Max Planck-UBC-UTokyo School@Hongo (2018/2/18) Thermal Hall effect of magnons Hosho Katsura (Dept. Phys., UTokyo) Related papers: H.K., Nagaosa, Lee, Phys. Rev. Lett. 104, 066403 (2010). Onose et al., Science 329, 297 (2010). Ideue et al., Phys. Rev. B 85, 134411 (2012). 1/25 Outline 1. Spin Hamiltonian • Exchange and DM interactions • Microscopic origins 2. Elementary excitations 3. Hall effect and thermal Hall effect 4. Main results 5. Summary Coupling between magnetic moments 2/25 Classical v.s. Quantum • Dipole-dipole interaction Usually, too small (< 1K) to explain transition temperatures… • Exchange interaction ( Si: spin at site i ) Direct exchange: J < 0 ferromagnetic (FM) Super-exchange: J > 0 antiferromagnetic (AFM) Anisotropies Spin-orbit int. breaks SU(2) symmetry. Dzyaloshinskii-Moriya (DM) int.: D Spin tend to NOTE) Inversion breaking is necessary. be orthogonal (Crude) derivation 3/25 2-site Hubbard model U • Hamiltonian 1 2 • 2nd order perturbation at half-filling, Pauli’s exclusion ↑↑ ↑↓ ↓↑ ↓↓ Origin of exchange int. = electron correlation! Can explain AFM int. What about FM int.? (Multi-orbital nature, Kanamori-Goodenough, …) Origin of DM interaction (1) 4/25 Spin-dependent hopping Inversion symmetry broken, but Time-reversal symmetry exists. Due to spin-orbit 1 2 θ=0 reduces to the • Hopping matrix spin-independent case Unitary transformation One can ``absorb” the spin-dependent hopping! New fermions satisfy the same anti-commutation relations. Number ops. remain unchanged. • Hamiltonian in terms of f Origin of DM interaction (2) 5/25 Effective Hamiltonian • How does it look like in original spins? Ex.) Prove the relation. Hint: express in terms of σs. Heisenberg int. Dzyaloshinskii- Kaplan-Shekhtman-Aharony Moriya (DM) int. -Entin-Wohlman (KSAE) int. NOTE) One can eliminate the effect of the DM interaction if there is no loop. 6/25 Outline 1. Spin Hamiltonian 2. Elementary excitations • What are magnons? • From spins to bosons • Diagonalization of BdG Hamiltonian 3. Hall effect and thermal Hall effect 4. Main results 5. Summary What are magnons? 7/25 FM Heisenberg model in a field • Ground state: spins are aligned in the same direction. z: coordination number Elementary excitations -- Intuitive picture -- Ground state Excitation =NG mode Cf.) non-relativistic Nambu-Goldstone bosons Watanabe-Murayama, PRL 108 (2012); Hidaka, PRL 110 (2013). The picture is classical. But in ferromagnets, ground state and 1-magnon states are exact eigenstates of the Hamiltonian. 1-Magnon eigenstates 8/25 ``Motion” of flipped spin |i> is not an 1 2 N eigenstate! Flipped spin hops to the neighboring sites. Bloch state Ex.） 1D is an exact eigenstate with energy E(k) What about DM int.? D vector // z-axis Magnon picks up a phase factor! From spins to bosons 9/25 Holstein-Primakoff transformation • Bose operators Number op.: • Spins in terms of b Obey the commutation relations of spins Often neglect nonlinear terms. (Good at low temperatures.) b raises Sz • Magnetic ground state = vacuum of bosons Sublattice structure AFM int. Approximate 1-magnon state • Spins on the other sublattice: a lowers Sz One needs to introduce more species for a more complex order. Diagonalization of Hamiltonian 10/25 Quadratic form of bosons h, Δ : N×N matrices • Ferromagnetic case Problem reduces to the diagonalization of h. Most easily done in k-space (Fourier tr.). • AFM (or more general) case Para-unitary Transformation leaves the boson commutations unchanged. are e.v. of Involved procedure. See, e.g., Colpa, Physica 93A, 327 (1978). 11/25 Outline 1. Spin Hamiltonian 2. Elementary excitations 3. Hall effect and thermal Hall effect • Hall effect and Berry curvature • Anomalous and thermal Hall effects • General formulation 4. Main results 5. Summary Hall effect and Berry curvature 12/25 Quantum Hall effect (2D el. Gas) y x TKNN formula PRL, 49 (1982) Integer n is a topological number! • Bloch wave function • Berry connection • Berry curvature Chern number Kubo formula relates Chern # and σxy Anomalous Hall effect 13/25 QHE without net magnetic field • Onsager’s reciprocal relation Time-reversal symmetry (TRS) must be broken for nonzero σxy • Haldane’s model (PRL 61, 2015 (1988), Nobel prize 2016) Local magnetic field can break TRS! n.n. real and n.n.n. complex hopping Integer QHE without Landau levels Spontaneous symmetry breaking TRS can be broken by magnetic ordering. • Anomalous Hall effect Review: Nagaosa et al., RMP 82, 1539 (2010). ：magnetization Itinerant electrons in ferromagnets. (i) Intrinsic and (ii) extrinsic origins. Anomalous velocity by Berry curvature in (i). Thermal Hall effect 14/25 Thermal current Cs are matrix, in general. Onsager relation: Absence of J: • Wiedemann-Franz law Universal for weakly interacting electrons Righi-Leduc effect Transverse temperature gradient is produced in response to heat current In itinerant electron systems from Wiedemann-Franz What about Mott insulators? Hall effect without Lorentz force? Berry curvature plays the role of magnetic field! General formulation 15/25 TKNN-like formula for bosons Still well defined for • Bloch w.f. , Berry curvature 1-magnon Hamiltonian • Earlier work without paring term - Fujimoto, PRL 103, 047203 (2009) - H.K., Nagaosa & Lee, PRL 104, 066403 (2010) - Onose et al., Science 329, 297 (2010) Δ: energy separation Bose distribution Terms due to the orbital motion of magnon are missing… • Modified linear-response theory - Matsumoto & Murakami, PRL 106, 197202; PRB 84, 184406 (2011) Universally applicable to (free) bosonic systems! Magnons, phonons, triplons, photons (?) … NOTE) No quantization. 16/25 Outline 1. Spin Hamiltonian 2. Elementary excitations 3. Hall effect and thermal Hall effect 4. Main results • Kagome-lattice FM • Pyrochlore FM • Comparison of theory and experimement 5. Summary Magnon Hall effect 17/25 Theory Magnons do not have charge. They do not feel Lorentz force. Nevertheless, they exhibit thermal Hall effect (THE)! Keys: 1. TRS is broken spontaneously in FM 2. DM interaction leads to Berry curvature ≠ 0 NOTE) Original theory concerned the effect of scalar chirality. Experiment Magnon THE was indeed observed in FM insulators! Onose et al., Science 329, 297 (2010). Lu2V2O7 1 1.5 k (a) xy 0.3T /V) kxy 7T 0.8 B M 0.3T m 1 0.6 W/Km) M 7T 3 - 0.4 0.5 (10 0.2 xy k 0 0 Magnetization ( 0 50 100 150 Role of DM interaction 18/25 DM vectors Kagome model Bosonic ver. of Ohgushi-Murakami-Nagaosa (PRB 62 (2000)) Scalar chirality order there ( ) DM. Nonzero Berry curvature! is expected to be nonzero. MOF material Cu(1-3, bdc) FM exchange int. b/w Cu2+ moments - Hirschberger et al., PRL 115, 106603 (2015) - Chisnell et al., PRL 115, 147201 (2015) Nonzero THE response. Sign change consistent with theories: Mook, Heng & Mertig PRB 89, 134409 (2014), Lee, Han & Lee, PRB 91, 125413 (2015). 19/25 Pyrochlore ferromagnet Lu2V2O7 Y. Onose et al., Science 329, 297 (‘10). Lu2V2O7 1 1 A H || [111] D T=5K 0.8 H=0.1T 0.8 0.6 Isotropic /V)) B 0.6 /V) m B ( 0.4 m M ( 0.4 0.2 Tc=70K M H || [100] H || [111] 0.2 0 H || [110] B 104 cm) 0 0.2 0.4 0.6 0.8 1 m H (T) 103 0 2 Highly 0.8 10 E 4+ 1 resistive V : (t ) , S=1/2 101 H||[111] 2g Resistivity( 0.6 0T 100 C 5T • Trigonal crystal field 9T 1.5 0.4 (J/molK) 1 C (W/Km) 0.2 xx • Origin of FM: orbital pattern k 0.5 Magnon & phonon Polarized neutron diffraction 0 10 20 0 50 100 150 1.5 1.5 (Ichikawa et al., JPSJ 74 (‘03)) T(K) T (K ) Observed thermal Hall conductivity 20/25 Lu2V2O7 H||[100] 2 80K 70K 60K 50K 1 0 -1 -2 W/Km) -3 2 40K 30K 20K 10K (10 xy k 1 0 -1 -2 -5 0 5 -5 0 5 -5 0 5 -5 0 5 Magnetic Field (T) Anomalous? Related to TRS breaking? Model Hamiltonian 21/25 FM Heisenberg + DM • Allowed DM vectors Elhajal et al., PRB 71; Kotov et al., PRB 72 (2005). • Stability of FM g.s. against DM Spin-wave Hamiltonian Band structure ( ) Only is important. • Hamiltonian in k-space Λ： 4x4 matrix. 4 bands Comparison of theory and experiment 22/25 Formula (at H=0+) Berry curvature around k=0 can be obtained analytically - Explains the observed isotropy - D/J is the only fitting parameter Fitting The fit yields |D/J| ~ 0.38 Observed in other pyrochlore FM insulators Ho2V2O7: D/J ~ 0.07, In2Mn2O7: D/J ~ -0.02 Reasonable! Cf.) D/J ~ 0.19 in pyrochlore AFM CdCr2O4 Chern, Fennie & Tchernyshov, PRB 74 (2006). What about other lattices 23/25 YTiO3, S=1/2, Tc=30K Provskite-like lattices • Absence of THE in La2NiMnO6 and YTiO3 - Ideal cubic perovskite No DM - In reality, it’s distorted nonzero DM What’s the reason? Flux pattern staggered Berry curvature is zero because of pseudo TRS in • Presence of THE in BiMnO3 - The origin is unclear… Tc ~100K May be due to complex orbital order 24/25 Summary Thermal Hall effect in FM insulators • Mechanism Heat current is carried by magnons. Driven by Berry curvature due to DM int. • TKNN-like formula • Observation in pyrochlore FMs Lu2V2O7, Ho2V2O7, In2Mn2O7 Consistency - Below FM transition - Isotropy of - Reasonable D/J Agreement is excellent! Mysteries - Nonzero in BiMnO3 - Effect of int. between magnons 25/25 Other directions Thermal Hall effects of bosonic particles • Phonon: (Exp.) Strohm, Rikken, Wyder, PRL 95, 155901 (2005) (Theory) Sheng, Sheng, Ting, PRL 96, 155901 (2006) Kagan, Maksimov, PRL 100, 145902 (2008) • Triplon: (Theory) Romhányi, Penc, Ganesh, Nat. Comm. 6, 6805 (2015) • Photon: (Theory) Ben-Abdallah, PRL 116, 084301 (2016) Topological magnon physics • Dirac magnon Honeycomb: Fransson, Black-Schaffer, Balatsky, PRB 94, 075401 (2016) • Weyl magnon Pyrochlore AFM: F-Y.