Heisenberg-Kitaev physics in magnetic fields

Transcription

1 Heisenberg-Kitaev physics in magnetic fields Lukas Janssen & Eric Andrade, Matthias Vojta L.J., E. Andrade, and M. Vojta, Phys. Rev. Lett. 117, (2016) L.J., E. Andrade, and M. Vojta, Phys. Rev. B 96, 0630 (2017) A. Wolter, L. Corredor, L.J., et al., Phys. Rev. B 96, 0105(R) (2017) SFB 113

2 α-rucl3 in field: [Sears et al., PRB(R) 17] [Baek et al., PRL 17] [Johnson et al., PRB 15] [Wolter et al., PRB(R) 17] T N (K) (K) Smag( J mol K ) [Banerjee et al., NPJQM 18] T (K) µ 0 H (T) Questions: Reason for anisotropy? Nature of field-induced phases? g factors or intrinsic? quantum paramagnet/topological spin liquid? Appropriate spin model? sign of K1, role of J3, Γ1?

3 <latexit sha1_base6="oj31nb3shz07glefsoqs3fiqlg=">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</latexit> α-rucl3 in zero field: Zigzag antiferromagnet Neutron diffraction: [Johnson et al., PRB 15] +~m stagg ~m stagg y +~e z +~m stagg x z Moment direction: [Cao et al., PRB 16] or ~S k ±[001] 2 ac ; ](~S;a) ' 35 ~S k ±[xxz] 2 ac ; ](~S;a) ' +35

4 Mechanisms to stabilize zigzag Extended Heisenberg-Kitaev models: h i J 1 ~S i ~S j +K 1 S i S j +Γ 1 (S i S j +S i S j ) H = X hiji + X hhijii J 2 ~ Si ~S j +K 2 S i S j + X hhhijiii J 3 ~ Si ~S j ~ h X i ~S i neglect trigonal distortion J 1 K 1 Ŵ 1 J 2 K 2 J 3 Set Material (mev) (mev) (mev) (mev) (mev) (mev) Method Ref. Year 1 α-rucl Fittoneutron scattering [35,36] Na 2 IrO Fit to susceptibility and neutron scattering [30] Ŵ α-rucl DFT + t/u expansion [] Na 2 IrO DFT+ exact diagonalization [32] Ŵ Na 2 IrO DFT+ t/u expansion, direction of moments [0,5] 2016 (2 + Ŵ) Na 2 IrO MRCI,fittoθ CW [7] 201 (2 + Ŵ) α-rucl MRCI, fit to magnetization [13] /3 α-rucl DFT+ exact diagonalization [32] α-rucl Fit to neutron scattering [33] α-rucl DFT+ t/u expansion [3] α-rucl DFT+ t/u expansion [38] J 3 α-rucl Fit to neutron scattering [39] 2017 [LJ, Andrade, Vojta, PRB 17]

5 Mechanisms to stabilize zigzag J 1 K 1 Ŵ 1 J 2 K 2 J 3 Set Material (mev) (mev) (mev) (mev) (mev) (mev) Method Ref. Year 1 α-rucl Fittoneutron scattering [35,36] Na 2 IrO Fit to susceptibility and neutron scattering [30] Ŵ α-rucl DFT + t/u expansion [] Na 2 IrO DFT+ exact diagonalization [32] Ŵ Na 2 IrO DFT+ t/u expansion, direction of moments [0,5] 2016 (2 + Ŵ) Na 2 IrO MRCI,fittoθ CW [7] 201 (2 + Ŵ) α-rucl MRCI, fit to magnetization [13] /3 α-rucl DFT+ exact diagonalization [32] α-rucl Fit to neutron scattering [33] α-rucl DFT+ t/u expansion [3] α-rucl DFT+ t/u expansion [38] J 3 α-rucl Fit to neutron scattering [39] 2017 Three scenarios: (1) Antiferromagnetic K1, ferromagnetic J1 [Chaloupka et al., PRL 13]

6 Mechanisms to stabilize zigzag J 1 K 1 Ŵ 1 J 2 K 2 J 3 Set Material (mev) (mev) (mev) (mev) (mev) (mev) Method Ref. Year 1 α-rucl Fittoneutron scattering [35,36] Na 2 IrO Fit to susceptibility and neutron scattering [30] Ŵ α-rucl DFT + t/u expansion [] Na 2 IrO DFT+ exact diagonalization [32] Ŵ Na 2 IrO DFT+ t/u expansion, direction of moments [0,5] 2016 (2 + Ŵ) Na 2 IrO MRCI,fittoθ CW [7] 201 (2 + Ŵ) α-rucl MRCI, fit to magnetization [13] /3 α-rucl DFT+ exact diagonalization [32] α-rucl Fit to neutron scattering [33] α-rucl DFT+ t/u expansion [3] α-rucl DFT+ t/u expansion [38] J 3 α-rucl Fit to neutron scattering [39] 2017 Three scenarios: (1) Antiferromagnetic K1, ferromagnetic J1 [Chaloupka et al., PRL 13] (2) Ferromagnetic K1, antiferromagnetic J3 [Winter et al., PRB 16] [Katukuri et al., NJP 1; ]

7 Mechanisms to stabilize zigzag J 1 K 1 Ŵ 1 J 2 K 2 J 3 Set Material (mev) (mev) (mev) (mev) (mev) (mev) Method Ref. Year 1 α-rucl Fittoneutron scattering [35,36] Na 2 IrO Fit to susceptibility and neutron scattering [30] Ŵ α-rucl DFT + t/u expansion [] Na 2 IrO DFT+ exact diagonalization [32] Ŵ Na 2 IrO DFT+ t/u expansion, direction of moments [0,5] 2016 (2 + Ŵ) Na 2 IrO MRCI,fittoθ CW [7] 201 (2 + Ŵ) α-rucl MRCI, fit to magnetization [13] /3 α-rucl DFT+ exact diagonalization [32] α-rucl Fit to neutron scattering [33] α-rucl DFT+ t/u expansion [3] α-rucl 3 S DFT+ t/u expansion [38] J 3 α-rucl Fit to neutron scattering [39] 2017 Three scenarios: (1) Antiferromagnetic K1, ferromagnetic J1 [Chaloupka et al., PRL 13] (2) Ferromagnetic K1, antiferromagnetic J3 [Winter et al., PRB 16] (3) Ferromagnetic K1, positive Γ1 [Rau et al., PRL 1; Ran et al., PRL 17] we will show that finite J 1 < 0 is also needed [Katukuri et al., NJP 1; ]

8 Mechanisms to stabilize zigzag J 1 K 1 Ŵ 1 J 2 K 2 J 3 Set Material (mev) (mev) (mev) (mev) (mev) (mev) Method Ref. Year 1 α-rucl Fittoneutron scattering [35,36] Na 2 IrO Fit to susceptibility and neutron scattering [30] Ŵ α-rucl DFT + t/u expansion [] Na 2 IrO DFT+ exact diagonalization [32] Ŵ Na 2 IrO DFT+ t/u expansion, direction of moments [0,5] 2016 (2 + Ŵ) Na 2 IrO MRCI,fittoθ CW [7] 201 (2 + Ŵ) α-rucl MRCI, fit to magnetization [13] /3 α-rucl DFT+ exact diagonalization [32] α-rucl Fit to neutron scattering [33] α-rucl DFT+ t/u expansion [3] α-rucl DFT+ t/u expansion [38] J 3 α-rucl Fit to neutron scattering [39] 2017 Three scenarios: (1) Antiferromagnetic K1, ferromagnetic J1 [Chaloupka et al., PRL 13] (2) Ferromagnetic K1, antiferromagnetic J3 [Winter et al., PRB 16] (3) Ferromagnetic K1, positive Γ1 [Rau et al., PRL 1; Ran et al., PRL 17] we will show that finite J 1 < 0 is also needed [Katukuri et al., NJP 1; ] This talk: Magnetic-field behavior allows to distinguish scenarios!

9 Warm-up: SU(2) Heisenberg antiferromagnet in field Minimal model: H = X hiji J 1 ~S i ~S j ~h X ~S i i ~h ~h =0

10 Warm-up: SU(2) Heisenberg antiferromagnet in field Minimal model: H = X hiji J 1 ~S i ~S j ~h X ~S i i ~h ~h =0

11 Warm-up: SU(2) Heisenberg antiferromagnet in field Minimal model: H = X hiji J 1 ~S i ~S j ~h X ~S i i ~h 0 < ~h J 1 S ~S i ~h (largest susceptibility)

12 Warm-up: SU(2) Heisenberg antiferromagnet in field Minimal model: H = X hiji J 1 ~S i ~S j ~h X ~S i i ~h 0 < ~h < 6J 1 S homogenous canting h ~m i =S ~h =(6J 1 S)

13 Warm-up: SU(2) Heisenberg antiferromagnet in field Minimal model: H = X hiji J 1 ~S i ~S j ~h X ~S i i ~h 6J 1 S ~h fully polarized h~m i =S =1 since one-magnon state eigenstate of H h ~m i =S ~h =(6J 1 S)

14 α-rucl3: Field directions

15 Scenario 1: Antiferromagnetic K1, ferromagnetic J1 Minimal model: H = X hiji h J 1 ~S i ~S j +K 1 S i S j [Chaloupka et al., PRL 10; PRL 13] i Zero field: cubic-axes zigzag ~S k ±~e z along Ru-Cl bonds consistent with one of the two options in α-rucl 3 : [Cao et al., PRB 16]

16 Scenario 1 (K1 > 0, J1 < 0) in field: ~h k [ 110] k b Spin-wave spectrum in high-field phase: 3 Set 1 ~ h k [ 110] k b h = h + c K M 2 K M 1 M 3 K Γ K along Ru-Ru bonds ε~q /(AS) K Γ M 1 K M 2 K M 3 K (J 1, K 1 ) = (-.6, +7.0) mev [Banerjee et al., Nat. Mat. 16] gap closes precisely at z zigzag ordering wavevector ~ Q = M 2 consistent with a direct continuous transition to canted zigzag

17 Scenario 1 (K1 > 0, J1 < 0) in field: ~h k [ 110] k b Magnetization curve: M1 along Ru-Ru bonds Γ M2 Γ 1.0 M3 ~m ĥ/s M1 M2 M uniform h ~ Sik ~ h 0.2 cubic-axis z-zigzag h/ K 1 S Set 1 (K 1 -J 1 ) ~ h k [ 110] k b MC (S ) anal.(s ) SW (S =1/2) 2 (J 1, K 1 ) = (-.6, +7.0) mev but: high-field state not fully polarized: h~m i =S < 1

18 Scenario 1 (K1 > 0, J1 < 0) in field: ~h k [ 110] k b Classical phase diagram in [1 10] field: h/(as) canted Néel π π 2 canted zigzag 3π polarized ' = ' = 1.85 ' =2 5π 3π 2 canted stripy Néel zigzag FM stripy Néel 7π ~ h [ 110] canted Néel J 1 = Acos K 1 =2Asin

19 Scenario 1 (K1 > 0, J1 < 0) in field: ~h k [111] k c Zero field: ~S i along cubic axes simple canting impossible! cubic-axes zigzag canted zigzag will compete with other states Spin-wave spectrum in high-field phase: ε~q/(as) ~ h k [111] ' =0.62 h =2.61AS M 1 K K Γ magnon gap closes at K points! 0.0 K Γ M K different from zigzag ordering wavevector

20 Scenario 1 (K1 > 0, J1 < 0) in field: ~h k [111] k c Zero field: ~S i along cubic axes simple canting impossible! cubic-axes zigzag canted zigzag will compete with other states Spin-wave spectrum in high-field phase: ε~q/(as) ~ h k [111] ' =0.62 h =2.61AS Continuous transition to canted zigzag impossible! magnon gap closes at K points! 0.0 K Γ M K M 1 K K Γ different from zigzag ordering wavevector

21 Scenario 1 (K1 > 0, J1 < 0) in field: ~h k [111] k c K 1 > 2J 1 : intermediate phases 2J 1 >K 1 > J 1 : first-order transition (a) (b)

22 Scenario 1 (K1 > 0, J1 < 0) in field: ~h k [111] k c K 1 > 2J 1 : intermediate phases h = 0: (a) [111] [ 110] [11 2] zigzag

23 ~ Scenario 1 (K1 > 0, J1 < 0) in field: h k [111] k c K1 > 2J1 : intermediate phases 0 < h/(as) < 1.8: h = 0.18AS (a) AF star

24 ~ Scenario 1 (K1 > 0, J1 < 0) in field: h k [111] k c K1 > 2J1 : intermediate phases 1.8 < h/(as) < 2.0: h = 1.8AS (a) [111] zigzag star cf. [Chern et al., PRB 17]

25 ~ Scenario 1 (K1 > 0, J1 < 0) in field: h k [111] k c K1 > 2J1 : intermediate phases 2.0 < h/(as) < 2.6: (a) AF vortex consistent with spin-wave analysis

26 Scenario 1 (K1 > 0, J1 < 0) in field: ~h k [111] k c Classical phase diagram in [111] field: [LJ, Andrade, Vojta, PRL 16] h/(as) AF vortex polarized diluted star canted zigzag star canted Néel vortex Néel canted stripy FM AF star canted zigzag star Néel zigzag FM stripy Néel π π 2 3π ' = ' = 1.85 ' =2 5π 3π 2 7π ~ h [111] J 1 = Acos K 1 =2Asin similarly complicated phase diagram for [112 ] direction (a axis)

27 Scenario 1 (K1 > 0, J1 < 0) in field: ~h k [111] k c Classical phase diagram in [111] field: [LJ, Andrade, Vojta, PRL 16] h/(as) AF vortex polarized diluted star canted zigzag star canted Néel vortex Néel canted stripy FM AF star canted zigzag star Néel zigzag FM stripy Néel π π 2 3π ' = ' = 1.85 ' =2 5π 3π 2 7π ~ h [111] zigzag/stripy & star Klein points intermediate phases also for S = 1/2 J 1 = Acos K 1 =2Asin

28 <latexit sha1_base6="qtqawdc3kifkuqqnawuodq5hq0m=">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</latexit> <latexit sha1_base6="qtqawdc3kifkuqqnawuodq5hq0m=">aaacnxicbvhdbtmwfhbdz7bw13l3vguebessiambrdi1ucaakgbkj1uv9wjc7jatz3idgzvlbfgabifn+ftcnmg0y0jw/r8nr+fc76kkmk6kprdca5dv3fzy3mrvhx7zt173e2dlzyvdcchz2vuzhkwkixgornollhefqicztmxyz9ows0vut6s1suoffwrkumodhptbsp305j+pg+p/eea1nmgdiwvmu5vf0vn+32on7ugl0khb0sgun0+1owtkclwq1xkshcdrsyvgcexdpkpcuc+bzoceyhbov2ujxj1psrz1ka5czf7wjd/ptrgbj2orifqcdn7gxfkvyfb1y67ghscv2udjvffzsvkrqclnddu2gqo7nwalgrvlfkz2cao7/bmgqgnx7luvkg0ploircpjhbkv1dmzv9xwsnwdbnr8tzbb2unzreimk2w39btr8yx8wl2byql2dmj7leuvr0+qsudukn/sshx75o6fwjl2twfqz3+8f96ooz3mdqyrnjdskd8ote5jamybtysoaek+/kb/ljfgu0ebw8dz6sqonom3ofrfkw+gof+cvl</latexit> <latexit sha1_base6="qtqawdc3kifkuqqnawuodq5hq0m=">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</latexit> Scenario 1 (K1 > 0, J1 < 0) in field: ~h k [111] k c Classical phase diagram in [111] field: [LJ, Andrade, Vojta, PRL 16] h/(as) AF vortex polarized diluted star canted zigzag star canted Néel vortex Néel canted stripy FM AF star canted zigzag star Néel zigzag FM stripy Néel π π 2 3π ' = ' = 1.85 ' =2 5π 3π 2 7π ~ h [111] DMRG: magnetic field h α Heisenberg model canted Neel AFM h coupling parameter α polarized phase canted stripy AFM topological phase Kitaev model J 1 =1 K 1 = 2 [Jiang, Gu, Qi, Trebst, PRB 11]

29 0 α coupling parameter α Kitaev model h canted Neel AFM topological phase canted stripy AFM magnetic field h Heisenberg model <latexit sha1_base6="qtqawdc3kifkuqqnawuodq5hq0m=">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</latexit> <latexit sha1_base6="qtqawdc3kifkuqqnawuodq5hq0m=">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</latexit> <latexit sha1_base6="qtqawdc3kifkuqqnawuodq5hq0m=">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</latexit> Scenario 1 (K1 > 0, J1 < 0) in field: ~h k [111] k c Classical phase diagram in [111] field: [LJ, Andrade, Vojta, PRL 16] h/(as) AF vortex polarized diluted star canted zigzag star canted Néel vortex Néel canted stripy FM AF star canted zigzag star Néel zigzag FM stripy Néel π π 2 3π ' = ' = 1.85 ' =2 5π 3π 2 7π ~ h [111] DMRG: polarized phase J 1 =1 K 1 = 2 [Jiang, Gu, Qi, Trebst, PRB 11]

30 Scenario 1 (K1 > 0, J1 < 0) in field: ~h k [111] k c Classical phase diagram in [111] field: [LJ, Andrade, Vojta, PRL 16] h/(as) AF vortex polarized diluted star canted zigzag star canted Néel vortex Néel canted stripy FM AF star canted zigzag star Néel zigzag FM stripy Néel π π 2 3π ' = ' = 1.85 ' =2 5π 3π 2 7π ~ h [111] coupling parameter α magnetic field h Heisenberg model 0 α Kitaev model h polarized phase canted Neel AFM canted stripy AFM topological phase DMRG: [Jiang, Gu, Qi, Trebst, PRB 11]

31 Summary: Scenario 1 (K1 > 0, J1 < 0) Magnetization process strongly anisotropic typically no simple canting but only weak angle dependence of susceptibility and h c does not explain anisotropy in α-rucl 3 unless field along high-symmetry direction exotic intermediate phases large unit cell, multi-q, vortex

32 Scenario 2: Ferromagnetic K1, antiferromagnetic J3 Minimal model: H = X hiji K 1 S i S j + X J 3 ~S i ~S j [Winter et al., PRB 16] Zero field: hhhijiii cubic-planes zigzag ~S k ±(c~e x +s~e y ); c 2 +s 2 =1 i.e., within Ru 2 Cl 2 planes degeneracy will be lifted in real materials

33 Scenario 2 (K1 < 0, J3 > 0) in field For Γ 1 = 0: uniform h ~ Sik ~ h 0.8 uniform h ~ Sik ~ h ~m ĥ/s cubic-plane x/y/z-zigzag h/ K 1 S Set 2 (K 1 -J 3 ) ~ h k [ 110] k b MC (S ) anal.(s ) SW (S =1/2) 5 ~m ĥ/s cubic-plane x/y/z-zigzag h/ K 1 S Set 2 (K 1 -J 3 ) ~ h k [111] k c MC (S ) anal.(s ) SW (S =1/2) classical magnetization curve independent of field direction! (K 1, J 3 ) = (-17, +6.8) mev [Winter et al., PRB 16] and angle dependence of quantum corrections very weak

34 Scenario 2 (K1 < 0, J3 > 0) in field: Perturbations 3 O(2) degeneracy lifted by: (i) quantum fluctuations: [Sizyuk et al., PRB 16] [Chaloupka et al., PRB 16] ~S i k ±~e x ;±~e y ;±~e z cubic-axes zigzag (ii) off-diagonal Γ 1 > 0: ( ±[110] or symmetry-equivalent; if 0 < Γ 1 K 1 ~S i k ±[11 1] or symmetry-equivalent; if Γ 1 K 1 face-center zigzag (iii) magnetic field: ~S i ~h cubic-plane zigzag competition between different effects can be tuned by h

35 Scenario 2 (K1 < 0, J3 > 0) in field: Perturbations Metamagnetic transitions between canted zigzag states: ~h k [ 110] k b ~h k [111] k c uniform h ~ Sik ~ h 0.8 uniform h ~ Sik ~ h ~m ĥ/s face-center z-zigzag h/ K 1 S Set 2+Γ ~ h k [ 110] k b MC (S ) anal.(s ) SW (S =1/2) 2 ~m ĥ/s cubic-plane x/y/z-zigzag 0.2 face-center 0.0 x/y/z-zigzag h/ K 1 S Set 2+Γ ~ h k [111] k c MC (S ) anal.(s ) SW (S =1/2) 2 (J 1, K 1, Γ 1, J 2, K 2, J 3 ) = (+3, -17, +1, -3, +6, +1) mev [Sizyuk et al., PRB 16]

36 Summary: Scenario 2 (K1 < 0, J3 > 0) Magnetization process mostly isotropic very weak angle dependence of susceptibility and h c does not explain anisotropy in α-rucl 3 transitions between different canted zigzag states possible no exotic intermediate phases for some field directions high-field transition continuous

37 Scenario 3: Ferromagnetic K1, positive Γ1 Minimal model: H = X hiji h i J 1 ~S i ~S j +K 1 S i S j +Γ 1 (S i S j +S i S j ) [Ran et al., PRL 17] Zero-field phase diagram? [Rau et al., PRL 1]

38 Scenario 3: Ferromagnetic K1, positive Γ1 Minimal model: H = X hiji h i J 1 ~S i ~S j +K 1 S i S j +Γ 1 (S i S j +S i S j ) [Ran et al., PRL 17] Zero-field phase diagram? [Rau et al., PRL 1] (a) (b) (c) K M 1 K (K 1, Γ 1 ) = (-6.8, +9.5) mev [Ran et al., PRL 17] Γ M 2 Γ Γ M 3 0 J 1 K J 1 K J 1 K 1 multi-q incommensurate ferromagnet failure of Luttinger-Tisza

39 Scenario 3: Ferromagnetic K1, positive Γ1 Minimal model: H = X hiji h i J 1 ~S i ~S j +K 1 S i S j +Γ 1 (S i S j +S i S j ) [Ran et al., PRL 17] Zero-field phase diagram? [Rau et al., PRL 1] (a) (b) (c) K M 1 K (K 1, Γ 1 ) = (-6.8, +9.5) mev [Ran et al., PRL 17] Γ M 2 Γ Γ M 3 0 J 1 K J 1 K J 1 K 1 multi-q incommensurate ferromagnet failure of Luttinger-Tisza

40 Scenario 3: Ferromagnetic K1, positive Γ1 Minimal model: H = X hiji h i J 1 ~S i ~S j +K 1 S i S j +Γ 1 (S i S j +S i S j ) [Ran et al., PRL 17] Zero-field phase diagram? [Rau et al., PRL 1] (a) (b) (c) in J 1 -Γ 1 model M 1 M 1 M 2 M 2 Γ M 3 M 3 J 1 =0 0 < J 1 Γ J 1 Γ classical SL zigzag incommensurate [Rousochatzakis & Perkins, PRL 17] zigzag can be recovered for Γ 1 K 1 failure of Luttinger-Tisza

41 Scenario 3: Ferromagnetic K1, positive Γ1 Minimal model: H = X hiji h i J 1 ~S i ~S j +K 1 S i S j +Γ 1 (S i S j +S i S j ) [Ran et al., PRL 17] Zero-field phase diagram? [Rau et al., PRL 1] (a) (b) (c) in J 1 -Γ 1 model M 1 M 1 M 2 M 2 Γ M 3 M 3 J 1 =0 0 < J 1 Γ J 1 Γ classical SL zigzag incommensurate [Rousochatzakis & Perkins, PRL 17] zigzag can be recovered for Γ 1 K 1 failure of Luttinger-Tisza

42 Scenario 3: Ferromagnetic K1, positive Γ1 Minimal model: H = X hiji h i J 1 ~S i ~S j +K 1 S i S j +Γ 1 (S i S j +S i S j ) [Ran et al., PRL 17] Zero-field phase diagram? [Rau et al., PRL 1] (a) (b) (c) in J 1 -Γ 1 model M 1 M 1 M 2 M 2 Γ M 3 M 3 J 1 =0 0 < J 1 Γ J 1 Γ classical SL zigzag incommensurate [Rousochatzakis & Perkins, PRL 17] zigzag can be recovered for Γ 1 K 1 failure of Luttinger-Tisza

43 Scenario 3: Ferromagnetic K1, J1, positive Γ1 Minimal model: Zero field: H = X hiji h i J 1 ~S i ~S j +Γ 1 (S i S j +S i S j ) i.e., take the limit K 1 /Γ 1 0 face-center zigzag ~S k ±[11 1] towards faces of RuCl 6 octahedra

44 <latexit sha1_base6="fas0fz6ozma6opgr0j5sbc9j5mk=">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</latexit> Scenario 3: Ferromagnetic K1, J1, positive Γ1 (J 1, K 1, Γ 1, J 3 ) = (-0.5, -5.0, +2.5, +0.5) mev [Winter et al., Nat. Comm. 17] (a) m ĥ/s (c) Set 3+J 3 h [111] c MC (S ) anal.(s ) SW (S =1/2) face-center x/y/z-zigzag uniform S h (b) m ĥ/s (d) face-center x/y-zigzag uniform S h Set 3+J 3 h [001] ac MC (S ) anal.(s ) m ĥ/s Observations: uniform S h m ĥ/s Set 3+J 3 Set 3+J 3 h [ 110] b 0. h [11 2] a MC (S ) MC (S ) anal.(s ) 0.2 face-center face-center anal.(s ) SW (S =1/2) z-zigzag x/y-zigzag SW (S =1/2) h/ K 1 S h/ K 1 S uniform S h 2 (1) Out-of-plane h c much larger than in-plane h c as in α-rucl 3 (2) h~m k i =S < 1 in polarized phase already in classical limit

45 Scenario 3 (K1, J1 < 0, Γ1 > 0): Consequences of strong Γ1 Toy model: H = X hiji Γ 1 (S i S j +S i S j ) x z y In polarized state S ~ i S 8 >< +Γ 1 if Si ~ k [111] k c he 0 =(NS 2 )i = Γ 1 =2 if Si ~ k [ 110] k b >: Γ 1 =2 if Si ~ k [11 2] k a Γ 1 > 0 means antiferromagnetic for out-of-plane spins ferromagnetic for in-plane spins similarly: components of S i in out-of-plane vs. in-plane direction naturally explains in-plane vs. out-of-plane anisotropy

46 Scenario 3 (K1, J1 < 0, Γ1 > 0): Field-angle dependence Susceptibility: Critical field h c : [LJ, Andrade, Vojta, PRB 17] (a) a b a [11 2] [01 1] [ 12 1] [ 110] [ 211] [ 101] [ 1 12] (b) a b a [11 2] [01 1] [ 12 1] [ 110] [ 211] [ 101] [ 1 12] 0 π/ π/2 3π/ 2.0 π ϕ 0 π/ π/2 3π/ 2.0 π ϕ h ab Set 3+J h ac d( m ĥ) d(h/ K1 ) 1.0 h ac h0/ K1S 1.0 h ab 0.5 Set 3+J 3 h h π/2 ϑ 0 π/ π/2 3π/ π [11 2] [11 1] [110] [111] [112] [001] [ 1 12] a c a π/2 ϑ 0 π/ π/2 3π/ π [11 2] [11 1] [110] [111] [112] [001] [ 1 12] a c a (J 1, K 1, Γ 1, J 3 ) = (-0.5, -5.0, +2.5, +0.5) mev immediately testable in α-rucl 3

47 <latexit sha1_base6="adgygkubgvqdhjuxjuigqnqpz6i=">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</latexit> Scenario 3 (K 1, J 1 < 0, Γ 1 > 0): High-field transversal magnetization High fields: expect ground state w/o spontaneous symmetry breaking i.e., expect high-field state to be adiabatically connected to fully polarized state If Γ 1 = 0: S x i 7! S x i ; S y i 7! S y i ; Sz i 7! S z i is an accidental higher symmetry For ~ h k [001]: (a) Γ1 = 0: h~s i ik[001] ) h~m k i =S =1 (b) Γ1 0: h~s i ik[xxz]withx6= 0 not forbidden ) finite h~m i? ~ h possible ~m is a direct measure of Γ 1 independent of K 1, J 1, J 3, h~m? i = h~m k i ' 0:51 at h = h c + for (J1, K 1, Γ 1, J 3 ) = (-0.5, -5.0, +2.5, +0.5) mev

48 Summary: Scenario 3 (K1, J1 < 0, Γ1 > 0) Magnetization process strongly anisotropic χ ab much larger than χ c even without g-factor anisotropy χ ab /χ c 3 for (J 1, K 1, Γ 1, J 3 ) = (-0.5, -5.0, +7.0, +0.5) mev consistent with small trigonal distortion [Agrestini et al., PRB(R) 17] high-field state with transversal magnetization for field in intermediate direction, e.g., [001]

49 Conclusions: Magnetization process crucially dependent on zigzag stabilization: Scenario 1 (K 1 > 0, J 1 < 0) Weak anisotropy in χ and hc Various exotic intermediate phases Scenario 2 (K 1 < 0, J 3 > 0) Nearly no anisotropy in χ and hc Metamagnetic transitions between different canted zigzag Scenario 3 (K 1, J 1 < 0, Γ 1 > 0) Strong anisotropy in χ and hc Finite transversal magnetization in high-field state Intermediate phases? natural explanation for anisotropy in α-rucl 3 α-rucl 3 : expect strong Γ 1 > 0, K 1 < 0, and small J 3 > 0 possibly measurable?