topology in magnetism - École polytechnique
TRANSCRIPT
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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