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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

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

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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)

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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

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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

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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 )

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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)

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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.

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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

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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

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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)

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

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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

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

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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

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

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

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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.)

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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

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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

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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

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Numerics (DMRG) Staggered part of the dimer operators

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

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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:

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

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J 1 -J 2 spin chain with antiferromagnetic J 1

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

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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 ……..

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

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

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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)

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J 1 -J 2 spin chain with ferromagnetic J 1

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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

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

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

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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

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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

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

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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

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

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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)

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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)

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

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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.

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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

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Dimer order parameter Different dimer order dimer 123

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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)

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

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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

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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

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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)

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