Thermal Hall effect of magnons Hosho Katsura (Dept. Phys., UTokyo) - - PowerPoint PPT Presentation

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Thermal Hall effect of magnons Hosho Katsura (Dept. Phys., UTokyo) - - PowerPoint PPT Presentation

Feb 29, 2024 157 likes •424 views

Max Planck-UBC-UTokyo School@Hongo (2018/2/18) Thermal Hall effect of magnons Hosho Katsura (Dept. Phys., UTokyo) Rel ated papers : H.K., Nagaosa, Lee, Phys. Rev. Lett . 104 , 066403 (2010). Onose et al., Science 329 , 297 (2010).

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SLIDE 1

Max Planck-UBC-UTokyo School@Hongo (2018/2/18)

Thermal Hall effect of magnons

Hosho Katsura

(Dept. Phys., UTokyo)

  • 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).

Related papers:

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SLIDE 2

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

1/25

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SLIDE 3

Coupling between magnetic moments

2/25

 Classical v.s. Quantum

  • Dipole-dipole interaction
  • Exchange interaction

Usually, too small (< 1K) to explain transition temperatures…

 Anisotropies

Direct exchange: J < 0  ferromagnetic (FM) Super-exchange: J > 0  antiferromagnetic (AFM) ( Si: spin at site i ) Spin-orbit int. breaks SU(2) symmetry.  Dzyaloshinskii-Moriya (DM) int.: D

NOTE) Inversion breaking is necessary. Spin tend to be orthogonal

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SLIDE 4

(Crude) derivation

3/25

 2-site Hubbard model

Origin of exchange int. = electron correlation! Can explain AFM int. What about FM int.? (Multi-orbital nature, Kanamori-Goodenough, …)

  • Hamiltonian

U 1 2

  • 2nd order perturbation at half-filling,

Pauli’s exclusion

↑↑ ↑↓ ↓↑ ↓↓

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SLIDE 5

Origin of DM interaction (1)

4/25

 Spin-dependent hopping

1 2 Inversion symmetry broken, but Time-reversal symmetry exists.

  • Hopping matrix

θ=0 reduces to the 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

Due to spin-orbit

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SLIDE 6

Origin of DM interaction (2)

5/25

 Effective Hamiltonian

  • How does it look like in original spins?

Heisenberg int. Dzyaloshinskii- Moriya (DM) int. Kaplan-Shekhtman-Aharony

  • Entin-Wohlman (KSAE) int.

Ex.) Prove the relation. Hint: express in terms of σs. NOTE) One can eliminate the effect of the DM interaction if there is no loop.

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SLIDE 7

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

6/25

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SLIDE 8

What are magnons?

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 FM Heisenberg model in a field

Ground state Excitation =NG mode

 Elementary excitations -- Intuitive picture --

  • Ground state: spins are aligned in the same direction.

z: coordination number

The picture is classical. But in ferromagnets, ground state and 1-magnon states are exact eigenstates of the Hamiltonian.

Cf.) non-relativistic Nambu-Goldstone bosons Watanabe-Murayama, PRL 108 (2012); Hidaka, PRL 110 (2013).

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SLIDE 9

1-Magnon eigenstates

8/25

 ``Motion” of flipped spin

1 2 N

|i> is not an eigenstate!

Flipped spin hops to the neighboring sites.

Ex.） 1D

 Bloch state

is an exact eigenstate with energy E(k)

 What about DM int.?

D vector // z-axis Magnon picks up a phase factor!

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SLIDE 10

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.)

  • Magnetic ground state = vacuum of bosons

 Sublattice structure

AFM int.  Approximate 1-magnon state

  • Spins on the other sublattice:

One needs to introduce more species for a more complex order.

b raises Sz a lowers Sz

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SLIDE 11

are e.v. of

Diagonalization of Hamiltonian

10/25

 Quadratic form of bosons

  • Ferromagnetic case

h, Δ : N×N matrices

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. Involved procedure. See, e.g., Colpa, Physica 93A, 327 (1978).

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SLIDE 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
  • 4. Main results
  • 5. Summary

11/25

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SLIDE 13

x y

Hall effect and Berry curvature

12/25

 Quantum Hall effect (2D el. Gas)  TKNN formula

Integer n is a topological number!

  • Bloch wave function
  • Berry connection
  • Berry curvature

Chern number

Kubo formula relates Chern # and σxy PRL, 49 (1982)

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SLIDE 14

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

：magnetization

Itinerant electrons in ferromagnets. (i) Intrinsic and (ii) extrinsic origins. Anomalous velocity by Berry curvature in (i). TRS can be broken by magnetic ordering.

  • Anomalous Hall effect Review: Nagaosa et al., RMP 82, 1539 (2010).
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Thermal Hall effect

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 Thermal current

Cs are matrix, in general.

  • 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!

Onsager relation: Absence of J:

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General formulation

15/25

 TKNN-like formula for bosons

  • Earlier work
  • 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

  • Bloch w.f. , Berry curvature

Still well defined for 1-magnon Hamiltonian without paring term 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.

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SLIDE 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
  • 5. Summary

16/25

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SLIDE 18

Magnon Hall effect

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 Theory  Experiment

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

Magnon THE was indeed observed in FM insulators!

Onose et al., Science 329, 297 (2010). NOTE) Original theory concerned the effect of scalar chirality.

Lu2V2O7

50 100 150 0.5 1 1.5 0.2 0.4 0.6 0.8 1 (a)

kxy 0.3T kxy 7T M 0.3T M 7T

kxy (10

  • 3 W/Km)

Magnetization (mB/V)

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SLIDE 19

Role of DM interaction

18/25

 Kagome model

DM vectors

 MOF material Cu(1-3, bdc)

Bosonic ver. of Ohgushi-Murakami-Nagaosa (PRB 62 (2000))

Scalar chirality order there ( )  DM. Nonzero Berry curvature! is expected to be nonzero.

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).

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SLIDE 20

Pyrochlore ferromagnet Lu2V2O7

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V4+: (t2g)1, S=1/2

  • Origin of FM: orbital pattern

Polarized neutron diffraction (Ichikawa et al., JPSJ 74 (‘03))

  • Trigonal crystal field

0.2 0.4 0.6 0.8 1

Lu2V2O7

H || [111] H=0.1T M(mB/V)) A 100 101 102 103 104 Resistivity(cm) B 50 100 150 0.5 1 1.5 kxx(W/Km) T(K) C 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 H || [100] H || [111] H || [110] m0H (T) M (mB/V) T=5K D 10 20 0.2 0.4 0.6 0.8 T1.5 (K1.5) C (J/molK) 0T 5T 9T H||[111] E

  • Y. Onose et al., Science 329, 297 (‘10).

Isotropic

Magnon & phonon

Highly resistive Tc=70K

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SLIDE 21

Observed thermal Hall conductivity

20/25

  • 5

5 20K Magnetic Field (T)

  • 5

5 30K

  • 5

5

  • 2
  • 1

1 2 40K 50K

  • 5

5 10K 60K 70K

  • 2
  • 1

1 2 80K kxy (10-3 W/Km) Lu2V2O7 H||[100]

Anomalous? Related to TRS breaking?

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Model Hamiltonian

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  • Allowed DM vectors

Elhajal et al., PRB 71; Kotov et al., PRB 72 (2005).

  • Stability of FM g.s. against DM

 FM Heisenberg + DM  Spin-wave Hamiltonian

Only is important.

Band structure ( )

  • Hamiltonian in k-space

Λ： 4x4 matrix.  4 bands

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SLIDE 23

Comparison of theory and experiment

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 Formula (at H=0+)

Berry curvature around k=0 can be obtained analytically

 Fitting

The fit yields |D/J| ~ 0.38

  • Explains the observed isotropy
  • D/J is the only fitting parameter

Chern, Fennie & Tchernyshov, PRB 74 (2006).

Reasonable! Cf.) D/J ~ 0.19 in pyrochlore AFM CdCr2O4 Observed in other pyrochlore FM insulators Ho2V2O7: D/J ~ 0.07, In2Mn2O7: D/J ~ -0.02

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SLIDE 24

What about other lattices

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 Provskite-like lattices

  • Absence of THE in La2NiMnO6 and YTiO3

YTiO3, S=1/2, Tc=30K

  • Ideal cubic perovskite  No DM
  • In reality, it’s distorted  nonzero DM

Flux pattern  staggered Berry curvature is zero because of pseudo TRS in

What’s the reason?

  • Presence of THE in BiMnO3

Tc ~100K

  • The origin is unclear…

May be due to complex

  • rbital order
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SLIDE 25

Summary

24/25

 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
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SLIDE 26

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)

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 Topological magnon physics

  • Dirac magnon

Honeycomb: Fransson, Black-Schaffer, Balatsky, PRB 94, 075401 (2016)

  • Weyl magnon

Pyrochlore AFM: F-Y. Li et al., Nat. Comm. 7, 12691 (2016) Pyrochlore FM: Mook, Henk, Mertig, PRL 117, 157204 (2016)

  • Topological magnon insulators

Nakata, Kim, Klinovaja, Loss, PRB 96, 224414 (2017)