A05: Quantum crystal and ring exchange. Novel magnetic states induced by ring exchange

Transcription

1 A05: Quantum crystal and ring exchange Novel magnetic states induced by ring exchange Members: Tsutomu Momoi (RIKEN) Kenn Kubo (Aoyama Gakuinn Univ.) Seiji Miyashita (Univ. of Tokyo) Hirokazu Tsunetsugu (ISSP, Univ. of Tokyo) Takuma Ohashi (RIKEN Osaka Univ.) Masahiro Sato (RIKEN) 1

2 Multiple-spin exchange model on the triangular lattice Mott transition in frustrated electron systems 2D solid 3 He (T. Momoi, K. Kubo) Ring exchange Spin nematic/quadrupolar phases reentrant behavior (T. Ohashi, T. Momoi, H. Tsunetsugu, N. Kawakami) S=1/2 frustrated ferromagnets spin-triplet RVB state (T. Momoi) S=1 bilinear-biqurdratic model (H. Tsunetsugu) Spin dynamics, spin crossover (S. Miyashita) Supersolid Magnetism in cold atoms (S. Miyashita) 2

3 Magnon pairing and crystallization in triangular lattice multiple-spin exchange model Tsutomu Momoi (RIKEN) Collaborators: Philippe Sindzingre (Univ. of P. & M. Curie) Kenn Kubo (Aoyama Gakuinn Univ.) Nic Shannon (Bristol Univ.) Ryuichi Shindou (RIKEN) Outline 1. Introduction: Spin nematic order BEC of bound magnon pairs Spin-triplet RVB state 2. Multiple-spin exchange model: J-J 4 -J 5 -J 6 model Spin nematic phase, 1/2 magnetization plateau 3. Summary

4 Introduction: Competition between FM and AF orders Nearest-neighbor FM interaction J 1 + competing antiferromagnetic interaction J 2 FM order AF order 0 Weak AF interaction J 2 Strong AF interaction Emergence of new quantum phase 4

5 Frustrated magnets with 1 st neighbor FM interaction Triangular lattice Square lattice 2D solid He3 Pb 2 VO(PO 4 ) 2 E. Kaul et al. (CuCl)LaNb 2 O 7, (CuBr)A 2 Nb 3 O 10 H. Kageyama et al. Ring exchange 1D zigzag lattice edge-sharing chain cuprates LiCuVO 4, LiCu 2 O 2, Rb 2 Cu 2 Mo 3 O 12, Li 2 ZrCuO 4 Kanamori-Goodenough Rule O2 Cl Cu 2+ Cu 2+ FM nearest neighbor J 1 AF next nearest neighbor J 2 5

6 Spin nematic phase in between FM and AF phases J 1 -J 2 model H = J S S + J 1 N.N. S S i j 2 i j N.N.N, Square lattice N. Shannon, TM, and P. Sindzingre, PRL 96, (2006). 1D zigzag lattice T. Hikihara, L. Kecke, TM, and A. Furusaki, PRB (2008) M. Sato, TM, and A. Furusaki, PRB (2009) Poster P40 Nematic phase FM J 1, AF J 2 FM (π,0) AF Neel AF FM nearest neighbor J 1 AF next nearest neighbor J 2 6

7 Characteristics of spin nematic order in spin-1/2 frustrated ferromagnets N. Shannon, TM, and P. Sindzingre, PRL 96, (2006). uniform state, i.e. no crystallization no spin order S i = 0 at h=0 x y or no transverse spin order Si = Si = 0 for h>0 spin liquid-like behavior gapless excitations spin quadrupolar order 2 2 Q = S S S S, Q = S S + S S x y x x y y xy x y y x ij i j i j ij i j i j Bond-nematic order 7

8 Spin nematic order can be regarded as BEC of bound magnon pairs with k =(0,0) h A. V. Chubukov, PRB (1991) N. Shannon, TM, and P. Sindzingre, PRL (2006). phase coherence S S i j = Qe 2iθ S S = S S S S i S S + S S = Qe x x y y x y y x 2iθ i j i j i j i j i j x y 2 2 xy spin quadrupolar order 8

9 Why bound magnon pairs are stable in frustrated FM? Near saturation field, 1. Individual magnons are nearly localized In square-lattice J 1 -J 2 model, zero line modes at J 2 / J 1 = ½. 2. Two (or three) magnon bound states are mobile and stable In square-lattice J 1 -J 2 model, d-wave two-magnon bound states with k =(0,0) are most favored. Coherent motion 9

10 Bond-nematic ordered state in S=1/2 magnets Roughly speaking,.. Linear combination of all possible configurations of S z = ±1 dimers dimer configuration ( ) z # of vertical S =1 dimers z 1 dimers with S = ± 1 = entangled state cf. Spin quadrupolar order state in S = 1 bilinear-biquadratic model wave function i φ product state i 2 2 Q = S S S S x y x x y y i i i i i Q = S S + S S xy x y y x i i i i i Site-nematic order 10

11 Slave boson formulation of spin nematic states in frustrated ferromagnets R. Shindou and TM, PRB (2009) Fermion representation μ 1 S α σ j = fj μ f β ( μ =,, ) 2 αβ j x y z f, f fermion operators j j Local constraint Using Hubbard-Stratonovich transformation, we can decouple FM interaction into triplet pairing, 2 t tri ij, μ 4 Si Sj Dij + [ ψi Uij, μψ jτμ ] Uij, μ = μ = xyz,, Dij, μ 0 0 D where D ij denote d-vectors of triplet pairing f f f f x y z j l j l + Δ = D ˆ jl idjl Djl jl = z x y + f f f f Djl Djl idjl j l j l ψ j f f j, j, = f f j, j, In mean-field approximation, FM interaction prefers triplet pairing. 11

12 Theoretical description of bond-nematic states When triplet pairing appears, spin space becomes anisotropic. Quadrupolar order parameter in mean-field approximation δ μν μ ν 2 Q = μν jl Djl Djl Djl + H.c. 3 2 For example, 2 2 ( ) ( ) i( ) x y x y y x j l = = = + + j j l l j l j l jl jl jl jl jl jl S S f f f f f f f f D D D D D D cf. nematic order in liquid crystals, d (r): director vectors ( ) ( ) ( ) d ( ) δ μν μ ν Q r d r d r μν r = 3 director D-vector correspondence rod

13 N. Shannon, TM and P. Sindzingre ( 06) Mean-field approximation of square lattice J 1 -J 2 model New phase triplet-pairing on FM interactions and hopping amplitude on AF interactions spin-triple resonating valence bond state (spin-triplet RVB state) NN bond (triplet pairing) Balian-Werthamer (BW) state π-flux states Nematic phase FM J 1, AF J 2 flat-band state

14 This mean-field solution has the same magnetic structure as d-wave bond nematic state. BW state d-wave bond nematic state N. Shannon, TM and P. Sindzingre ( 06) D D ( δ, + δ, ) = i x jl j l e j l e ( δ, + δ, ) y y = i y jl j l e j l e x x

15 Low energy excitations around the BW state Spin fluctuation has gapless Nambu-Goldstone modes Individual spinon excitations have a full gap Gauge fluctuation also has a gap. (a gapped Z 2 state) Perspectives Variational Monte Carlo simulation 15

16 Magnetism of two-dimensional solid 3 He on graphite 4/7 phase in 2 nd layer of 2D solid 3 He on graphite gapless spin liquid magnetization plateau at 1/2 specific heat K. Ishida, M. Morishita, H. Fukuyama, PRL (1997) linear specific heat (cf. 2D FM) double peak structure No drop of susceptibility down to 10μK R. Masutomi, Y. Karaki, and H. Ishimoto, PRL (2004). H. Nema, A. Yamaguchi, T. Hayakawa, and H. Ishimoto, PRL (2009). 16

17 Theoretical model: multiple spin exchange model Ring-exchange interactions Dirac, Roger, Hetherington, Delrieu, RMP 55, 1 (1983) Three spin exchange is dominant and ferromagnetic P P P (, i j) P ( j, k) P ( k,) i = effective two spin exchange is ferromagnetic (J=J 2-2J 3 ) Frustrated ferromagnet Parameter fitting Collin et al., PRL 86, 2447 (2001). J=-2.8, J 4 =1.4, J 5 =0.45, J 6 =1.25 (mk) 17

18 In case of two- and four-spin exchange model (J-J 4 model) In a strong J 4 regime J 4 / J = ½ At zero field, the ground state doesn t have any order and it has a large spin gap. G.Misguich, B.Bernu, C.Lhuillier, and C.Waldtmann, PRL (1998) Magnetization process has a wide plateau at m/m sat = ½, which comes from uuud spindensity wave structure TM, H. Sakamoto, and K.Kubo, PRB (1999) h/j 4 18

19 In case of two- and four-spin exchange model (J-J 4 model) Near the border of FM phase 0.24 < K/ J < 0.28 TM, P. Sindzingre, N. Shannon, PRL (2006) m > 0, condensation of 3 magnon bound states Triatic order (octupolar order) S S S ϕe S i i+ e i+ e = x i 1 2 y = S = 0 i 3iϑ m = 0, strong competition between nematic and triatic correlations J 4 / J cf. J 4 / J =0.5, J 5 / J =0.16, J 6 / J =0.44 Collin et al. 19

20 J-J 4 -J 5 -J 6 ring-exchange model We aim at giving a quantitative comparison with experiments. In the classical limit (S ) Mean-field phase diagram In the quantum case (S=1/2) One magnon excitations ( k) h 2( J2 4J4 10J5 2J6) { 3 cosk e cosk e cosk e } ε = have zero flat mode at mean-field phase boundary. Individual magnons are localized! 20

21 Magnon instability to the FM (fully polarized) state at saturation field J-J 4 -J 5 J-J 2 -J 64 4 model (case J= 2 of J 5 =J 6 =0) space rotation R π/3-1 Antiferro-triatic state space rotation R π/3 d+id wave exp( ±i 2 π / 3) Chiral (?) nematic state space rotation R π/3 1 3 sublattice structure k y Ferro-triatic state k x FM 3 magnon 1 magnon Canted AF 23

22 Numerical results Instability at saturation magnetization process ΔS z = 2 ΔS z = 3 Condensation of d+id-wave magnon pairs (BEC) 24

23 Condensation of bosons with two spices d±id-wave magnon pairs wave number k = (0,0) double-fold degeneracy with chirality j = exp ± i 2π 3 (±: chirality) density imbalance n + >n - chiral nematic order equal density n + =n - non-chiral nematic order ( 2 ) O = S S + js S + j S S Od+ id O d+ id i i+ e i i+ e i i+ e i d id 25

24 Chiral symmetry breaking? R π /3 = 1 R j j π /3 =, 2 Chiral symmetry breaking acquires double-fold degeneracy in the low-lying states. R j j π /3 =, R π /3 =, 2 1, ( σ = 1) j j 2 1, ( σ = 1) Answer: No. However, some of them are not degenerate no chiral symmetry breaking 27

25 Possible nematic orders induced by d+id-wave magnon pairs Chiral nematic state Non-chiral nematic state I Non-chiral nematic state II order parameters: Q +, Q - order parameter: Q + +Q - order parameter: Q + Q - Symmetries in low-lying states

26 Magnetization plateau at m/m sat =1/2 m/ m = 1/2 sat Symmetries are consistent with BEC of bound magnon pairs No magnon bound state SDW (uuud structure) at m/m sat =1/2 29

27 Crossover from FM interaction dominant system to AF ring exchange dominant system m/ m = 1/2 m/ m sat 1 sat FM interaction is dominant magnon bound states 4 spin ring exchange is dominant magnon crystallization SDW (uuud structure) cf. Effective two spin exchange is renormalized by J 5, J 6 J eff = J-10J 5 +2J 6 30

28 Phase diagram 36 spins 27, 28 spins still large size dependence remains too large J 6? 31

29 Another magnetization plateau? SDW at m/m sat =5/9 9-fold degeneracy Unit vectors (3, 0), (3/2, 3*sqrt{3}/2) Reciprocal vectors (2π/3, 2π/3*sqrt{3}), (0,4π/3*sqrt{3}) 36 spins 6 9 sublattice SDW particle = two magnon bound state 6 32

30 Conclusions Spin nematic phase appears in spin-1/2 frustrated ferromagnets BEC of bound magnon pairs spin-triplet RVB state Multiple-spin exchange model on the triangular lattice The 4/7 phase of solid 3 He film is in the proximity to the edge of 1/2-plateau. Non-plateau states show condensation of d+id wave magnon pairs, which leads to a non-chiral nematic phase Low magnetization region seems to support magnon pairing, but there are still large finite-size effects

31 How it looks in experiments. uniform no spin order gapless excitations magnon pairing (spin-triplet pairing) no lattice distortion no Bragg peak in Neutron scattering specific heat -- possibly double peak structure -- finite susceptibility Unusual magnon excitations in S(k,ω) h // z 1d case ω S ZZ ( k, ω) ω S + ( k, ω) Mπ Mπ π /2 0 π /2 k π /2 0 Mπ Mπ π /2 k rapid decay of NMR relaxation rate 1/T 1 P. 40 M. Sato, TM, and A. Furusaki, PRB 80, (2009) 36