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
Jan 08, 2016
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.