PHYSICAL REVIEW B 107, 024408 (2023)

Topological magnons on the triangular kagome lattice

1 1,2,*
Meng-Han Zhang () and Dao-Xin Yao ()
1
State Key Laboratory of Optoelectronic Materials and Technologies, Center for Neutron Science and Technology, Guangdong Provincial Key
Laboratory of Magnetoelectric Physics and Devices, School of Physics, Sun Yat-Sen University, Guangzhou 510275, China
2
International Quantum Academy, Shenzhen 518048, China

(Received 12 July 2022; revised 11 December 2022; accepted 14 December 2022; published 9 January 2023)

We present the topology of magnons on the triangular kagome lattice (TKL) by calculating its Berry
curvature, Chern number, and edge states. In addition to the ferromagnetic state, the TKL hosts ferrimagnetic
ground state as its two sublattices can couple with each other either ferromagnetically or antiferromagnetically.
Using Holstein-Primakoff (HP) boson theory and Green’s function approach, we find that the TKL has a rich
topological band structure with added high Chern numbers compared with the kagome and honeycomb lattices.
The magnon edge current allows a convenient calculation of thermal Hall coefficients and the orbital angular
momentum gives correlation to the Einstein–de Haas effect. We apply the calculations to the TKL and derive
the topological gyromagnetic ratio showing a nonzero Einstein–de Haas effect in the zero-temperature limit.
Our results render the TKL as a potential platform for quantum magnonics applications including high-precision
mechanical sensors and information transmission.

DOI: 10.1103/PhysRevB.107.024408

I. INTRODUCTION response than the kagome and honeycomb lattices both in

the thermal Hall effect and the EdH effect. Our calculations
The discovery of gyromagnetism [1], the interconversion
are applicable to the magnon transport theory which makes
between spin and mechanical rotational motions, revealed that
remarkable progress in coding and processing information
the origin of magnetism was the intrinsic angular momen-
[23] due to the small dissipation significantly reducing the
tum of electrons. By determining the gyromagnetic ratio [2],
energy consumption [24,25].
the Einstein–de Haas (EdH) effect provides a more accurate
Distinct from the ordinary bipartite lattices, the TKL with
measurement of the rotational motion rather than electron-
nine spins in the unit cell can produce magnetic long-range
spin resonance or ferromagnetic resonance [3]. Recent studies
order in both ferro- and ferrimagnetic states [26–30]. It is
show that circularly polarized phonons can absorb the angular
worthwhile to study the topological properties of magnons
momentum of the spin system, which provide an atomistic
and related effects for these ordered states on the TKL which
picture of the EdH effect [4]. Indicating the transfer between a
has been realized experimentally in a two-dimensional metal
magnetic moment and a macroscopic mechanical rotation, the
organic framework halide series, Cu9 X2 (cpa)6 (X = F, Cl, Br;
EdH technique attracts increasing attention and has important
cpa = anion of 2-carboxypentonic acid) [31,32]. It is best
consequences in the fields of quantum thermal transport [5],
described as a spin frustrated TKL on a layered metal organic
nano-magneto-mechanical systems [6–9], spintronics [10],
framework formed by inserting an extra set of triangles inside
magnonics [11], and ultrafast magnetism [12,13].
of the kagome triangles [33,34]. With an odd number of spins
As the bosonic analog of the electron system, the orbital
in the unit cell, the TKL gives rise to three times the unit cell
motions of magnons are driven by the Berry curvature in mo-
of the kagome lattice [35], and hence a new platform to ex-
mentum space from the topological band structure [14–17].
plore topological magnon effects. The Dzyaloshinskii-Moriya
These orbital motions cause the thermal Hall effect arising
(DM) interaction induces a fictitious magnetic flux and leads
from the edge current of magnons. It has been observed ex-
to the existence of nonzero Berry curvature. With different
perimentally in a number of three-dimensional ferromagnetic
Heisenberg exchange couplings, the nonzero DM interaction
pyrochlores (Lu2 V2 O7 , Ho2 V2 O7 , and In2 Mn2 O7 ) [18,19].
on the TKL induces a rich phase diagram accompanied by the
According to the linear response theory, there is a reduced
topologically protected gapless edge modes. As the inversion
angular momentum generated by the orbital motion of the
symmetry breaking can eliminate the degeneracy of energy
magnon [20,21]. The reduced angular momentum per unit
bands, the TKL provides a promising avenue for realizing
cell consists of two components, the edge current and the
exotic quantum phenomena [36,37], magnon thermal devices
self-rotation, and is related to the EdH effect [22]. We ap-
[38,39], and magnon mechanical devices [22].
ply the calculation of angular momentum on various lattices
In this work, we theoretically study the topological magnon
finding that the triangular kagome lattice (TKL) has a larger
excitations on the TKL proposing effective realizations for
both the thermal Hall effect and the EdH effect. We track
the corresponding density of states (DOS) of edge states
*
Corresponding author: yaodaox@mail.sysu.edu.cn by using the real-space Green’s function. The thermal Hall

2469-9950/2023/107(2)/024408(10) 024408-1 ©2023 American Physical Society

MENG-HAN ZHANG AND DAO-XIN YAO PHYSICAL REVIEW B 107, 024408 (2023)

conductance behavior κ xy provides useful insights of the

magnon transport as it can detect the charge-neutral quasi-
particles that would not directly couple to electromagnetic
probes. The EdH physics is properly captured by the thermal
dependence of the gyromagnetic ratio as a function of the
different material parameters. This behavior is inherited from
the topology of the magnon bulk bands and further confirms
the sign change behavior of thermal Hall response. Through
estimating the gyromagnetism, we find that the TKL has a
larger EdH response than the kagome and honeycomb lattices.
This paper is organized as follows. In Sec. II we introduce
the model (Sec. II A) and present the equations for spin-wave
Hamiltonians using the spin-wave theory and HP boson the-
ory. Chern number and thermal Hall conductance are defined FIG. 1. Schematics of the TKL with shaded regions that repre-
in Sec. II C. Then, we present edge state geometry, the formal- sent the unit cell. a1 , a2 are basis vectors of the primitive unit cell
ism of Green’s function (Sec. II B), and angular momentum and the arrows within the single triangular blocks indicate the config-
expressions (Sec. II D). In Sec. III we present our results urations of the DM-induced flux, highlighted by black solid arrows.
on topological energy bands (Sec. III A), density of states The red arrows are A sites and others are B sites. (a) Ferromagnetic
(Sec. III B), thermal Hall effect (Sec. III C), and finally we ground state with Ja > 0 and Jb > 0. (b) Ferrimagnetic ground state
discuss the Einstein–de Haas effect of our results (Sec. III D). with Ja > 0 and Jb <0.
In Sec. IV we discuss and conclude our findings.

The TKL has a ferromagnetic ground state for Ja > 0

II. MODEL AND METHODS
and Jb > 0 in Fig. 1(a), while a ferrimagnetic ground state
A. Triangular kagome spin model for Ja > 0 and Jb < 0 is shown in Fig. 1(b). Here we use
To present the method of approach in a concrete back- the Holstein-Primakoff (HP) representation to study the mag-
ground we consider a Heisenberg model on the TKL with nine netic excitations for the ordered states. The original spin
spins in the unit cell, where the total Hamiltonian is given by Hamiltonian can be mapped to a bosonic tight-binding model
following the HP transformation:
H = H0 + HDM + HK + HB . (1) 
Our model Hamiltonian contains the nearest-neighbor Heisen- Sm+ = Smx + iSmy = 2S − αm† αm αm ,
berg exchange interactions, where the H0 is 
  Sm− = Smx − iSmy = αm† 2S − αm† αm ,
H0 = −Ja Sm · Sn − Jb Sm · Sn , (2)
mn mn Smz = S − αm† αm , (6)

and Ja , Jb are two types of the nearest-neighbor exchange where αm† (αm ) is the bosonic magnon creation (annihi-
couplings within the sublattice  (A trimers indicated with red lation) operator at site m. Within the approximation of
sites) and ∇ (B trimers indicated with green sites) as shown  √
2S − αm† αm → 2S, the Hamiltonian has the form
in Fig. 1. The HDM term represents the nearest-neighbor DM
interaction which is usually dominant perturbative anisotropy 
 
to the Heisenberg exchange interactions. Therefore, it could H=− (Ja + iνmn D)Sαm† αn + (Jb + iνmn D)Sαm† αn
be considered as mna mnb
 
HDM = Dmn · (Sm × Sn ). (3) 
mn + H.c. + (2K + h) αm† αm + E0 , (7)
m
Here we introduce the anisotropy term and the Zeeman term
to have the magnetic order even at finite temperature based on where D is the z component of the nearest-neighbor DM
the Mermin-Wagner theorem [40,41]. The anisotropy term is interaction, E0 is ground state energy, and νmn = ±1 corre-
given by sponding to the direction of DM interaction. Subsequently, we
  2 perform the Fourier transformation using the definition
HK = −K Smz , (4)
m 1  ik·Rm †
αk† = √ e αm . (8)
where K is the easy-axis anisotropy along the z axis. And the N m
external Zeeman magnetic field term is given by
 Thus, in the reciprocal space the Hamiltonian is given by
HB = −h Smz , (5)
m

H= ψk† H (k)ψk , (9)
where h = gμB B, B is the external magnetic field. k

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TOPOLOGICAL MAGNONS ON THE TRIANGULAR KAGOME … PHYSICAL REVIEW B 107, 024408 (2023)

where ψk† = (α1,k

†
, α2,k
†
, α3,k
†
, α4,k
†
, α5,k
†
, α6,k
†
, α7,k
†
, α8,k
†
, α9,k
†
).
The spin-wave Hamiltonian matrix is
⎡ ⎤
E1 I3×3 Ak Bk
⎢ ⎥
S ⎣ A†k Ck 03×3 ⎦, (10)
†
Bk 03×3 Dk
matrix Ak is
⎡ ⎤
−γ1 e−ik·a1 −γ1 e−ik·a2 0
⎣ −γ1 eik·a1 0 −γ1 eik·(a1 −a2 ) ⎦, (11)
0 −γ1 eik·a2 −γ1 eik·(−a1 +a2 )
matrix Bk is
⎡ ⎤
−γ1 eik·a1 −γ1 eik·a2 0
⎣−γ1 e−ik·a1 0 −γ1 eik·(−a1 +a2 ) ⎦, (12)
0 −γ1 e−ik·a2 −γ1 eik·(a1 −a2 )
matrix Ck is
FIG. 2. The TKL ribbon with periodic boundary condition along
⎡ ⎤
E2 −γ2 eik·(a1 −a2 ) −γ2 e−ik·a2 y axis and open boundary condition along the x axis. The ribbon has
⎣−γ2 eik·(−a1 +a2 ) E2 −γ2 e−ik·a1 ⎦, (13) W periodic one-dimensional chains; the numbers nearing sites are x
−γ2 eik·a2 −γ2 eik·a1 E2 indices.

and matrix Dk is
⎡ ⎤ Hamiltonian matrix can be written as
E2 −γ2 eik·(−a1 +a2 ) −γ2 eik·a2
⎣−γ2 eik·(a1 −a2 ) ⎡ ⎤
E2 −γ2 eik·a1 ⎦, (14) G(k) F (k)† 0 ··· 0
−γ2 e−ik·a2 −γ2 e−ik·a1 E2 ⎢ ... .. ⎥
⎢F (k) G(k) F (k)† . ⎥
⎢ ... ... ⎥
where E1 = 4Jb + 2K + h, E2 = 2Ja + 2Jb + 2K + h, γ1 = H (k) = ⎢ 0 ⎥
⎢ 0 F (k) ⎥, (17)
Jb + iνmn D, and γ2 = Ja + iνmn D. The√lattice vectors are ⎢ .. ... ... ... ⎥
given by a1 = 41 (1, 0)a and a2 = 18 (−1, 3)a with the lattice ⎣ . F (k) †⎦

constant chosen as a = 0.1 nm. The energy bands obtained via 0 ··· 0 F (k) G(k)
diagonalizing the bilinear spin-wave Hamiltonian are shown
in Fig. 4. where G(k) and F (k) are 9 × 9 matrices with G(k)ii =
E0 (i = {1, 2, 3}), G(k)ii = E1 (i = {4, 5, 6, 7, 8, 9}),
B. Green’s functions in a ribbon sample
G(k)i j = G(k)†ji , G(k)14 = G(k)27 = −γ1 e−ika3 , G(k)15 =
G(k)29 =G(k)38 =F (k)36 = − γ1 e−(1/2)ika3 , G(k)17 =G(k)24 =
For a nontrivial topology of the bulk band structure, the − γ1 eika3 , G(k)18 =G(k)26 =G(k)39 =F (k)35 = − γ1 e(1/2)ika3 ,
edge states of the TKL appear in the DM-induced gaps for G(k)45 = G(k)79 = −γ2 e(1/2)ika3 , G(k)46 = G(k)78 =
this ribbon sample. Due to the bulk-edge correspondence, −γ2 e−(1/2)ika3 , G(k)56 = −γ2 e−ika3 , G(k)89 = −γ2 eika3 ,
the topological chiral gapless edge modes are related to the G(k)i j = 0 (otherwise), F (k)i j = 0 (otherwise), a3 = 0.25a.
nonzero Chern numbers. We rewrite the Hamiltonian in the We choose W = 20 to ensure that the results are convergent
(x, ky ) space as our ribbon sample is expanded to an open with W . There are mainly two types of edges for the TKL:
boundary condition along the x direction and a periodic the zigzag edge and the armchair edge. In our case, we
boundary condition along the y direction choose the armchair edge because the high-symmetry points
1  ikRm ·ey † K and K  in the Brillouin zone overlap with each other along
αkx
†
= e αmx , (15) the ky direction [42]. Thus, the top and bottom edges are
Ny m
perpendicular to the x direction shown in Fig. 2.
where x can run from i1 to 9(W − 1) + i1 (i1 = For the purpose of calculating transport properties of
{1, 2, 3, 4, 5, 6, 7, 8, 9}) and W denotes the number of magnons, we introduce the retarded and advanced Green’s
periodic one-dimensional chains along the x direction. We functions
replace ky by k. The formalism for calculating the band
structure of the ribbon geometry is a 9W × 9W matrix-form  αk,n
†
(r  )αk,n (r)
Hamiltonian which is given by GR (r, r  ) = , GA (r, r  ) = [GR (r, r  )]† ,
ε + iη − H
 k,n
H= ϕk† H (k)ϕk , (16) (18)
k
where η is a positive infinitesimal, ε is the excitation energy,
where ϕk† = (αi†1 ,k , αi†1 +1,k , . . . , α9(W
†
−1)+i1 ,k ) in the open and r and r  represent excitation and response, respectively.
boundary condition α0,k |0 = α9W
† †
+1,k |0 = 0. The The spectral representation of the Green’s function can be

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FIG. 3. Berry curvature of magnon bands with Ja = 0.5, Jb = 1, and D = 0.3. The (a), (b), (c), (d), (e), (f), (g), (h), and (i) figures corre-
spond to the first, second, third, fourth, fifth, sixth, seventh, eighth, and ninth bands (from lower to higher), respectively. The Chern numbers
are given by {1, 0, 0, 2, −2, −1, 0, 1, −1}.

written as [43] The associated Chern number assigned to the nth band is
 defined by
2η
A= αk,n (r)αk,n
†
(r  ) . (19) 
(ε − H )2 + η2 1
k,n Cn = d 2 k nk . (23)
2π BZ
And the DOS can also be defined as
The Chern number is always a quantized integer in the Bril-
 h̄Tr(A) louin zone. When the gap between two bands is finite but very
ρ(ε) = αk,n αk,n
†
δ(ε − H) = . (20)
2π small, in general the Berry curvature is mostly concentrated
k,n
around the point of direct gap between the bands. We show
With the above Green’s functions, we can calculate the spec- the Berry curvature of magnon bands in Fig. 3 with D = 0.3,
tral function and the DOS of this ribbon sample. Both of them Ja = 0.5, and Jb = 1.
reflect the magnetic and topological properties of the TKL, Being charge neutral particles, magnons are not affected
which can solidify our proposal for the thermal Hall effect by external electric field and conventional electric field driven
and the EdH effect. Hall effect cannot be observed directly. Based on the semi-
classical theory, the thermal gradient along the topological
magnon system would drive a transverse magnon current
C. Berry curvature and thermal Hall conductance known as the thermal Hall effect. In our TKL system, the
In our model, nontrivial band topology can be character- transverse current is understood as a consequence of the pres-
ized by a nonzero Berry curvature defined via the eigenstates ence of chiral edge states induced by the DM interaction.
of the system [44]. And a nontrivial band topology arises We calculate the thermal Hall conductivity κxy by the Kubo
only when the system exhibits the nontrivial gap and edge formula. It can be expressed as a weighted summation of the
state modes in the spin-wave excitation spectra. In the case Berry curvature [20,45]
of two-dimensional noninteracting magnons, generally topo-
logical invariant like Chern number denotes the topological kB2 T 
κxy = − c2 [ρ(εnk )]nk , (24)
nature of reciprocal space. We calculate the Berry connection 4π 2 h̄a
n,k
in the reciprocal space of the TKL as
where kB is the Boltzmann constant, T is the temperature, and
Aλn = iψλ |∇kn |ψλ , (21) ρ(εnk ) = [eεnk /kB T − 1]−1 is the Bose function. We choose the
with |ψλ  being a normalized wave function of the λth Bloch lattice constant a = 0.1 nm as the typical layer spacing for
band such that H (k)|ψλ  = Eλ (k)|ψλ . The Berry connection practical calculation. The c2 (x) is defined as
is not a gauge invariant quantity but the Berry curvature is  
1+x 2
gauge invariant. The form of Berry curvature is given by c2 = (1 + x) ln − (ln x)2 − 2Li2 (−x), (25)
x
 [λ|∇k H (k)|n × n|∇k H (k)|λ]z
λk = i . (22) where Li2 (x) is the polylogarithmic function. Considering the
n=λ
(Eλ − En )2
thermal fluctuation, we calculate the deviation of sublattice

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TOPOLOGICAL MAGNONS ON THE TRIANGULAR KAGOME … PHYSICAL REVIEW B 107, 024408 (2023)

FIG. 4. The magnon bands of the TKL with various DM interactions. The Dirac points are located at the K point ( 23 π , − 23 π ) and the K 
point (− 23 π , 23 π ) in the first Brillouin zone. The parameters of ferromagnetic coupling with K = 0.1, h = 0.1: (a) Ja = 0.5, Jb = 1, D = 0.
(b) Ja = 0.5, Jb = 1, D = 0.1. (c) Ja = 0.5, Jb = 1, D = 0.2. (d) Ja = 0.5, Jb = 1, D = 0.3. The parameters of ferrimagnetic ground state with
K = 0.1, h = 0.1 are set as (e) Ja = 0.2, Jb = −1, D = 0. (f) Ja = 0.2, Jb = −1, D = 0.1.

magnetization from the saturation value where the γe is given by 2me /(ge), g is the Landé factor, and
   e and me are the charge and mass of the electron, respectively.
m = S − Smz = αm† αm  = ρ(εnk ), (26) Then we define a differential gyromagnetic ratio response
n,k γm∗ as
where the Curie temperature Tc is determined by m (Tc ) =  
∂Ltot /∂T
S. γm∗ = . (31)
∂m/∂T h

D. Angular momentum and gyromagnetic ratio Different from the electron systems, the gyromagnetic ratio
response of topological magnons cannot be measured simply
There are correction terms to the thermal Hall conductivity in experiment, but from a response to a temperature change.
in the linear response theory, by noting that the temperature
gradient is not a dynamical force but a statistical force. Thus,
III. RESULTS
the transport coefficients for magnons consist of the deviations
of a particle density operator and the current operators. The A. Topological magnon bands
current operators are expressed in terms of the reduced orbital Here we target the ferrromagnetic and ferrimagnetic
angular momentum of magnons ground states of the TKL. As shown in Fig. 4, the DM in-
  ∂ψn  


T c1 [ρ(εnk )] − ρ(εnk )εnk  ∂ψn ,
2kB teraction which breaks the time-reversal symmetry can open
ledge = 2Im
4π 2 h̄ ∂k  x k B
 ∂k
y
the gap at the Dirac points. Thus, we study the topological
n,k magnon bands on the TKL and take |Jb | as the unit of energy
(27) while Ja = 0.5, Jb = 1, K = 0.1, and h = 0.1. For ferromag-
where c1 (x) = (1 + x) ln(1 + x) − x ln x is another weight netic Jb , we consider the DM value at D = 0, 0.1, 0.2, and
function. In addition, the magnon wave packet carries an ad- 0.3 while the numerical solutions of the energies at the high-
ditional self-rotation motion originating from Berry curvature symmetry point  are given in Table I.
[20,21] The Dirac points are located at the K point ( 23 π , − 23 π ) and
  ∂ψn  ρ(εnk )   the K  point (− 23 π , 23 π ) in the first Brillouin zone. Hence, we
2kB  ∂ψn
lself =  −  also calculate the numerical solutions of the high-symmetry
2Im
∂k  2k
(εnk H )  ∂k . (28)
4π 2 h̄
n,k
x B y point K in Table II while K  is equivalent. Additionally, for
Ja = 1, Jb = 1 the top band becomes threefold degenerate. As
We calculate the total angular momentum per unit cell by an analog of a spin-orbit interaction in electronic topologi-
summing the edge current and the self-rotation cal insulators, DM interactions can introduce nonzero Berry
Ltot = m∗ (ledge + lself ), (29)
TABLE I. Energy of each band at  point with Ja = 0.5, Jb = 1.
where Ltot represents the total angular momentum. Within the
low-temperature approximation, the mass of the magnon can
D Energy (from lower to higher)
be approximated as the effective mass m∗ at the  point of the
first band. Thus, the gyromagnetic ratio of magnons can be 0 {0.30, 2.30, 2.61, 2.61, 3.80, 3.80, 5.49, 5.49, 6.30}
expressed as 0.1 {0.30, 2.30, 2.47, 2.75, 3.63, 3.97, 5.18, 5.80, 6.30}
0.2 {0.30, 2.30, 2.32, 2.86, 3.45, 4.15, 4.89, 6.13, 6.30}
γe Ltot
γm = , (30) 0.3 {0.30, 2.16, 2.30, 2.94, 3.28, 4.32, 4.64, 6.30, 6.53}
h̄m

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TABLE II. Energy of each band at K point with Ja = 0.5, Jb = 1. TABLE IV. Chern numbers with Ja = 0.2, Jb = −1.

D Energy (from lower to higher) D Chern number of each band (from lower to higher)
0 {0.96, 0.96, 2.62, 3.55, 3.55, 3.80, 5.49, 5.89, 5.89} 0 {0, 0, 0, 0, 0, 0, 0, 0, 0}
0.1 {0.93, 0.99, 2.61, 3.47, 3.61, 3.75, 5.54, 5.77, 6.04} 0.1 {1, 0, −1, −1, 2, −1, −1, 0, 1}
0.2 {0.90, 1.01, 2.61, 3.36, 3.61, 3.65, 5.68, 5.68, 6.20}
0.3 {0.86, 1.04, 2.61, 3.24, 3.42, 3.67, 5.61, 5.87, 6.29}

curvature and change the Chern numbers of some magnon

bands. In Table III, we numerically check the Chern numbers
of different magnon bands, which can distinguish various
topological phases.
If Ja > 0 and Jb < 0, the ground state is the ferrimagnetic
state and a small DM interaction can change the band struc-
ture significantly. From our calculations the antiferromagnetic
coupling is unfavorable for energy band topology. Here we
choose the DM value at D = 0, 0.1 for antiferromagnetic cou-
pling with Ja = 0.2, Jb = −1, K = 0.1, and h = 0.1. In this
case, the bands resemble three copies of magnon bands on the
kagome lattice ferromagnet with a flat band and three disper-
sive Dirac magnon bands in each copy as shown in Figs. 4(e)
and 4(f) [46]. For nonzero DM interaction, the magnon bands
are separated by a finite energy gap proportional to the DM
interaction in all the parameter regions [47,48]. And the Chern
numbers are shown in Table IV.
The Berry curvature and the Chern number can be posi-
tive or negative. Both of them become zero when we adjust
some of the nine bands to topologically trivial phases, and the
summation of Chern numbers for all bands is always zero.

B. Armchair edge states

According to the bulk-edge correspondence, the summa-
tion of Chern numbers up to the jth band is equal to the
number of pairs of edge states in the gap. We calculate the
bulk-edge energy spectrum which corresponds to the surface
property of the ribbon sample. The gapless edge states and
the DOS are shown in Fig. 5. We choose a 9 × 20 lattice and
introduce the Green’s functions to calculate the armchair edge
states of our ribbon sample. The emerging peaks of the DOS
are dependent on the topological band structure. We expect to
derive the value of the Chern number for each distinct band
from the edge state pattern itself.
As a result, the dispersion of armchair edge states in the
one-dimensional Brillouin zone is shown in Fig. 5 with Ja =
0.5, Jb = 1, K = 0.1, h = 0.1, and D = 0.3. We also calculate
the DOS of a two-dimensional TKL system with ferrimag-
netic ground state for Ja = 0.2, Jb = −1, K = 0.1, h = 0.1,
and D = 0.05. It can be written as a sum of Dirac-δ functions

TABLE III. Chern numbers with Ja = 0.5, Jb = 1.

FIG. 5. Magnon density of states with energy bands on the TKL.
D Chern number of each band (from lower to higher)
(a) The magnon band structure and DOS corresponding to topologi-
0 {0, 0, 0, 0, 0, 0, 0, 0, 0} cal edge states for a TKL ribbon (Ja = 0.5, Jb = 1, K = 0.1, h = 0.1,
0.1 {−1, 1, −1, 2, 0, −1, 1, 0, −1} and D = 0.3). The dispersions of the edge states in gaps are shown
0.2 {−1, 1, −1, 2, −2, 1, 0, 1, −1} by red curves. (b) The magnon band structure and DOS of the TKL
0.3 {−1, 0, 0, 2, −2, 1, −1, 0, 1} in ferrimagnetic state with Ja = 0.2, Jb = −1, K = 0.1, h = 0.1, and
D = 0.05.

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TABLE V. Peak and convergence values of thermal Hall conduc-

tivity for different parameters.

Parameters Peak value Convergence value

Ja = 0.2, Jb = −1, D = −0.1 0.124 −0.030
Ja = 0.5, Jb = 1, D = −0.1 0.091 −0.050
Ja = 0.5, Jb = 1, D = −0.2 0.301
Ja = 0.5, Jb = 1, D = −0.3 0.832

excitations and becomes negative (κ xy < 0) due to the fact that

c2 (x) > 0. At high temperature, higher-energy bands carrying
FIG. 6. Low-temperature thermal Hall conductivity on the TKL. opposite Berry flux contribute significantly and a strong Zee-
In ferromagnetic coupling the DM interactions are −0.1, −0.2, −0.3 man field diminishes the thermal population difference among
with Ja = 0.5, Jb = 1, K = 0.1, and h = 0.1, respectively. For an- the bands by creating a large gap for all the bands. These
tiferromagnetical couplings between sublattices A and B, the DM behaviors of κ xy are inherited from the topology of the bulk
interaction is −0.1 with Ja = 0.2, Jb = −1, K = 0.1, and h = 0.1. magnon bands [49,50]. The peak values and convergence val-
ues of thermal Hall conductivity for the different parameters
with energies corresponding to the set of eigenvalues of the on the TKL are listed in Table V.
Hamiltonian. The appearance of edge modes leads to nonzero It is noted that the energy bands with high Chern numbers
DOS in each DM-induced gap. And the DOS is no longer have large weights in the calculation of thermal Hall effect
symmetric about the Dirac point. The topological structure and the dominant contribution comes from the Dirac points
of energy bands is described by the magnon transport of K (K  ). With the enhancement of DM interaction, the bands
armchair edge states and the corresponding DOS. with high Chern numbers appear. Thus, both the sign change
and the peak vanish in all the parameter regions and κ xy
C. Thermal Hall effect increases significantly. In real materials, the Curie temperature
can be increased significantly due to the presence of single-ion
Thermal Hall effect is a key experimental signature to anisotropies, interlayer couplings, and so on. Here we show
detect the magnon transport arising from the edge current of the results of thermal Hall conductivity on the TKL in a
topological excitations. The DM-induced Berry curvature acts large temperature range to illustrate the behaviors of κ xy for
as an effective magnetic field that deflects the propagation different parameters.
of magnons in the system. The nonzero Chern numbers are
associated with topological chiral gapless edge modes which
D. Einstein–de Haaseffect
appear in the DM-induced gaps. And the nontrivial topology
of the Berry curvature leads to magnon edge states which The Einstein–de Haas effect is the ultimate macroscopic
carry a transverse heat current upon the application of a lon- manifestation originating from a subtle microscopic exchange
gitudinal temperature gradient. Unlike electrons the magnons of spin angular momentum [51]. According to the linear re-
have no charge and the rotation is not due to “Lorentz force.” sponse theory, the magnon wave packet undergoes two types
Thus, the DM interaction plays the role of an effective mag- of orbital motions and the total angular momentum is defined
netic field by altering the propagation of magnons in the as the summation of these two types of rotational motions. We
system [49].
The plot of κ xy (T ) vs T /|Jb | is displayed with different
DM interactions, respectively, in Fig. 6. We take h = 0.1
and K = 0.1 for both ferromagnetic state and ferrimagnetic
state. Because of the opposite Berry curvatures of the higher
magnon bands, the κ xy (T ) for D = −0.1 changes its sign upon
raising the temperature. We observe that the thermal Hall
conductivity decreases with the emergence of antiferromag-
netic coupling. Especially when the magnon is excited to the
energy band possessing a high Chern number, the κ xy can be
effectively changed with a positive peak at low temperatures
followed by a long negative tail in the high-temperature re-
gion.
As shown in Fig. 6, we also calculate the thermal Hall con-
ductivity coefficients from HP theory with D = −0.2, −0.3. FIG. 7. Temperature dependence of the topological gyromag-
The values keep increasing upon raising the temperature and netic ratio γm on the TKL. The gyromagnetic ratio contributions,
have not reached saturation at the phase transition points. compared to the electronic value, for individual topological edge
At low temperature and weak field, the lowest-lying magnon current, self-rotation of the magnon wave packet, and the total an-
band dominates thermal transport. The thermal Hall conduc- gular momentum contribution to γm are shown. Parameter choices
tivity vanishes at zero temperature as there are no thermal are D = 0.1, 0.3 with Ja = 0.5, Jb = 1, K = 0.1, and h = 0.1.

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TABLE VI. Curie temperature Tc /|Jb | for different parameters. ture variation of the topological gyromagnetic ratio compared
to the electronic value. As the γe∗ is equal to γe for electrons,
Lattice Parameter field Tc /|Jb | the γm∗ /γe∗ can be simplified as γm∗ /γe . Hence, the differential
TKL Ja = 0.5, Jb = 1, D = −0.1, K = 0.1 0.884 gyromagnetic ratio is renormalized from the γm response.
TKL Ja = 0.5, Jb = 1, D = −0.2, K = 0.1 0.878 From our calculations we find that the ferrimagnetic frustrated
TKL Ja = 0.5, Jb = 1, D = −0.3, K = 0.1 0.867 structure suppresses the band topology by reducing the γm /γe
TKL Ja = 0.2, Jb = −1, D = −0.1, K = 0.1 0.816 and γm∗ /γe . Considering the differential gyromagnetic ratio
response, the magnon system also has a peak value before
descending as seen in Fig. 8(b). Thus, there is an optimal
temperature of the differential gyromagnetic ratio at which the
define the gyromagnetic ratio as the angular momentum di- magnon insulator will have the strongest response.
vided by the magnetic moment of magnons, which is related to In Stewart’s apparatus, the EdH effect was observed from
the magnetization change of the system. Each magnon mode the amount of the transient angular momentum change. This
can be excited or annihilated and has its own gyromagnetic experimental setup can be explored to measure the differential
response. In Fig. 7, we calculate the transport properties from gyromagnetic ratio via being exposed to an external heat bath
the gyromagnetic ratio, finding that the self-rotation motion with a temperature gradient [52]. From an experimental point
and the edge current on the TKL are opposite in directions. of view, there is an optimal temperature zone in which our the-
However, the self-rotation has a larger part in the negative ory can be tested well. The values of optimal temperature for
region, which results in a negative value of the total angular various TKL systems are all around T = 0.20|Jb |. Addition-
momentum in all temperature regions. Especially in the zero- ally, the anisotropy and external magnetic field can enhance
temperature limit, the self-rotation motion of the TKL has a the EdH effect; for instance, the peak of γm∗ /γe reaches 0.223
finite response at the  point which sets it apart from the other for the ferromagnetic TKL with D = −0.3, Ja = 0.5, Jb = 1,
usual lattice candidates that have been explored before. K = 0.1, and h = 0.1. Our formalism, analytical approach,
Our results infer that the total gyromagnetic contribution and eventual conclusions will hold not only for the TKL
increases significantly at first and reaches a peak value at system, but also for a wider variety of ferrimagnetic systems.
about T = 0.20|Jb |. The value stabilizes when it approaches
the Curie temperature Tc . Since the Tc is in the ballpark
IV. CONCLUSIONS
of |Jb |, the HP representation is valid until T  T ∗ (T ∗ ∼
0.5|Jb |), providing a quantitative description of the thermal In summary, we have investigated the topological magnons
Hall conductivity and the gyromagnetic ratio in the tempera- on the TKL, which can give detectable results on the thermal
ture range [0, T ∗ ]. Above T ∗ , the results obtained from the HP Hall conductance and the Einstein–de Haas effect. In the
representation illustrate the trends that one would expect from presence of armchair edges for a ribbon sample, we find that
a more accurate calculation. Since the relevant topological the nonzero summation of Chern numbers for different bands
features of the EdH response happen in the ballpark of T = below the gap leads to a magnon current transport along the ky
0.2|Jb |, the HP representation is enough to describe them. The direction of this gap [46,53]. By using the real-space Green’s
Curie temperatures for the TKL are listed in the Table VI. To function approach, we have studied the armchair edge modes
further analyze the physical content of Fig. 7, we compare to calculate the DOS in our sample. Theoretical and exper-
the gyromagnetic ratio of the TKL system with respect to the imental studies have shown that thermal Hall conductance
kagome lattice system. We show the results of our calculation can have a sign change as temperature or magnetic field is
in Fig. 8. The γm /γe shown in Fig. 8(a) represents the tempera- varied [54]. Our results show that the sign change behaviors

FIG. 8. Comparison of the Einstein–de Haas effect response on the TKL and kagome lattice. (a) The gyromagnetic ratio variation with
temperature is shown. Parameter choices on the kagome lattice are J = Jb = 1, D = −0.1, and h = 0.3. Parameter choices on the TKL with
K = 0.1 and h = 0.1 are D = −0.1, −0.2, −0.3 for Ja = 0.5, Jb = 1, and D = −0.1 for Ja = 0.2, Jb = −1. (b) The differential gyromagnetic
ratio response is shown. Parameter choices are the same as before.

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emerge on the TKL when the topological features are reduced tations, the thermal Hall effect [55], and the EdH effect. In
by the antiferromagnetic coupling. At the Curie temperature, real materials, the observed results are influenced by other
the thermal Hall conductivities are always convergent for all kinds of effects, but these may not be a concern for systems
selected parameters. We further find that there is a peak for in which topological magnons already dominate the thermal
the thermal Hall conductance when the low magnon bands Hall effect and the EdH effect. The TKL structure has been
dominate and the peak vanishes when the DM interaction is found in Cu9 X2 (cpa)6 (X = F, Cl, Br; cpa = anion of 2-
strong enough. The influence of a nonzero Berry curvature and carboxypentonic acid) which has tunable magnetic couplings
its underlying topological identity is preserved even though [56]. The thermal Hall effect of spin excitations arises in
the lattice structure changes. the usual way via the breaking of inversion symmetry of the
We show the calculations for the EdH effect of topologi- lattice by a nearest-neighbor DM interaction [57]. It is also
cal magnons for both the ferro- and ferrimagnetic states and possible to realize the TKL in cold atom systems and higher-
propose that the TKL is a suitable lattice for the observation order topology of magnons [58,59]. Our study provides a new
of the EdH effect. Especially in the low-temperature region, vision to realize the thermal Hall effect and the EdH effect.
the magnon description is more effective. Comparing with the The thermal Hall effect that arises from the edge current of
traditional kagome and honeycomb lattices, this compound magnons is useful to control the magnon transport; then the
lattice has a better topological magnon structure with added EdH effect can produce a potential mechanical effect which
high Chern numbers to produce stronger EdH effect. We has potential applications in quantum informatics and topo-
investigate the angular momentum for topological edge cur- logical magnon spintronics [12].
rent and self-rotation originating from the Berry curvature in
momentum space. These two angular momentum components
ACKNOWLEDGMENTS
with opposite signs offset each other, but the self-rotation has
a larger part which ensures that the total angular momen- We would like to thank Trinanjan Datta and
tum contribution has a nonzero value. The EdH effect is a Jun Li for helpful discussions. This project is
macroscopic mechanical manifestation caused by the angular supported by NKRDPC-2022YFA1402802, NKRDPC-
momentum conservation principle and can be detected by a 2018YFA0306001, NSFC-11974432, NSFC-92165204,
mechanical experimental setup [22,52]. GBABRF-2019A1515011048, Leading Talent Program of
We have studied various TKL systems with different Guangdong Special Projects (201626003), and Shenzhen
coupling parameters to explore the topological magnon exci- International Quantum Academy (Grant No. SIQA202102).

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