新田宗土/Muneto Nitta (慶應義塾大学/Keio U.)

Brane-anti-brane Annihilations in Field Theory and BEC

基研研究会「場の理論と弦理論」@京大基礎物理学研究所

2012年7月24日

① MN, Phys.Rev.D85 (2012) 101702 [arXiv:1205.2442 [hep-th]]

② MN, Phys.Rev.D85 (2012) 121701 [arXiv:1205.2443 [hep-th]]

③ MN, arXiv:1206.5551 [hep-th]

Hiromitsu Takeuchi (Hiroshima U.)

Kenichi Kasamatsu(Kinki U.), Makoto Tsubota (Osaka City U.)

① Phys.Rev.A85(2012)053639[arXiv:1203.4896 [cond-mat.quant-gas]]

② arXiv:1205.2330 [cond-mat.quant-gas]

③ J.Low.Temp.Phys.162(2011)243 [arXiv:1205.2328 [cond-mat.quant-gas]]

④ JHEP 1011 (2010) 068 [arXiv:1002.4265 [cond-mat.quant-gas]]

Field Theory

Bose-Einstein Condensates(BEC)

Pair annihilations of particle and anti-particle (hole)

turn to energy.

In string theory, pair annihilations of D-brane and

anti-D-brane were studied extensively. A.Sen etc

Dp-brane + anti-Dp-brane → D(p-2)-branes

How about extended objects?

How about extended objects in field theory (& cond-mat)?

closed string production by brane pair annihilation

brane

anti-brane

2nd component inside vortex p -p

Brane-anti-brane

annihilation in BEC Simulation

by Takeuchi

Takeuchi-Ksamatsu-MN-Tsubota,

J.Low.Temp.Phys.162(2011)243 [arXiv:1205.2328 [cond-mat.quant-gas]]

Plan of my talk

§1 Introduction (2p)

§2 Domain wall annihilation (12+3p)

§3 Monopole-string annihilation (6+1p)

§4 Knot/Vorton/Knotted instanton (4+3p)

§5 Conclusion (1p)

Plan of my talk

§1 Introduction (2p)

§2 Domain wall annihilation (12+3p)

§3 Monopole-string annihilation (6+1p)

§4 Knot/Vorton/Knotted instanton (4+3p)

§5 Conclusion (1p)

O(3) sigma model

equivalent to

CP1 model

Vacuum

Vacuum

13 n

13 n

Stereographic

coordinate u z

yx

n

innu

1

1. (Truncated model of) 2component BECs

2. Ferromagnet

Target space = S2

n(x)=(n1,n2,n3)

N

S u

u

0u

domain

wall

Single domain wall

x1

x2(3)

un ,13

0 ,13

un

Wall solution U(1) phase

Arrows

viewed from N

WT

u

uuuum

u

muudx

u

umudxE

22

*

11

*

22

2

11

22

222

1

1

2

1

2

1

Bogomol’nyi completion for domain wall

1

1

2

2

2

2

1

1

22

**1

1

1

1

1

1

2

x

x

zzW

u

um

u

udxm

u

uuuumdxT

01 muu

BPS equation Topological charge

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

phase p

x1

x2(3)

un ,13

0 ,13

un

Approximate solution

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

Unwinding

Fix

x1

x2(3)

phase p

un ,13

0 ,13

un

Approximate solution

? ?

4 possibilities of

domain wall ring

Unstable to decay

Domain wall

rings

Topologically

stable lump Z )(1 2

2 Sp Z )(1 2

2 Sp

Domain wall

rings

L

xyyxyx

T

u

uuuui

u

uiud

u

uMudE

22

**

22

2

22

222

11

1

r

r

k

u

uuuuixdT

xyyx

L

p

2

122

**

2

Lump topological charge

Bogomol’nyi completion for lumps

Z )( 2

2 Sk p

0uiu yx

BPS equation

Wall annihilations in 3+1 dimensions

Vortex-loops formed

Mass matrix

Fayet-Illiopoulos parameter

U(1) Gauge theory with Nf =2 (extended Abelian-Higgs model)

Lumps = Vortices

This can be extended to U(Nc), Nf

T-dual

Hanany-Witten(‘97)

Embedding into String theory Eto,MN,Ohashi,Ohta&Sakai,

Phys.Rev. D71 (2005) 125006

[hep-th/0412024]

Kinky D-brane

for a domain wall Kinky D-brane

for wall-anti-wall Vortex!!

Eto,MN,Ohashi,Ohta&Sakai(th/0412024)

Kinky D-brane

for wall-anti-wall Vortex!!

Hanany&

Tong(‘03) Kinky D-brane

for a domain wall

Eto,MN,Ohashi,Ohta&Sakai(th/0412024)

D1

Plan of my talk

§1 Introduction (2p)

§2 Domain wall annihilation (12+3p)

§3 Monopole-string annihilation (6+1p)

§4 Knot/Vorton/Knotted instanton (4+3p)

§5 Conclusion (1p)

Non-Abelian Gauge theory (Non-Abelian-Higgs model)

U(2) gauge theory NF =2

in d=3+1, 4+1 or 5+1

real adjoint Higgs

NF =2 fundamental Higgs H

Mass matrix

Fayet-Illiopoulos parameter

2v

vacuum 2

21vH

MColor-flavor

locked vacuum

U(2)C ×SU(2)F

→SU(2)C+F

Non-Abelian vortex

This solution breaks SU(2)C+F → U(1)

The moduli space of Nambu-Goldstone modes:

0

)( ,

)( 0

ANO

12

12

0

ANO zzFF

v

zzHH

)( ),( ANO

12

ANO zFzH

21FC

)1(

)2(SP

U

SU CCCC

Translation 0z Internal symmety

0m Hanany-Tong,

Konishi et.al (’03)

We can embed the ANO solution

The effective theory is the model 1PC

“vacuum state” fluctuation of zero modes

vm 0

2

0z

Tong(‘03), Hanany-Tong,

Shifman-Yung (‘04)

Eto,Isozumi,MN&Sakai(‘04)

Confined monopole

Kink on a NA vortex = monopole

Domain wall

tension Monopole

mass

Numerical solution

by Fujimori

d=4+1

Monopole string

in vortex membrane

Monopole-anti-monopole

in vortex membrane

Twisted monopole-rings are created in d=4+1

What are

they??

Answer: Yang-Mills instantons (particles)

Lumps in NA vortex

= YM instantons in bulk

Eto-Isozumi-MN

-Ohashi-Sakai

PRD72 (2005)

[hep-th/0412048]

Embedding into String theory

monopole

anti-monople

Instanton!!

vortex

vortex

vortex

Domain wall and Anti-domain wall

Monopole and Anti-monopole

Strings (anti-)vortices (anti-)Yang-Mills instantons

Sheets (membranes)

Closed vortex strings

Closed instanton strings

Summary up to here

String theory

Plan of my talk

§1 Introduction (2p)

§2 Domain wall annihilation (12+3p)

§3 Monopole-string annihilation (6+1p)

§4 Knot/Vorton/Knotted instanton (4+3p)

§5 Conclusion (1p)

Brane-anti-brane with stretched string

Approximate analytic solution

A B C

Untwisted loop

Twisted loop

Knot (n=1)

Twisted loop

Knot (n=2)

Twisted loop

Faddeev-Niemi, Nature 387 (1997) 58

Knot soliton (Hopfion)

Linking number = 1

Knots in Spin 1 BEC

Kawaguchi-MN-Ueda

PRL(‘08) [arXiv:0802.1968

[cond-mat.other]]

Y2

Y1

brane

anti-brane

-p p Phase of

Pair anihilation

stable vortex ring(vorton)

Y2

Y1 Y2 has superfluid flow inside a vortex ring of

Fundamental string

Y1

Simulation

by Takeuchi

Vorton creation in BEC

MN-Takeuchi-Kasamatsu-Tsubota

Phys.Rev.A85(2012)053639 [arXiv:1203.4896 [cond-mat.quant-gas]]

= 3D Skyrmion

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)

Dark soliton

removing (Untwisted)

vortex ring

Experiments in BEC

decay

Vorton!!

Vorton creation in BEC

3D Skyrmion unstable or at most metastable

Artificial “SU(2) gauge field”

stabilizes 3D Skyrmion Kawakami,Mizushima,MN&Machida

Phys. Rev. Lett. 109, 015301 (2012)

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Plan of my talk

§1 Introduction (2p)

§2 Domain wall annihilation (12+3p)

§3 Monopole-string annihilation (6p)

§4 Knot/Vorton/Knotted instanton (4+3p)

§5 Conclusion (1p)

Domain wall and Anti-domain wall

Monopole and Anti-monopole

Strings (anti-)vortices (anti-)Yang-Mills instantons

Sheets (membranes)

Closed vortex strings

Closed instanton strings

Sheets with stretched strings

Knots (Hopfions) Knotted instantons

① Brane-anti-brane annihilations in field theory

massive O(3) sigma model

2221

4

1zSMSdE

r

stereographic coordinate

u Sx iSy

1 Sz

u for Sz 1

u 0 for Sz 1

E dru

2M 2 u

2

1 u2

2

Reu

Imu

×

S

D-brane-like object in Field Theory

Bogomol’nyi-Prasad-Sommerfield bound

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

r

r

TV = 2 p NV

vortex charge

TW = ± M, 0

domain wall

charge

00

2

1

0 2

0

1

)2(

1

)1(

izzM

Wz

N

j j

N

j j

Vyx

euMuu

uuiuk

k

BPS equations

u ,z emM zz0 i 0

j

(1) j1

Nk1

j

(2) j1

Nk2

x iy

vortices

domain wall

vortices in

vortices in

y1

y2

¼ BPS

solution

Y.Isozumi, M.Nitta, K.Ohashi, N.Sakai

Phys.Rev. D71 (2005) 065018 moduli (z0, 0) ∈ R x S1

localized U(1) Nambu-Goldstone mode

½ BPS solutions

I 2 d3 det gij Fij

i0 ijk jAk DBI action endpoints of vortices are electric charges

BIon

duality

J.P.Gauntlett, R.Portugues,

D. Tong, P.K. Townsend

Phys.Rev. D63 (2001) 085002

O(3) sigma model

CP1 model

Vacuum N

Vacuum S

un ,13

0 ,13

unAdding 4 derivative term

= Faddeev-Skyrme model

All exact(analytic) solutions

of ¼ BPS wall-vortex states Y.Isozumi, MN, K.Ohashi, N.Sakai

Phys.Rev. D71 (2005) 065018

[hep-th/0405129]

Exact analytic solutions

-2

-1.5

-1

-0.5

0

0.5

1

0 2 4 6 8

z

r

Y1 (z > 0)

Sz

vortex

domain wall (z = 0)

domain

wall

Y2 (z<0)

monopole

(boojum)

Sz

z

x

y vortex

Wall position

(log bending)

sigma model

BEC

Sz =0

Kasamatsu-Takeuchi-MN-Tsubota

JHEP 1011:068,2010[arXiv:1002.4265]

D-brane in a laboratory

(a)

Y1

Y2

z

x

y

(b) (c)

y = 0 plane

Sz

z = 0 plane

monopole

anti-monopole

Sz

z

x

y

z

x

y

vortex lattice

Analytic (approx) solution of brane-anti-brane with strings

Untwisted loop Unstable to decay

Phase of 1Y

Phase of 2Y

Vorton

Vorton creation in BEC

MN-Takeuchi-Kasamatsu-Tsubota

Phys.Rev.A85(2012)053639 [arXiv:1203.4896 [cond-mat.quant-gas]]

Inside a NA vortex in d=5+1

4 derivative correction Eto-Fujimori-MN-Ohashi-Sakai

PTP(2012)

[arXiv:1204.0773[hep-th]]

Knotted instanton MN,arXiv:1206.5551 [hep-th]