By definition, for any non-negative $\psi\in H^{1}(Q_{T})$ such that
$\psi(\cdot,T)=0$,

$\begin{split}&\;\int_{0}^{T}\int_{\mathbb{R}^{d}}\nabla\psi\cdot\nabla\big{(}p^{0}-p^{1}\big{)}-\big{(}\rho^{0}-\rho^{1}\big{)}\partial_{t}\psi\,dx\,dt\\\
=&\;\int_{\mathbb{R}^{d}}\psi(x,0)\big{(}\rho_{0}^{0}(x)-\rho_{1}^{0}(x)\big{)}\,dx+\int_{0}^{T}\int_{\mathbb{R}^{d}}\psi\big{(}f^{0}-f^{1}\big{)}\,dx\,dt\leq
0.\end{split}$

Hence, for any $R>0$ and any non-negative $\psi\in
H^{1}(\mathbb{R}^{d}\times[0,T])$ supported in $B_{R}\times[0,T]$ such that
$\psi(\cdot,T)=0$,

(A.1)
$\int_{0}^{T}\int_{B_{R}}(\rho^{0}-\rho^{1})\partial_{t}\psi+(p^{0}-p^{1})\Delta\psi\,dx\,dt\geq\int_{0}^{T}\int_{\partial
B_{R}}\frac{\partial\psi}{\partial\nu}\cdot\big{(}p^{0}-p^{1}\big{)}\,d\sigma(x)\,dt.$

Define

$A=\frac{\rho^{0}-\rho^{1}}{\rho^{0}-\rho^{1}+p^{0}-p^{1}},\quad
B=\frac{p^{0}-p^{1}}{\rho^{0}-\rho^{1}+p^{0}-p^{1}}.$

We define $A=0$ whenever $\rho^{0}=\rho^{1}$ (even when $p^{0}=p^{1}$), and
$B=0$ whenever $p^{0}=p^{1}$ (even when $\rho^{0}=\rho^{1}$). Since $p^{0}\in
P_{\infty}(\rho^{0})$ and $p^{1}\in P_{\infty}(\rho^{0})$, we have
$A,B\in[0,1]$. Then (A.2) can be written as

(A.2)
$\int_{0}^{T}\int_{B_{R}}\big{(}\rho^{0}-\rho^{1}+p^{0}-p^{1}\big{)}\big{(}A\partial_{t}\psi+B\Delta\psi\big{)}\,dx\,dt\geq\int_{0}^{T}\int_{\partial
B_{R}}\frac{\partial\psi}{\partial\nu}\cdot\big{(}p^{0}-p^{1}\big{)}\,d\sigma(x)\,dt.$

Let $G$ be a compactly supported non-negative smooth function in $Q_{T}$.
Assume it is supported in $B_{R_{0}}\times[0,T]$ for some $R_{0}>0$. Take an
arbitrary $R\geq 2R_{0}$. As in [PQV14], we introduce smooth positive
approximations of $A$ and $B$, denoted by $A_{n,R}$ and $B_{n,R}$, such that
for some universal $C_{R}>0$ that depends on $R$,

$A_{n,R},B_{n,R}\in\left[\frac{1}{n},1\right],\quad\|A_{n,R}-A\|_{L^{2}(B_{R}\times[0,T])}\leq\frac{C_{R}}{n},\quad\|B_{n,R}-B\|_{L^{2}(B_{R}\times[0,T])}\leq\frac{C_{R}}{n}.$

In the view of (A.2), let $\psi_{n,R}$ solve the (mollified) dual equation

$\begin{split}&\partial_{t}\psi_{n,R}+\frac{B_{n,R}}{A_{n,R}}\Delta\psi_{n,R}=-G\mbox{
in }B_{R}\times[0,T],\\\ &\psi_{n,R}|_{\partial
B_{R}\times[0,T]}=0,\quad\psi_{n,R}(\cdot,T)=0\mbox{ in }B_{R}.\end{split}$

$\psi_{n,R}$ is then a smooth function on $B_{R}\times[0,T]$. Plugging it into
(A.2) as the test function, we find that

$\int_{0}^{T}\int_{B_{R}}\big{(}\rho^{0}-\rho^{1}\big{)}(-G)\,dx\,dt+\mathcal{E}_{n,R}\geq\int_{0}^{T}\int_{\partial
B_{R}}\frac{\partial\psi_{n,R}}{\partial\nu}\cdot\big{(}p^{0}-p^{1}\big{)}\,d\sigma(x)\,dt,$

where

$\mathcal{E}_{n,R}:=\int_{0}^{T}\int_{B_{R}}\big{(}\rho^{0}-\rho^{1}+p^{0}-p^{1}\big{)}\left(B-\frac{AB_{n,R}}{A_{n,R}}\right)\Delta\psi_{n,R}\,dx\,dt.$

We can argue as in [PQV14, GKM22] to show that $\psi_{n,R}$ is non-negative
and uniformly bounded on $B_{R}\times[0,T]$, whose bound only depends on $G$
and $T$, but not on $n$ or $R$. We can also prove that $\mathcal{E}_{n,R}\to
0$ as $n\to+\infty$. Moreover,

(A.3) $\frac{\partial\psi_{n,R}}{\partial\nu}\leq CR^{-(d-1)}\mbox{ on
}\partial B_{R}\times[0,T],$

where $C$ only depends on $d$, $T$, and $G$, but not on $n$ or $R$. To prove
(A.3), we recall that $R\geq 2R_{0}$ and $\psi_{n,R}\leq C_{*}$, where
$C_{*}=C_{*}(G,T)$. Let $\tilde{\psi}_{R}$ solve

$\Delta\tilde{\psi}_{R}=0\mbox{ on
}B_{R}\backslash\overline{B_{R_{0}}},\quad\tilde{\psi}_{R}\big{|}_{\partial
B_{R_{0}}}=C_{*},\quad\tilde{\psi}_{R}\big{|}_{\partial B_{R}}=0.$

Then we find

$\begin{split}&\partial_{t}\big{(}\tilde{\psi}_{R}-\psi_{n,R}\big{)}+\frac{B_{n,R}}{A_{n,R}}\Delta\big{(}\tilde{\psi}_{R}-\psi_{n,R}\big{)}=0\mbox{
in }(B_{R}\backslash\overline{B_{R_{0}}})\times[0,T],\\\
&\big{(}\tilde{\psi}_{R}-\psi_{n,R}\big{)}\big{|}_{\partial(B_{R}\backslash\overline{B_{R_{0}}})\times[0,T]}\geq
0,\quad\big{(}\tilde{\psi}_{R}-\psi_{n,R}\big{)}=0\mbox{ in
}B_{R}\backslash\overline{B_{R_{0}}}.\end{split}$

By the maximum principle, $\psi_{n,R}\leq\tilde{\psi}_{R}$ on
$(B_{R}\backslash\overline{B_{R_{0}}})\times[0,T]$, and thus (A.3) follows. In
fact, when $d=1,2$, the bound can be improved.

Combining the above estimates yields that, whenever $R\geq 2R_{0}$,

$\begin{split}\int_{0}^{T}\int_{B_{R}}\big{(}\rho^{0}-\rho^{1}\big{)}(-G)\,dx\,dt\geq&\;-CR^{-(d-1)}\int_{0}^{T}\int_{\partial
B_{R}}\big{|}p^{0}-p^{1}\big{|}\,d\sigma(x)\,dt\\\
\geq&\;-CR^{-\frac{d-1}{2}}\left(\int_{0}^{T}\int_{\partial
B_{R}}\big{|}p^{0}-p^{1}\big{|}^{2}\,d\sigma(x)\,dt\right)^{\frac{1}{2}},\end{split}$

where $C$ only depends on $d$, $T$, and $G$. By Definition 3.9, $p^{i}\in
L^{2}(Q_{T})$. Sending $R\to+\infty$, we obtain that

$\begin{split}\int_{0}^{T}\int_{\mathbb{R}^{d}}\big{(}\rho^{0}-\rho^{1}\big{)}(-G)\,dx\,dt\geq&\;-C\liminf_{R\to+\infty}R^{-\frac{d-1}{2}}\left(\int_{0}^{T}\int_{\partial
B_{R}}\big{|}p^{0}-p^{1}\big{|}^{2}\,d\sigma(x)\,dt\right)^{\frac{1}{2}}=0.\end{split}$

Since $G$ is an arbitrary compactly supported non-negative smooth function in
$Q_{T}$, we conclude that $\rho^{0}\leq\rho^{1}$ almost everywhere. ∎

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# Anti-chiral edge states in Heisenberg ferromagnet on a honeycomb lattice

Dhiman Bhowmick  Pinaki Sengupta  School of Physical and Mathematical
Sciences, Nanyang Technological University, Singapore

###### Abstract

We demonstrate the emergence of anti-chiral edge states in a Heisenberg
ferromagnet with Dzyaloshinskii–Moriya interaction (DMI) on a honeycomb
lattice with inequivalent sublattices. The DMI, which acts between atoms of
the same species, differs in magnitude for the two sublattices, resulting in a
shifting of the energy of the magnon bands in opposite directions at the two
Dirac points. The chiral symmetry is broken and for sufficiently strong
asymmetry, the band shifting leads to anti-chiral edge states (in addition to
the normal chiral edge states) in a rectangular strip where the magnon current
propagates in the same direction along the two edges. This is compensated by a
counter-propagating bulk current that is enabled by the broken chiral
symmetry. We analyze the resulting magnon current profile across the width of
the system in details and suggest realistic experimental probes to detect
them. Finally, we discuss about possible materials that can potentially
exhibit such anti-chiral edge states.

PACS numbers

85.75.-d, 75.47.-m, 73.43.-f, 72.20.-i

###### pacs:

Valid PACS appear here

### Introduction.-

Quantum magnets have emerged as a versatile platform for realizing magnetic
analogues of the plethora of topological phases that have been predicted,
analysed, classified and observed in electronic systems over the past decade.
Haldane’s paradigmatic modelHaldane (1988) of tight binding electrons on a
honeycomb lattice with complex next-nearest neighbor hopping – that
constitutes the foundation of many of the electronic topological phases – has
a natural realization in (quasi-) 2D insulating ferromagnets such as CrI3Chen
_et al._ (2018) and AFe2(PO4)2 (A=Ba,Cs,K,La)Kim and Kee (2017). In many of
these materials, the dominant Heisenberg exchange is supplemented by a next-
nearest neighbour anti-symmetric DMI. Magnetic excitations in these systems
are described by two species of quasi-particles – spinons with up and down
spins. The Kane-Mele-Haldane model – analogous to the Kane Mele model for
electrons – has been proposed to describe the spinons over a wide range of
temperatures.Kim  _et al._ (2016) The spinon bands acquire a non-trivial
dispersion due to Berry phase arising from the DMI. This results in a spin
Nernst effect (SNE) where a thermal gradient drives a transverse spin current,
a spinon version of the spin Hall effectKim  _et al._ (2016); Zyuzin and
Kovalev (2016); Cheng  _et al._ (2016); Zhang  _et al._ (2018). In a finite
sample, the two spinon species generate two counterpropagating spin currents
along the edges that are protected by chiral symmetry of the Hamiltonian –
analogous to two copies of the thermal Hall effect (THE) of magnons that has
been observed in many insulating magnetsOnose  _et al._ (2010); Hirschberger
_et al._ (2015); Ideue  _et al._ (2012); Hentrich  _et al._ (2019). .

Recently, there has been growing interest in engineering systems with co-
propagating edge currentsColomés and Franz (2018); Mandal  _et al._ (2019);
Vila  _et al._ (2019), through an ingenious, yet physically unrealistic,
modification of the Haldane model. The conservation of net current is
satisfied by counter propagating bulk current. That is, the bulk is not
insulating, in contrast to conventional topological insulators. In this work,
we demonstrate that anti-chiral states arise naturally in spinons on a
honeycomb magnet comprised of two different magnetic ions, with unequal DMI
for the two sublattices. In the absence of DMI, the spinon dispersion consists
of two doubly degenerate bands with linear band crossings at K and
K′Pershoguba  _et al._ (2018). A finite DMI lifts the degeneracy between the
two spinon branches and opens up a gap in the spectrumOwerre (2016a, b, 2017);
Pantaleón  _et al._ (2019). For asymmetric DMI, the two bands for each spinon
species are shifted in opposite directions relative to each other at the K and
K′ points in the Brillouin zone. This results in similar dispersion for the
gapless modes at both edges, giving rise to co-propagating edge states. This
is shown to yield effective anti-chiral edge states for the spinons in
addition to normal chiral ones. We present a detailed characterization of the
nature of the edge and bulk spinon states and suggest suitable experimental
signatures to detect these novel topological states.

### Model.-

We consider a Heisenberg ferromagnet on the honeycomb lattice with unequal DMI
($D_{A}$ and $D_{B}$) on the two sub-lattices. Introducing the symmetric and
anti-symmetric combinations of $D_{A}$ and $D_{B}$ as,
$D~{}=~{}\frac{1}{2}\left(D_{A}+D_{B}\right)$,
$D^{\prime}~{}=~{}\frac{1}{2}\left(D_{A}-D_{B}\right)$ – termed chiral and
anti-chiral DMI respectively for reasons that will become clear later – the
Hamiltonian is given by,

$\displaystyle\pazocal{H}=$ $\displaystyle-J\sum_{\left\langle
i,j\right\rangle}\mathbb{S}_{i}\cdot\mathbb{S}_{j}+D\sum_{\left\langle\left\langle
i,j\right\rangle\right\rangle}\nu_{ij}\hat{z}\cdot\left(\mathbb{S}_{i}\times\mathbb{S}_{j}\right)$
$\displaystyle+D^{\prime}\sum_{\left\langle\left\langle
i,j\right\rangle\right\rangle}\nu_{ij}^{\prime}\hat{z}\cdot\left(\mathbb{S}_{i}\times\mathbb{S}_{j}\right)-B\sum_{i}S_{i}^{z},$
(1)

where, $J>0$ is the nearest neighbor Heisenberg interaction and $\nu_{ij}=+1$
when $i$ and $j$ are along the cyclic arrows shown in Fig.1(b). Finally,
$\nu_{ij}^{\prime}=+\nu_{ij}$ for sublattice-A and
$\nu_{ij}^{\prime}=-\nu_{ij}$ for sublattice-B. The magnetic field $B$ is
introduced in a Zeeman coupling term to stabilize the ferromagnetic ground
state at finite temperature. The energy scale is set by choosing $J=1$ – all
other parameters in the Hamiltonian are in units of $J$.

Figure 1: (color online)(a) The honeycomb lattice structure. (b) The
directions along which $\nu_{ij}=+1$ has been shown, otherwise $\nu_{ij}=-1$.
(c) The directions along which $\nu_{ij}^{\prime}=+1$ has been shown,
otherwise $\nu_{ij}^{\prime}=-1$.

The ground state of the hamiltonian (Eq.1) is ferromagnetic for
$J>-{3\sqrt{3}\over 2}\sum_{s}\left|D+sD^{\prime}\right|,\quad s=\pm 1$. We
apply the Schwinger Boson mean field theory (SBMFT) to study the topological
character of the low energy magnetic excitations at a finite temperature. The
Schwinger Boson representation consists of the mapping the spin operators into
spinons as, $S_{i}^{+}=c_{i,\uparrow}^{\dagger}c_{i,\downarrow}$,
$S_{i}^{-}=c_{i,\downarrow}^{\dagger}c_{i,\uparrow}$,
$S_{i}^{z}=\frac{1}{2}\left(c_{i,\uparrow}^{\dagger}c_{i,\uparrow}-c_{i,\downarrow}^{\dagger}c_{i,\downarrow}\right),$
where $c_{i,s}$ and $c_{i,s}^{\dagger}$ are the annihilation and creation
operators of spin-1/2 up or down spinons respectively. The constraint
$\sum_{s}c_{i,s}^{\dagger}c_{i,s}=2S$, $\forall i$ on the bosonic operators
ensures the fulfillment of the spin-S algebra.

After applying Schwinger Boson transformation along with the constraint, and
using a mean field approximation to reduce the 4-body operators to bilinear
forms, the spin model Eq.1 is mapped to the the mean field hamiltonian,

$\displaystyle\pazocal{H}=$ $\displaystyle-\eta J\sum_{\left\langle
i,j\right\rangle,s}\left[\hat{c}^{\dagger}_{i,s}\hat{c}_{j,s}+\mathtt{H.c.}\right]+\sum_{i,s}\left(\lambda-\frac{sB}{2}\right)\hat{c}^{\dagger}_{i,s}\hat{c}_{i,s}$
$\displaystyle+\frac{D}{2}\sum_{\left\langle\left\langle
ij\right\rangle\right\rangle,s}\left[\left(i\nu_{ij}s\zeta_{-s}+s\xi_{-s}\right)\hat{c}^{\dagger}_{i,s}\hat{c}_{j,s}+\mathtt{H.c.}\right]$
$\displaystyle+\frac{D^{\prime}}{2}\sum_{\left\langle\left\langle
ij\right\rangle\right\rangle,s}\left[\left(i\nu^{\prime}_{ij}s\zeta_{-s}+s\xi^{\prime}_{-s}\right)\hat{c}^{\dagger}_{i,s}\hat{c}_{j,s}+\mathtt{H.c.}\right]$
(2)

where the mean field parameters are defined as,
$\eta=\sum_{s}\left\langle\hat{\chi}_{ij,s}\right\rangle\equiv\sum_{s}\left\langle\hat{c}^{\dagger}_{i,s}\hat{c}_{j,s}\right\rangle$
evaluated on the nearest neighbour-bonds, and
$\zeta=\zeta^{\prime}=\frac{1}{2}\left\langle\hat{\chi}_{ij,s}+\hat{\chi}_{ji,s}\right\rangle$,
$\xi=\frac{\nu_{ij}}{2i}\left\langle\hat{\chi}_{ij,s}-\hat{\chi}_{ji,s}\right\rangle$
and
$\xi^{\prime}=\frac{\nu^{\prime}_{ij}}{2i}\left\langle\hat{\chi}_{ij,s}-\hat{\chi}_{ji,s}\right\rangle$,
evaluated on next nearest neighbour bonds. The terms associated with the
parmeters $\eta,\nu_{ij}\zeta_{-s}$ of spinon Hamiltonian Eq2 constitute the
Kane-Mele-Haldane modelKim  _et al._ (2016). The term with parameter
$\nu^{\prime}_{ij}\zeta_{-s}$ corresponds to the anti-chiral hopping term
introduced in Ref.Colomés and Franz, 2018. The terms with the parameters
$\xi_{s}$ and $\xi^{\prime}_{s}$ have no effect on the energy or the
topological character of the bands, as the parameters are found to be much
smaller compared to other mean field parameters. $\lambda$ is the Lagrange
undetermined multiplier introduced to implement the local constraint. The mean
field parameters are obtained by solving a set of self-consistent equations,
derived by minimizing the Helmholtz free energy at a particular temperatureSM
.

Figure 2: (color online)(a)band along symmetry lines $\Gamma K$,
$KM$,$MK^{\prime}$,$K^{\prime}\Gamma$ for
$J=1.0,B=0.1,D_{ch}=0.1,D_{Ach}=0.05,T=0.25$. The blue band is for down-spinon
band and the red band is for up-spinon band. (b) The schematic of the
antichiral contribution of the current from edge-states(blue-arrows) and bulk-
states(red-arrows). (c) Nernst-conductivity v.s. temperature plot for
different DMIs. The inset shows magnified figure of the rectangular portion of
the figure.

###  Results.-

Band structure for spinons at a temperature T=0.25J is shown in Fig.2(a). In
the absence of DMI, the two bands cross linearly at the Dirac points K and
K′Pershoguba  _et al._ (2018). A finite DMI opens up a gap with magnitude
$\Delta_{s}=3\sqrt{3}\left|D\zeta_{-s}\right|$ in each spinon sector at K and
K′Owerre (2016a, b, 2017); Pantaleón  _et al._ (2019); Kim  _et al._ (2016).
For anisotropic system ($D_{A}\neq D_{B}$) considered here, the gap opening is
not symmetric and leads to a tilting of the spinon bands near the Dirac
momenta. The band tilting for each band in each spinon sector, defined as the
energy difference between two Dirac-points in the same band, is given by,
$T_{s}^{\tau}=3\sqrt{3}\left|D^{\prime}\zeta_{-s}\right|$. While the anti-
chiral DMI drives the tilting of the bands, it has no effect on the magnitude
of the band gap. Crucially, the tilting is opposite for the two species of
spinons. For the parameters chosen in Fig.2(a), the gap and tilting for the
up-spinon bands are smaller than those for the down spinon bands. This is
because in the presence of positive magnetic field $B=0.1$ considered here,
there are fewer down spinons and consequently,
$\zeta_{\downarrow}<\zeta_{\uparrow}$.

The bands in each spinon sector carry non-zero Berry curvature. Spin Nernst
effect has been proposed as a physical phenomenon to identify Berry curvature
of spinon bands when there are comparable numbers of up and down spinons. Here
we explore whether it can detect the existence of anti-chiral DMI. The Nernst
conductivity has been calculated using the expression
$\mathbin{\alpha^{s}_{xy}}{=\frac{1}{2V}\sum_{\mathbb{k},s,\tau}sc_{1}[\rho^{\tau}_{s}(\mathbb{k})]\Omega^{\tau}_{s}(\mathbb{k})}$Kim
_et al._ (2016),Kovalev and Zyuzin (2016), where
$\mathbin{\rho^{\tau}_{s}(\mathbb{k})}{=1/(\exp(E^{\tau}_{s}(\mathbb{k})/T)-1)}$
is the Bose-Einstein distribution and
$\mathbin{\Omega^{\tau}_{s}(\mathbb{k})}{=i\sum_{\tau^{\prime}\neq\tau}\frac{\left\langle
u^{\tau}_{s}(\mathbb{k})\middle|\frac{\partial\pazocal{H}}{\partial
k_{x}}\middle|u^{\tau^{\prime}}_{s}(\mathbb{k})\right\rangle\left\langle
u^{\tau^{\prime}}_{s}(\mathbb{k})\middle|\frac{\partial\pazocal{H}}{\partial
k_{y}}\middle|u^{\tau}_{s}(\mathbb{k})\right\rangle-(k_{x}\leftrightarrow
k_{y})}{(E^{\tau}_{s}(\mathbb{k})-E^{\tau^{\prime}}_{s}(\mathbb{k}))^{2}}}$.
The results are plotted in Fig.2(c) for different $D$ and $D^{\prime}$.
Increase in $D$ increases the band gap as well as the Berry curvature away
from the Dirac-points. As a result, the Nernst conductivity is substantially
affected by $D$ (Fig.2(c)). Conversely, since the Berry curvature is
independent of $D^{\prime}$, the anti-chiral DMI has very little effect in the
Nernst conductivity. The effect of $D^{\prime}$ on Nernst conductivity can be
observed at low temperature due to tilting of the band structure(inset of
Fig.2(c)). But at higher temperature, the $D^{\prime}$ has no influence in
Nernst conductivity, because the contributions from higher bands overshadows
the effects of band tilting. So, the presence of $D^{\prime}$ in the system is
very hard to detect using Nernst conductivity. Instead, we suggest an
alternative way to detect the presence of antichiral DMI.

The gapped bands are topologically non-trivial with Chern numbers
$C_{\uparrow}^{-}=+1,C_{\uparrow}^{+}=-1,C_{\downarrow}^{-}=-1,C_{\downarrow}^{+}=+1$Matsuoka
_et al._ (2005). Due to bulk-edge correspondence, we expect to observe edge
states in a finite system. In the isotropic limit ($D_{A}=D_{B}$), the edge
states are topologically protected by a chiral symmetry. The spinon currents
along the two edges are equal and opposite for the up and down spinons. This
results in a net flow of spins along the two edges in opposite directions –
any scattering to the bulk states is prevented by symmetry constraints. For
the asymmetric system considered here, $D^{\prime}$ induces an anti-chiral
edge current of spinons where each species of spinon flows in the same
direction along the two edges. This is balanced by couterflow current of
spinons in the opposite direction carried by the bulk modes. The anti-chiral
DMI breaks the chiral symmetry protecting the edge states and enables
scattering between edge and bulk states. This edge-to-bulk scattering produces
the bulk current that balances the anti-chiral edge current. In the following
we discuss how the bulk and edge state dispersion changes due to interplay
between the chiral and anti-chiral DMI.

Fig. 3 shows the spinon bands for a honeycomb nano-ribbon with dimension
$200\times 500$ lattice sites with zigzag edges, together with the spin
current profile along the width of the ribbon. Three different sets of
$(D,D^{\prime})$ are chosen to illustrate the evolution of band dispersion and
spin currents with changing DMI. For clarity of presentation, only one species
of spinons is illustrated. Along with the total spin current, the
contributions from the bulk and two edge modes are calculated separately to
identify the effects of $D^{\prime}$ on each component. The spinon bands and
the individual spin currents are color coded for easy identification. Green
represents the bulk bands and their contribution to the spin current at each
position along the width of the ribbon; red (blue) denotes the localized
spinon mode and the associated spin current at the top (bottom) edge. A
negative (positive) value of the spin current denotes spinon transport to the
left (right) along the length of the ribbon.

Figure 3: (color online)(a) A honeycomb ribbon. The encircled sites are the
basis of unit cell. The figure sets
$\left\\{(b)-(d)\right\\}$,$\left\\{(e)-(g)\right\\}$,$\left\\{(h)-(j)\right\\}$
represents result for down-spinon from $200\times 500$ stripe with DMIs
$\left\\{D=0.1,D^{\prime}=0.001\right\\}$,$\left\\{D=0.001,D^{\prime}=0.1\right\\}$,
$\left\\{D=0.1,D^{\prime}=0.1\right\\}$ respectively. The other parameters are
$J=1.0,B=0.1,T=0.5$ for all the plots. The figure sets
$\left\\{(b),(e),(h)\right\\}$ shows the band structure and inset of the
figures shows the magnified dispersion of the edge states. The figure sets
$\left\\{(c),(f),(i)\right\\}$ gives the spatial current distribution along
width of the stripe. The figure sets $\left\\{(d),(g),(j)\right\\}$ shows the
average of spatial current distribution over four sites respectively. In the
figure sets $\left\\{(b),(c),(e),(f),(h),(i)\right\\}$ the green, red and blue
plots corresponds to bulk-state, upper edge edge-state and lower-edge edge-
state respectively. Moreover, the results for up-spinon is qualitatively same,
the only difference is the dispersions are opposite for the bulk and edge
states.

For $D>D^{\prime}$, the tilting of the bands is small and the dispersion of
edge states at upper and lower edges are opposite, as shown in the Fig.3(c).
The edge states are predominantly chiral in nature, and the spin current at
the two edges are opposite in direction (though not equal in magnitude due to
$D^{\prime}\neq 0$, which breaks chiral symmetry). For large $D^{\prime}$
($D^{\prime}~{}\gg~{}D$), the tilting of the bands at the Dirac points is much
greater and yields identical dispersion for the two edge states (Fig. 3(e)).
This results in anti-chiral edge states where the spin current is in the same
direction along both edges of the ribbon (Fig. 3(f)). Finally, when
$D~{}\approx~{}D^{\prime}$, one of the edge states (the top edge in the
present case) acquires a dispersionless character (Fig. 3(h)). In other words,
the edge state at the top is localized with no spinon transport while the
bottom edge has a finite dispersion with a finite edge current (Fig. 3(i)).
Because of $U(1)$-symmetry of each spinon sector, there is a counter-
propagating bulk current to compensate the imbalance between edge states.
However, the bulk current is not uniform across the width of the ribbon.
Instead, it is primarily confined to a small region near the edges. At each
edge, the bulk current opposes the edge current, with its magnitude decreasing
rapidly away from the edges.

We suggest that magnetic force microscopy(MFM) offers a promising experimental
technique to measure the spinon current across the nano-ribbon and hence can
detect the presence of anti-chiral edge states. Current MFM techniques can
probe the local spin current in a finite sample to a resolution of a few nm.
Since the topological character of the spinon bands for the different ranges
of anisotropic DMI is reflected in distinct current profile across the ribbon,
we believe MFM provides a promising experimental technique to identify anti-
chiral edge states in real quasi-2D materials. Additionally, inelastic neutron
scattering spectra can also indirectly detect the presence of anti-chiral edge
modes, by probing the magnon band structure. If the bands are tilted or the
energy at $K$ and $K^{\prime}$-point are unequal, it will suggest the presence
of anti-chiral edge modes.

Finally, we show that the dynamical-spin structure factor(DSSF) at the edge of
the materialSM  defined as
$\chi(\Omega)=i\sum_{l\in\text{edge}}\left[\chi_{ll}^{xx}(\Omega)+\chi_{ll}^{zz}(\Omega)\right],$
where,
$\mathbin{\chi_{ij}^{\alpha\alpha}(\Omega)}{=-i\int^{\infty}_{-\infty}dte^{-i\Omega
t}\left\langle\hat{S}^{\alpha}_{i}(t)\hat{S}^{\alpha}_{j}(0)\right\rangle_{0}}$
, offers a promising route to detecting anti-chiral edge states. The DSSF,
shown in the figure Fig.4(a)-(b) for different edges, can be interpreted as
the number of edge magnons present in a given energy level, and is
proportional to the product of the density of states and Bose-Einstein
distribution for the corresponding energy level. The signature of the edge
states is reflected in the features of the DSSF near $\Omega=\frac{3J}{2}+B$,
which is the energy of the edge states in the absence of any DMI. Our results
show that $\chi(\Omega)$ is dramatically different for the two edges, when
$\left|D\right|\approx\left|D^{\prime}\right|$(equivalently $D_{A}\gg D_{B}$
or $D_{B}\gg D_{A}$). The DSSF can be measured using the experimental set-up
shown in Fig.4(c), according to referenceJoshi  _et al._ (2018). The quantum
fluctuation of the spins at the edges gives rise to spin current in metal and
as a consequence the spin current gives rise to the charge current in
transverse direction due to inverse spin Hall effect. Measurement of the
noise-spectrum in the charge current gives the information of the DSSF.

The presence of anti-chiral DMI requires two in-equivalent sub-lattices in the
2D-honeycomb lattice, as shown in Fig.4(d). The presence of two different
types of atoms will result in asymmetric DMI, leading to a broken inversion
symmetry and non-zero $D^{\prime}$. Mirror symmetry along the dotted lines
prevents any non-zero perpendicular DMI on nearest-neighbour bonds, whereas
in-plane mirror symmetry suppresses any in-plane DMI. While we

Figure 4: (color online) (color online) (a)-(b) Dynamical spin structure
factor for two edges at temperature $T=0.4J$, with parameters
$D=0.1,D^{\prime}=0.09,B=0.01$. Red and blue colors denote upper and lower
edges. (c) The experimental setup for spin Hall noise spectroscopy. (d) A
ferromagnetic material with two different sub-latices with a mirror symmetric
plane along the dashed lines. (e) A proposed material based on real materials
CrGeTe3 and CrSiTe3 to realize antichiral edge states.

are not aware of any such material at present, the recent discovery of
ferromagnetic order in 2D limit of several Cr-based compounds including CrI3
Chen  _et al._ (2018), CrBr3 Samuelsen  _et al._ (1971); Tsubokawa (1960);
Williams  _et al._ (2015a), CrSrTe3Williams  _et al._ (2015b) and CrGeTe3Gong
_et al._ (2017) as well as the Fe-based family of compounds AFe_2(PO_4)_2
(A=Ba,Cs,K,La)Kim and Kee (2017) offer great promise. These quasi-2D materials
consist of weakly Van Der Waals-coupled honeycomb ferromagets. Presence of
chiral DMI in some members of this family Chen  _et al._ (2018)has been
established using inelastic neutron scattering spectroscopy. In materials like
CrSrTe3 and CrGeTe3, presence of inversion center at the center of honeycomb
cell makes the two sub-lattices equivalent. The inversion symmetry can be
removed by replacing every other Ge atom by an Si atom as depicted in
Fig.4(e). In a similar vein, replacement of P atom by another Group V element
in AFe_2(PO_4)_2 (A=Ba,Cs,K,La)Kim and Kee (2017) will break the inversion
symmetry of the lattice. The breaking of inversion symmetry may, in principle,
give rise to additional interactions in these materials, e.g., nearest
neighbor DMI. However, we have verified that inclusion of additional
interactions, including nearest neighbor DMI as well as 2nd and 3rd nearest
neighbor Heisenberg interactions only modifies the linear dispersion of the
edge states, and does not suppress the appearance of anti-chiral edge statesSM
.

In conclusion, we have studied a Heisenberg ferromagnet with additional next
nearest neighbor DMIs on a honeycomb lattice with broken sublattice symmetry.
The unequal DMI between atoms on different sublattices, together with the
broken chiral symmetry results in the emergence of anti-chiral edge states, in
addition to the normal chiral modes. This is manifested in unique spin current
distribution across the width of a finite system with ribbon geometry. We
propose experimental probes to detect the presence of anti-chiral edge states
as well as a potential material where such states may be realized
experimentally.

Financial support from the Ministry of Education, Singapore, in the form of
grant MOE2016-T2-1-065 is gratefully acknowledged.

##  I Schwinger Boson Mean Field Theory

The parent spin Hamiltonian is,

$\displaystyle\pazocal{H}=$ $\displaystyle-J\sum_{\left\langle
i,j\right\rangle}\bm{S}_{i}\cdot\bm{S}_{j}+D\sum_{\left\langle\left\langle
i,j\right\rangle\right\rangle}\nu_{ij}\hat{z}\cdot\left(\bm{S}_{i}\times\bm{S}_{j}\right)$
$\displaystyle+D^{\prime}\sum_{\left\langle\left\langle
i,j\right\rangle\right\rangle}\nu_{ij}^{\prime}\hat{z}\cdot\left(\bm{S}_{i}\times\bm{S}_{j}\right)-B\sum_{i}S_{i}^{z},$
(3)

After implementing the Schwinger boson transformation and imposing the
constraints, the spinon Hamiltonian reads,

$\displaystyle\pazocal{H}_{sp}=$ $\displaystyle-\frac{J}{2}\sum_{\left\langle
i,j\right\rangle}\left(\hat{c}^{\dagger}_{i,\uparrow}\hat{c}^{\dagger}_{j,\downarrow}\hat{c}_{i,\downarrow}\hat{c}_{j,\uparrow}+\hat{c}^{\dagger}_{i,\downarrow}\hat{c}^{\dagger}_{j,\uparrow}\hat{c}_{i,\uparrow}\hat{c}_{j,\downarrow}\right)$
$\displaystyle-\frac{J}{4}\left(\hat{c}^{\dagger}_{i,\uparrow}\hat{c}_{i,\uparrow}\hat{c}^{\dagger}_{j,\uparrow}\hat{c}_{j,\uparrow}+\hat{c}^{\dagger}_{i,\downarrow}\hat{c}_{i,\downarrow}\hat{c}^{\dagger}_{j,\downarrow}\hat{c}_{j,\downarrow}+\hat{c}^{\dagger}_{i,\uparrow}\hat{c}_{i,\uparrow}\hat{c}^{\dagger}_{j,\uparrow}\hat{c}_{j,\uparrow}+\hat{c}^{\dagger}_{i,\downarrow}\hat{c}_{i,\downarrow}\hat{c}^{\dagger}_{j,\downarrow}\hat{c}_{j,\downarrow}\right)$
$\displaystyle+\frac{iD}{2}\sum_{\left\langle\left\langle
i,j\right\rangle\right\rangle}\nu_{ij}\left(\hat{c}^{\dagger}_{i,\uparrow}\hat{c}_{i,\downarrow}\hat{c}^{\dagger}_{j,\downarrow}\hat{c}_{j,\uparrow}-\hat{c}^{\dagger}_{i,\downarrow}\hat{c}_{i,\uparrow}\hat{c}^{\dagger}_{j,\uparrow}\hat{c}_{j,\downarrow}\right)$
$\displaystyle+\frac{iD^{\prime}}{2}\sum_{\left\langle\left\langle
i,j\right\rangle\right\rangle}\nu^{\prime}_{ij}\left(\hat{c}^{\dagger}_{i,\uparrow}\hat{c}_{i,\downarrow}\hat{c}^{\dagger}_{j,\downarrow}\hat{c}_{j,\uparrow}-\hat{c}^{\dagger}_{i,\downarrow}\hat{c}_{i,\uparrow}\hat{c}^{\dagger}_{j,\uparrow}\hat{c}_{j,\downarrow}\right)$
$\displaystyle-\frac{B}{2}\sum_{i}\left(\hat{c}^{\dagger}_{i,\uparrow}\hat{c}_{i,\uparrow}-\hat{c}^{\dagger}_{i,\downarrow}\hat{c}_{i,\downarrow}\right)+\sum_{i}\lambda_{i}\left(\hat{c}^{\dagger}_{i,\uparrow}\hat{c}_{i,\uparrow}+\hat{c}^{\dagger}_{i,\downarrow}\hat{c}_{i,\downarrow}-2S\right)$
$\displaystyle+3NJS^{2}-4SN\lambda$ (4)

The bond operators have to be choosen such that the total number of spinon is
conserved in the mean field Hamiltoian which is equivalent to
$S_{z}$-conservation in terms of spinLee  _et al._ (2015). Defining the bond-
operators, $\hat{\chi}_{ij}=\hat{c}^{\dagger}_{i,s}\hat{c}_{j,s}$ and
$\hat{\chi}_{ij}=\left(\hat{\chi}_{ij,\uparrow}+\hat{\chi}_{ij,\downarrow}\right)/2$,
we can re-write the Hamiltonian as,

$\displaystyle\pazocal{H}_{sp}=$ $\displaystyle-2J\sum_{\left\langle
i,j\right\rangle}:\hat{\chi}_{ij}^{\dagger}\hat{\chi}_{ij}:$
$\displaystyle-\frac{D}{2}\sum_{\left\langle\left\langle
i,j\right\rangle\right\rangle}i\nu_{ij}\left(:\hat{\chi}_{ij,\uparrow}^{\dagger}\hat{\chi}_{ij,\downarrow}:-:\hat{\chi}_{ij,\downarrow}^{\dagger}\hat{\chi}_{ij,\uparrow}:\right)$
$\displaystyle-\frac{D^{\prime}}{2}\sum_{\left\langle\left\langle
i,j\right\rangle\right\rangle}i\nu^{\prime}_{ij}\left(:\hat{\chi}_{ij,\uparrow}^{\dagger}\hat{\chi}_{ij,\downarrow}:-:\hat{\chi}_{ij,\downarrow}^{\dagger}\hat{\chi}_{ij,\uparrow}:\right)$
$\displaystyle-\frac{B}{2}\sum_{i}\left(\hat{c}^{\dagger}_{i,\uparrow}\hat{c}_{i,\uparrow}-\hat{c}^{\dagger}_{i,\downarrow}\hat{c}_{i,\downarrow}\right)$
$\displaystyle+\lambda\sum_{is}\hat{c}^{\dagger}_{i,s}\hat{c}_{i,s}-4SN\lambda+3NJS^{2}$
(5)

New bond operators are defined so that corresponding mean field parameters are
real:
$\hat{A}_{ij,s}=\frac{1}{2}\left(\hat{\chi}_{ij,s}+\hat{\chi}_{ji,s}\right)$,
$\hat{B}_{ij,s}=\frac{\nu_{ij}}{2i}\left(\hat{\chi}_{ij,s}-\hat{\chi}_{ji,s}\right)$,
$\hat{B^{\prime}}_{ij,s}=\frac{\nu^{\prime}_{ij}}{2i}\left(\hat{\chi}_{ij,s}-\hat{\chi}_{ji,s}\right)$.
Defining mean field parameters
$\eta=\left\langle\hat{\chi}_{ij}^{\dagger}\right\rangle=\left\langle\hat{\chi}_{ij}\right\rangle$,
$\zeta_{s}=\left\langle\hat{A}_{ij,s}^{\dagger}\right\rangle=\left\langle\hat{A}_{ij,s}\right\rangle$,
$\xi_{s}=\left\langle\hat{B}_{ij,s}^{\dagger}\right\rangle=\left\langle\hat{B}_{ij,s}\right\rangle$,
$\xi^{\prime}_{s}=\left\langle\hat{B^{\prime}}_{ij,s}^{\dagger}\right\rangle=\left\langle\hat{B^{\prime}}_{ij,s}\right\rangle$,
the quartic terms of Hamiltonian can be decoupled into quadratic and the mean-
field spinon Hamiltonian takes the form as,

$\displaystyle\pazocal{H}_{sp}^{mf}=$ $\displaystyle-\eta J\sum_{\left\langle
i,j\right\rangle,s}\left[\hat{c}^{\dagger}_{i,s}\hat{c}_{j,s}+\mathtt{H.c.}\right]$
$\displaystyle+\frac{D}{2}\sum_{\left\langle\left\langle
ij\right\rangle\right\rangle,s}\left[i\nu_{ij}s\zeta^{ch}_{-s}\hat{c}^{\dagger}_{i,s}\hat{c}_{j,s}+\mathtt{H.c.}\right]$
$\displaystyle+\frac{D}{2}\sum_{\left\langle\left\langle
ij\right\rangle\right\rangle,s}\left[s\xi^{ch}_{-s}\hat{c}^{\dagger}_{i,s}\hat{c}_{j,s}+\mathtt{H.c.}\right]$
$\displaystyle+\frac{D^{\prime}}{2}\sum_{\left\langle\left\langle
ij\right\rangle\right\rangle,s}\left[i\nu^{\prime}_{ij}s\zeta^{Ach}_{-s}\hat{c}^{\dagger}_{i,s}\hat{c}_{j,s}+\mathtt{H.c.}\right]$
$\displaystyle+\frac{D^{\prime}}{2}\sum_{\left\langle\left\langle
ij\right\rangle\right\rangle,s}\left[s\xi^{Ach}_{-s}\hat{c}^{\dagger}_{i,s}\hat{c}_{j,s}+\mathtt{H.c.}\right]$
$\displaystyle+\sum_{i,s}\left(\lambda-\frac{sB}{2}\right)\hat{c}^{\dagger}_{i,s}\hat{c}_{i,s}$
$\displaystyle+6NJ\eta^{2}-6DN\sum_{s}s\zeta_{s}\xi_{-s}-6D^{\prime}N\sum_{s}s\zeta_{s}\xi^{\prime}_{-s}-4SN\lambda+3NJS^{2}$
(6)

Fourier transformation of the mean field Hamiltonian to momentum-space yields,

$\pazocal{H}_{sp}^{mf}=\sum_{\bm{k}\in\mathtt{B.Z.},s}\Psi^{\dagger}_{\bm{k},s}\left[g_{s}(\bm{k})I+\bm{h}_{s}(\bm{k})\cdot\bm{\sigma}\right]\Psi_{\bm{k},s}+E_{0},$
(7)

where,
$\Psi_{\bm{k},s}^{\dagger}=\left(\hat{a}_{\bm{k},s}^{\dagger},\hat{b}_{\bm{k},s}^{\dagger}\right)$.
$\hat{a}_{\bm{k},s}^{\dagger}$ and $\hat{b}_{\bm{k},s}^{\dagger}$ are the
creation operators for Schwinger bosons on sublattice-A and sublattice-B (see
Fig.1(a) of main text), respectively. $\bm{\sigma}_{\alpha}$($\alpha=x,y,z$)
represents the Pauli matrices. The other terms are given by,

$\displaystyle g_{s}(\bm{k})=$
$\displaystyle-\frac{sB}{2}+\lambda+sD\xi_{-s}\gamma_{c}^{\beta}+sD^{\prime}\left(\xi^{\prime}_{-s}\gamma_{c}^{\beta}-\zeta_{-s}\gamma_{s}^{\beta}\right),$
$\displaystyle h_{s}(\bm{k})=$
$\displaystyle\begin{pmatrix}-J\eta\gamma_{c}^{\alpha}\\\
J\eta\gamma_{s}^{\alpha}\\\ -Ds\zeta_{-s}^{c}\gamma_{s}^{\beta}\end{pmatrix},$
$\displaystyle E_{0}=$ $\displaystyle
6NJ\eta^{2}-6DN\sum_{s}s\zeta_{s}\xi_{-s}-6D^{\prime}N\sum_{s}s\zeta_{s}\xi^{\prime}_{-s}-4SN\lambda+3NJS^{2},$
(8)

where,
$\gamma_{c}^{\beta}=\sum_{j}\cos(\bm{k}\cdot\bm{\beta}_{j}),\gamma_{s}^{\beta}=\sum_{j}\sin(\bm{k}\cdot\bm{\beta}_{j}),\gamma_{c}^{\alpha}=\sum_{j}\cos(\bm{k}\cdot\bm{\alpha}_{j}),\gamma_{s}^{\alpha}=\sum_{j}\sin(\bm{k}\cdot\bm{\alpha}_{j})$
and the vectors $\bm{\beta}_{j}$ and $\bm{\alpha}_{j}$ are shown in figure
Fig.1(a). $N$ is the number of unit cells in lattice. $E_{0}$ is the energy of
the ground state and the energies of spinons are considered with respect to
the ground state energy.

After diagonalizing the k-space Hamiltonian we get,

$\pazocal{H}_{sp}^{mf}=E_{0}+\sum_{\bm{k},s,\tau}E_{s}^{\tau}(\bm{k})\hat{c}^{\dagger}_{\bm{k},\tau,s}\hat{c}_{\bm{k},\tau,s},$
(9)

where, the relative energies,

$E_{s}^{\tau}(\bm{k})=g_{s}(\bm{k})+\tau\left|h_{s}(\bm{k})\right|,$ (10)

refer to the upper ($\tau=+1$) and the lower ($\tau=-1$) band for each spinon
sectors $s=\pm 1$.

From this we get the internal energy and the entropy of the non-interacting
system as,

$\displaystyle U$
$\displaystyle=E_{0}+\sum_{\bm{k},s,\tau}\rho_{s}^{\tau}(\bm{k})E_{s}^{\tau}(\bm{k}),$
$\displaystyle S$
$\displaystyle=k_{B}\sum_{\bm{k},s,\tau}[\left(1+\rho_{s}^{\tau}(\bm{k})\right)\ln\left(1+\rho_{s}^{\tau}(\bm{k})\right)-\rho_{s}^{\tau}(\bm{k})\ln\rho_{s}^{\tau}(\bm{k})],$
(11)

where,
$\rho_{s}^{\tau}(\bm{k})=\left[\exp\left(E^{\tau}_{s}(\bm{k})\right)-1\right]^{-1}$
is the Bose-Einstein distribution of spin-s spinons in the $\tau$-band. The
Helmohltz-free-energy is given by,

$\displaystyle G$ $\displaystyle=U-TS$
$\displaystyle=E_{0}-{k_{B}T}\sum_{\bm{k},s,\tau}\ln\left(\frac{1}{1-\exp\left(\frac{-E_{s}^{\tau}(\bm{k})}{k_{B}T}\right)}\right)$
(12)

After minimizing the Helmholtz free energy with respect to the mean field
parameters, we get six self consistent equations, given by,

$\displaystyle 2S=$
$\displaystyle\frac{1}{2N}\sum_{\bm{k},\tau,s}\rho_{s}^{\tau}(\bm{k})$
$\displaystyle 1=-$
$\displaystyle\frac{J}{12N}\sum_{\bm{k},s,\tau}\tau\frac{\rho_{s}^{\tau}(\bm{k})}{\left|h_{s}\right|}\left|\sum_{j}e^{i\bm{k}\cdot\bm{\alpha}_{j}}\right|^{2}$
$\displaystyle D\xi_{s}+D^{\prime}\xi^{\prime}_{s}=$
$\displaystyle\frac{1}{6N}\sum_{\bm{k},\tau}\left(D^{\prime}-\tau
s\frac{D^{2}\zeta_{-s}\gamma}{h_{s}}\right)\rho_{s}^{\tau}(\bm{k})\gamma_{s}^{\beta}$
$\displaystyle\zeta_{s}=$
$\displaystyle\frac{1}{6N}\sum_{\bm{k},\tau}\rho^{\tau}_{s}(\bm{k})\sum_{j}\cos(\bm{k}\cdot\bm{\beta}_{j}),$
(13)

The mean field parameters are obtained by solving these six self-consistent
equations. It is notable that the mean field parameter $\eta$ can be chosen as
real, absorbing the complex phase factor into operator $\hat{c}_{i,s}$. All
other mean field parameters already chosen to be real. Using the parameters,
we plot the band structure and evaluate corresponding topological information.
For a fixed set of $J,D,D^{\prime},B$, the mean field parameters are solved
and plotted against temperature $T$ in Fig5. The parameters $\eta$ and
$\zeta_{s}$ represent short range correlations identifying magnetic ordering
and serve as order parameters for the ferromagnetic to paramegnetic transition
at higher temperaturesSarker  _et al._ (1989). The constraint $\lambda$ is
considered uniform throughout the lattice to retain the translational symmetry
of the lattice.

At low temperatures, finite, non-zero values of $\eta$ and $\zeta_{\uparrow}$
denote ferromagnetic ordering. A positive $B$ determines that the spins are
all aligned along the +ve x-direction at $T=0$. In other words, the system is
populated with up-spinons. As the temperature increases, thermally excited
down-spinons are generated, resulting in a finite, non-zero
$\zeta_{\downarrow}$. Finally at high temperatures, a vanishing of all the
mean field parameters denote a transition to the paramagnetic phase. The
paramagnetic phase transition with all zero correlations to be expected to be
an outcome of large-N expansion. It has been shown for Heisenberg model that
taking into account of the quantum fluctuations in the mean field parameter
removes the phase transitionTchernyshyov and Sondhi (2002).

Figure 5: (color online)Plot of mean-field parameters, for
$J=1.0,D=0.1,D^{\prime}=0.05,B=0.1$

##  II Calculation of the edge-state and velocity distribution on a stripe
geometry

Figure 6: (color online)The honeycomb lattice structure of size ($20\times
14$) with periodic boundary condition along x-axis and open boundary condition
along y-axis. The encircled sites are the basis of unit cell.

The Hamiltonian in tight binding Hamiltonian can be written as,

$\pazocal{H}=\sum_{ij}t_{ij}\ket{i}\bra{j},$ (14)

where, i and j are the sites of lattice and $t_{ij}$ is the hopping amplitude.
More explicitly the Hamiltonian can be written as a matrix form asGuclu  _et
al._ (2014),

$\pazocal{H}=\sum_{m,s}\left[\Psi_{m,s}^{\dagger}U_{s}\Psi_{m,s}+\Psi_{m,s}^{\dagger}T_{s}\Psi_{m+1,s}+\Psi_{m+1,s}^{\dagger}T_{s}^{\dagger}\Psi_{m,s}\right],$
(15)

where,
$\Psi_{m,s}^{\dagger}=(\hat{b}_{1,m,s}^{\dagger},\hat{a}_{2,m,s}^{\dagger},...,\hat{a}_{N,m,s}^{\dagger},)$
and $N$ is the number of sites along the stripe(the sites inside the red
circle of Fig.6 makes one stripe) and $m$ denotes the stripe index and
$s$-denotes up or down spinon type. $U$ and $T$ are $N\times N$ matrices.
Matrix-$U$ contains all the onsite and intra-stripe hopping elements(inside
the red circle of Fig.6) and matrix-$T$ contains all the inter-stripe hopping
elements.

Imposing periodic boundary condition along the longitudinal direction as shown
in the figure6, one can Fourier transform the Hamiltonian with a 1D Bloch-wave
vector, given by $\Psi_{m}=(1/\sqrt{M})\sum_{k=0}^{M-1}\Psi_{k}e^{-i2\pi
km/M}$, where $M$ is the number of stripes along x-axis in Fig.6. After
Fourier transform the Hamiltonian can be written as,

$\displaystyle H$
$\displaystyle=\sum_{k}\Psi^{\dagger}_{k}\left[U+\left(Te^{i\frac{2k\pi}{M}}+\text{H.c.}\right)\right]\Psi_{k}$
$\displaystyle=\sum_{k}\Psi^{\dagger}_{k}H_{k}\Psi_{k},$ (16)

The Hamiltonian can be diagonalized as,

$\epsilon_{k}=P^{\dagger}H_{k}P,$ (17)

where, $P$ is an unitary matrix and the corresponding eigenvector is given by,

$\Psi_{k}^{d}=P^{\dagger}\Psi_{k}$ (18)

Diagonalizing the momentum space Hamiltonian, we obtain the bands for the
stripe geometry, as shown in the figures Fig.3(b), 3(e), 3(h) of the main
text.

The velocity operator, used to evaluate the spinon transport properties, are
expressed in terms of the k-space eigenstates asD. Mahan (2000)Laurent ,

$\hat{v}=\sum_{ij}v_{ij}\ket{i}\bra{j},$ (19)

where the coefficients $v_{ij}$ is given by,

$v_{ij}=\bra{i}\hat{v}\ket{j}=-\frac{i}{\hbar}\bra{i}[\hat{r},\pazocal{H}]\ket{j}=-\frac{i}{\hbar}(\bm{r}_{i}-\bm{r}_{j})t_{ij}.$
(20)

We have calculated the distribution of the velocity component along x-axis
across the cross section of the ribbon, which are shown in figure
Fig.3(c),3(f),3(i) of the main text.

##  III Calculation of the dynamical spin structure factor

To calculate the dynamical spin structure factor, we have used the Holstein-
Primakoff transformation, given by,

$\displaystyle\hat{S}^{x}_{m}=\frac{1}{2}\left(\hat{a}_{m}+\hat{a}_{m}^{\dagger}\right),$
$\displaystyle\hat{S}^{y}_{m}=\frac{i}{2}\left(\hat{a}_{m}-\hat{a}_{m}^{\dagger}\right),$
$\displaystyle\hat{S}^{z}_{m}=S-\hat{a}_{m}^{\dagger}\hat{a}_{m},$ (21)

where, $\hat{a}_{m}^{\dagger}$ and $\hat{a}_{m}$ are the creation annihilation
operators of Holstein-Primakoff bosons. For the case of ferromagnet with up
spin at each site, the Holstein-Primakoff boson represents the down-spinons in
Schwinger boson picture at low temperature. The diagonalized Hamiltonian can
be written as,

$H_{k}=\sum_{n}\epsilon_{n}(k)\hat{a}_{n,k}^{d\dagger}\hat{a}_{n,k}^{d},$ (22)

where, $\hat{a}^{d}_{n,k}$ is the bosonic operator after diagonalization.
Using the above relation and Heisenberg’s equation of motion it can be proved
that,

$\displaystyle\hat{a}_{n,k}^{d\dagger}(t)$
$\displaystyle=\hat{a}_{n,k}^{d\dagger}(0)e^{i\epsilon_{n}(k)t}$
$\displaystyle\hat{a}_{n,k}^{d}(t)$
$\displaystyle=\hat{a}_{n,k}^{d}(0)e^{-i\epsilon_{n}(k)t}$ (23)

The dynamical spin structure factor in terms of Holstein-Primakoff Boson, is
given by,

$\displaystyle\chi(\Omega)$
$\displaystyle=\sum_{m\in\text{edge}}\left[\chi^{xx}_{mm}(\Omega)+\chi^{zz}_{mm}(\Omega)\right]$
$\displaystyle=\sum_{m\in\text{edge}}\left[\int^{\infty}_{-\infty}dte^{-i\Omega
t}\left(-i\left\langle\hat{S}^{x}_{1,m}(t)\hat{S}^{x}_{1,m}(0)\right\rangle\right)+\int^{\infty}_{-\infty}dte^{-i\Omega
t}\left(-i\left\langle\hat{S}^{z}_{1,m}(t)\hat{S}^{z}_{1,m}(0)\right\rangle\right)\right]$
$\displaystyle=-i\sum_{m}\int^{\infty}_{-\infty}dte^{-i\Omega
t}\left[\frac{S}{2}\left\langle\hat{a}_{1,m}(t)\hat{a}_{1,m}(0)+\hat{a}_{1,m}(t)\hat{a}_{1,m}^{\dagger}(0)+\hat{a}_{1,m}^{\dagger}(t)\hat{a}_{1,m}(0)+\hat{a}_{1,m}^{\dagger}(t)\hat{a}_{1,m}(0)\right\rangle\right.$
$\displaystyle\qquad\qquad\qquad\left.+\left\langle
S^{2}-S\hat{a}_{1,m}^{\dagger}(0)\hat{a}_{1,m}(0)-S\hat{a}^{\dagger}_{1,m}(t)\hat{a}_{1,m}(t)\right\rangle\right]\qquad\qquad\text{[Neglecting
the higer order terms]}.$ (24)

Transforming the boson operator $\hat{a}_{1,m}(t)$ into diagonalized boson
operator $\hat{a}^{d}_{1,m}(t)$ and using the relation Eq.23, we derive the
spin-structure factor.

##  IV Modulation of edge state dispersion in presence of other interactions
in a honeycomb ferromagnet

Figure 7: (color online) DM-interactions on nearest neighbour bonds.

The spin Hamiltonian studied in theis work is an idealized model. For example,
the long range Heisenberg interactions are neglected. Moreover, the breaking
of inversion symmetry required for anti-chiral DM-term $D^{\prime}$ might give
rise the nearest neighbour out of plane DM-interactions and also in plane DM-
interactions. At low temperature, the in plane interactions can be neglected,
which gives rise to three magnon interactions in terms of Holstein Primakoff
Bosons. Neglecting, any presence of in plane DM-interaction at low
temperature, we can re-write a more general Hamiltonian of the material as,

$\displaystyle\pazocal{H}=J_{1}\sum_{\left\langle i,j\right\rangle}\bm{S}_{i}$
$\displaystyle\cdot\bm{S}_{j}+J_{2}\sum_{\left\langle\left\langle
i,j\right\rangle\right\rangle}\bm{S}_{i}\cdot\bm{S}_{j}+J_{3}\sum_{\left\langle\left\langle\left\langle
i,j\right\rangle\right\rangle\right\rangle}\bm{S}_{i}\cdot\bm{S}_{j}$
$\displaystyle+D_{1}\sum_{\left\langle
ij\right\rangle_{A}}\nu_{ij}\hat{z}\cdot(\bm{S}_{i}\times\bm{S}_{j})+D_{2}\sum_{\left\langle
ij\right\rangle_{B}}\nu_{ij}\hat{z}\cdot(\bm{S}_{i}\times\bm{S}_{j})$
$\displaystyle+D\sum_{\left\langle\left\langle
ij\right\rangle\right\rangle_{B}}\nu_{ij}\hat{z}\cdot(\bm{S}_{i}\times\bm{S}_{j})+D^{\prime}\sum_{\left\langle\left\langle
ij\right\rangle\right\rangle_{B}}\nu^{\prime}_{ij}\hat{z}\cdot(\bm{S}_{i}\times\bm{S}_{j})$
$\displaystyle+B\sum_{i}\bm{S}_{i},$ (25)

where, the DM-interactions $D_{1}$ and $D_{2}$ are defined on the nearest
neighbour bonds $\left\langle ij\right\rangle_{A}$ and $\left\langle
ij\right\rangle_{B}$ as shown in Fig.7. Any single ion anisotropy terms acts
as chemical potential for the spin-excitation, and is accounted for by a
renormalization of the magnetic field. As a more realistic model, we have
fixed the Heisenberg interactions present in the material CrI3Chen  _et al._
(2018), $J_{1}=2.09$ meV, $J_{2}=0.16$ meV, $J_{3}=0.18$ meV. The nearest
neighbour DM-terms and magentic field are fixed as $D_{1}=0.1$ meV,
$D_{2}=0.15$ meV, $B=g\mu_{B}B_{z}=0.01$ meV. We transformed the spin
Hamiltonian into magnon Hamiltonian using Holstein Primakoff transformation
defined in Eq.21. Then, plotted the band structure and dynamical spin
structure factor in Fig8. Our results confirm that the qualitative behaviour
is same as the ideal model considered in the test and the behaviour of edge
states mostly depends on the DM-interactions $D$ and $D^{\prime}$. The
presence of other interaction terms in the Hamiltonian just distorts the
linear dispersion of the edge states to a non-linear dispersion.

Figure 8: (color online) Band structure of magnons of a stripe geometry for
parameters (a) $D=0.31meV$, $D^{\prime}=0.01meV$, (b) $D=0.31meV$,
$D^{\prime}=0.28meV$, (c) $D=0.01meV$, $D^{\prime}=0.31meV$. The dynamical
spin structure factor for upper edge at $T=0.4$ for parameters, (d)
$D=0.31meV$, $D^{\prime}=0.01meV$, (e) $D=0.31meV$, $D^{\prime}=0.28meV$, (f)
$D=0.01meV$, $D^{\prime}=0.38meV$. The dynamical spin structure factor for
lower edge at $T=0.4$ for parameters, (g) $D=0.31meV$, $D^{\prime}=0.01meV$,
(h) $D=0.31meV$, $D^{\prime}=0.28meV$, (i) $D=0.01meV$, $D^{\prime}=0.38meV$.

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