Thermal Hall effect of magnons

Transcription

1 Max Planck-UBC-UTokyo (2018/2/18) Thermal Hall effect of magnons Hosho Katsura (Dept. Phys., UTokyo) Related papers: H.K., Nagaosa, Lee, Phys. Rev. Lett. 104, (2010). Onose et al., Science 329, 297 (2010). Ideue et al., Phys. Rev. B 85, (2012).

2 Outline 1. Spin Hamiltonian Exchange and DM interactions Microscopic origins 1/25 2. Elementary excitations 3. Hall effect and thermal Hall effect 4. Main results 5. Summary

3 Coupling between magnetic moments Classical v.s. Quantum Dipole-dipole interaction 2/25 Usually, too small (< 1K) to explain transition temperatures Exchange interaction ( S i : 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 NOTE) Inversion breaking is necessary. Spin tend to be orthogonal

4 (Crude) derivation 2-site Hubbard model Hamiltonian 1 2 3/25 U 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, )

5 Origin of DM interaction (1) Spin-dependent hopping 1 2 Hopping matrix Unitary transformation One can ``absorb the spin-dependent hopping! Inversion symmetry broken, but Time-reversal symmetry exists. Due to spin-orbit 4/25 θ=0 reduces to the spin-independent case New fermions satisfy the same anti-commutation relations. Number ops. remain unchanged. Hamiltonian in terms of f

6 Origin of DM interaction (2) Effective Hamiltonian 5/25 How does it look like in original spins? Ex.) Prove the relation. Hint: express in terms of σs. Heisenberg int. Dzyaloshinskii- Moriya (DM) int. Kaplan-Shekhtman-Aharony -Entin-Wohlman (KSAE) int. NOTE) One can eliminate the effect of the DM interaction if there is no loop.

7 Outline 1. Spin Hamiltonian 2. Elementary excitations What are magnons? From spins to bosons Diagonalization of BdG Hamiltonian 6/25 3. Hall effect and thermal Hall effect 4. Main results 5. Summary

8 What are magnons? FM Heisenberg model in a field 7/25 Ground state: spins are aligned in the same direction. Elementary excitations -- Intuitive picture -- Ground state z: coordination number 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.

9 1-Magnon eigenstates ``Motion of flipped spin 1 2 N i> is not an eigenstate! 8/25 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!

10 From spins to bosons Holstein-Primakoff transformation Bose operators Spins in terms of b Number op.: 9/25 Obey the commutation relations of spins Often neglect nonlinear terms. (Good at low temperatures.) Magnetic ground state = vacuum of bosons Sublattice structure AFM int. Approximate 1-magnon state Spins on the other sublattice: b raises S z a lowers S z One needs to introduce more species for a more complex order.

11 Diagonalization of Hamiltonian Quadratic form of bosons 10/25 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).

12 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 11/25 4. Main results 5. Summary

13 Hall effect and Berry curvature Quantum Hall effect (2D el. Gas) 12/25 y x TKNN formula Integer n is a topological number! Bloch wave function Berry connection Berry curvature PRL, 49 (1982) Chern number Kubo formula relates Chern # and σ xy

14 Anomalous Hall effect QHE without net magnetic field Onsager s reciprocal relation Time-reversal symmetry (TRS) must be broken for nonzero σ xy 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 13/25 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).

15 Thermal Hall effect Thermal current 14/25 Onsager relation: Absence of J: Wiedemann-Franz law Righi-Leduc effect Cs are matrix, in general. Universal for weakly interacting electrons 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!

16 General formulation TKNN-like formula for bosons Bloch w.f., Berry curvature Earlier work - Fujimoto, PRL 103, (2009) - H.K., Nagaosa & Lee, PRL 104, (2010) - Onose et al., Science 329, 297 (2010) Δ: energy separation Terms due to the orbital motion of magnon are missing 15/25 Still well defined for 1-magnon Hamiltonian without paring term Modified linear-response theory - Matsumoto & Murakami, PRL 106, ; PRB 84, (2011) Bose distribution Universally applicable to (free) bosonic systems! Magnons, phonons, triplons, photons (?) NOTE) No quantization.

17 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 16/25 5. Summary

18 Magnon Hall effect Theory Magnetization ( m B /V) 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). 17/25 Lu 2 V 2 O 7 k xy (10-3 W/Km) (a) k xy 0.3T k xy 7T M 0.3T M 7T

19 Role of DM interaction Kagome model DM vectors 18/25 Bosonic ver. of Ohgushi-Murakami-Nagaosa (PRB 62 (2000)) Scalar chirality order there ( Nonzero Berry curvature! MOF material Cu(1-3, bdc) ) DM. is expected to be nonzero. FM exchange int. b/w Cu 2+ moments - Hirschberger et al., PRL 115, (2015) - Chisnell et al., PRL 115, (2015) Nonzero THE response. Sign change consistent with theories: Mook, Heng & Mertig PRB 89, (2014), Lee, Han & Lee, PRB 91, (2015).

20 Pyrochlore ferromagnet Lu 2 V 2 O 7 19/25 V 4+ : (t 2g ) 1, S=1/2 Trigonal crystal field Origin of FM: orbital pattern Polarized neutron diffraction (Ichikawa et al., JPSJ 74 ( 03)) M(m B /V)) Resistivity( cm) k xx (W/Km) A B 10 0 C Highly resistive Y. Onose et al., Lu Science 2 V 2 O 329, 297 ( 10). 7 H [111] H=0.1T Tc=70K Magnon & phonon M (m B /V) T(K) C (J/molK) D E T=5K H [100] H [111] H [110] m 0 H (T) H [111] 0T 5T 9T Isotropic T 1.5 (K 1.5 )

21 Observed thermal Hall conductivity Lu 2 V 2 O 7 H [100] 2 80K 70K 60K 50K 20/ k xy (10-3 W/Km) K 1 30K 20K 10K Magnetic Field (T) Anomalous? Related to TRS breaking?

22 Model Hamiltonian FM Heisenberg + DM 21/25 Allowed DM vectors Elhajal et al., PRB 71; Kotov et al., PRB 72 (2005). Stability of FM g.s. against DM Spin-wave Hamiltonian Only is important. Band structure ( ) Hamiltonian in k-space Λ: 4x4 matrix. 4 bands

23 Comparison of theory and experiment Formula (at H=0 + ) 22/25 - Explains the observed isotropy - D/J is the only fitting parameter Fitting The fit yields D/J ~ 0.38 Berry curvature around k=0 can be obtained analytically Observed in other pyrochlore FM insulators Ho 2 V 2 O 7 : D/J ~ 0.07, In 2 Mn 2 O 7 : D/J ~ Reasonable! Cf.) D/J ~ 0.19 in pyrochlore AFM CdCr 2 O 4 Chern, Fennie & Tchernyshov, PRB 74 (2006).

24 What about other lattices Provskite-like lattices 23/25 YTiO 3, S=1/2, Tc=30K Absence of THE in La 2 NiMnO 6 and YTiO 3 - 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 BiMnO 3 - The origin is unclear May be due to complex orbital order Tc ~100K

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 24/25 Observation in pyrochlore FMs Lu 2 V 2 O 7, Ho 2 V 2 O 7, In 2 Mn 2 O 7 Consistency - Below FM transition - Isotropy of - Reasonable D/J Agreement is excellent! Mysteries - Nonzero in BiMnO 3 - Effect of int. between magnons

26 Other directions Thermal Hall effects of bosonic particles Phonon: (Exp.) Strohm, Rikken, Wyder, PRL 95, (2005) (Theory) Sheng, Sheng, Ting, PRL 96, (2006) Kagan, Maksimov, PRL 100, (2008) 25/25 Triplon: (Theory) Romhányi, Penc, Ganesh, Nat. Comm. 6, 6805 (2015) Photon: (Theory) Ben-Abdallah, PRL 116, (2016) Topological magnon physics Dirac magnon Honeycomb: Fransson, Black-Schaffer, Balatsky, PRB 94, (2016) Weyl magnon Pyrochlore AFM: F-Y. Li et al., Nat. Comm. 7, (2016) Pyrochlore FM: Mook, Henk, Mertig, PRL 117, (2016) Topological magnon insulators Nakata, Kim, Klinovaja, Loss, PRB 96, (2017)