chiral order in frustrated magnetsold.apctp.org/conferences/2008/multiferroics/img/kawamura.pdf ·...
TRANSCRIPT
Chiral Order in Frustrated Magnets
Hikaru Kawamura
Osaka University
APCTP workshop on MultiferroicsDec.11, 2008 POSTECH
1. Frustration and chirality2. Spin-chirality decoupling (I)
Regularly frustrated XY magnets in low dimensions
3. Chirality-induced novel critical behaviorStacked-triangular antiferromagnets in 3D
4. Spin-chirality decoupling (II)spin glass ordering and the chirality mechanism
5. Spin-chirality decoupling (III)chiral glass order in granular high-Tc superconductors
Outline
What is frustration?Various elements compete
with each otherorder destabilized
novel fluctuations enhanced
Frustration inducednovel properties
?
Origin of frustration researchー frustrated magnetism
Spin remains disorderedResidual entropy appears (violate the 3rd law?)
2D antiferrom
agneticIsing m
odel
(Wannier 1951)(Shoji 1955)
triangular latticekagome lattice
New phenomena!
Magnetic ordering supressed
Frustration
Weak interactiondominates
The 3rd law
New paradigmNew concept
Spins cant with each other, partly relieving frustration
Example of geometrical frustration (B)
Three vector spins coupled antiferromagnetically
Chirality (I) --- vector chirality
chirality: κ = Σ Si × Sj(spin current) = Σ sin(θi - θj)
chirality ーchirality +
XY spin
right-handed left-handed
Z2 × SO(2)
chiralityspin
rotation
multiferro
Spin super-structure and chirality induced by frustration
[chirality +] [chirality -]
「right」 「left」
frustration induced superstructureemergence of the chirality
Electric polarization!
spiral(helical)magnet
Electricpolarization
chirality induced ferroelectricity
spin frustration
spin spiralemergence of chirality
ferroelectricity
RMnO3 , RMn2O5 etc
Chirality controle by electric field
Polarization controle by magnetic field
[chirality +] [chirality -]
「right」 「left」
Spin current mechanism of ferroelectricity[H. Katsura, N. Nagaosa and A.V. Balatsky, ’05]
Electric polarization is generated by the spinCurrent (vector chirality)
Inverse Dzyaloshinskii-Moriya interaction
HDM=Dij・Si×SjDzyaloshinskii-Moriya interaction
vectorchirality
Chirality controle by electric fields
Couter-clockwise spiralClockwise spiral
temperature
electric polarization
chiralitychirality
polarization
Electric field
Electric field
chirality chirality
polarization polarizatin
[T.Arima et al ‘07]
χ = Si・Sj×Sk
Heisenberg spin
chirality -
right-handedchirality +
left-handed
Z2 × SO(3)
chiralityspin
rotation
Chirality (II) --- scalar chirality
Unconventional anomalous Hall effect
Aharonov-Bohm effect
phase factor ofconduction electrons:
effective flux Φ= 1/2 ×chirlaity χ
χ = S1 ・ S2×S3Gigantic effective field (104T)due to the spin chirality
Φ = B S
e-
S
anaology
Chirality induced gigantic effective magnetic field and novel transport phenomena
- Berry phase and anomalous Hall effect
Normal Hall effect due to magnetic fields
Chirality induced Hall effect
S1
S2
S3
Φ
Locally, chirality is a composite operator of spins, NOT independent of spins
spin - chirality decouplingBeyond a crossover length scale, L > L*,
chiral correlations outgrow spin correlations, i.e.,
ξ(chiral) >> ξ(spin)
eventually leading to two separate transitions,
Tc (spin) < Tc (chiral)
Fluctuations important !
Spin-chirality decoupling leads to
Chiral spin liquid (chiral phase)chirality LROspin disordered
possibleFerroelectric spin liquid ??
Ferroelectricity of magnetic originbut without a magnetic LRO
Spin-chirality decouplingin certain frustrated vector spin systems
Chiral order in regularly frustrated XY magnetsin low dimensions
* 1D triangular-ladder XY antiferromagnet* 2D triangular-lattice XY antiferromagnet
Chiral order in 3D Heisenberg spin glasses* 3D isotropic Heisenberg SG* chirality scenario of experimental
spin-gass transitionsChiral order in ceramic high-Tc superconductors
Novel States of Matter Induced by Frustration
Monbu-Kagaku-Syo Project
Term of Project: 2007-2011 (5 years)
Head: H. KawamuraGroup leaders: S.Maegawa, H.Tsunetsugu,H.Kageyama, T.Arima, K. Hirota, H. KatoriNumber of Core Researchers : 36
Budget: 1.2 billion Yen (~ 12 million $)
Grant-in-Aid for Scientific Research on Priority Areas
A01 Fundamental Properties of Frustrated Systems
Novel Order in Geometrically Frustrated Magnet
Frustration and Chirality
Quantum Frustration
Frustration and Quantum Transport
Frustration and Multiferroics
Frustration and Relaxor
Geometrical frustration and functions in spin-charge-lattice coupled system
A02 Frustation-induced Novel Phenomena and their Applications
Organization http://www.frustration.jp
HK
T. Arima
2. Spin-chirality decoupling (I)
Chiral order in regularly frustrated XY magnets
in low dimensions
--- 1D triangular-ladder and 2D trianguar-lattice XY antiferromagnet
1D XY AF on the triangular ladderexact solution [Horiguchi et al ’90]
* Both the spin and the chirality order at Tc= 0* Spin and chirality correlation length exponents
are mutually different.νs = 1 < νk = ∞
→ two different diverging length scales at the T= 0 transition, i.e.,
“spin-chirality decoupling” ( rigorous ! )
2D triangular-lattice XY AF
H = J Σij Si・Sj (J > 0) Si = (Six, Siy)
ordered state: 120 structure with chirality
Specific heat of the 2D triangular XY AF (MC)
[D.H. Lee et al ’84] [S. Miyashita & H. Shiba ’84]
divergent specific heat ← accompanied with the chiral order
Successive spin and chirality transitions are likely to be realized, i.e., the spin-chiralitydecoupling with
0 < TXY (spin) < TI (chiral)
T= TI : chiral transitionchirality exhibits a LRO while spin decaysexponentially.
T=T2 : Kosterlitz-Thouless transition of spin (phase)spin exhibits a quasi-LRO with power-lay decay.
T
0 TXY TI
[H.J. Xu et al (’96) (~2%) , L. Capriotti et al (’98) (~2%), S. Lee et al (’98) (~2%), D. Loisson et al (~0.3%), Y.Ozeki and N.Ito (’03) (~1%) ]
TT*TI
τ
t∗
chirality
spin
spin-chirality decoupling
spin-chirality couplingl*
ξ
TXY
Length and time scales at the phase transitions of the 2D triangular-lattce XY antiferromagnet
3. Chirality-induced novel critical behavior
3D stacked-triangular XY and Heisenberg antiferromagnets
3D stacked-triangular-lattice XY AF
Spin and chirality order simultaneously withonly one diverging length scale, i.e.,no spin-chirality decoupling,
whereas,chiral degrees of freedom
leads to the possiblenew chiral universality class
α∼.35 , β~0.26 , γ ~ 1.14 , ν ~ 0.55c.f. [ α∼0, β~1/3 , γ ~ 1.3 , ν ~ 0.7 ]
[H.K. ’85- ] symmetry analysis, RG, MC[review: H.K. J. Phys. Condens. Matter 10 (1998) 4707]
Landau-Ginzburug-Wilson Hamiltonianof chiral magnets for RG analysis
H = (grada)2 + (gradb)2 + r (a2 + b2)+ u ( a2 + b2 )2 + v { (a・b)2 - a2 b2}
a=(a1,a2,a3,…,an), b=(b1,b2,b3,…, bn)n : number of spin componentsv > 0
spin order at a wavevector Q
S(r) = a(r) cosQr + b(r) sinQr
a ⊥ b ⇔ v > 0
Continous transition of new universality
vs.first-order transition
Is the new chiral fixed point stable ?For larger n, yes !For physical cases of n=2 and 3, still under debate.
Higher-order loop expansion yields a stable fixed point accompanied with exotic oscillating RG flows
Specific heat of the stacked-triangular-lattice XY antiferromagnet CsMnBr3
α[exp.] ~ 0.39 (5) ; A+/A- ~ 0.32 (20)α[theory] ~ 0.34 (6) ; A+/A- ~ 0.36 (20)
[R.Deutschmann et al ‘92]
Order parameter of the stacked-triangular-lattice XY antiferrmagnet CsMnBr3
[T. Mason et al ’89]
[Y. Ajiro et al ’88]
β [exp.] = 0.25(1)
β [theory] = 0.25(2)
Critical properties of the chiralityChirality orders simultaneously with the spin,
with a common correlation length→ “one-length scaling”
νspin = νchiral ~ 0.55(3)
Other exponents differ between for the spin and the chirality:
βchiral = 0.45(2) βspin = 0.25(1)φchiral = 1.22(6) φspin = 1.38(8)
How to measure the chirality ?
Measurements of the vector chirality
Polarized neutron scattering
I ( - ) – I ( - ) = ΔI(q)
q : momentum transfer
q - integral of I(q) ∝ “chirality”
[V.P. Plakhty et al ‘00]
φc=1.29(7)βc=0.44(7)
βc=0.42(7)
Chirality of the stacked-triangular-lattice XY antiferromagnet CsMnBr3
In multiferroic materials, chiral critical properties might bemeasurable by the standard polarization measurements !
P ∝ κ = Si × Sj
Use multiferroic properties as a probe of the chirality
4. Spin-chirality decoupling (II)
Chiral order in 3D Heisenberg spin glasses
and chirality scenario of experimental SG ordering
Canonical SG CuMn, AuFe, etc.
Heisenberg system with weak random magnetic anisotropy
Experimentally, a thermodynamic SGtransition and a SG ordered state has been established.
The true nature of the SG transition and the SG order state ?→ still at issue
Numerical results on the 3D EA SG model* 3D Ising SGThe existence of a finite-temperature SG transition established in zero field.
*3D isotropic Heisenberg SGEarlier studies suggested no finite-T transition
[Olive,Young, Sherrington, ’86, F. Matsubara et al ‘91]
The possibility of a finite-T transition in the chiral sector was suggested [H.K., ’92] --- chiral glass state
spin-chirality decoupling TSG < TCG
Chirality scenario of experimental SG transition[H.K. ’92]
Chirality scenario of SG transiton [H.K. ’92~]
* Isotropic Heisenberg SG in 3D exhibits a spin-chirality decoupling, with the chiral-glass ordered phase not accompanying the standard SG order.
* Chirality is a hidden order parameter of real SG transitions. Experimental SG transition is a “disguized” chiral-glass traqnsition: The spin is mixed into the chirality via the random magnetic anisotropy.
χijk = Si・(Sj×Sk)k j
iright
k
j
ileftScalar chirality
Spin-chiralitydecouplingin the 3D isotropicHeisenberg SG
How the spin and chiral correlatins grow ?
TTCG
τ
T*
chirality
spin
spin-chirality decoupling
spin-chirality couplingl*
ξ
TSG
Chiral glassphase
Recent controversy on the 3D Heisenberg SGDue to the progress in the computer ability and simulationteqnique, significant numerical study now becomes possiblefor the 3D Heisenberg SG.
Consensus in recent numerical studies: The 3D Heisenberg SG exhibits a finite-T transition.
However, its nature has still been largely controversial.Spin & chirality are decoupled or not ?
* Yes, decoupling occurs (TSG < TCG)
H.K.’98, K.Hukushima & H.K. ’00 ‘05
* No decoupling (TSG = TCG)
F.Matsubara, T.Shirakura et al,
B.W.Lee & A.P.Young ’03; ’07I.Campos et al ’06 (comment: I.Campbell & H.K. ‘07)
Chirality scenariohas been contested !
Model
3D Edwards-Anderson Heisenberg SG model
H = - Jij Σ ij Si ・ Sj
Si=(Six,Siy,Siz) : classical Heisenberg spin
Jij : Nearest-neighbor random Gaussian coupling with zero mean and variance J2
Simple cubic lattice of size N=L3
Periodic boundary conditions applied
new MC simulation [D.X. Viet and H.K. to appear in PRL]
Spin and chirality correlation length ratio ξ/L
The transition temperature can be estimated from the sizeand temperature dependence of the dimensionless ratio ξ/L
T T T
ξ/Lξ/L ξ/L
Tg
L larger L larger L larger
T>0 transition subtle caseT=0 transition
--- a quantity most intensively studied
Correlation length ratios ξ/L [D.X. Viet and H.K., 2008]
chirality spin
Tcross extrapolated to L=∞
TCG=0.145±0.004TSG=0.120±0.006
Binder ratio; g
g = (1/2) {3-2<q4>/<q2>2}Dimensionless quantity representing the ratiobetween the fourth and the second moments of the overlap.
T TT
Tg
L larger L largerL larger
Standard T>0 transition Special T>0 transitionT=0 transition
gg g
Binder ratio of the 3D Gaussian Heisenberg SG[D.X. Viet and H.K., ‘08]
growing negative dip→ consistent with 1-step RSB
TSG < 0.12Tcross
Tdip
chirality spin
~
chirality spin
Glass order parameters q(2) [D.X. Viet and H.K., 2008]
TCG ~ 0.148±0.005 TSG < 0.13 < TCG~
Critical properties of the chiral-glass transition of the 3D Heisenberg SG
Finite-size scaling plot of the chiral-glass order parameter qCG
(2)
chiral-glass exponents
νCG= 1.4(2)ηCG= 0.6(2)
differ from the 3DIsing values !
νCG~ 2.5ηCG~ - 0.35
0
0.5
1
1.5
2
2.5
3
3.5
4
-0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
q(2)
CG
L1+
η
(T-TCG)L1/ν
TCG=0.145νCG=1.2ηCG=0.8
L=32L=24L=16L=12
L=8L=6
Our MC Results on the 3D isotropic Heisenberg SG
1. The spin and the chirality are decoupled, i.e.,the chiral-glass order occurs at least 15% higherthan the spin-glass order
2. The chiral-glass criticality is different fromthat of the 3D Ising spin glass, characterized bythe exponents;
νCG= 1.4 ± 0.2, ηCG = 0.6 ± 0.2
Chirality hypothsis [H.K. 1992]
Isotropic (ideal) system→ “spin-chirality decoupling”
Real system is weakly anisotropic !Spin is “recoupled ” to the chirality due to the weak random magnetic anisotropy D.
Z2 [chiral] × SO(3) [spin-rotation]
The chiral-glass transition now appears as the SG transition.
“spin-chirality recoupling”
T
DSG
CG SG(CG)phase
para phase
Prediction from the chirality hypothesis for the SG transition of canonical SG
1. SG transition temperature T=Tg is of O(J)、depending on the anisotropy D as
Tg(D) ~ TCG(0) + cD + … .Only one fixed point (chiral-glass fixed point) governs the SG order both for zero and nonzero D.
2. SG critical exponents of canonical SG differ fromthe 3D Ising exponents, and are β~1, γ~2 , δ~3etc. There is no crossover in the standard sense.
Anisotropy (d ) dependence of the transitiontemperature of canonical SG
[A. Fert et al ’88]
AuFePt & AgMnAu
d
Critical properties of canonical SG
CanonicalSG Exp.
3D Isingtheory
3D IsingExp.
Chiralβ~1γ~2ν~1.2η~0.8
Direct test of the chirality scenario
→ needs to measure the chirality directly
Use an anomalous Hall effect as a probe of chiral order !
Measurements of linear and nonlinear chiral susceptibilities, Xχ & Xχnl, becomes possible viameasurements of Hall coefficient Rs.
[G. Tatara & H.K. ’02, H.K. ’03]
Rs = ρxy /M = -Aρ –Bρ2 – CD [Xχ + Xχnl (DM)2 + ...]
* Approach from the strong coupling[Ye et al, ‘99; Ogushi et al, ’00]
* Approacj from the weak coupling[G. Tatara and H.K., '02]
Perturbation calculation of the Hall conductivity based on the s-d model and the linear response
→ The third-order term with respect to the exchange interaction J gives a finite contribution to ρxy which is proportional to the uniform chirality of the system χ0 .
○ To get finiteρxy(chiral), one nees a finite total chirality χ0>0
○ In SG, however, χ0 is canceled out to be zero in the bulk so that no netρxy(chiral) is expected to be realized ?
○ In fact, if the system possesses s a uniform net magnetization M , a uniform component of the total chirality is induced due to the spin-orbit interaction λ.
[Ye et al, ‘99]From weak-coupling approach, one has
<Hso> = D’Mχ0 = -Hχχ0
D’ = Dλ(J/εF)2(Jτ)2
which breaks the chiral symmetry and serves as a conjugate field to the chirality.
Hχ= - D’M (chiral field)
Anomalous Hall effect as a chiral order in spin glasses [H.K., PRL (2003)]
Linear and nonlinear chiral susceptibilities
Xχ = (dχ0/ dHχ)Hχ=0 ,
Xχnl = (1/6) (d3χ0/ dHχ3 )Hχ=0
χ0 = - Xχ (DM) - Xχnl (DM)3 + ...
ρxy[chiral] = Cχ0= -CDM [Xχ + Xχnl (DM)2 + ... ]
Anomalous Hal coeffeicient Rs
Rs = ρxy /M = -Aρ –Bρ2 –CD [Xχ + Xχnl (DM)2 + ...]
Prediction [H.K. ’02]
For canonical spin glasses,(i) Linear part of Rs exhibits a cust at T=Tg .
(ii) Nonlinear part of Rs exhibits a divergence with an exponent ~2 atT=Tg .
(iii) Chiral part of Rs exhibits a following scaling form
Rs[chiral] ~ |t|βχ G(M2/|t|βχ+γχ)βχ~1 , γχ~2
Tg
T
FC
ZFC
Rs
small Mlarge M
FC
Tg
Anomalous Hall resistivity ρxy and Hall coefficient Rs of canonical SG; AuMn
[T. Taniguchi et al ’04]
Kageyama et al (‘03) FeAl (RSG) T.Taniguchi et al (’04) AuFeP.Pureur et al (’04) AuMn; T. Taniguchi et al (’06) AuMn
Critical property of the chirality
Chiral susceptibility from Rs
[T. Taniguchi et al ’06 ]
δχ=3.3
δχ=1.7
SG ordered state might possess a complex phase-space structure
RSB (replica-symmetry breaking)
Possible multi-valley structure in SG
Replica symmetry breaking (RSB)
Ergodicity is broken in the phase space into many “pure components” unrelated to globalsymmetry of the Hamiltonian
[ex.] mean-field SG model (SK model)
Overlap distribution function: P(q)
overlap: qq=(1/N) Σi Si
(a)Si(b) (a,b replica index )
P(q’) = [< δ(q’-q) >]
P(q) is an indicator of RSB.
Overlap distribution function P(q)
(a) No RSB (b) full-step RSB (e.g., SK model)(c) 1-step RSB (d) Combination of (b) & (c)
Droplet pictureFisher & Huse
Hierarchical RSBParisi
qEA‐qEA
0
0.005
0.01
0.015
0.02
0.025
0.03
-0.6 -0.4 -0.2 0 0.2 0.4 0.6
P(q
diag
)
qdiag
T/J=0.133
L=32L=24L=16L=12
L=8L=6
0
0.01
0.02
0.03
0.04
0.05
0.06
-0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04
P(q
χ)
qχ
T/J=0.133
L=32L=24L=16L=12
L=8L=6
Overlap distribution of the isotropic modelchirality spin [T = 0.133 < TCG ]
Chiral P(qχ) :central peak & ±qEA peak→ 1-step-like RSB ?
qEA peak-qEA peak
central peak
±J ; [K.Hukushima & HK ’05]
In equilibrium, FDT holdsR(t1,t2); response functionC(t1,t2); correlation function T; heat bath temperature
R(t1,t2)=(1/kBT) [dC(t1,t2) / dt1]
In off-equilibrium, FDT does not hold, but there is an off-equilibrium counterpart
R(t1,t2) = (X(t1,t2) / kBT) [dC(t1,t2) / dt1]In the limit of t1,t2 → ∞,
X(t1,t2) → X(C(t1,t2))
P(q) = d X(q) / dq P(q); overlap distribution function
Spin glass in off-equilibrium
[L.F. Kugliandolo and J. Kurchan ‘93]
1 / kBTeff
(a) No RSB(b) full-step RSB(c) 1-step RSB(d) combination of (b)+(c)
Susceptibility χ(t1,t2) vs. Correlation C(t1,t2)
FDT line
Off-equilibrium MC simulation of the isotropic 3D Gaussian Heisenberg SG
[HK, ’98]
Chiral autocorrelations Cχ(t,tw) at T=0.05
Chiral EA parameter qEA
qEA plateau
T
qEA
TCG ~ 0.15β ~ 1
superaging
log t
log t/tw
quasi-equilibriumregime
aging regime
tw: waiting time
Off-equilibrium simulation of the weaklyanisotropic 3D ±J Heisenberg SG [HK, ’03]
D=0.01χ-C plot for the spin
FDT line
1-step behavior is observed in the aging regime with Teff ~ 2Tg irrespective of T !!
Tg ~ 0.21
CdCr1.7In0.3S4 [D.Herisson and M.Ocio, ‘02]
Teff ~ 1.9 Tg (measured at T=0.8Tg)
Experiment on Heisenberg-like SG
1-step-like ?
SG (chiral-glass) ordered state of canonical SG exhibits a one-step-like RSB
Analogy to molecular glasses ?Dynamical equations describing structural glass aresimilar to MF SG models exhibiting a 1-step RSB
[T.E. Kirkpatrick, D. Thirumalai, P.G. Wolynes ’87]
1. Discontinuous 1-step RSB (discontinuous qEA at T=Tg)p>2-spin MF SG, p>4 state MF Potts SGdynamical TD and static Tg (TD > Tg)
2. Continuous 1-step RSB (continuous qEA at T=Tg)2<p<4 state MF Potts SGNo dynamical TD
structural glass
Heisenberg-like SG
Chiral order in ceramic high-Tcsuperconductors
Complete isotropy of the gauge spaceprovides a unique opportunity to test the spin(phase)-chirality decoupling
5. Spin-chirality decoupling (III)
Chiral-glass state in ceramic high-Tc superconductors [H.K. ‘95]
Chirality is hard to measure in magnets.But, in superconductors, the superconducting order parameter is given by
exp(iθ) = cosθ + i sinθ
where the chirality
sin(θi – θj) = Si × Sj
just corresponds to the supercurrent.
θ1 θ2
[Josephson junction]
Josephson current I = J sin(θ1-θ2)
Cuprate high-Tc superconductors : d –wave
π junction is possible (phase reversal; J < 0)
Loop containing odd number of π junctionsodd ring (π ring)
frustration in “phase” of superconductorspontaneous loop current
Couter-clockwise current
chirality +
Clockwise current
chirality ―
granular(ceramic)
superconductors
Cuprate granular (ceramic)superconductors
Random Josephson network consisting of both π and 0 junctions → anaology to XY SG
Chiral-glass state
spontaneous generation of Random flux in zero field (random freezing of chirality)
negative divergence of the nonlinear susceptibility at TCG (γ~4.4)
aging and rejuvenation-memory effect
AC susceptibilities of YBa2Cu4O8 ceramic superconductos[M. Hagiwara et al, ‘05]
Nonlinear susceptibility near Tc
Critical phenom
ena of nonlinear susceptibility
Linear susceptibility
slope=4.4
Intra-graintransition
Intrer-graintransition
Ideal system to detect the possible spin(phase)-chirality decoupling
Gauge space is completely isotropic !
T [chirality] > T [phase] ?
Linear resistivity ρ should remain nonzero,i.e., an ohmic behavior is expected
in the chiral-glass phase, if there occurs the spin(phase) –chirality
decoupling
Experimental work now in progress !
Summary
* Frustarion causes many interesingphenomena. It might provide us a new paradigm in condensed matter physics.
* Chirality, both the scalar and the vector ones, causes various intriguing phenomena, includingmultiferroic phenomena.
* A novel “spin-chirality decoupling” phenomenon occurs in certain frustrated systems, including spin glasses and even granular superconductors.