Topological Magnetic Excitations on the Distorted Kagome Antiferromagnets: Applications to Volborthite, Vesignieite, and Edwardsite

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by  S. A. Owerre

Topological Magnetic Excitations on the Distorted Kagomé Antiferromagnets:
Applications to Volborthite, Vesignieite, and Edwardsite.
S. A. Owerre1,2
1Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada.
2African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg, Cape Town 7945, South Africa.
In this Letter, we identify a noncoplanar chiral spin texture on the distorted kagomé-lattice
antiferromagnetsinducedbythepresenceofamagneticfieldappliedperpendiculartothedistorted
kagomé plane. The noncoplanar chiral spin texture has a nonzero scalar spin chirality similar to
a chiral quantum spin liquid with broken time-reversal symmetry. In the noncoplanar regime we
observenontrivialtopologicalmagneticexcitationsandthermalHallresponsethroughtheemergent
field-induced scalar spin chirality in contrast to collinear ferromagnets in which the Dzyaloshinskii-
7
1 Moriya(DM)spin-orbitinteractionisthedrivingforce. Ourresultsshedlightonrecentobservation
0 ofthermalHallconductivityindistortedkagomévolborthiteatnonzeromagneticfieldwithnosignal
2 ofDMspin-orbitinteraction,andtheresultsalsoapplytootherdistortedkagoméantiferromagnets
such as vesignieite and edwardsite.
r
a
M
I. INTRODUCTION uum of spinon excitations in inelastic neutron scattering
4 experiment [31, 32]. Nevertheless, an applied magnetic
fieldorpressuredestabilizestheQSLnatureofthesema-
A chiral spin liquid (CSL) is a particular type of
] terials and leads to an induced magnetic order at low
l spin liquids in which time-reversal symmetry (TRS) is
e temperatures [31, 33, 34].
broken macroscopically [1–3]. The scalar spin chirality
-
r χ = S (S S ) measures broken TRS and de- Theemergingexperimentaltechniquetoprobethena-
t h ijki h i · j × k i
s fines the hallmark of a CSL. It is also generally believed ture of magnetic excitations in quantum magnets with
.
t thatthemagneticexcitationsinanyquantumspinliquid possible QSL ground states is the measurement of ther-
a
(QSL)arebosonicspin-1/2spinonsinwhichtheconven- mal Hall effect. For quantum magnets with magnetic
m
tional bosonic spin-1 magnons fractionalize. According long-range order the thermal Hall effect has been real-
d- to Kalmeyer and Laughlin [1] a CSL can be thought of ized in the collinear kagomé ferromagnet Cu(1-3, bdc)
n as a bosonic version of ν = 1/2 Laughlin state in frac- [35] and a number of collinear pyrochlore ferromagnets
o tional quantum Hall effect. In recent years, a plethora [36,37]. InthiscasethetransversethermalHallconduc-
c of theoretical models has been proposed in which a CSL tivity(κxy)canbeexplainedintermsofBerrycurvature
[ phase can be stabilized. The recent theoretical numeri- induced by the DM spin-orbit interaction [38–40] lead-
5 cal work shows that a CSL can emerge in a kagomé lat- ing to topological magnons [41–49] and Weyl magnons
v tice Mott insulator in which a magnetic field induces an [50, 51] similar to spin-orbit coupling electronic systems
9 explicit spin chirality interaction in a t/U perturbative [52–56]. In a recent experiment [57], a nonzero κxy has
9 expansion of the Hubbard model at half filling [4], hence been observed in a frustrated distorted kagomé volbor-
1 TRS is broken explicitly. In other numerical works a thite at a strong magnetic field of 15 T with no signs
5
spontaneously broken TRS CSL is found away from the ofthe DMspin-orbitinteractionandno discerniblether-
0
. isotropic kagomé-lattice antiferromagnet by adding ad- mal Hall signal was observed at zero magnetic field [58].
1 ditional second and third nearest neighbour interactions The authors attributed the presence of κxy to nontrivial
0
[5–9] or Ising anisotropy [10]. elementary excitations in the gapless QSL phase, how-
7
ever a strong field of 15 T causes low-temperature mag-
1 In real materials, however, a CSL has remained elu-
: sive because material synthesis usually comes with de- netic phases in volborthite [26–28]. Nevertheless, the ex-
v
act nature of the low-temperature magnetic phases at
i fects such as structural distortion and perturbative
X 15 T is poorly understood, but the intrinsic DM spin-
anisotropy like the intrinsic DM spin-orbit interaction
orbit anisotropy suggests a Q=0 coplanar/noncollinear
r [11, 12], which tend to destabilize disordered QSL
a Néel order.
and induce a magnetic long-range order. For quan-
tum kagomé antiferomagnets (QKAF), various exper- Inpreviousexactdiagonalizationstudy,ithasbeenes-
imentally accessible materials such as iron jarosites tablishedthattheout-of-planeDMspin-orbitanisotropy
[13–16], vesignieite BaCu V O (OH) [17–23], edward- can induce a quantum critical point (QCP) in spin-1/2
3 2 8 2
site Cd Cu (SO ) (OH) .4H O [24], and volborthite QKAF at Dc/J 0.1 [59]. A moment free QSL phase
2 3 4 2 6 2
Cu V O (OH) 2H O [25–28] have long-range magnetic ispredictedf⊥orD∼<Dc andaQ=0Néelphaseexists
3 2 7 2 2
orders below ce·rtain temperatures and they are gener- for D > Dc. Fr⊥om diff⊥erent experimental inspections,
ally attributed to the presence of DM spin-orbit inter- it is g⊥enerall⊥y believed that both edwardsite and vesig-
action. The recent proposals of QSL materials are her- nieite are located in the ordered regime below certain
bertsmithite ZnCu (OH) Cl [29, 30] and the calcium- temperature with 0.1<D /J <0.16 [19, 20], and there
3 6 2
⊥
chromium oxide Ca Cr O [31] which show a contin- is a possibility that volborthite also belongs to the or-
10 7 28
2
dered regime at low temperatures. But herbertsmithite
ZnCu3(OH)6Cl2 has D /J 0.08 [30], thus remain a A
⊥ ∼
QSL. An applied magnetic field (H 2D plane) can in-
⊥
ducenoncoplanarspintextureswithanonzeroscalarspin
ϑ
chirality. The emergent scalar spin chirality due to non-
coplanar spins might be different from the spontaneous
scalar spin chirality in the CSL, but their effects on the
magnetic excitations should be the same. In this Letter (cid:12) (cid:12) B C
we show that a nonzero Berry curvature can be induced
by a chiral spin texture in real space rather than the (cid:12) (cid:12) (cid:12)
DM spin-orbit interaction. The Berry curvature stems
from the emergent scalar spin chirality which measures (cid:12) (cid:12) (cid:12) (cid:12)
the solid angle subtended by three noncoplanar spins in
T
a unit triangular plaquette of the kagomé lattice. The 2
Berry curvature of the chiral spin texture is the driving y
M
force of the thermal Hall effect in frustrated distorted 6
C
QKAF with/without magnetic long-range order.
J J
(cid:12) (cid:12) (cid:12) (cid:12)
II. MODEL
(cid:12) (cid:12) (cid:12)
J
0
y
We consider the microscopic spin Hamiltonian on the
T
distorted QKAF subject to a magnetic field perpendicu- 1
lar to the kagomé plane. The Hamiltonian is given by (cid:12) (cid:12)
(cid:12)z x
= J S S +D S S H S , (1)
ij i j ij i j i
H · · × − ·
Xhiji(cid:2) (cid:3) Xi
where S is the magnetic spin moment at site i, and the
i
summationrunsovernearest-neighbourspins. J =J > FIG.1. Coloronline. Thecantednoncollinear/coplanarmag-
ij
netic order with positive vector chirality on the distorted
0 on the diagonal bonds, and Jij = J0 = Jδ on the
kagomé lattice and the symmetry group of the lattice. The
horizontal bonds with δ = 1 as shown in Fig. 1. The
6 out-of-planeDMinteractionliesatthemidpointsbetweentwo
inversion symmetry breaking between the bonds on the
magnetic ions as indicated by small dotted circles. The spin
kthaegyomliée alattttihceemalildopwosinatsDbMetwveeecntotrwDo imja=gne−tiDc⊥ioˆznsanads ttrriiaadngalerewsitehpaJra=tedJ0baysashnowannglaetϑth6=e to1p20fi◦gounre.each isosceles
6
shown in Fig. 1. The last term is the out-of-plane exter-
nal magnetic field H = hˆz and h = gµ H. The Hamil-
B
tonian (1) is applicable to volborthite [60–64] and other
formsofHamiltonianhavealsobeenproposedforvolbor-
thite [65–67]. The out-of-plane DM interaction is always
present on the kagomé lattice and it stabilizes the copla-
nar/noncollinearspinconfiguration[13]. Insomekagomé
materialsanin-planeDMinteractionD maybepresent
k
duetolackofmirrorplanes. Itleadstoweakout-of-plane
ferromagnetismwithsmallferromagneticmoment. How-
ever,itisusuallynegligiblecomparedtotheout-of-plane
component for most spin-1/2 kagomé antiferromagnetic
materials[30,59]andcanberemovedbygaugetransfor-
mation. For kagomé volborthite[57], therewas no signal
of both in-plane and out-of-plane DM interaction on the
observed κ . Also, for potassium Fe-jarosite with spin-
xy
5/2 the in-plane DM interaction (or the DM interaction
in general) does not necessarily induce topological mag-
neticexcitations[16]. Therefore,wewillneglectthesmall
FIG. 2. Color online. Magnon dispersion of distorted QKAF
in-planeDMcomponentatthemomentandcommenton
at zero magnetic field and several values of distortion with
its effects in the subsequent sections.
D /J =0.2.
Let us start from what is known for distorted kagomé ⊥
antiferromagnets at D = h = 0 [62]. In this case, the
ij
3
is no degeneracy except a global spin rotation about the
1 1
z-axis, and for δ >1/2 the classical ground state is non-
collinear but coplanar. The collinear phases are triv-
0.8 0.8
ial. In the present study, we focus on the canted copla-
nar/noncollinear regime (δ > 1/2) as the ground state
0.6 0.6
Γ) K) properties of volborthite, vesignieite, and edwardsite are
ω(00.4 ω(00.4 believed to live in this regime.
0.2 D⊥/J0=.20.1
D⊥/J=0.15 III. CLASSICAL GROUND STATE
D⊥/J=0.2
0 0
0 1 2 0 1 2 Now, we consider the classical ground state of the full
δ δ Hamiltonian (1). In the classical limit, the spin opera-
tors can be approximated as classical vectors, written as
FIG. 3. Color online. The lifted zero mode ω0 varies in the Si = Sni, where ni = (sinφcosθi,sinφsinθi,cosφ) is
Brillouin zone as function of the distortion for zero magnetic a unit vector and θ labels the spin oriented angles on
i
field φ = 0. The dash lines connect isotropic point δ = 1. the spin triad and φ is the magnetic-field-induced cant-
H
The coplanar/noncollinear order is valid for δ>0.5. ing angle. For δ > 1/2 the ground state is the canted
coplanar/noncollinear spin configuration in Fig. 1. The
classical energy is given by
e (φ)=2J(2+δ) (1 cosϑ)cos2φ+cosϑ (3)
0
−
4D sin2φ(cid:2)sinϑ(1 cosϑ) 3hcosφ(cid:3),
− ⊥ − −
where e (φ) = E(φ)/NS2, and N is the number of sites
0
per unit cell. The magnetic field is rescaled in unit of
S. The minimization of e (φ) yields the magnetic-field-
0
induced canting angle cosφ=h/h where
s
(1 cosϑ)
h = − 4J(2+δ)+8D sinϑ . (4)
s
3 ⊥
(cid:2) (cid:3)
As can be seen from Eq. 3 the classical energy depends
ontheDMinteractionasitcontributestothestabilityof
thecoplanar/noncollinearspinconfiguration. Incollinear
FIG. 4. Color online. Topological magnon dispersion of dis-
spin configurations, the DM interaction does not con-
tortedQKAFatfinitemagnetic-field-inducedscalarspinchi-
rality. The parameters chosen are h/hs = 0.3 and D /J = tributetotheclassicalenergy,insteaditprovidesTMDs,
0.2. ⊥ e.g. in kagomé ferromagnets [35–39, 41–46, 50, 51].
Hamiltonian can be written as IV. TOPOLOGICAL MAGNETIC EXCITATIONS
2
Jδ 1
= SA+SB +SC const., (2) The magnetic excitations of frustrated QKAF can be
H 2 δ −
∆ (cid:18) (cid:19) studied in different formalisms. In real materials the in-
X
trinsicDMspin-orbitinteractionandanexternalapplied
where S form a triad as depicted in Fig. 1. It
A,B,C magnetic field are very likely to induce a magnetic long-
is easily seen that the classical energy is minimized for
range order in frustrated QKAF [26–28, 33, 34]. There-
1δSA+SB+SC =0, yielding ϑ=arccos(−1/2δ)6=120◦ fore, we will employ the Holstein-Primakoff (HP) trans-
for δ = 1. The classical configuration is now a canted
6 formation [68] valid in the low temperature regime. In
coplanar/noncollinearspinsforδ >1/2[62]. However, it
the Supplemental Material (SM), we have shown that a
also has an extensive degeneracy as in the ideal kagomé
magnetic field (H 2D plane) can induce a chiral non-
Heisenberg antiferromagnets. There are several limiting ⊥
coplanar spin texture with an emergent scalar spin chi-
casesinwhichthesystemiseitherbipartitewithcollinear
rality
magnetic order or non-bipartite with non-collinear mag-
netic order. The limiting case δ 0 maps to a bi-
→ φ =Jφ χijk, (5)
partite square lattice with collinear magnetic order and H
i,j,k,∆
δ maps to a decoupled antiferromagnetic chains. X
→ ∞
For δ < 1/2 it has been established that the classical where J = cosφ h. As we previously mentioned, the
φ
∝
groundstateiscollinear(up-down-downstate)andthere scalar spin chirality breaks time-reversal ( ) symmetry
T
4
δ=0.95 δ=1.005 δ=1.01 δ=0.95 δ=1.01
0.3 0.18
0.16
0.25
S 0.14
J
2
)/ 0.2 0.12
x
k
ω( T T 0.1
/0.15 /
xy xy0.08
κ κ
- -
0.1 0.06
0.04
S 0.05 h/hs=0.05 0.02 h/hs=0.05
/2J 0 h/hs=0.06 0 h/hs=0.06
)x 0 5 10 0 5 10
k
ω( T/JS T/JS
FIG. 6. Color online. Low temperature dependence of the
0 0.5 11 0.5 00 0.5 1 topological thermal Hall conductivity with D /J =0.2.
k /2π k /2π k /2π ⊥
x x x
δ=0.95 δ=1.01
1 0.8
FIG.5. Coloronline. Chiralmagnonedgemodes(redlines)of 0.6
distorted QKAF for a strip geometry with D /J =0.2. Top
panel: φ = 0, h/hs = 0. Bottom panel: ⊥φ = 0, h/hs = 0.5 0.4
H H 6
0.3. 0.2
xy 0 xy 0
κ κ
macroscopically and can be spontaneously developed in - -
-0.2
CSL [1–8]. Its effects on the magnetic excitations should
be the same in the ordered and disordered phases. In -0.5 -0.4
the magnetic-field-induced noncoplanar regime the DM T =2J -0.6 T =2J
interaction is not necessarily needed. Its main effect on T =2.5J T =2.5J
-1 -0.8
QKAF is to induced a long-range magnetic order. The
-0.1 0 0.1 -0.1 0 0.1
systemexhibitsnontrivialtopologicaleffectsthroughreal gµ H gµ H
B B
spaceBerrycurvatureofthechiralnoncoplanarspintex-
ture.
FIG. 7. Color online. Field dependence of the topological
The magnetic excitations at zero magnetic field and thermalHallconductivityatlowtemperatureswithD /J =
zero DM interaction with δ = 1 still exhibit a zero en- 0.2. ⊥
6
ergy mode ω = 0 (not shown). Lattice distortion is
0
unable to lift the zero energy mode. In addition, there
are two non-degenerate dispersive modes ω = ω (not phase. For instance, -symmetry is a good symmetry
1 6 2 M
shown)inducedbylatticedistortionincontrasttoundis- of the kagomé lattice, but it flips the in-plane spins and
torted kagomé antiferromagnets with degenerate modes -symmetry flips the spins again and leaves the mag-
T
ω =ω . Althoughtheexactparametervaluesofvolbor- netic order invariant. Hence is a symmetry of the
1 2 TM
thitearenotknown,anout-of-planeDMinteractionisin- coplanar/noncollinear magnetic order. Since the lattice
trinsictothekagomélattice. AmoderateDMinteraction distortion and the out-of-plane DM interaction preserve
(D /J = 0.2) lifts the zero energy mode and stabilizes this combined symmetry the system should be topologi-
the⊥coplanar/noncollinearspinconfigurationasshownin cally trivial as we will show below. Therefore, we expect
Fig. 2. We find that the lifted zero mode is no longer a this model to be topologically nontrivial when either -
T
constant flat band but varies in the BZ (see SM), that is symmetry or -symmetry is broken.
M
ω (Γ)=ω (K), where Γ=(0,0) and K=(2π/3, 0) see The combined symmetry can be broken in two ways:
0 0
6
Fig. 3. This is one of the differences between distorted (1) If the kagomé lattice lacks -symmetry, an in-
M
and undistorted kagomé antiferromagnets at finite DM plane DM component D can be allowed according to
k
interaction. the Moriya rules [12]. The in-plane DM component pre-
The symmetry group of the kagomé lattice is gener- serves -symmetrybutbreaks -symmetry,hence
T M TM
ated by lattice translation T, 6-fold rotation ( π/3 will be broken. This leads to weak out-of-plane ferro-
6
C
rotation about the center of the hexagonal plaquettes), magnetism or noncoplanar spin canting with scalar spin
and mirror reflection symmetry (see Fig. (1)). Sev- chiralityandweakferromagneticmoment[13,69]. Thus,
M
eral combinations of the space group symmetry and - for kagomé antiferromagnetic materials with D D
T ⊥ (cid:28) k
symmetry can lead to identity in the coplanar order we expect the out-of-plane ferromagnetism to be domi-
5
nant and hence the (in-plane) DM interaction should be (in units of kB/~) for two values of the lattice distortion
the primary source of topological spin excitations as we and the magnetic field. The topological Hall conductiv-
alreadyknowincollinearferromagnetswithout-of-plane ity captures a negative value in both regimes δ < 1 and
DM interaction. However, most kagomé antiferromag- δ > 1 and the thermal Hall conductivity is suppressed
netic materials have a dominant intrinsic out-of-plane in the latter. As we previously noted the thermal Hall
component D D and the weak ferromagnetism in- conductivity does not originate from the DM spin-orbit
duced by D c⊥an(cid:29)be nkegligible. In fact, there was no sig- interaction but from the chiral spin texture of the non-
nalofanyDkMinteractionontheobservedκ inkagomé coplanar spins induced by the magnetic field. The DM
xy
volborthite[57]. Therefore,theremustbeanothersource spin-orbit interaction is only necessary to stabilize the
of topological spin excitations apart from the DM inter- coplanar order but a second-nearest neighbour antifer-
actions. romagnetic interaction or an easy-plane anisotropy also
(2) An alternative way to break symmetry is stabilizes the coplanar/noncollinear Néel order. There-
TM
by applying an external magnetic field perpendicular to fore, kagomé antiferromagnetic materials with a weak
the kagomé plane (H 2D plane). A finite out-of- (negligible) DM interaction can possess a thermal Hall
⊥
plane magnetic field breaks -symmetry but preserves conductivity from the real space Berry curvature of the
T
-symmetry. Itinducesanoncoplanarspintexturewith chiralspintextureandthisshouldbepossibleintheCSL
M
anonzeroscalarspinchirality(seeSM).Themagnondis- phase even at zero magnetic field. This is an analog of
persions are also gapped as depicted in Fig. 4 similar to topological Hall effect in electronic systems [70–73]. On
the case without magnetic field. However, the gap exci- the other hand, Fig. 7 shows the magnetic field depen-
tations with and without the magnetic field are different denceofκxy whichshowsasymmetricsignchangeasthe
whichcanbeshownbysolvingforthechiraledgemodes. magnetic field is reversed as can be understood from the
The magnon edge modes for a strip geometry with open rotation of the spins and the corresponding sign change
boundary conditions along the y-direction and infinite in the spin chirality. We note that the role of magnetic
along x-direction are shown in Fig. 5 at zero magnetic field in the QKAF is different from the triplon bands in
field (top panel). We see that the system does not pos- a dimerized quantum magnet [74], where no scalar spin
sess topologically protected gapless edge modes and the chiralitywasinducedχijk =0,thereforetopologicalmag-
Chern number vanishes (see SM). On the other hand, netic excitations and thermal Hall effect stem from the
the edge modes at finite magnetic field shown in bottom DM interaction [74].
panel of Fig. 5 are gapless — an indication of a topo-
logical system. The magnetic-field-induced noncoplanar
chiral spin texture does not require a DM interaction in VI. CONCLUSION
generalaslong-rangeordercanbeinducedthroughother
interactionssuchasasecond-nearestneighbour. Besides,
In summary, we have studied topological magnetic ex-
thescalarspinchiralityχ shouldhavethesameeffects
ijk citations and thermal Hall effect induced by the real
on magnetic excitations in the CSL.
space Berry curvature of the chiral spin texture on the
distorted kagomé antiferromagnets applicable to vesig-
nieite,edwardsite,andvolborthite. Thelackofinversion
V. TOPOLOGICAL THERMAL HALL EFFECT symmetry on the kagomé lattice allows a Dzyaloshinskii-
Moriya (DM) interaction. However, in contrast to fer-
The mechanism that gives rise to a nonzero thermal romagnets the DM interaction is only necessary to sta-
Halleffectininsulatingfrustratedmagnetshasremained bilizethecoplanar/noncollinearmagneticspinstructure.
an open question. In electronic (metallic) systems there We showed that lattice distortion and DM interaction
is a concept of the topological Hall effect in which an introduce gap magnetic excitations, but the system re-
unconventional anomalous Hall conductivity is induced mains topologically trivial with neither protected chiral
by the scalar spin chirality resulting from the configu- edge modes nor finite thermal Hall conductivity. By
ration of the chiral spin textures rather than the spin- turning on an out-of-plane external magnetic field, we
orbit interaction as reported in several electronic ma- showed that a finite scalar spin chirality is induced from
terials [70–73]. In this section, we will show that this the noncoplanar chiral spin texture. This gives rise to a
mechanism can be observed also in charge-neutral mag- real space Berry curvature which measures the solid an-
neticexcitationsonthefrustratedQKAF.Inthiscasethe glesubtendedbythreenoncoplanarspinsindependentof
appropriate transport measurement is the thermal Hall the DM interaction. We note that the real space Berry
effect in which a temperature gradient T induces a curvature of the chiral spin texture should persist in the
heat current Q. From linear response−th∇eory, one ob- chiral spin liquid (CSL) phase due to the presence of the
J
tains Q = κ T, where κ is the thermal spontaneous scalar spin chirality.
Jα − β αβ∇β αβ
conductivityandthetransversecomponentκxy isassoci- TheresultsofthisLettersuggestthattheexperimental
P
ated with the thermal Hall conductivity given explicitly resultofthermalHallresponseinvolborthitecanbebest
in Ref. [40]. describedusingadistortedHeisenbergmodelasopposed
Figure6showsthelow-temperaturedependenceofκ to a frustrated alternating spin chain. It is noted that
xy
6
the DM interaction can be removed by a gauge trans- magnetic field dependence of κ and possibly the topo-
xy
formationforfrustratedone-dimensional(1D)spinchain logicalmagneticexcitationsatvariousmagneticfieldand
[75]andanideal1Dsystemshouldnothaveachiralnon- temperature ranges. Furthermore, experiment should
coplanar magnetic order and scalar spin chirality should also try to measure the spontaneous or magnetic-field-
be absent. An alternative approach is the Schwinger bo- inducedscalarspinchiralitythatgivesrisetotopological
son formalism, however the scalar spin chirality does not magnetic excitations and thermal Hall response. We be-
appear explicitly in this formalism. Instead the DM in- lieve that the results of this Letter have shed some light
teraction generates a magnetic flux leading to topolog- onrecentobservationofthermalHallconductivityinvol-
ical spin excitations even in the absence of an applied borthite [57].
magnetic field [43, 76]. This is reminiscent of collinear
ferromagnets and thus sharply contrast with the present
results. ACKNOWLEDGMENTS
Theseresultsalsocallforfurtherinvestigationofther-
mal Hall effect in the kagomé volborthite and other dis- The author would like to thank M. Yamashita, for
torted quantum kagomé antiferromagnets such as vesig- elaborating on the experimental result of thermal Hall
nieite[17–23]andedwardsite[24]. Althoughwecaptured response in kagomé volborthite. Research at Perime-
a negative thermal Hall conductivity (κ ) as seen in ex- ter Institute is supported by the Government of Canada
xy
periment [57], it would be interesting to determine the throughIndustryCanadaandbytheProvinceofOntario
parameter values of volborthite and also measure the through the Ministry of Research and Innovation.
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Topological Magnetic Excitations on the Distorted Kagomé Antiferromagnets
Supplemental Material
S. A. Owerre1,2
1Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada.
2African Institute for Mathematical Sciences, 6 Melrose Road, Muizenberg, Cape Town 7945, South Africa.
(Dated: March 7, 2017)
I. MODEL HAMILTONIAN experimentally, the noninteracting magnon model suffi-
ciently describes the system. The corresponding Hamil-
7 WeconsidertheHamiltonianforkagoméantiferromag- tonian that contribute to noninteracting magnon model
1 nets with DM interaction and Zeeman magnetic field is given by
0
given by
2
r = JijSi Sj +Dij Si Sj H Si, (1) HJ = Jij cosθijSi·Sj +sinθijcosφzˆ·(Si×Sj)
Ma H hXi,ji(cid:2) · · × (cid:3)− ·Xi hXi,ji (cid:2) (6)
4 where H = hˆz, h = gµBH and Dij = −D⊥ˆz. This par- +2sin2 θij (sin2φSxSx+cos2φSzSz) ,
ticular case of out-of-plane DM interaction is very com- 2 i j i j
mon in most distorted kagomé antiferromagnetic mate- (cid:18) (cid:19)
el] rials. It is also interesting because topological magnetic HDM =D⊥ sinθij(cos2φSixSjx+SiyS(cid:3)jy (7)
- excitations are not induced by the DM interaction as we hXi,ji(cid:2)
str wbriellaskhsotwh.eTchome pprleetseenScUe(o2f)oruott-aotfi-opnlansyemDmMetirnytedraowctniontos +sin2φSizSjz)−cosθijcosφzˆ·(Si×Sj) ,
at. U(1) symmetry about the z-axis. The classical energy is Hh =−hcosφ Siz, (cid:3) (8)
m given by Xi
- e (φ)=2J(2+δ) (1 cosϑ)cos2φ+cosϑ (2)
d 0 − where θij = θi θj. At zero magnetic field, i.e., φ =
−
n 4D sin2φ(cid:2)sinϑ(1 cosϑ) 3hcosφ(cid:3), π/2,thechiralinteractionisabsentinthemagnonmodel
o − ⊥ − − despite the the presence of DM interaction. Hence, the
c where e0(φ) = E(φ)/NS2, N is the number of sites per systemistopologicallytrivialatzerofield. Themagnetic-
[
unit cell, and ϑ = arccos( 1/2δ) which is not exactly field-induced scalar spin chirality
−
5 120◦ for δ = 1. The magnetic field is rescaled in unit
6
v of S. The minimization of e (φ) yields the field-induced
0
9 canting angle cosφ=h/h where χ = χijk, (9)
s H
9 i,j,k=∆
X
1 (1 cosϑ)
5 hs = − 4J(2+δ)+8D sinϑ . (3)
0 3 ⊥ originates from noncoplanar spin texture formed by the
1. The excitations above (cid:2)the classical ground sta(cid:3)te are ob- spin triad Si, Sj, Sk, where χijk = Si ·(Sj ×Sk). It
0 tained as follows. The procedure involves performing a is well-known that hχijki can be nonzero even in the
7 rotationaboutthez-axisonthetriadbythespinoriented absence of magnetic ordering Sj = 0, e.g. in chiral
h i
1 angles ϑ in order to achieve the coplanar configuration. spin liquid phase. This is a very crucial difference be-
v: As the out-of-plane magnetic field is turned on, we have tweencollinearferromagnetsandantiferromagnetsonthe
i toalignthespinsalongthenewquantizationaxisbyper- kagomé lattice. To obtain the magnon dispersions, we
X forming a rotation about the y-axis by the field canting proceed as usual by introducing the Holstein Primakoff
r angle φ. The total rotation matrix takes the form spin bosonic operators
a
Si =Rz(θi)·Ry(φ)·S0i, (4) Siz =S−a†iai, (10)
where
S
cosθ cosφ sinθ cosθ sinφ Siy =i 2(a†i −ai), (11)
i i i r
−
z(θi) y(φ)= sinθicosφ cosθi sinθisinφ , S
R ·R  −sinφ 0 cosφ  Six =r2(a†i +ai), (12)
 (5)
and θi =ϑA,B,C. In the following, we drop the prime in wherea†i(ai)arethebosoniccreation(annihilation)oper-
the rotated coordinate. At low temperatures accessible ators. The noninteracting magnon tight binding Hamil-
2
tonian is given by In the undistorted limit δ 1, ζ = ζ = ζ . The
AA BC CA
→
resulting magnon bands at zero field = 0 has a flat
H=S Gzij(a†iai+a†jaj)+Gdij(e−iΦija†iaj +h.c.) constantmode. ForthedistortedcaseδH=χ 1thesituation
6
hXi,ji(cid:2) is different. We have ζAA > ζBC, ζCA for δ < 1 and
(13) ζ < ζ , ζ for δ > 1. The magnon bands at zero
AA BC CA
+Goij(a†ia†j +h.c.) +hφ a†iai, field Hχ = 0 no longer have a flat constant mode in the
entire Brillouin zone see main text.
i
(cid:3) X
where
Gz = J [cosθ +2cos2φsin2(θ /2)] (14)
ij − ij ij ij II. MATRIX DIAGONALIZATION
sin2φsinθ ,
ij
−D⊥
Gd = (GR)2+(GM)2, (15) The Hamiltonian is diagonalized by the generalized
ij ij ij Bogoliubov transformation Ψ = Q , where is a
k k k k
GRij =Jqij cosθij + 2sin2φsi2n2(θij/2) (16) 2QN†k ×=2(Nβk†pAaβrak†uBnβitk†aCr)ybmeaintrgixthaendquQPa†ksip=ar(tQic†kle, Qop−ePkr)atwoirtsh.
(cid:2) sin2φ (cid:3) The matrix k satisfies the relations,
+D sinθ 1 , P
ij
⊥ − 2
(cid:18) (cid:19)
GGoMij ==scions2φφ(J2ijJsinsiθnij2(−θD/⊥2)cosDθij)s,inθ , ((1187)) PPkk††Hτ3(Pkk)P=kτ=3,Ek ((2256))
ij 2 ij ij − ⊥ ij where =diag(ω , ω ),τ =diag(I , I ),
k kα kα 3 N N N N
and hφ = hcosφ(cid:0). The fictitious magnetic flu(cid:1)x or solid and ωE are the energy−eigenvalues. From× E−q. 26×we
kα
atannglΦe su=btGenMd/eGdRby. WtherecelenaorlnycosepelatnhaartsΦpinsisinsognivzeernobiny get Pk† = τ3Pk−1τ3, and Eq. 25 is equivalent to saying
ij ij ij ij that we need to diagonalize the Hamiltonian 0(k) =
the absence of DM interaction. It should be noted that H
τ (k), whose eigenvalues are given by τ and the
3 3 k
the Q=0 magnetic structure that can be stabilized by H E
columns of are the corresponding eigenvectors. The
k
other means as mentioned in the text. In momentum P
eigenvalues of this Hamiltonian cannot be obtained ana-
space we obtain
lytically except at zero field. The paraunitary operator
H=S 2Mzαβδαβ +2Mdαβ a†kαakβ (19) Pk defines a Berry curvature given by
k,α,β
+MoαβXa†k(cid:0)αa†−kβ +akαa−kβ ,(cid:1) Ωij;α(k)=−2Im[τ3(∂kiPk†α)τ3(∂kjPkα)]αα, (27)
(cid:16) (cid:17) withi,j = x,y and arethecolumnsof . Inthis
where α,β =A, B, C and the coefficients are given by { } Pkα Pk
form, the Berry curvature simply extracts the diagonal
components which are the most important. From Eq. 25
z =diag(ζAA,ζBB,ζCC), (20) the Berry curvature can be written alternatively as
M
with ζ = Gz +Gz +h /2, ζ = ζ = Gz +
AA AB CA φ BB CC AB
Gz +h /2.
BC φ Im[ v v ]
0 γAdBe−iΦAB γCdAeiΦCA Ωij;α(k)=−2 hPkα|(ωi|Pkα0iωhPk)α20| j|Pkαi ,
Md =γγC∗A∗dAdBee−iΦiΦACBA γB∗dCe0iΦBC γBdCe0−iΦBC, (21) αX06=α kα− kα0 (28)
 0 γo γo 
Mo =γγA∗ooB γA0oB γBCo0CA, (22) wThheerBeevrry=cu∂rHva0t(ukr)e/i∂skredlaetfiendestotthheevseolloidcitayngolepeΩra(kto)r∝s.
C∗A B∗C Φ and the Chern number is defined as,
 
where γd = Gd cosk , γd = Gd cosk , γd =
GdCAcosAkB3; γAoABB =1 BGCoABcosBkC1, γ2BoC CA = α = 1 dkxdky Ωxy;α(k). (29)
GoBCcosk2,γCoA = GoCAcosk3. ki = ki · ei, and C 2π ZBZ
e =( 1/2, √3/2), e =(1,0), e =( 1/2, √3/2).
1 2 3
− − − It can be shown that the Chern numbers are related
The Hamiltonian can be written as
to the scalar-chirality-induced fictitious flux Φ as =
1
H=S Ψ†kH(k)Ψk−const., (23) 0, C2 =−sgn(sin(Φ)), C3 =sgn(sin(Φ)), where C
k
where Ψ†k =(b†kA,Xb†kB, b†kC, b kA, b kB, b kC), and 1
− − − sin(Φ)= χ . (30)
z + d o 2 ijk
(k)= M M M . (24)
H o z+ d
(cid:18) M M M (cid:19)