# Unpaired topological triply degenerate point for spin-tensor-momentum-
coupled ultracold atoms

Zhoutao Lei  Guangdong Provincial Key Laboratory of Quantum Metrology and
Sensing $\&$ School of Physics and Astronomy, Sun Yat-Sen University (Zhuhai
Campus), Zhuhai 519082, China  State Key Laboratory of Optoelectronic
Materials and Technologies, Sun Yat-Sen University (Guangzhou Campus),
Guangzhou 510275, China  Yuangang Deng<EMAIL_ADDRESS>Guangdong
Provincial Key Laboratory of Quantum Metrology and Sensing $\&$ School of
Physics and Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082,
China  Chaohong Lee<EMAIL_ADDRESS>Guangdong Provincial Key
Laboratory of Quantum Metrology and Sensing $\&$ School of Physics and
Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China  State
Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen
University (Guangzhou Campus), Guangzhou 510275, China

(August 27, 2024)

###### Abstract

The realization of triply degenerate points (TDPs) with exotic fermionic
excitations has opened a new perspective for the understanding of our nature.
Here we explore the coexistence of single unpaired TDP and multiple twofold
Weyl points (WPs) and propose an experimental scheme with ultracold
pseudospin-1 atomic gases trapped in optical lattices. We show that the
predicted single unpaired TDP emerged by the interplay of quadratic spin-
vector- and spin-tensor-momentum-coupling could possess a topological
nontrivial middle band. This exotic TDP with mirror symmetry breaking is
essential different from the recently observed TDPs that must appear in pairs
due to the Nielsen-Ninomiya theorem and host the topological trivial middle
band. Strikingly, the topologically protected Fermi-arc states directly
connect the unpaired TDP with additional WPs, in contrast to the conventional
Fermi-arc states that connect the same degeneracies of band degenerate points
with opposite chirality. Furthermore, the different types of TDPs with unique
linking structure of Fermi arcs can be readily distinguished by measuring spin
texture along high-symmetry lines of the system. Our scheme provides a
platform for emerging new fermions with exotic physical phenomena and
versatile device applications.

##  I introduction

Topological semimetal, a kind of gapless topological phase of matter, is
characterized by bulk band degenerate points (BDPs) and supports topologically
protected Fermi-arc surface states [1, 2]. The Weyl (Dirac) semimetal [3, 4,
5] acts as a paradigm that hosts pairs of twofold (fourfold) linear BDPs. The
excitations near these two types of BDPs provide analogs for relativistic
fermions with half-integer spins in quantum field theory. Unlike Weyl points
(WPs) carrying singly quantized topological charge, Dirac points can be
regarded as the overlap of two WPs with opposite chirality. Besides the
experimentally realized two- and fourfold BDPs for half-integer spin particles
[6, 7, 8, 9, 10, 11, 12, 13, 14], the new types of BDPs with different
degeneracies have been studied ranging from condensed matter physics to
ultracold quantum gases [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26].
These novel fermionic excitations that underlie exotic physical properties are
only constrained by the space-group symmetries of the crystal but are not
bound by the Poincaré symmetry imposed in high-energy physics [27, 28, 29, 30,
31].

Remarkably, the triply degenerate point (TDP), which represents a new fermion
for integer pseudospins without high-energy counterparts [19, 20, 22, 21, 23,
24, 25, 26], has led to tremendous advances in experiments [32, 33, 34]. In
general, the typical Hamiltonian of TDPs carrying a double topological charge
is made up of the linear order of spin-vector-momentum coupling (SVMC), e.g.,
$H=\mathbf{q}\cdot\mathbf{F}$, where $\mathbf{{q}}$ is the momentum deviation
from the TDPs and $\mathbf{F}$ is the spin-1 matrix. Meanwhile, a different
type of spin-tensor-momentum coupling (STMC) can be constructed in high-spin
ultracold atoms, which could offer a flexible and controllable platform for
exploring exotic topological quantum matters [35, 36, 37, 38, 39, 40, 41, 42].
Recently advances demonstrate that STMC [23, 43] and higher-order dispersion
[24, 25] spin-1 systems could generate TDPs with different topological
charges.

In their pioneer explorations, the TDPs appear in pairs which carry opposite
topological charges due to the “no-g” theorem [44, 45] and host a topological
trivial middle band as well. These properties lead to TDPs for integer-spin
systems possessing an even number of topological nontrivial bands, which is
similar to WPs for half-integer spin. Based on currently available
experimental techniques of spin-orbit coupling (SOC) [46, 47, 48, 49, 50] and
Raman-assisted tunneling in optical lattices [51, 52, 53, 54, 55, 56, 57], it
is of great interest to explore _whether single unpaired TDP with a
topological nontrivial middle band can be realized._ An affirmative answer
will open up unprecedented opportunities for exploring fundamental particles
and provide a broad physics community of applications [58, 59, 60].

In this work we investigate the coexistence of single unpaired TDP and
multiple twofold WPs in a spin-1 system and propose an experimental scheme to
realize these exotic TDPs with ultracold atoms trapped in a cubic optical
lattice. There appear different types of TDPs induced by quadratic SVMC and
STMC. In particular, a single unpaired TDP hosting a nontrivial middle band is
realized with mirror symmetry breaking. We should emphasize that this
unconventional TDP has never been observed before and distinguishes the
recently observed TDPs appearing in pairs or hosting a topological trivial
middle band [22, 61, 62, 63, 64, 32, 34, 33, 23, 24, 21, 25, 34]. Moreover,
several twofold WPs with single or double topological charge appear, which
demonstrates the coexistence of TDP and WPs on certain surfaces. The
coexistence of different types of fermions provides an unprecedented
opportunity to significantly enhance our understanding of the properties of
basic particles and explore exotic physical phenomena [65, 66, 67, 68, 69, 62,
63, 70, 71, 64, 72, 73, 74]. Remarkably, the nontrivial, topologically
protected Fermi arcs connecting different types of fermions (e.g., TDP and
WPs) are emerged. In particular, the proposed scheme has the advantage that
the generated exotic TDPs can be unambiguously distinguished via spin texture
measurements in experiments [49, 75, 76, 43].

This paper is organized as follows. In Sec. II we introduce the model and
Hamiltonian for construction of Raman-assisted STMC. Section III is devoted to
study the topology of the single unpaired TDPs. In Sec. IV we discuss the
topologically protected Fermi-arc surface states. In Sec. V we present an
accessible measurement method to distinguish the different types of TDPs.
Finally, a brief summary is given in Sec. VI.

##  II Model and Hamiltonain

We consider an ultracold gas of $N$ fermionic atoms subjected to a bias
magnetic field $\mathbf{B}$ along the quantization $z$ axis. The involved
atomic level structure is shown in Fig. 1(a), where five relevant magnetic
Zeeman sublevels with three ground states and two electronic excited states
are involved. Take 40K as an example [40]; the atomic transition from the
ground state $|\sigma\rangle(\sigma=\\{\uparrow,0\\})$ to the excited state
$|e_{\sigma}\rangle$ is driven by a pair of $\pi$-polarized standing-wave
lasers with Rabi frequencies $\Omega^{\prime}_{\pi}\sin(k_{L}x-k_{L}y)$ and
$i\Omega^{\prime}_{\pi}\sin(k_{L}x+k_{L}y)$, as shown in Fig. 1(b). Here
$k_{L}$ is the wave vector of the laser for the optical lattice with
$a=\pi/k_{L}$ being the lattice constant. Then the total Rabi frequency
$\Omega_{1}({\bf
r})=\Omega_{\pi}[\sin(k_{L}x)\cos(k_{L}y)+i\cos(k_{L}x)\sin(k_{L}y)]$ is
generated with $\Omega_{\pi}=\sqrt{2}e^{i\pi/4}\Omega^{\prime}_{\pi}$ [77]. To
generate SOC, the atomic transition
$|0\rangle\leftrightarrow|e_{\uparrow}\rangle$ and
$|e_{0}\rangle\leftrightarrow|\downarrow\rangle$ are driven by two
$\sigma$-polarized plane-wave lasers propagating along $z$ direction with the
same Rabi frequency $\Omega_{2}({\bf r})=\Omega_{3}({\bf r})$.

Figure 1: (a) Involved level structure. (b) Laser configurations for
generating quadratic SVMC and STMC.

For large atom-light detuning $\Delta$, two excited states can be
adiabatically eliminated to yield a pseudospin-1 system. In the tight-binding
regime with a sufficiently strong lattice potential, the lattice Hamiltonian
which dominates by the lowest $s$ orbit and nearest-neighbor hopping is given
by (see Appendix A for more details)

$\displaystyle{H_{0}}=\\!-\\!\\!$
$\displaystyle\sum_{\mathbf{j},\alpha=x,y}[it_{0}\hat{\psi}^{{\dagger}}_{\mathbf{j}}\hat{F}_{\alpha}\hat{\psi}_{\mathbf{j}+\mathbf{1}_{\alpha}}\\!+t\hat{\psi}^{{\dagger}}_{\mathbf{j}}(2\hat{F}^{2}_{z}-\hat{I})\hat{\psi}_{\mathbf{j}+\mathbf{1}_{\alpha}}\\!+{\rm
H.c.}]$ $\displaystyle-$ $\displaystyle
t\sum_{\mathbf{j}}(\hat{\psi}^{{\dagger}}_{\mathbf{j}}\mathcal{R}^{\sigma\sigma}_{z}\hat{\psi}_{\mathbf{j}+\mathbf{1}_{z}}+{\rm
H.c.})$ $\displaystyle+$
$\displaystyle\sum_{\mathbf{j}}[(\delta+\delta_{2})\hat{n}_{\mathbf{j},\uparrow}-(\delta-\delta_{2})\hat{n}_{\mathbf{j},\downarrow}],$
(1)

where
$\hat{\psi}_{\mathbf{j}}=[\hat{a}_{\mathbf{j},\uparrow},\hat{a}_{\mathbf{j},0},\hat{a}_{\mathbf{j},\downarrow}]^{T}$
and
$\hat{n}_{\mathbf{j},\sigma}=\hat{a}^{{\dagger}}_{\mathbf{j},\sigma}\hat{a}_{\mathbf{j},\sigma}$,
with $\hat{a}_{\mathbf{j},\sigma}$ being the annihilate operator for
spin-$\sigma$ component in the $\mathbf{j}$ site, $t_{0}$ is Raman-assisted
spin-flip hopping, $t$ is spin-independent hopping,
$\mathcal{R}_{z}=\exp(-i\phi\hat{F}_{z})$ is induced by the Peierls
substitution with Peierls phase $\phi=\sqrt{2}k_{L}a=\sqrt{2}\pi$, and
$\delta$ ($\delta_{2}$) is the effective tunable linear (quadratic) Zeeman
shift. Here the unit vectors along three directions are defined by:
$\mathbf{1}_{x}=(1,0,0)$, $\mathbf{1}_{y}=(0,1,0)$, and
$\mathbf{1}_{z}=(0,0,1)$.

Under the periodic boundary condition with translational invariance, the
Hamiltonian (II) in momentum space takes the form

$\displaystyle{H}_{0}(\mathbf{k})=\sum_{\alpha=x,y,z}d_{\alpha}(\mathbf{k})\hat{F}_{\alpha}+d^{\prime}_{z}(\mathbf{k})\hat{F}^{2}_{z}+\epsilon(\mathbf{k})\hat{I},$
(2)

where $\mathbf{k}=(k_{x},k_{y},k_{z})$ is in the first Brillouin zone (FBZ),
$d_{x}(\mathbf{k})=2t_{0}\sin(k_{x}a)$,
$d_{y}(\mathbf{k})=2t_{0}\sin(k_{y}a)$, $d_{z}(\mathbf{k})=\delta-
t_{1}\sin(k_{z}a)$,
$d^{\prime}_{z}(\mathbf{k})=\delta_{2}-4t\cos(k_{x}a)-4t\cos(k_{y}a)-t_{2}\cos(k_{z}a)$,
$\epsilon(\mathbf{k})=2t\cos(k_{x}a)+2t\cos(k_{y}a)-2t\cos(k_{z}a)$, and
$t_{1}=2t\sin(\sqrt{2}\pi)$ and $t_{2}=2t[\cos(\sqrt{2}\pi)-1]$ are introduced
for shorthand notation. Obviously, the first term
$d_{\alpha}(\mathbf{k})\hat{F}_{\alpha}$ represents the conventional SVMC, and
the second term $d^{\prime}_{z}\hat{F}^{2}_{z}$ denotes the interesting STMC
for our high-spin system.

Remarkably, the Hamiltonian (2) possesses a fourfold rotation symmetry around
the $z$ axis
$\mathcal{C}_{4z}H_{0}(k_{x},k_{y},k_{z})\mathcal{C}^{-1}_{4z}=H_{0}(-k_{y},k_{x},k_{z})$
with $\mathcal{C}_{4z}=e^{-i\pi/2\hat{F}_{z}}$. As a result, the emerged BDPs
will appear at high-symmetry lines due to $\mathcal{C}_{4z}$ symmetry. After
ignoring the identity term of $\epsilon(\mathbf{k})\hat{I}$, the interplay
between the SVMC $d_{z}(\mathbf{k})\hat{F}_{z}$ and the STMC
$d^{\prime}_{z}(\mathbf{k})\hat{F}^{2}_{z}$ will break mirror symmetry, which
is necessary to guarantee the appearance of a single unpaired topological TDP.
In fact, the separate term of SVMC and STMC is satisfying the different mirror
symmetry with ${H}_{0}(k_{x},k_{y},k_{z})={H}_{0}(k_{x},k_{y},\pi-k_{z})$ and
${H}_{0}(k_{x},k_{y},k_{z})={H}_{0}(k_{x},k_{y},-k_{z})$, respectively. For
simplicity, we set $t_{0}=t$, since the topological phase boundary is
independent of nonzero $t_{0}/t$. Thus the free parameters in our model reduce
to the tunable linear Zeeman shift $\delta$ and quadratic Zeeman shift
$\delta_{2}$.

##  III Triply Degenerate Point

To investigate the topology of our system, we calculate the energy spectrum by
diagonalizing Hamiltonian (2). The system has three energy bands, whose
eigenstates and eigenenergies are respectively denoted as
$|\psi_{n}(\mathbf{k})\rangle$ and $E_{n}$ with $n=\pm 1,0$ for the $n$th
band. In contrast to the previously found TDPs which appear in pairs [e.g.,
located at ${\bf K}_{+}=(0,0,k_{0})$ and ${\bf K}_{-}=(0,0,\pi/a-k_{0})$] with
opposite chirality [32, 22, 33, 34, 23, 24, 21, 25, 34] under the “no-go”
theorem [44, 45], we emphasize that the single TDP in our model is unpaired
due to the mirror symmetry breaking. It can be shown that the single unpaired
TDP is located at momentum ${\bf K}=(0,0,k_{0})$ when the linear and quadratic
Zeeman fields satisfy $\delta=t_{1}\sin(k_{0}a)$ and
$\delta_{2}=8t+t_{2}\cos(k_{0}a)$. Then the effective TDP Hamiltonian with
expanding Eq. (2) in the vicinity of ${\bf K}$ takes the form

$\displaystyle H(\mathbf{q})$ $\displaystyle=$
$\displaystyle\sum_{\alpha=x,y}v_{\alpha}q_{\alpha}\hat{F}_{\alpha}+(v_{z}q_{z}+v^{\prime}_{z}q^{2}_{z})\hat{F}_{z}$
(3)
$\displaystyle+(u_{z}q_{z}+u^{\prime}_{z}q^{2}_{z})\hat{F}^{2}_{z}+v_{0}(\mathbf{q})\hat{I},$

where $\mathbf{q}$ indicates the wave vector with respect to ${\bf K}$,
$v_{x}=v_{y}=2t_{0}a$, $v_{z}=-t_{1}\cos(k_{0}a)a$,
$u_{z}=t_{2}\sin(k_{0}a)a$, $v^{\prime}_{z}=({t_{1}/2t_{2}})au_{z}$,
$u^{\prime}_{z}=-({t_{2}}/{2t_{1}})av_{z}$, and
$v_{0}(\mathbf{q})=4t-2t\cos(k_{0}a)+2t\sin(k_{0}a)q_{z}a$. Remarkably, the
different types of SOC are generated in our model. Besides the linear
dispersion along the $q_{z}$ axis, both the quadratic dispersion of SVMC and
STMC terms emerge. Here, the linear and quadratic Zeeman fields can be treated
as the control knobs for SOC strengths $v_{z}$, $u_{z}$, $v^{\prime}_{z}$, and
$u^{\prime}_{z}$.

In general, the quantized topological charge of a TDP can be characterized by
the first Chern number,

$\displaystyle\mathcal{C}_{n}=\frac{1}{2\pi}\oint_{\mathbf{S}}\mathbf{\Omega}_{n}\cdot
d\mathbf{S},$ (4)

where $\mathbf{S}$ indicates a closed surface enclosing the TDP and
$\mathbf{\Omega}_{n}(\mathbf{k})=\mathbf{\nabla}_{\mathbf{k}}\times\langle\psi_{n}(\mathbf{k})|i\mathbf{\nabla}_{\mathbf{k}}|\psi_{n}(\mathbf{k})\rangle$
is the Berry curvature of the $n$th band.

Figure 2: (a) The phase diagram in $v_{z}$-$u_{z}$ parameter plane. (b)-(e)
The energy dispersion near TDP as a function of $q_{x}$ and $q_{z}$ with
$q_{y}=0$ for different phases. The black circle and blue square (magenta
diamond and red triangle) in (a) mark the parameters for (b) and (c) [(d) and
(e)], corresponding to different types of conventional (unconventional) TDPs.

In Fig. 2(a) we show the phase diagram in the $v_{z}$-$u_{z}$ parameter plane.
Each distinct TDP is characterized by a Chern number configuration,
$\\{\mathcal{C}_{-1},\mathcal{C}_{0},\mathcal{C}_{1}\\}$. As can be seen, the
Chern number for the $n$th band is $\mathcal{C}_{n}=-2nv_{z}/|v_{z}|$ when the
SOC strengths in Eq. (3) satisfy $|v_{z}/u_{z}|>1$. While for
$|u_{z}/v_{z}|>1$, the Chern number for the $n$th band is satisfying
$\mathcal{C}_{n}=-nv_{z}/|v_{z}|$, which is analogous to generation of
topological TDPs induced by STMC [23]. We should note that the SOC strength of
$v_{z}$ and $u_{z}$ are highly controlled by $\delta$ and $\delta_{2}$.

Figure 3: (a)–(d) The energy dispersion along the high-symmetry line
$k_{x}=k_{y}=0$ for the same parameters as in Figs. 2(b)–2(e), respectively.
(e–h) The corresponding in-plane Berry curvature of the lowest band
$\mathbf{\Omega}_{-1}$ in the $k_{y}=0$ plane. (i–l) The corresponding surface
spectral density on the $(0\overline{1}0)$ surface with Fermi energy shown as
a green dashed line in (a–d), respectively. Here the red- (black-) circle
(square) and triangle (diamond) denote the single (double) WPs and TDP which
hosts a positive (negative) topological charge, respectively. The white
numbers represent topological charge of BDPs.

To process further, we use the Chern number of lowest band
$\mathcal{C}=\mathcal{C}_{-1}$ to characterize the topological charge of TDP.
Therefore the single unpaired TDP carries a double (single) topological charge
with $\mathcal{C}=\pm 2$ ($\mathcal{C}=\pm 1$) in the region $|v_{z}/u_{z}|>1$
($|u_{z}/v_{z}|>1$). In Figs. 2(b) and 2(c), we plot the typical band spectra
for the double and single TDP phase, respectively. We show that the middle
bands for conventional TDPs are topological trivial with $\mathcal{C}_{0}=0$.
These results can be understood as follows. The quadratic terms in Hamiltonian
(3) do not influence the band closing and opening of bulk gap when
$|u_{z}/v_{z}|\neq 1$ and $v_{z}\neq 0$, which indicates the system topology
is captured by an effective Hamiltonian with only including linear terms,
$\widetilde{H}(\mathbf{q})=\sum_{\alpha=x,y,z}v_{\alpha}q_{\alpha}\hat{F}_{\alpha}+u_{z}q_{z}\hat{F}^{2}_{z}$.
The Hamiltonian $\widetilde{H}(\mathbf{q})$ preserves a linear relation
dependence of the momenta with
$\widetilde{H}(\mathbf{q})=-\widetilde{H}(-\mathbf{q})$. Thus the Berry
curvature is constrained by the relation
$\mathbf{\Omega}_{n}(\mathbf{q})=\mathbf{\Omega}_{-n}(\mathbf{-q})$, which
yields a topological trivial middle band with $\mathcal{C}_{0}=0$ and
$\mathcal{C}_{-1}=-\mathcal{C}_{1}$. Compared with conventional TDPs, we find
three types of unconventional TDPs exhibiting a topological nontrivial middle
band (${\cal C}_{0}\neq 0$) at $v_{z}=\pm u_{z}^{\prime}$ or $v_{z}=0$ [dashed
lines in Fig. 2(a)]. To further characterize these phases, we plot the typical
energy spectra $E({\bf q})$ for two of them, as shown in Figs. 2(d) and 2(e).
We emphasize that these unconventional TDPs with ${\cal C}_{0}\neq 0$ are
induced by the interplay of quadratic SVMC and STMC. Explicitly, the quadratic
SVMC ($v^{\prime}_{z}q^{2}_{z}\hat{F}_{z}$) breaks the relation
$\widetilde{H}(\mathbf{q})=-\widetilde{H}(-\mathbf{q})$ at the boundary of
topological phase transitions, which yields
$\mathcal{C}_{-1}\neq-\mathcal{C}_{1}$. Then the topological nontrivial middle
band with nonzero Chern number can be expected since
$\sum_{n}\mathcal{C}_{n}=0$, which is in contrast to the conventional TDPs
proposed in Refs. [22, 61, 62, 63, 64, 32, 33, 34, 23, 24, 21, 25, 34] that
possess topological trivial middle band. In the absence of the STMC
($u_{z}q_{z}\hat{F}^{2}_{z}$), we find that the topological nontrivial TDP
will reduce to a topological trivial TDP with ${\cal C}_{n}=0$.

##  IV Fermi Arcs of TDPs

Besides the unconventional and conventional TDPs, two types of twofold WPs
with different topological charges are observed in our system. Figures
3(a)–3(d) show the energy spectrum for different TDPs along the high-symmetry
line with $k_{x}=k_{y}=0$. There appear three or two WPs as well as a single
unpaired conventional or unconventional TDP in the FBZ, respectively.
Furthermore, we plot the Berry curvature of the lowest band
$\mathbf{\Omega}_{-1}$ for the four types of TDP phases in the $k_{z}$-$k_{x}$
plane, see Figs. 3(e)–3(h). The topological charges of WPs can be well defined
by the lower touching band topology. Explicitly, WPs with single
($\mathcal{C}=\pm 1$) and double ($\mathcal{C}=\pm 2$) topological charges are
called single WPs and double WPs, respectively. The emergence of twofold WPs
can be characterized by an effective two-band Hamiltonian as discussed in
Appendix B. As can be seen, TDP or WPs with positive (negative) Chern number
displays as a source (sink) of Berry curvature, which unambiguously
demonstrates its nontrivial topological properties. We also check that the
total topological charge of all BDPs for the $n$th energy band in the FBZ is
zero, that is, $\sum_{BDPs}\mathcal{C}_{n}=0$ [44, 45].

To gain more insight into TDPs, we explore the topological protected Fermi
arcs, which are the surface states crossing the Fermi energy. Figures
3(i)–3(l) display the typical surface spectra density for different types of
TDPs on the $(0\overline{1}0)$ surface by using the recursive Green’s function
method [78] (see Appendix C). Compared to earlier realized TDPs appearing in
pairs [32, 22, 33, 34, 23, 24, 21, 25, 34], the emerged single TDP is unpaired
due to mirror symmetry breaking. Interestingly, in our system the Fermi-arc
states will link the single unpaired TDP and multiple WPs, which is
essentially different from the observed coexistence of paired TDPs or WPs
where Fermi arcs link the same types of fermions in materials [32, 79]. It
clearly shows that each single unpaired TDP has a unique linking structure of
Fermi arcs.

We find that the number of Fermi arcs linking to a single unpaired TDP is
determined by its topological charge $\mathcal{C}$. Moreover, the Fermi arcs
directly link single unpaired TDPs with two (one) WPs for conventional
(unconventional) TDP phases, as shown in Figs. 3(i) and 3(j) [Figs. 3(k) and
3(l)]. It is shown that the conventional TDPs with $\mathcal{C}=-2$ and
$\mathcal{C}=1$ are both exhibiting two separate Fermi arcs with three nodes.
These Fermi-arc patterns [Figs. 3(i) and 3(j)] linking single unpaired TDP
with two WPs are yet to be studied, which is in contrast to the recently study
in Refs. [66, 62, 67, 63, 68, 64, 61] that Fermi arcs link the same number of
conventional TDPs with fourfold BDPs with opposite chirality. Of particular
interest, the unconventional TDP with single (double) topological charge
connects with one WP with opposite chirality and hosts one (two) Fermi arcs as
well, as displayed in Fig. 3(k) [Fig. 3(l)]. In fact, an unconventional TDP
can be treated as a combination of one conventional TDP and one WP (see
Appendix D), which could provide a new insight into unconventional TDP with a
topological nontrivial middle band. Remarkably, the observed exotic Fermi arcs
could act as the unique surface fingerprints for different types of TDPs.

Figure 4: The spin textures for different energy bands near a TDP along
$k_{x}=k_{y}=0$ for spin-$\uparrow$ (red$\bigtriangleup$), spin-$0$
($\bigcirc$), and spin-$\downarrow$ (blue$\Box$). The parameters are the same
as in Figs. 2(b)–(e).

##  V Determination of TDPs

In order to further reveal the topology of different TDPs, we calculate their
spin textures. We should emphasize that the eigenstates of helicity branches
for Hamiltonian (2) are always reduced to bare spin states at the high-
symmetry line $k_{x}=k_{y}=0$ due to $\mathcal{C}_{4z}$ symmetry. These
properties provide a unique opportunity to experiment measure the Chern number
of single unpaired TDPs through spin-resolved time-of-flight images in
ultracold quantum gases [49, 75, 76]. Figure 4 shows the typical spin textures
around four types of TDP phases at $k_{x}=k_{y}=0$. As can be seen, the bare
spin state of the middle band does not cause spin flips at the band crossing
for conventional double [Fig 4(a)] and single [Fig 4(b)] TDP, respectively,
which is consistent with the results of predicting a topological trivial
middle band, as displayed in Figs. 2(b) and 2(c). Interestingly, the bare spin
state of the lowest band will cause spin flips during the band crossing
process. The double (single) spin flips corresponding to the spin-angular-
momentum transition from $|\downarrow\rangle$ ($|0\rangle$) to
$|\uparrow\rangle$ ($|\downarrow\rangle$) is in connection with topological
double (single) TDPs with $\mathcal{C}=-2$ ($\mathcal{C}=1$).

Compared with the conventional TDPs, the bare spin state of the middle band
for unconventional TDPs is observed in the spin flips process, as shown in
Figs. 4(c) and 4(d). Remarkably, the Chern number for the topological
nontrivial middle band is equivalent to changing values of spin angular
momentum after spin flips. Analogously, the single and double TDPs correspond
to single ($|0\rangle\rightarrow|\uparrow\rangle$) and double
($|\uparrow\rangle\rightarrow|\downarrow\rangle$) spin flips for the lowest
band, respectively. Remarkably, both conventional and unconventional TDPs with
nonzero topological charges can be directly extracted by measuring spin-flip
processes along the high-symmetry line in experiments [49, 75, 76, 43]. The
advantage of our proposal is that the characterization of these novel
topological TDPs do not need to measure the Chern number of bands [56, 57] and
Bloch state tomography over the FBZ for ultracold quantum gasses [80, 81]. In
addition, the generation of Raman-assisted STMC can utilize the self-ordering
dynamic SOC in cavity quantum electrodynamics [82, 83, 84], where there is an
additional collective-emission-induced cooling mechanism which will facilitate
the experimental realization of many-body topological quantum matters.
Finally, we should emphasize that the Raman-induced heating can be greatly
mitigated by employing the long-lived excited states in alkali-earth atoms
[85, 86, 87].

##  VI Conclusions

Based upon the currently available techniques of Raman-induced SOC, we explore
the generation of exotic TDPs beyond the Nielsen-Ninomiya theorem and discuss
its experimental realization via ultracold pseudospin-1 atoms. We give the
phase diagram via analyzing the topological properties of these exotic TDPs.
It has been shown that the interplay of the quadratic SVMC and STMC gives rise
to the coexistence of topological single unpaired TDP and multiple twofold
WPs. In particular, the unconventional single and double TDPs with a
topological nontrivial middle band are observed. Interestingly, there emerge
four different patterns of topologically protected Fermi-arc states directly
linking TDPs and WPs, which corresponds to the coexistence of TDPs and WPs. As
to the experimental detection, an accessible measurement method to distinguish
different types of TDPs was proposed, which could facilitate the experimental
feasibility for studying topological semimetal in cold atoms. Finally, we
point out that our scheme can be extended to study exotic topological Fulde-
Ferrell superfluid in high-spin systems [77, 88, 89, 90].

###### Acknowledgements.

This work is supported by the National Key R$\&$D Program of China (Grant No.
2018YFA0307500), the NSFC (Grants No. 11874433, No. 12135018, No. 11874434,
and No. 12025509), the Key-Area Research and Development Program of GuangDong
Province under Grant No. 2019B030330001, and the Science and Technology
Program of Guangzhou (China) under Grant No. 201904020024.

##  Appendix A Lattice Hamiltonian

In this section we derive the lattice Hamiltonian in detail with the level
diagram and laser configuration given in Fig. 1 of the main text. As shown in
Fig. 1 of main text, the atomic transition from the ground state
$|\sigma\rangle(\sigma=\\{\uparrow,0\\})$ to the excited state
$|e_{\sigma}\rangle$ is driven by a pair of $\pi$-polarized standing-wave
lasers with Rabi frequencies $\Omega^{\prime}_{\pi}\sin(k_{L}x-k_{L}y)$ and
$i\Omega^{\prime}_{\pi}\sin(k_{L}x+k_{L}y)$, corresponding the atom-light
detuning $\Delta=\omega_{a}-\omega_{L}$, with $\omega_{L}$ being the laser
detuning. Then the total Rabi frequency $\Omega_{1}({\bf
r})=\Omega_{\pi}[\sin(k_{L}x)\cos(k_{L}y)+i\cos(k_{L}x)\sin(k_{L}y)]$ is
generated with $\Omega_{\pi}=\sqrt{2}e^{i\pi/4}\Omega^{\prime}_{\pi}$. To
generate spin-orbit coupling (SOC), the atomic transitions
$|0\rangle\leftrightarrow|e_{\uparrow}\rangle$ and
$|e_{0}\rangle\leftrightarrow|\downarrow\rangle$ are driven by
$\sigma$-polarized plane-wave lasers propagating along the $z$ direction with
the same Rabi frequency $\Omega_{2}({\bf r})=\Omega_{3}({\bf
r})=\Omega_{\sigma}e^{i\sqrt{2}k_{L}z}$, where the laser frequency is
$\omega_{L}+\Delta\omega_{L}$ and $\omega_{L}-\Delta\omega_{L}^{\prime}$,
respectively. Then the single-particle Hamiltonian is given by

$\displaystyle\bm{h}_{in}=$ $\displaystyle\omega_{Z}$
$\displaystyle\hat{b}^{{\dagger}}_{\uparrow}\hat{b}_{\uparrow}+\omega_{q}\hat{b}^{{\dagger}}_{0}\hat{b}_{0}-\omega_{Z}\hat{b}^{{\dagger}}_{\downarrow}\hat{b}_{\downarrow}$
(5) $\displaystyle+$
$\displaystyle(\omega_{a}+\omega^{\prime}_{Z})\hat{e}^{{\dagger}}_{\uparrow}\hat{e}_{\uparrow}+(\omega_{a}+\omega^{\prime}_{q})\hat{e}^{{\dagger}}_{0}\hat{e}_{0}$
$\displaystyle+$
$\displaystyle[e^{i\omega_{L}t}(\mathbf{\Omega^{*}_{1}}\hat{b}^{{\dagger}}_{\uparrow}\hat{e}_{\uparrow}+\mathbf{\Omega^{*}_{1}}\hat{b}^{{\dagger}}_{0}\hat{e}_{0})+{\rm
H.c.}]$ $\displaystyle+$
$\displaystyle[\mathbf{\Omega^{*}_{2}}e^{i(\omega_{L}+\Delta\omega_{L})t}\hat{b}^{{\dagger}}_{0}\hat{e}_{\uparrow}+{\rm
H.c.}]$ $\displaystyle+$
$\displaystyle[\mathbf{\Omega^{*}_{3}}e^{i(\omega_{L}-\Delta\omega_{L}^{\prime})t}\hat{b}^{{\dagger}}_{\downarrow}\hat{e}_{0}+{\rm
H.c.}],$

where $\hat{b}_{\sigma=\uparrow,0,\downarrow}$ and
$\hat{e}_{\sigma=\uparrow,0}$ are, respectively, the annihilation operators
for the ground and excited states. The atomic transition frequency for ground
states to excited states is indicated by $\omega_{a}$. $\omega_{Z}$ and
$\omega_{q}$ ($\omega^{\prime}_{Z}$ and $\omega^{\prime}_{q}$) are the linear
and quadratic Zeeman shift for the ground states (excited states),
respectively.

By introducing the rotating frame that is defined by the unitary
transformation $\tilde{\mathcal{U}}=e^{-iUt}$ with

$\displaystyle U=$
$\displaystyle(\omega_{q}+\Delta\omega_{L})\hat{b}^{{\dagger}}_{\uparrow}\hat{b}_{\uparrow}+\omega_{q}\hat{b}^{{\dagger}}_{0}\hat{b}_{0}$
(6)
$\displaystyle+(\omega_{q}+\Delta\omega_{L}^{\prime})\hat{b}^{{\dagger}}_{\downarrow}\hat{b}_{\downarrow}$
$\displaystyle+(\omega_{q}+\Delta\omega_{L}+\omega_{L})\hat{e}^{{\dagger}}_{\uparrow}\hat{e}_{\uparrow}+(\omega_{q}+\omega_{L})\hat{e}^{{\dagger}}_{0}\hat{e}_{0},$

then the Hamiltonian (5) reduces to

$\displaystyle\bm{h}_{in}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{\mathcal{U}}^{\dagger}\bm{h}_{in}\tilde{\mathcal{U}}-i\tilde{\mathcal{U}}^{\dagger}\frac{\partial}{\partial_{t}}\tilde{\mathcal{U}}$
(7) $\displaystyle=$
$\displaystyle\delta_{\uparrow}\hat{b}^{{\dagger}}_{\uparrow}\hat{b}_{\uparrow}-\delta_{\downarrow}\hat{b}^{{\dagger}}_{\downarrow}\hat{b}_{\downarrow}$
$\displaystyle+$
$\displaystyle(\Delta+\delta_{\uparrow}-\omega_{z}+\omega^{\prime}_{z})\hat{e}^{{\dagger}}_{\uparrow}\hat{e}_{\uparrow}+(\Delta-\omega_{q}+\omega^{\prime}_{q})\hat{e}^{{\dagger}}_{0}\hat{e}_{0}$
$\displaystyle+$
$\displaystyle[\mathbf{\Omega^{*}_{1}}\hat{b}^{{\dagger}}_{\uparrow}\hat{e}_{\uparrow}+\mathbf{\Omega^{*}_{1}}\hat{b}^{{\dagger}}_{0}\hat{e}_{0}+{\rm
H.c.}]$ $\displaystyle+$
$\displaystyle[\mathbf{\Omega^{*}_{2}}\hat{b}^{{\dagger}}_{0}\hat{e}_{\uparrow}+\mathbf{\Omega^{*}_{3}}\hat{b}^{{\dagger}}_{\downarrow}\hat{e}_{0}+{\rm
H.c.}].$

Here $\delta_{\uparrow}=\omega_{z}-\omega_{q}-\Delta\omega_{L}$ and
$\delta_{\downarrow}=\omega_{z}+\omega_{q}+\Delta\omega^{\prime}_{L}$ are the
two-photon detuning. As a result, we can get the Heisenberg equations about
different internal states

$\displaystyle i\frac{\partial\hat{b}_{\uparrow}}{\partial t}$
$\displaystyle=$
$\displaystyle\delta_{\uparrow}\hat{b}_{\uparrow}+\mathbf{\Omega^{*}_{1}}\hat{e}_{\uparrow},$
$\displaystyle i\frac{\partial\hat{b}_{0}}{\partial t}$ $\displaystyle=$
$\displaystyle\mathbf{\Omega^{*}_{1}}\hat{e}_{0}+\mathbf{\Omega^{*}_{2}}\hat{e}_{\uparrow},$
$\displaystyle i\frac{\partial\hat{b}_{\downarrow}}{\partial t}$
$\displaystyle=$
$\displaystyle-\delta_{\downarrow}\hat{b}_{\downarrow}+\mathbf{\Omega^{*}_{3}}\hat{e}_{0},$
$\displaystyle i\frac{\partial\hat{e}_{\uparrow}}{\partial t}$
$\displaystyle=$
$\displaystyle(\Delta+\delta_{\uparrow}-\omega_{z}+\omega^{\prime}_{z}-i\gamma)\hat{e}_{\uparrow}+\mathbf{\Omega_{1}}\hat{b}_{\uparrow}+\mathbf{\Omega_{2}}\hat{b}_{0},$
$\displaystyle i\frac{\partial\hat{e}_{0}}{\partial t}$ $\displaystyle=$
$\displaystyle(\Delta-\omega_{q}+\omega^{\prime}_{q}-i\gamma)\hat{e}_{0}+\mathbf{\Omega_{1}}\hat{b}_{0}+\mathbf{\Omega_{3}}\hat{b}_{\downarrow},$
(8)

where $\gamma$ is the spontaneous emission rate for the excited states. For
the large detuning limit, i.e., $|\Omega_{\sigma,\pi}/\Delta|\ll 1$,
$|\delta_{\uparrow,\downarrow}/\Delta|\ll 1$, and $|\gamma/\Delta|\ll 1$, we
can adiabatically eliminate the exited states, i.e.,
$i\dot{\hat{e}}_{\uparrow,0}=0$, which yields

$\displaystyle\hat{e}_{\uparrow}\approx$
$\displaystyle-\frac{\mathbf{\Omega_{1}}\hat{b}_{\uparrow}+\mathbf{\Omega_{2}}\hat{b}_{0}}{\Delta},$
$\displaystyle\hat{e}_{0}\approx$
$\displaystyle-\frac{\mathbf{\Omega_{1}}\hat{b}_{0}+\mathbf{\Omega_{3}}\hat{b}_{\downarrow}}{\Delta}.$
(9)

After substituting Eq. (A) to Eq. (A), we obtain the effective Heisenberg
equation about ground states:

$\displaystyle i\frac{\partial\hat{b}_{\uparrow}}{\partial t}=$
$\displaystyle\delta_{\uparrow}\hat{b}_{\uparrow}-\frac{1}{\Delta}\mathbf{\Omega^{*}_{1}}\mathbf{\Omega_{2}}\hat{b}_{0},$
$\displaystyle i\frac{\partial\hat{b}_{0}}{\partial t}=$
$\displaystyle-\frac{1}{\Delta}[\mathbf{\Omega_{1}}\mathbf{\Omega^{*}_{2}}\hat{b}_{\uparrow}+|\mathbf{\Omega_{2}}|^{2}\hat{b}_{0}+\mathbf{\Omega^{*}_{1}}\mathbf{\Omega_{3}}\hat{b}_{\downarrow}],$
$\displaystyle i\frac{\partial\hat{b}_{\downarrow}}{\partial t}=$
$\displaystyle-\delta_{\downarrow}\hat{b}_{\downarrow}-\frac{1}{\Delta}[\mathbf{\Omega_{1}}\mathbf{\Omega^{*}_{3}}\hat{b}_{0}+|\mathbf{\Omega_{3}}|^{2}\hat{b}_{\downarrow}].$
(10)

Here we assume the Raman coupling strength of the standing wave is
sufficiently weak with satisfying $\Omega_{\pi}/\Omega_{\sigma}\ll 1$.
Therefore the $\pi$-polarized laser-induced position-dependent Stark shift can
be safely ignored. Then the effective Hamiltonian for ground-states space is
given by

$\displaystyle\bm{h}_{in}=$
$\displaystyle\left(\begin{array}[]{ccc}\delta+\delta_{2}&-\frac{1}{\Delta}\mathbf{\Omega^{*}_{1}}\mathbf{\Omega_{2}}&0\\\
-\frac{1}{\Delta}\mathbf{\Omega_{1}}\mathbf{\Omega^{*}_{2}}&0&-\frac{1}{\Delta}\mathbf{\Omega^{*}_{1}}\mathbf{\Omega_{3}}\\\
0&-\frac{1}{\Delta}\mathbf{\Omega_{1}}\mathbf{\Omega^{*}_{3}}&-\delta+\delta_{2}\end{array}\right),$
(14)

where
$\delta=\frac{\delta_{\uparrow}+\delta_{\downarrow}}{2}+\frac{|\Omega_{\sigma}|^{2}}{2\Delta}$
($\delta_{2}=\frac{\delta_{\uparrow}-\delta_{\downarrow}}{2}-\frac{3|\Omega_{\sigma}|^{2}}{2\Delta}$)
is the effective tunable linear (quadratic) Zeeman shift.

Incorporating the center-of-mass motion, the system Hamiltonian reads

$\displaystyle\bm{h}_{0}=\left[\frac{\mathbf{p}^{2}}{2M}+\mathcal{U}_{ol}(\mathbf{r})\right]\hat{I}+\bm{h}_{in},$
(15)

where $M$ is the atomic mass, and $\hat{I}$ is the identity matrix. After
applying the gauge transformation $\\{|\uparrow\rangle\rightarrow
e^{-i\sqrt{2}k_{L}z}|\uparrow\rangle,|\downarrow\rangle\rightarrow
e^{i\sqrt{2}k_{L}z}|\downarrow\rangle\\}$, the effective single-particle atom-
light Hamiltonian becomes

$\displaystyle{\bm{h}}_{0}=$
$\displaystyle\frac{({\mathbf{p}}-{\mathbf{A}})^{2}}{2M}+M_{x}\mathbf{(r)}\hat{F}_{x}+M_{y}\mathbf{(r)}\hat{F}_{y}$
(16)
$\displaystyle+\delta\hat{F}_{z}+\delta_{2}\hat{F}^{2}_{z}+\mathcal{U}_{\rm{ol}}(\mathbf{r})\hat{I},$

where $\hat{F}_{x,y,z}$ is the spin-1 matrix, ${\mathbf{A}}=\sqrt{2}\hbar
k_{L}\hat{F}_{z}$ is the vector potential introduced by the gauge
transformation, $M_{x}(r)=\Omega\sin(k_{L}x)\cos(k_{L}y)$, and
$M_{y}(r)=\Omega\cos(k_{L}x)\sin(k_{L}y)$ with
$\Omega=-\sqrt{2}\Omega_{\pi}\Omega_{\sigma}/\Delta$.

Furthermore, we consider the atoms are deeply trapped by a spin-independent
red-detuned three-dimensional ($3$D) optical lattice $\mathcal{U}_{\rm
ol}(\mathbf{r})=-U_{\rm
ol}[\cos^{2}(k_{L}x)+\cos^{2}(k_{L}y)+\cos^{2}(k_{L}z)]$. Here $U_{\rm ol}$ is
the depth of lattice, and $k_{L}$ is the wave vector of the optical trapping
laser with the lattice constant $a=\pi/k_{L}$. When the lattice potential is
sufficiently strong, the system enters the tight-binding regime. By only
considering the lowest $s$ orbits and nearest-neighbor hopping, the lattice
Hamiltonian takes the form as

$\displaystyle{H_{0}}=\leavevmode\nobreak\ $
$\displaystyle\sum_{\sigma=\uparrow,\downarrow}\sum_{\mathbf{j},\zeta=\pm
1}(-1)^{(m+n)}\frac{t_{0}}{\sqrt{2}}(\zeta\hat{a}^{{\dagger}}_{\mathbf{j},\sigma}\hat{a}_{\mathbf{j}+\zeta\mathbf{1}_{x},0}-i\zeta\hat{a}^{{\dagger}}_{\mathbf{j},\sigma}\hat{a}_{\mathbf{j}+\zeta\mathbf{1}_{y},0}+{\rm
H.c.})+\sum_{\mathbf{j}}[(\delta+\delta_{2})\hat{n}_{\mathbf{j},\uparrow}-(\delta-\delta_{2})\hat{n}_{\mathbf{j},\downarrow}]$
$\displaystyle-$
$\displaystyle\sum_{\sigma=\uparrow,0,\downarrow}\sum_{\mathbf{j}}t(\hat{a}^{{\dagger}}_{\mathbf{j},\sigma}\hat{a}_{\mathbf{j}+\mathbf{1}_{x},\sigma}+\hat{a}^{{\dagger}}_{\mathbf{j},\sigma}\hat{a}_{\mathbf{j}+\mathbf{1}_{y},\sigma}+\hat{a}^{{\dagger}}_{\mathbf{j},\sigma}\mathcal{R}^{\sigma\sigma}_{z}\hat{a}_{\mathbf{j}+\mathbf{1}_{z},\sigma}+{\rm
H.c.}),$ (17)

where $\hat{a}_{\mathbf{j},\sigma}$ is the annihilate operator for the
$\sigma$ component in the $\mathbf{j}$ site, and
$\hat{n}_{\mathbf{j},\sigma}=\hat{a}^{{\dagger}}_{\mathbf{j},\sigma}\hat{a}_{\mathbf{j},\sigma}$.
Here, we define the 3D lattice index $\mathbf{j}\equiv{(m,n,l)}$ and unit
vectors along three directions: $\mathbf{1}_{x}=(1,0,0)$,
$\mathbf{1}_{y}=(0,1,0)$, and $\mathbf{1}_{z}=(0,0,1)$. Furthermore, $t=-\int
d{\mathbf{r}}w^{*}_{\mathbf{i}}({\mathbf{r}})[{\mathbf{p^{2}}}/(2M)+\mathcal{U}_{ol}(\mathbf{r})]w_{\mathbf{i}+\mathbf{1}_{\alpha}}({\mathbf{r}})$
is the spin-independent hopping along the $\alpha$ direction with
$\alpha=x,y,z$, while $t_{0}=\Omega\int
d{\mathbf{r}}w^{*}_{\mathbf{i}}({\mathbf{r}})|M_{\alpha^{\prime}}(r)|w_{\mathbf{i}+\mathbf{1}_{\alpha^{\prime}}}({\mathbf{r}})$
is the strength of spin-flipping hopping with $\alpha^{\prime}=x,y$, where
$w_{\mathbf{j}}({\mathbf{r}})\equiv w({\mathbf{r}}-{\mathbf{r}}_{\mathbf{j}})$
is the localized Wannier function of the lowest $s$ orbit. And
$\mathcal{R}_{z}=\exp(-i\sqrt{2}\pi\hat{F}_{z})$ is induced by the Peierls
substitution with Peierls phase $\sqrt{2}k_{L}a=\sqrt{2}\pi$.

After the gauge transformation,
$\hat{a}_{\mathbf{j},0}\rightarrow(-1)^{(m+n)}i\hat{a}_{\mathbf{j},0}$, we
obtain the lattice Hamiltonian provided in the main text,

$\displaystyle{H_{0}}=\\!-\\!\\!$
$\displaystyle\sum_{\mathbf{j},\alpha=x,y}[it_{0}\hat{\psi}^{{\dagger}}_{\mathbf{j}}\hat{F}_{\alpha}\hat{\psi}_{\mathbf{j}+\mathbf{1}_{\alpha}}\\!+t\hat{\psi}^{{\dagger}}_{\mathbf{j}}(2\hat{F}^{2}_{z}-\hat{I})\hat{\psi}_{\mathbf{j}+\mathbf{1}_{\alpha}}\\!+{\rm
H.c.}]$ $\displaystyle-$ $\displaystyle
t\sum_{\mathbf{j}}(\hat{\psi}^{{\dagger}}_{\mathbf{j}}\mathcal{R}^{\sigma\sigma}_{z}\hat{\psi}_{\mathbf{j}+\mathbf{1}_{z}}+{\rm
H.c.})$ $\displaystyle+$
$\displaystyle\sum_{\mathbf{j}}[(\delta+\delta_{2})\hat{n}_{\mathbf{j},\uparrow}-(\delta-\delta_{2})\hat{n}_{\mathbf{j},\downarrow}],$
(18)

with
$\hat{\psi}_{\mathbf{j}}=[\hat{a}_{\mathbf{j},\uparrow},\hat{a}_{\mathbf{j},0},\hat{a}_{\mathbf{j},\downarrow}]^{T}$.

##  Appendix B The effective Hamiltonian of WPs

To reveal the topological features of twofold WPs more directly, we derive
their effective two-band Hamiltonian in this section. We will demonstrate that
the effective two-band Hamiltonian of single WPs is made up of linear spin-
vector-momentum coupling, which is quadratic dispersion along the $x$ and $y$
directions for double WPs. Without loss of generality, we just calculate the
results about the single WP with momentum $(0,0,k_{SW}=0.08\pi/a)$ and double
WP with momentum $[0,0,k_{DW}=2\pi/(3a)]$ for the conventional single TDP
phase, respectively.

The Hamiltonian of the single WP in the low-energy limit with neglecting the
higher-order dispersion takes the form as (with ignoring a constant term)

$\displaystyle H_{SW}(\mathbf{q})$
$\displaystyle=\sum_{\alpha=x,y}2t_{0}aq_{\alpha}\hat{F}_{\alpha}-t_{1}\cos(k_{SW}a)aq_{z}\hat{F}_{z}$
$\displaystyle+t_{2}\sin(k_{SW}a)aq_{z}\hat{F}^{2}_{z}$
$\displaystyle+t_{1}[\sin(\pi/3)-\sin(k_{SW}a)](\hat{F}_{z}-\hat{F}^{2}_{z}),$
(19)

where $\mathbf{q}$ indicates the wave vector with respect to the location of
single WP $(0,0,k_{SW}=0.08\pi/a)$, $t_{1}=2t\sin(\sqrt{2}\pi)$, and
$t_{2}=2t[\cos(\sqrt{2}\pi)-1]$. Correspondingly, the Heisenberg equation
reads

$\displaystyle i\frac{\partial\hat{b}_{\uparrow}}{\partial t}$
$\displaystyle=$
$\displaystyle[t_{2}\sin(k_{SW}a)-t_{1}\cos(k_{SW}a)]aq_{z}\hat{b}_{\uparrow}$
$\displaystyle+\sqrt{2}t_{0}a({q_{x}-iq_{y}})\hat{b}_{0},$ $\displaystyle
i\frac{\partial\hat{b}_{0}}{\partial t}$ $\displaystyle=$
$\displaystyle\sqrt{2}t_{0}a[({q_{x}+iq_{y}})\hat{b}_{\uparrow}+({q_{x}-iq_{y}})\hat{b}_{\downarrow}],$
$\displaystyle i\frac{\partial\hat{b}_{\downarrow}}{\partial t}$
$\displaystyle=$ $\displaystyle\sqrt{2}t_{0}a({q_{x}+iq_{y}})\hat{b}_{0}$ (20)
$\displaystyle+[t_{2}\sin(k_{SW}a)+t_{1}\cos(k_{SW}a)]aq_{z}\hat{b}_{\downarrow}$
$\displaystyle-2t_{1}[\sin(\pi/3)-\sin(k_{SW}a)]\hat{b}_{\downarrow}.$

Obviously, the energies of spin-$\uparrow$ and spin-$0$ components equal zero
when $\mathbf{q}=0$, so these two spin components degenerate at this single
WP. Then we can adiabatically eliminate the spin-down component to get the
effective two-band Hamiltonian, and one may set
$i\dot{\hat{b}}_{\downarrow}=0$. As a result, we have

$\displaystyle\hat{b}_{\downarrow}\approx\frac{\sqrt{2}t_{0}a({q_{x}+iq_{y}})}{2t_{1}[\sin(\pi/3)-\sin(k_{SW}a)]}\hat{b}_{0},$
(21)

where we ignore the term in proportion to $q_{z}$ in the denominator under the
limit $\mathbf{q}\rightarrow 0$. By applying the condition $t_{0}=t$, Eq. (21)
can be simplified to

$\displaystyle\hat{b}_{\downarrow}\approx-0.49({q_{x}+iq_{y}})a\hat{b}_{0}.$
(22)

After substituting Eq. (22) to Eq. (B), we obtain the effective Heisenberg
equation about degenerating states,

$\displaystyle i\frac{\partial\hat{b}_{\uparrow}}{\partial t}$
$\displaystyle=$
$\displaystyle[t_{2}\sin(k_{SW}a)-t_{1}\cos(k_{SW}a)]aq_{z}\hat{b}_{\uparrow}$
$\displaystyle+\sqrt{2}t_{0}a({q_{x}-iq_{y}})\hat{b}_{0},$ $\displaystyle
i\frac{\partial\hat{b}_{0}}{\partial t}$ $\displaystyle=$
$\displaystyle\sqrt{2}t_{0}a[({q_{x}+iq_{y}})\hat{b}_{\uparrow}-0.49({q_{x}+iq_{y}})a\hat{b}_{0}].$
(23)

After neglecting the unit term and higher-order dispersions, we can get the
effective two-band Hamiltonian describing the low-energy excitation near this
single WP,

$\displaystyle H(\mathbf{q})=$
$\displaystyle\sqrt{2}t_{0}a\sum_{\alpha=x,y}q_{\alpha}\hat{\sigma}_{\alpha}$
(24)
$\displaystyle+\frac{[t_{2}\sin(k_{SW}a)-t_{1}\cos(k_{SW}a)]aq_{z}}{2}\hat{\sigma}_{z},$

where $\hat{\sigma}_{\alpha}$ indicates the Pauli matrix.

Obviously, the Hamiltonian (24) is dominated by the linear dispersions and
describes a singly WP with topological charge $\mathcal{C}=1$ considering the
coefficient of the second term
$[t_{2}\sin(k_{SW}a)-t_{1}\cos(k_{SW}a)]/{2}\approx 0.62>0$. We should note
that the other singly WPs have similar results.

In contrast, the Hamiltonian of the double WP with momentum
$[0,0,k_{DW}=2\pi/(3a)]$ in the low-energy limit is

$\displaystyle H_{DW}(\mathbf{q})=$ $\displaystyle
2t_{0}a\sum_{\alpha=x,y}q_{\alpha}\hat{F}_{\alpha}-t_{1}\cos(k_{DW}a)aq_{z}\hat{F}_{z}$
(25)
$\displaystyle+t_{2}\sin(k_{DW}a)aq_{z}\hat{F}^{2}_{z}+t_{2}(\hat{F}^{2}_{z}-\hat{I}),$

where $\mathbf{q}$ indicates the wave vector with respect to this double WP.
Then the corresponding effective Heisenberg equation reads

$\displaystyle i\frac{\partial\hat{b}_{\uparrow}}{\partial t}$
$\displaystyle=$
$\displaystyle[t_{2}\sin(k_{DW}a)-t_{1}\cos(k_{DW}a)]aq_{z}\hat{b}_{\uparrow}$
$\displaystyle+\sqrt{2}t_{0}a({q_{x}-iq_{y}})\hat{b}_{0},$ $\displaystyle
i\frac{\partial\hat{b}_{0}}{\partial t}$ $\displaystyle=$
$\displaystyle\sqrt{2}t_{0}a[({q_{x}+iq_{y}})\hat{b}_{\uparrow}+({q_{x}-iq_{y}})\hat{b}_{\downarrow}]-t_{2}\hat{b}_{0},$
$\displaystyle i\frac{\partial\hat{b}_{\downarrow}}{\partial t}$
$\displaystyle=$ $\displaystyle\sqrt{2}t_{0}a({q_{x}+iq_{y}})\hat{b}_{0}$ (26)
$\displaystyle+[t_{2}\sin(k_{DW}a)+t_{1}\cos(k_{DW}a)]aq_{z}\hat{b}_{\downarrow}.$

In contrast to single WPs, the spin-$\uparrow$ and -$\downarrow$ components
degenerate at this double WP (the energies of these two spin components equal
to zero when $\mathbf{q}=0$). We can get the effective two-band Hamiltonian
through adiabatically eliminating the spin-$0$ component, i.e.,
$i\dot{\hat{b}}_{0}=0$, which yields

$\displaystyle\hat{b}_{0}=\frac{\sqrt{2}t_{0}a}{t_{2}}[(q_{x}+iq_{y})\hat{b}_{\uparrow}+(q_{x}-iq_{y})\hat{b}_{\downarrow}],$
(27)

and by using $t_{0}=t$, we have

$\displaystyle\hat{b}_{0}\approx-0.56a[(q_{x}+iq_{y})\hat{b}_{\uparrow}+(q_{x}-iq_{y})\hat{b}_{\downarrow}].$
(28)

Substituting Eq. (28) into Eq. (B), one can derive an effective Heisenberg
equation for the degenerating space:

$\displaystyle i\frac{\partial\hat{b}_{\uparrow}}{\partial t}$
$\displaystyle=$
$\displaystyle[t_{2}\sin(k_{DW}a)-t_{1}\cos(k_{DW}a)]aq_{z}\hat{b}_{\uparrow}$
$\displaystyle-0.79t_{0}a^{2}[(q^{2}_{x}+q^{2}_{y})\hat{b}_{\uparrow}+(q_{x}-iq_{y})^{2}\hat{b}_{\downarrow}],$
$\displaystyle i\frac{\partial\hat{b}_{\downarrow}}{\partial t}$
$\displaystyle=$
$\displaystyle-0.79t_{0}[(q_{x}+iq_{y})^{2}\hat{b}_{\uparrow}+(q^{2}_{x}+q^{2}_{y})\hat{b}_{\downarrow}]$
(29)
$\displaystyle+[t_{2}\sin(k_{DW}a)+t_{1}\cos(k_{DW}a)]aq_{z}\hat{b}_{\downarrow}.$

As a result, the effective two-band Hamiltonian for the double WP reads

$\displaystyle H(\mathbf{q})=$
$\displaystyle-0.79t_{0}[({q^{2}_{x}-q^{2}_{y}})\hat{\sigma}_{x}+2{q_{x}q_{y}}\hat{\sigma}_{y}]$
(30) $\displaystyle-t_{1}\cos(k_{DW}a)aq_{z}\hat{\sigma}_{z}.$

As can be seen, the obtained Hamiltonian is dominated by the quadratic
dispersions along $x$ and $y$ directions, in contrast to the Hamiltonian (24).
We find that this double WP carries a double topological charge with
$\mathcal{C}=-2$.

##  Appendix C Surface spectral function

In this section, we present how to get the surface spectral function by using
the recursive Green s function method [78]. To gain more insight into the
topologically protected Fermi arcs, we consider the surface states of the
system by imposing a hard-wall confinement along the $y$ direction (open
boundary condition) direction. After the Fourier transformation along $x$ and
$z$ direction (periodic boundary conditions), the effective one-dimensional
Hamiltonian can be obtained,

$\displaystyle{H_{y}}=$
$\displaystyle\sum_{j}h\hat{\varphi}^{{\dagger}}_{j}\hat{\varphi}_{j}+(u\hat{\varphi}^{{\dagger}}_{j}\hat{\varphi}_{j+1}+{\rm
H.c.}),$ (31)

where
$\hat{\varphi}_{j}=[\hat{a}_{j,\uparrow},\hat{a}_{j,0},\hat{a}_{j,\downarrow}]^{T}$
is the annihilate operator in the $j$th site along the $y$ direction, $h$ is
the effective local Hamiltonian, and $u$ represents the effective nearest-
neighbor intersite connection (see Fig. 5). Explicitly, we have

$\displaystyle h=$ $\displaystyle
2t_{0}\sin(k_{x}a)\hat{F}_{x}+[2t\cos(k_{x}a)-2t\cos(k_{z}a)]\hat{I}$
$\displaystyle+[\delta_{2}-4t\cos(k_{x}a)-t_{2}\cos(k_{z}a)]\hat{F}^{2}_{z}$
$\displaystyle+[\delta-t_{1}\sin(k_{z}a)]\hat{F}_{z},$ $\displaystyle u=$
$\displaystyle-it_{0}\hat{F}_{y}-2t\hat{F}^{2}_{z}+t\hat{I}.$

Figure 5: The structure describing the effective one-dimensional chain.

To get the surface spectral function about the $(0\overline{1}0)$ surface, we
first divide the system to the device region and semi-infinite lattices, which
are labeled by red dots and blue dots, as shown in Fig. 5. Explicitly, the
device Green function satisfies

$\displaystyle G_{D}(E)=[E^{+}-\hat{H}_{D}-\Sigma_{R}(E)]^{-1},$ (32)

where the $E^{+}=E+i\eta$, $\hat{H}_{D}$ is the Hamiltonian describing the
uncoupling device region and $\Sigma_{R}(E)$ is the self-energy matrix
describing the effects of the semi-infinite lattices, which can be obtained
through the surface Green function $G_{0,0}$,

$\displaystyle\Sigma_{R}(E)={\cal U}G_{0,0}{\cal U}^{{\dagger}},$ (33)

where the operator ${\cal U}$ describes the coupling between the device region
and the semi-infinite lattices. Once the device Green function is obtained,
the surface spectral function can be calculated,

$\displaystyle\rho(E)=-\frac{1}{\pi}\Im[\sum_{s}G_{D}(s,s;E)],$ (34)

where $s$ indicates sites belonging to the surface of the device region (left
end). Thus the essential step is to calculate the surface Green function in
our system.

Furthermore, we will use the recursive Green function method to calculate the
surface Green function $G_{0,0}$ [78]. The Green function for the semi-
infinite lattices satisfies the equation

$\displaystyle[(E^{+}-H_{R})G(E)]_{n,m}=\delta_{n,m},$ (35)

where $H_{R}$ is defined over semi-infinite lattices and can be divided to
local Hamiltonian $h$ and connection operators $u$. Then a sequence of results
can be obtained,

$\displaystyle(E^{+}-h)G_{0,0}=$ $\displaystyle I+uG_{1,0}$
$\displaystyle(E^{+}-h)G_{1,0}=$ $\displaystyle uG_{2,0}+u^{{\dagger}}G_{0,0}$
$\displaystyle\vdots$ $\displaystyle(E^{+}-h)G_{n,0}=$ $\displaystyle
uG_{n+1,0}+u^{{\dagger}}G_{n-1,0},$ (36)

which yields

$\displaystyle[E$
${}^{+}-h-u(E^{+}-h)^{-1}u^{{\dagger}}]G_{0,0}=I+u(E^{+}-h)^{-1}uG_{2,0},$
$\displaystyle[E^{+}-h-u(E^{+}-h)^{-1}u^{{\dagger}}-u^{{\dagger}}(E^{+}-h)^{-1}u]G_{n,0},$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
=u(E^{+}-h)^{-1}uG_{n+2,0}+u^{{\dagger}}(E^{+}-h)^{-1}u^{{\dagger}}G_{n-2,0}.$

To process further, a new recursion equation about the event site reads

$\displaystyle(E^{+}-\varepsilon^{s}_{1})G_{0,0}=$ $\displaystyle
I+\alpha_{1}G_{2,0}$ $\displaystyle(E^{+}-\varepsilon_{1})G_{2,0}=$
$\displaystyle\alpha_{1}G_{4,0}+\beta_{1}G_{0,0}$ $\displaystyle\vdots$
$\displaystyle(E^{+}-\varepsilon_{1})G_{2n,0}=$
$\displaystyle\alpha_{1}G_{2(n+1),0}+\beta_{1}G_{2(n-1),0},$ (38)

with

$\displaystyle\alpha_{1}=$ $\displaystyle u(E^{+}-h)^{-1}u,$
$\displaystyle\beta_{1}=$ $\displaystyle
u^{{\dagger}}(E^{+}-h)^{-1}u^{{\dagger}},$ $\displaystyle\varepsilon^{s}_{1}=$
$\displaystyle h+u(E^{+}-h)^{-1}u^{{\dagger}},$
$\displaystyle\varepsilon_{1}=$ $\displaystyle
h+u^{{\dagger}}(E^{+}-h)^{-1}u.$

By repeating this process $k$ times, one can derive the following equations:

$\displaystyle(E^{+}-\varepsilon^{s}_{k})G_{0,0}=$ $\displaystyle
I+\alpha_{k}G_{2^{k},0}$ $\displaystyle(E^{+}-\varepsilon_{k})G_{2^{k},0}=$
$\displaystyle\alpha_{k}G_{2^{k}\cdot 2,0}+\beta_{k}G_{0,0}$
$\displaystyle\vdots$ $\displaystyle(E^{+}-\varepsilon_{k})G_{2^{k}\cdot
n,0}=$
$\displaystyle\alpha_{k}G_{2^{k}\cdot(n+1),0}+\beta_{k}G_{2^{k}\cdot(n-1),0},$

with

$\displaystyle\alpha_{k}=$
$\displaystyle\alpha_{k-1}(E^{+}-\varepsilon_{k-1})^{-1}\alpha_{k-1},$
$\displaystyle\beta_{k}=$
$\displaystyle\beta_{k-1}(E^{+}-\varepsilon_{k-1})^{-1}\beta_{k-1},$
$\displaystyle\varepsilon^{s}_{k}=$
$\displaystyle\varepsilon_{k-1}+\alpha_{k-1}(E^{+}-\varepsilon_{k-1})^{-1}\beta_{k-1},$
$\displaystyle\varepsilon_{k}=$
$\displaystyle\varepsilon^{s}_{k-1}+\beta_{k-1}(E^{+}-\varepsilon_{k-1})^{-1}\alpha_{k-1}.$

When $||\alpha_{k}||$ and $||\beta_{k}||$ are sufficiently small, the surface
Green function can be obtained via

$\displaystyle G_{0,0}\approx(E^{+}-\varepsilon^{s}_{k})^{-1}.$ (40)

Figure 6: (a)–(d) The surface spectral density of the $(0\overline{1}0)$
surface at the $k_{z}$-$E$ plane with $k_{x}=0$ corresponding to the
parameters used in Figs. 2(b)–2(e) of the main text. (e)–(p) The surface
spectral density at the surface Brillouin zone with the Fermi energy from the
vicinity of TDP to WP shown as green dashed lines in (a)–(d). The red-
(black-) circle (square) and triangle (diamond) denote the singly (doubly)
quantized WPs and TDP with a positive (negative) topological charge,
respectively. The white numbers represent topological charge of BDPs.

In summary, the surface spectral function can be obtained through the
following steps: (I) Using the recursion process Eqs. (C)–(C) to get
$\varepsilon^{s}_{k}$. (II) Using Eq. (40) to obtain the surface Green
function. (III) Using Eqs. (32) and (33) to yield the device Green function.
(IV) Using Eq. (34) to calculate the surface spectral function of
$(0\overline{1}0)$ surface. In addition, the surface spectral function of the
$(010)$ can also be calculated by using the same methods with putting the
device region on the right end.

##  Appendix D Fermi arcs of TDPs

In this section we present the details of the surface spectral density for
different Fermi energies. Figures 6(a)–6(d) shows the surface spectral density
at the $k_{z}$-$E$ plane for four types of TDP phases using the recursive
Green’s function, respectively. Obviously, there are several surface states
with energy between the middle and upper band. In contrast to the surface
states below the middle band just depending on Chern number
$\mathcal{C}_{-1}$, the surface states above the middle band are determined by
the sum of the lower and middle bands’ Chern number
$(\mathcal{C}_{-1}+\mathcal{C}_{0})$. In particular, surface states always
link to the BDPs with opposite topological charges, corresponding to the Chern
number equal to $|\mathcal{C}_{-1}+\mathcal{C}_{0}|$. As to the doubly
conventional TDP, there are two surface states above the middle band which
both link the conventional TDP and the double WP with $\mathcal{C}=2$. By
contrast, there is only one surface state connecting the TDP and WP between
the middle and higher band for the singly conventional TDP and unconventional
TDP phases.

Figures 6(b)–6(d) display the evolution of the Fermi arc for different TDPs.
We begin with Fig. 6(b) corresponding to $k_{0}a=\pi/3$, where there is a
single conventional TDP with $\mathcal{C}=1$, one single WP with Chern number
$\\{1,-1,0\\}$, and a double WP with $\\{-2,2,0\\}$ for the middle and lowest
band. When $k_{0}$ becomes larger, corresponding to the coupling strength
$v_{z}$ ($u_{z}$) becoming large (small), the momentum difference between the
TDP and the double WP will become smaller. Finally, these two BDPs will be
converged when $k_{0}a=\pi/2$. Therefore a singly unconventional TDP appears,
corresponding with its Chern number $\\{-1,2,-1\\}=\\{1,0,-1\\}+\\{-2,2,0\\}$
and giving rise to the surface states shown in Fig. 6(c). Analogously, the
single conventional TDP will be converged into a single WP when $k_{0}a=\pi/3$
decreases to $k_{0}a=\arctan{t_{1}/t_{2}}$, which induces the double
unconventional TDP ($\\{2,-1,-1\\}=\\{1,0,-1\\}+\\{1,-1,0\\}$) with surface
states shown in Fig. 6(d). These results reveal that the number of BDPs is
different for conventional and unconventional TDP phases, which corresponds to
the distinctly linking structure of the Fermi arcs.

To complete the results of surface spectral density as shown in Figs.
3(i)–3(l) of the main text, we further plot the Fermi arcs with Fermi energies
closer to twofold WPs in Figs. 6(e)–6(p). As can be seen, when the Fermi
energy is far away from the TDP but close to the WPs, the bulk Fermi surface
projections near the TDP (WP) are growing (decreasing). Finally, we find that
the Fermi arcs will link to the twofold WPs directly with tuning the Fermi
energy, as shown in Figs. 6(m)–6(p).

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volume={71}, date={2019}, number={2}, pages={259–277}, issn={0041-5995}, }}
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