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KIAS workshop Sept 1, 2008
A tale of two spin chiralities
infrustrated spin systems
Jung Hoon Han (SungKyunKwan U, Korea)
KIAS workshop Sept 1, 2008
• Vector spin chirality in triangular lattice XY model
Part IPart I
KIAS workshop Sept 1, 2008
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
KIAS workshop Sept 1, 2008
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)
KIAS workshop Sept 1, 2008
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
KIAS workshop Sept 1, 2008
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
KIAS workshop Sept 1, 2008
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
KIAS workshop Sept 1, 2008
Villain’s idea of vSC in classical spin systems Villain’s idea of vSC in classical spin systems
Villain, JPhysC (1977)
KIAS workshop Sept 1, 2008
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) ?
KIAS workshop Sept 1, 2008
Cinti et al. PRL 100, 057203 (2008)
vCSL found ?vCSL found ?
KIAS workshop Sept 1, 2008
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
KIAS workshop Sept 1, 2008
J2
J1
KIAS workshop Sept 1, 2008
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
KIAS workshop Sept 1, 2008
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?
KIAS workshop Sept 1, 2008
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
KIAS workshop Sept 1, 2008
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
KIAS workshop Sept 1, 2008
Monte Carlo results for chiralityMonte Carlo results for chirality
Binder cumulant
Finite-SizeScaling
KIAS workshop Sept 1, 2008
• 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)
KIAS workshop Sept 1, 2008
• Scalar spin chirality in Heisenberg spin-1/2 model on pyrochlore lattice
Part IIPart II
KIAS workshop Sept 1, 2008
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)
KIAS workshop Sept 1, 2008
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
KIAS workshop Sept 1, 2008
1 2
3
““Rokhsar’s rule”Rokhsar’s rule”
Rokhsar PRL 65, 1506 (1990)
KIAS workshop Sept 1, 2008
Connection of MFT to chiralityConnection of MFT to chirality
In the fermion representation of spins,
KIAS workshop Sept 1, 2008
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
KIAS workshop Sept 1, 2008
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
KIAS workshop Sept 1, 2008
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
KIAS workshop Sept 1, 2008
Energy after Energy after projectionprojection
We consider a number of different flux configurations
The two chiral states stand out among others
KIAS workshop Sept 1, 2008
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
KIAS workshop Sept 1, 2008
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KIAS workshop Sept 1, 2008
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KIAS workshop Sept 1, 2008
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KIAS workshop Sept 1, 2008
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KIAS workshop Sept 1, 2008
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?
KIAS workshop Sept 1, 2008
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