Knotted field distributions of order parameters

in pseudogap phase states

L. Martina

Dipartimento di Fisica, Università del Salento

Sezione INFN - Lecce

• A. Protogenov, V. Verbus , RAS - Nizhny Novgorod, Russia•EINSTEIN – RFBR •cond-mat.str-el/0706.0639 Nonlinear Physics V

2-components Ginzburg – Landau Model

3D

•Two – Higgs doublet model (T.D. Lee, Phys. Rev. D 8 (1973) 1226)

•Spin – Charge decomposition in Yang – Mills (L. Faddeev A. Niemi (2006)•Spin – density waves in cuprate •two charged condensates .•two charged condensates of tightly bounded fermion pairs, • two-band superconductor •(Nb, T , V , Nb-doped SrT iO3, hT MgB2 ) (E. Babaev, L.V. Faddeev, A.J. Niemi,

Phys. Rev. B 65 (2002) 10051)

221, C

Mermin – Ho vorticity

the densities of the Cooper pairs

paramagnetic current

Gauge-invariant vector field

mass

Nonlinear Physics V

the magnetic order (Néel) vector

Group Theoretical Classification

of the Local Minima of V(, n) I.P. Ivanov, cond-mat/0802.2107

b d

Phases

Skyrme – Faddeev

1 component Ginzburg-Landau in E.M.

Inhomogeneous Superconductor

Quasi-1 dim distribution b

dL

24

bGL

2

Nonlinear Physics V

A.F. Vakulenko and L.V. Kapitanskii, Sov. Phys. Dokl. 24, 433 (1979)L. Faddeev, A. Niemi, Nature 387, 1 May (1997) 58..R.S. Ward, Nonlinearity 12 (1999) 241 V. M. H. Ruutu et al, Nature 382 (1996) 334.

Skyrme – Faddeev model

Hopf Invariant

Stability of large-Q configurations

/1min

L. Faddeev, Quantisation of Solitons, preprint IAS-75-QS70, 1975;

Nonlinear Physics V

/2,/1 knotknot RR

Q=1

M.F. Atiyah, N.S. Manton, Phys. Lett. A 222 (1989) 438

nvrin ,exp 22:, SSv

QvWinding ,

0,20

Trial function

r

tghr 12

5.4

min SFS 7.116 2

min

SFS

L. Faddeev, A.J. Niemi, Nature 387 (1997),59 Nonlinear Physics V

y

x

-2 -1 0 1 2

-2

-1

0

1

2

x

z

Q=1

Nonlinear Physics V

-2 -1 0 1 2

-2

-1

0

1

2

n-field

H-field

-2 -1 0 1 2

-2

-1

0

1

2

y

x

-2 -1 0 1 2

-2

-1

0

1

2

x

z

r

hr

Ok4

2

2

2

22 sec

2cos1

8n

,

/

/11

2g

r

rfb d

bb

Inhomogeneous Superconductor Vconst min

V. I. Arnold and B. A. Khesin: Topological methods in hydrodynamics..A. P. Protogenov Physics-Uspekhi 49, 667 (2006).

Hoelder

Ladyzhenskaya

2

4/32 132

Q

LQS h

QL

SFh SS Nonlinear Physics V

216

Quasi 1- dim distribution

0,, ndbBB

Compressible fluid

X.G. Wen, A. Zee, Phys. Rev. B 46 (1992) 2290

Nonlinear Physics V

General Case

Closed quasi 1-dimdistribution

Packing parameter

cnh SSS 0 TOROID STATE

V.M. Dubovik, V.V. Tugushev Phys. Rep. 187, 145 (1990).

Dense packing, anti-chirality Nonlinear Physics V

Toroid Moment

T

Toroid distributions:Near inhomogeneous superconductorQuasi – planar knotsAntiferromagnetic ordering

Topological phase transition : hom. SuperC. Toroid order

mrdxTmdx 33

2

1,0

Nonlinear Physics V

Conclusions

2-component Ginzburg – Landau ModelSpecial class of phasesTopological classificationEstimate of parametersAppearence of nets of toroi solutionsAnalogy with Dimeric system on the Lattice

Open problemsExplicit construction of solutions (approximated)Discretization schemes based on group invarianceFractional – Statistics of toroid distributionRoksar-Kivelson type Hamiltonian