アトミックGS 12

場の理論と物性論における
トポロジカル量子現象
～スカーミオンを中心に ～
2012/8/23
Muneto Nitta (新田宗土)
Keio U. (慶應義塾大学)
① スピノールBEC
共同研究者 cond-mat.
川口由紀，小林信吾，上田正仁（東大本郷），
小林未知数（東大駒場）， 内野瞬（スイス）
② 多成分BEC
笠松健一 (近畿大)，竹内宏光（広島大），
坪田誠（大阪市大），衛藤稔（山形大）
③ BECにおける人工ゲージ場
川上巧人, 水島健, 町田一成（岡山大）
④ フェルミ気体・超伝導
高橋大介(東大駒場)，土屋俊二（東京理大），
吉井涼輔（京大基研）, Giacomo Marmorini（理研）
⑤ 非可換統計
安井繁宏, 板倉 数記（KEK)，広野雄士（東大/理研）
ボゾン系
フェルミオン系
Plan of my talk
§1 Introduction(BEC and Vortices) (13p)
§2 Skyrmions (7p)
§3 Multi-component BECs (7p+3p)
§4 3D Skyrmions in BECs
§4-1 Brane annihilation (4p+22p)
§4-2 Non-Abelian gauge field (7p)
§5 Conclusion (1p)
Plan of my talk
§1 Introduction(BEC and Vortices) (13p)
§2 Skyrmions (7p)
§3 Multi-component BECs (7p+3p)
§4 3D Skyrmions in BECs
§4-1 Brane annihilation (4p+22p)
§4-2 Non-Abelian gauge field (7p)
§5 Conclusion (1p)
1924 Bose Einstein Condensation(BEC), Bose & Einstein
BEC occurs when de Broglie wave length λ of particles
is comparable with the mean distance.
E=
2

λT−2 ≈ k BT
2m
λT =
h2
2πmk BT
Transition temperature
T0 =
3.31  N 


mk B  V 
2
Number of condensates
T
N0
= 1− 
T
N
 0




3/ 2
2/3
``Pure” BEC (99% is BEC)
Cold atomic gases
1995 cold atomic bose gas
87Rb, 23Na, 7Li
Cornell (Colorado), Ketterle(MIT)
& Wieman (Colorado)
2003 cold atomic fermion gas
JILA(Colorado)， MIT
doppler laser cooling
magneto-optical trap
evaporative cooling
Temperature ~ 10- 6,10- 7 K
Number ~ 106, Size ~ 10- 3cm
trapping potential
1
V = Mω 2 r 2
2
ω : frequency
M : mass of atoms
Mω 2 R 2
p2
3
≅
≅ k BT
2
2M
2


R
≅
≈ 1/ 3
λT =
p
MωR
N
de Broglie
wave length
transition temperature
Ψ
R
R≅

N 1/ 6
Mω
mean particle
distance
ωN 1 / 3
3
6
T ≅
≈ 10 −6 [ K ] for ω ≈ 10 [ Hz ], N ≈ 10
kB
Scalar BEC, 4He superfluid
Bogoliubov theory for weakly interactive Bose gas
(with point interaction)
g
V ( r ) = gδ ( r ) point interaction
mean field approximation
ψ ( x ) = Ψ( x ) + φ ( x )
wave function
for condensation
fluctuation (phonon)
: non-condensed component
Scalar BEC, 4He superfluid
Gross-Pitaevskii (nonlinear Schrödinger) Equation
∂ψ   2 2
i
= −
∇ + Vext − µ + g ψ
∂t  2 M
µ
2

δE
ψ = δψ *

4π 2 a S
g≡
M
M : mass of atoms a S : s-wave scattering length
1
Vext(r) : trapping potential Vext = Mω 2 r 2
2
: chemical potential
Gross-Pitaevskii energy functional
 2
g 4
2
2
∇ψ + (Vext − µ )ψ + ψ 
E [ψ ] = ∫ d r 
2
 2M

3
For d=1 with Vext=0, it is integrable. [Zakharov-Manakov (‘74)]
It is used in optics and water waves. Examples are bright soliton and dark soliton.
vortex
y
real
Order Parameter Space(OPS)
space
= U(1)
Superconductors under magnetic field
Flux quantization k ∈ π [U (1)] ≅ Z
1
hc
hc
Φ = k = Φ 0k Φ 0 = = 2.07 × 10−15 [ weber ]
2e
2e
x
U. Essmann and H. Trauble
Max-Planck Institute, Stuttgart
@ Oslo Superconductivity Lab http://www.fys.uio.no/super/vortex/
Physics Letters 24A, 526 (1967)
Gallery of Abrikosov Lattices in Superconductors
quantization of circulation
k ∈ π 1[U (1)] ≅ Z
∫ dr ⋅ v
ψ = f ( r )e
eff

=
k
M
v eff
1
=
2i
Ψ ∗∇Ψ − Ψ∇Ψ ∗
Ψ ∗Ψ
ikθ
energy
k2
1− 2 2
mφ r
system size Λ
T = 2πv 2 k 2 log Λ
2
4
π
v
Inter-vortex F =
distance R
R
force
tension
Vortex nucleation under rotation
M
Rotation in rotating frame ∇ → ∇ − i Ω × r

2
2

M
g
2


3  
E [ψ ] = ∫ d r 
Ω × r ψ + (V − µ )ψ + ψ
∇ − i

2


 2M 
4





Abo-Shaeer, Raman, Vogels, Ketterle, Science 292, 476-479 (2001)
A proof of superfluidity
BEC/BCS Crossover
Fermions with
pseudo spin
Zwierlein, Abo-Shaeer,
Schirotzek, Schunck
& Ketterle
Nature 435, 1047-1051
(23 June 2005)
A proof of superfluidity
in all range of BES/BCS
Artificial Gauge Field A review: J.Dalibard et.al.,
Rev. Mod. Phys. 83, 1523–1543 (2011)
Two-state model
e
Two states { g , e }
Hamiltonian
g
coupling
Eigenstates of U = Dressed states
eigenvalues
Full state
Born-Oppenheimer
approximation
Gauge
A jl = i χ j ∇χ l
field
Neglecting ψ 2 , EOM of ψ 1
Gauge fields
as Berry phase
Synthetic magnetic fields for ultracold neutral atoms
Lin, Compton, Jimenez-Garcia, Porto & Spielman,
Nature 462, 628-632 (3 December 2009)
3 states
interaction
1 dark state
2 bright states
Adiabatic approx
Synthetic magnetic fields for ultracold neutral atoms
Lin, Compton, Jimenez-Garcia, Porto & Spielman,
Nature 462, 628-632 (3 December 2009)
A proof of artificial magnetic field
Plan of my talk
§1 Introduction(BEC and Vortices) (13p)
§2 Skyrmions (7p)
§3 Multi-component BECs (7p+3p)
§4 3D Skyrmions in BECs
§4-1 Brane annihilation (4p+22p)
§4-2 Non-Abelian gauge field (7p)
§5 Conclusion (1p)
What is a Skyrmion?
1D Skyrmion
=Sine-Gordon kink
1
1
π (S ) = Z
2D Skyrmion
Model of
nucleon
in HEP
π 2 (S 2 ) = Z
3D Skyrmion
π 3(S 3 ) = Z
T.H.R. Skyrme
3dim
hedgehog
A Nonlinear theory of strong interactions
Proc.Roy.Soc.Lond. A247 (1958) 260-278
A Unified Field Theory of Mesons and Baryons
Nucl.Phys. 31 (1962) 556-569
1D Skyrmion O(2) model (=sine-Gordon model)
Bogomol’nyi completion
2E =
[
] ∫ dx (∂ θ )

∫ dx (∂ xθ ) + cos θ =
2
2
x

θ 
+ sin 2   − 1
2

2


θ

 
θ 
= ∫ dx  ∂ xθ  sin    ± 2∂ xθ sin   − 1 ≥ T1D
 2 
2


Bogomol’nyi-Prasad-Sommerfield
(BPS) equation
SG Topological charge
Sine-Gordon T1D = ± ∫ dx∂ xθ sin (θ / 2 )
θ 
∂ xθ  sin   = 0
kink
= 2 ∫ dx∂ x cos(θ / 2 )
2
k ∈π1(S1) = Z
= 2[cos(θ / 2 )]x = −∞
x = +∞
O(3) sigma model 1. (Truncated model of) 2component BECs
2. Ferromagnet
1
2
E = (∇S ) S(x)=(S1,S2,S3)
S 3 = +1
N
2
2
S =1
u=∞
2
Target space = S2
S
Stereographic
S − iS2
u= 1
coordinate u
1 − S3
equivalent to
CP1 model
E=
∫ dr
∑α
∂αu
(1 + u )
2
2 2
u
S S 3 = −1
u=0
(=lump, sigma model instanton)
2D Skyrmion
Bogomol’nyi completion
2
∂
∑α α u
2
E=
∫d
x
(1 + u )
2 2
 ∂ u  i∂ u 2 i (∂ u *∂ u − ∂ u *∂ u )
x
y
x
y
y
x
2

 ≥ TL
= ∫d x
±
2 2
2 2
 1+ u

u
1
+


(
)
BPS equation
∂ x u  i∂ y u = 0
∂ zu = 0
z ≡ x + iy
(
)
2D Skyrme topological charge
TL = ± ∫ d 2 x
= 2πk
i (∂ x u *∂ y u − ∂ y u *∂ x u )
(1 + u )
2 2
k ∈π 2 (S 2 ) = Z
2D Skyrmion
BPS equation
∂ z u = 0 z ≡ x + iy
u
−1
k
λi
i =1
z − zi
=∑
∂ x u  i∂ y u = 0
N S 3 = +1
u=∞
S
u → ∞ (| z |→ ∞ )
u → 0 ( z → zi )
k =1
TL = 2πk
k ∈π 2 (S ) = Z
2
S S 3 = −1
u=0
2
2D Skyrmion
Cond-mat examples:
Ferromagnet, quantum Hall systems
Spin 1 BEC, Polar phase
G
U (1)Φ × SO (3) F
≅
H P ( Ζ 2 )Φ + Fx × U (1) Fz
S 1Φ × S 2 F
≅
( Ζ 2 )Φ + F
 G 
 ≅ Ζ
π 2 
 HP 
Choi, Kwon, and Shin,
PRL 108, 035301 (2012)
3D Skyrmion
 φ1 ( x ) 
2


∈
C
φ ( x) 

 2
S ≅ SU ( 2)
3
O(4) sigma model~Skyrme model
π 3(S ) = Z
3
| φ1 |2 + | φ2 |2 = 1
S
3
U†U = 1,
det U =| φ1 |2 + | φ2 |2 = 1
N:
 φ1 − φ2* 

U ≡
 φ φ *  ∈ SU ( 2)
 2 1 
Skyrmion ansatz
 f ( r )r ⋅ σ 
U ( x ) = exp i

r


→ +12 , f ( r ) → 0 ( r → ∞ )
→ −12 , f ( r ) → 1 ( r → 0)
U = 12
(φ1 , φ2 ) = (1,0)
3
S
S2
S:
U = −12
(φ1 , φ2 ) = ( −1,0)
Plan of my talk
§1 Introduction(BEC and Vortices) (13p)
§2 Skyrmions (7p)
§3 Multi-component BECs (7p+3p)
§4 3D Skyrmions in BECs
§4-1 Brane annihilation (4p+22p)
§4-2 Non-Abelian gauge field (7p)
§5 Conclusion (1p)
2 component BEC/superfluid
Gross-Pitaevskii energy functional in rotating frame
  2
gij
2
2
2
2
∇ψ i + (Vext − µi )ψ i  + ∑ ψ i ψ j 
E [ψ ] = ∫ d r ∑ 
 i, j 2
 i  2mi

2

∂ψ i   2 2
2
Ω
i
= −
∇ + Vext − µi + gii ψ i + gij ψ j − Ω ⋅ Lψ i
∂t
Ψ2
 2mi

Ψ1
3
Atomic interaction
g11 = g 22 ≡ g
Trapping potential
aij: s-wave scattering length
- 87Rb
S. B. Papp et al., Phys. Rev. Lett. 101, 040402 (2008) 85Rb - 87Rb
174Yb - 176Yb
T. Fukuhara et al., Phys. Rev. A. 79, 021601 (2009)
G.Modugno et al., Phys. Rev. Lett. 89, 190404 (2002)
41K
Sigma model representation
K.Kasamatsu., M.Tsubota, M. Ueda, Phys. Rev. A 71, 043611 (2005)
 Ψ1 

=
Ψ 
 2
 φ1 
nT 

φ 
2
 
| φ1 |2 + | φ2 |2 = 1
S3
(
S 2 = 1 S2
†
pseudo-spin: S = φ σφ = S x , S y , S z
)
T
σ : Pauli matrix
E=
(
)
[nT (g11 + g 22 + 2 g12 ) − 4(µ1 + µ2 )],
8
nT
nT2
[nT (g11 − g 22 ) − 2(µ1 − µ2 )], c2 =
(g11 + g 22 − 2 g12 )
c1 =
4
8
c0 =
Ψ1 ≠ 0, Ψ2 = 0
2
 
nT
2
(
)
∇
+
∇
+ V j nT
d
r
n
S
∑
T
α

∫  2m 
4 α

mnT
(v eff − Ω × r )2 + c0 + c1S z + c2 S z2 
+
2

n
2
arg Ψ1 / Ψ2
T
Ψ1 = 0, Ψ2 ≠ 0
phase structure
g>g12
g=g12
g<g12
Anti-ferromagnetic
SU(2) symmetric
Ferromagnetic
Ψ1 ≠ 0, Ψ2 = 0
| Ψ1 |=| Ψ2 |≠ 0
Sz
Sy
Sx
2 comp coexist
Ψ1 = 0, Ψ2 ≠ 0
2 comp are separated
0.01
0.02
0.008
0.015
0.006
0.01
0.004
0.002
0.005
0
-12
-8
-4
0
4
8
12
0
|Ψ1|2
|Ψ2|2
SU(2)symmetric
g=g12
SU(2) symmetric
Massless
O(3) model
Coreless vortex
= lump，2D Skyrmion
π 2 (S ) = Z
2
Kasamatsu, Tsubota, Ueda
Integer vortex
1 U(1) winding
(Ψ1 , Ψ2 ) = ( f (r )eiθ , f (r )eiθ ) ~ eiθ (1,1)
g12<0 attraction
singular vortex(~1comp)
g12>0 repulsion -> splitting
Vortex molecule
Repulsion balanced with
internal coherent coupling
(Rabi frequency)
∆E = − ( Ψ2 Ψ1e − i∆t + c . c .)
(0,1)
(1,0)
*
SineGordon kink Son-Stephanov(‘02)
Kasamatsu-Tsubota-Ueda(‘05)
3D Skyrmion = vorton in two comonent BECs
π 3(S ) = Z
3
 Ψ1 

=
Ψ 
 2
 φ1 
nT 

φ 
 2
Khawaja & Stoof, Nature (‘01)
Ruostekoski & Anglin (‘01)
Battye, Cooper & Sutcliffe (‘02)
| φ1 |2 + | φ2 |2 = 1 Herbut & Oshikawa (‘06)
S 3 ≅ SU ( 2)
 φ1 − φ2* 
 f ( r )r ⋅ σ 

=
U ≡
exp
i
 ∈ SU ( 2)
φ φ * 
r


 2 1 
U†U = 1,
det U =| φ1 |2 + | φ2 |2 = 1
Topological equivalence to 3D skyrmion
Phase of Ψ1
 Ψ1   1 

 0

=
Ψ 
 2  
@boundary
Vorton
3D skyrmion
3 component BEC/superfluid Eto-MN,
Gross-Pitaevskii energy functional
Phys.Rev. A85 (2012) 053645
  2
gij
2
2
2
2
∇ψ i + (Vext − µi )ψ i  + ∑ ψ i ψ j 
3 ∑ 

E [ψ ] = ∫ d r  i  2mi

 i, j 2
*


−
ω
ψ
ij i ψ j


Vortex
internal coherent coupling (Rabi frequency)
trimer = CP2 Skyrmion
(1,0,0)
(0,1,0)
(0,0,1)
enegy density
(ω12 , ω23 , ω31 ) =
(0.01,0.05,0.05)
asymmetric
(ω12 , ω23 , ω31 ) =
(0.05,0.05,0.05)
symmetric
(ω12 , ω23 , ω31 ) =
(0.2,0.05,0.05)
asymmetric
BEC
Vortex trimer
Y-junction of domain walls
Eto-MN, PRA85 (2012) 053645
Baryon = q-q-q QCD
Y-junction of fluxes
(not Δ)
Ichie-Suganuma et.al (‘03)
Plan of my talk
§1 Introduction(BEC and Vortices) (13p)
§2 Skyrmions (7p)
§3 Multi-component BECs (7p+3p)
§4 3D Skyrmions in BECs
§4-1 Brane annihilation (4p+22p)
§4-2 Non-Abelian gauge field (7p)
§5 Conclusion (1p)
§4-1 Brane annihilation
Creating vortons and three-dimensional skyrmions from domain wall
annihilation with stretched vortices in Bose-Einstein condensates
Phys. Rev. A85 (2012) 053639
e-Print: arXiv:1203.4896 [cond-mat.quant-gas]
Hiromitsu Takeuchi (Hiroshima U.)
Kenichi Kasamatsu(Kinki U.), Makoto Tsubota (Osaka City U.)
Related papers:
①Tachyon Condensation in Bose-Einstein Condensates
e-Print: arXiv:1205.2330 [cond-mat.quant-gas]
②Analogues of D-branes in Bose-Einstein condensates
JHEP 1011 (2010) 068
e-Print: arXiv:1002.4265 [cond-mat.quant-gas]
Brane-anti-brane annihilation in BEC
closed string production by brane pair annihilation
Simulation
by Takeuchi
brane
anti-brane
2nd component inside vortex
-π
π
Experiments
Watching Dark Solitons Decay into Vortex Rings
in a Bose-Einstein Condensate
B. P. Anderson et.al., Phys. Rev. Lett. 86, 2926–2929 (2001)
(JILA, National Institute of Standards and Technology and Department of Physics,
University of Colorado, Boulder, Colorado)
removing
decay
Dark soliton
Untwisted
vortex ring
Our proposal
Experiments
Watching Dark Solitons Decay into Vortex Rings
in a Bose-Einstein Condensate
B. P. Anderson et.al., Phys. Rev. Lett. 86, 2926–2929 (2001)
(JILA, National Institute of Standards and Technology and Department of Physics,
University of Colorado, Boulder, Colorado)
removing
decay
Dark soliton
Twisted
vortex ring
Vorton!!
Brane annihilation with stretched string
Ψ2has superfluid flow inside a vortex ring of Ψ1
stable vortex ring(vorton)
brane
Ψ1
Simulation
by Takeuchi
Fundamental string
Ψ2
Ψ1
Pair
anihilation
anti-brane
Phase of
Ψ2 -π
π
g<g12
Massive O(3) sigma model
ferromagnetic
1
2
E = (∇S ) + m 2 (1 − S32 )
2
S(x)=(S1,S2,S3)
equivalent to
CP1 model
E=
∫ dr
G.S. N n3 = +1
u=∞
S2=1
Target space = S2
∑α
∂ α u + m2 u
2
(1 + u )
2 2
S − iS2
Stereographic
u= 1
coordinate u
1 − S3
domain
wall
2
V = m 2 (1 − S32 )
u
G.S. S n3 = −1
u=0
Single domain wall
n3 = +1,
u=∞
Arrows
viewed from N
Phase
separation
n3 = −1,
Wall solution U(1) phase u = 0
Bogomol’nyi completion for domain wall
2
2
2
∂
u
m
u
+
α
1 ∑α
E=
∫ dx
(1 + u )
2 2
 ∂ u  2mu 2
*
* 
(
)
∂
∂
2
m
u
u
u
u
+
1
1
= ∫ dx1  1
±
2
2 2

 1+ u 2
1+ u


≥ TW
(
)
(
)
Topological charge
BPS equation
∂ 1u  mu = 0
TW = ± ∫ dx1
2m (u *∂ z u + u∂ z u * )
(1 + u )
2 2
uw = e ± mx
1
+ iϕ
x1 = +∞
1 − u 
1− u 


= ±m 
= ± m ∫ dx ∂1
2 
1+ u 2 
1 + u  x1 = −∞


2
1
2
A pair of a domain wall and an anti-domain wall
Ψ1
Ψ2
Ψ1
π phase
(dark soliton)
Fix
π
n3 = +1,
u=∞
phase
n3 = −1,
u=0
Approximate solution
A pair of a domain wall and an anti-domain wall
Ψ1
Ψ2
π
Ψ1
phase
(dark soliton)
Fix
π
n3 = +1,
u=∞
phase
Unwinding
Approximate solution
n3 = −1,
u=0
A pair of a domain wall and an anti-domain wall
Ψ1
Ψ2
π
Ψ1
phase
(dark soliton)
Fix
π
n3 = +1,
u=∞
phase
Unwinding
Approximate solution
n3 = −1,
u=0
?
?
4 possibilities of
domain wall ring
Domain wall
rings
Ψ1
Ψ2
Unstable to decay
Ψ1
Ψ2
Domain wall
rings
Ψ1
Ψ2
+ 1∈π 2 (S 2 ) = Z
Topologically
Stable
2D Skyrmion
Ψ1
Ψ2
− 1∈π 2 (S 2 ) = Z
Wall annihilations in 3 dimensions
Vortex-loops formed
Brane-anti-brane
with stretched string
ψ1
ψ2
ψ1
Phase &
amplitude
Spin structure
Exact analytic solutions
All exact(analytic) solutions
of ¼ BPS wall-vortex states
Y.Isozumi, MN, K.Ohashi, N.Sakai
Phys.Rev. D71 (2005) 065018
Bogomol’nyi-Prasad-Sommerfield (BPS) bound for vortex-domain wall
E =
∫ dr
∑α
∂αu
2
2
(1 + u )
u
2
2 2
 ∂ u  i∂ u
x
y
= ∫ dr 
 1+ u 2 2

(
+
+M
2
)
∂ z u  2 Mu
≥ TW + TV
(1 + u )
2 2
±
2
(
i ∂ xu ∂ yu − ∂ yu ∂ xu
*
*
(1 + u )
)
2 2
(
)

2 M u * ∂ z u + u∂ z u * 
±
2 2

1+ u

(
)
TV = 2 π ΝV
vortex
(2d Skyrmion)
charge
TW = ± M, 0
domain wall
charge
D-brane in a laboratory
Kasamatsu-Takeuchi-MN-Tsubota
JHEP 1011:068,2010[arXiv:1002.4265]
Ψ1 (z > 0)
z
Wall position
(log bending) Sz
1
sigma model
0.5
0
domain wall (z =
0)
-0.5
Ψ2 (z<0)
vorte
x
-1.5
Sz
z
x
vorte
x
BEC
-1
monopol
e
domai (boojum
n wall )
y
=0
Sz
-2
0
2
4
r
6
8
Analytic (approximate) solution
Untwisted loop
B
C
A
Twisted loop
Vorton (n=1)
Twisted loop
Vorton (n=2)
Untwisted loop
Unstable to decay
Twisted loop
Phase of Ψ1
Phase of Ψ2
Twisted loop
Vorton
Phase of Ψ1
Phase of Ψ2
Twisted loop Knot soliton (Hopfion)
Linking number = 1
Vorton
Plan of my talk
§1 Introduction(BEC and Vortices) (13p)
§2 Skyrmions (7p)
§3 Multi-component BECs (7p+3p)
§4 3D Skyrmions in BECs
§4-1 Brane annihilation (4p+22p)
§4-2 Non-Abelian gauge field (7p)
§5 Conclusion (1p)
§4-2 Non-Abelian
gauge field
Artificial “SU(2) gauge field”
stabilizes 3D Skyrmion
Kawakami,Mizushima,MN & Machida
Phys. Rev. Lett. 109, 015301 (2012)
Non-Abelian gauge fields
Non-Abelian gauge fields is induced on
degenerate states by Berry phase.
N+1 states
N-1 dark states { DA }
+ 2 bright states
Juzeliūnas, Ruseckas & Dalibard
Phys. Rev. A 81, 053403 (2010)
(N-1)x(N-1) gauge fields
SU(2) gauge fields
Ai =
∑
a ,i
Aiaσ a
We use A i =
∑
a ,i
ˆσ x + y
ˆ σ x ) + κ zz
ˆσ x
Aiaσ a = κ ⊥ (x
Crossover of Skyrmions
Crossover of Skyrmions
3D
2D
1D
Plan of my talk
§1 Introduction(BEC and Vortices) (13p)
§2 Skyrmions (7p)
§3 Multi-component BECs (7p+3p)
§4 3D Skyrmions in BECs
§4-1 Brane annihilation (4p+22p)
§4-2 Non-Abelian gauge field (7p)
§5 Conclusion (1p)
§5 Conclusion
• 位相的励起、特に渦やスカーミオンは、物性物理で広
く現れ、系の相やダイナミクスを支配する重要な自由
度である。
• 位相的励起を観測することで、系の自由度、対称性、
超流動性、超伝導性などがわかる(こともある)。
• 基礎物理（素粒子物理、ハドロン物理(QCD)、宇宙論）
でも現れ重要。
渦やスカーミオンの物理学の構築に向けて
両分野の交流が不可欠