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Major ideas:. Collaborators:. Texts:. J. M. Leinaas, J. Myrheim, R. Jackiw, F. Wilczek,. J. R. Schrieffer, F. Wilczek, A. Zee,. Geometric Phases in Physics. Fractional Statistics and Anyon Superconductivity. T. Einarsson, S. L. Sondhi, S. M. Girvin,.

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slide1

Major ideas:

Collaborators:

Texts:

J. M. Leinaas, J. Myrheim, R. Jackiw, F. Wilczek,

J. R. Schrieffer, F. Wilczek, A. Zee,

Geometric Phases in Physics

Fractional Statistics and Anyon Superconductivity

T. Einarsson, S. L. Sondhi, S. M. Girvin,

M. V. Berry, Y-S. Wu, B. I. Halperin, R. Laughlin,

S. B. Isakov, J. Myrheim, A. P. Polychronakos

(both World Scientific Press)

F. D. M. Haldane, N. Read, G. Moore

Fractional Statistics

of Quantum Particles

D. P. Arovas, UCSD

7 Pines meeting, Stillwater MN, May 6-10 2009

slide3

- condensation

- classical limit:

- no classical analog:

bosons

fermions

breaks U(1)

gauge

quarks

matter

leptons

- antisymmetric wavefunctions

- symmetric wavefunctions

- real or complex quantum fields

- Grassmann quantum fields

- must pair to condense

Two classes of quantum particles:

slide4

++

e

e

e

e

p

p

p

p

n

n

n

boson

fermion

4He

3He

++

slide5

Only two one-dimensional representations of SN :

Bose:

Fermi:

Eigenfunctions of Ĥ classified by unitary representations of SN :

Hamiltonian invariant under label exchange:

Is that you, Gertrude?

where

i.e.

Quantum Mechanics

of Identical Particles

slide6

Paths on M are classified by homotopy :

QM propagator:

{

and

homotopic if

with

smoothly deformable

(manifold)

NO

YES

Path integral description

slide7

In order that the composition rule be preserved,

the weights χ(μ) must form a unitary representation of π1(M) :

Think about the Aharonov-Bohm effect :

Path composition ⇒ group structure : π1(M) = “fundamental group”

The propagator is expressed as a sum over homotopy classes μ :

weight for class μ

slide8

Then :

?

one-particle “base space”

:

disconnected

for indistinguishables? But...not a manifold!

:

simply connected

configuration space for N distinguishable particles

:

multiply connected

N-string braid group

how to fix :

Y-S. Wu (1984)

Laidlaw and DeWitt (1971) : quantum statistics and path integrals

slide9

- unitary one-dimensional representations of

:

:

generated by

=

- topological phase :

=

change in relative angle

- absorb into Lagrangian:

with

slide10

Particles see each other as a source of geometric flux :

Gauge transformation :

multi-valued

physical

statistical

single-valued

Anyon wavefunction :

Charged particle - flux tube composites :

(Wilczek, 1982)

exchange phase

slide11

Low density limit :

F

F

{

bosons fermions

B

B

B

F

F

B

B

B

Johnson and Canright (1990)

DPA (1985)

How do anyons behave?

Anyons break time reversal symmetry when

i.e. for values of θ away from the Bose and Fermi points.

What happens at higher densities??

slide12

Integrate out the statistical gauge field

via equations of motion:

lazy HEP convention:

linking

metric

So we obtain an effective action,

statistical b-field particle density

Given any theory with a conserved particle current, we can transmute statistics:

Wilczek and Zee (1983)

minimal coupling

Chern-Simons term

Examples: ordinary matter, skyrmions in O(3) nonlinear σ-model, etc.

Chern-Simons Field Theory and Statistical Transmutation

slide13

The many body anyon Hamiltonian contains only statistical interactions:

Total energy

sound mode :

Mean field Ansatz :

Landau levels :

But absence of low-lying particle-hole excitations ⇒ superfluidity! (?)

filling fraction

The magnetic field experienced by fermion i is

filled Landau levels

Anyon Superconductivity

fermions plus residual statistical interaction

slide14

(n+1)th Landau level partially filled

system prefers B=0

+

nth Landau level partially empty

+

Meissner effect confirmed by RPA calculations

A. Fetter et al. (1989)

Anyons in an external magnetic field :

Y. Chen et al. (1989)

slide15

Signatures of anyon superconductivity

Unresolved issues

p even

p odd

(not much work since early 1990’s)

Y. Chen et al. (1989)

B/F

B

q even

- route to anyon SC doesn’t hinge on broken U(1) symmetry

- Zero field Hall effect

- reflection of polarized light

Wen and Zee (1989)

B/F

F

q odd

“spontaneous violation of fact” (Chen et al.)

- local orbital currents

- charge inhomogeneities at vortices

- Pairing? BCS physics? Josephson effect?

statistics of parent

duality treatments of Fisher, Lee, Kane

slide16

The Hierarchy

Laughlin state at

:

(1983)

- Haldane / Halperin

Quasihole excitations:

(1983 / 1984)

- condensation of quasiholes/quasiparticles

Quasihole charge

deduced from plasma analogy

- Halperin : “pseudo-wavefunction” satisfying fractional statistics

Fractional Quantum Hall Effect

slide17

Evolution of degenerate levels ➙ nonabelian structure :

Adiabatic evolution

solution to SE

(projected)

adiabatic WF

where

Path :

where

Complete path :

Wilczek and Zee (1984)

Geometric phases

M. V. Berry (1984)

slide18

- Compute parameters in adiabatic effective Lagrangian

quasihole charge

For statistics, examine two quasiholes:

Exchange phase is then

from Aharonov-Bohm phase :

This establishes

in agreement with Laughlin

Adiabatic quasihole statistics

DPA, Schrieffer, Wilczek (1984)

slide19

Numerical calculations of e* and θ

- good convergence for quasihole states

- quasielectrons much trickier ; convergence better for Jain’s WFs

- must be careful in defining center of quasielectron

Laughlin quasielectrons

Jain quasielectrons

Jain quasielectrons

statistics

statistics

charge

Kjo̸nsberg and Myrheim (1999)

Sang, Graham and Jain (2003-04)

slide20

Extremize the action :

incompressible quantum liquid with

,

,

Solution :

Effective field theory for the FQHE

Girvin and MacDonald (1987) ; Zhang, Hansson, and Kivelson (1989); Read (1989)

Basic idea : fermions = bosons +

slide21

- ‘duality’ transformation to quasiparticle variables

reveals fractional statistics with new CS term!

- quasiparticles are vortices in the bosonic field

,

Quasiparticle statistics in the CSGL theory

slide22

S

D

S

D

- dependence ⇒

Mach-Zehnder

Fabry-Perot

fractional statistics

relative phase :

relative phase :

phase interference depends on number of

quasiparticles which previously tunneled

changing B will nucleate bulk quasiholes,

resulting in detectable phase interference

Statistics and interferometry :

Stern (2008)

slide23

Moore and Read (1991)

Nayak and Wilczek (1996)

- This leads to a very rich braiding structure, involving higher-dimensional

representations of the braid group

- The degrees of freedom are essentially nonlocal, and are

associated with Majorana fermions

states with

quasiholes :

quasihole creator

- At

,

there are

with

Read and Green (2000)

Ivanov (2001)

- There is a remarkable connection with vortices in

(px+ipy)-wave superconductors

- For M Laughlin quasiholes, one state :

- These states hold promise for fault-tolerant quantum computation

Nonabelions

slide24

FQHE quasiparticles obey fractional exclusion statistics :

= # of quasiparticles of species

= # of states available to

qp

Model for exclusion statistics :

Exclusion statistics

Haldane (1991)

slide25

✸ The anyon gas at

✸ Exotic nonabelian statistics at

is believed to be a superconductor

Key Points

✸ In d=2, a one-parameter (θ) family of quantum statistics exists

between Bose (θ=0) and Fermi (θ=π), with broken T in between

✸ Anyons behave as charge-flux composites (phases from A-B effect)

✸ Two equivalent descriptions :

(i) bosons or fermions with statistical vector potential

(ii) multi-valued wavefunctions with no statistical interaction

✸ Beautiful effective field theory description via Chern-Simons term

✸ FQHE quasiparticles have fractional charge and statistics

✸ Related to exclusion statistics (Haldane), but phases essential