Hidden Multiple-spin Interactions as an Origin of Spin Scalar Chiral Order in Frustrated Kondo Lattice Models

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by  Yutaka Akagi

Hidden Multiple-spin Interactions as an Origin of Spin Scalar Chiral Order in
Frustrated Kondo Lattice Models
Yutaka Akagi1, Masafumi Udagawa1,2, and Yukitoshi Motome1
1Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
2Max-Planck-Institut fu¨r Physik komplexer Systeme, 01187 Dresden, Germany
(Dated: January 17, 2012)
We reveal the significance of kinetic-driven multiple-spin interactions hidden in geometrically-
frustrated Kondo lattice models. Carefully examining the perturbation in terms of thespin-charge
couplinguptothefourthorder,wefindthatapositivebiquadraticinteractioniscriticallyenhanced
2 andplaysacrucialroleonstabilizingaspinscalarchiralordernear1/4fillinginatriangularlattice
1 case. This is a generalized Kohn anomaly, appearing only when the second-order perturbation is
0 inefficient because of the degeneracy under frustration. The mechanism is potentially common to
2 frustrated spin-chargecoupled systems, leading to emergence of unusualmagnetic orders.

n
a
J Localized spins in metal interact with each other via eralpointsontheFermisurface,whichdevelopscontinu-

4 effective interactions mediated by the kinetic motion of ously to a full gapasincreasingthe spin-chargecoupling
1 itinerantelectrons. Anunderlyingmechanismistheoret- and approaching 1/4 filling. This naturally explains the
icallyestablishedastheRuderman-Kittel-Kasuya-Yosida robustness of the chiral ordered phase.
]
l (RKKY)interaction[1–3]. The preciseformofthe inter- We consider the Kondo lattice model on a triangular
e
action sensitively depends on the electronic structure of lattice, following the previous studies [6, 7, 9, 10]. The
-
r the system as well as the coupling between charge and essentialresultiscommonto otherfrustratedlattices,as
t
s spin. This offers the possibility of various magnetic or- mentioned later. The Hamiltonian is given by
.
t derings in the spin-charge coupled systems.
a
m Recently, unusual magnetic orderings have been of in- H=−t (c†i,αcj,α+h.c.)−JH c†i,ασαβci,β·Si, (1)
tensiveresearchinterestingeometrically-frustratedspin- hiX,ji,α iX,α,β
-
d charge coupled systems. In particular, a spin scalar chi-
n ral ordering has attracted considerable attention, since where c†i,α (ci,α) is a creation (annihilation) operator of
o it underlies numerous fascinating phenomena, such as conduction electron at site i with spin α, σ is the Pauli
c
[ the unconventional anomalous Hall effect [4, 5]. As a matrix, and Si is a localized moment. The sum in the
prominent example, a scalar chiral ordering with a four- first term is taken over the nearest-neighbor sites on the

1 sublattice noncoplanar spin configuration was discussed triangular lattice. The sign of the spin-charge coupling
v
3 for the Kondolattice model on a triangular lattice [6–9]. JH does not matter, since we assume Si to be a classical
5 Theparticularorderappearsintworegions,near3/4fill- spin.
0 ingand1/4filling[7]. Whiletheformerwaspredictedon Given a specific spin configuration {Si}, it is possi-
3 the basis of the perfect nesting of the Fermi surface [6],
.
1 the latter is unexpected from the nesting scenario. Sur-
0 prisingly,the1/4fillingphaseisstableuptomuchlarger
2 spin-chargecoupling than the 3/4filling one [7]. The ro-
1
bustnessofthe noncoplanarordernear1/4fillingcastsa
:
v fundamental question on the mechanism of chiral order-
i ing.
X
In this Letter, we point out the significance of kinetic-
r
a driven multiple-spin interactions for the noncoplanar or-
dering under geometrical frustration. Carefully exam-
ining the perturbation in terms of the spin-charge cou-
pling up to the fourth order, we find that the second-
orderRKKYcontributionsaredegenerateamongseveral FIG. 1. (Color online) Ordering patterns of localized spins
considered in the perturbation calculations: (1a) ferro-
different magnetic orderings because of the frustration,
magnetic, (2a) two-sublattice collinear stripe, (3a) three-
whereas the degeneracy is lifted by fourth-order contri- sublattice 120◦ noncollinear, (4a) four-sublattice all-out, and
butions. Among several terms, a positive biquadratic (4f) four-sublattice coplanar90◦ fluxorders. Theright panel
interactionis critically enhanced and plays a crucialrole is the phase diagram of Hamiltonian (1) near 1/4 filling.
in the emergence of the noncoplanar order. We ascribe Arrows (i) and (ii) indicate the parameter changes used in
the mechanism of the critical enhancement to a Fermi Fig. 4(d). PS represents a phase-separated region and the
surfaceeffect —a generalizedKohnanomalyata higher bolddottedlineat1/4fillingdenotesaninsulating(4a)state.
Seealso Figs. 1 and 3 in Ref. [7].
level of perturbation. It opens a local energy gapat sev-
2
ble to obtain the exact free energy F = −T log(1+ (4a), and (4f) states have the same energy within the
a
exp(−βEa)) from the eigenvalues of H, {Ea}P(T is tem- second-order perturbation.
perature and β = 1/T). In the previous study, from The fourth-order term in eq. (2) is calculated in a
the comparison of the ground state energies, two of the similar manner. For example, the result for the four-
authors discovered a scalar chiral phase with a four- sublattice orders is summarized into a form of effective
sublattice noncoplanar spin order [Fig. 1(4a)] near 1/4 multiple-spin interactions as
filling [7]. As shown in the right panel of Fig. 1, the chi-
ral order sets in by an infinitesimal JH from slightly less F(4) = T JH 4
than 1/4 filling, and extended to a large J insulating
H 2 (cid:18) 4 (cid:19)
region at 1/4 filling [7]. Xωp Xk j1,j2,jX3,j4=1∼4Ql,QXm,Qn
Inordertoelucidatetheoriginoftherobustnoncopla- eiδj1·(Ql+Qm+Qn)e−iδj2·Qle−iδj3·Qme−iδj4·Qn×
nar ordering, we consider the perturbation in terms of G0(ω )G0 (ω )G0 (ω )G0 (ω )
J . The analysisprovidescoherentunderstandingofthe k p k+Qn p k+Qm+Qn p k+Ql+Qm+Qn p
H
×[(S ·S )(S ·S )+(S ×S )·(S ×S )]
phasediagramnotonlyinthesmallJH regionbutinthe j1 j2 j4 j3 j1 j2 j4 j3
large JH insulating region, as we will discuss later. Tak- =JH4 JSi·Sj +B(Si·Sj)2+B∗(Si×Sj)2
ingthesecondtermineq.(1)astheperturbation(H′)to 1≤Xi<j≤4(cid:2) (cid:3)
thefirstterm(H ),weusethestandardperturbationthe-
0 + M(S ·S )(S ·S )+M∗(S ×S )·(S ×S )
oryto calculatethe energyforvariousorderingpatterns. i j i k i j i k
Wehereconsideracollectionoforderedstatesuptofour- i6=Xj6=k(cid:2) (cid:3)
sublatticeordering,whichappearedinthephasediagram + L(S ·S )(S ·S )+L∗(S ×S )·(S ×S ) .
i j k l i j k l
in the small JH region; namely, (1a) ferromagnetic, (2a) i6=Xj6=k6=l(cid:2) (cid:3)
two-sublatticestripe,(3a)three-sublattice120◦coplanar, (4)
(4a) four-sublattice all-out, and (4f) four-sublattice flux
orders, as shown in Fig. 1. The free energy is expanded Here the sum of Q runs over (0,0), (π,0)(= Q ),
l,m,n a
in the form (0,π)(= Q ), and (π,π)(= Q ) (Q , Q , and Q are
b c a b c
four-sublatticeorderingwavevectors[6]). Theindicesi,j
β
F −F =−Tlog T exp − H′(τ)dτ denote the four sublattices, and δi =(0,0), (1,0), (0,1),
0
D (cid:16) Z0 (cid:17)Econ and (1,1), respectively [see inset of Fig. 2(a)]. G0k(ωp)=
=−T βdτ βdτ TH′(τ )H′(τ ) {iωp−(ǫk−µ)}−1isthenoninteractingGreen’sfunction,
2!Z 1Z 2 1 2 con whereωp istheMatsubarafrequencyandµisachemical
0 0 (cid:10) (cid:11) potential. Thus, the fourth-order contribution includes
T β β
− dτ ··· dτ TH′(τ )···H′(τ ) +··· three-spinandfour-spineffectiveinteractionsinaddition
4!Z 1 Z 4 1 4 con
0 0 (cid:10) (cid:11) to two-spin interactions. Each coefficient is given by the
=F(2)+F(4)+··· , (2) sum of four Green’s functions products.
First let us discuss the result of second-order pertur-
where F0 is the free energy obtained from H0, τ is the bation. The second-order term is proportional to χ0, as
imaginary time, and T represents time-ordered product. shown in eq. (3). The comparison of χ as a function
0
h···icon meansthe averageoverthe connecteddiagrams. of the electron density n is presented in Fig. 2(a). Since
From the spin rotational symmetry, odd order terms do the magnetic order with largestχ is stabilized when J
0 H
not appear in the expansion. is turned on, the result indicates that the ferromagnetic
By decoupling the brackets h···icon with the Wick’s order appears for 0.0≤n.0.15 and 0.8.n≤1.0, and
theorem, each term in eq. (2) can be written in a the three-sublattice 120◦ coplanar order is realized for
compact form. The second-order term gives the well- 0.3 . n . 0.55. See also Fig. 3 in Ref. [7]. Meanwhile,
known RKKY form [1–3]; F(2) = −JH2 q|Sq|2χ0(q), χ0(π,π) is the largestin the remaining regions, in which
where the bare magnetic susceptibilityPis defined as we have to consider the next fourth-order terms to lift
χ0(q) = −N1 k{fF(ǫk−q)−fF(ǫk)}/(ǫk−q −ǫk), and the degeneracy between (2a), (4a), and (4f) states [11].
Sq = N1 jSPjeiq·rj (fF is the Fermi distribution func- The fourth-order free energies in the remaining re-
tion, ǫk iPs an eigenvalue of H0 in the momentum basis, gions for (2a), (4a), and (4f) are indicated in Figs. 2(b)
andN isthenumberofsites). Specifically,foreachmag-
and 2(c). The results show that (2a) two-sublattice
netic order, F(2) can be rewritten as
stripe phase has the lowest fourth-order contribution in
a wide range of n, while (4a) four-sublattice all-out or-
χ (0,0) for (1a)
0
F(2) =−J2 ×χ (2π/3,2π/3) for (3a) (3) der becomes lowest in narrow windows at n≃0.225 and
H 0
χ (π,π) for (2a)(4a)(4f). n ≃ 0.75. The stability of (4a) state at n ≃ 0.75 is ex-
0
plained by the perfect nesting of the Fermi surface at

Here,wetakethetriangularlatticeintheformofasquare 3/4 filling [6]. On the other hand, the reason for the
lattice,asshownininsetofFig.2(a). Notethatthe(2a), stabilization at n≃0.225 is not obvious.
3
(a) the coefficient of each term, the product of the Green’s
0.6
③ ④ ③ ④ χ0(0,0) functions, as a function of n in Fig. 3. The results
0.5 χ(2π/3,2π/3) showthatallthecoefficientsshowdrasticchangesaround
0.4 ① ② ① ② χ00(π,π) n≃0.225: Inparticular,thecoefficientofbiquadraticin-
teraction, B, shows a pronounced peak with the largest
χ0 0.3 δ3③δ4 ④ ③ ④ contribution. Thisindicatesthattheeffectivepositivebi-
① ② ① ② quadraticinteractionplaysadominantroleatn≃0.225.
0.2 δ2
In general, the positive biquadratic interaction favors a
0.1 noncollinear spin configuration. Particularly, among the
degenerate(2a),(4a),and(4f)states,the (4a)noncopla-
0.0
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 narall-outorderisnaturallyselectedbythisinteraction.
n
The emergence of the positive biquadratic interaction
(b) 0.30 (c) 10 is remarkable, because the effective biquadratic interac-
0.25 (2a) (2a) tion has a negative coefficient in most cases. For ex-
8
0.20 (4a) (4a) ample, in the Hubbard-type Hamiltonian, a fourth-order
4(4)/(J/t)FH 0000....00110505 (4f) 4(4)/(J/t)FH 246 (4f) ptcmoaealrialtz/ueqnrdubeagasnatpittiouinvnmesfrybflosiumtqecumttauhsdae,rtasiaottrincsospniiangnl-t-scelooarautgtcpiitvcliieeonngrc.iolsiuemMptilotoinruegnsoeuvagaenalrltd,yiviltneehaebldoris---
0
-0.05 quadratic interactions. Our result indicates that an un-
-0.10 -2 usual positive biquadratic interaction is induced by the
-0.15 -4 kinetic motion of electrons at a special filling.
0.18 0.20 0.22 0.24 0.26 0.28 0.64 0.68 0.72 0.76
n n Why is the positive biquadratic interaction enhanced
here? In order to elucidate the origin, let us look at the
electronicstateatn≃0.225indetail. Figure4(a)shows
FIG. 2. (Color online) (a) χ0(Q) in eq. (3) as a function of theFermisurfaceatn=0.225,whichiscircular,withno
n. The inset represents a square lattice with diagonal bonds substantial nesting tendency. At first sight, this feature-
whichistopologicallyequivalenttothetriangularlattice. (b), lessFermisurfacedoesnotleadtoanytypeofsingularity.
(c)Thefourth-orderfreeenergiesof(2a),(4a),and(4f)states It, however, conceals an unexpected instability toward
in the region near (b) n ∼ 1/4 and (c) n ∼ 3/4. In the
four-sublattice ordering. The key observationis that the
calculation, we take 800×800 grid points in the extended
ordering vectors of four sublattice order, Q = (π,0),
first Brillouin zoneand T =0.03t. a
Q = (0,π), and Q = (π,π), connect the particular
b c
points on the Fermi surface when n reaches ≃0.225;the
interaction/(JH/t)4 tangents of the Fermi surface are parallel between the
0.025
connected pairs — see Fig. 4(a). These connections give
J
0.020 B rise to a singularity analogous to the 2kF-singularity or
B* Kohnanomaly,observedinanisotropicelectrongas[12].
0.015
M Itisworthytonotethat,incontrasttotheKohnanomaly
0.010 M* usually discussed in the second order perturbation [12],
L
the singularity considered here arises in the fourth-order
L*
0.005
perturbation [13].
0.000 The scenario is confirmed by numerically calculating
the contributions only from the parts of Fermi surface
-0.005 connected by Q , Q , and Q . We calculate the coef-
a b c
ficients of the effective four-spin interactions in eq. (4)
-0.010
0.18 0.20 0.22n 0.24 0.26 0.28 by limiting the k summation within the small hatched
patches shown in Fig. 4(a). As shown in Fig. 4(b), the
contributions from the small patches with δ =π/40 lead
FIG. 3. (Color online) Coefficients of the different terms in toapositiveandlargestbiquadraticinteractionB,which
the fourth-order effective Hamiltonian [eq. (4)] as a function wellaccountsforthe resultinFig.3. This indicates that
of the electron density n. the anomalous behaviors in Fig. 3 can be attributed to
the particle-hole process within such small area in the
momentum space.
In order to gain an insight into how the (4a) state is The critical enhancement is, in fact, a divergence
stabilized at n ≃ 0.225, we carefully analyze the fourth- in the limit of T → 0 in the perturbative calcula-
order contributions in eq. (4) term by term. We plot tions. The T dependence is analytically calculated by
4
on,the Fermisurfaceismodifiedattheconnectedpoints
reflecting the local gap opening. The gap extends to
shrink the Fermi surface as J increases and n becomes
H
closer to 1/4 filling; eventually, the Fermi surface dis-
appears in the insulating state at 1/4 filling. This con-
tinuouschangeclearlydemonstratesthatthegeneralized
Kohn anomaly revealed by the perturbative argument
provides an essential ingredient for the robust chiral or-
dering including the insulating region.
Let us emphasize the importance of the geometrical
frustration in this generalized Kohn anomaly. Frustra-
tionleavesdegeneracyatthelevelofsecond-orderpertur-
bation, and prepares the stage for the fourth-order term
to select the type of magnetic order. In this sense, ge-
ometrical frustration is an indispensable prerequisite for
thepositivebiquadraticinteractiontoplayanactiverole.
In fact, when the frustrationis reduced by introducing a
spatialanisotropyinthetriangularlatticestructure,non-
coplanarorderisnotstabilizedwithinthefour-sublattice
FIG. 4. (Color online) (a) The Fermi surface at n = 0.225 order, even though the positive biquadratic interaction
in the extended Brillouin zone scheme. The connections is present in the fourth-order terms: the effect of posi-
of the Fermi surfaces by the four-sublattice wave vectors tive biqadratic interaction is completely masked by the
Qa =(π,0),Qb =(0,π),Qc =(π,π)areshown. Sixhatched second-order RKKY interaction. Meanwhile, for other
squaresinthefirstBrillouinzoneindicatethepatchesusedin
frustrated lattices, such as checkerboard, face-centered-
thecalculationsofthedatain(b). Thepatchesareδ×δeach,
cubic,andpyrochlorelattices,wefoundthatsimilarfour-
and centered at the special k points, (±π/2,0), (0,±π/2),
and (±π/2,∓π/2), which are connected with each other by sublatticeorderiscommonlystabilizednear1/4fillingby
thefour-sublatticewavevectors. (b)Coefficientsof theeffec- the positive biqiuadratic interaction [14]. This suggests
tive four-spin interactions in eq. (4) obtained by taking the thatthepresentmechanismisrobustwidelyinfrustrated
k summation only within the patches in (a) with δ = π/40. spin-charge coupled systems.
(c) The peak valuesof B˜ =TPkPωp[G0k(ωp)]2[G0k+Q(ωp)]2 To summarize, we have carried out the perturbation
with patch δ = π/8 as a function of T. The peak value of
expansionuptothefourthorderinJ /tandconstructed
B calculated bytheintegration overtheentire first Brillouin H
zoneisalsoplotted. TwostraightlinesarethefittingtoT−1.5. the effective multiple-spin interactions for the ferromag-
(d) The Fermi surface evolution in the (4a) phase along the neticKondolatticemodelonatriangularlattice. Among
arrows in the right panel of Fig. 1; (i) from JH = 0 to 0.75 the kinetic-driven four-spin interactions, we have found
(every 0.25) at a fixed n = 0.2255 and (ii) from n = 0.231 thatabiquadraticinteractiontakesalargepositivecoeffi-
to 0.249 (every 0.006) at a fixed JH = 0.75. Only the first cientandincreasesdivergentlyas T →0 near1/4filling.
quadrant of thefolded Brillouin zone is shown.
This signals an instability toward a noncoplanar chiral
phase near the 1/4 filling with a local gap formation.
The originof largepositive biquadratic interactionis as-
approximatingtheconnectedfragmentsoftheFermisur- cribed to the Fermi surface connection by the ordering
faces by simple biquadratic functions centered at the wave vectors of four sublattice order, which we call the
connected k points. The dominant contribution to generalized Kohn anomaly. Our result provides compre-
the divergence comes from the terms proportional to hensiveunderstandingofnotonlytheemergencebutalso
T k ωp[G0k(ωp)]2[G0k+Q(ωp)]2 (≡ B˜) in eq. (4); it the robustness of the chiral ordering. The mechanism is
leaPds Pto quick divergence ∝ T−1.5, reflecting the fact universalinfrustratedspin-chargecoupledsystems,pos-
that the tangents of the Fermi surface are parallel to sibly even when electron correlationbetween conduction
each other at the connected points. The details of the electronsisincludedunlesstheFermisurfaceisdestroyed
calculaions will be reported elsewhere. Indeed, the nu- by electron correlation.
merically calculated values of B˜ as well as B increase
We acknowledgehelpfuldiscussionswithTakahiroMi-
with ∝T−1.5 as T →0, as shown in Fig. 4(c).
sawa and Youhei Yamaji. This work was supported by
The divergence leads to a local gap formation at the Grants-in-Aid for Scientific Research (No. 19052008,
connectedpointsontheFermisurface. Figure4(d)shows 21340090, and 21740242), Global COE Program “the
the Fermi surface evolution in the (4a) phase, as we Physical Sciences Frontier”, the Strategic Programs for
change the parameters (n,J ) along the arrows (i) and Innovative Research (SPIRE), MEXT, and the Compu-
H
(ii)ontherightpanelofFig.1. AssoonasJ isswitched tational Materials Science Initiative (CMSI), Japan.
H
5
[10] Y. Akagi and Y. Motome, J. Phys.: Conf. Ser. 320,
012059 (2011).
[11] Atageneralfilling,weneedtoconsideralargermagnetic
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[2] T. Kasuya, Prog. Theor. Phys.16, 45 (1956). wave vector. At n ≃ 0.225, however, χ0(q) has maxima
[3] K.Yosida, Phys.Rev.106, 893 (1957). at the four-sublattice ordering wave vectors, q = Qa,
[4] K. Ohgushi, S. Murakami, and N. Nagaosa, Phys. Rev. Qb, and Qc. Therefore, our argument on the basis of
B 62, R6065 (2000). four-sublattice unit cell is plausible.
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(2008). anomaly: As shown in Fig. 2(a), a shoulder-like singu-
[7] Y.Akagiand Y.Motome, J. Phys.Soc.Jpn.79, 083711 larity of χ0(π,π) at n ≃ 0.225, analogous to the polar-
(2010). ization functionofatwo-dimensionalelectron gas—the
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