Transcript A tale of two spin chiralities in frustrated spin systems

A tale of
two spin chiralities
in
frustrated spin systems
Jung Hoon Han
(SungKyunKwan U, Korea)
KIAS workshop Sept 1, 2008
Part I
• Vector spin chirality in triangular
lattice XY model
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From spin chirality to ferroelectricity
Connection of spin chirality (for noncollinear magnetism) to local dipole
moment, or ferroelectricity, was noticed after some key neutron
experiment on TbMnO3
T
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Vector spin chirality (vSC)
It was soon realized that the relevant physics was in the coupling of the
local dipole moment to the local 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)
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vSC-driven multiferroics
Material
TbMnO3
(Kimura et al Nature 2003;
Kenzelman et al PRL 2005)
Ni3V2O8
(Lawes et al PRL 2005)
Ba0.5Sr1.5Zn2Fe12O22
(Kimura et al PRL 2005)
CoCr2O4
(Yamasaki et al. PRL 2006)
MnWO4
(Taniguchi et al. PRL 2006)
CuFeO2
(Kimura et al PRB 2006)
LiCuVO4
(Naito et al JPSJ 2007)
LiCu2O2
(Park et al PRL 2007)
d-electron
Polarization
(C/m2); Q
Specifics
d4 (t2g)3 (eg)1
800; q~0.27
Orbital order
d8 (t2g)6 (eg)2
100; q~0.27
Kagome
d5 (t2g)3 (eg)2
150 (B=1T); N/A
N/A
Co2+ : d7 (e)4 (t2)3
Cr3+ : d3 (t2g)3
2; [qq0] q~0.63
ferrimagnetic
d5 (t2g)3 (eg)2
50;
q=(-.214, .5, .457)
N/A
d5 (t2g)3 (eg)2
400 (B>10T);
1/5<q<1/4
2D triangular;
Field-driven
d9 (t2g)6 (eg)3
N/A; q~0.532
1D chain
d9 (t2g)6 (eg)3
<10; q~0.174
1D chain
RED = magnetic ions
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Existing Experiments
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
Spiral
Magnetic
Collinear
Magnetic
Paramagnetic
T, frustration
Ferroelectric
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What’s possible
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)
Chiral spin states !
Magnetic
Chiral
Paramgnetic
T, frustration
Ferroelectric
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Villain’s idea of vSC in classical spin
systems
Villain, JPhysC (1977)
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Vector Chiral Spin Liquid (vCSL) ?
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 nonmagnetic, 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
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vCSL found ?
Cinti et al. PRL 100, 057203 (2008)
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1D model of vCSL (quantum)
XXZ spin chain (S=1) with nearest and next-nearest neighbor exchange
J1 , 
J2 , 
Hikihara et al. JPSJ 69, 259 (2000)
vCS correlation is long-ranged
vCS found for XY-like, J2-dominant
regime of the model
=J2/J1
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J2
J1
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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
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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?
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J1-J2 model on triangular lattice
PONH, arXiv:0804.4034
PM = paramagnetic
aM = (algebraic ordered) magnetic
aN = (algebraic ordered) nematic
C = chirality-ordered
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Chirality for J1-J2 model
Introduce a vector potential
From the corresponding free energy define the spin current
Staggered sum of the spin current is the vSC
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Monte Carlo results for chirality
FiniteSize
Scaling
Binder
cumulant
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Summary (Part I)
• 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
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Part II
• Scalar spin chirality in Heisenberg
spin-1/2 model on pyrochlore lattice
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Background 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)
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VMC 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
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“Rokhsar’s rule”
Rokhsar PRL 65, 1506 (1990)
3
1
2
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Connection of MFT to chirality
In the fermion representation of spins,
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VMC ground state found by MIT group
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
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Our 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
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Mean-field analysis
Self-consistent mean-field solution
gives chiral spin liquid ground state
pi/2-flux through triangles,
0-flux through hexagons
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Energy after
projection
We consider a number of different flux configurations
The two chiral states stand out among others
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Flux after projection
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
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
2
1
3
4
1
3
2
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
12
11
9
16
10
4
15
13
3
1
14
8
2
5
7
6
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2
1
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4
1
3
2
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12
11
9
16
10
4
15
13
14
3
1
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5
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6
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Effective theory?
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
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Summary
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
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