Coexistence of composite-bosons and composite
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Transcript Coexistence of composite-bosons and composite
Disordering of a quantum Hall
superfluid
M.V. Milovanovic, Institute of Physics, Belgrade, Serbia
The quantum Hall bilayer
1 1
2 2
A fundamental problem of:
Superfluid disordering in 2+1 dimensions!
d (distance between layers) small
(
z
z
)
(
w
w
)
(
z
w
)
111
i
j
k
l
p
q
i
j
k
l
p
,
q
Superfluid
Appropriate quasiparticles CBs (composite bosons)
Th: Wen and Zee,PRL 69, 1811(1992) …,
Exp: Spielman et al., PRL 84, 5808 (2000)…
Theory of Moon et al.
PRB 51, 5138 (1995)
ei
Quantum Mechanical view of electron – spinor states,
1
Ground state is a condensate of same spin states –
phase coherence
N
1
i
(
c
e
c
)
|0
k
k
0
k = angular momentum in disc geometry
essentially XY model physics – physics of superfluid
expect:
(1) Goldstone mode
(2) elementary charged vortices-merons
(3) finite T BKT transition
fixed relative number of particles state =
111
d large
1/ 2 1/ 2
2
P
Det
(
e)
(
z
z
1
/
2
i
j)
i
j
i
k
r
i
j
Fermi-liquid-like state
Approprite quasiparticles CFs (composite fermions)
Th: B.I. Halperin, P.A. Lee, and N. Read, PRB 47, 7312 (1993),…,
Exp: R.L. Willett et al., PRL 71, 3846 (1993),…
Experiments
Spielman et al., PRL 87, 036803 (2001)
Kellogg et al., PRL 93, 036801 (2004)
Experiments
Discrepancies from ideal superfluid – “imperfect superfluid”
Kellogg et al., PRL 93, 036801 (2004)
Experiments
drag – evolution with d
Kellogg et al., PRL 90, 246801 (2003)
persistance of intercorrelations
for large d
Experiments – transition at finite T
Phase boundary at νT = 1
Conductance at zero bias G(0) vs. T, d
G0 Kd / l c d / l
p
Champagne et al., PRL 100, 096801 (2008)
What about intermediate distances,
how transition proceeds?
What is the superfluid disordering
that results in 1/ 2 1/ 2 ?
Two paradigms of superfluid disordering:
(1) BKT (2D XY) dipole unbinding
(2) λ (3D XY) condensation of loops
N ~ Ne
~
(z z )
A
B
A
A
,B
M.V.Milovanovic, Bull. Am. Phys. Soc. 48 (2003);
S.H. Simon, E.H. Rezayi, and M.V. Milovanovic, PRL 91, 046803 (2003)
B
(I)
(II)
(a) and (c)
superfluid
(b) dsf., com. – vortex metal
(d) dsf., incom. – top. phase?
M.V.Milovanovic, PRB 75, 035314(2007), Z. Papic and M.V. Milovanovic, PRB 75, 195304(2007)
Chern-Simons linear response
Re 00 : Q
(a)
nf
x
, d 0.5,
1 / 10
kk f
b
k
,
kf
(b)
b
d 1.5,
1 / 10, p 1, p 1 / 10
nf
(a)
(b): vortex metal – (I) universality class
(a): neutral fermion pairs
in dual (Laughlin plasma) picture
|
|
|
|
k
l
k
l
p
q
p
q
|
|
F
(
)
F
(
)
s
s
i
j
i
,
j
M.V.Milovanovic and Z.Papic, PRB 79, 115319 (2009)
(a)
(b) BKT unbinding or dipole dissociation
(a):
z
,
z
w
w
111
1
/
2
1
/
2
z wn
A
ziw
q m
m
,n
p,q
z w
zkw
i
j
l
i,j
k,l
d
d
exactly rewritten as:
2
1
2
k
l
n
k
l
p
p
q
i
q
j
i,j
z
F
F
,z
s
s
1
n
111
2
z
i
i
Fock space of neutral fermions:
|
exp
i
,
,
|
1
n
11
1
n
1
n
Stern and Halperin proposal with phase separation
(fermi liquid puddles inside superfluid)
explains drag experiments
by deriving semicircle law - A.Stern and B.I. Halperin, PRL 88, 106801 (2002)
But also (a) and (b) (homogenous wave functions)
in a Chern-Simons response conform to semicircle law!
casecase
(b):
ˆ
E
2Jf
2
J
f
b
2
ˆ
where
2i
y
e
b
,
f
b
,
fJ
b
,
f
b
,
f dia
ˆ
E
2Jf J
J
b
b
b
(
J Jb Jf
J
0J
0
and
look
for
D
E
J
e
e
2
2
D
xx
D
xy 2
2
2
semicircle law
S.H.Simon et al., PRL 91, 046803(2003)
Z. Papic and M.V. Milovanovic, PRB 75, 195304(2007)
(c):(c)spin-wave (phonon) contribution- in (II) universalty class
(©
Bogoliubov:
MM
Chern-Simons:
Lopez, Fradkin PRB 51, 4347
(1995);Jiang,Ye PRB74, 245311(2006)
f d
exp
k k 111
k k
B
expc k k 111 111
k
from wave functions:
(I)
d
(II)
d 1 d 2
2
1 d 2
2
2
2
1
1 2 0
1 2
1
1 2
2
1 2 0
~
exp f d lnklB k k
k
1
pairing ~
z
~
~
1+1 neutral
fermion
1
back to Bogoliubov
z
z
Chern Sim ons( CDW )
z
(c)
(d): topological phase? -(II) univesality class
(d):
d
1
2
2
Det
(
z
z
)
(
z
z
)
i
j
k
l
z
z
i
j
k
l
i
j
1
1
Det
Det
11
z
z
z
z
i
j
k
l
M.V.Milovanovic and Z.Papic, PRB 79,115319 (2009)
|z
z |
|z
z|
|z
z|
2
k
l
k
l
p
q
i
p
2
q
2
j
i,j
1
exp
i
4
z
,
z
exp
i
4
z
,
z
2
2
|
z
z
|
1
2
1
1
2
2
use bosonic CFT analogies:
exp
i
4
z
,
z
exp
i
4
z
,
z
,
,
exp
i
4
z
,
z
1
1
2
2
N
N
1
z
,
z
z
z
,
z
,
z
z
z
Excitations:
1
1
2
|
z
w
|
V
exp
i
4
w
,
w
|
z
w
|
2
i
i
2
i
i
z
w
z
w
W
exp
i
4
w
,
w
z
w
z
w
i
i
i
i
i
i
i
i
V
w
,
w
V
w
,
w
W
w
,
w
W
w
,
w
11
22
33
44
|
w
w
||
w
w
|
w
w
w
w
w
w
w
w
w
w
w
w
w
1
w
w
w
2
2
2
2
1 3
2 3
2 4
1 4
2
2
2
2
2
2
2
2
1 2
3 4
1 4
2 3
1 3
2 4
1
by CFT analogies:
topological phase is of the kind described by
BF Chern-Simons theory
1
~
a
(
b
)
ja
jb
But!
But
But
can be any real number, also zero
There must be also a branch of gapless excitations
(1) impurities in exps. on bilayer cause BKT disordering
via pairs of neutral fermions (they lock charged
elementary merons)
(2) we may hope that sufficiently clean bilayer systems
may serve as generators (via loop condensation)
of (quasi) topological phases
described by doubled Chern-Simons theories