A quest for Pfaffian
(Talk at Physics Faculty, Belgrade, 2010)
Milica V. Milovanović
Institute of Physics Belgrade
Scientific Computing Laboratory
Hall experiment:
J.P.Eisenstein and H.L.Stormer, Science 248,1461(1990)
T= 85mK
Plateaus ! Rigidity !
filling factor =
  Ne / N
In rotationally symmetric gauge in two dimensions:
z  x  iy
Single particle wave functions:
m
z e
Orbits at radius:
1 2
 |z|
4
m  0,, N 1
 r 2   2m
Imagine that we are at the middle of the plateau at 1/3 How the ground state of the system would look like?
Laughlin answer:
R.B. Laughlin, PRL 50, 1995 (1983)
 (z  z )
i
j
m
e

1
4

|z i |2
i j
m3
Ne 1

N 3
antisymmetry
and in the cases of other “hierarchical constructions” odd denominator expected!
FQHE at 5/2 !
R. Willet et al., PRL 59, 1776, 1987
W. Pan et al.,PRL 83, 3530 ,1999.
Theoretical Moore-Read answer:
G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991)


1
1


2
(z i  z j )  A 



i j
(
z

z
)
(
z

z
)


1
2
N

1
N
e
e


Pfaffian
Pfaffian part describes a pairing among
particles as in a superconductor =
BCS pairing of spinless fermions
Pfaffian for 4 particles:
1
1
1
1
1
1


(z1  z 2 ) (z 3  z 4 ) (z1  z 3 ) (z 2  z 4 ) (z1  z 4 ) (z 2  z 3 )
p-wave superconductor (p-ip)
1
lim|k|0  k ~ k x  i k y  lim|z| g (z) ~
z
pairing function
wave function of a pair
Effective theory of a p-wave superconductor

| k | 0
N. Read and D.Green, PRB 61,10267(2000)
i.e. BCS mean field theory for
K eff


1 
 

   k c k c k   k c k c k   k c k c k 
2

k 
k2
k   k  ,  k 
2 m
k 
eigenfunction of rotations in
k ~ k x  i k y
for eigenvalue

k
  1
Excitations by Bogoliubov:
Ek 
0
Ground state
0
 k    |  k |
2
2
a gapped system
e
1
2

k
g k ck ck
| 0

1
1
gk ~
 g( r ) ~
kx  i ky
z
“weak pairing”

should not be too large:
|  k |2
Ek    k 
2
If

large: (a)
then likely: (b)
k0
k  kF
local maximum
local minimum
i.e. Fermi liquid phase
FQHE systems
(a) 5/2 : numerics favorable for Pfaffian in 2nd LL
Pfaffian is the most simple ansatz if not only
explanation of plateau,
R.H. Morf, PRL 80, 1505 (1998), E.H. Rezayi and F.D.M. Haldane, PRL 84, 4685 (2000)
(b) 1/2 : exps. and numerics find Fermi-liquid-like phase
(no plateau), E. Rezayi and N. Read, PRL 72, 900 (1994)
at 1/2 (1/4) in WQWs (wide quantum wells):
signatures of FQHE – minima in
 xx !
J. Shabani et al., Phys. Rev. Lett. 103, 256802 (2009)
likely nature of these states is multi-component
(two-component)
theory (mathematical identity)
A(331 )  Pf
two-component:
331   (zi  z j )3  (w k  w l )3  (z p  w q )
i j
k l
p q
Pf state can lead to a first topological quantum computer!
We want to know how to make Pfaffian!
331  Pf ?
tunneling( t )
FL
F
 eff
BCS formalism
of 331:
331
Pf
with tunneling
t
k  k  t
chemical potentials of parts:
even:  
t
odd:   t

eff    t
grows with
tunneling!
FL
F
 eff
BCS formalism
of 331:
331
Pf
t
likely outcome: Fermi liquid

 

 




If  t(c c  c c )  t(c  c )(c  c )
i.e. an open system then we may have a path: eff
with Pfaffian outcome

How to recognize Pfaffian?
Pfaffian makes
a topological phase!
X.-G. Wen, Int. J. Mod. Phys. B 6, 1711 (1992)
What are the signatures of a topological phase?
(a) gap
(b) characteristic degeneracy of ground state
on higher genus surfaces like torus
Torus
Create a qp-qh pair,
separate and drag
in opposite directions along
one of the two distinct
paths of torus and
annihilate:
a global process
Cylinder:
To go to the other side
requires energy (gapped excitations)
and we may not end up
with the same ground state but
a new sector
FQH state:
Filling factor:
Degeneracy on
torus:
Laughlin
1/3
Moore-Read
Pfafian
1/2
2

3
(331)
1/2
2

4
3
3 – number connected with quasiparticles of Pfaffian:
neutral fermions and vortices of the underlying
superconductor, M. Milovanovic and N.Read, PRB53, 13559 (1996)
Numerics with tunneling, Z. Papic et al., arxiv:0912.3103
331  Pf ?  FL
Sphere; overlap with tunneling:
Sphere is biased for Pfaffian.
in a bilayer
Torus; ground states with tunneling:
No (clear) signatures of 3(2) Pf degeneracy
(2 – trivial degeneracy in a translatory invariant QH system at ½)
The quest for Pfaffian goes on!