magnonic quantum hall effect & the wiedemann-franz law

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)]

Magnon Carries 𝜇B & 𝑘B

≤ ≪

Magnon 𝜇B 𝑘B

Low-energy collective mode in insulating magnet

Yes !

QUESTION

Can magnon 𝜇B (boson) transport be similar to electron 𝑒 (fermion) transport ?

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]

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 current 2010 Kajiwara et al., Nature

Magnon-BEC current 2016 Hillebrands-group, Nat. Phys.

2016

Aharonov & Casher, PRL (1984) Aharonov-Casher effect on magnon 2014 Yale-group, PRL: Observation.

2006

Onose et al., Science Katsura et al., PRL (2010) Matsumoto & Murakami, PRL (2011)

Magnon thermal Hall effect

Spin-Seebeck effect 2008 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., APL Inverse 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., PRL 2000 (See remark by Bunkov & Volovik, arXiv:1003.4889)

80 years

76 years

10 years

≈ ≈

BACKGROUND: Experimental Progress

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.

QUESTION

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 ?

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

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)

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: Δ𝐸𝑛 = 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)]

Magnonic Hall conductances ≠ Chern #

NOTE: Generally,

𝑛 = 0

𝑛 = 1

𝐸0𝒌

𝐸1𝒌

𝑘

~ℏ𝜔c

~ℏ𝜔c

𝑛 = 2

Magnon Hall Conductance 𝐺𝑦𝑥 at Clean Bulk

Magnonic Bloch w.f.： &

Magnon Hall conductance:

Periodic lattice potential:

Periodic electric vector potential： 𝐴m 𝒓 = 𝐴m(𝒓 + 𝑹𝑞) 𝑞 ∈ ℕ+ 𝑹𝑞 = 𝑞𝐑 Bloch wave-vector 𝒌

Chern number: Topological invariant

Magnonic Hall conductances ≠ Chern #

NOTE: Generally,

𝑛 = 0

𝑛 = 1

𝐸0𝒌

𝐸1𝒌

𝑘

~ℏ𝜔c

~ℏ𝜔c

𝑛 = 2

Magnon Hall Conductance 𝐺𝑦𝑥 at Clean Bulk

Magnonic Bloch w.f.： &

𝐸 𝐸F

1

𝑛F

0

Fermion:

Magnon Hall conductance:

Periodic lattice potential:

Periodic electric vector potential： 𝐴m 𝒓 = 𝐴m(𝒓 + 𝑹𝑞) 𝑞 ∈ ℕ+ 𝑹𝑞 = 𝑞𝐑 Bloch wave-vector 𝒌

Chern number: Topological invariant

Quantized

𝑛 = 0

𝑛 = 1

𝐸0𝒌

𝐸0∗

Almost flat band 𝐸𝑛𝒌:

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 𝐺𝑦𝑥 at Clean Bulk

Magnonic Bloch w.f.： &

Periodic lattice potential:

Periodic electric vector potential： 𝐴m 𝒓 = 𝐴m(𝒓 + 𝑹𝑞) 𝑞 ∈ ℕ+ 𝑹𝑞 = 𝑞𝐑 Bloch wave-vector 𝒌

See also [Xu, Ohtsuki, and Shindou, PRB (2016)]

𝐸0∗

𝑘B𝑇

Magnonic WF law in quantum Hall system：

Magnonic WF law

Thermal Hall Conductance 𝐾𝑦𝑥 ∝ 𝜈0

Matsumoto & Murakami, PRL (2011)

NOTE: 𝐾𝑦𝑥 ≠ 𝐿22/𝑇 for magnon 𝑦𝑥

𝑦𝑥 𝐿𝑖𝑗 ∝ 𝜈0: Quantized in almost flat band

Universal at low temperature (𝑘B𝑇 ≪ 𝐸0∗):

KN, Klinovaja & Loss, arXiv:1611.09752 (2016)

NOTE: Broken in classical Hall regimes due to 𝐿𝑖𝑗 𝜇𝜇

: 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

Last Question

Q. Chiral edge magnon state: Still exist in `periodic’ electric vector potential 𝑨𝐦 ?

ANSWER: YES. KN, Klinovaja & Loss, arXiv:1611.09752 (2016)

𝑞 ≫ 1 𝑞 = 6

𝑞 = 4 𝑞 = 3

Isotropic case: 𝐽𝑥 = 𝐽𝑦

(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(𝑘𝑦)

cf., Spin Hamiltonian:

< 𝜋

𝑎𝑦

𝑈

𝑈

𝑈

𝑈

𝑈

𝑈 𝑈

𝑈

𝑈

𝑎𝑥

𝐴m

𝐴m 𝐴m

𝑞 = 3

𝑞 ≫ 1 𝑞 = 6

𝑞 = 4 𝑞 = 3

Isotropic case: 𝐽𝑥 = 𝐽𝑦

(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(𝑘𝑦)

cf., Spin Hamiltonian:

< 𝜋

Tight-binding model:

AC phase:

Landau gauge:

Periodicity:

Spectrum: 𝐸 = 𝐸(𝑘𝑦)

𝑞 ≫ 1 𝑞 = 6

𝑞 = 4 𝑞 = 3

Anisotropic case: 𝐽𝑥 ≠ 𝐽𝑦

(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

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)

Appendix

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