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-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 iO 3, hT MgB 2 ) (E. Babaev, L.V. Faddeev, A.J. Niemi, Phys. Rev. B 65 (2002) 10051)

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/

bd Phases Skyrme – Faddeev 1 component Ginzburg-Landau in E.M. Inhomogeneous Superconductor Quasi-1 dim distribution 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 L. Faddeev, Quantisation of Solitons, preprint IAS-75-QS70, 1975; Nonlinear Physics V

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

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

H-field y x x z

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

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

General Case Closed quasi 1-dim distribution Packing parameter 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 superconductor Quasi – planar knots Antiferromagnetic ordering Topological phase transition : hom. SuperC. Toroid order Nonlinear Physics V

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