Transcript Slide 1

Fractional Statistics
of Quantum Particles
D. P. Arovas, UCSD
7 Pines meeting, Stillwater MN, May 6-10 2009
Collaborators:
Major ideas:
Texts:
J. R. Schrieffer, F. Wilczek, A. Zee,
T. Einarsson, S. L. Sondhi, S. M. Girvin,
S. B. Isakov, J. Myrheim, A. P. Polychronakos
J. M. Leinaas, J. Myrheim, R. Jackiw, F. Wilczek,
M. V. Berry, Y-S. Wu, B. I. Halperin, R. Laughlin,
F. D. M. Haldane, N. Read, G. Moore
Geometric Phases in Physics
Fractional Statistics and Anyon Superconductivity
(both World Scientific Press)
Two classes of quantum particles:
bosons
gauge
matter
fermions
quarks
leptons
- symmetric wavefunctions
- antisymmetric wavefunctions
- real or complex quantum fields
- Grassmann quantum fields
- condensation
- must pair to condense
- classical limit:
breaks U(1)
- no classical analog:
4He
3He
boson
e
p
fermion
n
n
e
p
n
p
++
++
e
e
p
Quantum Mechanics
of Identical Particles
Is that you,
Gertrude?
Hamiltonian invariant under label exchange:
i.e.
where
Eigenfunctions of Ĥ classified by unitary representations of SN :
Only two one-dimensional representations of SN :
Bose:
Fermi:
Path integral description
QM propagator:
(manifold)
Paths on M are classified by homotopy :
and
homotopic if
YES
with
{
NO
smoothly
deformable
Path composition ⇒ group structure : π1(M) = “fundamental group”
The propagator is expressed as a sum over homotopy classes μ :
weight for class μ
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 :
Laidlaw and DeWitt (1971) : quantum statistics and path integrals
one-particle “base space”
configuration space for N distinguishable particles
?
for indistinguishables? But...not a manifold!
how to fix :
Then :
:
disconnected
:
simply connected
:
multiply connected
N-string braid group
Y-S. Wu (1984)
:
generated by
=
=
- unitary one-dimensional representations of
:
- topological phase :
change in relative angle
- absorb into Lagrangian:
with
Charged particle - flux tube composites :
(Wilczek, 1982)
exchange
phase
Particles see each other as a source of geometric flux :
physical
Gauge transformation :
statistical
multi-valued
single-valued
Anyon wavefunction :
How do anyons behave?
Low density limit :
{
F
bosons
fermions
F
B
B
F
B
B
B
F
B
DPA (1985)
Johnson and Canright (1990)
Anyons break time reversal symmetry when
i.e. for values of θ away from the Bose and Fermi points.
What happens at higher densities??
Chern-Simons Field Theory
and Statistical Transmutation
lazy HEP convention:
metric
Given any theory with a conserved particle current, we can transmute statistics:
minimal coupling
Chern-Simons term
Examples: ordinary matter, skyrmions in O(3) nonlinear σ-model, etc.
Integrate out the statistical gauge field
via equations of motion:
⇒
linking
statistical b-field
So we obtain an effective action,
Wilczek and Zee (1983)
particle density
Anyon Superconductivity
fermions plus residual
statistical interaction
The many body anyon Hamiltonian contains only statistical interactions:
The magnetic field experienced by fermion i is
Mean field Ansatz :
⇒ Landau levels :
filling fraction
⇒
Total energy
filled Landau levels
⇒ sound mode :
But absence of low-lying particle-hole excitations ⇒ superfluidity! (?)
Anyons in an external magnetic field :
+
nth Landau level partially empty
+
(n+1)th Landau level partially filled
Y. Chen et al. (1989)
system prefers B=0
Meissner effect confirmed by RPA calculations
⇒
A. Fetter et al. (1989)
Signatures of anyon superconductivity
Y. Chen et al. (1989)
- Zero field Hall effect
- reflection of polarized light
- local orbital currents
- charge inhomogeneities at vortices
Unresolved issues
Wen and Zee (1989)
(not much work since early 1990’s)
- route to anyon SC doesn’t hinge on broken U(1) symmetry
“spontaneous violation of fact” (Chen et al.)
- Pairing? BCS physics? Josephson effect?
statistics of parent
p even
p odd
q even
B/F
B
q odd
duality treatments of Fisher, Lee, Kane
B/F
F
Fractional Quantum Hall Effect
Laughlin state at
:
(1983)
Quasihole excitations:
Quasihole charge
deduced from plasma analogy
The Hierarchy
- Haldane / Halperin
(1983 / 1984)
- condensation of quasiholes/quasiparticles
- Halperin : “pseudo-wavefunction” satisfying fractional statistics
Geometric phases
M. V. Berry (1984)
Adiabatic evolution
solution to SE
(projected)
Complete path :
adiabatic WF
where
Evolution of degenerate levels ➙ nonabelian structure :
Path :
where
Wilczek and Zee (1984)
Adiabatic quasihole statistics
DPA, Schrieffer, Wilczek (1984)
- Compute parameters in adiabatic effective Lagrangian
quasihole charge
from Aharonov-Bohm phase :
This establishes
in agreement with Laughlin
For statistics, examine two quasiholes:
⇒
Exchange phase is then
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
statistics
Kjo̸nsberg and Myrheim (1999)
Jain quasielectrons
charge
Jain quasielectrons
statistics
Sang, Graham and Jain (2003-04)
Effective field theory for the FQHE
Girvin and MacDonald (1987) ; Zhang, Hansson, and Kivelson (1989) ; Read (1989)
Basic idea : fermions = bosons +
Extremize the action :
Solution :
,
,
incompressible quantum liquid with
Quasiparticle statistics in the CSGL theory
- quasiparticles are vortices in the bosonic field
,
- ‘duality’ transformation to quasiparticle variables
reveals fractional statistics with new CS term!
Statistics and interferometry :
Stern (2008)
Fabry-Perot
relative phase :
D
S
changing B will nucleate bulk quasiholes,
resulting in detectable phase interference
Mach-Zehnder
D
relative phase :
phase interference depends on number of
quasiparticles which previously tunneled
S
- dependence ⇒
fractional statistics
Nonabelions
Moore and Read (1991)
Nayak and Wilczek (1996)
Read and Green (2000)
Ivanov (2001)
quasihole creator
- For M Laughlin quasiholes, one state :
- At
, there are
states with
quasiholes :
with
- 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
- There is a remarkable connection with vortices in
(px+ipy)-wave superconductors
- These states hold promise for fault-tolerant quantum computation
Exclusion statistics
Haldane (1991)
= # of quasiparticles of species
= # of states available to
qp
Model for exclusion statistics :
FQHE quasiparticles obey fractional exclusion statistics :
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
✸ The anyon gas at
is believed to be a superconductor
✸ FQHE quasiparticles have fractional charge and statistics
✸ Exotic nonabelian statistics at
✸ Related to exclusion statistics (Haldane), but phases essential