Electronic IQHE: The quantum Hall conductance equals the Chern integer which characterizes the number of edge modes. Magnonic Hall effect: The Hall conductance (in the clean limit) may or may not equal the Chern integer characterizing the number of edge modes, generally. However, in an almost flat band, the Hall conductance equals the Chern integer. KN, J. Klinovaja & D. Loss, arXiv:1611.09752 (2016) show that a magnonic quantum Hall effect and Wiedemann-Franz law can occur in such an almost flat band at lower temperatures than the Landau gap.

1. University of Basel
Magnonic Quantum Hall Effect & the Wiedemann-Franz Law
Kouki Nakata
KN, J. Klinovaja & D. Loss, arXiv:1611.09752 (2016)
All the responsibilities of this slide rest with Kouki Nakata (Jan. 2017)
See also [KN, P. Simon, and D. Loss: Phys. Rev. B 92, 134425 (2015)]

2. Magnon Carries 𝜇B & 𝑘B
≤ ≪
Magnon𝜇B𝑘B
Low-energy collective mode in insulating magnet

3. Yes !
QUESTION
Can magnon 𝜇B (boson) transport be similar to electron 𝑒 (fermion) transport ?

4. Electron 𝑒
= Fermion
Magnon 𝜇B
= Boson
Wiedemann-Franz (WF) law
Franz and Wiedemann, Annalen der Physik (1853)
Magnonic Wiedemann-Franz law
KN, P. Simon & D. Loss, PRB (2015)
Superconductors
Onnes (1911)
Quasi-equilibrium magnon condensate
Demokritov et al., Nature (2006)
Magnon-BEC current
Hillebrands-group, Nat. Phys. (2016)
Josephson effect
Josephson, Phys. Lett. (1962)
Magnonic Josephson effect
KN, K. A. van Hoogdalem, P. Simon & D. Loss, PRB (2014)
KN, P. Simon & D. Loss, PRB (2015)
Integer quantum Hall effect (IQHE)
Klitzing et al., PRL (1980)
TKNN, PRL (1982) / Kohmoto, Ann. Phys. (1985)
Magnonic IQHE & the WF law
KN, J. Klinovaja & D. Loss (2016), arXiv:1611.09752
QUESTION
Can magnon 𝜇B (boson) transport be similar to electron 𝑒 (fermion) transport ?
See review article [KN, P. Simon & D. Loss, arXiv:1610.08901]

5. Dirac & Weyl magnon
Spin-wave： Magnon F. Bloch, Z. Physik. Holstein & Primakoff, Phys. Rev. (1940)1930
Li et al., Nat. Commun. (2016)Balatsky-group, PRB (2016).
Quasi-equilibrium magnon-BEC Demokritov et al. (Hillebrands-group), Nature
Spin-wave spin current： Magnon current2010 Kajiwara et al., Nature
Magnon-BEC current2016 Hillebrands-group, Nat. Phys.
2016
Aharonov & Casher, PRL (1984)Aharonov-Casher effect on magnon2014 Yale-group, PRL: Observation.
2006
Onose et al., Science
Katsura et al., PRL (2010)
Matsumoto & Murakami, PRL (2011)
Magnon thermal Hall effect
Spin-Seebeck effect2008 Uchida et al.(`08, `10, `11), Nature. Adachi et al., PRB (2011)
2014
- 2016 Magnon WF law
Magnon Josephson effect
Magnon IQHE
KN et al.
Saitoh et al., APLInverse spin-Hall effect
cf. Magnonic Hall effect in frustrated magnets:
Fujimoto, PRL (2009)
Topological magnonic insulators:
Shindou et al., PRB (2013-2014),
Zhang et al. (2013), & Mook et al. (2014).
Equilibrium magnon-BEC Nikuni et al., PRL2000 (See remark by Bunkov & Volovik, arXiv:1003.4889)
80 years
76 years
10 years
≈
≈
BACKGROUND: Experimental Progress

6. Magnonic Hall Effects + …
1995: Haldane & Arovas, PRB
2009: Fujimoto, PRL
2010: Onose et al., Science
2010: Katsura et al., PRL
2011: Matsumoto & Murakami, PRL & PRB
2013: Shindou et al., PRB etc. (2013, 2013, 2014, 2016)
2013: Zhang et al., PRB
2014: Mook et al., PRB (2014, 2014, 2015)
Quantum Hall Effects
1982: Thouless, Kohmoto, Nightingale, and Nijs, PRL
1985: Kohmoto, Ann. Phys.
1985: Niu, Thouless, and Wu, PRB
・・・
2010: Xiao, Chang, and Niu, RMP
Observation of the magnon Hall effect
& the theories
Topological magnonic insulators
Phase twist & Berry curvature
in magnonic system
Picture from Google search
BACKGROUND: Magnonic Topological Insulator
NOTE: See [Haldane and Arovas, PRB (1995)] & [Xu, Ohtsuki, and Shindou, PRB (2016)] for disordered quantum Hall systems,
and [Matsumoto & Murakami, PRL & PRB (2011)], [Shindou et al., PRB (2013, 2014)], & their review [Murakami & Okamoto, JPSJ (2017)}
for chiral edge states in dipolar int. and the bulk-edge correspondence.

7. QUESTION

8. QUESTION
Electronic IQHE:
[Quantum Hall conductance] = [Chern integer] = [# of edge modes]
TKNN, PRL (1982)
Kohmoto, Ann. Phys. (1985)
Hatsugai, PRL (1997)
Halperin, PRB (1982)
Magnonic Hall effect:
[Hall conductance (clean)] = or ≠ [Chern integer] = [# of edge modes]
Shindou et al., PRB (2013)
Bulk-edge correspondence
Zhang et al., PRB (2013)
Mook et al., PRB (2015)Matsumoto & Murakami, PRL & PRB (2011)
∝ [Berry curvature]
?
Klitzing et al., PRL (1980)
A. Yes ! Only under a certain condition: KN, Klinovaja & Loss, arXiv:1611.09752 (2016).
Q. Does the relation hold also for magnons ?
Q. Magnonic Hall conductance： Is it really characterized by Chern integer ?

9. Meier & Loss, PRL (2003)
Magnonic quantum Hall effect & the WF law
KN, Klinovaja & Loss, arXiv:1611.09752 (2016)
Magnonic classical Hall effect in Aharonov-Casher phase
TKNN, PRL (1982)
Kohmoto, Ann. Phys. (1985)
Topological description:
STRATEGY

10. Geometric Phases
(Electrically) charged particle：
Magnetic vector potential
Magnon = Magnetic dipole:
Aharonov-Bohm phase Aharonov-Casher (AC) phase
Electric vector potential ~
Meier & Loss, PRL (2003). Mignani, J. Phys. A (1991)
Aharonov and Bohm, Phys. Rev. 115, 485 (1959) Aharonov and Casher, PRL, 53, 319 (1984)
𝑩 = 𝜵 × 𝑨
= A pair of oppositely charged magnetic monopoles
NOTE) Katsura et al., PRL (2005):
DM int.  Aharonov-Casher effect
Hoogdalem et al., PRB (2013)
Mook et al., PRB (2014, `15, `16). Zhang et al., PRB (2013)

11. Aharonov-Casher Effect & Landau Quantization
Electric field gradient 𝜀:
Electric vector potential：
 Cyclotron motion: Chiral edge state
Effective mass of magnon:
KN, Klinovaja & Loss, arXiv:1611.09752 (2016)
ℛ
DM int.  Vector potential analogous to 𝑨m
Landau gauge: + …
Landau gap: Δ𝐸 𝑛 = 2.5 meV = 18 K
e.g., 𝐽 = 80meV, 𝐷DM = 0.7meV, ℛ = 15nm etc.
Within experimental reach: Nagaosa & Tokura, Nat. Nanotech. (2013)
2) Skyrmion lattice induced by DM int.
1) External electric field gradient
Hoogdalem, Tserkovnyak, and Loss, PRB (2013)
Average fictitious field (textured magnetization)
Landau level：
[Katsura et al., PRL (2005)]
AC [Meier & Loss, PRL (2003)]
AB [Kohmoto, Ann. Phys. (1985)]

12. Magnonic Hall conductances ≠ Chern #
NOTE: Generally,
𝑛 = 0
𝑛 = 1
𝐸0𝒌
𝐸1𝒌
𝑘
~ℏ𝜔c
~ℏ𝜔c
𝑛 = 2
Magnon Hall Conductance 𝐺 𝑦𝑥
at Clean Bulk
Magnonic Bloch w.f.： &
Magnon Hall conductance:
Periodic lattice potential:
Periodic electric vector potential： 𝐴m 𝒓 = 𝐴m(𝒓 + 𝑹 𝑞) 𝑞 ∈ ℕ+𝑹 𝑞 = 𝑞𝐑
Bloch wave-vector 𝒌
Chern number: Topological invariant

13. Magnonic Hall conductances ≠ Chern #
NOTE: Generally,
𝑛 = 0
𝑛 = 1
𝐸0𝒌
𝐸1𝒌
𝑘
~ℏ𝜔c
~ℏ𝜔c
𝑛 = 2
Magnon Hall Conductance 𝐺 𝑦𝑥
at Clean Bulk
Magnonic Bloch w.f.： &
𝐸𝐸F
1
𝑛F
0
Fermion:
Magnon Hall conductance:
Periodic lattice potential:
Periodic electric vector potential： 𝐴m 𝒓 = 𝐴m(𝒓 + 𝑹 𝑞) 𝑞 ∈ ℕ+𝑹 𝑞 = 𝑞𝐑
Bloch wave-vector 𝒌
Chern number: Topological invariant

14. Quantized
𝑛 = 0
𝑛 = 1
𝐸0𝒌
𝐸0
∗
Almost flat band 𝐸 𝑛𝒌:
Band width Still
𝐸1𝒌
𝑘
~ℏ𝜔c
Chern number: Topological invariant
Magnon Hall conductance:
e.g., Almost flat band in skyrmion lattice induced by DM int.
[Hoogdalem, Tserkovnyak, and Loss, PRB (2013)]
Magnon Hall Conductance 𝐺 𝑦𝑥
at Clean Bulk
Magnonic Bloch w.f.： &
Periodic lattice potential:
Periodic electric vector potential： 𝐴m 𝒓 = 𝐴m(𝒓 + 𝑹 𝑞) 𝑞 ∈ ℕ+𝑹 𝑞 = 𝑞𝐑
Bloch wave-vector 𝒌
See also [Xu, Ohtsuki, and Shindou, PRB (2016)]

15. 𝐸0
∗
𝑘B 𝑇
Magnonic WF law in quantum Hall system：
Magnonic WF law
Thermal Hall Conductance 𝐾 𝑦𝑥
∝ 𝜈0
Matsumoto & Murakami, PRL (2011)
NOTE: 𝐾 𝑦𝑥
≠ 𝐿22/𝑇 for magnon
𝑦𝑥
𝑦𝑥
𝐿𝑖𝑗 ∝ 𝜈0: Quantized in almost flat band
Universal at low temperature (𝑘B 𝑇 ≪ 𝐸0
∗
):
KN, Klinovaja & Loss, arXiv:1611.09752 (2016)
NOTE: Broken in classical Hall regimes due to 𝐿𝑖𝑗
𝜇𝜇

16. : With off-diagonal elements
: Without off-diagonal elements
Off-diagonal Elements: Thermal Hall Conductance
Magnonic WF law
(a) (a’)
(b) (b’)
With off-diagonal:
Without off-diagonal:
Thermal conductance: The ratio: WF law
Satisfied
Broken
Magnonic WF law

17. Last Question
Q. Chiral edge magnon state:
Still exist in `periodic’ electric vector potential 𝑨 𝐦 ?
ANSWER: YES.
KN, Klinovaja & Loss, arXiv:1611.09752 (2016)

18. 𝑞 ≫ 1 𝑞 = 6
𝑞 = 4 𝑞 = 3
Isotropic case: 𝐽 𝑥 = 𝐽 𝑦
(a)-(d): Chiral edge states
(a)-(b): NOT flat bulk gap
Chiral Edge Magnon State: Isotropy
Tight-binding model:
AC phase:
Landau gauge:
Periodicity:
 Spectrum: 𝐸 = 𝐸(𝑘 𝑦)
cf., Spin Hamiltonian:
< 𝜋
𝑎 𝑦
𝑈
𝑈
𝑈
𝑈
𝑈
𝑈 𝑈
𝑈
𝑈
𝑎 𝑥
𝐴m
𝐴m𝐴m
𝑞 = 3

19. 𝑞 ≫ 1 𝑞 = 6
𝑞 = 4 𝑞 = 3
Isotropic case: 𝐽 𝑥 = 𝐽 𝑦
(c)-(d): Bulk gap ``closed’’
 Gapless
(a)-(d): Chiral edge states
(a)-(b): NOT flat bulk gap
~ Weyl systems
cf., Weyl magnon in AF [Li et al., Nat. Comm.(2016)]
NOTE) Weak disorder:
Edge mode will not couple to bulk
Chiral Edge Magnon State: Isotropy
Tight-binding model:
AC phase:
Landau gauge:
Periodicity:
 Spectrum: 𝐸 = 𝐸(𝑘 𝑦)
cf., Spin Hamiltonian:
< 𝜋

20. Tight-binding model:
AC phase:
Landau gauge:
Periodicity:
 Spectrum: 𝐸 = 𝐸(𝑘 𝑦)
𝑞 ≫ 1 𝑞 = 6
𝑞 = 4 𝑞 = 3
Anisotropic case: 𝐽 𝑥 ≠ 𝐽 𝑦
(c)-(d): Bulk gap ``closed’’
 Gapless
(a)-(d): Chiral edge states
(b): NOT flat bulk gap
~ Weyl systems
cf., Weyl magnon in AF [Li et al., Nat. Comm.(2016)]
Chiral Edge Magnon State: Anisotropy
cf., Spin Hamiltonian:
NOTE) Weak disorder:
Edge mode will not couple to bulk
< 𝜋

21. Q. Magnonic quantum Hall systems ?： WF law ?
Q. Magnonic Hall conductance： Is it really characterized by Chern integer in clean limit ?
SUMMARY
Electronic IQHE:
[Quantum Hall conductance] = [Chern integer] = [# of edge modes]
TKNN, PRL (1982)
Kohmoto, Ann. Phys. (1985)
Hatsugai, PRL (1997)
Halperin, PRB (1982)
Bulk-edge correspondence
Zhang et al., PRB (2013)
Mook et al., PRB (2015)
∝ [Berry curvature] Shindou et al., PRB (2013)
Matsumoto & Murakami, PRL & PRB (2011)
Magnonic Hall effect:
[Hall conductance (clean)] = or ≠ [Chern integer] = [# of edge modes]
≠
=
Generally
Almost flat
band
A. Yes, only in the almost flat band.
Klitzing et al., PRL (1980)
A. Yes, at lower temperature than the Landau gap in the almost flat band.
Magnonic Quantum Hall Effect & the Wiedemann-Franz Law
KN, J. Klinovaja & D. Loss, arXiv:1611.09752 (2016)

23. Appendix

24. Hall Currents vs Longitudinal Currents
A edge mode Many bulk modes  Longitudinal currents:
𝑞 = 4 𝑞 = 3
(c)-(d): Bulk gap ``closed’’
 Gapless
~ Weyl systems
cf., Weyl magnon in AF [Li et al., Nat. Comm.(2016)]
NOTE) Weak disorder:
Edge mode will not couple to bulk