Knotted field distributions of order parameters in pseudogap phase states

1. L. Martina Dipartimento di Fisica, Università del Salento Sezione INFN - Lecce Knotted field distributions of order parameters in pseudogap phase states • A. Protogenov, V. Verbus , RAS - Nizhny Novgorod, Russia • EINSTEIN – RFBR • cond-mat.str-el/0706.0639 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. 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) Nonlinear Physics V I.P. Ivanov, cond-mat/0802.2107

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

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

12. 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 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