1. Competing phases of XXZ spin chain with frustration Akira Furusaki (RIKEN) July 10, 2011 Symposium on Theoretical and Mathematical Physics, The Euler International Mathematical Institute

2. Collaborators: Shunsuke Furukawa (U. Tokyo) Toshiya Hikihara (Gunma U.) Shigeki Onoda (RIKEN) Masahiro Sato (Aoyama Gakuin U.) Thanks: Sergei Lukyanov (Rutgers)

3. Outline 1.Introduction: frustrated spin-1/2 J 1 -J 2 XXZ chain 2.XXZ chain (J 2 =0): review of bosonization approach 3.Phase diagram of J 1 -J 2 XXZ spin chain a.J 1 >0 (antiferromagnetic) b.J 2 <0 (ferromagnetic)

4. J1J1 J 2 > 0(AF) If J 2 is antiferromagnetic, spins are frustrated regardless of the sign of J 1. J 1 -J 2 spin chain is the simplest spin model with frustration. frustrated spin-1/2 J 1 -J 2 chain

5. Materials: Quasi-1D cuprates (multi-ferroics) Rb 2 Cu 2 Mo 3 O 12 LiCuVO 4 LiCu 2 O 2 NaCu 2 O 2 Cu (3dx 2 -y 2 ) O(2p x ) J1J1 J2J2 PbCuSO 4 (OH) 2 b a c Cu 2+ spin S=1/2

6. Quasi-1D spin-1/2 frustrated magnets with ferro J 1 J 1 <0 J 2 >0 Cu O b a c 0-2-3-4 LiCuVO 4 LiCu 2 O 2 Li 2 ZrCuO 4 Enderle et al. Europhys.Lett.,2005 Masuda et al. PRL,2004; PRB,2005 Drechsler et al. PRL,2007  CuO 2 chain: edge-sharing network ferromagnetic J 1 (Kanamori-Goodenough rule)  Multiferroicity Observation of chiral ordering through electric polarization P Seki et al., PRL, 2008 (LiCu 2 O 2 )

7. Model J1J1 J 2 (>0, antiferro) x y easy-plane anisotropy -4 4 Ferro Antiferro Spin spiral 0 finite chirality Classical ground state Frustration occurs when J 2 >0, irrespective of the sign of J 1. Frustrated spin-1/2 J 1 -J 2 XXZ chain The phase diagram is symmetric. J 1 >0 and J 1 <0 are equivalent under pitch angle (applies only in the classical case)

8. Quantum case S=1/2  Antiferromagnetic case (J 1 >0,J 2 >0) is well understood. - Singlet dimer order is stabilized (J 2 /J 1 >0.24). Haldane, PRB1982 Nomura & Okamoto, J.Phys.A 1994 White & Affleck, PRB 1996 Eggert, PRB 1996 - Vector chiral ordered phase (quantum remnant of the spiral phase) is found for small Nersesyan,Gogolin,& Essler, PRL 1998 Hikihara,Kaburagi,& Kawamura, PRB 2001  Classical spiral (chiral) order is destroyed by strong quantum fluctuations in 1D.

9. Previous study of spin-1/2 J 1 -J 2 chain Ground-state phase diagram for AF-J 1 case Two decoupled J 2 chains Majumdar-Ghosh line T. Hikihara, M. Kaburagi, and H. Kawamura, PRB (2001), etc. K. Okamoto and K. Nomura, Phys. Lett. A (1992). J 1 chain

10. Quantum case S=1/2  Antiferromagnetic case (J 1 >0,J 2 >0) is well understood. - Singlet dimer order is stabilized (J 2 /J 1 >0.24). Haldane, PRB1982 Nomura & Okamoto, J.Phys.A 1994 White & Affleck, PRB 1996 Eggert, PRB 1996 - Vector chiral ordered phase (quantum remnant of the spiral phase) is found for small Nersesyan,Gogolin,& Essler, PRL 1998 Hikihara,Kaburagi,& Kawamura, PRB 2001  Classical spiral (chiral) order is destroyed by strong quantum fluctuations in 1D.  The ferromagnetic-J 1 case (J 1 0) is less understood. Goal: to determine the ground-state phase diagram

11. Our strategy perturbative RG analysis around J 1 =0 or J 2 =0. XXZ spin chain: exactly solvable low-energy effective theory (bosonization) numerical methods density matrix renormalization group (DMRG) time evolving block decimation for infinite system (iTEBD)

12. XXZ spin chain: brief review mostly standard textbook material, plus some relatively new developments

13. XXZ spin chain Exactly solvable: Bethe ansatz gapless phase １ antiferromagnetic Ising order ferromagnetic Ising order Tomonaga-Luttinger liquid energy gap gapless excitations power-law correlations LRO

14. Effective field theory: bosonization Luther, Peschel : bosonic field is relevant for irrelevant for For marginally irrelevant for

15. In the critical phase The cosine term is irrelevant in the low-energy limit : Gaussian model : bosonic field : exactly determined by Bethe ansatz not directly obtained from Bethe ansatz Spin operators

16. T. Hikihara & AF (1998) Lukyanov & Zamolodchikov (1997) Lukyanov (1998) more recently, Maillet et al.

17. exact numerics from S. Lukyanov, arXiv:cond-mat/9809254

18. dimer correlation (staggered) dimer correlation: as important as the spin correlations NN bond (energy) operators ( ) unknown: we have determined numerically using DMRG [cf. Eggert-Affleck (1992)] scaling dimension = 1/2 at AF Heisenberg point known (can be obtained from energy density etc.)

19. Analytic results for uniform components Uniform part of dimer operators = energy density in uniform chain We can evaluate the coefficients of the uniform comp. from the exact results of the energy density

20. Dimer operators in finite open chain Dimer order induced at open boundaries penetrates into bulk decaying algebraically Dirichlet b.c. for boson field : Open boundary condition mode expansion

21. DMRG results Calculate the local dimer operator for a finite open chain using DMRG fit the data to the form obtained by bosonization to determine excellent agreement between DMRG data and bosonization forms

22. Numerics (DMRG) Staggered part of the dimer operators

23. coefficient Exact formulas for are not known. Hikihara, AF & Lukyanov, unpublished

24. Effective field theory Lukyanov & Zamolodchikov NPB (1997) 1 st order perturbation in gives the leading boundary contribution to free energy of semi-infinite (or finite) spin chains for for Dirichlet b.c., Boundary specific heat: Boundary susceptibility:

25. Boundary energy of open XXZ chain L spins AF & T. Hikihara, PRB 69, 094429 (2004) lowest energy of a finite open chain with

26. J 1 -J 2 spin chain with antiferromagnetic J 1

27. Ground-state phase diagram for AF-J 1 case Majumdar-Ghosh line J 1 chain Two decoupled J 2 chains

28. dimer phase J 2 >0 changes and scaling dimension of If is relevant and, then is pinned at. dimer LRO If is relevant and, then is pinned at. Neel LRO Haldane ‘82 White & Affleck ’96 ……..

29. Ground-state phase diagram for AF-J 1 case Majumdar-Ghosh line J 1 chain Two decoupled J 2 chains

30. Perturbation around J 1 =0 Two decoupled J 2 chains dimer order vector chiral order

31. When relevant → Characteristics of the vector chiral state power-law decay, incommensurate Nersesyan-Gogolin-Essler (1998) Vector chiral order  Vector chiral order A quantum counterpart of the classical helical state  S x S x & S x S y spin correlation Opposite sign Vector chiral phase no net spin current flow p-type nematic Andreev-Grishchuk (1984)

32. J 1 -J 2 spin chain with ferromagnetic J 1

33. Phase diagram & chiral order parameter ferromagnetic J 1 antiferromagnetic J 1 The vector chiral order phase is large in the ferromagnetic J1 case and extends up to the vicinity of the isotropic case

34. Perturbation around J 1 =0 Two decoupled J 2 chains dimer order vector chiral order dimension

35. Phase diagram & chiral order parameter ferromagnetic J 1 antiferromagnetic J 1

36. Ground-state phase diagram for Ferro-J 1 case Two decoupled J 2 chains J 1 chain LiCu 2 O 2 LiCuVO 4 PbCuSO 4 (OH) 2 NaCu 2 O 2 Rb 2 Cu 2 Mo 3 O 12 Li 2 ZrCuO 4

37. Sine-Gordon model for spin-1/2 J 1 -J 2 XXZ chain with ferromagnetic coupling J 1 We begin with the J 2 =0 limit. J 1 chain Ferromagnetic Easy-plane Effective Hamiltonian (sine-Gordon model) TL-liquid (free-boson) partirrelevant perturbation TL-liquid parameter velocity Ferromagnetic SU(2) Heisenberg

38. Spin and dimer operators If the cosine term becomes relevant, then Neel order dimer order J 1 chain BKT-type RG equation

39. Exact coupling constant in the J 1 chain (J 2 =0) It vanishes and changes its sign at i.e., Relation between and excitation gaps of finite-size systems from perturbation theory for cosine term estimated by numerical diagonalization Exact value is known in J 1 chain S. Lukyanov, Nucl. Phys. B (1998). “Dimer” gap “Neel” gap

40. We can check the position of =0 from numerical-diagonalization result. J 2 =0

41. This relation is stable against perturbations conserving symmetries. Generally the exact value of is not known in the presence of such perturbations (J 2 ). However, the position of =0 is determined by the equation which can be numerically evaluated. J 2 perturbation makes the term relevant. Neel and dimer phases are expected to emerge. Neel order dimer order Gaussian phase transition point (c=1)

42. Phase diagram and Neel/dimer order parameters Ground-state phase diagram of easy-plane anisotropic J 1 -J 2 chain J 1 chain Curves of =0 Irrelevantrelevant dimer <0 Neel >0 dimer <0 Neel >0 Furukawa, Sato & AF PRB 81, 094410 (2010)

43. Direct calculation of order parameters from iTEBD method  >0  <0 XY component of dimer Z component of dimer Neel operator (Z component of spin)

44. Neel phase The emergence of the Neel phase is against our intuition: ferromagnetic & easy-plane anisotropy Spin correlation functions in the Neel phase Short-range behavior is different from that of the standard Neel order.

45. Dimer phase FM-J 1 case AF-J 1 case dimer phase in the AF-J 1 region dimer phase in the FM-J 1 region Neel On the XY line (  =0) “triplet” dimer“singlet” dimer J 1 -J 2 XY chain with FM J 1 J 1 -J 2 XY chain with AF J 1  rotation at every even site

46. Dimer order parameter Different dimer order dimer 123

47. zoom weak dimer order string order parameter dimer order parameter 123 long-range string order Sato, Furukawa, Onoda & AF Mod. Phys. Lett. 25, 901 (2011)

48. Summary ferromagnetic J 1 antiferromagnetic J 1 Sato, Furukawa, Onoda & AF Mod. Phys. Lett. 25, 901 (2011) Furukawa, Sato & AF PRB 81, 094410 (2010)

49. Construction of ground-state wave function of J 1 -J 2 chain Neel order! projection to single-spin space a trimer state in every triangle

50. multi-magnon instability Phase diagram in magnetic field (h>0, J 1 0, ) Vector-chiral phase Antiferro-triaticAntiferro-nematic SDW 2 SDW 3 Hikihara, Kecke, Momoi & AF PRB 78, 144404 (2008); Sudan et al. PRB 80, 140402 (2009) Nematic Nematic (IC) 1

51. J 1 -J 2 Heisenberg spin chain in magnetic field J 1 <0J 1 >0 J 2 >0 Okunishi & Tonegawa (2003); McCulloch et al. (2008); Okunishi (2008); Hikihara, Momoi, AF, Kawamura (2010)