KIAS workshop Sept 1, 2008 A tale of two spin chiralities in frustrated spin systems Jung Hoon Han (SungKyunKwan U, Korea)

KIAS workshop Sept 1, 2008

A tale of two spin chiralities

infrustrated spin systems

Jung Hoon Han (SungKyunKwan U, Korea)


• Vector spin chirality in triangular lattice XY model

Part IPart I


Connection of spin chirality (for noncollinear magnetism) to local dipole moment, or ferroelectricity, was noticed after some key neutron experiment on TbMnO3

T

From spin chirality to ferroelectricityFrom spin chirality to ferroelectricity


It was soon realized that the relevant physics was in the coupling of the local dipole moment to the local vector spin chirality (vSC)

Vector spin chirality (vSC)Vector spin chirality (vSC)

Noncollinear magnetic states possess a nonzero vSC

vSC breaks inversion symmetry, preserves time-reversal, that’s the same symmetry as the local dipole moment

Mostovoy PRL96, 067601 (2006)


Material d-electronPolarization(C/m2); Q

Specifics

TbMnO3

(Kimura et al Nature 2003;Kenzelman et al PRL 2005)

d4 (t2g)3 (eg)1 800; q~0.27 Orbital order

Ni3V2O8

(Lawes et al PRL 2005)d8 (t2g)6 (eg)2 100; q~0.27 Kagome

Ba0.5Sr1.5Zn2Fe12O22

(Kimura et al PRL 2005)d5 (t2g)3 (eg)2 150 (B=1T); N/A N/A

CoCr2O4 (Yamasaki et al. PRL 2006)

Co2+ : d7 (e)4 (t2)3

Cr3+ : d3 (t2g)32; [qq0] q~0.63 ferrimagnetic

MnWO4

(Taniguchi et al. PRL 2006)d5 (t2g)3 (eg)2

50; q=(-.214, .5, .45

7)N/A

CuFeO2

(Kimura et al PRB 2006)

d5 (t2g)3 (eg)2 400 (B>10T); 1/5<q<1/4

2D triangular;

Field-driven

LiCuVO4

(Naito et al JPSJ 2007)d9 (t2g)6 (eg)3 N/A; q~0.532 1D chain

LiCu2O2 (Park et al PRL 2007)

d9 (t2g)6 (eg)3 <10; q~0.174 1D chainRED = magnetic ions

vSC-driven multiferroicsvSC-driven multiferroics


Often, there is first a magnetic transition to COLLINEAR spin states, for which no polarization is induced

A second transition at a lower temperature to spiral spin states cause

nonzero polarization

Existing Existing ExperimentsExperiments

T, frustrationSpiralMagnetic

CollinearMagnetic

Paramagnetic

Ferro-electric


T, frustrationMagnetic Chiral Paramgnetic

Can we envision a phase without magnetic order, but still has the remnant of vSC (vector spin chirality) ?

Theoretically certainly possible.

vCSL (vector chiral spin liquid)

Ferroelectric

Chiral spin states !

What’s possibleWhat’s possible


Villain’s idea of vSC in classical spin systems Villain’s idea of vSC in classical spin systems

Villain, JPhysC (1977)


Can we have an example of non-magnetic, chirality-ordered phase?

The idea is motivated by recent trends in multiferroics, but an analogous one has been around for a few decades

No serious efforts for quantum spin cases until recently

It is entirely possible that coupling to ferroelectric moment occurs in non-magnetic, yet vSC-ordered phase (a exotic new matter?)

Perhaps low-D, small-S (highly quantum), highly frustrated spin systems are a good place to look for vCSL

Vector Chiral Spin Liquid (vCSL) ?Vector Chiral Spin Liquid (vCSL) ?


Cinti et al. PRL 100, 057203 (2008)

vCSL found ?vCSL found ?


XXZ spin chain (S=1) with nearest and next-nearest neighbor exchange

,1J

,2J

1D model of vCSL (quantum)1D model of vCSL (quantum)

vCS correlation is long-ranged

vCS found for XY-like, J2-dominant regime of the model

Hikihara et al. JPSJ 69, 259 (2000)

=J2/J1


J2

J1


Furukawa et al. arXiv:0802.3256v1

A recent calculation of Furukawa et al. confirmed existence of vCSL phase in the same model with S=1/2


2D model of vCSL (classical)2D model of vCSL (classical)

We recently re-examined AFM XY model on triangular lattice with huge bi-quadratic exchange

+ + +

+++- - -

---Magnetic ordering naturally leads to vSC, but can the converse be also true?


JJ11-J-J22 model on triangular lattice model on triangular lattice

PM = paramagneticaM = (algebraic ordered) magneticaN = (algebraic ordered) nematicC = chirality-ordered

PONH, arXiv:0804.4034


Chirality for JChirality for J11-J-J22 model model

Introduce a vector potential

From the corresponding free energy define the spin current

Staggered sum of the spin current is the vSC


Monte Carlo results for chiralityMonte Carlo results for chirality

Binder cumulant

Finite-SizeScaling


• We have come a long way since the initial discovery of multiferroicity in understanding the coupling of vCS and local electric dipoles

• An interesting possibility of purely vCS-ordered liquid phase is opening up (GL theory, 1D quantum spin models and compounds)

• A 2D classical model which supports vCSL phase seems feasible (2D AFM XY on triangular lattice)

• 2D quantum model with vCSL ground state will be exciting

Summary (Part Summary (Part I)I)


• Scalar spin chirality in Heisenberg spin-1/2 model on pyrochlore lattice

Part IIPart II


Background on scalar spin chiralityBackground on scalar spin chirality

A non-magnetic state with nonzero averages of scalar spin chirality was proposed by Wen, Wilczek, Zee in connection with high-Tc cuprates (sCSL)

sCSL breaks T (Time reversal symmetry) while vCSL breaks I (Inversion symmetry)

Both spin chiralities arise due to some sort of frustration, including one of geometric nature (triangular, kagome, pyrochlore, etc)


VMC work of MIT group on KagomeVMC work of MIT group on Kagome

Solve Heisenberg model on kagome lattice by

(1) Re-writing spin as a fermion bilinear (2) Solve the mean-field theory with (3) Refine the state with Gutzwiller projection of doubly occupied sites

Ran et al. PRL 98, 117205 (2008); Hermele et al. arXiv:0803.1150v2


1 2

3

““Rokhsar’s rule”Rokhsar’s rule”

Rokhsar PRL 65, 1506 (1990)


Connection of MFT to chiralityConnection of MFT to chirality

In the fermion representation of spins,


VMC ground state found by MIT VMC ground state found by MIT groupgroup

Nonchiral SL with pi flux through hexagons had lowest VMC energy

Effective field theory is one of Dirac spinons

Had Rokhsar’s rule prevailed, one would arrive at sCSL


Our VMC work on pyrochloreOur VMC work on pyrochlore

Looking at the same type of theory on the pyrochlore lattice seems natural

We looked at fMFT+VMC of S=1/2 Heisenberg spins on pyrochlore lattice

Kim JH & Han JH, arXiv:0807.2036v1


Mean-field Mean-field analysisanalysis

Self-consistent mean-field solution gives chiral spin liquid ground state

pi/2-flux through triangles,0-flux through hexagons


Energy after Energy after projectionprojection

We consider a number of different flux configurations

The two chiral states stand out among others


Flux after Flux after projectionprojection

Average flux through each triangle can be calculated within VMC scheme

Amount of flux is reduced from mean-field value pi/2, but remains nonzero

It is a sCSL state with long-range ordered chiralities


1

2

3

4

1

2

3


1

2

3

4

5

6

7

8

9 11

10

12

13

14

15

16


2

1 3

4

1

2

3


1

2

3

4

5

6

7

8

9 11

10

12

13

14

15

16


Construction of effective field theory difficult due to

The presence of a flat band at the Fermi level for [pi/2,pi/2,0] (uniform) chiral state

The presence of a gap for [pi/2,-pi/2,0] (staggered) chiral state

Effective theory?Effective theory?


SummarySummary

Combined fMFT+VMC suggests sCSL state as potential (proximate) ground state of Heisenberg model on pyrochlore lattice (maybe first time sCSL was discovered in FMFT+VMC theory)

Further work with ED on pyrochlore may be fruitful

Documents

KIAS workshop Sept 1, 2008 A tale of two spin chiralities in frustrated spin systems Jung Hoon Han (SungKyunKwan U, Korea)