1. Dipartimento di Fisica, Università del Salento
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/
Nonlinear Physics V

2. 2-components Ginzburg – Landau Model
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) 3D

3. Gauge-invariant vector field
the densities of the Cooper pairs
the magnetic order (Néel) vector
paramagnetic current
Gauge-invariant vector field
mass
Mermin – Ho vorticity
Group Theoretical Classification
of the Local Minima of V(r, na)
I.P. Ivanov, cond-mat/
Nonlinear Physics V

4. b d Phases Skyrme – Faddeev 1 component Ginzburg-Landau in E.M.
Inhomogeneous Superconductor
Quasi-1 dim distribution
Nonlinear Physics V

5. Skyrme – Faddeev model Hopf Invariant
L. Faddeev, Quantisation of Solitons,
preprint IAS-75-QS70, 1975;
Hopf Invariant
Stability of large-Q configurations
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.
Nonlinear Physics V

6. M.F. Atiyah, N.S. Manton, Phys. Lett. A 222 (1989) 438
Trial function
Q=1
M.F. Atiyah, N.S. Manton, Phys. Lett. A 222 (1989) 438
L. Faddeev, A.J. Niemi, Nature 387 (1997),59
Nonlinear Physics V

7. y x x z
n-field
Q=1
Nonlinear Physics V

8. y x x z
H-field

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

11. Quasi 1- dim distribution
Compressible fluid
X.G. Wen, A. Zee, Phys. Rev. B 46 (1992) 2290
Quasi 1- dim distribution
Nonlinear Physics V

12. Dense packing, anti-chirality
General Case
Closed quasi 1-dim
distribution
Packing parameter
V.M. Dubovik, V.V. Tugushev
Phys. Rep. 187, 145 (1990).
TOROID STATE
Dense packing, anti-chirality
Nonlinear Physics V

13. Toroid Moment T Toroid distributions:
Nonlinear Physics V T
Toroid distributions:
Near inhomogeneous superconductor
Quasi – planar knots
Antiferromagnetic ordering
Topological phase transition : hom. SuperC. Toroid order

14. Conclusions
2-component Ginzburg – Landau Model
Special class of phases
Topological classification
Estimate of parameters
Appearence of nets of toroi solutions
Analogy with Dimeric system on the Lattice
Open problems
Explicit construction of solutions (approximated)
Discretization schemes based on group invariance
Fractional – Statistics of toroid distribution
Roksar-Kivelson type Hamiltonian