A tale of two spin chiralities in frustrated spin systems

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A tale of two spin chiralitiesinfrustrated
spin systems
Jung Hoon Han (SungKyunKwan U, Korea)
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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, thats the same symmetry as the
local dipole moment
Mostovoy PRL96, 067601 (2006)
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vSC-driven multiferroics
Material d-electron Polarization (?C/m2) Q Specifics
TbMnO3 (Kimura et al Nature 2003 Kenzelman et al PRL 2005) d4 (t2g)3 (eg)1 800 q0.27 Orbital order
Ni3V2O8 (Lawes et al PRL 2005) d8 (t2g)6 (eg)2 100 q0.27 Kagome
Ba0.5Sr1.5Zn2Fe12O22 (Kimura et al PRL 2005) d5 (t2g)3 (eg)2 150 (B1T) N/A N/A
CoCr2O4 (Yamasaki et al. PRL 2006) Co2 d7 (e)4 (t2)3 Cr3 d3 (t2g)3 2 qq0 q0.63 ferrimagnetic
MnWO4 (Taniguchi et al. PRL 2006) d5 (t2g)3 (eg)2 50 q(-.214, .5, .457) N/A
CuFeO2 (Kimura et al PRB 2006) d5 (t2g)3 (eg)2 400 (Bgt10T) 1/5ltqlt1/4 2D triangular Field-driven
LiCuVO4 (Naito et al JPSJ 2007) d9 (t2g)6 (eg)3 N/A q0.532 1D chain
LiCu2O2 (Park et al PRL 2007) d9 (t2g)6 (eg)3 lt10 q0.174 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
T, frustration
Spiral Magnetic
Collinear Magnetic
Paramagnetic
Ferro- electric
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Whats 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 !
T, frustration
Magnetic
Chiral
Paramgnetic
Ferroelectric
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Villains 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
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
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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 (S1) with nearest and
next-nearest neighbor exchange
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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Furukawa et al. arXiv0802.3256v1
A recent calculation of Furukawa et al. confirmed
existence of vCSL phase in the same model with
S1/2
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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, arXiv0804.4034
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
Finite- Size 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.
arXiv0803.1150v2
Terms of 4 interacting fermions
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Rokhsars rule
Rokhsar PRL 65, 1506 (1990)
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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 Rokhsars 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
fMFTVMC of S1/2 Heisenberg spins on pyrochlore
lattice
Kim JH Han JH, arXiv0807.2036v1
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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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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 fMFTVMC suggests sCSL state as
potential (proximate) ground state of Heisenberg
model on pyrochlore lattice (maybe first time
sCSL was discovered in FMFTVMC theory) Further
work with ED on pyrochlore may be fruitful