Pauli-like principle for Abelian and non-Abelian FQHE ...haldane/talks/fdmh_aps2006.pdf · Pauli-like principle for Abelian and non-Abelian FQHE quasiparticles F. D. M. Haldane ,

Pauli-like principle for Abelian and non-Abelian

FQHE quasiparticles F. D. M. Haldane , Princeton University.

P 46.7, APS March Meeting, Baltimore, March 15, 2006

Supported in part by NSF MRSEC DMR0213706 at Princeton Center for Complex Materials

* Generalization of the notion of Fock space to fractional statistics particles.

* Simple and transparent rules for counting states, and a powerful new technique for practical many-body calculations with fractional statistics.

• Holes in the filled Landau level ( ν = 1 ) are h/e vortices that obey Fermi statistics, and can be also be described in a standard fermion Fock space with a Pauli principle.

!(r1, . . . rN ) =N!

i=1

Nh!

j=1

(zi ! wj)!

i<j

(zi ! zj)!

i

!R(ri)

• expand in an (angular momentum) zm occupation number basis

complex coordinate of the j’th vortex/hole

Gaussian state centered at R.

|1111111111111100000000000!

|1110111101111111000000000!edge of (circular) droplet

droplet swells as holes are added

This is a Slater determinant !

what about the 1/m Laughlin states?

• wavefunction looks very similar, but is not a Slater determinant. Is there an alternative “occupation number” description?

!(r1, . . . rN ) =N!

i=1

Nh!

j=1

(zi ! wj)!

i<j

(zi ! zj)m!

i

!R(ri)

• In fact, the Laughlin state (circular droplet) is given by

P0|1001001001001001001000000!

Projection into a certain subspace; without this, the state is the (incorrect) Tao-Thouless Slater Determinant state

Beyond “standard” occupation number formalism

• k-particle 1/m Laughlin droplet creation operator:

!km(R)†|vac! "!

i>j

(zi # zj)mk!

i=1

"R(ri)

• For k = 1, this is just the standard lowest Landau-level single-particle creation operator

c(R)†|vac! " !R(r1)

Fundamental property of 1/m Laughlin state (and of Laughlin state + fractional charge/statistics quasiholes):

• There are NO pairs of particles with relative angular momentum (z-z′)m′ where m′ < m. The Laughlin state is annihilated by these two-body destruction operators!

• Define P0 (2,m) as the projection into the subspace of null states of (all) η2,m′(R), m′ ≤ m.

• 1/m Laughlin state with quasiholes at ri obeys

!2,m!(R)|!! = 0, m! = m" 2,m" 4, . . . all R

P0(2,m! 2)|!" = 0, c(ri)|!" = 0.

A linearly-independent basis of the 1/m Laughlin null space:

• Choose a subset of the set of all Slater determinant states where

• NO GROUP OF m CONSECUTIVE ORBITALS CONTAINS MORE THAN 1 PARTICLE

• then project on these with P0 (2,m-2) ! The Laughlin state is the incompressible highest-density state obeying this rule.

• The 1/3 Laughlin state is

* (all statements here have been confirmed numerically)

P0(2, 1)|1001001001001001001000000!

Essentially the same principle works for the (non-Abelian) Moore-Read (”Pfaffian”) and

Read-Rezayi (”parafermion”) states!!!

• Moore-Read state (FQHE analog of a BCS state):

|!(m)MR〉 ∝ Pf

ij

!1

zi − zj

" #

i<j

(zi − zj)m#

i

!R(ri)

Pf(A2n×2n) =1

2nn!

2n!!

P=1

!(P )n"

i=1

AP (2n−1)P (2n)

• 1/m+1 = 1, 1/2, 1/3, ... Moore-Read states are the highest density null states of the 3-body operator.

!3,m(R)|!! = 0 all R!2,m!(R)|!! = 0, m! = m" 2,m" 4, . . . all Rplus (for m > 1)

• Moore-Read basis set: Not more than 2 particles in 2m+2 consecutive orbitals, plus not more than 1 in m consecutive orbitals.

• ν = 1/2 fermionic MR state (m=1) :P0(3, 1)|1100110011001100110011000000!

• h/2e (fractionalized) “nonabelian” vortices at Ri:

!2,m(Ri)|!! = 0 Can’t destroy a 2-particle pair centered at vortex position Ri

(can be done at all other locations

• Moore-Read state is k=2 member of ν = k/mk+2 Read-Rezayi states. Everything generalizes to k > 2.

Topological degeneracy on the torus:

P0| . . . 1010101010 . . .!P0| . . . 0101010101 . . .!

P0| . . . 100100100 . . .!P0| . . . 010010010 . . .!P0| . . . 001001001 . . .!

P0| . . . 110011001100 . . .!P0| . . . 001100110011 . . .!

P0| . . . 100110011001 . . .!P0| . . . 011001100110 . . .!

Abelian 1/3 Laughlin state, 3 states related by center-of-mass translations

non-Abelian 1/2 Moore-Read state, 3 distinct groups of 2 states related by center-of-mass translations (total 3x2 = 6)

examples of defects:

• Laughlin 1/3 state, with a charge -1/3 quasihole (h/e vortex) at the origin:

• Moore-Read 1/2 state, with a charge -1/2 double (Abelian) quasihole (h/e vortex)

• Fractionalization of this into two non-abelian charge-1/4 quasiholes (h/2e vortices)

P0|01001001001001001001 . . .!

P0|01100110011001100110011 . . .!

P0|101010101010011001100110011 . . .!AB

A

fractionalization of the h/e vortex:

P0| . . . 11001100110001100110011 . . .!P0| . . . 11001100101010100110011 . . .!P0| . . . 11001010101010101010011 . . .!

dimer dimer“Haldane gap/AKLT”

Note: a (not accidental!) similarity to a spin-1 quantum chain at the critical point separating “Haldane gap” and dimerized phases

Classification of polynomial occupation numer states as a partially-ordered set (POSET)

• Three classes:

• (a) “allowed configurations” that satisfy the Pauli-like k-particle selection rule (includes TOP) |1001001001......>

• (b) “not allowed , but generated by “squeezing” action of P0 on “allowed configurations” (includes BOTTOM) |0110001001....>

• (c) “excluded” (not (a) or (b)) |1010101001...>

Leads to a highly efficient method for numerical construction of null space of η(k+1,m) and η(2,m′<m)

Two ingredients of a calculation:

• “Null space” (such as lowest Landau level) defines a low energy Hilbert space.

• Need a second ingredient: a Hamiltonian that acts in this Hilbert space (e.g. Coulomb interaction, background potential, probes to manipulate vortex positions, etc.)

Numerical study of Moore-Read h/2e vortices

• Project the Coulomb and substrate potential into the null space (a generalization of projection into a Landau level).

• Null space basis states are computed accurately with a (new?) variant of Lanczos, (annihilation is accurate to floating-point machine precision, 64bit).

• Spherical geometry (sphere surrounding a magnetic monopole) is used: eliminates edges, Wigner-Eckert simplifies calculations.

• basis set size for 14/28 Moore-Read :140,116,60. But, after construction of the “fractional statistics generalization of Fock space” this is reduced to 540!

• Energy levels found, local single-particle and m=1 pair densities imaged.

!1 !0.5 0 0.5 1!1

!0.8

!0.6

!0.4

!0.2

0

0.2

0.4

0.6

0.8

1

!1 !0.5 0 0.5 1!1

!0.8

!0.6

!0.4

!0.2

0

0.2

0.4

0.6

0.8

1

!1 !0.5 0 0.5 1

!1

!0.8

!0.6

!0.4

!0.2

0

0.2

0.4

0.6

0.8

1

!1 !0.5 0 0.5 1

!1

!0.8

!0.6

!0.4

!0.2

0

0.2

0.4

0.6

0.8

1

single-particle density

m=1 two-particle density

Tetrahedral arrangement of 4 MR h/2e vortices, (14 electrons, 28 orbitals)

Sphere is mapped to unit disk.

the qubit doublet is split by the Coulomb interaction, both states are shown

One qubit is left after positions of vortices are fixed.

Summary.• A very simple recipe for counting and constructing

basis set of quasiholes in Laughlin, Moore-Read, and Read-Rezayi FQHE states.

• Practical method for previously-impossible inhomogenous finite-size calculations; “Fock space” for fractional statistics

• (also, for m=0 bosons) A remarkable correspondence to integrable critical spin chains: (k=1 Laughlin to S=1/2 Haldane-Shastry, k>1 to spin-k/2 generalizations of HS) (not described here)