1. Vector spin chirality in classical & quantum spin systems
Jung Hoon Han (SKKU)

2. Collaboration Jung Hoon Kim, Jin Hong Park (SKKU)
Shigeki Onoda (RIKEN)
Naoto Nagaosa (U. Tokyo)
Chenglong Jia (Germany)
Raoul Dillenschneider (Augsburg)

3. Motivation
A class of materials with strong coupling of spin & lattice (orbital) degrees of freedom were (re)discovered.
One stark manifestation of the coupling is the control of ferroelectric polarization using only the magnetic field.
Nature 426, 55 (2003)

4. Exchange-striction-driven (RMn2O5) Spin-chirality-driven
Two types of improper multiferroics
Exchange-striction-driven (RMn2O5)
Spin-chirality-driven

5. 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
Idea of non-magnetic, but vSC-ordered magnetic state goes back as early as Villain, J. Phys. C 10, 4793 (1977). Other theoretical works on vSC:
S. Miyashita and H. Shiba, J. Phys. Soc. Jpn. 53, 1145 (1984).
H. Kawamura and M. Tanemura, Phys. Rev. B 36, 7177 (1987).
H. Kawamura, J. Phys.: Condens. Matter 10, 4707 (1998).

6. Connection to ferroelectricity
Connection of vSC to local dipole moment, or ferroelectricity, was noticed after some key neutron experiment T

7. Same symmetry !
With spiral magnetism one can define a SENSE of ROTATION, which is equivalent to the vSC
The forward-rotating spins produce UP polarization, the backward-rotating spins produce DOWN polarization, or vice versa.
Symmetry arguments for multiferroic materials are worked out in
A. B. Harris, Phys. Rev. B 76, (2007).
T. Arima, JPSJ 76, (2007) – CuFeO2
INVERSION

8. 400 (B>10T); 1/5<q<1/4 2D triangular; Field-driven
Bookkeeping of chiral 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; 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)3
2; [qq0] q~0.63
ferrimagnetic
MnWO4
(Taniguchi et al. PRL 2006)
50;
q=(-.214, .5, .457)
CuFeO2
(Kimura et al PRB 2006)
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)
<10; q~0.174
RED = magnetic ions

9. Microscopic Theories (mean-field)
TbMnO3
Mn d orbitals are orbitally ordered
Magnetic interaction contain both ferro and antiferro
Exchanges; fertile ground for frustrated magnetism

10. Microscopic Theories (mean-field)
For general d-electron configurations

11. Microscopic Theories
H. Katsura, N. Nagaosa, and A. V. Balatsky, PRL 95, (2005)
JONH, PRB 74, (2006)
JONH, PRB 76, (2007)

12. Microscopic Theories (LDA)
Mean-field calculation reflects distortion of electronic wave functions
due to spiral magnetic order
The mechanism is the spin-orbit coupling
The wave function distortion would generically lead to internal electric field,
which would tend to displace ions, and generate dipole moments
LDA calculation reflects the atomic movement better than MF calculation
(Xiang & Whangbo, PRL (2007)

13. Microscopic Theories (LDA)

14. Existing Experiments
Often, there is a first 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
Paramagnetic
Ferro-
electric

15. What’s possible Can we envision a phase without magnetic order,
but still has the remnant of vSC (vector spin chirality) ?
Theoretically certainly possible.
vSCL (vector spin chiral liquid)
Chiral spin states !
T, frustration
Magnetic
Chiral
Paramgnetic
Ferroelectric

16. Villain’s idea of vSC
Villain, JPhysC (1977)

17. Ginzburg-Landau theory of vSCL
Recent work: Ginzburg-Landau calculation showed possibility of a chiral spin liquid phase with sufficient frustration
Onoda & Nagaosa PRL 99, (2007)

18. Quantum spin S=1/2 Multiferroic
Seki et al. arXiv:
Park et al. PRL (2007)

19. Quantum spin S=1/2 Multiferroic
Enderle et al. EPL (2005)
Naito et al JPSJ (2007)

20. Search for models of vSCL
Both materials are exciting due to quantum nature of S=1/2 spins and the 1D character of spin network
However, the ferroelectricity is concomitant with spiral magnetic ordering
Not a true vSCL yet

21. vSCL found ?
Cinti et al. PRL 100, (2008)

22. Search for models of vSCL
In light of recent developments, search for model Hamiltonians (both classical & quantum) where the ground state supports ordered vSC but no ordering of magnetic moments
There is already some evidence from GL theory
A 1D microscopic calculation exists for a 1D spin model
We know little about 2D vSCL
A classical 2D model with ordered vSC but no magnetic moment may have been found (arXiv: )

23. 1D model of vSCL (quantum)
XXZ spin chain (S=1/2) with nearest and next-nearest neighbor exchange
Hikihara et al. JPSJ 69, 259 (2000)
vSCL found for XY-like, J2-dominant
regime of the model

24. Furukawa et al. arXiv: v1
A recent calculation of Furukawa et al. confirmed existence of vCSL phase in the same model

25. 2D model of vSCL (quantum) ??
In 2D we do not seem to have any idea how to write down a quantum spin model with long-range vSC correlation
2D vSC solid state can be generated easily with Dzyaloshinskii-Moriya interaction
1D example:

26. Analogy with persistent current
For S=1/2, Jordan-Wigner mapping gives
Spin chirality maps onto bond current
A current-flowing ground state in the fermion picture corresponds to vSC-ordered state in the spin picture (Dillenschneider et al. arXiv: )

27. - + 2D model of vSCL (classical)
A similar manipulation can be made in 2D to obtain vSCS
A true vSCL state is still out of reach…
We recently re-examined AFM XY model on triangular lattice with huge bi-quadratic exchange + -
Magnetic order -> vSC order
But can vSC order first before magnetism does?

28. J1-J2 model on triangular lattice
PONH, arXiv:
vSC ordering occurs below RED curve
Magnetic ordering occurs below BLACK & BLUE curve
The region x>xc supports vSCL phase with condensation of vSC, but no magnetic ordering has yet occurred

30. Summary
We have come a long way since the initial discovery of multiferroicity in
understanding the coupling of vSC and local electric dipoles
An interesting possibility of purely vSC-ordered liquid phase is opening up
(GL theory, 1D quantum spin models and compounds)
A 2D classical model which supports vSCL phase seems feasible
(2D AFM XY on triangular lattice)
2D quantum model with vSCL ground state will be exciting