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/0706.0639 Nonlinear Physics V

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

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/0802.2107 Nonlinear Physics V

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

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

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

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

y x x z H-field

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 Quasi 1- dim distribution Nonlinear Physics V

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

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

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