ORCiD: 0000-0003-4370-473X

# Topological magnons in one-dimensional ferromagnetic Su-Schrieffer-Heeger
model with anisotropic interaction

Peng-Tao Wei1,2 Jin-Yu Ni1,2 Xia-Ming Zheng3,1 Da-Yong Liu3,1<EMAIL_ADDRESS>Liang-Jian Zou1<EMAIL_ADDRESS>1 Key Laboratory of Materials Physics,
Institute of Solid State Physics, HFIPS, Chinese Academy of Sciences, Hefei
230031, China  2 Science Island Branch of Graduate School, University of
Science and Technology of China, Hefei 230026, China  3 Department of Physics,
School of Sciences, Nantong University, Nantong 226019, China

###### Abstract

Topological magnons in a one-dimensional (1D) ferromagnetic (FM) Su-
Schrieffer-Heeger (SSH) model with anisotropic exchange interactions are
investigated. Apart from the intercellular isotropic Heisenberg interaction,
the intercellular anisotropic exchange interactions, i.e. Dzyaloshinskii-
Moriya interaction (DMI) and pseudo-dipolar interaction (PDI), also can induce
the emergence of the non-trivial phase with two degenerate in-gap edge states
separately localized at the two ends of the 1D chain, while the intracellular
interactions instead unfavors the topological phase. The interplay among them
has synergistic effects on the topological phase transition, very different
from that in the two-dimensional (2D) ferromagnet. These results demonstrate
that the 1D magnons possess rich topological phase diagrams distinctly
different from the electronic version of the SSH model and even the 2D
magnons. Due to the lower dimensional structural characteristics of this 1D
topological magnonic system, the magnonic crystals can be constructed from
bottom to top, which has important potential applications in the design of
novel magnonic devices.

##  I Introduction

Since the discovery of quantum Hall effect PRL45-494 ; PRL48-1559 ; PRL49-405
; RMP58-519 , the topological phase of matter has attracted extensive
attention. Although the topological phase of matter is first found in
electronic systems, it is later extended to the field of quasi-particles, such
as phonons PRL105-225901 , photons PRL100-013904 ; NM12-233  and even magnons
Sci329-297 , etc. Katsura et al. PRL104-066403  theoretically predicted the
thermal Hall effect of magnons. Subsequently, Onose et al. Sci329-297
experimentally confirms the existence of magnon Hall effect in Lu2V2O7 with
pyrochlore structure, which sets off an upsurge of studying the topological
properties of magnons. So far, various non-trivial topological phases of
magnons for different crystal structures and magnetic interactions had been
presented. In two-dimensional (2D) system, it is found that the anisotropic
magnetic interaction, such as the magnetic dipole-dipole interaction
PRB87-174402 ; PRB90-104417 ; PRB98-224409 , Dzyaloshinskii-Moriya interaction
(DMI) PRB97-174413 ; PRB99-214424 ; PRB99-174412 ; JPCM29-185801  and pseudo-
dipolar interaction (PDI) PRB95-014435 ; PRApp9-024029  rather than the
isotropic magnetic interaction (Heisenberg-type exchange interaction), usually
produce the non-trivial topological phase of magnon in different types of
lattices, e.g. honeycomb lattice, Kagome lattice and triangular lattice, etc.
These anisotropic interactions often open an energy gap and lead to
topological magnon insulator, and can be seen as effective spin-orbit
couplings in electronic topological insulator. Due to the inherent topological
protection, the topological magnons generally have a wider application
prospect than the traditional ones. Due to the abundance of magnetic materials
and magnetic structures EPL103-47010 , as well as the manipulation of various
magnetic interactions, magnons have become an important platform for
topological physics and quantum technology PR915-1 ; ARCMP13-171 . As a
consequence, it has great potential application for designing novel quantum
devices in magnonics JPD43-264001 ; NP11-453 .

As we know, there also are abundant topological phases in one-dimensional (1D)
electronic systems, e.g. Su-Schrieffer-Heeger (SSH) model and Kitaev chain,
etc. Indeed unlike electrons, the magnon topological phases are easier to be
realized by regulating magnetic interactions. Although the 2D magnons are
extensively studied in previous years and numerous topological magnon states
(e.g. magnonic topological insulator, magnonic topological semimetal with
Dirac or Weyl magnon, etc.) are discovered, the topological properties of
magnons in 1D system remain unclear. We can expect more abundant topological
phases in 1D magnon systems than in electronic systems due to the rich
regulation of magnetic interactions. We notice that recently Pirmoradian et
al. PRB98-224409  constructed a 1D magnetosphere chain PRL42-1698 , in which
there is only dipole-dipole interaction rather than Heisenberg-type
interaction between macrospins on the magnetospheres, and they discovered
topological magnons by manipulating the external magnetic field. As a matter
of fact, the magnetosphere chain they studied is a classical magnetic
macrosystem. However, the mechanism to induce intrinsic topological magnons
without external field is still lacking, especially the topological properties
in the lattice model of the realistic magnetic materials is deserved to
explore.

In this paper, here we investigate the possible magnonic topological phases
and the related topological phase transitions in a 1D ferromagnetic (FM) SSH
model by manipulating both the isotropic Heisenberg exchange interaction and
the anisotropic exchange interactions, including DMI and PDI. To characterize
the topological phase, we adopt the Zak phase for the infinite system and the
real-space topological number for the finite system, respectively. We first
investigate the pure Heisenberg-type interaction, and find that the
topological phase can be realized by adjusting the ratio of intracellular
interaction $J_{1}$ and intercellular interaction $J_{2}$, i.e.
$J_{2}$$/$$J_{1}$. This result is very different from that in the 2D honeycomb
ferromagnet, in which there is no topological phase transition for only
isotropic Heisenberg-type interaction. Moreover, the introduction of
anisotropic interactions, i.e. DMI (intracellular interaction $D_{1}$ and
intercellular interaction $D_{2}$) and PDI (intracellular interaction $F_{1}$
and intercellular interaction $F_{2}$), also plays an important role in the
realization of the topological phases. We demonstrate that the intercellular
interactions ($J_{2}$, $D_{2}$, and $F_{2}$) favor topological non-trivial
phase, while the intracellular interactions ($J_{1}$, $D_{1}$, and $F_{1}$)
hinder it, in the 1D FM SSH model. This paper is organized as follows: the
theoretical model and method are introduced in Sec. II; the numerical results
and discussions are presented in Sec. III; and the last section is the remarks
and conclusions.

##  II Model and Method

Analogous to the electronic SSH model PRL42-1698 , as shown in Fig. 1(a), we
construct a 1D FM SSH model including the intracellular and intercellular
interactions, i.e. $J_{1}$ and $J_{2}$, which presents a pure Heisenberg-type
model. In general, there are two types of FM ground states, that is, the spin
orientation is perpendicular to or along the direction of the 1D chain, as
shown in Fig. 1(c), and thus these two quantization axes are considered,
respectively.

Figure 1: (Color online) (a) 1D electronic SSH model, (b) 1D FM SSH model.
Each unit cell contains two lattice sites A and B. The solid and dotted lines
indicate the strong and weak exchange interactions, respectively. $J_{1}$ and
$J_{2}$ denote the intracellular and intercellular interactions, respectively.
(c) Two types of FM ground states: the spin quantization axes are $z$-axis
(top panel) and $x$-axis (bottom panel), respectively.

The Hamiltonian of the 1D spin chain with anisotropic exchange interactions
can be written as

$\displaystyle H$ $\displaystyle=$ $\displaystyle H_{0}+H_{AE}+H_{DM}+H_{PD}.$
(1)

$H_{0}$ is the Heisenberg exchange interaction, reading

$\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle-
J_{1}\sum_{i=1}^{N}{\mathbf{S}_{i,A}\cdot\mathbf{S}_{i,B}}-J_{2}\sum_{i=1}^{N-1}{\mathbf{S}_{i,B}\cdot\mathbf{S}_{i+1,A}},$
(2)

where $J_{1}$ and $J_{2}$ represent the exchange interaction in the unit cell
(intracellular) and between the two unit cells (intercellular), respectively.
And $H_{AE}=-\frac{1}{2}\sum_{i}{K_{i}{{(S}_{i}^{z})}^{2}}$ is the easy-axial
anisotropy term for the quantization $z$-axis, where $K_{i}$ is the axial
anisotropy energy ($K_{i}$=10$J_{1}$ is used for all the calculations).
$H_{DM}$ denotes the DMI term JPCS4-241 ; PR120-91 , originating from the
spin-orbit coupling, while $H_{PD}$ represents the PDI PR52-1178 ;
PRL102-017205 , mainly contributed from both the strong spin-orbit coupling
and the orbital degree of freedom.

The DMI and PDI are covered by the following forms:

$\displaystyle H_{DM}$ $\displaystyle=$
$\displaystyle\mathbf{D}_{1}\cdot\sum_{i}\left(\mathbf{S}_{i,A}\times\mathbf{S}_{i,B}\right)$
$\displaystyle+\mathbf{D}_{2}\cdot\sum_{i,i+1}\left(\mathbf{S}_{i,B}\times\mathbf{S}_{i+1,A}\right),$
$\displaystyle H_{PD}$ $\displaystyle=$ $\displaystyle-
F_{1}\sum_{i=1}^{N}{\left(\mathbf{S}_{i,A}\cdot\mathbf{e}_{i,AB}\right)\left(\mathbf{S}_{i,B}\cdot\mathbf{e}_{i,AB}\right)}$
$\displaystyle-
F_{2}\sum_{i=1}^{N-1}{\left(\mathbf{S}_{i,B}\cdot\mathbf{e}_{i,i+1,BA}\right)\left(\mathbf{S}_{i+1,A}\cdot\mathbf{e}_{i,i+1,BA}\right)},$

where $\mathbf{D}_{1}$ ($F_{1}$) and $\mathbf{D}_{2}$ ($F_{2}$) represent
intracellular and intercellular nearest neighbor DMIs (PDIs), respectively.
$\mathbf{e}_{i,AB}$ is the unit vector between lattice sites A and B in the
$i$-th unit cell, and $\mathbf{e}_{i,i+1,BA}$ is the unit vector between
lattice site B of the $i$-th unit cell and lattice site A of the $(i+1)$-th
unit cell. $\mathbf{e}_{i,AB}$ and $\mathbf{e}_{i,i+1,BA}$ point in the
direction of the chain.

We firstly analyze the magnetic ground state before calculating the magnon
spectrum. The Heisenberg term with $-J<0$ favors the FM state, and the easy-
axial anisotropy term with $K_{z}$ tends to align the spins parallel along the
$z$ axis. However, in the presence of anisotropic magnetic exchange
interactions, the nearest-neighboring DMI and PDI, the magnetic ground state
may change. In the classical spin limit, the derivation process with respect
to the spin canting angles PRB97-094412  is performed to obtain the minimal
energy of the total Hamiltonian $H$ (Eq.(1)). Two possible ground states are
found to be the FM states along $z$-axis and $x$-axis with the total energies
per unit cell $-(J_{1}+J_{2})-K$ and $-(J_{1}+J_{2})-(F_{1}+F_{2})$,
respectively. Hence the phase boundary is $K=F_{1}+F_{2}$, that is, when
$K>F_{1}+F_{2}$, it is FM state with spins oriented along the $z$ axis ($z$-FM
phase), otherwise, it is FM state with the spins oriented along the $x$ axis
($x$-FM phase). Through the analysis of the total energy, it is found that the
component of the DMI $D_{z}$, will not affect the $z$-FM phase because it does
not contribute to the total energy. While the PDI $F>0$ prefers each pair of
spins to be parallel to the bond between them ($x$ axis), so it can change the
magnetic ground state. Note that all the parameter values used ($K=10J_{1}$,
$F=J_{1}$, $0<F_{2}/F_{1}\leq 5$, for the convenience of theoretical analysis)
fall into the $z$-FM phase, even for the large values of $D_{z}$ or $F$.

By the linear Holstein-Primakoff (HP) transformation PR58-1098  with bosonic
generation (annihilation) operator (for A sublattice, $a^{{\dagger}}$ ($a$);
for B sublattice, $b^{{\dagger}}$ ($b$)), the Hamiltonian of magnon in
momentum $k$-space can be expressed as
$H=\frac{1}{2}\sum_{k}{\Psi_{k}^{{\dagger}}H_{k}\Psi_{k}}$, where
$\Psi_{k}=\left(a_{k},a_{-k}^{{\dagger}},b_{k},b_{-k}^{{\dagger}}\right)^{T}$.
Using the linear HP transformation, the spin operators for the A (or B)
sublattice are given by

$\begin{cases}S_{A}^{{\dagger}}=\sqrt{2S}a\\\
S_{A}^{-}=\sqrt{2S}a^{{\dagger}}\\\ S_{A}^{z}=S-a^{{\dagger}}a.\\\
\end{cases}$

By the Bogoliubov transformation INC7-794 , we can diagonalize the magnon
Hamiltonian and obtain the energy spectrum of the magnon. The magnon
Hamiltonian $H_{k}$ satisfies the generalized eigenvalue problem PR139-A450 ,

$\displaystyle\eta H_{k}\Gamma_{k}$ $\displaystyle=$
$\displaystyle\Gamma_{k}\eta E_{k},$ (5)

where $\Gamma_{k}$ is the transformation matrix that diagonalizes the magnon
Hamiltonian, $\eta$ is a metric matrix, $\eta=I_{2\times 2}\otimes\sigma_{z}$,
$I$ is the identity matrix, and $\sigma_{z}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}$ is the Pauli matrix. In the generalized eigenvalue problem
of boson, $E(k)=-E(-k)$ is artificially introduced. As a result, we only need
to consider the non$-$negative eigenvalues of the energy spectrum. On the
other hand, for a finite 1D system with $N$ unit cells, the generalized
eigenvalue problem in real space is $\eta H\Gamma=\Gamma\eta E$, where $H$ is
the $2N\times 2N$ real space matrix, and $\eta=I_{N\times
N}\otimes\sigma_{z}$.

As is known that, the electronic 1D SSH model possesses the chiral ($\Gamma$),
spatial ($P$) and temporal ($T$) inversion symmetries PE119-113973  defined as
$\Gamma H(k)\Gamma^{-1}=-H(k)$, $PH(k)P^{-1}=H(-k)$, $TH(k)T^{-1}=H(-k)$,
respectively, in which $\Gamma=I_{2\times 2}\otimes\sigma_{z}$, $P=I_{2\times
2}\otimes\sigma_{x}$, $T=I_{2\times 2}\otimes\begin{pmatrix}0&i\\\
i&0\end{pmatrix}$. The chiral symmetry requires on-site energy is 0, while the
spatial and temporal inversion symmetry only require the identical on-site
energies PRL127-147401 . In consequence, for the 1D FM SSH model with pure
Heisenberg-type interaction, the chiral symmetry is broken, while the spatial
and temporal inversion symmetries are preserved. When the anisotropic DMI and
PDI exist in the system, the spatial inversion symmetry is broken while only
the temporal inversion symmetry is preserved. Consequentially, the 1D FM SSH
model of magnons has no direct correspondence to the electronic systems
because of the different symmetries; thus, the topological magnons of the
magnetic SSH model are not superficial extensions of the electronic SSH
system. The most important difference is that the topological magnons
involving anisotropic interaction features do not have direct correspondence
to any electronic systems, though the isotropic Heisenberg term ($J_{1}$ and
$J_{2}$) corresponds to the electronic interactions ($t_{1}$ and $t_{2}$). In
addition, it should be pointed out that the change of quantization axis does
not affect the symmetry of the system.

For a 1D infinite system, Zak phase PRL62-2747  is often used to describe its
topological properties within the momentum $k$-space,

$\displaystyle\varphi_{Zak}$ $\displaystyle=$ $\displaystyle-i\int_{BZ}^{\
}{\Psi_{k}^{{\dagger}}\frac{d}{dk}\Psi_{k}{dk}}.$ (6)

When $\varphi_{Zak}$ is not 0, the topological phase is non-trivial, and the
system has topological edge state; otherwise, the topological number is 0, and
the topological phase is trivial, which means no topological edge state in the
system.

For a 1D finite system, we extend the real-space topological number of the
electronic (fermionic) version AP356-383  to the magnonic (bosonic) case. The
topological number of real space $N_{R}$ is described as

$\displaystyle N_{R}$ $\displaystyle=$
$\displaystyle\frac{1}{2}Sig((X+iH)\eta),$ (7)

where $X$ is a matrix composed of rescaled coordinates consistent with the
size of $H$. For the $i$-th unit cell, its reduced coordinates are
$X_{i,j}=\frac{i\delta_{ij}}{N}$. $Sig(Q)$ is equal to the number of positive
eigenvalues of $Q=(X+iH)\eta$.

##  III Results and Discussions

###  III.1 Heisenberg-type interaction

We first investigate the 1D FM SSH model with only pure Heisenberg-type
isotropic exchange interaction. Through the linear HP transformation mentioned
above, the Heisenberg interaction term $H_{0}$ can be expressed as

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\left(J_{1}+J_{2}+K\right)S\sum_{i}\left(a_{i}^{{\dagger}}a_{i}+b_{i}^{{\dagger}}b_{i}\right)$
$\displaystyle-
J_{1}S\sum_{i}\left(a_{i}b_{i}^{{\dagger}}+a_{i}^{{\dagger}}b_{i}\right)-J_{2}S\sum_{i}\left(b_{i}a_{i+1}^{{\dagger}}+b_{i}^{{\dagger}}a_{i+1}\right).$

The matrix of $H_{0}$ can be seen in the Appendix.

The band structures dependent on the ratio $J_{1}/J_{2}$ are presented in Fig.
2(a)-(c). It is obviously found that a topological phase transition occurs at
a critical ratio $J_{1}/J_{2}$=1, accompanied by an energy gap close. This is
very similar to the case of the electronic 1D SSH model TI2017 , that is the
phase transition between topologically trivial and non-trivial phases can be
realized by regulating the ratio of the intracellular and intercellular
interactions, distinctly different from that in the 2D honeycomb ferromagnet
in which the topological phase transition is mainly induced by the anisotropic
interaction rather than the isotropic interaction JPCM29-185801 . Notice that
for the critical ratio $J_{1}/J_{2}$=1, the system exhibits a linear Dirac-
like dispersion, but it is trivial owning to the band accidental degeneracy.
Therefore, for $J_{1}/J_{2}$$\neq$1, although the energy band opens a gap, it
is not necessarily topologically non-trivial. It is demonstrated that the
intercellular (intracellular) interaction $J_{2}$ ($J_{1}$) favors (unfavors)
the topological non-trivial phase. Indeed, for $J_{1}<J_{2}$ (in comparison
with $\delta t=t_{1}-t_{2}>0$ for the electronic SSH model), the magnon state
is a 1D topological magnon insulator, as shown in Fig. 2(c). In addition, we
note that the position of the gap close at $k=\pi$ or $-\pi$ is different from
that of the electronic SSH model at $k=0$.

Figure 2: (Color online) Band structures dependent on the ratio $J_{1}/J_{2}$
for 1D FM SSH model and the real-space energy spectrum of finite 1000 lattice
sites of 1D chain, respectively: $J_{1}/J_{2}$=1.5 (a) and (d), 1.0 (b) and
(e), 0.5 (c) and (f).

Owning to the finite energy gap of 1D topological magnon insulator, a
topologically protected edge state can be expected according to the bulk-edge
correspondence PRB95-035421 . In order to investigate the edge state of this
topological magnon insulator, the energy spectrums of finite lattice sites in
real space are also given, as displayed in Fig. 2(d)-(f). Accordingly, two
degenerate edge states are found located within the band gap for the
topological magnon insulating phase, which can be seen in Fig. 2(f).

Through the analysis of the real-space-resolved wave functions for the two
degenerate edge states, it is found that the two edge states are separately
localized at the two ends of the 1D chain, as seen in Fig. 3. In fact, this
degenerate edge states can be simply lifted by introducing an inequivalence
between A and B sites in the SSH model.

Figure 3: (Color online) Wave-function amplitude dependent on the lattice
index. The two degenerate edge states in Fig. 2(f) are separately localized at
both ends of the 1D chain.

Considering the topological characteristic of this in-gap edge states, it can
be regarded as zero-energy mode of magnon at finite frequency in 1D quantum
magnets, which is very like the Majorana zero mode in 1D chain (nanowire) of
electronic system Science336-1003 . It should be emphasized that the in-gap
edge state as a novel type of bound state is distinctly different from the
two-magnon bound state in 1D Heisenberg spin chain Nature502-76 .

The corresponding topological invariants in both the momentum $k$-space (Zak
phase $\varphi_{Zak}$) and the real space ($N_{R}$) are shown in Fig. 4. It
can be found that the topological number changes from 0 to 1 at the critical
point $J_{1}$=$J_{2}$ after the energy gap is closed and reopened, indicating
the phase transition process between the topological trivial and non-trivial
phases. Notice that it can be seen that the topological number of $k$-space
and real space are consistent with each other in describing the topological
phase transition, implying our methods in calculating the topological
invariants are effective.

Figure 4: (Color online) Momentum $k$-space Zak phase $\varphi_{Zak}$ and
real-space topological number $N_{R}$ in the 1D FM SSH model with the
Heisenberg-type interaction.

For the quantization $x$-axis, as displayed in Fig. 1(c), the Heisenberg
interaction term $H_{0}^{{}^{\prime}}$ can be rewritten as

$\displaystyle H_{0}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\left(J_{1}+J_{2}\right)S\sum_{i}\left(a_{i}^{{\dagger}}a_{i}+b_{i}^{{\dagger}}b_{i}\right)$
$\displaystyle-
J_{1}S\sum_{i}\left(a_{i}^{{\dagger}}b_{i}+h.c.\right)-J_{2}S\sum_{i}\left(b_{i}^{{\dagger}}a_{i+1}+h.c.\right)$

Obviously, it is consistent with the form of the quantization $z$-axis which
indicates that the change of the quantization axis does not affect our results
for the pure Heisenberg-type interaction.

###  III.2 In the presence of DMI

Although the isotropic Heisenberg interaction can induce the topological phase
in 1D spin chain, the anisotropic interactions play an essential role in
topological phase transition in 2D lattices, such as honeycomb PRL117-227201 ;
PRB95-014435 ; PRB97-174413 ; PRB99-214424  and Kagome PRB87-144101 ;
PRB90-024412  lattices, etc. Therefore the influence of the anisotropic
exchange interactions deserves investigation. Here we first focus on the DMI
originating from the spin-orbit coupling. Using the linear HP transformation,
the DMI term $H_{DM}$ (more details are shown in Appendix) can be written as

$\displaystyle H_{DM}$ $\displaystyle=$ $\displaystyle-
iD_{1z}S\sum_{i}\left(a_{i}^{{\dagger}}b_{i}-a_{i}b_{i}^{{\dagger}}\right)$
$\displaystyle-
iD_{2z}S\sum_{i,i+1}\left(b_{i}^{{\dagger}}a_{i+1}-b_{i}a_{i+1}^{{\dagger}}\right),$

where $D_{1z}$ and $D_{2z}$ are the components of intracellular
$\mathbf{D}_{1}$ and intercellular $\mathbf{D}_{2}$ on the quantization
$z$$-$axis of spin, respectively. For simplicity, we fix $D_{1z}=J_{2}=1$ for
the $J_{1}/J_{2}$ case and $D_{1z}=J_{1}=1$ for the $J_{2}/J_{1}$ case in all
the calculations. Note that due to the chiral characteristic of DMI, the
values of both $D_{1z}$ and $D_{2z}$ can be positive or negative for 1D chain,
and all cases of four possible signs are considered. It is found that the
change of sign does not change the topological phase diagram, but only the
position of the closing gap in the energy spectrum during the phase
transition. For $D_{2z}/D_{1z}>0$, the energy gap closes at $k=\pi/2$ (Note
that for the pure DMI alone, the gap closes at $k=0$, not shown here,
different from the case of the pure Heisenberg interaction.); otherwise, the
position where energy gap closes is associated with $D_{2z}/D_{1z}$.
Therefore, we only consider two cases with the relative sign of the
$D_{2z}/D_{1z}$ being positive or negative.

Figure 5: (Color online) The $J$-$D$ phase diagram of $J_{1}/J_{2}$ (a) and
$J_{2}/J_{1}$ (b) for the 1D FM SSH model with the anisotropic DMI. The white
dotted lines represent the phase transition boundaries between the trivial and
non-trivial phases given by the relations
$\left|D_{2z}/D_{1z}\right|=J_{1}/J_{2}$ ($D_{1z}=J_{2}=1$ is fixed) and
$\left(J_{2}/J_{1}\right)^{2}+\left(D_{2z}/D_{1z}\right)^{2}=2$
($D_{1z}=J_{1}=1$ is fixed), respectively. (c) and (d) are $J_{1}-J_{2}$ phase
diagrams at $D_{1z}=1.0>D_{2z}=0.5$ and $D_{1z}=1.0<D_{2z}=1.5$, respectively.
The black dashed lines denote the original phase transition boundaries of the
pure Heisenberg interactions.

Compared with the case of pure Heisenberg interaction, the phase transition
condition (more details are shown in Appendix) in the presence of the
anisotropic DMI is $J_{1}^{2}+{D}_{1z}^{2}-J_{2}^{2}-D_{2z}^{2}=0$. Thus, the
phase transition occurs at $\left|D_{2z}/D_{1z}\right|=J_{1}/J_{2}$ for the
fixed $D_{1z}=J_{2}=1$ case, and
$\left(J_{2}/J_{1}\right)^{2}+\left(D_{2z}/D_{1z}\right)^{2}=2$ for the fixed
$D_{1z}=J_{1}=1$ case. Obviously, the intercellular $D_{2}$ (intracellular
$D_{1}$) DMI boosts (weakens) and even induces (eliminates) the topologically
non-trivial phase, though the Heisenberg interaction alone can lead to the
topological magnon phase, which obviously can be seen in Figs. 5-7.

Figure 6: (Color online) Band structures dependent on the ratio
$D_{2z}/D_{1z}$ at $J_{1}/J_{2}$=1.5: (a) $D_{2z}/D_{1z}$=1.0 (b)
$D_{2z}/D_{1z}$=1.5 (c) $D_{2z}/D_{1z}$=2.0. The vertical black dotted line in
(b) indicates the position where the energy gap is closed. Figure 7: (Color
online) Momentum $k$-space Zak phase $\varphi_{zak}$ and real-space
topological number $N_{R}$ in the 1D FM SSH model with the Heisenberg and DMI
interactions for $J_{1}/J_{2}$=1.5.

For the quantization $x$-axis, as displayed in Fig. 1(c), the DMI term
$H_{DM}^{{}^{\prime}}$ can be rewritten as

$\displaystyle H_{DM}^{{}^{\prime}}$ $\displaystyle=$ $\displaystyle-
iD_{1x}S\sum_{i}\left(a_{i}^{{\dagger}}b_{i}-a_{i}b_{i}^{{\dagger}}\right)$
$\displaystyle-
iD_{2x}S\sum_{i,i+1}\left(b_{i}^{{\dagger}}a_{i+1}-b_{i}a_{i+1}^{{\dagger}}\right),$

where $D_{1x}$ and $D_{2x}$ are the components of $\mathbf{D}_{1}$ and
$\mathbf{D}_{2}$ on the quantization $x$-axis of spin, respectively. From the
analysis of the DMI, we can find that only the components in the direction of
quantization axis will affect the topological properties of the system.
Therefore, the conclusions drawn above are still valid.

###  III.3 In the presence of PDI

Apart from the DMI, the PDI is another essential anisotropic exchange
interaction, which is generally from the combination of the strong spin-orbit
coupling and the orbital degree of freedom PRL102-017205 . Consequently, the
PDI has the different influence on magnons due to its origin distinctly
different from that of the DMI. For the quantization $z$-axis, using the
linear HP transformation, the PDI term $H_{PD}$ (more details are shown in
Appendix) are described as

$\displaystyle H_{PD}$ $\displaystyle=$
$\displaystyle-\frac{F_{1}}{2}S\sum_{i}\left(a_{i}b_{i}+a_{i}b_{i}^{{\dagger}}+a_{i}^{{\dagger}}b_{i}+a_{i}^{{\dagger}}b_{i}^{{\dagger}}\right)$
$\displaystyle-\frac{F_{2}}{2}S\sum_{i,i+1}\left(b_{i}a_{i+1}+b_{i}a_{i+1}^{{\dagger}}+b_{i}^{{\dagger}}a_{i+1}+b_{i}^{{\dagger}}a_{i+1}^{{\dagger}}\right).$

In this 1D chain case, the direction of the PDI is along the $x$-axis and
perpendicular to the quantization $z$-axis. For simplicity, $F_{1}=J_{2}=1$
for the $J_{1}/J_{2}$ case and $F_{1}=J_{1}=1$ for the $J_{2}/J_{1}$ case is
fixed in all the calculations. The phase transition condition (more details
are shown in Appendix) in the case of the Heisenberg interaction and PDI is
$F_{1}+2J_{1}-F_{2}-2J_{2}=0$. Thus the boundary between the trivial and non-
trivial phases is determined by the relation $2J_{1}/J_{2}-1=F_{2}/F_{1}$ for
$F_{1}/J_{2}=1$, and $-2J_{2}/J_{1}+3=F_{2}/F_{1}$ for $F_{1}/J_{1}=1$.
Similar to the Heisenberg interaction and DMI cases, the intercellular $F_{2}$
(intracellular $F_{1}$) PDI favors (unfavors) the topologically non-trivial
phase, as demonstrated in Figs. 8-10. Compared with the cases of the
Heisenberg and DMI interactions, the position of gap close for the pure PDI
case is at $k=\pi$, same as the Heisenberg one. This is a result of the
different symmetries of these three magnetic interactions.

Figure 8: (Color online) Band structures dependent on the ratio $F_{2}/F_{1}$
for $J_{1}/J_{2}$=1.5: $F_{2}/F_{1}$=1.5 (a), 2.0 (b), and 2.5 (c). The
vertical black dotted lines in (b) denote the positions of the gap closed for
the PDI case, different from that of the DMI case. Figure 9: (Color online)
The $J$-$F$ phase diagram of $J_{1}/J_{2}$ (a) and $J_{2}/J_{1}$ (b) for the
1D FM SSH model with the anisotropic PDI. The white dotted lines represent the
phase transition boundaries between the trivial and non-trivial phases given
by the relations $2J_{1}/J_{2}-1=F_{2}/F_{1}$ ($F_{1}=J_{2}=1$ is fixed) and
$-2J_{2}/J_{1}+3=F_{2}/F_{1}$ ($F_{1}=J_{1}=1$ is fixed), respectively. (c)
and (d) are $J_{1}-J_{2}$ phase diagrams at $F_{1}=1.0>F_{2}=0.5$ and
$F_{1}=1.0<F_{2}=1.5$, respectively. The black dashed lines denote the
original phase transition boundaries of the pure Heisenberg interactions.
Figure 10: (Color online) Momentum $k$-space Zak phase $\varphi_{zak}$ and
real-space topological number $N_{R}$ in the 1D FM SSH model with the
Heisenberg and PDI interactions for $J_{2}/J_{1}=1.0$.

For the quantization $x$-axis, as displayed in Fig. 1(c), the PDI term
$H_{PD}^{{}^{\prime}}$ can be rewritten as

$\displaystyle H_{PD}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle-\left(F_{1}+F_{2}\right)S\sum_{i}\left(a_{i}^{{\dagger}}a_{i}+b_{i}^{{\dagger}}b_{i}\right),$
(13)

and then the PDI becomes a term similar to the easy-axial anisotropy $K$ term,
equivalent to the on-site energy of magnon. Therefore, the PDI has no
influence on the topological properties in this case.

###  III.4 In the presence of both DMI and PDI

Finally, we turn to the case of coexistence of both the anisotropic DMI and
PDI. In fact, the Heisenberg interaction, DMI and PDI may possibly coexist in
the same realistic magnetic materials due to the the existence of multiple
degrees of freedom, especially in 3$d$, 4$d$ and even 5$d$ correlated
materials, such as magnetic materials in iridates PRL105-027204 ; PRB84-054409
, etc. However, the phase transition condition is no longer linear. For
simplicity, both $D_{1z}=J_{1}$, $F_{1}=J_{1}$ and $J_{1}=1$ are fixed in all
the calculations. Then the phase transition boundary can be determined
numerically.

As shown in Figs. 11, comparing the $J$-$D$, $J$-$F$ and $D$-$F$ phase
diagrams, it is found that the coexistence of the isotropic Heisenberg
interaction, anisotropic DMI and PDI show different topological phase diagrams
from these in Fig. 5(a) and Fig. 9(a) as a result of the interplay among them.

Figure 11: (Color online) The $J$-$D$, $J$-$F$ and $D$-$F$ phase diagrams for
the 1D FM SSH model with both the anisotropic DMI and PDI.

Evidently, these three interactions have synergistic effects on the generation
of topologically non-trivial phase. In essence, it is the intercellular
interactions ($J_{2}$, $D_{2}$ and $F_{2}$) rather than the intracellular
interactions ($J_{1}$, $D_{1}$ and $F_{1}$) that contribute to the emergence
of the topological phases. As a consequence, these different magnetic exchange
interactions introduce a rich topological phase diagram in the 1D magnon
system, suggesting a realization of multiple manipulation of topological
magnonics. As matter of fact, the relative strength between the isotropic and
anisotropic exchange interactions, can be changed by the strain and pressure
regulations in the realistic materials, subsequently the topological phase
transition of magnons can be achieved.

##  IV Remarks and Conclusions

We notice that different from our local spins, topological magnons in 1D
itinerant flatband ferromagnet are also proposed PRB97-245111 . Indeed, the 1D
topological magnons can be explored in realistic one-dimensional (1D) magnets
PRB104-085429 , quasi-1D magnetic materials NJP15-093043 , or 1D organic
magnetic systems NM17-308 ; NRM5-87 . As can be expected, more 1D magnetic
models rather than the SSH model can be extended to topological magnons.
Another way to implement the 1D ferromagnetic SSH model is to strip an edge
from the magnonic 2D SSH model in honeycomb ferromagnets PRB103-014407 . On
the other hand, our results also imply that the topological magnons in the 1D
systems are distinctly different from these in the 2D ones, introducing a new
platform for the topological magnons.

In summary, we construct a one-dimensional (1D) ferromagnetic (FM) SSH model
with the anisotropic interactions, and find a topological magnon non-trivial
phase described by the Zak phase, induced not only by the pure strong
intercellular isotropic Heisenberg interaction $J_{1}/J_{2}<1$, but also by
the strong intercellular anisotropic DMI and PDI. The intercellular
anisotropic DMI and PDI in combination with the intercellular isotropic
Heisenberg-type interaction manifest a synergistic effect on the topological
phase transition. In addition, the quantization axis of spin also
substantially affect the topological magnon phase diagram owing to the
anisotropic interactions. Of particular interest, the existence of
topologically protected in-gap edge states (magnon bound states) in 1D magnets
with FM SSH model provides a possible route to Majorana-like particle
realization. Due to the rich manipulated magnetic interactions and the 1D
structural characteristics, abundant topological magnon states and magnonic
crystals can be designed from bottom to top, suggesting potential applications
in the field of topological magnonics.

###### Acknowledgements.

This work was supported by the National Sciences Foundation of China under
Grant Nos. 11974354, 11774350 and 11574315. The calculations were performed in
Center for Computational Science of CASHIPS, the ScGrid of Supercomputing
Center and Computer Network Information Center of Chinese Academy of Science.

##  V Appendix

For the quantization $z$-axis, the explicit matrix form of $H_{0}$, $H_{DM}$
and $H_{PD}$ are

$\displaystyle H_{0}=\begin{pmatrix}\varepsilon_{0}&0&h(k)&0\\\
0&\varepsilon_{0}&0&h^{*}(k)\\\ h^{*}(k)&0&\varepsilon_{0}&0\\\
0&h(k)&0&\varepsilon_{0}\end{pmatrix}$ (14) $\displaystyle
H_{DM}=\begin{pmatrix}0&0&f(k)&0\\\ 0&0&0&f^{*}(k)\\\ f^{*}(k)&0&0&0\\\
0&f(k)&0&0\end{pmatrix}$ (15) $\displaystyle
H_{PD}=\begin{pmatrix}0&0&g(k)&g(k)\\\ 0&0&g^{*}(k)&g^{*}(k)\\\
g^{*}(k)&g(k)&0&0\\\ g^{*}(k)&g(k)&0&0\end{pmatrix}$ (16)

where $\varepsilon_{0}=\left(J_{1}+J_{2}+K\right)S$,
$h(k)=-J_{1}S-J_{2}Se^{-ika}$, $f(k)=-iD_{1}S+iD_{2}Se^{-ika}$,
$g(k)=\left(-F_{1}S-F_{2}Se^{-ika}\right)/2$.

For $H_{1}=H_{0}+H_{DM}$, we assume $J_{1}/J_{2}=\alpha>0$,
$D_{1z}/J_{2}=\beta_{1}>0$, $D_{2z}/D_{1z}=\gamma_{1}$,
$D_{2z}/J_{2}=\gamma_{1}\beta_{1}$. When the topological phase transition
occurs at the closed energy gap, and the parameters are satisfied by

$\displaystyle\begin{cases}-\alpha-\cos(ka)+\gamma_{1}\beta_{1}\sin(ka)=0\\\
\sin(ka)-\beta_{1}+\gamma_{1}\beta_{1}\cos(ka)=0\\\ \end{cases}$ (17)

Cancelling $sin(ka)$ and $cos(ka)$, we find
$\beta_{1}^{2}+\alpha^{2}=1+\gamma_{1}^{2}\beta_{1}^{2}$. Here we take
$\beta_{1}=1$ and $a=1$. As a result, the condition of the phase transition
(i.e. the gap closed) is $\alpha^{2}=\gamma_{1}^{2}$. When
$\gamma_{1}=\alpha$, $\sin(ka)=1$, that is, $k=\pi/2a$; when
$\gamma_{1}=-\alpha$, $\sin(ka)=(1-\alpha^{2})/(1+\alpha^{2})$,
$\cos(ka)=-2\alpha/(1+\alpha^{2})$.

For $H_{2}=H_{0}+H_{PD}$, we assume $F_{1}/J_{2}=\beta_{2}>0$,
$F_{2}/F_{1}=\gamma_{2}>0$, $F_{2}/J_{2}=\gamma_{2}\beta_{2}$. The parameters
are satisfied by

$\displaystyle(2\alpha+\beta_{2})^{2}+(2+\gamma_{2}\beta_{2})^{2}+2(2\alpha+\beta_{2})(2+\gamma_{2}\beta_{2})\cos(ka)=0$
(18)

Obviously, $-1\leq\cos(ka)<0$. Assuming $n=\cos(ka)-(-1)$, it can be rewritten
as

$\displaystyle((2\alpha+\beta_{2})-(2+\gamma_{2}\beta_{2}))^{2}+2n(2\alpha+\beta_{2})(2+\gamma_{2}\beta_{2})=0$
(19)

We can see that it is valid only for $n=0$, i.e. $\cos(ka)=-1$. We take
$\beta_{2}=1$, and then $\gamma_{2}=2\alpha-1$. Then the condition of
topological phase transition can be determined analytically.

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