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Decomposition of Variables and Duality in non-Abelian Models A . P. Protogenov Institute of Applied Physics of the RAS , N . Novgorod V. A. Verbus Institute for Physics of Microstructures of the RAS , N . Novgorod. Outline. Phase diagram SU(2) and U(1) mean field theory states

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  1. Decomposition of Variables and Duality in non-Abelian Models A. P. ProtogenovInstitute of Applied Physics of the RAS, N. Novgorod V. A. VerbusInstitute for Physics of Microstructures of the RAS, N. Novgorod

  2. Outline • Phase diagram • SU(2) and U(1) mean field theory states • Knots of the order parameter distributions • Current pseudogap phases • SU(2)decomposition of variables • Conclusion

  3. Standard t-J model

  4. Two-componentorder parameter P.A. Lee, N. Nagaosa, X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006) Iwasawa decomposition

  5. SU(2) mean field theory

  6. Two-componentGinzburg-Landau-Wilson functional E. Babaev, L.D. Faddeev, A.J. Niemi, PR B ‘02

  7. Some useful identities

  8. О(3) Skyrme-Faddeev sigma model

  9. Hopf invariant, Q, for a map is the linking number in S3 of the preimages of two generic points in S2.

  10. Examples of knots

  11. Knot scales

  12. Packing degree, α of the knot filaments is a small parameter of the model α = Vknot/ V~ ξ2R / R3 ~ ξ2/R2 < 1 α ~ æ-2

  13. The result of this “surgical cut” is the following structure of phase distributions on a crystal surface

  14. Gain in current pseudogap states(V. Verbus,А. P., JETP Lett. 76, 60 (2002))

  15. Current pseudogap states

  16. SU(2) decomposition of variables: Hamiltonian in the infrared limit

  17. SU(3) case Flag manifold F2 = SU(3)/(U(1)×U(1)) instead ofCP1 = SU(2)/U(1) = S2 dimF2 = 6 instead of dimCP1= 2 How does Hopf invariant Q for the flag maniford F2 look like? Problem:2-form F does not exact! Note thatπ3(F2)=Zas well asπ3(CP1)=Z

  18. Conclusion 1. The origin of the internal inhomogeneity and universal character of the phase stratification is the multi-vacuum structure in the form of the knotted vortex-like order parameter distributions. 2. As a result of phase competition, we have a natural window: α/ξ < c < 1/ξ , for the existence of the free energy gain due to supercurrent with large value of the momentum, c. Here, α = ξ2/R2 < 1 is a knot packing degree, ξ is the correlation length.

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