1. Decomposition of Variables and Duality in non-Abelian Models A. P
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

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

5. Standard t-J model

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

7. SU(2) mean field theory

8. Two-component Ginzburg-Landau-Wilson functional
E. Babaev,
L.D. Faddeev,
A.J. Niemi, PR B ‘02

9. Some useful identities

10. О(3) Skyrme-Faddeev sigma model

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

12. Examples of knots

13. Knot scales

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

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

17. Gain in current pseudogap states (V. Verbus, А. P. , JETP Lett

18. Current pseudogap states

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

20. SU(3) case Problem: 2-form F does not exact!
Flag manifold F2 = SU(3)/(U(1)×U(1))
instead of CP1 = 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 that π3(F2)=Z as well as π3(CP1)=Z

21. Conclusion universal character of the phase stratification
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.