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References�

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R.A.Battye,P.M.Sutcliffe Phys.Rev.Lett.81,4798(1998)

Lagrangian�

StaWc#hamiltonian�

PosiWve#definite#for�

Hopf#charge#density�

#

Some#applicaWons#of#the#SF#model##

�•  superconductors###########E.#Babaev:#PRL#(2002)##

•  BoseBEinstain#condensate############E.#Babaev,#L.#D.#Faddeev#and#A.#J.#Niemi:#PR#B#(2002)#

##########T.#Kawakami,#T.#Mizushima,#M.#Nica#and#K.#Machida:##

#######################################################################################PRL#(2012)##

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configuraWons)#B#ChoBFaddeevBNiemi#decomposiWon#

###########L.#D.#Faddeev#and#A.#J.#Niemi:#PRL#(1999)##

#

Modified#SkyrmeBFaddeev#model#

�•  Recently#became#popular#some#modificaWons#

of#the#original#SkyrmeBFaddeev#model##

•  The#result#obtained#by#H.#Gies#[Phys.#Rev#D#(2001)]#########in#calculaWon#of#the#Wilsonian#effecWve#acWon#

####of#the#SU(2)#YangBMills#theory#suggest#

####unavoidable#appearance#of#quarWc#term##

•  The#extended#SkyrmeBFaddeev#model##given#

####by#L.#A.#Ferreira#[JHEP#(2009)]##

#

The#extended#SkyrmeBFaddeev#model#

�Lagrangian#�

Sterographic#project�

StaWc#hamiltonian�

L.A.Ferreira, JHEP05(2009)001, L.A.Ferreira, et.al., JHEP11(2009)124�

PosiWve#definite#for#�

EquaWon#of#moWon�

Zero#curvature#condiWon:�

Vortex#in#the#Extended#Skyrms#Faddeev#model�

L.A.Ferreira, JHEP05(2009)001�

AddiWonal#condiWon:�

SoluWons� Sta$c&vortex�

Traveling&wave&for&x3�

Ansatz�

(the#infinite#set#of)#conserved#currents#given#by##�

with#any#funcWon####�

One#can#see�

SoluWon#ansatz�

Boundary#condiWons�

Toroidal#coordinates:�

Hopfions#in#the#extended#Skyrme#Faddeev#model�

(Dimensionless)#staWc#energy�

Corresponds#to#the#zero##

curvature#condiWon#

Numerically-one-finds-:-�

The#soluWons#approximately#

saWsfy#the#zeroBcurvature#

(submodel#)condiWon!�

Topological#(Hopf)#charge#

Hopf#map�

####��������������

Lower#energy#bound#formula#

A.F.Vakulenko,L.V.Kapitanskii, Sov.Phys.Dokl. 24,433(1979) R.S.Ward, Nonlinearity 12,241 (1999)G�

(Sovolev#inequality)�

Vakulenko#

Ward�

EquaWon#for#g�

EquaWon#for#)�

EquaWons#of#moWon�

QH� &&(m,n)� """"""""a� E [M / |e|]�

E/Ebound��

<r2>1/2

[-1/|e|M]�

1� (1,1)� 0.53� 236.2� 1.18� 0.55�2� (1,2)� 0.79� 390.6� 1.16� 0.95�

(2,1)� 0.66� 466.7� 1.39� 1.04�3� (1,3)� 1.09� 548.7� 1.21� 1.45�

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•  Now#we#have#obtained#similar#results#of#

CPN-version-of#the#extended#SF#model.#

• We#are#invesWgaWng#some#physical#

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#

RenormalizaWon#group#discussion#strongly#suggests#

that#�-and-e2-have-same-sign�

�� e2�

RenomalizaWon#group� ±� ±�

SkyrmeBFaddeev#(SF)� 0� s� succeeded�

GladikowskiBHellmund#(GH)� u� s� succeeded�

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Gies#model#with#2nd#derivaWve� s� s� poten>ally-succeeded�

Ours� u� u�