Topology in Magnetism - École Polytechnique

Topology in Magnetism

André THIAVILLE

Laboratoire de Physique des Solides UMR CNRS 8502

Université Paris-Sud, Orsay

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Topology (Listing 1847)

Greek: place, position Ex: topography biotope

Greek: science

Analysis situs (Leibniz, 1679)

(Latin) (Latin)

(Leibniz, Euler, Listing, Möbius, Riemann, Klein, Betti, Poincaré)

www.analysis-situs.math.cnrs.fr

The study of those properties of geometrical objects which remain unchanged under continuous transformations of the object (Ian Stewart, Concepts of modern mathematics, Dover, 1975)

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1 - TOPOLOGY & STATICS OF MAGNETIC TEXTURES

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Mapping on the unit sphere (Feldtkeller indicatrix) Role of the coverage of the sphere Singular point (Bloch point)

E. Feldtkeller Z. angew. Phys. 19, 530-536 (1965) [Z. angew. Phys. 17, 121-130 (1964)]

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E. Feldtkeller Z. angew. Phys. 19, 530-536 (1965)

Reversal of a magnetic core

S=1

S=0 S=2

Topologically stable configuration

Ferrite core memory

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Topological theory of defects: Looking for defects

In three-dimensional space : d=3

To catch a point (d’=0) take a sphere (r=2)

To catch a line (d’=1) take a circle (r=1)

To catch a surface (d’=2) take two points(r=0)

The hunter’s recipe

d’ + r + 1 = d

prey lasso real space

G. Toulouse, M. Kléman, J. Phys. Lett. 37, L149-L151 (1976). M. Kléman, Points, lines and walls (Wiley, Chichester, 1983).

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Recognizing if a defect is present : homotopy

Order parameter space V

(sphere)

(torus)

mapping

Physical space

21 S trivial

ZZT 21

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The homotopy groups

V1Fundamental group First homotopy group

Classes of equivalence of closed paths drawn on V

V2 Second homotopy group Classes of equivalence of « closed surfaces » drawn on V

.

.

.

V3 Third homotopy group

V0 Zeroth homotopy group Set of connected components of V

Classes of equivalence of « closed volumes » drawn on V

Computation of the : algebraic topology / geometrical intuition Vn

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1) The defects in standard magnetism

2SV 3D spins

20 S trivial 21 S trivial ZS 22

Number of spin components

1 nSV

Space dim.

Points

Lines

Walls 1nr S

trivial

Znr

nr

1

1

Ising XY Heisenberg

d’ + (n-1) + 1 = d Bloch point

2D vortex

Ising walls

V

M. Kléman, Points, lines and walls (Wiley, Chichester, 1983).

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2a) Topologically stable configurations in infinite samples

ASSUME magnetization uniform at infinity

THEN d

d SR ~2R

2S

ZSn 22:3Skyrmions are topologically stable configurations Bubbles (except if S=0) are topologically stable also

ZSn 11:2

Winding domain walls, for 2D spins

Mapping of the full configuration on V

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2b) Topologically stable configurations in finite samples

ASSUME magnetization is fixed on the sample boundary

Example : vortex in a disk, 3D spins, 2D space

+/- integer times the sphere is covered

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Topological defects vs. Topologically stable configurations

A topological defect cannot disappear by itself; need - expulsion from the sample - annihilation with opposite defect

hedgehog BP combed to embed in

BP

Sk tube

Annihilation of a topologically stable configuration: inject a topological defect from one surface

Creation of topological defects in the interior: in pairs with opposite signs

Bloch Sk tube

2 BP’s

Exchange energy diverges at the core Exchange energy density does not diverge

Back to ferrite core switching

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H 200 nm

400 nm

1000 nm

diameter :

T. Okuno et al., J. Magn. Magn. Mater. 240, 1 (2002)

Example: the vortex core reversal

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Ni80Fe20 « permalloy » disks, 50 nm thickness

240nm 240nm

Topography Magnetic image

Images by magnetic force microscopy (MFM)

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T. Shinjo et al., Science 289, 930 (2000)

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q

Large angle: vortex expulsion

Low angle: vortex core reversal

Two reversal processes

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Vortex (Disk diameter=200 nm, thickness=50 nm, mesh=2.5 nm; image : 60nm)

Vortex reversing its core : a Bloch point is involved

z= 28 nm z= 26 nm z= 24 nm z= 22 nm z= 0 nm z= 50 nm

z= 28 nm z= 26 nm z= 24 nm z= 22 nm z= 0 nm z= 50 nm

color code

A vortex core reversing by a travelling Bloch point

A. Thiaville, J.M. Garcia, R. Dittrich, J. Miltat, T. Schrefl, Phys. Rev. B 67, 094410 (2003)

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Topological numbers for a continuous texture

1nr Strivial

Znr

nr

:1

:1

: number of times V is covered

dSSn 1

1

Example : the skyrmion topological number

4),( 22 SSyxm

dxdymy

m

x

mdS

m

x

m

y

m

Positive surface m

x

m

y

m

Negative surface

dxdym

y

m

x

mNSk

4

1

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The winding number S of a magnetic « bubble »

Domain wall :

0

)(sin

)(cos

q

q

m

CS q

SS ,2)()2( qq

Periodicity requirement

integer

Generic case

q

0C 2/CCcase

1S

middle top Thick film: bottom X-ENS-UPS 2017 - A. Thiaville 25

The ancestors : magnetic bubbles (1970-1990)

H = 0 H > 0

Sample courtesy of Jamal Ben Youssef, Univ. Bretagne Occidentale (Brest)

200 mm

Epitaxial monocrystalline garnet film with perpendicular anisotropy, magneto-optical microscopy

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A.P. Malozemoff, J.C. Slonczewski Magnetic domain walls in bubble materials (Academic Press, 1979)

Bubbles convention: core is down (bias field applied)

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dxdymmm yx

)(

0C 2/C C

Surface covered on the unit sphere

4 4 4

mx

( red: blue: ) my

4pSNpS Sk ;4

polarity of the core winding number

For 1 domain

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1

1

1

SkN

p

S

1

1

1

SkN

p

S

Same topology, but no relation !

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1

1

1

SkN

p

S

1

1

1

SkN

p

S

Same topology, and some relation

Rotate spins by 180°

Same exchange energy Different demag energy

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2 – TOPOLOGY & DYNAMICS OF MAGNETIC TEXTURES

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2a – DYNAMICS OF MAGNETIC TEXTURES

MICROMAGNETICS

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0

0.2

0.4

0.6

0.8

1

m

my

mx

position x

atomic spins continuous distribution

Founding assumptions of Micromagnetics

x

y

mTMM s

)(

1) Fixed magnetization modulus

2) Slow variations at the atomic scale -> continuous model trm ,

1m

« micromagnetization »

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i j jiSSJE

Basic magnetic energy terms

H

Exchange

Anisotropy Demagnetising field

Applied field

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Micromagnetic equations : statics

Dss HmMHmMmKGmAE

00

2

2

1)()( mm

exchange anisotropy applied field demagnetizing field

+ boundary conditions

Statics : minimise Brown equations

effective field exchangeanisodemagappliedeff HHHHH

mM

A

s

0

2

mm

E

MH

s

eff

m 0

1

0

mxH eff

0

nm

rdEV

3

NB « functional derivative »

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Magnetization dynamics

/ML

Angular momentum dynamics

gyromagnetic ratio (>0)

meg

gB

2

m

dtLd

m

H

HmMs

0

m

mHdtmd

0

..102.25

00IS m

28 GHz/ T

Can be found directly from quantum mechanics

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Micromagnetic equations : dynamics

dtmdmmH

dtmd

eff 0

Landau-Lifshitz-Gilbert (LLG)

mHmmH effeff

2

0

1

Effective field exchangeanisotropydemagappliedeff HHHHH

mM

A

s

0

2

mm

E

MH

s

eff

m0

1

: Gilbert damping parameter

(solved form of LLG)

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Properties of the magnetization dynamics

0.2)(

2

dtmdm

dt

md

Conservation of the magnetization modulus

2

00

00

)/(.

..

dtmdMmxH

dtmdM

dtmdxmHM

dtmdHM

dtdE

seff

s

effs

effs

mm

mm

Decrease of the energy with time : the magnetic system is not isolated

1)

2)

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2b –TOPOLOGY & DYNAMICS OF MAGNETIC TEXTURES

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Topology and magnetization dynamics: the Thiele equation

LLG equation mmmHm tefft

0

`solved’ form mmmmH tteff

0/

ASSUME a magnetization structure in rigid translation

))((),( 0 tRrmtrm

Force on the structure

i

effs

i

effs

i

ix

mHM

R

mHM

dR

dEF 0

00

mm

j j

jtx

mVm 0

j ijj

js

ix

m

x

m

x

mmV

MF 000

0

0

0

m

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Topology and magnetization dynamics: the Thiele equation

A.A. Thiele, Phys. Rev. Lett. 30, 230 (1973); J. Appl. Phys. 45, 377 (1974)

0

FFF dissipgyro

Gyrotropic force

VGFg

Sk

ssz Nh

Mdxdydzm

y

m

x

mMG

m

m4

0

00

00

0

0

Dissipation force

VDF

dxdydz

x

m

x

mMD

ji

sij

00

0

0 .

m

j j

j

ji

sj

ji

si V

x

m

x

mMVm

x

m

x

mMF 00

0

00

00

0

0

m

m

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Applications of the Thiele equation

0

FVDVG

Simple wall (Gz=0) 0

FVD

hM

dxdzx

mMD

T

ssxx

2

0

0

2

0

0

0

m

m

hHMF sx 02mHV T

x

0

(both per unit length)

Magnetic bubble or skyrmion under a field gradient

Defines the Thiele domain wall width T

A.A. Thiele, J. Appl. Phys. 45, 377 (1974)

xz

y VD

GV

A.A. Thiele, Phys. Rev. Lett. 30, 230 (1973)

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Topological dynamics # 1: Skew propagation of “bubble domains”

Improved setup: rotating gradient

Vella-Coleiro 1972

Patterson 1975

Tabor 1972

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Topological dynamics # 2 : gyrotropic propagation of vertical lines in bubble garnets

A. Thiaville, J. Miltat, Europhys. Lett. 26, 57 (1994)

t= 0 200 400 ns

with BP, Gz=0

Hz= 5.4 Oe 500 ns 36

mm

2 2

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Topological dynamics # 3 : gyrotropic vortex motion

2

1

1

p

S

2

1

1

p

S

2

1

1

p

S

2

1

1

p

S

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Topological dynamics # 3 : gyrotropic vortex motion

B. Van Waeyenberge et al., Nature 444, 461 (2006) X-ENS-UPS 2017 - A. Thiaville 46

Topological dynamics # 4 : skyrmions transverse deflection under current pulses

W. Jiang et al., Nat. Phys. 13, 162 (2017)

2.8 1010 A/m2, 50 ms

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A. Thiaville et al., Europhys. Lett. 69, 990 (2005)

More applications of the Thiele equation

0

FuVDuVG

Thiele equation under CIP STT

Free structure (F=0), no gyrovector uV

/

Free structure (F=0), with gyrovector, non nonadiabatic term (=0)

22

2

)( DG

uGuGDV

z

z

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Spin transfer torque (CIP geometry)

electrons

before after

CPP spin transfer between successive x slices

L. Berger, J. Appl. Phys. 49, 2156 (1978)

Adiabatic limit (walls are wide): carrier spins always along local magnetization -> angular momentum given per unit time in the slab dx

))()(( dxxsxsPe

J

dxx

mP

e

J

2dx

dt

mdM s

=

s

m

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mux

mu

dt

md

transferspin

)(_

s

B

Me

gPJu

2

m

Permalloy : Cm

Me

g

s

B /1072

311

m

1 x 1012 A/m2 & P = 0.5 u = 35 m/s

u : a velocity that expresses the spin transfer (spin drift velocity) (Zhang & Li : bJ)

Spin transfer torque in continuous form (CIP)

« adiabatic » term

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Full LLG equation under CIP-STT

mmumummmHm xxtefft

0

"adiabatic term" "non-adiabatic term"

Solved form

mmumu

mHmmHm

xx

effefft

1

1

1002

Initial velocity for step current

A. Thiaville et al., Europhys. Lett. 69 990 (2005)

uV20

1

1

m

E

MH

s

eff

m0

1 effective field of other micromagnetic terms

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M. Kläui et al.,

Phys. Rev. Lett. 95, 026601 (2005)

10 nm thick 500 nm wide Ni80Fe20

2.2 1012 A/m2

10 ms

Vortex walls move easily by spin-transfer torque

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T. Schulz et al. Nat. Phys. 8, 301 (2012)

Topological Hall effect in the skyrmions phase of MnSi

A. Neubauer et al. Phys. Rev. Lett. 102, 186602 (2009)

Emergent electric field when skyrmions are moving (by DC current)

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L. Berger, Phys. Rev. B 33, 1572 (1986) G.E. Volovik, J. Phys. C 20, L83 (1987)

Spin electro-motive force by a moving magnetic texture

i

ix

m

t

mm

eE

2

Effective electric field acting on majority (+) or minority (-) spin electrons when going everywhere in the moving spin frame

w

V

eP

dt

d

ePU

y

x2

vortex wall

w

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S.A. Yang et al., Phys. Rev. Lett 102, 067201 (2009)

Spin emf detection for domain wall motion

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K. Tanabe et al., Nat. Commun. 3: 845 (2012)

Spin emf detection for vortex gyration

mz mx,y V

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3 – BREAKING THE TOPOLOGY

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`Topological protection’ of a bubble

Cannot collapse continuously

Can collapse continuously

Cannot collapse continuously

Cannot collapse continuously

lowest energy

No bubble: S = 0

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The Bloch point has a finite energy

r

rm

r

Aeexc

2

2for + rotations

RAEexc 8 R radius where BP profile applies

Exchange Energy density : Total energy:

hedgehog > circulating > spiraling

Demag energy

W. Döring, J. Appl. Phys. 39, 57 (1968)

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Field switching of vortex cores in NiFe disks: calculations for perfect samples

BP injection is assisted by defects

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0 10 20 30 40 50 60

-500

-450

-50

0

50

100

En

erg

y (

me

V)

Distance along path (rad)

Path 1 ref. [2]

Path 2 ref. [2]

this work

0 10 20 30 40 50 60-600

-400

-200

0

200

400

600

800

E-E

uniform

(m

eV

)

Distance along path (rad)

Heisenberg Exchange

DMI K + demag. + DMI

Anisotropy (K)

Dipolar coupling (demag.) K + demag.

Total energy

1.2 1.4 1.6 1.80

50

100

150

200

250

Skyrm

ion

su

rfa

ce

(n

m²)

E from Ref. [2]

E (this work)

E

(m

eV

)

d (meV)

0

50

100

Skyrmion surface

S. Rohart, J. Miltat, A. Thiaville, Phys. Rev. B 93, 214412 (2016) & 95, 136402 (2017) I. Lobanov, H. Jonsson, V.M. Uzdin, Phys. Rev. B 94, 174418 (2016)

Skyrmion annihilation in 1 ML Co / Pt (111)

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4 – BEYOND TOPOLOGY

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Chirality matters also CS q

0C 2/C Ccase

1S

For z axis up Néel skyrmion Bloch skyrmion Néel skyrmion left-handed right-handed right-handed

N. Nagaosa, Y. Tokura, Nat. Nanotech. 8, 898 (2013) F. Hellman et al., arXiv: 1607:00435 (to appear in Rev. Mod. Phys.)

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Chirality enforced by antisymmetric exchange

Simplest bulk case D(u) // u Ultrathin film case

jiij

antisym

ij SSDE

jS

iS

ji SS

ijD

ijr

jS '

ji SS '

Favors an helix

substrate

ijr

ijD

jS

iS

ji SS

jS '

ji SS '

Favors a cycloid

Dzyaloshinskii-Moriya interaction (DMI)

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calculated

observed (stripes: Uchida et al., Science 311, 359 (2006))

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Domain wall motion by the spin Hall effect

Co 0.6 nm

Pt 3 nm electrons

CIP + spin Hall in Pt CPP with y polarized

(spin-orbit scattering) reference layer

)(1

ymmt

m

SHE

teM

gJ

teM

gJ

s

BHx

s

Bzspin

22

1 , mqm

Slonczewski

CPP-STT

First demonstration of the effect : P.P.J. Haazen, E. Murè, J.H. Franken, R. Lavrijsen, H.J.M. Swagten, B. Koopmans Nat. Mater. 12, 299 (2013)

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SHE force on a magnetic structure (Néel skyrmion)

LLG with SHE pmmmmmHm eff

10

Solve for Heff

(Thiele procedure) mpmmmmHeff

11

0

The forces are mHM

dX

dEF xeff

sx

0

0

m

pmmM

mpmM

F xs

xsSHE

x

m

m

0

0

0

0

DMI energy density (interfacial DMI)

xyyx

y

zz

yx

zz

x

DM

mmmmD

y

mm

y

mm

x

mm

x

mmDe

)()(

SHE: for j//x one has p//y

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LLG with SHE pmmmmmHm eff

10

Solve for Heff

(Thiele procedure) mpmmmmHeff

11

0

The forces are mHM

dY

dEF yeff

sy

0

0

m

pmmM

mpmM

F ys

ysSHE

y

m

m

0

0

0

0

DMI energy density (volumic DMI)

yyxx

zx

xz

y

zz

y

DM

mmmmD

y

mm

y

mm

x

mm

x

mmDe

)()(

SHE: for j//x one has p//y

SHE force on a magnetic structure (Bloch skyrmion)

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Conclusions

Topology : barrier of language and mathematics, but a powerful tool especially when geometrical vision insufficient Magnetism a rather simple case, as order parameter is simple Topology has measurable and visible consequences - dynamics of magnetic textures - electric transport in the presence of magnetic textures

Topological protection is not absolute; topological defects exist Topology does not describe everything: case of chirality

Next : references

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References : magnetism of matter and topology

E. Feldtkeller Mikromagnetisch stetige und unstetige Magnetisierungskonfigurationen Z. angew. Phys. 19, 530-536 (1965). A.A. Thiele Steady-state motion of magnetic domains Phys. Rev. Lett. 30, 230-233 (1973). A.A. Thiele On the momentum of ferromagnetic domains J. Appl. Phys. 47, 2759-2760 (1976). J.C. Slonczewski Force, momentum and topology of a moving magnetic domain J. Magn. Magn. Mater. 12, 108-122 (1979). H.-B. Braun Topological effects in nanomagnetism Adv. Phys. 61, 1-116 (2012). N. Nagaosa, Y. Tokura Topological properties and dynamics of magnetic skyrmions Nat. Nanotech. 8, 899-911 (2013).

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References : topological theory of defects and structures in condensed matter

G. Toulouse, M. Kléman Principles of a classification of defects in ordered media J. Phys. Lett. 37, L149-L151 (1976). G.E. Volovik, V.P. Mineev Line and point singularities in superfluid 3He JETP Lett. 24, 561-563 (1976). M. Kléman Points. Lignes. Parois. (Editions de Physique, Orsay, 1977); Points, lines and walls (Wiley, Chichester, 1983). N.D. Mermin The topological theory of defects in ordered media Rev. Mod. Phys. 51, 591-648 (1979). L. Michel Symmetry defects and broken symmetry. Configurations hidden symmetry Rev. Mod. Phys. 52, 617-651 (1980). H.R. Trebin The topology of non-uniform media in condensed matter physics Adv. Phys. 31, 195-254 (1982).

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