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Competing phases of XXZ spin chain with frustration

Competing phases of XXZ spin chain with frustration. Akira Furusaki (RIKEN). Collaborators: Shunsuke Furukawa (U. Tokyo) Toshiya Hikihara (Gunma U.) Shigeki Onoda (RIKEN) Masahiro Sato (Aoyama Gakuin U.). Thanks: Sergei Lukyanov (Rutgers). Outline.

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Competing phases of XXZ spin chain with frustration

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  1. Competing phases of XXZ spin chain with frustration Akira Furusaki (RIKEN) 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 • Introduction: frustrated spin-1/2 J1-J2 XXZ chain • XXZ chain (J2=0): review of bosonization approach • Phase diagram of J1-J2 XXZ spin chain • J1>0 (antiferromagnetic) • J2<0 (ferromagnetic)

  4. frustrated spin-1/2 J1-J2 chain J2> 0(AF) J1 If J2 is antiferromagnetic, spins are frustrated regardless of the sign of J1. J1-J2 spin chain is the simplest spin model with frustration.

  5. a b c Materials: Quasi-1D cuprates (multi-ferroics) Cu2+ spin S=1/2 NaCu2O2 LiCu2O2 PbCuSO4(OH)2 LiCuVO4 Cu (3dx2-y2) J1 J2 O(2px) Rb2Cu2Mo3O12

  6. Multiferroicity Observation of chiral ordering through electric polarization P J2 >0 a J1 <0 Cu c b O Seki et al., PRL, 2008 (LiCu2O2) Quasi-1D spin-1/2 frustrated magnets with ferro J1 -4 -3 -2 -1 0 Li2ZrCuO4 LiCu2O2 LiCuVO4 Drechsler et al. PRL,2007 Masuda et al. PRL,2004; PRB,2005 Enderle et al. Europhys.Lett.,2005 • CuO2 chain: edge-sharing network ferromagnetic J1 (Kanamori-Goodenough rule)

  7. Model Frustrated spin-1/2 J1-J2 XXZ chain y J2 (>0, antiferro) easy-plane anisotropy Frustration occurs when J2>0, irrespective of the sign of J1. x J1 Classical ground state The phase diagram is symmetric. J1>0 and J1<0 are equivalent under 4 -4 0 Antiferro Spin spiral Ferro pitch angle (applies only in the classical case) finite chirality

  8. Quantum case S=1/2 • Classical spiral (chiral) order is destroyed by strong quantum fluctuations in 1D. • Antiferromagnetic case (J1>0,J2>0) is well understood. - Singlet dimer order is stabilized (J2/J1>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

  9. Previous study of spin-1/2 J1-J2 chain Ground-state phase diagram for AF-J1case K. Okamoto and K. Nomura, Phys. Lett. A (1992). T. Hikihara, M. Kaburagi, and H. Kawamura, PRB (2001), etc. J1 chain Two decoupled J2 chains Majumdar-Ghosh line

  10. Quantum case S=1/2 • Classical spiral (chiral) order is destroyed by strong quantum fluctuations in 1D. • Antiferromagnetic case (J1>0,J2>0) is well understood. - Singlet dimer order is stabilized (J2/J1>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 • The ferromagnetic-J1 case (J1<0,J2>0) is less understood. Goal: to determine the ground-state phase diagram

  11. Our strategy • perturbative RG analysis around J1=0 or J2=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 reviewmostly standard textbook material, plus some relatively new developments

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

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

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

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

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

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

  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 Open boundary condition Dirichlet b.c. for boson field : 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) 1st 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. L spins lowest energy of a finite open chain with • Boundary energy of open XXZ chain AF & T. Hikihara, PRB 69, 094429 (2004)

  26. J1-J2 spin chain with antiferromagnetic J1

  27. Ground-state phase diagram for AF-J1case J1 chain Two decoupled J2 chains Majumdar-Ghosh line

  28. dimer phase Haldane ‘82 White & Affleck ’96 …….. J2>0 changes and scaling dimension of If is relevant and , then is pinned at . If is relevant and , then is pinned at . dimer LRO Neel LRO

  29. Ground-state phase diagram for AF-J1case J1 chain Two decoupled J2 chains Majumdar-Ghosh line

  30. Perturbation around J1=0 vector chiral order dimer order Two decoupled J2 chains

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

  32. J1-J2 spin chain withferromagnetic J1

  33. Phase diagram & chiral order parameter ferromagnetic J1 antiferromagnetic J1 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 J1=0 vector chiral order dimer order dimension Two decoupled J2 chains

  35. Phase diagram & chiral order parameter ferromagnetic J1 antiferromagnetic J1

  36. Ground-state phase diagram for Ferro-J1 case PbCuSO4(OH)2 Li2ZrCuO4 NaCu2O2 Rb2Cu2Mo3O12 LiCu2O2 LiCuVO4 J1 chain Two decoupled J2 chains

  37. Sine-Gordon model for spin-1/2 J1-J2 XXZ chain with ferromagnetic coupling J1 J1 chain We begin with the J2=0 limit. Ferromagnetic Easy-plane Ferromagnetic SU(2) Heisenberg Effective Hamiltonian (sine-Gordon model) TL-liquid (free-boson) part irrelevant perturbation velocity TL-liquid parameter

  38. Spin and dimer operators If the cosine term becomes relevant, then Neel order dimer order BKT-type RG equation J1 chain

  39. Exact coupling constant in the J1 chain (J2=0) S. Lukyanov, Nucl. Phys. B (1998). 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 J1 chain “Dimer” gap “Neel” gap

  40. We can check the position of l=0 from numerical-diagonalization result. J2=0

  41. This relation is stable against perturbations conserving symmetries. Generally the exact value of l is not known in the presence of such perturbations (J2). However, the position of l=0 is determined by the equation which can be numerically evaluated. J2 perturbation makes the l term relevant . Neel and dimer phases are expected to emerge. Neel order Gaussian phase transition point (c=1) dimer order

  42. Phase diagram and Neel/dimer order parameters Ground-state phase diagram of easy-plane anisotropic J1-J2 chain Furukawa, Sato & AF PRB 81, 094410 (2010) Neel l>0 dimer l<0 Neel l>0 Curves of l=0 dimer l<0 Irrelevant relevant J1 chain

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

  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-rangebehavior is different from that of the standard Neel order.

  45. Dimer phase AF-J1 case FM-J1case Neel dimer phase in the AF-J1 region dimer phase in the FM-J1 region On the XY line (D=0) J1-J2 XY chain with FM J1 J1-J2 XY chain with AF J1 p rotation at every even site “triplet” dimer “singlet” dimer

  46. Dimer order parameter 1 2 3 dimer Different dimer order dimer

  47. Sato, Furukawa, Onoda & AF Mod. Phys. Lett. 25, 901 (2011) dimer order parameter string order parameter 1 2 3 weak dimer order zoom long-range string order

  48. Summary ferromagnetic J1 antiferromagnetic J1 Furukawa, Sato & AF PRB 81, 094410 (2010) Sato, Furukawa, Onoda & AF Mod. Phys. Lett. 25, 901 (2011)

  49. Construction of ground-state wave function of J1-J2 chain a trimer state in every triangle projection to single-spin space Neel order!

  50. multi-magnon instability Phase diagram in magnetic field (h>0, J1<0, J2>0, ) Hikihara, Kecke, Momoi & AF PRB 78, 144404 (2008); Sudan et al. PRB 80, 140402 (2009) Antiferro-triatic Antiferro-nematic Nematic SDW2 SDW3 SDW2 SDW3 1 Vector-chiral phase Nematic (IC)

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