{2}=(-p_{0},-p_{0})$ are related by
$k\leftrightarrow-k$. The colliding point $\Lambda=(\pi/2,\pi/2)$ is the
mirror symmetric point which is invariant under both $M_{1}$ and $M_{2}$.

It is constructive to compare the mirror symmetric point
$\Lambda=(\pi/2,\pi/2)$ in the momentum space here with that in the SOC
parameter space $\beta=\pi/2$ discussed in SOCM . In the former, the
Hamiltonian which is the sum over all momenta has the same symmetry, the
Doppler shit term still exists at a quadratic order, but non-vanishing. While
in the latter, the Hamiltonian has an enlarged symmetry which dictates the
exact vanishing of the Doppler shift term.

Figure S1: Illustration of the special k-points used in this Letter:
$\Gamma=(0,0),~{}\Lambda=(\pi/2,\pi/2),~{}R=(\pi,\pi),~{}X=(\pi,0),~{}Y=(0,\pi)$.
The $K_{1}$ and $K_{2}$ are the two minima of $\omega_{3}(k)$ when
$n_{0}U/t<8t_{s}/(\sqrt{2}t-t_{s})$; the $P_{1}$ and $P_{2}$ are the two
minima of $\omega_{3}(k)$ when $n_{0}U/t>8t_{s}/(\sqrt{2}t-t_{s})$. The arrows
show that: as $n_{0}U/t$ increases, the two minima of $\omega_{3}(k)$ approach
each other along the X-Y line, then collide at $\Lambda$-point, then bounce
off into two opposite perpendicular directions. The thick square is the
lattice Brillouin Zone (BZ), the thin one is the reduce Brillouin Zone (RBZ).

It is interesting to compare the scattering process of the two bosonic Dirac
points with the same chirality in Fig.S1 with the annihilation process of the
two fermionic Dirac points with the opposite chirality discussed in TQPT . As
shown in this work, the former happens in the high energy modes, so it is not
a QPT. It is not a TPT either. While the latter is both where various physical
quantities satisfy various scaling functions. The collision of two bosonic
vortices with opposite vorticities in an expanding curved space-time was
analyzed in gravity  where two surface waves also emerge after the collision
and head in opposite direction along the normal direction.

##  II  B. Determination of the band structures induced by the OFQD at $h=0$.

Solving the band touching condition $\omega_{2}(k)=\omega_{3}(k)$ within the
regime (I) $n_{0}U/t<8t_{s}/(\sqrt{2}t-t_{s})$, we obtain the two solutions
$K_{1}=(k_{0x},k_{0y})$ and $K_{2}=(k_{0y},k_{0x})$, where

$\displaystyle
k_{0y}=\arcsin\Big{[}\frac{\sqrt{2}tn_{0}U}{t_{s}(8t+n_{0}U)}\Big{]}=k_{0},\quad
k_{0x}=\pi-k_{0}.$ (S4)

The two solutions are related by the remaining $M_{1}$ symmetry. The band
touching happened at high energy
$\omega_{2}(K_{1})=\omega_{3}(K_{1})=8t(4t+n_{0}U)/(8t+n_{0}U)=\omega_{0}>4t$.
Expanding the $\omega_{2,3}(k)$ near $K_{1,2}$ with $k=K_{1,2}+q$, we obtain

$\displaystyle\omega_{2,3}(K_{1}+q)=\omega_{0}\mp\sqrt{v_{1}^{2}(q_{x}-q_{y})^{2}+v_{2}^{2}(q_{x}+q_{y})^{2}}+c(q_{x}-q_{y})\>,$
$\displaystyle\omega_{2,3}(K_{2}+q)=\omega_{0}\mp\sqrt{v_{1}^{2}(q_{x}-q_{y})^{2}+v_{2}^{2}(q_{x}+q_{y})^{2}}-c(q_{x}-q_{y})\>,$
(S5)

where

$\displaystyle v_{1}$ $\displaystyle=\sqrt{2}\cos
k_{0}\,t_{s}(8t+n_{0}U)/\sqrt{\smash[b]{16t(4t+n_{0}U)}},$ $\displaystyle
v_{2}$ $\displaystyle=\sqrt{2}\cos
k_{0}\,t_{s}(8t+n_{0}U)^{2}/[16t(4t+n_{0}U)],$ $\displaystyle c$
$\displaystyle=\sqrt{2}\cos k_{0}\,t_{s}n_{0}^{2}U^{2}/[16t(4t+n_{0}U)].$ (S6)

The two solutions are related by the remaining $M_{1}$ symmetry. The Doppler
shift term is along the $[1\bar{1}]$ direction, which is nothing but the
“collision” direction in Fig.S1. Note that the relative sizes of $v_{1}$ and
$c$ is not important here, due to the high energy $\omega_{0}$ which has
direct experimental consequences as stressed in Sec.M9, so it will not drive
any instabilities. In a sharp contrast, in the fermionic QAH under an
injecting current discussed in QAHinj , due to the absence of the $\omega_{0}$
term, the sign change of the relative size drives a TPT.

Solving the band touching condition $\omega_{2}(k)=\omega_{3}(k)$ within the
regime (II) $n_{0}U/t>8t_{s}/(\sqrt{2}t-t_{s})$, we obtain the two solutions
$P_{1}=+(k_{0x},k_{0y})$ and $P_{2}=-(k_{0x},k_{0y})$, where

$\displaystyle
k_{0x}=k_{0y}=\arcsin\Big{[}\frac{4t}{8t+n_{0}U}\sqrt{{\frac{4t(4t+n_{0}U)}{4t^{2}-2t_{s}^{2}}}}\Big{]}=p_{0}.$
(S7)

The two solutions are related by the remaining $M_{2}$ symmetry. The band
touching happens at the high energy
$\omega_{2}(P_{1})=\omega_{3}(P_{1})=8t(4t+n_{0}U)/(8t+n_{0}U)=\omega_{0}>4t+2\sqrt{2}t_{s}$.
Expanding the $\omega_{2,3}(k)$ near $P_{1,2}$ with $k=P_{1,2}+q$, we obtain

$\displaystyle\omega_{2,3}(P_{1}+q)=\omega_{0}\mp\sqrt{v_{1}^{2}(q_{x}+q_{y})^{2}+v_{2}^{2}(q_{x}-q_{y})^{2}}+c(q_{x}+q_{y})\>,$
$\displaystyle\omega_{2,3}(P_{2}+q)=\omega_{0}\mp\sqrt{v_{1}^{2}(q_{x}+q_{y})^{2}+v_{2}^{2}(q_{x}-q_{y})^{2}}-c(q_{x}+q_{y})\>,$
(S8)

where

$\displaystyle v_{1}$ $\displaystyle=[2t(8t+n_{0}U)/(n_{0}U)]\cot p_{0},$
$\displaystyle v_{2}$
$\displaystyle=[t_{s}(8t+n_{0}U)/\sqrt{8t(4t+n_{0}U)}]\cos p_{0},$
$\displaystyle c$ $\displaystyle=[2tn_{0}U/(8t+n_{0}U)]\cot p_{0}$ (S9)

The Doppler shift term is along the $[11]$ direction, which is nothing but the
“bouncing-off” direction in Fig.S1.

In summary, near the band touching point, we always have

$\displaystyle\omega_{2,3}(K_{i}+q\text{ or
}P_{i}+q)=\omega_{0}\mp\sqrt{\smash[b]{v_{\parallel}^{2}(q_{\parallel})^{2}+v_{\perp}^{2}(q_{\perp})^{2}}}-(-1)^{i}c_{\parallel}q_{\parallel}\>,$
(S10)

where $i=1,2$,
$q_{\parallel}=(q_{x}-q_{y})/\sqrt{2},q_{\perp}=(q_{x}+q_{y})/\sqrt{2}$ for
regime (I),
$q_{\parallel}=(q_{x}+q_{y})/\sqrt{2},q_{\perp}=(q_{x}-q_{y})/\sqrt{2}$ for
regime (II), and $v_{\parallel},v_{\perp},c_{\parallel}$ are easy to deduce
from the previous expressions Eq.S6 and Eq.S9.

Figure S2: The OFQD induced band structure of Bogoliubov excitation bands
$\omega_{2}$ and $\omega_{3}$ at $h=0$. The parameters are $t=1$, $t_{s}=1/2$,
$n_{0}U=1$, and $\theta=\pi/2$, $\phi=-\pi/4$. The bottom plane is the contour
plot of $\omega_{3}-\omega_{2}$ as a function of $k_{x}$ and $k_{y}$. It is
clear to see there are two Dirac points at the two in-commensurate momenta
denoted by $+$ sitting on the Reduced BZ boundary.

##  III  C. The folding of the two lower bands and the two upper bands and the
physical meanings of $C_{I}=C_{1+2}$ and $C_{I\\!I}=C_{3+4}$ at $h<h_{c}$.

Now we look at the band structures of the two lower bands and the two upper
bounds. At $h=0$, in the XY-AFM phase with $c_{0}=1/\sqrt{2}$ and
$c_{\pi}=e^{-i\pi/4}/\sqrt{2}$, solving the band touching condition
$\omega_{1}(k)=\omega_{2}(k)$ or $\omega_{3}(k)=\omega_{4}(k)$, we find the
solutions $k=(k_{0x},k_{0y})$ form a straight line
$\\{(k_{0x},k_{0y})|k_{0x}-k_{0y}=\pi\\}$. Thus, there is a line degeneracy
between $\omega_{1}$ and $\omega_{2}$. The Fig.S3 shows that the $\omega_{1}$
and $\omega_{2}$ indeed touch along one reduced BZ edge.

Figure S3: The band structure of the Bogoliubov excitation bands $\omega_{1}$
and $\omega_{2}$ at $h=0$. The bottom plane is the contour plot of
$\omega_{2}-\omega_{1}$ as a function of $k_{x}$ and $k_{y}$. It is clear to
see the two bands merge along one reduced BZ edge from $(0,-\pi)$ to
$(\pi,0)$. The parameters are the same as Fig.S2.

The line degeneracy between $\omega_{1,2}$ and $\omega_{3,4}$ suggests that we
can unfold the two bands along $[1\,\bar{1}]$ direction, by defining
$\Omega_{I}(k)=\omega_{2}(k)$ if $k$ in reduced BZ, otherwise
$\Omega_{I}(k)=\omega_{1}(k+Q^{\prime})$ where $Q^{\prime}=(+\pi,-\pi)$.
Similarly, we define $\Omega_{I\\!I}(k)=\omega_{3}(k)$ if $k$ in the reduced
BZ, otherwise $\Omega_{I\\!I}(k)=\omega_{4}(k+Q^{\prime})$.

The unfolded band structures are plotted in Fig.S4. Thus it is more nature to
consider $\omega_{1,2}$ as one band and $\omega_{3,4}$ as the other band; this
is the physical reason we always consider a combination of band Chern numbers
$C_{I}=C_{1+2}$ and $C_{I\\!I}=C_{3+4}$.

Figure S4: The unfold Bogoliubov excitation bands $\Omega_{I}$ and
$\Omega_{I\\!I}$ at $h=0$. There are two conic band crossings between
$\Omega_{I}$ and $\Omega_{I\\!I}$. The green dashed line represents the reduce
Brillouin zone. The new Brillouin zone is a rectangular shape, and the two
Dirac points denoted by $+$ sit on the longer side. The parameters are the
same as Fig.S2. When adding a small $0<|h|<h_{c}$, the line degeneracy still
exists, so $\Omega_{I}(k)$ and $\Omega_{I\\!I}(k)$ are still well-defined. Due
to the symmetry $[C_{4}\times C_{4}]_{D}$ restoration of the Z-FM above
$h>h_{c}$, the line degeneracy remains. Of course, any small $h$ opens a gap
between $\Omega_{I}$ and $\Omega_{I\\!I}$ around the two Dirac points and Lead
to the two effective Hamiltonians Eq.M13.

## References

  * (1) Fadi Sun, Xiao-Lu Yu, Jinwu Ye, Heng Fan, W. M. Liu, Topological Quantum Phase Transition in Synthetic Non- Abelian Gauge Potential: Gauge Invariance and Experimental Detections, Scientific Reports 3, 2119 (2013).
  * (2) Fadi Sun, Jinwu Ye, Wu-Ming Liu, Quantum incommensurate skyrmion crystals and commensurate to in-commensurate transitions in cold atoms and materials with spin–orbit couplings in a Zeeman field, New J. Phys. 19, 083015 (2017).
  * (3) Fadi Sun and Jinwu Ye, Topological phase transitions, invariants and enriched bulk-edge correspondences in fermionic gapless systems with extended Fermi surface, arXiv. Long version 45 PRB page.
  * (4) J. Ye and R. Bradenberger, The Formation and evolution of vortices in an expanding universe, Mod.Phys.Lett A, 157 (1990). The formation and evolution of U(1) gauged vortices in an expanding universe, Nucl. Phys. B346, 149 (1990).


11institutetext:  Harbin Institute of Technology, Shenzhen
11email<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS>
# MSDF: A General Open-Domain Multi-Skill Dialog Framework

Yu Zhao ** Xinshuo Hu ** Yunxin Li  Baotian Hu (()) Dongfang Li
Sichao Chen  Xiaolong Wang

###### Abstract

Dialog systems have achieved significant progress and have been widely used in
various scenarios. The previous researches mainly focused on designing dialog
generation models in a single scenario, while comprehensive abilities are
required to handle tasks under various scenarios in the real world. In this
paper, we propose a general Multi-Skill Dialog Framework, namely MSDF, which
can be applied in different dialog tasks (e.g. knowledge grounded dialog and
persona based dialog). Specifically, we propose a transferable _response
generator_ pre-trained on diverse large-scale dialog corpora as the backbone
of MSDF, consisting of BERT-based encoders and a GPT-based decoder. To select
the response consistent with dialog history, we propose a _consistency
selector_ trained through negative sampling. Moreover, the flexible copy
mechanism of external knowledge is also employed to enhance the utilization of
multiform knowledge in various scenarios. We conduct experiments on knowledge
grounded dialog, recommendation dialog, and persona based dialog tasks. The
experimental results indicate that our MSDF outperforms the baseline models
with a large margin. In the Multi-skill Dialog of 2021 Language and
Intelligence Challenge, our general MSDF won the 3rd prize, which proves our
MSDF is effective and competitive.

###### Keywords:

Multi-skill dialog Knowledge grounded dialog Conversational recommendation
Persona based dialog

††footnotetext: * Y. Zhao and X. Hu contribute equally to this
work.††footnotetext: ✉ Corresponding author

##  1 Introduction

Figure 1: Overall Multi-Skill Dialog Framework (MSDF). The response generator
first generates diverse responses through a sampling-based decoding algorithm
conditioned on the dialog history and optional extra information(e.g.
knowledge, user profile and/or machine persona). Then, the consistency
selector selects the most contextually consistent response as the final
response.

Propelled by the acquisition of large-scale dialog corpora and the advance of
pre-training technology, dialog generation models have made great progress,
and dialog systems have been applied in various scenarios, such as chit-chat,
knowledge grounded dialog, and conversational recommendation. However, most of
the previous works only focused on modeling the dialog system within a single
scenario, which is difficult to be applied in other scenarios directly. It
does not satisfy the requirements of practical application in the real world,
where the dialog model needs to generate responses in various scenarios. To
handle different tasks in various scenarios, multiple dialog skills are
requested, such as knowledge utilization, commodities recommendation, and
persona understanding skills. It is of great necessity to model a general
multi-skill dialog framework that can be flexibly applied in various
scenarios.

How to model the general multi-skill dialog systems that can effectively use
information from diverse sources still remains challenging. On the one hand,
the model needs to use various information (e.g. structured and unstructured
knowledge, persona, conversation topics), and the previous works usually
design complex models [11][6] to utilize specific information in a single
scenario, which results in lacking universality. On the other hand, the
previous works used complicated data processes and training processes [9] to
optimize models on specific dialog corpus, thus, the model is difficultly
transferred to other scenarios.

In this work, we propose the general multi-skill dialog framework MSDF to
address the above problems, which consists of a pre-trained dialog _response
generator_ and a _consistency selector_. As depicted in Fig.1, MSDF generates
responses in two stages: 1) generating diverse responses as the candidate
_response pool_ , via _dialog history_ (and _extra information_); 2) selecting
the _final response_ by consistency selector. Specifically, we first pre-train
a universal and transferable encoder-decoder based model on various diverse
dialog corpora, including chit-chat, knowledge dialog, and recommendation
dialog, to obtain a general model with multiple coarse-grained skills
enhanced. Then, we apply multiple identical encoders to encode different
source information and equip the decoder with the multi-source information
fusion module, which can be flexibly transferred to different application
scenarios. Moreover, a dialog history can be mapped into multiple acceptable
responses, which is also known as the one-to-many mapping [13], especially in
the open domain. Thus, we introduce a BERT-based consistency selector to
choose the most contextually consistent response with dialog history from the
response pool, which is trained via negative sampling to distinguish
consistent and inconsistent responses with dialog history.

Experiments are conducted to evaluate the performance of our MSDF in knowledge
grounded dialog, recommendation dialog, and persona dialog tasks. The
experimental results indicate that our MSDF outperforms the baseline models
with a large margin in terms of F1 and BLEU scores. In the Multi-skill Dialog
of 2021 Language and Intelligence Challenge, our MSDF won the third prize,
with 6th rank in the human evaluation of the finals. Both automatic and human
evaluation results indicate that our MSDF is effective and competitive.

Our contributions are as bellows.

  * •

We propose a general multi-skill dialog framework that solves various tasks in
different scenarios, namely MSDF, consisting of a dialog _response generator_
with multi-source information encoders and a _consistency selector_.

  * •

We pre-train an encoder-decoder based dialog generation model on various types
of large-scale open-domain dialog corpora, which can be effectively
transferred into our MSDF to solve different tasks.

  * •

The experimental results indicate that our MSDF outperforms the baseline
models with a large margin in knowledge dialog, recommendation dialog, and
persona dialog tasks, demonstrating the effectiveness of MSDF.

##  2 Related Work

The Multi-Skill Dialog of 2021 Language and Intelligence Challenge focuses on
three kinds of dialog generation tasks, including the knowledge grounded
dialog, recommendation dialog, and persona dialog. The previous works always
modeled dialog systems to solve a single task in a scenario. For knowledge
grounded dialogs, the model generates responses conditioned on dialog history
and the given external knowledge. [8] employed posterior probability
distribution of knowledge to guide the learning of prior distribution during
training. [6] proposed the sequential decision making method to select
knowledge used in response. And [12] modeled the knowledge selection as
walking over knowledge graphs. For the persona dialogs, the model is required
to understand and utilize the given persona to generate personalized
responses. [9] improved the performance of persona understanding through
persona negative sampling. [22] trained a VAE(Variational Auto-encoder) model
to produce persona-related topic words jointly generating responses. For
recommendation dialogs, the model needs to make recommendations through
conversations, which is usually separated to two tasks: recommendation and
dialog generation. [4] [27] incorporated the common sense knowledge graph to
improve the user profile understanding and recommendation dialog generation.
[28] incorporated topic planning to enhance the recommendation dialog.
Following the work of [3], we also propose a general multi-skill dialog
framework that can handle various tasks.

##  3 Method

###  3.1 Multi-skill Dialog

The multi-skill dialog task aims to construct a general dialog generation
model with multiple skills to solve various tasks in various scenarios. In our
work, we focus on the dialog modeling task with multiple skills, including
three sub-tasks: 1) knowledge dialog, 2) recommendation dialog, and 3) persona
dialog. The descriptions of the three sub-tasks are as follows111Details in:
https://aistudio.baidu.com/aistudio/competition/detail/67.

Figure 2: Responses generation process in MSDF. The top-right corner is the
detail of decoder block, and the top-left corner shows the process of copy
mechanism.

####  3.1.1 Knowledge Dialog

The inputs of the knowledge dialog include dialog goals
$\mathbf{G}=[\mathbf{g}_{1},\mathbf{g}_{2},…,\mathbf{g}_{r}]$, knowledge
information $\mathbf{K}=[\mathbf{k}_{1},\mathbf{k}_{2},…,\mathbf{k}_{s}]$, and
dialog history $\mathbf{H}=[\mathbf{x}_{1},\mathbf{x}_{2},…,\mathbf{x}_{n}]$,
where $\mathbf{g}_{i}$ is a description of dialog target or a conversation
topic, $\mathbf{k}_{i}$ is the form of a triple or an unstructured sentence,
and $\mathbf{x}_{i}$ consists of a sequence of words. The knowledge grounded
dialog generation task requires the model to use appropriate knowledge to
generate the response $\mathbf{R}=[r_{1},r_{2},…,r_{m}]$, where $r_{i}$
denotes the $i$-th word in response, $m$ denotes the length of response.

####  3.1.2 Recommendation Dialog

The inputs of the recommendation dialog include user profiles
$\mathbf{U}=[\mathbf{u}_{1},\mathbf{u}_{2},…,\mathbf{u}_{t}]$, conversation
scenario $\mathbf{S}$, dialog targets $\mathbf{G}$, knowledge information
$\mathbf{K}$, and dialog history $\mathbf{H}$, where $\mathbf{u}_{i}$ is an
aspect of the user profile in a key-value form. The recommendation dialog
generation task requires the model to make recommendations based on the user
profile and the scenario through conversation.

####  3.1.3 Persona Dialog

The inputs of the persona dialog include persona information of machine
$\mathbf{P}=[\mathbf{p}_{1},\mathbf{p}_{2},…,\mathbf{p}_{z}]$, and dialog
history $\mathbf{H}$, where $\mathbf{p}_{i}$ is a sentence describing the
persona. The persona dialog generation task requires the model to generate
natural, fluent and informative machine response in line with dialogue history
and persona information.

###  3.2 Overall Framework

We propose a general multi-skill dialog framework MSDF, mainly consisting of a
response generator and a history-response consistency selector, as shown in
Fig.1. MSDF generates the response in two steps: response generator is applied
to generate diverse responses as the response pool, and then consistency
selector chooses the most consistent response from the response pool as the
final output.

####  3.2.1 Response Generator

The previously proposed generation models, such as GPT2 [14], PLATO2 [1], and
BART [7], have shown their great generation performance obtaining from large-
scale corpora. To obtain a universal pre-trained dialog model that can be
flexibly transferred to the three sub-tasks, we pre-train an encoder-decoder-
based model, which consists of a BERT-based encoder and a GPT-based decoder,
and the optimization objective is to generate a target response conditioned on
dialog history. The six large-scale dialog datasets, including LCCC [19],
Weibo[17], Douban[21], DuConv [20], KdConv [26] and DuRecDial [11], are used
to pre-train the model. Specifically, we process all the data to a general
dialog form: for the multi-turn conversations, we split them into history-
response pairs; for the conversations with extra knowledge or user profiles,
we ignore them during pre-training. Moreover, to accelerate the pre-training
process, we use the parameters of DialoGPT [24] to initialize both the encoder
and decoder. We find that the model pre-trained with all data tends to
generate short responses, thus, we additionally select the half of data with
longer responses to pre-train it in the latter part of pre-training phase. In
each sub-task, we duplicate the encoder of pre-trained model to encode
specific information, and equip the decoder with an information fusion module.

Denote $\mathbf{H}_{l-1}^{D}\in\mathbb{R}^{L_{d}\times d}$ as the output of
($l-1$)-th decoder block, where $L_{D}$ is the length of decoding sequence and
$d$ is the hidden size of decoder blocks, the calculating process in each
decoder block is described as follows:

$\mathbf{H}_{l}^{SA}=SelfAttention\left(LayerNorm(\mathbf{H}_{l-1}^{D})\right)+\mathbf{H}_{l-1}^{D},$
(1)
$\mathbf{H}_{l}^{FA}=Fusion\left(\mathbf{H}_{l}^{SA},\mathbf{H}_{history},\mathbf{H}_{knowledge},\mathbf{H}_{persona}\right),$
(2)
$\mathbf{H}_{l}^{D}=FFN\left(LayerNorm(\mathbf{H}_{l}^{FA})\right)+\mathbf{H}_{l}^{FA},$
(3)

where $\mathbf{H}_{history}\in\mathbb{R}^{n\times d}$,
$\mathbf{H}_{knowledge}\in\mathbb{R}^{s\times d}$ and
$\mathbf{H}_{persona}\in\mathbb{R}^{z\times d}$ denotes the encoded
information, $\mathbf{H}_{l}^{SA}\in\mathbb{R}^{L_{d}\times d}$ denotes the
result of self-attention, and $\mathbf{H}_{l}^{FA}\in\mathbb{R}^{L_{d}\times
d}$ denotes the fused multi-source information. The $Fusion$ function is
calculated in the following way:

$\mathbf{A}_{l}(\mathbf{H}^{E})=softmax\left(\frac{\left(LayerNorm(\mathbf{H}_{l}^{SA})\mathbf{W}_{l}^{Q}\right)(\mathbf{H}^{E}\mathbf{W}_{l}^{K})^{T}}{\sqrt{d}}\right)(\mathbf{H}^{E}\mathbf{W}_{l}^{V}),$
(4)

$\mathbf{H}_{l}^{FA}=[\mathbf{A}_{l}(\mathbf{H}_{history});\mathbf{A}_{l}(\mathbf{H}_{knowledge});\mathbf{A}_{l}(\mathbf{H}_{persona})]\mathbf{W}_{l}^{P}+\mathbf{H}_{l}^{SA},$
(5)

where [;] indicates concatenation on the last dimension, $\mathbf{W}_{l}^{Q}$,
$\mathbf{W}_{l}^{K}$ and $\mathbf{W}_{l}^{V}$ are learnable projection
matrices and $\mathbf{W}_{l}^{P}\in\mathbb{R}^{3d\times d}$ is a learnable
attention fusing matrix (we get $\mathbf{W}_{l}^{P}\in\mathbb{R}^{2d\times d}$
when there are only two cross-attention to be fused).

Finally, the hidden state $\mathbf{H}_{l}^{D}\in\mathbb{R}^{L_{D}\times d}$ of
the $l$-th decoder layer is output by the $FFN$ layer and residual layer:

$\mathbf{H}_{l}^{D}=FFN(LayerNorm(\mathbf{H}_{l}^{FA}))+\mathbf{H}_{l}^{FA}.$
(6)

With the great performance of hybrid pointer-generator network [15] in
summarization, we utilize the attention-based copy mechanism to generate
knowledge-enhanced response. Decoder hidden states are put into linear
language model to get original vocabulary distribution
$\mathbf{P}_{vocab}\in\mathbb{R}^{L_{D}\times L_{V}}$,

$\mathbf{P}_{vocab}=softmax(\mathbf{H}^{D}\mathbf{W}^{LM}),$ (7)

where $W^{LM}\in\mathbb{R}^{d\times L_{V}}$ is the learnable language model
head and $L_{V}$ denotes the vocabulary length. And then attention-based copy
mechanism is utilized to generate extra knowledge enhanced response. We obtain
knowledge copy attention $\mathbf{A}_{copy}\in\mathbb{R}^{L_{D}\times L_{K}}$,
via cross-attention of decoded hidden states and encoded knowledge (or
persona) hidden state:

$\mathbf{A}_{copy}=softmax((\mathbf{H}^{D}\mathbf{W}^{Q})(\mathbf{H}_{knowledge}\mathbf{W}^{K})^{T}),$
(8)

where $\mathbf{W}^{Q}$ and $\mathbf{W}^{K}$ are learnable projection matrices,
and $L_{K}$ denotes the length of the knowledge sequence. Then generation
probability $\mathbf{p}_{gen}\in[0,1]$ can be calculated:

$\mathbf{p}_{gen}=sigmoid([\mathbf{A}_{copy}\mathbf{H}_{knowledge};\mathbf{H^{D}}]\mathbf{W}^{mlp}),$
(9)

where $W^{mlp}\in R^{2d\times 1}$ is learnable matrices. We obtain the
following probability distribution over the merged vocabulary to predict word
w:

$\mathbf{P}(w)=\log(\mathbf{p}_{gen}\mathbf{P}_{vocab}(w)+(1-\mathbf{p}_{gen})\mathbb{I}(w_{i}=w)A_{copy}(w)),$
(10)

where $\mathbb{I}(\cdot)$ is an indicator function, and $A^{copy}(w)$ is the
element at $w$-th column in the copy attention matrix.

####  3.2.2 Consistency Selector

Inspired by the BERT next sentence prediction [5], we propose a history-
response consistency selector, since the response should be the next sentence
by dialog history. The consistency selector is a binary classifier to
distinguish the consistent and inconsistent response with dialog history. We
construct the consistency selector by a pre-trained model RoBBERTa [10], plus
a linear head layer. We first concatenate the dialog history and response as
the pre-trained model input to get context representation, and then put the
first token (usually known as [CLS] token) to the linear head to get a binary
classification score. The consistency selector is trained on positive and
negative sampling examples from training data, where inconsistent responses
are randomly sampled. During inference, we get the positive score from the
consistency selector output as the consistency score.

###  3.3 Data Processing

We separate different input resources into four categories: dialog history,
knowledge information, persona information, and current reply (or previously
generated response during inference). The data preprocessing and reprocessing
are demonstrated as follows.

The knowledge dialog generation model consists of a dialog history encoder, a
knowledge encoder and a decoder. During data preprocessing of DuConv, we
reformat the knowledge graph to a pseudo unstructured knowledge sentences, by
concatenating the subject, predicate and object, and join all knowledge
sentence with special token “[SEP]”. Since the dialog topics are limited in
movies and film stars and there are at most two topics during a dialog, we
introduce special tokens to format the subject in the knowledge graph,
including “[movie1]”, “[movie2]”, “[star1]”, and “[star2]”, which will be
restored in data reprocessing after response generated. And we also add
speaker tokens “[speaker1]” and “[speaker2]” to distinguish speakers in the
dialog history. We also reprocess personal pronouns and figures due to the
knowledge, such as outcome date of movies, and birthday of stars.

The recommendation dialog generation model consists of a dialog history
encoder, a knowledge encoder, a persona encoder and a decoder. To facilitate
optimization, we make the knowledge encoder and persona encoder share
parameters. The knowledge graph in DuRecDial is also reformatted in the same
way as DuConv, with all the candidate recommendation goal planning
concatenated at the end. Since the recommendation goal is labeled with each
response, we introduce a new special token “[goal]” as a separator, and join
the golden goal with response， e.g. “[goal]问User性别[goal]我该称呼您是先生还是女士”(“[goal]
Asking about the user’s gender [goal] Should I call you Mr. or MS”). This
golden goal prefix in the response performs as the semantic guidance, which is
typically like conditional generation. The dialog situation is viewed as extra
knowledge concatenated before knowledge information. We also replace the user
name with a new special token “[uname]” and add speaker tokens. Personal
pronouns and figures will be corrected and goal information will be removed
during data reprocess.

The persona dialog generation model consists of a dialog history encoder, a
persona encoder and a decoder. We trained a simple Word2Vec by open-source
gensim implementation 222https://radimrehurek.com/gensim/intro.html, on CPC
sentences to estimate the similarity between the response and persona by
cosine similarity of average word vector, and randomly drop training examples
with persona information similarity less than 0.7. It is observed that the
generator prefers to generate longer sentences and there are too many
responses whose topic is about the user job, so we drop some examples with too
long responses and “工作”(“work”) in dialog utterances. Considering that
conversations in CPC are more like chit-chat than task-oriented dialog, we
abandon the copy mechanism in this generator.

##  4 Experiment

###  4.1 Experimental Settings

####  4.1.1 Dataset

For pre-training, we use the Weibo dataset [17], Douban multi-round
conversation [21], LCCC dataset [19], Emotional Conversational Dataset [25],
retrieval-assisted conversational dataset [2], and Kdconv dataset [26]. For
fine-tuning, we use DuConv [20] for knowledge grounded dialog, DuRecDial [11]
for the conversational recommendation, and Chinese persona chat (CPC) for
persona dialog. Statistics of all dataset are summarized in Table. 1.

Table 1: Statistics of all datasets. † denotes the datasets for fine-tuning, and the others are for pre-training. |  Train |  Dev |  Test
---|---|---|---
DuConv† [20] |  19858 |  2000 |  5000
DuRecDial† [11] |  6618 |  946 |  4645
Chinese Persona Chat† |  23000 |  1500 |  3000
Weibo dataset [17] |  3103764 |  443394 |  886790
Douban multi-round conversation [21] |  5000000 |  25001 |  1186
LCCC dataset [19] |  11987759 |  20000 |  10000
Emotional Conversational Dataset [25] |  899207 |  110000 |  110000
retrieval-assisted conversational dataset [2] |  5498480 |  107332 |  156706
Kdconv dataset [26] |  3000 |  300 |  2751

####  4.1.2 Implementation Details

For the response generator, the encoders and decoder settings follow the
DialoGPT [24], where the hidden size is 768 for 12 layers, the maximum input
length is 512, and there are 12 heads in multi-head attention. For all dropout
layers, the dropout rate is set to 0.1. We adopt cross-entropy loss as our
loss function, and the parameters are saved in term of the minimum cross-
entropy loss on the development datasets. The consistency selector is
implemented alike a NLI model as described in section 3.2.2, which is also
optimized through cross-entropy loss function.

###  4.2 Evaluation

The competitive baseline models are compared with the proposed MSDF on
different quantitative metrics. We use BLEU-1 and BLEU-2 to evaluate the
n-gram lexical similarity, F1 to evaluate the Chinese character level
similarity, and DISTINCT1 and DISTINCT-2 to evaluate the diversity of
generated responses. The total SCORE for the automatic evaluation is
calculated by averaging all the F1/BLEU1/BLEU2 scores for the subtasks. To
exhibit the improvement of our MSDF, we also implement and test four baseline
models: Seq2Seq [18], HRED [16], DialoGPT [24] and BERT-GPT [23]. All these
baseline models only take dialog history as inputs, without aquiring extra
information. Since test datasets are not released, we evaluate all the models
on LUGE platform 333LUGE:
https://aistudio.baidu.com/aistudio/competition/detail/48. The performance of
baseline models and our MSDF is presented in Table. 2, including ablation
experiments.

With respect to the human evaluation, we refer readers to the competition
leaderboard 444Leaderboard:
https://aistudio.baidu.com/aistudio/competition/detail/67 for the single-turn
or multi-turn dialog evaluation results.

Table 2: Automatic evaluation of test set B on baseline models and our MSDF (ranked by total score). All the results of F1, BLEU, and DISTINCT are average scores from three sub-tasks. |  SCORE |  F1 |  BLEU1/2 |  DISTINCT1/2
---|---|---|---|---
Seq2Seq [18] |  0.522 |  22.33 |  0.202/0.096 |  0.038/0.100
HRED [16] |  0.565 |  23.66 |  0.220/0.108 |  0.038/0.105
DialoGPT [24] |  0.373 |  17.05 |  0.141/0.061 |  0.079/0.313
BERT-GPT [23] |  0.573 |  24.51 |  0.215/0.113 |  0.061/0.214
MSDF |  0.934 |  38.62 |  0.333/0.215 |  0.057/0.183
-consistency selector |  0.872 |  36.00 |  0.314/0.198 |  0.051/0.173
-extra knowledge |  0.705 |  28.98 |  0.271/0.145 |  0.049/0.162

###  4.3 Discussion

Our MSDF outperforms the baseline models with a large margin, even though
without extra information, which strongly presents the effectiveness. The
consistency selector preferred to select the most common response. It reduces
the variance of generating performance and significantly improves automatic
evaluation results, without considering the limited diversity of responses in
human evaluation. Besides, the attention-based copy mechanism is of great
importance for generating knowledge-enhanced response. According to our
observation, DialoGPT and BERT-GPT still talk rubbish after fine-tuning,
despite resulting in higher DISTINCT scores. In the competition, our MSDF got
9-th rank in automatic evaluation and 6-th rank in human evaluation. It
increased by 3 ranks in human evaluation compared to the automatic evaluation.
We attribute this to the multi-skill pre-training, from which our model could
generate more human-like responses, in spite of the mismatch with the golden
references in automatic evaluation.

##  5 Conclusion

This paper describes our general multi-skill dialog framework MSDF, consisting
of a response generator and a dialog history consistency selector. We first
pre-train the basic encoder-decoder on diverse datasets and then fine-tune it
on the specific dataset to construct the response generator with a strong
specific skill. We won the third prize in Multi-task Dialog of 2021 Language
and Intelligence Challenge. Experiments are also conducted on several baseline
models. The vast improvement over the baseline models indicates that our
framework is effective and competitive. In future work, we will experiment
with our MSDF on more tasks to evaluate the comprehensive skills of our
framework and further improve its performance.

## Acknowledgement

We appreciate the insightful feedback from the anonymous reviewers and the
beneficial support from Baidu Inc. This work is jointly supported by grants:
Natural Science Foundation of China (No. 62006061 and 61872113), Strategic
Emerging Industry Development Special Funds of Shenzhen (JCYJ20200109113403826
and JCYJ20200109113441941) and Stable Support Program for Higher Education
Institutions of Shenzhen (No. GXWD20201230155427003-20200824155011001).

## References

  * [1] Bao, S., He, H., Wang, F., Wu, H., Wang, H., Wu, W., Guo, Z., Liu, Z., Xu, X.: Plato-2: Towards building an open-domain chatbot via curriculum learning. arXiv preprint arXiv:2006.16779 (2020)
  * [2] Cai, D., Wang, Y., Bi, W., Tu, Z., Liu, X., Shi, S.: Retrieval-guided dialogue response generation via a matching-to-generation framework. 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). pp. 1866–1875 (2019)
  * [3] Cao, Y., Bi, W., Fang, M., Tao, D.: Pretrained language models for dialogue generation with multiple input sources. In: Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: Findings. pp. 909–917 (2020)
  * [4] Chen, Q., Lin, J., Zhang, Y., Ding, M., Cen, Y., Yang, H., Tang, J.: Towards knowledge-based recommender dialog system. 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). pp. 1803–1813 (2019)
  * [5] Devlin, J., Chang, M.W., Lee, K., Toutanova, K.: Bert: Pre-training of deep bidirectional transformers for language understanding. In: Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers). pp. 4171–4186 (2019)
  * [6] Kim, B., Ahn, J., Kim, G.: Sequential latent knowledge selection for knowledge-grounded dialogue. In: International Conference on Learning Representations (2019)
  * [7] Lewis, M., Liu, Y., Goyal, N., Ghazvininejad, M., Mohamed, A., Levy, O., Stoyanov, V., Zettlemoyer, L.: Bart: Denoising sequence-to-sequence pre-training for natural language generation, translation, and comprehension. In: Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics. pp. 7871–7880 (2020)
  * [8] Lian, R., Xie, M., Wang, F., Peng, J., Wu, H.: Learning to select knowledge for response generation in dialog systems. In: IJCAI International Joint Conference on Artificial Intelligence. p. 5081 (2019)
  * [9] Liu, Q., Chen, Y., Chen, B., Jian-Guang, L., Chen, Z., Zhou, B., Zhang, D.: You impress me: Dialogue generation via mutual persona perception. In: Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics. pp. 1417–1427 (2020)
  * [10] Liu, Y., Ott, M., Goyal, N., Du, J., Joshi, M., Chen, D., Levy, O., Lewis, M., Zettlemoyer, L., Stoyanov, V.: Roberta: A robustly optimized bert pretraining approach. arXiv preprint arXiv:1907.11692 (2019)
  * [11] Liu, Z., Wang, H., Niu, Z.Y., Wu, H., Che, W., Liu, T.: Towards conversational recommendation over multi-type dialogs. In: Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics. pp. 1036–1049 (2020)
  * [12] Moon, S., Shah, P., Kumar, A., Subba, R.: Opendialkg: Explainable conversational reasoning with attention-based walks over knowledge graphs. In: Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics. pp. 845–854 (2019)
  * [13] Ni, J., Young, T., Pandelea, V., Xue, F., Adiga, V., Cambria, E.: Recent advances in deep learning-based dialogue systems. arXiv preprint arXiv:2105.04387 (2021)
  * [14] Radford, A., Wu, J., Child, R., Luan, D., Amodei, D., Sutskever, I.: Language models are unsupervised multitask learners. OpenAI blog 1(8), 9 (2019)
  * [15] See, A., Liu, P.J., Manning, C.D.: Get to the point: Summarization with pointer-generator networks. In: Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). pp. 1073–1083 (2017)
  * [16] Serban, I.V., Sordoni, A., Bengio, Y., Courville, A., Pineau, J.: Building end-to-end dialogue systems using generative hierarchical neural network models. In: Thirtieth AAAI Conference on Artificial Intelligence (2016)
  * [17] Shang, L., Lu, Z., Li, H.: Neural responding machine for short-text conversation. In: Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing (Volume 1: Long Papers). pp. 1577–1586 (2015)
  * [18] Sutskever, I., Vinyals, O., Le, Q.V.: Sequence to sequence learning with neural networks. Advances in Neural Information Processing Systems 27, 3104–3112 (2014)
  * [19] Wang, Y., Ke, P., Zheng, Y., Huang, K., Jiang, Y., Zhu, X., Huang, M.: A large-scale chinese short-text conversation dataset. In: CCF International Conference on Natural Language Processing and Chinese Computing. pp. 91–103. Springer (2020)
  * [20] Wu, W., Guo, Z., Zhou, X., Wu, H., Zhang, X., Lian, R., Wang, H.: Proactive human-machine conversation with explicit conversation goal. In: Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics. pp. 3794–3804 (2019)
  * [21] Wu, Y., Wu, W., Xing, C., Zhou, M., Li, Z.: Sequential matching network: A new architecture for multi-turn response selection in retrieval-based chatbots. In: Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). pp. 496–505 (2017)
  * [22] Xu, M., Li, P., Yang, H., Ren, P., Ren, Z., Chen, Z., Ma, J.: A neural topical expansion framework for unstructured persona-oriented dialogue generation. arXiv preprint arXiv:2002.02153 (2020)
  * [23] Zeng, G., Yang, W., Ju, Z., Yang, Y., Wang, S., Zhang, R., Zhou, M., Zeng, J., Dong, X., Zhang, R., et al.: Meddialog: A large-scale medical dialogue dataset. In: Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP). pp. 9241–9250 (2020)
  * [24] Zhang, Y., Sun, S., Galley, M., Chen, Y.C., Brockett, C., Gao, X., Gao, J., Liu, J., Dolan, W.B.: Dialogpt: Large-scale generative pre-training for conversational response generation. In: Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics: System Demonstrations. pp. 270–278 (2020)
  * [25] Zhou, H., Huang, M., Zhang, T., Zhu, X., Liu, B.: Emotional chatting machine: Emotional conversation generation with internal and external memory. In: Thirty-Second AAAI Conference on Artificial Intelligence (2018)
  * [26] Zhou, H., Zheng, C., Huang, K., Huang, M., Zhu, X.: Kdconv: A chinese multi-domain dialogue dataset towards multi-turn knowledge-driven conversation. In: Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics. pp. 7098–7108 (2020)
  * [27] Zhou, K., Zhao, W.X., Bian, S., Zhou, Y., Wen, J.R., Yu, J.: Improving conversational recommender systems via knowledge graph based semantic fusion. In: Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data Mining. pp. 1006–1014 (2020)
  * [28] Zhou, K., Zhou, Y., Zhao, W.X., Wang, X., Wen, J.R.: Towards topic-guided conversational recommender system. In: Proceedings of the 28th International Conference on Computational Linguistics. pp. 4128–4139 (2020)

##  Appendix 0.A Experimental Setting

Table 3: The experimental setting details for each task. Knowledge Dialog
---
Architecture | History encoder, Knowledge encoder, Decoder
Learning rate | $2.5\times 10^{-5}$
Batch size | $16$
Max epochs | $15$
Optimizer | AdamW($\beta_{1}=0.9,\beta_{2}=0.999,\epsilon=10\times 10^{-6}$)
Copy mechanism | Knowledge copy mechanism
Recommendation Dialog
Architecture | History encoder, Knowledge encoder, Persona encoder, Decoder
Learning rate | $2.5\times 10^{-5}$
Batch size | $2$
Max epochs | $10$
Optimizer | AdamW($\beta_{1}=0.9,\beta_{2}=0.999,\epsilon=10\times 10^{-6}$)
Copy mechanism | Knowledge copy mechanism
Persona Dialog
Architecture | History encoder, Persona encoder, Decoder
Learning rate | $2.5\times 10^{-5}$
Batch size | $16$
Max epochs | $10$
Optimizer | AdamW($\beta_{1}=0.9,\beta_{2}=0.999,\epsilon=10\times 10^{-6}$)
Copy mechanism | No copy mechanism

##  Appendix 0.B Samples

Table 4: Example of multi-turn knowledge dialog generation results. Due to the topic formatting, the generator performs well in capturing knowledge graph subject and object information. Knowledge graph |  [碟中谍,国家,美国]
(Mission: Impossible, country, USA)
[碟中谍,口碑,口碑一般]
(Mission: Impossible, public praise, common)
[碟中谍,类型,动作]
(Mission: Impossible, type, action)
[碟中谍,领域,电影]
(Mission: Impossible, field, movie)
[碟中谍,主演,强·沃特]
(Mission: Impossible, starring, Jon Voight)
[碟中谍,幕后花絮,电视剧版《谍中谍》的铁杆粉丝非常不满本片对吉姆的 描写]
(Mission: Impossible, behind the scenes, the hardcore fans of the TV series
“Mission: Impossible” are very dissatisfied with the film’s description of
Jim. )
[碟中谍,时光网评分,7.9]
(Mission: Impossible, Mtime score, 7.9)
[强·沃特,评论,又一老色鬼]
(Jon Voight, comment, another old pervert)
[强·沃特,星座,摩羯座]
(Jon Voight, constellation, capricorn)
[强·沃特,身高,191cm]
(Jon Voight, Height, 191cm)
[强·沃特,性别,男]
(Jon Voight, gender, male )
[强·沃特,职业,演员]
(Jon Voight, occupation, actor)
[强·沃特,领域,明星]
(Jon Voight, field, star)

---|---
Conversation |  Bot: 你喜欢看什么类型的电影？
(What kind of movies do you like to watch?)
User: 动作类
(Action)
Bot: 给你推荐一部电影，名字叫碟中谍 ， 有时间可以看一下。
(I’d like to recommend a movie called “Mission: Impossible”. You can watch it
when you have time.)
User: 好看吗
(Does it look good)
Bot: 这部电影的主演跟你一样都是摩羯座的，你知道是谁吗？
(The starring actors in this movie are Capricorns just like you. Do you know
who it is?)
User: 是谁呀
(Who is it)
Bot: 强·沃特，摩羯座的，你可以关注一下。
(Jon Voight, Capricorn, you can pay attention to him.)

Table 5: Example of multi-turn persona dialog generation results. The generation model can generate some persona-based response, while most of the time replied sentences are uncorrelated with persona, and the consistency selector prefers common response. We lay the blame on the open-domain dataset, although we filter training examples with more persona correlation. Bot persona |  persona0: 我最喜欢的歌手是毛不易
(My favorite singer is Buyi Mao)
persona1: 我真的很喜欢看《极限挑战》
(I really like watching “Extreme Challenge”)
persona2: 我喜欢吃脆饼干
(I like to eat crispy biscuits)
persona3: 我最喜欢的乐队是五月天
(My favorite band is Mayday)
---|---
Conversation |  User: 哈哈，是的还是第一部最好看了。
(Ha ha, yes, the first one is the best to watch.)
Bot: 是啊，我最喜欢的乐队是五月天，你知道这个乐队 吗？
(Yes, my favorite band is Mayday. Do you know this band?)
User: 我知道啊，我挺喜欢这个乐队的
(I know. I really like this band.)
Bot: 是吗？我也挺喜欢他们的，不过我更喜欢的是他们的歌 。
(Is it? I also like them very much, but I prefer their songs.)
User: 他们的歌很好听
(Their songs are very nice)
Bot: 是的，他们唱的歌都很好听，你有喜欢的歌手吗 ？
(Yes, they sing songs very well. Do you have any singers you like?)
User: 我喜欢毛不易
(I like Buyi Mao)
Bot: 是啊，毛不易的歌我都很喜欢。
(Yes, I like all Mao Buyi’s songs very much.)
User: 他唱歌很有感觉
(He sings very well)
Bot: 是的，我也是这么认为的，对了，你是做什么工作的啊？
(Yes, I think so. By the way, what do you do?)
User: 我是一名老师
(I am a teacher)
Bot: 老师也挺好的，可以教学生知识。
(The teacher is also very good and can teach students knowledge. )
User: 是的啊
(Yes)
Bot: 是啊，你平时喜欢做些什么啊？
(Yes, what do you like to do?)
User: 看看综艺，听听音乐啥的
(Watch variety shows, listen to music or something)
Bot: 综艺节目，《向往的生活》，你看过吗？
(Variety show, “The Longing for Life”, have you ever watched it?)

Table 6: Example of multi-turn recommendation dialog generation results. The generated responses are highly correlative with extra knowledge information. Note that the bot also follows the goal planning in this example, which we attribute to the goal joining ahead responses.  Situation |  聊天日期:2018-5-6，聊天时间:中午12:00，在学校
(Chat date: 2018-5-6, Chat time: 12:00 noon, at school )
---|---
Goal planning |  [1] 问日期 (User主动问 日期，Bot根据『参考知识』的『聊天日期』回答，然后User满足并好评)
……
[3] 电影推荐
(Bot主动，使用『娜娜的玫瑰战争』的某个评论当做推荐理由来推荐『娜娜的玫瑰战争』，User先问电影『国家地区、导演、类型、主演、口碑、评分』中的一个或多个，Bot回答，最终User接受)
[4]再见
([1] Ask the date(User actively asks the date, Bot answers according to the
”chat date” of ”Reference Knowledge”, and then User is satisfied and praised)
…… [3] Movie recommendation (Bot takes the initiative , Use a comment of
”Nana’s War of the Roses” as a recommendation reason to recommend ”Nana’s War
of the Roses”, the user first asks one or more of the movie ”country,
director, genre, starring, public praise, rating” , Bot answers, and finally
User accepts) –¿ [4] Goodbye )
User profile |  职业状态: 学生 (Occupation status: students)
年龄区间: 小于18 (Age range: less than 18)
……
喜欢的电影: 笑功震武林 (Favorite movies: Princess and Seven Kung Fu Masters)
拒绝: 音乐 (Rejection: music)
喜欢的明星: 谢娜 (Favorite star: Nana)
接受的电影: 大玩家 (Accepted movies: Super Player)
同意的新闻: 谢娜的新闻 (Agreed news: news about Nana)
Knowledge graph |  [ 聊天, 日期, 2018-5-6 ] (Chat, date, 2018-5-6)
[ 谢娜, 生日, 1981-5-6 ] (Nana, birthday, May 6, 1981)
[谢娜,主演,娜娜的玫瑰战争] (Nana, starring, Nana’s War of the Roses)
……
[娜娜的玫瑰战争,国家地区,中国] (Nana’s War of the Roses, country region, China)
[ 娜娜的玫瑰战争,类型,爱情喜剧] (Nana’s War of the Roses, type, romantic comedy)
[娜娜的玫瑰战争,主演,谢娜、李承铉、金伊彤] (Nana’s War of the Roses, starring, Nana、Lee Seung
Hyun、Yitong Jin)
[
娜娜的玫瑰战争,评论,《娜娜的玫瑰战争》不仅囊括种种恋爱难题，更通过谢娜的本色表演以及剧中的“痴心宝马男”李承铉帅炫造型，将生活中极为常见的爱情桥段以诙谐搞笑的形式释放出来。借力片中颇耐人寻味的爱情对白，和“拿得起放得下”的豁达爱情态度，宣告新时代年轻人“爱的保卫战”开始。
] (Nana’s War of the Roses, comments, “Nana’s War of the Roses” not only
covers all kinds of love problems, but also releases the most common love
scenes in life in the form of humor through the performance of Xie Na’s true
colors and the ”infatuated BMW man” Lee Seung Hyun’s handsome and dazzling
style With the thought-provoking love dialogue and the open-minded love
attitude of ”can afford to put it down”, the film announces the beginning of
the ”battle of love” for young people in the new era.)
Conversation |  User: 你好今天几号了啊？ (Hello, what’s the date today?)
Bot: 今天是2018年5月6日。 (Today is May 6, 2018.)
User: 知道啦，谢谢你 (I see. Thank you.)
Bot: 不客气，你知道吗？今天也是谢娜的生日哦，她是1986年5月6日出生的 。 (You’re welcome, you know? Today is
also the birthday of Nana. She was born on May 6th, 1986.)
User: 哇，真的吗，我最喜欢谢娜了 (Wow, really? I like Nana best)
Bot: 那我给你推荐一部她的电影吧，『娜娜的玫瑰战争』，不仅囊括种种恋爱难题，更通过谢娜的本色表演以及剧中的“痴心宝马男”李承铉
帅炫造型，将生活中极为常见的爱情桥段以诙谐搞笑的形式释放出来。 (I’d like to recommend her film, “Nana’s War
of the Roses”, which not only covers all kinds of love problems, but also
releases the most common love scenes in life in a humorous and funny way
through the natural performance of Nana and the ”infatuated BMW man” Lee Seung
Hyun’s handsome and dazzling style in the play.)
User: 这是什么类型的电影啊？ (What kind of movie is this?)
Bot: 是一部爱情喜剧片哦 。 (It’s a romantic comedy.)
User: 我喜欢看爱情喜剧片，那我就看这个了 (I like romantic comedies, so I will watch it.)
Bot: 你喜欢就好。 (I’m glad you like it.)