Keio U.soken.editorial/NewWebSite/so... · 2014. 2. 4. · T.H.R. Skyrme. A Nonlinear theory of strong interactions. Proc.Roy.Soc.Lond. A247 (1958) 260- 278. A Unified Field Theory

Muneto Nitta (新田宗土)Keio U. (慶應義塾大学)

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

① スピノールBEC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

共同研究者 cond-mat.

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.

Tkm

E BT ≈= −22

2λ

Tmkh

BT π

λ2

2

=

2/3

0

0 1

−=

TT

NN

Number of condensates

3/22

031.3

=

VN

mkT

B

Transition temperature

Cold atomic gases1995 cold atomic bose gas

87Rb, 23Na, 7LiCornell (Colorado), Ketterle(MIT)& Wieman (Colorado)

2003 cold atomic fermion gas JILA(Colorado)， MIT

Temperature ~ 10- 6,10- 7 KNumber ~ 106, Size ~ 10- 3cm

``Pure” BEC (99% is BEC)

doppler laser cooling magneto-optical trap evaporative cooling

TkMpRM

B23

22

222

≅≅ω

22

21 rMV ω=

trapping potentialω : frequency

6/1NM

Rω

≅3/1NR

RMpT ≈≅=ω

λ

de Broglie wave length

mean particledistance

][10 63/1

KkNTB

−≈≅ω

for 63 10],[10 ≈≈ NHzω

Ψ

R

M : mass of atoms

transition temperature

Bogoliubov theory for weakly interactive Bose gas (with point interaction)

)()( rgrV δ= point interaction

g

Scalar BEC, 4He superfluid

wave functionfor condensation

)()()( xxx φψ +Ψ=mean field approximation

fluctuation (phonon): non-condensed component

Gross-Pitaevskii (nonlinear Schrödinger) Equation

Gross-Pitaevskii energy functional

[ ] ∫

+−+∇= 42ext

22

3

2)(

2ψψµψψ gV

MdE r

*2

ext2

2

2 δψδψψµψ EgV

Mti =

+−+∇−=

∂∂

M

ag S24 π

≡

Sa : s-wave scattering lengthµ : chemical potential M : mass of atoms

Vext(r) : trapping potential 22ext 2

1 rMV ω=

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.

Scalar BEC, 4He superfluid

vortex

x

yreal space

Gallery of Abrikosov Lattices in Superconductors @ Oslo Superconductivity Lab http://www.fys.uio.no/super/vortex/

U. Essmann and H. TraubleMax-Planck Institute, Stuttgart Physics Letters 24A, 526 (1967)

Order Parameter Space(OPS)= U(1)

Superconductors under magnetic fieldZ≅∈ )]1([1 Uk πFlux quantization

kke

hc02

Φ==Φ ]weber[1007.22

150

−×==Φe

hc

quantization of circulation

θψ ikerf )(=energy

22

2

1rm

k

φ

−

Λ= log2 22kvT π Λsystem sizetension

Inter-vortexforce R

vF24π

= distance R

ΨΨΨ∇Ψ−Ψ∇Ψ

= ∗

∗∗

21

eff ivk

Md

=⋅∫ effvrZ≅∈ )]1([1 Uk π

[ ] ∫

+−+

×−∇= 42

223

2)(

2ψψµψψ gVMi

MdE rΩr

rΩ×−∇→∇

MiRotation in rotating frame

A proof of superfluidityAbo-Shaeer, Raman, Vogels, Ketterle, Science 292, 476-479 (2001)

Vortex nucleation under rotation

BEC/BCS Crossover

A proof of superfluidityin all range of BES/BCS

Zwierlein, Abo-Shaeer, Schirotzek, Schunck& KetterleNature 435, 1047-1051(23 June 2005)

Fermions with pseudo spin

eg ,

Artificial Gauge FieldTwo-state model

Two states

Hamiltonian

coupling

A review: J.Dalibard et.al., Rev. Mod. Phys. 83, 1523–1543 (2011)

Eigenstates of U = Dressed states

eigenvalues

g

e

Full state

Born-Oppenheimer approximation

ljjl i χχ ∇= AGauge field

Neglecting , EOM of2ψ 1ψ

Gauge fieldsas Berry phase

Synthetic magnetic fields for ultracold neutral atoms Lin, Compton, Jimenez-Garcia, Porto & Spielman,Nature 462, 628-632 (3 December 2009)

1 dark state2 bright states

3 statesinteraction

Adiabatic approx

A proof of artificial magnetic field

Synthetic magnetic fields for ultracold neutral atoms Lin, Compton, Jimenez-Garcia, Porto & Spielman,Nature 462, 628-632 (3 December 2009)

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)

1D Skyrmion=Sine-Gordon kink

2D SkyrmionZ=)( 1

1 Sπ

Z=)( 22 Sπ

3D SkyrmionZ=)( 3

3 Sπ

What is a Skyrmion?

T.H.R. SkyrmeA Nonlinear theory of strong interactionsProc.Roy.Soc.Lond. A247 (1958) 260-278A Unified Field Theory of Mesons and BaryonsNucl.Phys. 31 (1962) 556-569

Model of nucleon in HEP

3dimhedgehog

( )[ ] ( )

Dxx

xx

Tdx

dxdxE

1

2

222

12

sin22

sin

12

sincos2

≥

−

∂±

∂=

−

+∂=+∂=

∫

∫∫

θθθθ

θθθθ

SG Topological charge

Bogomol’nyi completion

Z=∈ )( 11 Sk π

1D Skyrmion

( )

( )( )[ ] +∞=

−∞==

∂=

∂±=

∫∫

xx

x

xD

dx

dxT

2/cos2

2/cos2

2/sin1

θ

θ

θθ

O(2) model (=sine-Gordon model)

Bogomol’nyi-Prasad-Sommerfield(BPS) equation

02

sin =

∂θθ x

Sine-Gordonkink

O(3) sigma model

equivalent toCP1 model

13 +=S

13 −=S

Stereographiccoordinate u 3

21

1 SiSSu

−−

=

1. (Truncated model of) 2component BECs2. Ferromagnet

Target space = S2

S(x)=(S1,S2,S3) N

Su

∞=u

0=u

( )2

21 SE ∇=

( )22

2

1 u

udE

+= ∑∫ α α∂

r

S2=12S

( )

( )( )

( ) Lxyyxyx T

u

uuuui

u

uiuxd

u

uxdE

≥

+

−

±

+=

+=

∫

∑∫

22

**

22

2

2

22

22

11

1

∂∂∂∂∂∂

∂α α

2D Skyrme topological charge

Bogomol’nyi completion

Z=∈ )( 22 Sk π

0=uiu yx ∂∂

BPS equation

2D Skyrmion

( )( )

ku

uuuuixdT xyyx

L

π

∂∂∂∂

21

22

**2

=+

−±= ∫

0=uz∂ iyxz +≡

(=lump, sigma model instanton)

kTL π2= Z=∈ )( 22 Sk π

0=uiu yx ∂∂ BPS equation

2D Skyrmion

0=uz∂ iyxz +≡ 13 +=S

13 −=S

N

S

∞=u

0=u

2S)|(| ∞→∞→ zu

)( 0 izzu →→

∑=

−

−=

k

i i

i

zzu

1

1 λ

1=k

2D Skyrmion

Choi, Kwon, and Shin, PRL 108, 035301 (2012)

F

F

FF

F

P

SS

USOU

HG

zx

+Φ

Φ

+Φ

Φ

×≅

××

≅

)(

)1()()3()1(

2

21

2

Ζ

Ζ

Spin 1 BEC, Polar phase

Ζ≅

PHG

2π

Cond-mat examples:Ferromagnet, quantum Hall systems

2

2

1

)()(

C∈

xx

φφ

3D Skyrmion

Z=)( 33 Sπ

)2(*1

*2

2

1 SUU ∈

−≡

φφ

φφ 1||||det

,12

22

1 =+==

φφUUU†

1|||| 22

21 =+ φφ

)2(SU3 ≅S

⋅

=r

rfixU σr)(exp)(

Skyrmion ansatz

3S

2S

21=U

3S

N:

S: ( ) )0,1(, 21 −=φφ

)( 0)( ,2 ∞→→+→ rrf1)0( 1)( ,2 →→−→ rrf1

( ) )0,1(, 21 =φφ

21−=U

O(4) sigma model~Skyrme model

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)

Gross-Pitaevskii energy functional in rotating frame

Trapping potential

aij: s-wave scattering length

Atomic interaction

Ψ1Ψ2

Ωijijiiii

i

i ggVmt

i ψψψµψ

⋅−++−+∇−=

∂∂ LΩ

22ext

22

2

2 component BEC/superfluid

[ ] ∫ ∑∑

+

−+∇=

jiji

ij

iiii

i

gV

mdE

,

222ext

22

3

2)(

2ψψψµψψ r

G.Modugno et al., Phys. Rev. Lett. 89, 190404 (2002) S. B. Papp et al., Phys. Rev. Lett. 101, 040402 (2008) T. Fukuhara et al., Phys. Rev. A. 79, 021601 (2009)

41K - 87Rb85Rb - 87Rb174Yb - 176Yb

ggg ≡= 2211

=

ΨΨ

2

1T

2

1

φφ

n 1|||| 22

21 =+ φφ S3

pseudo-spin: ( )Tzyx SSS ,,== φφ σS †σ : Pauli matrix

12 =S S2

K.Kasamatsu., M.Tsubota, M. Ueda, Phys. Rev. A 71, 043611 (2005)

( ) ( )

( )+++×−+

+

∇+∇= ∑∫2

2102

effT

T2T2

T

2

2

42

zz

j

ScSccmn

nVSnnm

dE

rΩv

r

αα

( ) ( )[ ]

( ) ( )[ ] ( )122211

2T

2212211TT

1

21122211TT

0

28

,24

,428

gggncggnnc

gggnnc

−+=−−−=

+−++=

µµ

µµ

Sigma model representation

0,0 21 =Ψ≠Ψ

0,0 21 ≠Ψ=Ψ

21 /arg ΨΨ

phase structureg<g12

Ferromagnetic

0

0.002

0.004

0.006

0.008

0.01

-12 -8 -4 0 4 8 12x

0,0 21 =Ψ≠Ψ

0,0 21 ≠Ψ=Ψ

0|||| 21 ≠Ψ=Ψ

g=g12

SU(2) symmetric

0

0.005

0.01

0.015

0.02

-12 -8 -4 0 4 8 12

|Ψ1|2 |Ψ2|2

Sz

Sx

Sy

2 comp are separated2 comp coexist

g>g12

Anti-ferromagnetic

Coreless vortex= lump，2D Skyrmion

Kasamatsu, Tsubota, Ueda

Z=)( 22 Sπ

Massless O(3) model

SU(2)symmetricg=g12

SU(2) symmetric

Integer vortex 1 U(1) winding

( ) )1,1(~))(,)((, 21θθθ iii eerferf=ΨΨ

g12>0 repulsion -> splitting

.)c.c( 1*

2 +ΨΨ−=∆ ∆− tieE

g12<0 attractionsingular vortex(~1comp)

(0,1)(1,0)

SineGordon kink

Repulsion balanced with internal coherent coupling

(Rabi frequency)

Vortex molecule

Kasamatsu-Tsubota-Ueda(‘05)Son-Stephanov(‘02)

=

ΨΨ

2

1T

2

1

φφ

n

3D Skyrmion = vorton in two comonent BECsZ=)( 3

3 Sπ

)2()(exp*1

*2

2

1 SUr

rfiU ∈

⋅

=

−≡

σrφφ

φφ

1||||det,1

22

21 =+=

=φφU

UU†

Khawaja & Stoof, Nature (‘01)Ruostekoski & Anglin (‘01)Battye, Cooper & Sutcliffe (‘02)Herbut & Oshikawa (‘06)1|||| 2

22

1 =+ φφ

)2(SU3 ≅S

Vorton

1ΨPhase of

3D skyrmion

Topological equivalence to 3D skyrmion

=

ΨΨ

01

2

1

@boundary

Gross-Pitaevskii energy functional3 component BEC/superfluid

[ ] ∫ ∑∑

−

+

−+∇

=

jiij

jiji

ij

iiii

i

gV

mdEψψω

ψψψµψψ

*,

222ext

22

3

2

)(2

r

internal coherent coupling (Rabi frequency)Vortex trimer = CP2 Skyrmion

enegy density(1,0,0) (0,1,0) (0,0,1)

Eto-MN, Phys.Rev. A85 (2012) 053645

( ))05.0,05.0,05.0(

,, 312312 =ωωω

( ))05.0,05.0,01.0(

,, 312312 =ωωω

( ))05.0,05.0,2.0(

,, 312312 =ωωω

symmetric

asymmetric

asymmetric

Ichie-Suganuma et.al (‘03)

Baryon = q-q-qY-junction of fluxes(not Δ)

QCDBEC Vortex trimerY-junction of domain walls

Eto-MN, PRA85 (2012) 053645

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 condensatesPhys. 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) 068e-Print: arXiv:1002.4265 [cond-mat.quant-gas]

closed string production by brane pair annihilation

brane

anti-brane2nd component inside vortex π-π

Brane-anti-brane annihilation in BECSimulation by Takeuchi

Watching Dark Solitons Decay into Vortex Rings in a Bose-Einstein CondensateB. 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)

Dark soliton

removing Untwistedvortex ring

Experiments

decay

Watching Dark Solitons Decay into Vortex Rings in a Bose-Einstein CondensateB. 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)

Dark soliton

removing Twistedvortex ring

Experiments

decay

Vorton!!

Our proposal

Ψ2

Ψ1

brane

anti-brane

-π πPhase of

Pairanihilation

stable vortex ring(vorton)

Ψ2

Brane annihilation with stretched stringΨ1Ψ2has superfluid flow inside a vortex ring of

Fundamental string

Ψ1

Simulation by Takeuchi

Massive O(3) sigma model

equivalent toCP1 model

G.S.

G.S.

13 +=n

13 −=nStereographiccoordinate u 3

21

1 SiSSu

−−

=

Target space = S2

S(x)=(S1,S2,S3)

N

Su

∞=u

0=u

domain wall

( ) ( )23

22 121 SmSE −+∇=

( )22

222

1 u

umudE

+

+= ∑∫ α α∂

r

( )23

2 1 SmV −=

S2=1

g<g12

ferromagnetic

Single domain wall∞=+=

un ,13

0 ,13

=−=

un

Wall solution U(1) phase

Arrows viewed from N

Phaseseparation

( )

( )( )

( )WT

u

uuuum

u

muudx

u

umudxE

≥

+

+±

+=

+

+=

∫

∑∫

22

*11

*

22

211

22

2221

1

2

1

2

1

∂∂∂

∂α α

Bogomol’nyi completion for domain wall

( )( )

+∞=

−∞=

+

−±=

+

−∂±=

+

+±=

∫

∫1

1

2

2

2

2

11

22

**1

11

11

1

2

x

x

zzW

uu

muu

dxm

u

uuuumdxT ∂∂

01 =muu ∂BPS equation

Topological charge

ϕimxeu +±=1

w

A pair of a domain wall and an anti-domain wall

phaseπ

∞=+=

un ,13

0 ,13

=−=

un

Approximate solutionπ phase(dark soliton)

Fix1Ψ

2Ψ

1Ψ

A pair of a domain wall and an anti-domain wall

phaseπ

Approximate solutionπ phase(dark soliton)

Unwinding

Fix∞=+=

un ,13

0 ,13

=−=

un

1Ψ

2Ψ

1Ψ

A pair of a domain wall and an anti-domain wall

phaseπ

Approximate solutionπ phase(dark soliton)

Unwinding

Fix∞=+=

un ,13

0 ,13

=−=

un

1Ψ

2Ψ

1Ψ

? ?

4 possibilities ofdomain wall ring

Unstable to decay

Domain wall rings

1Ψ 1Ψ

2Ψ2Ψ

Topologically Stable

2D Skyrmion

Z=∈+ )(1 22 Sπ Z=∈− )(1 2

2 Sπ

Domain wall rings

1Ψ1Ψ

2Ψ 2Ψ

Wall annihilations in 3 dimensions

Vortex-loops formed

Spin structurePhase &amplitude

Brane-anti-branewith stretched string

ψ1

ψ2

ψ1

All exact(analytic) solutions of ¼ BPS wall-vortex statesY.Isozumi, MN, K.Ohashi, N.SakaiPhys.Rev. D71 (2005) 065018

Exact analytic solutions

Bogomol’nyi-Prasad-Sommerfield (BPS) bound for vortex-domain wall

( )

( )( )

( )

( )( )

( )VW

zzz

xyyxyx

TT

u

uuuuM

u

Muu

u

uuuui

u

uiud

u

uMudE

+≥

+

+±

++

+

−±

+=

+

+=

∫

∑∫

22

**

22

2

22

**

22

2

22

222

1

2

1

2

11

1

∂∂∂

∂∂∂∂∂∂

∂α α

r

r

TV = 2 π ΝV

vortex(2d Skyrmion)charge

TW = ± M, 0domain wall charge

-2

-1.5

-1

-0.5

0

0.5

1

0 2 4 6 8

z

r

Ψ1 (z > 0)

Sz

vortex

domain wall (z = 0)

domain wall

Ψ2 (z<0)

monopole(boojum)

Sz

z

xy

vortex

Wall position(log bending)

sigma model

BEC

Sz =0Kasamatsu-Takeuchi-MN-TsubotaJHEP 1011:068,2010[arXiv:1002.4265]

D-brane in a laboratory

Analytic (approximate) solution

AB C

Untwisted loopTwisted loopVorton (n=1)

Twisted loopVorton (n=2)

Untwisted loopUnstable to decay

Twisted loop

Phase of 1Ψ

Phase of 2Ψ

VortonTwisted loop

Phase of 1Ψ

Phase of 2Ψ

VortonTwisted loop Knot soliton (Hopfion)

Linking number = 1

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)

Artificial “SU(2) gauge field”stabilizes 3D Skyrmion

Kawakami,Mizushima,MN & MachidaPhys. Rev. Lett. 109, 015301 (2012)

§4-2 Non-Abelian gauge field

Non-Abelian gauge fields

Non-Abelian gauge fields is induced on degenerate states by Berry phase.

Juzeliūnas, Ruseckas & DalibardPhys. Rev. A 81, 053403 (2010)N+1 states

(N-1)x(N-1) gauge fieldsN-1 dark states + 2 bright states

AD

∑=ia a

aii A

,σA

SU(2) gauge fields

( ) xzxxia aaii A σκσσκσ zyxA ˆˆˆ

,++== ⊥∑We use

Crossover of Skyrmions

Crossover of Skyrmions

3D

1D2D

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)

• 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

§5 Conclusion