Spatially anisotropic triangular antiferromagnet in magnetic field

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1 Spatially anisotropic triangular antiferromagnet in magnetic field Oleg Starykh, University of Utah Leon Balents, KITP Hosho Katsura, KITP Jason Alicea, Caltech Andrey Chubukov, U Wisconsin Christian Griset Shane Head EQUAM10, MPIPKS, Dresden, August

2 Spatially anisotropic triangular antiferromagnet in magnetic field Magnetization plateau in triangular lattice antiferromagnets Oleg Starykh, University of Utah Leon Balents, KITP Hosho Katsura, KITP Jason Alicea, Caltech Andrey Chubukov, U Wisconsin Christian Griset Shane Head EQUAM10, MPIPKS, Dresden, August

3 Outline experimental (Cs2CuBr4) and theoretical motivations classical antiferromagnet in a field: entropic selection spatial anisotropy - high-t stabilization of the plateau Quantum spins: zero-point fluctuations Large-S analysis of interacting spin waves Approach from one dimension sequence of plateaux and selection rules (attempt at) Unification

4 Experiment: M=1/3 magnetization plateau in Cs2CuBr4 Observed in Cs 2 CuBr 4 (Ono 2004, Tsuji 2007) J /J = 0.75 but not Cs 2 CuCl4 [J /J = 0.34] J S=1/2 J UUD (up-up-down) structure -- down-spins at the centers of hexagons; commensurate structure -- one down spin per every triangle first observation of up-up-down state in spin-1/2 triangular lattice antiferromagnet

5 Progress in one dimensional J1-J2 chain (zig-zag ladder) S=1/2 Okunishi, Tonegawa JPSJ (2003) Hikihara et al PRB (2010) M=1/3 plateau plateau is centered around J2 = J1/2 point for S > 1/2; semi-classical spin wave expansion is possible there (OS 2009) agrees with Oshikawa, Yamanaka, Affleck argument (PRL 2007): p S (1 - M) = integer S=1,3/2 p = period, S = spin, M = magnetization: M=1/3, p=3 possible for all S Heirich-Meisner et al PRB (2007) also seen in frustrated spin ladders, Michaud et al PRB 2010

6 D = 2 surprisingly complex phase diagram of spatially anisotropic triangular lattice antiferromagnet no definite conclusions from numerical studies yet... connections with interacting boson system Superfluids Mott insulators Supersolids Nikuni, Shiba 1995 Heidarian, Damle 2005 Wang et al 2009 Jiang et al 2009 Tay, Motrunich 2010

7 Outline experimental (Cs2CuBr4) and theoretical motivations classical antiferromagnet in a field: entropic selection spatial anisotropy - high-t stabilization of the plateau Quantum spins: zero-point fluctuations Large-S analysis of interacting spin waves Approach from one dimension sequence of plateaux and selection rules (attempt at) Unification

8 Classical isotropic Δ AFM in magnetic field No field: spiral (120 degree) state Magnetic field: accidental degeneracy H = J i,j S i S j i h Si H = 1 2 J ( i S i h 3J ) 2 S i1 + S i2 + S i3 = h 3J all states with form the lowest-energy manifold 6 angles, 3 equations => 2 continuous angles (upto global U(1) rotation about h) Planar Umbrella (cone) No plateau possible plateau is possible at collinear point

9 Phase diagram Head, Griset, Alicea, OS 2010

10 Thermal Order-from-Disorder mechanism RbFe(MoO4)2: S=5/2 Fe 3+ Svistov et al PRB (2003) Smirnov et al PRB (2007) plateau plateau is signaled by depression in dm/dh Two antiferromagnetically coupled layers Plateau width increases with T

11 Persistence of plateau in triangular structures cuboctahedron icosidodecahedron Mo 72 Fe 30 S=5/2 Heisenberg (kagome on a sphere) Schroder et al, PRL (2005) Rousochatzakis et al, PRB(2008)

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13 M dm/dh crossing UUD state at finite T

14 First order transition chirality Cv

15 Phase diagram of the classical anisotropic model Griset, Head, Alicea, OS 2010 Dual role of thermal fluctuations: promotes high-entropy states; destabilizes order

16 Outline experimental (Cs2CuBr4) and theoretical motivations classical antiferromagnet in a field: entropic selection spatial anisotropy - high-t stabilization of the plateau Quantum spins: zero-point fluctuations Large-S analysis of interacting spin waves Approach from one dimension sequence of plateaux and selection rules (attempt at) Unification

17 Order-from-Disorder via Quantum fluctuations fluctuation spectra of different spin structures are different: E = Eclass + ΔE sw quantum fluctuations prefer planar arrangement prefer collinear configuration even more, when possible: state with maximum number of soft modes wins. plateau is a quantum effect, width δh = 1.8 J/(2S) (hsaturation = 9 J) Chubukov, Golosov (1991) finite S effect E sw planar = S 2 plateau is the effect of interactions (hence, width ~ 1/S) between spin waves k ω planar (k) < E sw umbrella = S 2 ω umbrella (k) k Tsuji et al (2007) NMR spectra: Fujii et al (2004) Cs 2 CuBr 4 δh T-independent width The problem: spatial anisotropy stabilizes umbrella

18 Spatially anisotropic model: classical analysis fails H = J ij S S h i j ij i S z i J J J J 0 h sat h 0 h sat h 1/3-plateau Umbrella state: favored classically; energy gain (J-J ) 2 /J Planar states: favored by quantum fluctuations; energy gain J/S The competition is controlled by δ = S(J J ) 2 /J 2 dimensionless parameter Technical formulation: spatial anisotropy J-J causes softening of interacting (including 1/S correction) spin waves

19 Our semiclassical approach: treat spatial anisotropy (J-J ) as a perturbation to interacting spin waves single dimensionless parameter δ = S(1 - J /J) 2 : fully polarized state h h c2 planar distorted umbrella (2) BEC k = 0 h c1 commensurate planar UUD plateau zero-field spiral distorted umbrella (1) Alicea, Chubukov, Starykh PRL 102, (2009) incommensurate 2 low-energy gapped modes BEC k ! Cs 2 CuBr 4 Cs 2 CuCl 4 ~ S(1 - J /J) 2

20 More detailed phase diagram fully polarized state Exact dilute boson calculation (OS 2010) V incommensurate: fan h V commensurate planar distorted umbrella (2) h c2 BEC k = 0 h c1 commensurate planar incommensurate UUD plateau 2 low-energy gapped modes zero-field spiral distorted umbrella (1) Alicea, Chubukov, Starykh PRL 102, (2009) BEC k ! Cs 2 CuBr 4 Cs 2 CuCl 4 ~ S(1 - J /J) 2

21 Exact dilute boson calculation incomm. planar h h c2 commens planar distorted umbrella (2) h c1 planar incommen UUD plateau 2 low-energy distorted umbrella (1) BEC k ! ~ S(1 - J /J) 2 Variational wave function calculation Tay, Motrunich PRB 2010

22 Outline experimental (Cs2CuBr4) and theoretical motivations classical antiferromagnet in a field: entropic selection spatial anisotropy - high-t stabilization of the plateau Quantum spins: zero-point fluctuations Large-S analysis of interacting spin waves Approach from one dimension sequence of plateaux and selection rules (attempt at) Unification

23 Heisenberg spin chain via free Dirac fermions Spin-1/2 AFM chain = half-filled (1 electron per site, k F =π/2a ) fermion chain Spin-charge separation q=0 fluctuations: right- and left- spin currents 2k F (= π/a) fluctuations: charge density wave ε, spin density wave N Staggered Magnetization N Spin flip ΔS=1 -k F k F Susceptibility 1/q 1/q Staggered Dimerization ε = (-1) x S x S x+a ΔS=0 -k F k F 1/q Must be careful: often spin-charge separation must be enforced by hand

24 S=1/2 AFM Chain in a Field Field-split Fermi momenta: Uniform magnetization Half-filled condition S z component (ΔS=0) peaked at scaling dimension increases 1 Affleck and Oshikawa, 1999 S x,y components (ΔS=1) remain at π scaling dimension decreases 1/2 0 h/h sat 1/2 M 1 Derived for free electrons but correct always - Luttinger Theorem h sat =2J 0 h/h sat 1 XY AF correlations grow with h and remain commensurate Ising SDW correlations decrease with h and shift from π

25 Weakly coupled Heisenberg chains in magnetic field non - frustrated inter-chain coupling J S r S r N x r N x r + N y r N y r + N z r N z r most relevant less relevant 2πR 2 < 1/(2πR 2 ) spins order in the plane perpendicular to the direction of magnetic field (z): umbrella / cone / spin-flop states frustrated inter-chain coupling y+1 y S x,y ( S x,y+1 + S x+1,y+1 ) N x y x N x y+1+n y y x N y y+1 +sin(δ)sz π 2δ(y)S z π+2δ(y+1) less relevant 1+2πR 2 > 1/(2πR 2 ) frustration promotes collinear SDW order most relevant (small to intermediate fields)

26 Ideal J-J model in magnetic field OS, Balents 2007 Two important couplings for h>0 Quantum phase transition between SDW and Cone states Magnetic field relieves frustration! k F k F =2δ =2πM dim 1/2πR 2 : 1 -> 2 collinear SDW dim 1+2πR 2 : 2 -> 3/2 spiral cone state Critical point : 1+2πR 2 = 1/2πR 2 gives at M = T c cone sdw M 1/2 also: Kolezhuk, Vekua 2005 h/h sat

27 J-J model: magnetization plateaux via commensurate locking of SDW Collinear SDW state locks to the lattice at low-t - irrelevant (1d) umklapp terms become relevant once SDW order is present (when commensurate): multiparticle umklapp scattering -strongest locking is at M=1/3 M sat Observed in Cs2CuBr 4 (Ono 2004, Tsuji 07, Fortune 09) down-spins at the centers of hexagons T collinear SDW cone polarized uud h/h sat ( n Ψ R L) Ψ (π 2δ)n =2πm 2M =1 2m/n Cs2CuBr4 Fortune et al /3 2/3 n m M 1/3 1/2 3/5 1/5 2/3 naively thinking

28 Plateau more carefully Umklapp must respect triangular lattice symmetries translation along chain direction translation along diagonal spatial inversion H (n) umk = y Z n-th plateau width (in field) and OS, Katsura, Balents PRB 2010 M (n,m) = 1 1 2m 2 n φ y (x) φ y (x + 1) R(π 2δ) φ y (x) φ y+1 (x + 1/2) R(π 2δ)/2 φ y (x) πr φ y ( x) dx t n cos[ n R φ y] n = m (mod 2) width J /J n 2 /(4(4πR 2 1)) same parity condition n m M 1/3 1/2 3/5 1/5 2/3 large n leads to exponential suppression 1/3-plateau is most prominent, 3/5 is possible (if falls within the SDW region). Exponentially weak 1/2- and 2/3- plateaux, if any!

29 J << J limit collinear SDW cone polarized uud h/h sat

30 J << J limit collinear SDW cone polarized uud h/h sat h/h sat /5 1/3 collinear SDW cone polarized J /J = 1 J /J = 0

31 J << J limit to J = J point... h h c2 h c1 planar comm. planar UUD plateau 2 low-energy incomm. planar distorted umbrella (1) BEC k 0 distorted umbrella (2)???? cone longit. sdw h/h sat /5 1/3 collinear SDW cone polarized J /J = ! 0 J /J = 0

32 Global phase diagram Hypothesis: 1/3 plateau extends for all 0 < J /J < 1; other magnetization plateaux terminate above some critical J /J ratio. polarized h h c2 h c1 planar comm. planar UUD plateau incomm. planar distorted umbrella (1) distorted umbrella (2) cone = umbrella longitudinal sdw h/h sat /5 1/3 cone collinear SDW J /J = ! CAF 0 J /J = 0

33 Experimental relevance polarized h h c2 h c1 planar comm. planar UUD plateau incomm. planar distorted umbrella (1) distorted umbrella (2) cone = umbrella longitudinal sdw h/h sat /5 1/3 cone collinear SDW J /J = ! Cs2CuBr4 plateau - yes CAF 0 J /J = 0 inter-layer exchange J /J Cs2CuCl4 no plateau large J /J favors classical cone order!

34 Conclusions Magnetization plateau persists for all Jʼ/J semiclassical interacting spin waves near J - Jʼ << J 1d scaling + symmetry arguments near Jʼ << J Many other interesting phases Longitudinal SDW commensurate-incommensurate transitions Many open questions, excellent problem for numerical studies