A quest for Pfaffian

Milica V. MilovanovićInstitute of Physics Belgrade

Scientific Computing Laboratory

(Talk at Physics Faculty, Belgrade, 2010)

Hall experiment:

T= 85mK N/NePlateaus ! Rigidity ! filling factor =

J.P.Eisenstein and H.L.Stormer, Science 248,1461(1990)

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?

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)

W. Pan et al.,PRL 83, 3530 ,1999.

FQHE at 5/2 !R. Willet et al., PRL 59, 1776, 1987

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)

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

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

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”

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

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

likely nature of these states is multi-component (two-component)

J. Shabani et al., Phys. Rev. Lett. 103, 256802 (2009)

theory (mathematical identity)

Pf331)(A

two-component:

)wz()ww()zz( qqp

p3

llk

k3

jji

i331

Pf state can lead to a first topological quantum computer!

We want to know how to make Pfaffian!

?Pf)t(tunneling

331

eff

t

331

Pf

FLF

BCS formalism of :331

with tunneling tkk chemical potentials of parts:

tt teff

even:

odd:grows withtunneling!

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

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)

Torus

Create a qp-qh pair,separate and dragin opposite directions alongone of the two distinctpaths of torus andannihilate:

a global process

Cylinder:

To go to the other siderequires energy (gapped excitations)and we may not end upwith the same ground state but

a new sector

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)

Numerics with tunneling, Z. Papic et al., arxiv:0912.3103

FL?Pf331 in a bilayer

Sphere; overlap with tunneling:

Sphere is biased for Pfaffian.

Torus; ground states with tunneling:

No (clear) signatures of Pf degeneracy(2 – trivial degeneracy in a translatory invariant QH system at ½)

)2(3

The quest for Pfaffian goes on!