Kouki Nakata

University of Basel Switzerland

Magnon Transport Theory

All the responsibility of this slide rests with `Kouki Nakata' (2016)

10 pages digest of the works at Basel 2014-2015

Yes !

[Phys. Rev. B 90, 144419 (2014)] [Phys. Rev. B 92, 014422 (2015)] [Phys. Rev. B 92, 134425 (2015)]

We have established it: Magnon counterpart of electron transport

Q. Can we control magnon transport 𝝁𝑩 like electrons 𝒆 ?

FINAL GOAL Establish valid methods to control magnon transport 𝝁𝐁

Charge transport 𝑒 Magnon transport 𝜇B

Wiedemann-Franz (WF) law [R. Franz and G. Wiedemann,

Annalen der Physik 165, 497 (1853)]

Thermoelectric property

Magnon Wiedemann-Franz law [K. Nakata et al., Phys. Rev. B 92, 134425 (2015)]

Thermomagnetic property

Superconducting state [H. K. Onnes (1911)]

Persistent charge current [M. Buttiker et al. Phys. Lett. A, 96, 365 (1983)]

Magnon-BEC [S. O. Demokritov et al., Nature 443, 430 (2006)]

Persistent magnon-BEC current [K. Nakata et al., Phys. Rev. B 90, 144419 (2014)]

Josephson effect [B. D. Josephson, Phys. Lett. 1, 251 (1962)]

Magnon Josephson effect [K. Nakata et al., Phys. Rev. B 90, 144419 (2014)]

Quantum Hall effect [K. v. Klitzing et al., Phys. Rev. Lett. 45, 494 (1980)]

Magnon quantum Hall effect [K. Nakata & D. Loss, to be submitted (2016)]

Find the counterpart !!

Guiding Principle

Magnon Wiedemann-Franz Law

VS

(Free electron at low temp.) Low temp.：

Electron (metal) Magnon (FI)

R. Franz and G. Wiedemann [Annalen der Physik 165, 497 (1853)]

K. Nakata and D. Loss [Phys. Rev. B 92, 134425 (2015)]

Fermion Boson Statistics

Lorenz number

WF law （Low temp.）

𝑇-linear behavior = Universal

1853

[R. Franz and G. Wiedemann, Annalen der Physik 165, 497 (1853)]

Wiedemann-Franz Law for Electron Transport in Metal

Wiedemann-Franz Law for Magnon Transport in FI [K. Nakata and D. Loss, Phys. Rev. B 92, 134425 (2015)]

2015

Experiment [S. O. Demokritov et al., Nature 443, 430 (2006)]

Quasi-equilibrium Magnon-BEC

Experimental result by [A. A. Serga et al., Nat. commun. 5, 3452 (2014)]

Microwave pumping: Room temperature

Magnon VS Magnon-BEC

Incoherent spin precession Macroscopic coherent spin precession Macroscopic spins

= Sum of variety kinds of modes Macroscopic number of magnons occupies a single state

Quasi-equilibrium condensation

Part I: Magnon Part II: Magnon-BEC

= Spin-wave ~ Superconducting state of spin-wave

dc ac : Josephson effect

BEC

BEC

Magnetic field difference:

Part II: Condensed magnon (BEC) Part I: Non-condensed magnon

ac/dc Properties

J J J

J

J

J

J J

BEC

Magnon-BEC Ring Analogous to superconducting ring

A-C phase Persistent magnon-BEC current

E (A-C phase)

(Note; as long as magnons are in condensation)

Non-condensed magnon

Cylindrical wire

Condensed magnon

(1) Magnetic current (2) (3)

𝜇B

𝑉m 𝐼m

Electromagnetism by Magnon Current

[D. Loss and P. M. Goldbart, Phys. Lett. A 215, 197 (1996)] [F. Meier and D. Loss, Phys. Rev. Lett. 90, 167204 (2003)]

(Flow of magnetic dipole)

𝐸m

Magnon Transport Ferromagnetic Insulator

Universal thermomagnetic relation Magnon Seebeck & Peltier effects

III. Measurement II. Magnon-BEC Berry phase Josephson & persistent currents

Electromagnetic control Direct detection

I. Wiedemann-Franz Law for Magnon in FI

SUMMARY

[Phys. Rev. B 90, 144419 (2014)] [Phys. Rev. B 92, 014422 (2015)] [Phys. Rev. B 92, 134425 (2015)]

Supplemental Material

Appendix: Part I

Magnon & Heat Currents

Magnon current

Heat current

: Magnon lifetime (phenomenologically introduced)

Onsager relation:

Integrating over

Linear response:

Magnon & Heat Currents

Onsager Matrix Magnon current

Heat current

Onsager coefficient

Onsager relation

Polylogarithm function:

Exponential integral: Euler constant:

Cross-section area of the junction interface:

Thermal Conductance 𝑲 for Boson

Note: Definition of thermal conductance

with

WF law

Magnon current Heat current

Magnetic conductance: 𝑮

Thermal conductance: 𝑲

for fermions

for bosons (magnons)

Thermo-electric & –magnetic Effects VS

(Free electron at low temp.) Low temp.：

Electron (metal) Magnon (FI)

R. Franz and G. Wiedemann [Annalen der Physik 165, 497 (1853)]

K. Nakata, P. Simon, and D. Loss [Phys. Rev. B 92, 134425 (2015)]

Fermion Boson Statistics

Onsager relation

Thomson relation

Seebeck & Peltier

Lorenz number

WF law （Low temp.）

REMARK There was a possibility at low temp.：

Magnon WF law in FI:

𝐾

𝐺= (

𝑘B

𝑔𝜇B)2𝑇 ∙

𝑘B𝑇

𝑔𝜇B𝐵

𝑛−1

∝ 𝑇𝑛

𝐾

𝐺= (

𝑘B

𝑔𝜇B)2𝑇 ∝ 𝑇

Anisotropic spin 𝜂 ≠ 1 Magnon-magnon interactions

At such low temperatures: Phonon contributions are negligibly small

WF law & Onsager relations: Broken

Contributions of the breakings: Negligibly small at low temperatures 𝒪(10−1)K

WF law & Onsager relations: Approximately satisfied at such low temperatures [Note: Originally (𝜂 = 1), the WF law is realized at such low temperatures]

Broken Relations & Low Temperature

[H. Adachi et al., Appl. Phys. Lett. 97, 252506 (2010)]

Magnon VS Magnon-BEC

Incoherent spin precession Macroscopic coherent spin precession = Macroscopic spins

= Sum of variety kinds of modes = Macroscopic number of magnons occupies a single state

Quasi-equilibrium condensation

Number density： Number density：

Part I: Magnon Part II: Magnon-BEC

= Spin-wave ~ Superconducting state of spin-wave

Cooper pair: 𝑐𝒌↑𝑐−𝒌↓ BCS ≠ 0

Appendix: Part II

Microwave pumping Non-equilibrium steady state

Quasi-equilibrium magnon-BEC = [Metastable state]

≠ [Ground state]

Thermalization

Microwave: Switched off

FMR

B B

Quasi-equilibrium magnon-BEC

[C. D. Batista et al., Rev. Mod. Phys., 86, 563 (2014)]

= Dynamical condensation ≠ Thermal condensation

U(1)-symmetry: Broken U(1)-symmetry: Recovered

Quasi-equilibrium Magnon-BEC

Quasi-equilibrium Magnon-BEC [C. D. Batista et al., Rev. Mod. Phys., 86, 563 (2014)]

= Dynamical condensation ≠ Thermal condensation

Magnon-BEC Order Parameter [Textbook by Leggett] BEC: Einstein for free particles (i.e., no interactions)

Single-particle density matrix

𝜌1(𝒓, 𝒓′; 𝑡);

Probability amplitude 𝜌1(𝒓, 𝒓′; 𝑡) ≡ 𝜓 +(𝒓𝑡)𝜓(𝒓′𝑡)

(𝜓: Bose field)

Single eigenvalue Single BEC Several eigenvalues Fragmented BEC

Penrose & Onsager (1956)

lim𝒓−𝒓′→∞

𝜌1 𝒓, 𝒓′; 𝑡 = Ψ∗ 𝒓𝑡 Ψ (𝒓′𝑡)

Ψ(𝒓𝑡) ≡ 𝜓 (𝒓𝑡) : BEC order parameter = Off-diagonal long-range order (ODLRO) Widely used in BEC community

Yang (1962)

Extension of definition including interactions

Quasi-equilibrium magnon-BEC by microwave pumping satisfies this condition Experiment [S. O. Demokritov et al., Nature 443, 430 (2006)]

Quantum ? OR Classical ? [A. Ruckriegel and P. Kopietz, PRL 115, 157203 (2015)]

Magnetic field difference:

Period of ac Josephson effect:

Parameter values:

Josephson current:

Josephson magnon current:

Adjusting parameters:

10 ns

ac Josephson Effect

ac Josephson Effect: Nonlinear Effect

Magnon Josephson Eq.:

Josephson current:

Time-evolution of phase:

Nonlinear effect: 𝑧(0) ≠ 0 & Δ𝐵 = 0 ac Josephson effect Period 𝑇~10ns at weak 𝐽ex

Nonlinear effect

Linear effect

Experimental reach 1T/cm: Linear effect ≪ Nonlinear effect

Period 𝑻 of ac Josephson effect:

e.g.：

Within experimental reach

𝑧 0 = 0.6

Initial population imbalance:

dc Josephson Effect

Electric field Magnetic field

No A-C phase A-C phase

Electromagnetically realizable by applying an increasing magnetic field:

: dc effect ? : dc effect

Macroscopic Quantum Self-Trapping

MQST

(a)

(b)-(d)

MQST occurs when

``Direct Observation of Tunneling and Nonlinear Self-Trapping in a Single Bosonic Josephson Junction’’ [M. Albiez et al., PRL 95, 010402 (2005)]

MQST in Cold Atoms Already experimentally observed [M. Albiez et al., PRL 95, 010402 (2005)]

time (p = 50)

Destabilized deviation: 1/p << 1

Stable

Magnon-BEC Ring

Quantization：

time

1/p

1

Device for Direct Measurement To detect persistent quantized magnon-BEC current in the ring

Appendix: Others

3-dim Cubic Ferromagnet

Holstein-Primakoff (H-P) transformation

Heisenberg spin model:

Standard textbook [K. Kubo] on magnetism tells us:

Fourier transformation:

Parabolic dispersion: → 0 (𝑘 → 0)

Magnon = A kind of Nambu-Goldstone mode

= Massless particle = Non-relativistic magnon

𝜔𝑘

𝑘

Picture from google search

d-dim Cubic Anti-Ferromagnet (d≥3)

Bogoliubov transformation New Magnon Operators: 𝛼 & 𝛽

Diagonalization:

Linear dispersion: Relativistic magnon = Dirac magnon on d-dim AF

𝑘

𝜔𝑘

Standard textbook [K. Kubo] on magnetism tells us:

Nambu-Goldstone (NG) Theorem Magnon = A kind of NG mode (particle)

A continuous symmetry is spontaneously broken (SSB) Massless particles = NG boson

Heisenberg model: SSB of SO(3) Magnon = NG mode

B

A

A B

Magnon

[TEXTBOOK by Peskin]

``Rough & intuitive’’ correspondence

(massless)

Picture from wiki.

Picture from Google search

The relation between [# of broken symmetries] & [# of NG particles]: See [Watanabe-Murayama] & [Hidaka]

Mermin–Wagner–Hohenberg-Coleman theorem:

Continuous symmetries cannot be spontaneously broken (NO SSB):

- 𝑑 ≤ 2 - At finite temperature - Sufficiently short-range interactions

Absence of NG particles (e.g. magnons) on 𝑑 ≤ 2

Why 3-dim ?

See also recent development: [Phys. Rev. Lett. 107, 107201 (2011)] D. Loss, F. L. Pedrocchi, and A. J. Leggett

NOTE: The absence of SSB is valid only in the thermodynamic limit Ordering in finite size at finite temperatures is possible

Picture from Google search

Lattice structure

Band structure

Dirac Magnon on 2-dim Honeycomb Lattice

Magnon Dirac Eq.

[arXiv:1512.04902] J. Fransson, A. M. Black-Schaffer, and A. V. Balatsky

Dirac Magnon

AF Dirac Magnon

Dirac magnons are inherent to honeycomb lattice (geometric properties): Ferro or AF does not matter

In sharp contrast to cubic lattice

Ferromagnet Anti-ferromagnet

3-dim cubic lattice Since 1930

𝜔𝑘 ∝ 𝑘2 Non-relativistic

𝜔𝑘 ∝ 𝑘 Relativistic

2-dim honeycomb lattice [arXiv:1512.04902]

𝜔𝑘 ∝ 𝑘 Relativistic

𝜔𝑘 ∝ 𝑘 Relativistic

[arXiv:1512.04902] J. Fransson, A. M. Black-Schaffer, and A. V. Balatsky