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

2. KIAS workshop Sept 1, 2008 Vector spin chirality in triangular lattice XY model Part I

3. 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 TbMnO 3 T From spin chirality to ferroelectricity

4. 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) 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)

5. KIAS workshop Sept 1, 2008 Materiald-electron Polarization (  C/m 2 ); Q Specifics TbMnO 3 (Kimura et al Nature 2003; Kenzelman et al PRL 2005) d 4 (t 2g ) 3 (e g ) 1 800; q~0.27Orbital order Ni 3 V 2 O 8 (Lawes et al PRL 2005) d 8 (t 2g ) 6 (e g ) 2 100; q~0.27Kagome Ba 0.5 Sr 1.5 Zn 2 Fe 12 O 22 (Kimura et al PRL 2005) d 5 (t 2g ) 3 (e g ) 2 150 (B=1T); N/AN/A CoCr 2 O 4 (Yamasaki et al. PRL 2006) Co 2+ : d 7 (e) 4 (t 2 ) 3 Cr 3+ : d 3 (t 2g ) 3 2; [qq0] q~0.63ferrimagnetic MnWO 4 (Taniguchi et al. PRL 2006) d 5 (t 2g ) 3 (e g ) 2 50; q=(-.214,.5,.457) N/A CuFeO 2 (Kimura et al PRB 2006) d 5 (t 2g ) 3 (e g ) 2 400 (B>10T); 1/5<q<1/4 2D triangular; Field-driven LiCuVO 4 (Naito et al JPSJ 2007) d 9 (t 2g ) 6 (e g ) 3 N/A; q~0.5321D chain LiCu 2 O 2 (Park et al PRL 2007) d 9 (t 2g ) 6 (e g ) 3 <10; q~0.1741D chain RED = magnetic ions vSC-driven multiferroics

6. 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 Experiments T, frustration Spiral Magnetic Collinear Magnetic Paramagnetic Ferro- electric

7. KIAS workshop Sept 1, 2008 T, frustration MagneticChiralParamgnetic 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 possible

8. KIAS workshop Sept 1, 2008 Villain’s idea of vSC in classical spin systems Villain, JPhysC (1977)

9. 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) ?

10. KIAS workshop Sept 1, 2008 Cinti et al. PRL 100, 057203 (2008) vCSL found ?

11. KIAS workshop Sept 1, 2008 XXZ spin chain (S=1) with nearest and next-nearest neighbor exchange 1D model of vCSL (quantum) vCS correlation is long-ranged vCS found for XY-like, J 2 -dominant regime of the model Hikihara et al. JPSJ 69, 259 (2000) =J 2 /J 1

12. KIAS workshop Sept 1, 2008 J2J2 J1J1

13. 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

14. KIAS workshop Sept 1, 2008 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?

15. KIAS workshop Sept 1, 2008 J 1 -J 2 model on triangular lattice PM = paramagnetic aM = (algebraic ordered) magnetic aN = (algebraic ordered) nematic C = chirality-ordered PONH, arXiv:0804.4034

16. KIAS workshop Sept 1, 2008 Chirality for J 1 -J 2 model Introduce a vector potential From the corresponding free energy define the spin current Staggered sum of the spin current is the vSC

17. KIAS workshop Sept 1, 2008 Monte Carlo results for chirality Binder cumulant Finite- Size Scaling

18. 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 I)

19. KIAS workshop Sept 1, 2008 Scalar spin chirality in Heisenberg spin- 1/2 model on pyrochlore lattice Part II

20. KIAS workshop Sept 1, 2008 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)

21. KIAS workshop Sept 1, 2008 VMC work of MIT group on Kagome Terms of 4 interacting fermions 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

22. KIAS workshop Sept 1, 2008 1 2 3 “Rokhsar’s rule” Rokhsar PRL 65, 1506 (1990)

23. KIAS workshop Sept 1, 2008 Connection of MFT to chirality In the fermion representation of spins,

24. KIAS workshop Sept 1, 2008 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

25. KIAS workshop Sept 1, 2008 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

26. KIAS workshop Sept 1, 2008 Mean-field analysis Self-consistent mean-field solution gives chiral spin liquid ground state pi/2-flux through triangles, 0-flux through hexagons

27. KIAS workshop Sept 1, 2008 Energy after projection We consider a number of different flux configurations The two chiral states stand out among others

28. KIAS workshop Sept 1, 2008 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

29. KIAS workshop Sept 1, 2008  1 2 3 4 1 2 3

30.  1 2 3 4 5 6 7 8 9 11 10 12 13 14 15 16

31. KIAS workshop Sept 1, 2008  2 1 3 4 1 2 3

32.  1 2 3 4 5 6 7 8 9 11 10 12 13 14 15 16

33. 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?

34. KIAS workshop Sept 1, 2008Summary 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