Chiral Spin States in a Spin-charge Coupled System on the Shastry-Sutherland Lattice

1.2 MB·

by  Munir Shahzad

Chiral Spin States in a Spin-charge Coupled System on the Shastry-Sutherland
Lattice
Munir Shahzad and Pinaki Sengupta
School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371
(Dated: January 17, 2017)
We investigate the necessary conditions for the emergence of complex, non-coplanar magnetic
configurationsinaKondolatticemodelwithclassicallocalmomentsonthegeometricallyfrustrated
Shastry-Sutherland lattice, and their evolution in an external magnetic field. We demonstrate that
topologically non-trivial spin textures – including a new canted flux state – with non-zero scalar
chiralityarisedynamicallyfromrealisticshortrangeinteractions. Ourresultsestablishthatafinite
Dzyaloshinskii-Moriya(DM)interactionisnecessaryfortheemergenceofthesenovelmagneticstates
7 when the system is at half-filling for which the ground state is insulating. We identify the minimal
1
setofDMvectorsthatarenecessaryforthestabilizationofchiralmagneticphases. Furthermore,the
0
non-coplanarityofsuchstructurescanbetunedcontinuallybyapplyinganexternalmagneticfield.
2
Onceagain,thenatureoftheDMinteractionsdictatetheemergenceofmagnetizationplateausthat

n areubiquitousinthecanonicalShastry-Sutherlandmodel. Ourresultsarecrucialinunderstanding
a the magnetic and electronic properties of the rare earth tetraboride family of frustrated metallic
J magnets.

6
1
I. INTRODUCTION earth ions are arranged in a SSL in the layers. Due to

] strong spin-orbit coupling, the rare-earth ions in these
l
e Thestudyofstronglyinteractingquantummanybody compounds carry large magnetic moments and conse-
r- systems with independent spin and charge degrees of quently, can be treated as classical spins. They act as
st freedomonfrustratedlatticeshaveattractedheightened effective local fields that interact strongly with the elec-
. interest in the recent past. The interplay between geo- tron spin.22–29 In this study our goal is to construct
t
a metric frustration and strong interaction between itin- a minimal model where topologically non-trivial chiral
m erant electrons and localized moments in these systems magnetic phases can be realized from physically rele-
- results in novel quantum phases and phenomena that vant interactions and investigate their evolution in an
d
arenotobservedintheirnon-frustratedcounterparts1–8. external magnetic field. In particular, we explore the
n
o Frustrated interactions between the local moments, to- role of different components of the DM interaction in
c getherwithcrystalelectricfieldsandcouplingtotheitin- stabilizing different aspects of the local moment config-
[ erant electrons, often stabilize non-coplanar ordering of urations. What are the minimal DM vectors required

these moments1,9–12. When an electron moves through to stabilize a tunable non-coplanar spin configuration,
1
v suchbackgroundspintextures,itpicksupaBerryphase as well as magnetization plateaus? How does an ap-
7 whichunderliesseveralnoveltransportphenomenasuch plied field affect the non-coplanarity of the spin config-
9 as the topological (or geometric) Hall effect and uncon- uration? Does the nature of chiral spin state change in
2 ventional magnetoresistive properties13–16. The interest the presence of an external field? These are some of the
4 in these systems is driven both by the desire to under- questions we address in this work. Our results reveal
0 stand the underlying mechanism driving the novel phe- that multiple non-coplanar spin arrangements (charac-
.
1 nomenaaswellastocontroltheiremergencebyexternal terized by different values of the scalar spin chirality)
0 tuning fields in order to harness their unique functional- with long range magnetic order are stabilized over an
7 ities for practical applications. extended range of parameters. Not surprisingly, we find
1 In this paper, we study the Kondo Lattice that DM interactions play a crucial role in stabilizing
:
v Model (KLM) on the geometrically frustrated Shastry- chiral spin configurations. Furthermore, we are able to
Xi Sutherland lattice (SSL) with classical spins where the tune the non-coplanarity (equivalently, the topological
standard (antiferromagnetic) Heisenberg interaction be- character) of the spin textures – changing and suppress-
r
a tween the local moments is supplemented by an ad- ing the net chirality – by applying an external magnetic
ditional Dzyloshinskii-Moriya (DM) interaction. The field. This is in contrast to most previous studies where
SSL is a paradigmatic geometry to study the effects of the non-coplanar textures of the local moments is im-
competing interactions in the presence of frustration17. posedbyextraneousfactors(e.g., crystalelectricfieldin
TheShastry-SutherlandKondolatticemodel(SS-KLM) pyrochlores) and as such, cannot be changed easily.
has previously been studied with S = 1/2 local mo-
ments18–21 where quantum fluctuations of the local mo-
mentsplayacrucialroleindeterminingthecharacterof II. MODEL
the ground state. In the present study, we revisit this
model, but with the local moments treated as classical The Hamiltonian describing the SS-KLM with addi-
spins. Thisisnotsimplyofacademicinterest. Thereex- tional DM interactions is given by,
ist a complete family of rare-earth tetraborides (RB ,
4
R=Tm, Er, Ho, Dy) – quasi-two-dimensional metal-
lic magnets where the magnetic moment-carrying rare H=H +H +H +H (1)
e ex DM H
2
where H represents the electronic Hamiltonian,
e (cid:74) (cid:78)
(cid:88) (cid:88)
H =− t (c† c +h.c.)−J S ·s
e ij iσ jσ K i i y x
(cid:74) (cid:74) (cid:74)
(cid:104)i,j(cid:105),σ i
The first term is the kinetic energy of the itinerant elec- (cid:74)
z
trons – (cid:104)i,j(cid:105) represents the Shastry-Sutherland bonds (cid:74) (cid:78)
(viz., first neighbors along the principal axes and the al- (cid:74) D⊥out-of-plane
ternate diagonals) and t are the transfer integrals for
ij (cid:78) D⊥into-plane
these bonds. The second term is the on-site Kondo-like (cid:78) (cid:78) (cid:78)
interaction between the spin of the itinerant electrons si D(cid:107)
andlocalizedmomentsS . Theconductionelectronspin
i D(cid:48)
isdefinedass =c† σ c ,whereσ isthevectorele- (cid:74) (cid:78)
i i,α αβ i,β αβ
mentofusualPaulimatrices. Asmentionedintheintro-
duction we treat the localized spins as classical vectors
FIG. 1. (Color online) DM interaction defined on the unit
with unit length (|S |=1). In this limit, the sign of J
i K cell of SSL where directions of the arrow from site i to j
(ferromagnetic or antiferromagnetic) is irrelevant since
indicates the direction of cross product S ×S . The red
i j
eigenstatesthatcorrespondtodifferentsignsarerelated (cid:74)
arrows represent the parallel components of D while and
by a global gauge transformation. The states of the lo- (cid:78) represent out-of-plane and into components of D. Blue
calized spins are specified by the angular components as arrowsindicatethecomponentsofD(cid:48) onthediagonalbonds.
S = (sinθ cosφ ,sinθ sinφ ,cosθ ). Among the other The directions of x, y and z axis are also mentioned.
i i i i i i
terms in Eq. (1) H represents the classical Heisen-
ex
berg interaction between the localized spins, H =
ex
(cid:80) In order to explore the thermodynamic properties we
J S · S . H describes the Dzyaloshinskii-
(cid:104)i,j(cid:105) ij i j DM (cid:80) writethepartitionfunctionforthewholesystembytak-
Moriya (DM) interaction, H = D .(S ×S ),
DM (cid:104)i,j(cid:105) ij i j ing two traces,
where D is the DM vector which is determined by the
ij
crystal symmetry of the lattice. The precise values (di-
rections and magnitude) DM vectors will depend on the Z =Tr Tr exp[−β(He({x })−µNˆ )]·
S I r e
details of the crystal symmetry for each compound. In
exp[−β(H +H +H )] (2)
this study we choose a generic set of DM vectors and ex DM H
identify the minimal interactions that are necessary for
whereTr andTr representthetracesovertheclassical
S I
stabilizingnon-coplanarspintextures. Theexplicitform
localized spins denoted by {x } and the charge degrees
r
of the DM vectors on the different bonds is given in the
offreedomrespectively. Thetraceoveritinerantelectron
captionofFig.1. ThelasttermintheHamiltonian(1)is
degrees can easily be calculated by numerical diagonal-
theZeemantermfor thelocalizedspins duetoan exter- ization Hamiltonian matrix He for a fixed configuration
nal (longitudinal) magnetic field, H = −hz(cid:80) Sz. A
H i i of localized spins {xr},
Zeeman term for the itinerant electrons is not included
explicitly,sincetheinstantaneousspinorientationofthe
electrons are determined completely by the local mo- Tr exp[−β(He({x })−µNˆ )]=
I r e
mentsinthelargeJ limitthatweconsiderinthisstudy.
K (cid:89)
Hereafter the parameters with primes represent the in- (1+exp[−β(εν({xr})−µ)]) (3)
teractions on diagonal bonds while the unprimed are for ν
axial bonds.
where µ is the chemical potential, β =1/k T is the in-
B
verse temperature and Nˆ = 1 (cid:80) c† c is the num-
e 2N iσ iσ iσ
ber density of conduction electrons. The partition func-
III. METHOD AND OBSERVABLES
tion for the whole system then takes the form,
To investigate the above model, we use an unbiased
Monte Carlo (MC) method that has been used previ- Z =Tr exp[−S ({x })−β(H +H +H )] (4)
S eff r ex DM H
ously in the study of similar models5,30–32. A brief re-
The corresponding effective action is S ({x }) =
view of this method is presented here closely following eff r
(cid:80)
the references33,34. The dynamics of large localized mo- νF(y) where F(y)=−log[1+exp{−β(y−µ)}]. The
grand canonical trace over localized spin degrees of free-
mentsoftherareearthionsisslowcomparedtoitinerant
dom is evaluated by sampling the spin configuration
electronsand,accordingly,wecandecoupletheirdynam-
space using a classical Monte Carlo (MC) method. The
ics from that of the itinerant electrons. While studying
probability distribution for a particular configuration of
thelatter, wetreatthelocalmomentsasstatic, classical
localized spins {x }can be written as,
fields at each site. The electronic part of the Hamilto- r
nian is bilinear in fermionic operators. Using the single
P({x })∝exp[−S ({x })−β(H +H +H )] (5)
electron basis, He can be represented as 2N ×2N ma- r eff r ex DM H
trix for a fixed configuration of classical localized spins, The thermodynamic quantities that depend on local-
where N is the number of sites. izedspinsarecalculatedbythethermalaveragesofspin
3
configurations while the quantities that are associated
withitinerantelectronsarecalculatedfromtheeigenval-
ues and eigenfunctions of He({x }). We start the sim-
r
ulations with a random configuration of localized spins
{x } and calculate Boltzmann action S ({x }) for this
r eff r
configuration. The spin configuration is updated via
Metropolis algorithm based on the change in the effec-
tive actions of the configurations resulting from random
updates, ∆S = S ({x(cid:48)})−S ({x }). To identify
eff eff r eff r
magnetic orderings we calculate the magnetization per
unit site as well as spin structure factor which is the
Fourier transform of the spin-spin correlation function,
1
(a)
0.8
0.6
χ
0.4
L = 8
0.2 L = 12
L = 16
0
(b)
0.13
0.12
ms0.11
m/
0.1
FIG.3. (Coloronline)(a)Plotofspinstructurefactoralong
q and q indicating the two sharp peaks appearing at q =
0.09 x y
(0,π) and (π,0) and a small peak at q = (0,0) for 16×16
lattice. (b) The snapshot of real space localized spin config-
0.08
0 0.05 0.1 0.15 0.2 uration for 8×8 lattice showing the canted flux state. The
D MCcalculationsaredonekeepingT =0.02,t=1.0,t(cid:48) =1.2,
⊥ J = 0.1, J(cid:48) = 0.12, J = 3.0, D = 0.0, D(cid:48) = 0.0 and
K (cid:107)
hz =0.0.
FIG. 2. (Color online) (a) Chirality per unit cell and (b)
magnetization per unit site as a function of D for 8×8,
⊥
12×12and16×16latticesizes. Theresultsareobtainedat SSL)asthechiralorderparameter. Thisquantityiszero
T =0.02 while keeping t=1.0, t(cid:48) =1.2, J =0.1, J(cid:48) =0.12, for collinear or coplanar magnetic states such as ferro-
J =3.0, D =0.0, D(cid:48) =0.0 and hz =0.0.
K (cid:107) magnetic(FM),antiferromagnetic(AFM)andpureflux
states whereas it is non-zero for non-coplanar configura-
tions, e.g., all-out and 3-in 1-out states observed in py-
rochlores. Finally, as an additional characterization of
1 (cid:88)
S(q)= (cid:104)S ·S (cid:105)exp[iq·r ] (6) the chairal nature of the spin configurations, we mea-
N i j ij
sure the circulation of the in-plane components around
i,j
(cid:80)
each square palquette as f = S ·r , where S is
m (cid:3) i ij i
wherer isthepositionvectorfromith tojth siteand(cid:104)(cid:105) the spin at site i and r ’s are the vectors connecting
ij ij
represent the thermal averages over the grand canonical sites i and j around the square plaquette in a counter-
ensemble. To elucidate the difference between topolog- clockwise direction. A non-zero circulation identifies a
ical trivial and non-trivial states we evaluate the local flux configuration of the local moments.
scalar chirality. On a triangle the chirality is defined as,
χ =S ·(S ×S ) (7)
(cid:52) i j k
IV. RESULTS
Weusethetotalchiralityχ= 1 (cid:80) χ (wheresum
Nu (cid:52) (cid:52)
is over all the triangles formed on the plaquttes with Simulations of the hamiltonian (1) were performed in
diagonal bonds and N is the number of unit cells of lattices of dimension L × L with L = 8 − 16, over a
u
4
wide range of parameters. For smaller lattices, we used 1
(a)
exact diagonalization- Monte Carlo (ED-MC) method
where the full Hamiltonian is diagonalized to calculate
0.8
0.75
theeffectiveactionforeachMCstep. Forthelargerlat-
tices, we used travelling cluster approximation (TCA)
0.6
method35–38 – a 6×6 cluster of SSL is used to calculate ms 0.5
the effective action for one MC sweep. Once the system m/
is equilibrated then we calculated the thermal averages 0.4
by diagonalizing the full Hamiltonian matrix. To avoid
D = 0.0
||
getting trapped in local minima and to speed up the 0.2 D = -0.10
||
equilibration, we used simulated annealing. For this, we D = -0.15
||
start the simulations at a relatively high temperature 0
T = 0.1 with random localized spin configuration and 3 (b)
run the system for equilibration and then use final con-
2.5
figuration at this temperature to do the equilibration at
T =0.08. Werepeatthisprocesswithastepoftempera-
2
ture ∆T =0.02 finally calculating the thermal averages
χ
of the observables at temperature T = 0.02. For the 1.5
lattice sizes studied, the thermal gap to the lowest ex-
citation is greater than the finite size gap at T = 0.02. 1
In other words, T = 0.02 is sufficiently small such that
0.5
ground state estimates of the measured observables can
be reliably obtained. Measurements are done for 40,000
0
MC steps after 2000−60,000 steps for thermalization. 0 0.5 1 1.5 2
z
h
(a)
1.2
FIG. 5. (Color online) (a) Magnetization per unit site and
(b) chirality per unit cell as a function of external magnetic
1.1 field for different values of D . The results are obtained for
(cid:107)
12×12 at T = 0.02, t = 1.0, t(cid:48) = 1.2, J = 0.1, J(cid:48) = 0.12,
1 JK =3.0, D⊥ =0.10 and D(cid:48) =−0.05.
χ
0.9
L = 8
0.8 L = 12
L = 16
0.7
(b)
0.16
on the magnetic behavior at an electronic filling factor
(cid:104)N (cid:105) = 1/2, for which the system is in an insulating
e
0.14 state. The choice for the rest of the Hamiltonian pa-
ms rameters are guided by experimental observation in real
m/ materials. Theelectronichoppingmatrixelementsalong
0.12 the axial bonds are chosen as t=1.0 – this sets the en-
ergy scale for the problem. The diagonal hopping is set
to t(cid:48) = 1.2 and the values of the exchange interactions
0.1
alongtheaxialanddiagonalbondsaresetatJ =0.1and
J(cid:48) = 0.12. This choice is motivated by the experimen-
0 -0.05 -0.1 -0.15 -0.2
tal observation of approximately equal bond lengths in
D′
therareearthtetraboridefamilyofcompounds. Inmost
materials of relevance to the present model, there exist
FIG. 4. (Color online) (a) Chirality per unit cell and (b) strong DM interactions. While the exact nature of DM
magnetization per unit site as a function of D(cid:48) for 8×8, interaction depends on the crystal symmetries, we have
12×12and16×16latticesizes. Thecalculationsareobtained
chosen a generic form of DM interaction for our study.
at T =0.02, t=1.0, t(cid:48) =1.2, J =0.1, J(cid:48) =0.12, J =3.0,
K Indeed,investigatingtheroleofDMinteractioninstabi-
D =0.0, D =0.10 and hz =0.0.
(cid:107) ⊥ lizing non-coplanar spin configurations is a central goal
of the present study. Finally, following the experimental
With its multi-dimensional parameter space, the observationinotherfrustratedmetallicmagnetssuchas
Hamiltonian (1) is expected to support a rich array of the pyrochlores, the strength of the Kondo coupling is
ground state phases over different ranges of the param- chosen to be the strongest energy scale in the problem,
eters. In the present work, we restrict our attention J =3.0.
K
5
A. Effect of DM interaction of chirality as D is varied is shown in Fig. 2(a). At
⊥
D = 0, the ground state has predominantly longitudi-
⊥
Inthefirstpartofthestudy, asystematicvariationof nalAFMorderwithavanishinglysmallchirality. There
the different components of the DM vector is performed isasmall,butnon-zerouniformmagnetization–aconse-
to identify the minimal set of vectors necessary for non- quence of large Kondo-like coupling between the charge
coplanar configurations of the local moments. We study and spin degrees of freedom. With increasing the value
the effects of the DM vectors normal to the plane of the of D⊥, the ground state remains in the same phase with
lattice, D , and two in-plane components, D and D(cid:48) vanishing chirality up to a critical value ≈ 0.8 beyond
⊥ (cid:107)
(fig.1). It is found that while D is essential for the whichthereisadiscontinuoustransitiontoachiralstate
⊥
emergence of non-coplanarity, D drives the appearance characterized by a finite value of χ. This is accompa-
(cid:107)
of field induced magnetization plateaus. nied by a sharp drop in the uniform magnetization [see
Fig. 2(a)]. The sharp increase in chirality indicates a
non-coplanar component to the spin configuration and
1. Role of D establishes topologically non-trivial nature of the spin
⊥
configuration.
We start our discussion by analyzing the nature of
the magnetic ground state at zero field. The evolution
FIG. 6. (Color online) The real space configurations of local spins for 8×8 lattice shown at different values of magnetic field
(a) hz = 0.0 (b) hz = 0.8, just prior to a phase transition (c) hz = 1.2, after the phase transition point and (d) hz = 1.6,
when local spins are almost becoming polarized with magnetic field. The colorbar besides each plot indicates the out of plan
componentofthespinvector. TheMCsimulationsareperformedatT =0.02,t=1.0,t(cid:48) =1.2,J =0.1,J(cid:48) =0.12,J =3.0,
K
D =0.0, D =0.1 and D(cid:48) =−0.05.
(cid:107) ⊥
Additional insight into the magnetic ground state of-plane orientation of the spins, as well as the interac-
can be obtained from the momentum dependence of tionbetweenthelocalmomentsandthespinoftheitin-
the static spin structure factor which is illustrated in erant electrons. Nominally such features in the struc-
Fig. 3(a). It exhibits sharp peaks at (π,0) and (0,π). ture factor points towards a canted antiferromagnetic
Thereisanadditional,sub-dominantpeakat(0,0)indi- state. However, that is not the case here. The true na-
catinganetmagneticmomentevenatzeroexternalfield ture of the non-coplanar ground state is illustrated by
[see Fig. 2(b)], despite the AFM nature of the Heisen- a snapshot of the real space (periodic) equilibrium spin
berg interaction between the localized moments. This configuration obtained from the simulations and shown
is due to both the DM interaction which favors an out- schematically in Fig. 3(b). The in-plane components of
6
the local moments are arranged in a near-ideal flux pat- modify the plateau structure. Our results show that the
ternalongwithafiniteout-of-planecomponent–thatis, appearance and nature of the plateaus in the presence
the magnetic ground state is a canted flux state. Such ofsimultaneousDMandKondo-likeinteractionsdepend
complex spin textures are essential ingredients for the on the magnitude of D and D . For D > 0.1 neces-
⊥ (cid:107) ⊥
observation of topological Hall effect, chiral spin liquid sary to stabilize non-coplanar spin textures that we are
and other topologically interesting phases. In contrast interested in, the plateaus are completely suppressed in
to pyrochlores where the non-coplanar terahedral order- the absence of D . For small values (|D | (cid:46) 0.1), there
(cid:107) (cid:107)
ing of the local moments is fixed by the crystal field ef- arenosignaturesofanyplateau-likefeaturesinthefield
fects,thecantedfluxstateinourstudyarisedynamically dependence of the uniform magnetization – m/m in-
s
from the interplay between competing Heisenberg, DM creases monotonically with increasing field strength and
and Kondo-like interactions in the presence of geomet- reaches saturation around h ∼ 1.7. For D = −0.10
z (cid:107)
ric frustration. This enables us to control these complex signs of non-monotonicity begins to appear at m/m =
s
magneticorderingscontinuallyviaanexternalmagnetic 0.5 and 0.75. Further increasing the D component of
(cid:107)
field. DMinteractiontwomagnetizationplateausareobserved
at m/m = 0.5 and ∼ 0.75. The evolution of the uni-
s
form magnetization in a longitudinal magnetic field is
2. Role of D(cid:48) shown in Fig. 5(a) for 3 different values of D . The
(cid:107)
strongest plateau appears at m/m ∼ 0.75, in contrast
s
AfterfindingtheminimumvalueofD thatcausesthe to all other reported magnetization plateaus (theoreti-
⊥
topologicalnon-trivialphasetransitionwediscusstheef- calorexperimental)inthespin-onlyShastry-Sutherland
fectofD(cid:48) onthenon-coplanargroundstate. Theresults modelandassociatedquantummagnets. Thelargevalue
forchiralityandmagnetizationperunitsiteasafunction of magnetization plateau is a consequence of the addi-
of D(cid:48) are shown in Fig. 4 (a) and (b). The increase in tional canting out of the plane driven by the coupling
D(cid:48) results in an enlarged out-of-plane component of the of the local moments with the itinerant electrons as well
localized spins making the ground state more canted. as the strong DM interaction. With the increase of D(cid:107)
Hence with the increase of D(cid:48) the non-coplanarity of the canting of the localized spins increases even at zero
the magnetic ordered state increases [see Fig 4(a)]. The field as can be seen in Fig. 5(b) where we have plotted
same effect is observed in the behavior of the magneti- the chirality as a function of external field for different
zationperunitsite–magnetizationincreasesmonotoni- values of D(cid:107).
cally with D(cid:48) as the enlarged out-of-plane component of
Finally, we discuss the topological nature of the mag-
spins contributes to increase in zero field magnetization
netic ground state as it is tuned by an external field for
[seeshowninFig.4(b)]. Thestaticspinstructurefactor
a representative set of DM vectors where the zero-field
S(q) at D(cid:48) =−0.10 (not shown here) confirms that the
ground is in a canted flux state. Fig. 5(b) shows the
magnitude of the peak at q = (0,0) is higher than that
evolution of the net static spin chirality with increasing
without D(cid:48). Similarly, a plot of the real space snapshot
magnetic field. The ground state spin texture remains
ofthegroundstate(notshownhere)showsthatcanting
non-coplanarinnatureoveralargerangeofappliedfield
ofspinsincreasewiththeintroductionofD(cid:48). Itisworth
strength. The chirality increases monotonically up to a
mentioningthatD(cid:48) cannotinducethechiralphasetran-
critical field of h ≈ 1.0, above which there is a dis-
c1
sition on its own – one always needs a non-zero value of
continuoustransitiontoadifferentchiralspinstatewith
D for that.
⊥ small(butstillnon-zero)chirality. Eventually,whenthe
field strength is increased beyond a second critical field
h ≈1.5,thegroundstatelosesitsnon-coplanarityand
c2
3. Role of D (h =0)
(cid:107) z thechiralityiscompletelysuppressed. Onceagain,snap-
shots of local spin configurations and circulation eluci-
Like D(cid:48), the other in-plane component of the DM date the true nature of the magnetic ground state[see
vector (D(cid:107)) cannot induce non-coplanarity of the spin Fig. 6]. For hz =0 the circulation is equal in magnitude
configurations by itself. Instead, it simply reinforces and opposite in sign for the plaquettes with diagonal
such configurations driven by D⊥ (increased χ) as well bonds whereas it is vanishingly small in the other pla-
as increases the uniform magnetization by enhancing quettes. In other words, the in-plane components of the
the canting of the local moments away from the z-axis. local moments are arranged in a flux pattern on alter-
However, D(cid:107) plays a key role in driving magnetization nating plaquettes. Snapshots at h = 0.8(< hc1) reveal
plateaus in an external field, as detailed in the next sec- that the ground state remains a canted flux state for
tion. h ≤ h . The discontinuous transition at h is driven
c1 c1
by the partial breaking of the flux configuration of the
in-plane components–in other words, the transition is
B. Effects of an external magnetic field topologicalincharacter. Themomentsconnectedbythe
diagonalbondsinsidetheplaquetteswithpositivecircu-
One of the most intriguing features of the canonical lation still retain a large in-plane component and (π,0)
(purely magnetic) Shastry-Sutherland model is the ap- ordering, whereas the other spins are largely polarized.
pearance of magnetization plateaus in an applied mag- Thepartialbreakingofthefluxpatterncausesadropin
netic field. It is expected that DM interactions strongly the circulation across the transition [Fig. 7] whereas the
7
polarizationofhalfthespinsresultsinadropinthestag- Sutherland lattice. Our results show that complex, non-
geredmagnetization. Thetransitionath ismarkedby coplanar spin configurations can be generated dynami-
c2
the complete breaking of the flux pattern and polariza- cally from purely short range interactions and coupling
tion of the remaining spins. toitinerantelectrons. WeconcludethatDMinteractions
are necessary for the emergence of chiral spin configura-
tionswhentheelectronicspectrumisgapped,thatis,the
system is in an insulating state. We have carefully iden-
2 2
tified the minimal DM vectors necessary for the stabi-
0 0 lization of non-coplanar configurations as well as for the
appearance of magnetization plateaus in the presence of
−2 −2
suchtopologgicallynon-trivialconfigurationsofthelocal
moments.Furthermore,suchnon-coplanarstructurescan
(a) (b)
be tuned continually by applying an external magnetic

field. These results provide an insight into the origin
2 2
and nature of topologically non-trivial magnetic phases
0 0 in metallic magnets. They will also be crucial in under-
standing the magnetic and electronic properties of the
−2 −2 rareearthtetraboridefamilyofmetallicfrustratedmag-
nets.
(c) (d)
FIG. 7. (Color online) Snapshot depicting the circulation of
flux, clockwise or anticlockwise, on each plaquette of 8×8
lattice for different values of magnetic field (a) hz =0.0, (b)
hz = 0.8, just before the phase transition (c) hz = 1.2, just
after phase transition (d) hz = 1.6, when almost all spins
become parallel with h-field. The different parameters used
in the calculations are same as mentioned in Fig. 6. ACKNOWLEDGMENTS
It is a pleasure to thank G. Alvarez and Y. Kato for
V. SUMMARY useful discussions on the development of the numerical
method. The work is partially supported by Grant No.
To summarize, we have studied the Kondo lattice MOE2014-T2-2-112fromtheMinistryofEducation,Sin-
model with additional DM interaction on the Shastry- gapore.
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