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

Example of geometrical frustration (A)

Three Ising spins coupled antiferromagnetically

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 ! )

Temperature dependence of the spin and the chirality correlation lengths

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]

Instability points in k-space （Bragg points）

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

RG flow diagram in D=3 （XY; n=2）

[P. Carabrese et al ’02]

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

Experimental finding of spin glass (SG) [Canella and Mydosh, 1972]

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

Chiral-glass stateChiral glass paramagnetic

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

What is the nature of the chiral-glass ordered state ?

Is there an RSB ? If so, what type ?

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.