Magnetization plateau and other unusual phases of a spatially - PowerPoint PPT Presentation

Magnetization plateau and other unusual phases of a spatially anisotropic quantum antiferromagnet on triangular lattice Oleg Starykh, University of Utah GGI, Florence, May 25, 2012 Tuesday, May 29, 12

Collaborators Leon Balents Hong-Chen Jiang KITP KITP Hyejin Ju, UCSB Hyejin Ju Ru Chen UCSB UCSB Tuesday, May 29, 12

Outline • motivation: Cs 2 CuBr 4 , Cs 2 CuCs 4 • classical antiferromagnet in a field: entropic selection ‣ spatial anisotropy - high-T stabilization of the plateau • Quantum model in magnetic field ‣ DMRG (3-leg ladder) ‣ Various analytical limits - large-S analysis of interacting spin waves - weakly coupled spin chains - magnetization plateau(s) and selection rules ๏ Summary Tuesday, May 29, 12

Cs 2 CuBr 4 , Cs 2 CuCl 4 ✤ S=1/2 quantum triangular antiferromagnets with small exchange ✤ complex evolution in field! Cs 2 CuCl 4 Cs 2 CuBr 4 Y. Tokiwa et al, 2006 Fortune et al , 2009 Tuesday, May 29, 12

M=1/3 magnetization plateau in Cs 2 CuBr 4 ★ Observed in Cs 2 CuBr 4 (Ono 2004, Tsuji 2007) J’/J = 0.75 but not Cs 2 CuCl 4 [ J’/J = 0.34 ] S=1/2 J’ J ★ first observation of “ up-up-down ” state in spin-1/2 triangular lattice antiferromagnet ★ and 8 more phases (instead of 2 expected)! Both materials are spatially anisotropic triangular antiferromagnets Tuesday, May 29, 12

2D bosons on the lattice • connections with interacting boson system – Superfluids (XY order) – Mott insulators – Supersolids Andreev, Lifshitz 1969 Nikuni, Shiba 1995 Heidarian, Damle 2005 Wang et al 2009 Jiang et al 2009 Tay, Motrunich 2010 Tuesday, May 29, 12

Magnetization plateau in one dimensional J 1 -J 2 chain (zig-zag ladder) Okunishi, Tonegawa JPSJ (2003) Hikihara et al PRB (2010) S=1/2 M=1/3 plateau agrees with Oshikawa, Yamanaka, Affleck argument (PRL 2007): p S (1 - M) = integer p = period, S = spin, M = magnetization: M=1/3, p=3 S=1,3/2 possible for all S Heirich-Meisner et al PRB (2007) Tuesday, May 29, 12

Studies in 2D We will see many similarities with this study Variational Monte Carlo on 2D triangular lattice Tay, Motrunich (2010) Tuesday, May 29, 12

Outline • motivation: Cs 2 CuBr 4 , Cs 2 CuCs 4 • classical antiferromagnet in a field: entropic selection ‣ spatial anisotropy - high-T stabilization of the plateau • Quantum model in magnetic field ‣ DMRG (3-leg ladder) ‣ Various analytical limits - large-S analysis of interacting spin waves - weakly coupled spin chains - magnetization plateau(s) and selection rules ๏ Summary Tuesday, May 29, 12

Classical isotropic Δ AFM in magnetic field T=0 • Zero magnetic field: spiral (120 degree) state • Magnetic field: accidental degeneracy � S i · � � � � h · � H = J S j − S i i,j i � h � 2 H = 1 � � � � 2 J S i − 3 J △ i ∈△ � h • all states with form the lowest-energy manifold S i 1 + � � S i 2 + � S i 3 = 3 J – 6 angles, 3 equations => 2 continuous angles (upto global U(1) rotation about h ) Umbrella (cone) Planar No plateau possible Tuesday, May 29, 12

Phase diagram at finite T Head, Griset, Alicea, OS 2010 Finite T: minimize F = E - T S Planar states have higher entropy! Tuesday, May 29, 12

Phase diagram of the classical model: Monte Carlo finite size effect L=120 Z 3 U(1) para Z 3 Z 3 U(1) Seabra, Momoi, Sindzingre, Shannon 2011 Gvozdikova, Melchy, Zhitomirsky 2010 12 Tuesday, May 29, 12

Effect of spatial anisotropy J’ < J: energy vs entropy Umbrella state: favored classically, J’ = 0.765 J energy gain (J-J’) 2 /J similar with Pomeranchuk effect (He3): crystal-like UUD is stabilized at high temperature 1st order transition! Low T: energetically preferred umbrella High T: entropically preferred UUD 13 Y and V are less stable. Tuesday, May 29, 12

Outline • motivation: Cs 2 CuBr 4 , Cs 2 CuCs 4 • classical antiferromagnet in a field: entropic selection ‣ spatial anisotropy - high-T stabilization of the plateau • Quantum model in magnetic field ‣ DMRG (3-leg ladder) ‣ Various analytical limits - large-S analysis of interacting spin waves - weakly coupled spin chains - magnetization plateau(s) and selection rules ๏ Summary Tuesday, May 29, 12

Model Periodic boundary conditions along y in numerical studies J’ J ✤ Hamiltonian JS x,y · S x +1 ,y + J 0 ( S x,y · S x,y +1 + S x,y · S x � 1 ,y +1 ) X H = x,y X S z − h Cs 2 CuCl 4 : J’/J = 0.34 x,y x,y Cs 2 CuBr 4 : J’/J = 0.5-0.7 Tuesday, May 29, 12

Phase diagram (3 leg ladder) 0.6 Fully polarized Width of 4 Ms/3 plateau 0.4 0.2 IC planar Ising Q ≠ (4 π /3,0) 0.0 3 0.0 0.2 0.4 0.6 0.8 1.0 Magnetic field h Ising cone Q=(4 π /3,0) C planar 2 cone SDW Ms/3 plateau Q= ( 4 π /3,0 ) C planar Ising 1 SDW IC Ising Q ≠ (4 π /3,0) 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 R=1-J’/J Tuesday, May 29, 12

Isotropic case: quantum fluctuations select co- planar states 0.6 Fully polarized Width of 4 Ms/3 plateau 0.4 isotropic case: Chubukov+Golosov, 0.2 1991; IC planar Ising Q ≠ (4 π /3,0) 0.0 3 Alicea, Chubukov, 0.0 0.2 0.4 0.6 0.8 1.0 h Magnetic field h Starykh, 2008 distorted planar Ising cone umbrella (2) C planar Q=(4 π /3,0) h UUD preserves c2 2 cone SDW U(1) symmetry. UUD plateau Ms/3 plateau Gapped spin waves h c1 Q= ( 4 π /3,0 ) distorted C planar Ising 1 planar umbrella (1) SDW IC Ising Q ≠ (4 π /3,0) ! 1 2 3 4 Interacting magnons perturbed by spatial anisotropy 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 R=1-J’/J Tuesday, May 29, 12

Isotropic case 0.6 Fully polarized Width of 4 Ms/3 plateau 0.4 Slightly different take: 0.2 Monte Carlo on IC planar generalized classical Ising Q ≠ (4 π /3,0) 0.0 3 0.0 0.2 0.4 0.6 0.8 1.0 model Magnetic field h Ising cone Q=(4 π /3,0) C planar Modeling quantum 2 cone SDW spins by classical with biquadratic interaction Ms/3 plateau Griset, Head, Alicea, Q= ( 4 π /3,0 ) Starykh (2011) C planar Ising 1 SDW IC Ising Q ≠ (4 π /3,0) 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 R=1-J’/J Tuesday, May 29, 12

Plateau phase 0.6 Fully polarized Width of M/M s =1/3 plateau: 4 Ms/3 plateau 0.4 uud state 0.2 IC planar Ising Q ≠ (4 π /3,0) 0.0 3 0.0 0.2 0.4 0.6 0.8 1.0 Magnetic field h Ising cone Q=(4 π /3,0) C planar 2 cone SDW Ms/3 plateau Q= ( 4 π /3,0 ) C planar Ising 1 SDW Ising Q ≠ (4 π /3,0) 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 R=1-J’/J Tuesday, May 29, 12

High field: condensation of spin flips 0.6 Fully polarized Width of 4 Ms/3 plateau 0.4 0.2 IC planar Ising Q ≠ (4 π /3,0) 0.0 3 0.0 0.2 0.4 0.6 0.8 1.0 Magnetic field h Ising cone Q=(4 π /3,0) C planar 2 cone SDW Ms/3 plateau Q= ( 4 π /3,0 ) C planar Ising 1 SDW IC Ising Q ≠ (4 π /3,0) 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 R=1-J’/J Tuesday, May 29, 12

Spin-flip bosons ky 1st BZ Q kx 4 /3 π π ✤ Magnons at k=Q and k=-Q are degeneracy by inversion symmetry, but Q varies smoothly with R=1-J’/J ✤ Two “Bose condensates” h S + i = ψ 1 e iQx + ψ 2 e − iQx ✤ Free energy F = − µ ( | ψ 1 | 2 + | ψ 2 | 2 ) + 1 2 Γ 1 ( | ψ 1 | 4 + | ψ 2 | 4 ) + Γ 2 | ψ 1 | 2 | ψ 2 | 2 ✤ Γ 1 > Γ 2 : | ψ 1 | = | ψ 2 | Γ 1 < Γ 2 : ψ 1 ψ 2 = 0 | Tuesday, May 29, 12

Spin-flip bosons ✤ Quadratic parameters can be computed from single magnon spectra and quartic ones from exact solution of Bethe-Saltpeter equation k + p k + q k k + q k Γ Γ + q q − p = k � − q k � − q k � k � k � − p ✤ Results: ✤ In 2d, Γ 1 > Γ 2 for all R: incommensurate planar state near saturation for S=1/2 [planar-cone transition does appear for S > 1/2] ✤ For 3-leg ladder, Γ 1 = Γ 2 for R = 0.57: transition between cone and planar states (S=1/2) Tuesday, May 29, 12

High field: spin flip bosons ψ 1 = | ψ | e i θ 1 0.6 Fully polarized Two boson (c=2) Width of 4 Ms/3 plateau 0.4 ψ 2 = | ψ | e i θ 2 theory 0.2 IC planar Ising Q ≠ (4 π /3,0) 0.0 3 0.0 0.2 0.4 0.6 0.8 1.0 Magnetic field h Ising cone C planar Q=(4 π /3,0) 2 cone SDW Ms/3 plateau Q= ( 4 π /3,0 ) C planar Ising 1 SDW IC Ising Q ≠ (4 π /3,0) 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 R=1-J’/J Tuesday, May 29, 12

High field: spin flip bosons 0.6 Fully polarized Width of 4 Ms/3 plateau 0.4 0.2 IC planar Ising Q ≠ (4 π /3,0) 0.0 3 0.0 0.2 0.4 0.6 0.8 1.0 Magnetic field h Ising cone Q=(4 π /3,0) C planar ψ 1 = | ψ | e i θ 2 cone SDW ψ 2 = 0 Ms/3 plateau c=1 phase: Q= ( 4 π /3,0 ) C planar Ising 1 incommensurate SDW IC cone Ising Q ≠ (4 π /3,0) 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 R=1-J’/J Tuesday, May 29, 12

High field: spin flip bosons 0.6 Fully polarized Commensurate- Width of 4 Ms/3 plateau 0.4 Incommensurate 0.2 Transition IC planar Ising Q ≠ (4 π /3,0) 0.0 3 0.0 0.2 0.4 0.6 0.8 1.0 Magnetic field h Ising cone Q=(4 π /3,0) C planar 2 cone SDW Ms/3 plateau Q= ( 4 π /3,0 ) C planar Ising 1 SDW IC Ising Q ≠ (4 π /3,0) 0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 R=1-J’/J Tuesday, May 29, 12