Decomposition of Variables and Duality in non-Abelian Models

A. P. ProtogenovInstitute 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 • Knots of the order parameter distributions • Current pseudogap phases • SU(2) decomposition of variables • Conclusion

Standard t-J model

iii

ijjiji

ijji

nn

nnSScc

P

JPchtPH

1

4

1)..(

,

ccS iii

2

1

Two-component order parameter

,, hc

1 , , ,2

1

hhff

b

bh

f

f

f

f

ffff

ff

eUU

jijiij

jiij

isfd

ij

ijij

ijijij

aij

)(

P.A. Lee, N. Nagaosa, X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006)

Iwasawa decomposition

SU(2) mean field theory

ij

l

ij

l

i

l

iji

ijjiji

jijiijijij

mean

hhahh

hUh

UUUH

cht

chTrJ

0

..2

1

..2

1

8

3

)2(SUWWUWU

W

i

jijiij

iii

FTrg

LSU

2

2)2(4

1

ffff

ff

eUU

jijiij

jiij

isfd

ij

ijij

ijijij

aij

)(

Two-component Ginzburg-Landau-Wilson

functional E. Babaev, L.D. Faddeev, A.J. Niemi, PR B ‘02

82

21 242

2

2,1

3

2

cbA

c

eixdF kk

m

1 , ,2

2

2

12

iem

Some useful identities

2

1

)21(..*11

2

14

1

4212

222

n

ccia

nk

ninikli

acurl

na

О(3) Skyrme-Faddeev sigma model

1,,22

22

1 iem

ikikikkikk

cn

HFFcd

bHnxd

FFFF

216

1

2422242222

23

1 , , 2

2

1

nn

)21(.. , *112

cciaAaJ

c

ikkikiikikkiik aannnHccF

,

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

23: SS n

ikkikiik aannnH

SaaxdQ

23

3216

1

Examples of knots

6Q 6Q

Knot scales

R

Packing degree, α of the knot filaments is a small parameter of the model

R

α = Vknot / V ~ ξ2 R / R3 ~ ξ2/R2 < 1 α ~ æ-2

The result of this “surgical cut” is the following

structure of phase distributions on a crystal surface

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

ikkikiik aannnH

SaaxdQ

23

32

16

1

int

24

1 22223

F

ikik

F

ik

F

ikk HFFcHnxdF

cn

cn FForQLDualityQ

LQF

:1322

432

ZacxdL

3216

1

caAFHArotBikik

,

8

222

Current pseudogap states

rr

rH

rr

rri

ikk rdrdrT

332 3

16

1)(

i

ii mrTTrotrota

2

1

klikllkikli Hnnnacurl 21

21

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

mnVki

m

ikikkin

mmmnc

nmGFnnnH

ki

kkinMY

,8

1

3

][

),(][22

3

42222

2222

.lim.inf

cccF

mmmbb

nbnbbbnG

i

nm

i

nmnm

i

ikkiik

kiikki

kiikikkiik

2

][

2

3

333

SU(3) case

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

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.