Table Of ContentChiral Phase Transitions in a Metallic Frustrated Magnet
Munir Shahzad and Pinaki Sengupta
School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371
(Dated: January 24, 2017)
WeexplorethemagneticphasesinaKondolatticemodelonthegeometricallyfrustratedShastry-
Sutherland lattice at metallic electron densities, searching for topologically non-trivial chiral spin
textures. Motivated by experimental observations in many rare earth based frustrated metallic
magnets, we treat the local moments as classical spins and set the coupling between the itinerant
electrons and local moments as the largest energy scale in the problem. Our results show that a
canted flux state with non-zero static chirality is stabilized over an extended range of Hamiltonian
parameters. The chiral spin state can be quenched efficiently by external fields like temperature
7 andmagneticfieldaswellasbyvaryingthedegreeoffrustrationintheelectronicitinerancy. Inter-
1 estingly, unlike insulating electron densities, a Dzyaloshinskii-Moriya interaction between the local
0 moments is not essential for the emergence of their non-coplanar ordering.
2
n
I. INTRODUCTION tal work. In this work, we investigate the SS-KLM for
a
thestabilizationoftopologicallynon-trivialgroundstates
J
at one-quarter and three-quarter filling of itinerant elec-
1 Theinterplaybetweenchargeandspindegreesoffree-
trons, forwhichthegroundstateismetallic. Ourresults
2 domresultsinvariousnovelandexoticphasesinstrongly
correlatedelectronsystems. Aparadigmaticmodeltode- reveal that non-coplanar ground states are stabilized for
] scribe this interplay is the Kondo lattice model (KLM) a wide range of parameters involved in the Hamiltonian.
l
e ordoubleexchangemodel(DEM)inwhichlocalizedmo- Wediscussindetailthenatureoftopologicalphasetran-
r- ments are coupled to itinerant electrons1–5. The mobile sition as a function of thermal fluctuations, frustration
st electrons in these systems act as mediators to establish and external magnetic field for both number densities of
. the effective correlation between localized spins giving electrons.
t
a rise to magnetic behavior. On the other hand the scat-
m
tering of these electrons from localized moments may
II. MODEL
- affect the electronic and transport properties of these
d
systems. The study becomes more fascinating when we
n
have geometrical frustration as an extra degree of free- We consider the SS-KLM in the presence of a longi-
o
c dom like the hybrid systems in which localized spins tudinal magnetic field. The geometry of SSL along with
[ are arranged on geometrically frustrated lattices. These the first Brillouin Zone (BZ) is depicted in Fig. 1. The
charge-spin coupled systems on frustrated lattices ex- Hamiltonian describing the system under investigation
2
hibit chiral spin textures that drive novel and exotic is,
v
2 topological properties6–10. For these non-coplanar spin
54 caonngfileguχr(cid:52)atio=nsStihe· Sscja×larSskpiins cnhoirna-zlietryo.defiTnheed onnona-zterrio- Hˆ =− (cid:88) tij(c†iσcjσ+h.c.)−JK(cid:88)Si·si+
4 value of chirality breaks both time-reversal (T) and par- (cid:104)i,j(cid:105),σ i
0 ity (P) symmetries and drives some peculiar transport (cid:124) (cid:123)(cid:122) (cid:125)
1. phenomena such as geometrical or Topological Hall ef- Hˆe
(cid:88) (cid:88)
0 fect (THE)11–16. These chiral spin configurations act as JijSi·Sj −hz Siz (1)
7
a source of fictitious magnetic field–when an itinerant (cid:104)i,j(cid:105) i
1
electron moves over them in a closed path it picks up an (cid:124) (cid:123)(cid:122) (cid:125)
v: extra Berry phase which results in THE17–20. THE has Hˆc
Xi been studied for Ferromagnetic Kondo lattice model on The first two terms make up the electronic part of the
geometrically frustrated lattices such as triangular21–24, Hamiltonian Hˆ where we have a tight binding term for
r e
a kagome20,25,26, checkerboard27, pyrochlore28,29 and face- itinerant electron and a Kondo-like interaction between
centeredcubiclattice30. Chiralspintexturesareessential these electrons and localized spins. (cid:104)i,j(cid:105) denotes the
in realizing this novel transport phenomena. bonds on the SSL, where nearest neighbors (NN) are ax-
Thetopologicalnatureofchiralspintexturesemerging ial bonds and next nearest neighbors (NNN) are diag-
dynamically from the interplay between competing mi- onal bonds on the alternate plaquette and t describes
ij
croscopic interactions can be controlled with the help of the hopping matrix of itinerant electrons on these SSL
external magnetic field, temperature and pressure which bonds. We treat the localized spins S as classical vec-
i
make them a potential candidate for application in spin- tors of unit length so the sign of J becomes irrelevant
K
tronics. So it is very desirable to study the topologi- in the current model. The third term in Eq. (1) repre-
cal nature of these magnetic ordered states and associ- sents anti-ferromagnetic Heisenberg interaction between
ated phase transitions in order to stimulate experimen- thelocalizedspins. ThelasttermintheHamiltonian(1)
2
is the Zeeman term for the localized spins due to an fermionicoperatorswhereσ arethevectorelementsof
αβ
external (longitudinal) magnetic field. We consider the usual Pauli matrices. The localized spin are defined as
strong J coupling limit, in that case, the spin direction S = (sinθ cosφ ,sinθ sinφ ,cosθ ) with |S | = 1. The
K i i i i i i i
of itinerant electron is completely determined by the lo- one-electronbasiscanbeusedtoexpressHˆ as2N ×2N
e
calized moment. So we just consider the effect of an matrix for a particular arrangement of classical localized
external magnetic field on the localized moments only. spins.
From here onwards, we represent the interactions on the To study the thermodynamic properties the full par-
axialbondsasunprimedparameterswhileprimedfordi- tition function in the grand canonical ensemble is repre-
agonals bonds. We choose t = 1 as the energy unit. In sented in terms of two traces,
an accompanying work31, we have investigated the role
of Dzyaloshinskii-Moriya (DM) interactions on stabiliz-
ing chiral spin textures at half-filling, where there exists Z =Tr Tr exp[−β(Hˆ ({x })−µn )]·
c f e r e
a finite gap in the electronic spectrum. We found that
exp[−β(Hˆ )]. (2)
for the insulating state, DM interactions are essential in c
establishingnon-coplanarorderingofthelocalmoments.
In above equation, Tr and Tr are traces over the clas-
Inthiswork, wefocusonelectronicfillingfactorsn = 1 c f
and 34 – where ne = 21N (cid:80)ic†ici and N is the numbeer o4f fsriceaedlolomcarleiszpedectmivoemlye.nTtsh,e{exirg}e,nvaanlduefseormftihoeniHcadmegilrteoensiaonf
latticesites–forwhichtheelectronicspectrumisgapless, matrix Hˆ for a fixed configuration of the localized spins
e
i.e.,thegroundstateismetallic. Wedemonstratethatat are used to calculate the trace Tr ,
f
these parameter regimes, topologically non-trivial chiral
spin textures emerge even in the absence of DM inter-
actions. Here we report the results of our study on the Tr exp[−β(Hˆ ({x })−µn )]=
f e r e
effects of temperature, frustration and magnetic field on (cid:89)
(1+exp[−β(ε ({x })−µ)]) (3)
the emergent chiral spin orderings. ν r
ν
where β = 1/k T is the inverse temperature and µ is
B
ky the chemical potential. Using Eq. (3) the full partition
π
2 Y K function can be written as,
JJ,0,tt0 −π2 Γ Mπ2 kx Z =Trcexp[−Seff({xr})−β(Hˆc)] (4)
(cid:80)
where S ({x }) = F(y) is the effective action and
eff r ν
F(y) = −log[1+exp{−β(y −µ)}]. To calculate Tr a
c
−π2 classical MC method is used to sample the spin configu-
ration space. We can write the probability distribution
(a) (b)
for a fixed configuration of localized moments {x } as,
r
FIG.1. (Coloronline)(a)TheGeometryofSSL,wherethere
arediagonalbondsonalternateplaquetteinadditiontoaxial P({xr})∝exp[−Seff({xr})−β(Hˆc)] (5)
bondsalongxandy-axis. (b)1stBZofSSLlatticewithhigh
symmetric points mentioned. The eigenvalues and eigenfunctions of Hˆe({xr}) are
used to calculate the thermodynamic quantities related
toitinerantelectronswhilethequantitiesassociatedwith
thelocalizedspinsarecalculatedbythethermalaverages
III. METHOD AND OBSERVABLES of spin configurations. We select a random configuration
oflocalizedspin{x }andcalculatetheBoltzmannaction
r
S ({x }) for this configuration. Next, the random up-
The above model can be explored for thermodynamic eff r
dates performed to this spin configuration are accepted
properties by using an unbiased Monte Carlo (MC)
or rejected based on Metropolis algorithm. We use the
method. This approach has already been used in sim-
ilar type of studies21,32–35. We will present a brief in- magnetizationperunitsiteandstaticspinstructurefac-
tor to distinguish between different magnetic orders of
troduction of this method in accordance with the main
references36,37. We can separate and hence decouple the the localized spins,
slow dynamics of the localized spins of rare earth ions
from the fast dynamics of the itinerant electrons. In 1 (cid:88)
S(q)= (cid:104)S ·S (cid:105)exp[iq·r ] (6)
this case, the local moments at each site can be treated N i j ij
as classical static adiabatical fields. By replacing the i,j
itinerant electron spin as s = c† σ c , the elec-
i i,α αβ i,β where (cid:104)(cid:105) represent the thermal averages over the grand
tronic part of the Hamiltonian Hˆ becomes quadratic in canonical ensemble and r is the position vector from
e ij
3
2 2 IV. RESULTS AND DISCUSSIONS
(a) (c)
L = 8
1.5 L = 12 1.5
L = 16 We have done calculations for lattice size L = 8−16,
where L is the length of the lattice along one axis. The
χ 1 1χ
results for L = 8 are obtained by diagonalizing the full
Hamiltonian to calculate the Boltzmann factor as it is
0.5 0.5
faster for small lattice sizes. But for lattice size L > 8,
0 0 weusedthetravellingclusterapproximation(TCA)38–41,
(b) (d)
0.4 0.4 in this method, a cluster of 6×6 sites is moved sequen-
tially over the whole lattice and Boltzmann factor was
0.35 ne=1/4 ne=3/4 0.35 calculatedbyjustdiagonalizingthisclusterHamiltonian.
ms ms
m/ 0.3 0.3m/ Oncethesystemisequilibratedthenphysicalobservables
are calculated by diagonalizing the full Hamiltonian. To
0.25 0.25
approachequilibrationefficientlyandavoidgettingstuck
0.2 0.2 at any local minimum, we have used a simulated anneal-
0 0.02 0.04 0.06 0.08 0.1 0.02 0.04 0.06 0.08 0.1 ing method. At the beginning, we select a random con-
T/t T/t figuration of the localized spins at a relatively high tem-
perature T = 0.1 and equilibrate the system. Next, we
FIG. 2. (Color online) Plots of magnetization per unit site use the final configuration of this temperature to do the
(upper panels) and chirality (lower panels) as a function of
equilibration at T =0.08. This process is repeated with
temperature T at ne (cid:39) 1/4 (left) and 3/4 (right). The pa- a temperature step, ∆T = 0.02, finally measuring the
rameters used are t(cid:48) =0.25, J =0.1, J(cid:48) =0.025, hz =0 and thermalaveragesofthephysicalobservablesatT =0.02.
JK =8.0.
Measurementsaremadeafterevery1,000stepsofanMC
runof40,000stepsintotalafterdiscarding2000−60,000
MC steps for thermalization. We have divided the data
into 40 different bins to calculate the average value and
error from the standard deviation.
A. Role of Temperature
Throughout this work we choose the limit of strong
Kondocouplingbetweenitinerantelectronsandlocalized
spins(J =8.0)followingtheexperimentalobservations
K
in relevant metallic magnets. We start our discussion
by considering the thermal transition of the magnetic
ground state. Fig. 2 (a) and (b) show the temperature
FIG. 3. (Color online) (a) The momentum dependence of
dependence of magnetization and scalar spin chirality as
static spin structure factor exhibiting sharp peaks at q =
a function of temperature at n (cid:39)1/4 (here we have in-
(0,π), (π,0) and a small peak at (0,0). The results are ob- e
troduced (cid:39) sign because we are working ingrand canon-
tained using parameters t(cid:48) = 0.25, J = 0.1, J(cid:48) = 0.025,
hz =0,JK =8.0andT =0.02for12×12latticeatne (cid:39)1/4. icalensembleandelectrondensitiesarecontrolledbythe
(b)Electronicenergybandsforatightbindingmodelinclud- values of µ give number densities fluctuating within the
ingdoubleexchangeinteractionwithtriple-qorderstabilized error bars close to the required density). Initially, the
at q = (π,0), (0,π) and (0,0). The values of interaction magnetizationdecreasesveryslowlywithdecreasingtem-
parameters used are t=1.0, t(cid:48) =0.25 and JK =8.0 perature but around T (cid:39) 0.05 there is a discontinuous
drop in magnetization indicating a first order magnetic
phase transition. The nature of this transition becomes
muchclearerifwelookatFig.2(b)wherewehaveshown
site i to j. We use the total chirality χ= 1 (cid:80) χ to the behavior of chirality while varying T for n (cid:39) 1/4.
Nu (cid:52) (cid:52) e
differentiate between topological trivial and non-trivial The value of chirality is very small and it remains al-
magnetic states, where the sum is taken over all the tri- most flat at higher values of temperature indicating an
angles made on the alternate plaquettes with diagonal absence of non-coplanar ordering. The non-zero value of
bonds and N is the number of unit cells of SSL. The chiralityinthislimitcanbeattributedtothermalfluctu-
u
value of the chiral order parameter is non-zero for non- ations which are dominant at higher temperatures. But
coplanar configurations such as all-out, 2-in 2-out and as we approach T (cid:39) 0.05 a transition to a non-coplanar
3-in 1-out while it is zero for coplanar magnetic states spin ordering is indicated by a discontinuous increase in
such as anti-ferromagnetic (AFM), ferromagnetic (FM), the value of χ as seen in Fig. 2 (b). At lower temper-
stripe and pure flux states. atures, χ approaches a finite value – signalling a chiral
4
phase. These results clearly demonstrate the chiral sym- the thermal fluctuations drives the system from a topo-
metry breaking character of the transition. Quenching logicallytrivialparamagnettoatopologicallynon-trivial
state marked by finite chirality at low temperature.
FIG. 4. (Color online) Real space configurations of localized spins obtained from MC calculations on 8×8 lattice (a) when
T =0.03,afterthephasetransitionand(b)whenT =0.06,priortophasetransition. Thedataisobtainedforne (cid:39)1/4when
t(cid:48) =0.25, J =0.1, J(cid:48) =0.025, hz =0 and JK =8.0.
What is the nature of the ordered state? Insight into
(a) (d) 0.5
the nature of the magnetic ordering can be gained from 0.4 L = 8
a plot of the static spin structure factor, S(q) as a func- 0.35 L = 12 0.4
L = 16
tionofthemomentum[seeFig.3(a)]. Itconsistsofsharp ms0.3 ms
m/0.25 0.3m/
peaks at q = (π,0) and (0,π) and a sub-dominant peak
0.2
at q = (0,0). The dual peaks at (π,0) and (0,π) is a 0.2
0.15
signature of flux state in similar models. The in-plane
0.1 0.1
components of the local moments are arranged in a flux
pattern. The central peak indicates a canting of the mo- 1.5 n=1/4 n=3/4 1.5
e e
ments out of the plane. In other words, the magnetic
ground state is a canted flux state. This is confirmed χ 1 1χ
by snapshots of the local spin configurations above and
0.5 0.5
below the transition temperature [Fig. 4]. At higher
(b) (e)
temperatures, the local moments are in a paramagnetic
0 0
state. In the ordered state, the local moments are ori- (c)
S(π,0)
ented with their transverse components forming a flux 0.8 S(π,0) S(π,π) 0.8
patternaroundthesquareswithnodiagonal-withalter- N 0.6 S(π,π) S(0,0) 0.6N
nnaeitgehcbloorciknwgipselaaqnudetctoesu.nTtehr-ecrleociskwainseetcliorcnugliatutidoinnaalrocoumnd- |S(q)|/ 0.4 SSS(((ππ0,,,00π))) 0.4|S(q)|/
ponent as the double exchange interaction favors ferro- 0.2 S(0,0) (f) 0.2
magnetic ordering at this filling. The real space config-
0 0
urations of localized spins also established the fact that 0 1 2 3 1 2 3
the magnetic ground state is a canted flux phase. This t′/t t′/t
canted flux state is stabilized due the competition be-
tween AFM super-exchange interaction and the double FIG. 5. (Color online) Frustration dependence of magneti-
exchangeinteraction(thatfavorstheFMorderingatthis zation per site, chirality and static structure factor at q =
filling factor) in the presence of geometrical frustration. (0,0),(π,0)and(π,π)atne (cid:39)1/4and3/4. Thecalculations
aredonekeepingthevaluesofparametersJ =0.1,J(cid:48) =0.025,
Nextwemovetone (cid:39)3/4case,wehaveillustratedthe hz =0, JK =8.0 and T =0.02.
results obtained from MC calculations for magnetization
persiteandchiralitywhilevaryingT inFig.2(c)and(d)
respectively. Achiralphasetransitionsimilartothecase
for n (cid:39) 1/4 is also observed here. However, finite size ilar to the previous case. This similarity between the
e
effectsaremuchmorepronouncedforn (cid:39)3/4asseenin resultsforbothcasescanbeascribedtothesymmetryof
e
Fig.2(c)and(d). Theapproachtothermodynamiclimit band structure at one-quarter and three-quarter filling
occurs much more slowly with system size. The ground of itinerant electrons. In Fig. 3 (b) the electronic en-
state at the lower temperature is canted flux state sim- ergy bands along symmetric points in BZ are shown for
5
a tight-binding model with a double exchange term and beingunfrustratedatt(cid:48) =0tohighlyfrustratedatt(cid:48) (cid:38)t,
a triple-Q ordering of the local moments. The three val- whereas the degree of frustration in the direct exchange
ues of q used are that of canted flux state – (0,π), (π,0) between the local moments remain unaltered. We start
and (0,0). It is clear that the bands are symmetric for withn (cid:39)1/4–Fig.5(a)and(b)showtheresultsofMC
e
one-quarter and three-quarter filling of electron density. calculations for magnetization and chirality while vary-
ing the hopping integral on the diagonal bonds t(cid:48). At
t(cid:48)/t=0.0thegroundstatehasanon-zeromagnetization
B. Role of Frustration and finite chirality. The static spin structure factor for
this state exhibits a small peak at q = (0,0) in addi-
Nowweexploretheroleoffrustrationonthemagnetic tiontotwosharpandequalinmagnitudepeaksat(0,π)
properties of the current model. Since t(cid:48) and J(cid:48) are de- and (π,0). Actually this is the same canted flux state
termined by the overlap of different orbitals across the that we encountered earlier. The in-plane components
diagonal bonds, their ratio can, in principle, be differ- of the local moments are arranged in a pattern similar
ent for different materials. In this study, we choose to to a coplanar Flux state. Furthermore, the moments are
vary t(cid:48), keeping J(cid:48) constant. Both t(cid:48) and J(cid:48) induce frus- tiltedslightlyout-of-planeduetogeometricalfrustration
tration – the former in electron hopping and the latter (inHeisenberginteraction)andcompetitionbetweenFM
in exchange interactions between local moments. With double exchange interaction and AF Heisenberg interac-
varying t(cid:48), the nature of electron hopping changes from tion.
FIG.6. (Coloronline)TheequilibriumrealspacespinconfigurationsobtainedfromMCcalculationson8×8lattice(a)when
t(cid:48)/t = 0.2, before the 1st phase transition and (b) when t(cid:48)/t = 0.8, after 1st phase transition (c) when t(cid:48)/t = 1.0, just before
2nd phase transition and (d) when t(cid:48)/t = 2.2, after the 2nd phase transition. The data is obtained for ne (cid:39) 3/4 case when
J =0.1, J(cid:48) =0.025, hz =0.0 and JK =8.0 at T =0.02. See text for more details about these two phase transitions.
The flux pattern consists of columnar AFM arrange- of the spins. m/m jumps to ≈ 0.4 and remains con-
s
ments of the in-plane components with simultaneous stant for 0.75 (cid:46) t(cid:48)/t (cid:38) 1.0. In this intermediate range
(π,0)and(0,π)ordering. Withincreasingt(cid:48)/t, themag- of t(cid:48)/t, the flux ordering is broken and the transverse
netization remains constant at a value consistent with a components of the local moments are fully disordered.
canted flux state up to t(cid:48)/t ≈ 0.75, where there is a dis- The strength of the static structure factor S(q) is van-
continuous transition to a state with partial polarization ishingly small for the AFM ordering wave vectors (π,0),
6
(0,π) and (π,π) indicating the complete suppression of C. Role of Magnetic Field
AFM ordering. This breaking of in-plane flux ordering
results in an increase in the partial ordering of the lon-
Finally, we investigate the role of external magnetic
gitudinal component, reflected in the increased magne-
fieldinthecurrentmodelbothatone-quarterandthree-
tization. The chirality drops discontinuously to a very
quarterfillings ofelectrons. Applying anexternalfield is
small value as a result of the breaking of the flux pat-
the simplest and mostdirect way ofcontrolling the mag-
tern. Upon further increasing t(cid:48)/t there is a second dis-
netic character of a system. Our goal is to explore the
continuous spin-flop transition to a canted AFM state
tunability of the chiral spin states by applying a static,
with small magnetization. The in-plane components of
uniform, longitudinal external field. For n (cid:39) 1/4 the
e
the local moments are arranged in a staggered (π,π) or-
magnetic field dependence of magnetization and chiral-
der with a small out-of-plane component resulting from ityisshowninFig.7(a)and(b)respectively. Athz =0.0
the strong double-exchange interaction. The chirality is
the magnetic ground state is canted flux state which we
suppressed completely indicating a pure AFM character
have already discussed in detail. The uniform magne-
ofthegroundstatespinconfiguration. Thenatureofthe
tization is non-zero at zero field even though the direct
magnetic ground states in the different regimes is, once
exchangebetweenthelocalmomentsareAFMinnature.
again, confirmed by the snapshots of instantaneous spin
This is due to the strong coupling between the local mo-
configurations shown in Fig. 6 [shown here for n = 3/4
e ments and the itinerant electrons. The magnetization
case]. The in-plane components are seen to evolve from
m/m increasesmonotonicallyuptoh =0.2whenthere
s z
a flux pattern at small t(cid:48)/t to completely disordered in
isanabruptjumpinitsvaluefrom≈0.45to≈0.8,mark-
theintermediateregimetostaggeredAFMorderatlarge
ing a field driven discontinuous transition. The chirality,
values of t(cid:48)/t.
which is finite in the canted flux state (h <0.2), is bro-
z
The evolution of the magnetic ground state with kenatthetransition,establishingthechiralnatureofthe
changing t(cid:48)/t is qualitatively the same for n = 3/4, as field induced transition. Indeed the transition marks the
e
seen in Fig. 5 (d) and (e). The range of the intermediate breaking of the flux pattern of the in-plane components
state is much larger. We plan to investigate the nature of the local moments. This is confirmed by the behavior
of this intriguing state further in future studies. of the static structure factor at (0,0) and (π,0). At the
transition, the peak at (π,0) is completely suppressed,
whereas the (0,0) peak (proportional to the square of
1 1
(a) (d) the uniform magnetization) exhibits a discontinuous in-
crease in its value. With further increase in magnetic
0.8 0.8
field, the system approaches full polarization asymptot-
ms ms
m/ 0.6 0.6m/ ically. Once again the field dependence of the magnetic
ground state is qualitatively similar for n = 3/4, con-
0.4 LL == 812 0.4 firming the particle-hole symmetry of the seystem.
L = 16
0.2 0.2
2 ne=1/4 ne=3/4 2 V. SUMMARY
1.5 1.5
χ χ
1 1 To summarize we have studied in detail the magnetic
properties of the SS-KLM at filling factors of itinerant
0.5 0.5
(b) (e) electrons ne =1/4 and 3/4 at which the ground state is
0 0 metallic. We find that a chiral canted flux state is sta-
(c)
bilisedoveralargerangeofparametersatbothdensities.
0.8 0.8
Interestingly, in contrast to insulating ground states, a
q)|/N 0.6 SS((0π,,00)) 0.6q)|/N Dbizliyzainlogshnionns-kcioi-pMlaonraiyrasipnitnertaecxttiuorne.isTnohteecshseirnatliamlafogrnsettaic-
S( 0.4 S(π,0) 0.4S(
| S(0,0) | ground state can be suppressed via a discontinuous chi-
0.2 S(π,0) (f) 0.2 ral phase transition by external fields such as temper-
S(0,0)
0 0 ature, by a static, uniform longitudinal magnetic field,
0 0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4
as well as by increasing the magnitude of frustration in
h h
z z electroniccomponentoftheHamiltonian. Infact,tuning
the strength of diagonal hopping (which is responsible
FIG.7. (Coloronline)Themagneticfielddependenceofmag- for frustration) drives the magnetic ground state from
netization per site, chirality and static structure factor at
a canted flux phase at small frustration to a partially
q = (0,0),(π,0) and (π,π) at ne (cid:39) 1/4 and 3/4. The cal- polarized intermediate state at moderate frustration to
culationsaredonekeepingthevaluesofparameterst(cid:48) =0.25, a canted AFM state at large frustration. Interestingly,
J =0.1, J(cid:48) =0.025, JK =8.0 and T =0.02. while both the canted flux and AFM states have AFM
ordering in the transverse plane, the intermediate state
7
is characterized by a completely random distribution of ACKNOWLEDGMENTS
the in-plane components. We plan to investigate this in-
triguing state in the near future. The work was partially supported by Grant No.
MOE2014-T2-2-112fromtheMinistryofEducation,Sin-
gapore. We also acknowledge the use of HPC cluster at
NSSC, Singapore.
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