A quest for Pfaffian
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A quest for Pfaffian
Milica V. MilovanovićInstitute of Physics Belgrade
Scientific Computing Laboratory
(Talk at Physics Faculty, Belgrade, 2010)
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Hall experiment:
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T= 85mK N/NePlateaus ! Rigidity ! filling factor =
J.P.Eisenstein and H.L.Stormer, Science 248,1461(1990)
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In rotationally symmetric gauge in two dimensions: iyxz
Single particle wave functions:
2|z|41
m ez
1N,,0m
Orbits at radius: m2r2
Imagine that we are at the middle of the plateau at 1/3 -
How the ground state of the system would look like?
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Laughlin answer:
mj
jii )zz(
2i |z|
41
e
3m31
NNe
antisymmetry
and in the cases of other “hierarchical constructions” odd denominator expected!
R.B. Laughlin, PRL 50, 1995 (1983)
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W. Pan et al.,PRL 83, 3530 ,1999.
FQHE at 5/2 !R. Willet et al., PRL 59, 1776, 1987
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Theoretical Moore-Read answer:
2j
jii )zz(
)zz(1
)zz(1A
ee N1N21
PfaffianPfaffian part describes a pairing amongparticles as in a superconductor =
BCS pairing of spinless fermions
G. Moore and N. Read, Nucl. Phys. B 360, 362 (1991)
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Pfaffian for 4 particles:
)zz(1
)zz(1
)zz(1
)zz(1
)zz(1
)zz(1
324142314321
p-wave superconductor (p-ip)
z1~)z(glimkik~lim |z|yxk0|k|
pairing function wave function of a pair
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Effective theory of a p-wave superconductor N. Read and D.Green, PRB 61,10267(2000)
i.e. BCS mean field theory for 0|k|
kkkkkkkkkkeff cccc
21ccK
m2k,
2
kkk
k eigenfunction of rotations in k
yxk kik~ for eigenvalue 1
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Excitations by Bogoliubov:
2k
2kk ||E
0 a gapped system
Ground state
0|e kkkk ccg
21
0z1~)r(g
kik1~g
yxk
“weak pairing”
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should not be too large:
2
||E2
kkk
If large: (a) 0k local maximum
then likely: (b) Fkk local minimum
i.e. Fermi liquid phase
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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)
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at 1/2 (1/4) in WQWs (wide quantum wells):
signatures of FQHE – minima in !xx
likely nature of these states is multi-component (two-component)
J. Shabani et al., Phys. Rev. Lett. 103, 256802 (2009)
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theory (mathematical identity)
Pf331)(A
two-component:
)wz()ww()zz( qqp
p3
llk
k3
jji
i331
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Pf state can lead to a first topological quantum computer!
We want to know how to make Pfaffian!
?Pf)t(tunneling
331
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eff
t
331
Pf
FLF
BCS formalism of :331
with tunneling tkk chemical potentials of parts:
tt teff
even:
odd:grows withtunneling!
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eff
t
331
Pf
FLF
BCS formalism of :331
likely outcome: Fermi liquid
If )cc)(cc(t)cccc(t
i.e. an open system then we may have a path: effwith Pfaffian outcome
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How to recognize Pfaffian?
Pfaffian makes a topological phase!
What are the signatures of a topological phase?
(a) gap
(b) characteristic degeneracy of ground state on higher genus surfaces like torus
X.-G. Wen, Int. J. Mod. Phys. B 6, 1711 (1992)
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Torus
Create a qp-qh pair,separate and dragin opposite directions alongone of the two distinctpaths of torus andannihilate:
a global process
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Cylinder:
To go to the other siderequires energy (gapped excitations)and we may not end upwith the same ground state but
a new sector
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FQH state: Filling factor: Degeneracy on torus:
Laughlin 1/3 3
Moore-Read Pfafian
1/2 2 3
(331) 1/2 2 4
3 – number connected with quasiparticles of Pfaffian: neutral fermions and vortices of the underlying superconductor, M. Milovanovic and N.Read, PRB53, 13559 (1996)
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Numerics with tunneling, Z. Papic et al., arxiv:0912.3103
FL?Pf331 in a bilayer
Sphere; overlap with tunneling:
Sphere is biased for Pfaffian.
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Torus; ground states with tunneling:
No (clear) signatures of Pf degeneracy(2 – trivial degeneracy in a translatory invariant QH system at ½)
)2(3
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The quest for Pfaffian goes on!