A05: Quantum cr A05: Quantum crystal and ring e ystal and ring exchange change Novel magnetic stat No el magnetic states es induced b induced by ring e ring exchange change 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
� Multiple-spin exchange model � Mott transition in frustrated on the triangular lattice electron systems 2D solid 3 He (T. Momoi, K. Kubo) Ring exchange reentrant behavior (T. Ohashi, T. Momoi, � Spin nematic/quadrupolar phases H. Tsunetsugu, N. Kawakami) S=1/2 frustrated ferromagnets � Spin dynamics, spin crossover (S. Miyashita) S=1 bilinear-biqurdratic � Supersolid model (H. Tsunetsugu) � Magnetism in cold atoms (S. Miyashita) spin-triplet RVB state (T. Momoi) 2
Magnon pairing and crystallization in triangular lattice multiple-spin exchange model Collaborators: Philippe Sindzingre (Univ. of P. & M. Curie) Tsutomu Momoi （ RIKEN ） Kenn Kubo (Aoyama Gakuinn Univ.) Nic Shannon (Bristol Univ.) Ryuichi Shindou (RIKEN) 1. Introduction: Spin nematic order Outline 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
Introduction: Competition between FM and AF orders Nearest-neighbor FM interaction J 1 + competing antiferromagnetic interaction J 2 FM order AF order 0 J 2 Weak AF interaction Strong AF interaction Emergence of new quantum phase 4
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 Cu 2+ 1D zigzag lattice edge-sharing chain cuprates Cl LiCuVO 4 , LiCu 2 O 2 , Rb 2 Cu 2 Mo 3 O 12 , Li 2 ZrCuO 4 Kanamori-Goodenough Rule O 2 − Cu 2+ FM nearest neighbor J 1 AF next nearest neighbor J 2 5
Spin nematic phase in between FM and AF phases J 1 - J 2 model N. Shannon, TM, and P. Sindzingre, ∑ ∑ = + Square lattice ⋅ ⋅ J S S J S S , H 1 i j 2 i j PRL 96 , 027213 (2006) . N.N. N.N .N 1D zigzag lattice Nematic phase ( π ,0) AF FM J 1 , AF J 2 T. Hikihara, L. Kecke, TM, and A. Furusaki, PRB (2008) M. Sato, TM, and A. Furusaki, PRB (2009) Poster P40 FM Neel AF FM nearest neighbor J 1 AF next nearest neighbor J 2 6
Characteristics of spin nematic order in spin-1/2 frustrated ferromagnets N. Shannon, TM, and P. Sindzingre, PRL 96 , 027213 (2006) . spin � uniform state, i.e. no crystallization liquid-like � � � no spin order at h =0 = behavior S 0 i or no transverse spin order for h >0 = = x y S S 0 i i � gapless excitations � spin quadrupolar order 2 − 2 = − = + Bond-nematic order x y x x y y xy x y y x Q S S S S , Q S S S S ij i j i j ij i j i j 7
Spin nematic order can be regarded as “BEC of bound magnon pairs with k =(0,0)” A. V. Chubukov, PRB (1991) h N. Shannon, TM, and P. Sindzingre, PRL (2006) . phase − = − 2 θ i S S Qe coherence i j − = − − − + = 2 θ x x y y x y y x i S S S S S S i S S S S Qe i j i j i j i j i j spin quadrupolar order xy − 2 2 x y 8
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. T wo (or three) magnon bound states are mobile and stable 2. In square-lattice J 1 - J 2 model, d -wave two-magnon bound states with k =(0,0) are most favored. Coherent motion 9
Bond-nematic ordered state in S=1/2 magnets Roughly speaking,….. Linear combination of all possible configurations of S z = ± 1 dimers ∑ ( ) z # of vertical S dimers − =1 ± z 1 dimers with S = 1 dimer configuration = -- + -- + …….. entangled state cf. Spin quadrupolar order state in S = 1 bilinear-biquadratic model − 2 2 = − x y x x y y Q S S S S ≈ ⊗ i φ wave function i i i i i Site-nematic order i = + xy x y y x Q S S S S product state i i i i i 10
Slave boson formulation of spin nematic states in frustrated ferromagnets R. Shindou and TM, PRB (2009) Fermion representation 1 μ ⎡ ⎤ = † σ μ = , , S f f x y z ( ) ⎣ ⎦ α μ β j 2 j j αβ Local constraint , fermion operators f f ↑ ↓ j j Using Hubbard-Stratonovich transformation, we can decouple FM interaction into triplet pairing ⎡ ⎤ 0 D ∑ μ , − ⋅ → − 2 + ψ † ψ τ = ⎢ ij 4 | | [ t ] tri ⎥ S S D U U , μ μ μ − ∗ , , 0 i j ij i ij j ij D ⎣ ⎦ μ = μ , , , x y z ij ⎡ ⎤ f f where D ij denote d-vectors of triplet pairing ↑ ↓ , , j j ψ = ⎢ ⎥ − j † † ⎢ f f ⎥ ⎣ ⎦ ↓ ↑ ⎛ ⎞ , , j j ⎛ ⎞ f f f f − + x y z D iD D ⎜ ↑ ↑ ↑ ↓ ⎟ ˆ j l j l Δ = = ⎜ jl jl jl ⎟ ⎜ ⎟ ⎜ ⎟ + jl z x y D D iD ⎝ ⎠ f f f f ⎝ ⎠ jl jl jl ↓ ↑ ↓ ↓ j l j l In mean-field approximation, FM interaction prefers triplet pairing. 11
Theoretical description of bond-nematic states When triplet pairing appears, spin space becomes anisotropic. Quadrupolar order parameter δ 2 μν − μν = μ ν − + 2 . Q D D D H.c in mean-field approximation jl jl jl jl 3 ( ) ( ) ( ) ∗ 2 2 − − = = = − + − + † † x y x y y x S S f f f f f f f f D D i D D D D For example, ↓ ↑ ↓ ↑ ↑ ↑ ↓ ↓ j l jl jl jl jl jl jl j j l l j l j l cf. nematic order in liquid crystals, d ( r ): director vectors δ ( ) ( ) ( ) ( ) μν μν = μ ν − 3 Q r d r d r d r rod director – D-vector correspondence
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 “flat-band” state FM J 1 , AF J 2 N. Shannon, TM and P. Sindzingre (‘06)
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) ( ) = δ − δ x D i + − , , jl j l e j l e x x ( ) = δ − δ y D i + − , , jl j l e j l e y y
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
Magnetism of tw Magnetism of two-dimensional solid o-dimensional solid 3 He on graphit 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 double peak structure (cf. 2D FM) � No drop of susceptibility down to 10 μ K H. Nema, A. Yamaguchi, T. Hayakawa, R. Masutomi, Y. Karaki, and H. Ishimoto, and H. Ishimoto, PRL (2009). PRL (2004). 16
Theoretical model: multiple ‐ spin exchange model Ring-exchange interactions Dirac, Roger, Hetherington, Delrieu, RMP 55 , 1 (1983) Three spin exchange is dominant and ferromagnetic + − = + + 1 P P P i j ( , ) P j k ( , ) P k i ( , ) 3 3 2 2 2 � effective two spin exchange is ferromagnetic ( J = J 2 -2 J 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 (m K ) 17
In case of two- and four-spin exchange model ( J - J 4 model) 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 spin- density wave structure TM, H. Sakamoto, and K.Kubo, PRB (1999) h / J 4 18
In case of two- and four-spin exchange model ( J - J 4 model) model) Near the border of FM phase 0.24 < K/|J| < 0.28 TM, P. Sindzingre, N. Shannon, PRL (2006) � m > 0, − − − 3 ϑ = ϕ i S S S e + + i i e i e condensation of 3 magnon bound states 1 2 = = � “Triatic order” (octupolar order) x y S S 0 i i � 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
J - J 4 - J 5 - J 6 ring-e ring-exchange model hange model We aim at giving a quantitative comparison with experiments. � In In the classical limit (S the classical limit (S � ∞ ) � In the In the quantum case (S=1/2) uantum case (S=1/2) Mean-field phase diagram One magnon excitations ( ) ( ) ε = − + − + k h 2 J 4 J 10 J 2 J 2 4 5 6 { } × − ⋅ − ⋅ − ⋅ 3 cos k e cos k e cos k e 1 2 3 have zero flat mode at mean-field phase boundary. Individual magnons are localized ! 20
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