1. ISSP TASSP workshop Jun 19, 2008 Multiferroic behavior in spin-chirality- and exchange-striction-driven compounds Jung Hoon Han (SungKyunKwan U, Korea)

2. ISSP TASSP workshop Jun 19, 2008Collaboration Jung Hoon Kim, Jin Hong Park (SKKU) Kee Hoon Kim (SNU) Shigeki Onoda (RIKEN) Naoto Nagaosa (U. Tokyo) Chenglong Jia (Germany) Raoul Dillenschneider (Augsburg)

3. ISSP TASSP workshop Jun 19, 2008 A class of materials with strong coupling of spin & lattice or electronic degrees of freedom were (re)discovered. One stark manifestation of the coupling is the control of ferroelectric polarization using only the magnetic field.Motivation Nature 426, 55 (2003) TbMnO 3

4. ISSP TASSP workshop Jun 19, 2008 Magnetic field along a switches polarization from +b to -b axis in TbMn 2 O 5 Nature 429, 392 (2004) TbMn 2 O 5Motivation

5. ISSP TASSP workshop Jun 19, 2008 Exchange-striction-driven, symmetric spin exchange (e.g. TbMn 2 O 5 ) Spin-chirality-driven, anti-symmetric spin exchange (e.g. TbMnO 3 ) Two types of multiferroics

6. ISSP TASSP workshop Jun 19, 2008 Spin-chirality-driven, anti-symmetric spin exchange (e.g. TbMnO 3 ) Part I

7. ISSP TASSP workshop Jun 19, 2008 Connection of spin chirality (for noncollinear magnetism) to local dipole moment, or ferroelectricity, was noticed after some key neutron experiment T From spin chirality to ferroelectricity

8. ISSP TASSP workshop Jun 19, 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)

9. ISSP TASSP workshop Jun 19, 2008 For general d-electron configurations Microscopic Theories (mean-field) M O

10. ISSP TASSP workshop Jun 19, 2008 Microscopic Theories (mean-field) H. Katsura, N. Nagaosa, and A. V. Balatsky, PRL 95, 057205 (2005) JONH, PRB 74, 224444 (2006) JONH, PRB 76, 023708 (2007) Spiral, helical, conical spins give uniform polarization

11. ISSP TASSP workshop Jun 19, 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

12. ISSP TASSP workshop Jun 19, 2008 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) Recent LDA works on TbMnO 3

13. ISSP TASSP workshop Jun 19, 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

14. ISSP TASSP workshop Jun 19, 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. vSCL (vector spin chiral liquid) Ferroelectric Chiral spin states ! What’s possible

15. ISSP TASSP workshop Jun 19, 2008 Villain’s idea of vSC Villain, JPhysC (1977)

16. ISSP TASSP workshop Jun 19, 2008 Villain’s idea of vSC

17. ISSP TASSP workshop Jun 19, 2008 Can we have an example of non-magnetic, chirality-ordered phase? There had been discussions of vSCL in classical models of AFM with frustration No analogous 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 vSCL vSC liquid (vSCL) ?

18. ISSP TASSP workshop Jun 19, 2008 Cinti et al. PRL 100, 057203 (2008) vSCL found ?

19. ISSP TASSP workshop Jun 19, 2008 Quantum spin S=1/2 Multiferroic Seki et al. arXiv:0801.2533 Park et al. PRL (2007)

20. ISSP TASSP workshop Jun 19, 2008 Quantum spin S=1/2 Multiferroic Enderle et al. EPL (2005) Naito et al JPSJ (2007)

21. ISSP TASSP workshop Jun 19, 2008 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

22. ISSP TASSP workshop Jun 19, 2008 XXZ spin chain (S=1) with nearest and next-nearest neighbor exchange 1D model of vSCL (quantum) vSC correlation is long-ranged vSCL found for XY-like, J 2 -dominant regime of the model Hikihara et al. JPSJ 69, 259 (2000) =J 2 /J 1

23. ISSP TASSP workshop Jun 19, 2008 Furukawa et al. arXiv:0802.3256v1 A recent calculation of Furukawa et al. confirmed existence of vSCL phase in the same model with S=1/2

24. ISSP TASSP workshop Jun 19, 2008 2D model of vSCL (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?

25. ISSP TASSP workshop Jun 19, 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

26. ISSP TASSP workshop Jun 19, 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

27. ISSP TASSP workshop Jun 19, 2008 Monte Carlo results for chirality Binder cumulant Finite- Size Scaling

28. ISSP TASSP workshop Jun 19, 2008 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 Summary (Part I)

29. ISSP TASSP workshop Jun 19, 2008 Magnetic field along a switches polarization from +b to -b axis in TbMn 2 O 3 Nature 429, 392 (2004) TbMn 2 O 5 Exchange-striction-driven Multiferroics

30. ISSP TASSP workshop Jun 19, 2008 Two types of Mn ions: Mn 3+ ( oxygen tetrahedron ) Mn 4+ ( oxygen octahedron ) Exchange-striction-driven Multiferroics Radaelli et al. PRB (2005); PRL (2004,2006)

31. ISSP TASSP workshop Jun 19, 2008 Mn 3+ ( RED ) Mn 4+ ( BLUE ) Interactions along c-axis either FM or AFM (no frustration) Project down to 2D (ab plane) without loss of generality Looking at the Mn Lattice:

32. ISSP TASSP workshop Jun 19, 2008 Intra-chain interaction is AFM (no frustration) Inter-chain is also AFM, every other bond is frustrated Shifting the spin orientation by one lattice does not lower energy -> 2-fold degeneracy between every chains -> macroscopic degeneracy Frustrated Mn spin interactions

33. ISSP TASSP workshop Jun 19, 2008 According to pure Heisenberg exchange, the middle spin is frustrated It can choose one spin orientation and move toward atoms of opposite spins, lowering the overall exchange energy J J J-  J J+  J Exchange-striction Primer

34. ISSP TASSP workshop Jun 19, 2008 Exchange striction causes displacement of Mn 3+ resulting in net polarization along b axis (consistent with exp.) Lifts macroscopic degeneracy All this well known (S.-W. Cheong, Radaelli, etc.) before I got interested… OR Lifting of Degeneracy by Exchange-striction

35. ISSP TASSP workshop Jun 19, 2008 Electric polarization, initially along +b, reverses direction due to magnetic field (magneto-electric coupling) Effects of B field along a-axis on BiMn 2 O 5 Kee Hoon Kim et al. Submitted Dielectric constant along b-axis shows pronounced increases where P=0.

36. ISSP TASSP workshop Jun 19, 2008Scaling

37. P>0 and P<0 regions are separated by a line as if by a phase transition Genuine 2 nd order PT is impossible Phase diagram of BiMn 2 O 5 First-order PT

38. ISSP TASSP workshop Jun 19, 2008 Why P changes sign under H field? Why apparent critical behavior near H c ? What we need to know

39. ISSP TASSP workshop Jun 19, 2008 In an ordinary AFM, H field will cause spin flop. All spins will rotate either CW or CCW as to be orthogonal to B field The relative spin orientations will be the same after spin flop, hence no change in exchange striction force (~S i * S j ) -> Can’t explain experiment Suggestion: Perhaps each spin chain undergoes spin flop with different sense of rotations… (possible since inter-chain coupling is weak) Understanding Polarization Reversal

40. ISSP TASSP workshop Jun 19, 2008 Preliminary MC data The spins do rotate CW for even chains, CCW for odd chains; good pairs become bad pairs and vice versa -> explains polarization reversal Surprisingly, susceptibility (dielectric constant) shows a peak!

41. ISSP TASSP workshop Jun 19, 2008 A class of compounds RMn 2 O 5 are examples of exchange-striction-driven multiferroicity An interesting polarization reversal and sharp increase in dielectric susceptibility was observed in high-field experiment on BiMn 2 O 5 A model with both frustrated spin-spin interaction and exchange-striction coupling with reasonable agreement with experimental findings The precise critical nature of P remains to be understood Summary (Part II)

42. ISSP TASSP workshop Jun 19, 2008 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:

43. ISSP TASSP workshop Jun 19, 2008 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:0705.3993) Analogy with persistent current