Multiple spin exchange model on the triangular lattice

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1 Multiple spin exchange model on the triangular lattice Philippe Sindzingre, Condensed matter theory laboratory Univ. Pierre & Marie Curie Kenn Kubo Aoyama Gakuin Univ Tsutomu Momoi RIKEN T. Momoi, P. Sindzingre, K. Kubo Phys. Rev. Lett 108, (2012)

2 magnetism of 2D solid 3 He on graphite 4/7 phase in 2 nd layer of 3 He films on graphite gapless spin liquid magnetization M(H) heat capacity C(T) Ishida et al PRL 1997 C T cf 2D FM double peak structure plateau m=1/2 Nema et al PRL 2009 No drop of susceptibility below 10μK Matsutomi et al PRL 2004

3 Theoretical model: multiple-spin exchange model ring exchange interactions three spin exchange is dominant and ferromagnetic P 3 + P 1 3 = P 2 (i, j)+p 2 (k, l)+p 2 (k, i) effective two spin exchange (J=J 2-2J 3 ) is ferromagnetic Parameter fitting : Collin et al PRL (2001) J=-2.8, J 4 =1.4, J 5 =0.45, J 6 =1.25 (mk) frustrated ferromagnet

4 two and four spin exchange model (J-J 4 model) In a strong J 4 regime J 4 / J =1/2 at zero field the ground state does not show any order and has a large spin gap G. Misguich et al PRL 1998 magnetization process has a wide plateau at m/m sat =1/2 which comes from uuud spin density wave structure h/j 4 ring exchange interactions Momoi, Sakamoto, Kubo PRB 1999 uuud

5 two and four spin exchange model (J-J 4 model) Near the border of the FM phase 0.24<J 4 / J <0.28 m>0 condensation of 3 magnons bound states triatic (octopolar) order m=0 strong competition between nematic and triatic correlations T.Momoi, P.S., N. Shannon PRL 2006 S - i S - i+e1 S - i+e2 = φe 3i θ S x i = S y i = 0 cf J 4 / J =0.5, J 5 / J =0.16, J 6 / J =0.44 Collin et al

6 J-J 4 -J 5 -J 6 ring exchange model We aim at a quantitative comparison with experiments classical limit (S-> ) mean-filed phase diagram quantum case (S=1/2) one magnon excitation ɛ(k) =h 2(J 2 +4J 4 10J 5 +2J 6 ) {3 cos k e 1 + cos k e 2 + cos k e 3 } have zero flat mode at meanfield phase boundary individual magnons are localized

7 Magnon instability to the FM (fully polarized) state at saturation space rotation R π/3 1 antiferro triatic Instability at saturation space rotation R π/ J 2 = 2, 3J 6 =8J 5 J 2 =-2, 3J 6 =8J 5 FM 3 mag. (AF) 2 mag. quadrupolar nematic Ferro-triatic J state mag mag. (F) J 4

8 What is the nature of the quadrupolar phase? d±id : d x2-y2 ±id xy magnon pairs & c.c. j = e i2π/3 wave number k=(0,0) double-fold degeneracy with chirality Q + = i (S i S i+e1 + js i S i+e2 + j2 S i S i+e1 e2 ) e 3 e 2 Q = i (S i S i+e1 + j2 S i S i+e2 + js i S i+e1 e2 ) e 1 (±: chirality)

9 condensation of this system of two species of bosons if density imbalance n + >n - equal density n + =n - -> chiral nematic order -> non chiral nematic order Q + O U(1) = Q + Q = i 3 i (S i S i+e 1 S i S i+e 2 )

10 Exact diagonalization

11 chiral nematic state non chiral nematic state I non chiral nematic state I Q +, Q - Q + -Q - Q + -Q -

12 chiral nematic state non chiral nematic state I non chiral nematic state I Q +, Q - Q + -Q - Q + -Q - Symmetries in low-lying states

13 symmetries of the non-chiral Q + -Q - nematic state in applied field h h uniform state with k=0 Z3 lattice rotational symmetry breaking symmetry of the g.s

14 Nature of the ground-state in zero field (h=0)? exact diagonalization reveals a tower of states caracteristic of a bi-axial quadrupolar bond-nematic state with three mutually orthogonal directors order parameter : O SU(2) = i {Q xx i (e 1 )+Q yy i (e 2 )+Q zz i (e 1 e 2 )} Q αα i (r) =S α i S α i+r 1 3 S i S i+r

15 Short tour over the Anderson tower of quasi degenerate joint states (QDJS) uniaxial (collinear) Néel state uniaxial nematic biaxial (non-collinear) Néel state -> biaxial nematic

16 collinear Néel order (π,π) e.g square lattice : order parameter M staggered is a vector effective Hamiltonian -> quantum rotator Pisa tower: one state in every spin sector H = S2 2Nχ slope spin-gap ->0 as 1/N

17 uniaxial nematic as found in the J₁-J₂-K model (Shannon, Momoi, PS, PRL 2006) order parameter is a symmetric traceless rank-2 tensor Q αβ i (r) = 1 2 ( ) Si α S β i+r + Sβ i Sα i+r δ αβ 3 S i S i+r (headless) director no odd S spin states -> one state in every even spin sector N=36 J 1 =-1 J₂=0.6 J₃=0.2 gaps J 1 =-1 J₂=0.4 J₃=0

18 non-collinear Néel order: e.g. orthogonal Néel order parameter is a trihedron effective Hamiltonian -> quantum top H = S2 2Nχ + S2 3 ( ) 1 2Nχ 1 2Nχ 3 S! ! pr! 3 " > tower with (2S+1) states in every spin sector (a) (b)

19 present case : orthogonal (bi-axial) nematic 3 orthogonal directors z S order parameter SU(2)/D 2 (D 2 : dihedral group) -> states of the quantum top odd under reflexion do not enter into the tower -> S/2+1 states for even S, (S-1) states if S odd N=36 S Irreps Γ 2 Γ 3 Γ 1 Γ 2, Γ 3 Γ 3 Γ 1, Γ 2, Γ 3 Γ 1 (k =0,R 2π/3 =1, R π =1, σ = 1) Γ 2 (k =0,R 2π/3 =1, R π =1, σ = 1) Γ 3 (k =0,R 2π/3 = j, j 2, R π = 1) on the triangular lattice x y similar to the orthogonal site nematic Lauchli, Mila, Penc PRL 2006

20 orthogonal nematic (h=0)-> non-chiral nematic (h>0) two directors become to applied field (a) (b) (h>0) (a) h=0 SU(2) symmetric case S Irreps Γ 2 Γ 3 Γ 1 Γ 2, Γ 3 Γ 3 Γ 1, Γ 2, Γ 3 (b) h>0 U(1) symmetric case S (m) m s 2 m s 4n m s 2(2n + 1) Irreps Γ 3 Γ 1, Γ 3 Γ 2, Γ 3

21 magnetization plateau m=1/2 plateau disappear close to the FM phase 27,28spins uuud plateau non plateau FM non plateau J FM uuud plateau exp J 4 1 J= 2, J 4 = J= 2, J 4 =0.8 J 5 =J 6 /2=0.15 J 5 =J 6 /2= m/m s uuud state 0.2 (a) 0.2 (b) h h

22 Summary condensation of d+id magnon pairs gives rise to antiferro (orthognal) nematic order 5 and 6 spins exchange are important to understand the physics of solid 3 He film the 4/7 phase of solid 3 He film is in the proximity to the edge of the m=1/2 magetization plateau nematic states may appear in S=1/2 magnets with competing AF & FM interactions Thank for your attention

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