non-Abelian quantum statistics

38 Pages·2013·3.51 MB·English

Fractional quantum statistics
T. H. Hansson, Stockholm University
Thanks to:
Anders, Eddy, Emil, Jainendra, Jon-Magne, Juha, Maria, Susanne, .......
Outline: • What is fractional statistics?
• Where does the quantum Hall effect enter?
• What is non-Abelian fractional statistics?
• Anyons and Topological Field Theory
• Why does Microsoft care - topologically
protected quantum computing.
• What are the experiments?
NA quantum statistics Anyon School
T. H. Hansson Berlin, 2013
tisdag 24 september 13 1
Wave function for identical particles:
P !x . . .x! . . .x! . . .x! eiθ !x . . .x! . . .x! . . .x!
Ψ( ) = ij Ψ( )
1 1
ij i j N j i N
The particles are identical since an overall phase is not observable.
But changing back amounts to nothing!
2
P !x . . .x! . . .x! . . .x! !x . . .x! . . .x! . . .x!
Ψ( ) =Ψ( )
1 1
ij i j N i j N
2
P = 1 ⇒ eiθ = ±1
ij
So there are only two alternatives:
!
P = 1 ; θ = 0 Bosons!
T
ij ij
U
P − θ π B
= 1 ; = Fermions
ij ij
NA quantum statistics Anyon School
T. H. Hansson Berlin, 2013
tisdag 24 september 13 2
Permutations and braidings:
The basic idea:
⇥x ⇥x
The permutation : � , sgn(P ) = 1
i j ij
±
can also be viewed as a
!x !x
the exchange : i j
i�
e , � = 0, ⇥
�
!x !x
i j
For general θ, the particles are ANYONS!
Leinaas & Myrheim, 77
NA quantum statistics Anyon School
T. H. Hansson Berlin, 2013
tisdag 24 september 13 3
But what is meant by “exchange”?
In a path integral description, different classical
exchange paths corresponds to different braids.
All paths that are described by the same braid,
are assigned the same exchange phase!
In a Hamiltonian description, we can imagine
to “pin down” and move the particles around.
In such a real exchange process, the wave
function picks up a Berry phase!
NA quantum statistics Anyon School
T. H. Hansson Berlin, 2013
tisdag 24 september 13 4
Pathintegral for identical particles
(1) (2) (3) (4) (1) (2) (3) (4)
⇥x ⇥x ⇥x ⇥x ⇥x ⇥x ⇥x ⇥x
fi fi fi fi fi fi fi fi
!x !x
i j
!x !x
i j
i�
e
i�
e
�
i�
e
(1) (2) (3) (4) (1) (2) (3) (4)
⇥x ⇥x ⇥x ⇥x ⇥x ⇥x ⇥x ⇥x
in in in in in in in in
� = (3 4)� = � � = (6 1)� = 5 �
br br
� �
(�) (�)
�x =�x
fi
(�) (�) i
i⇥ (�) S
A(⌅x ⌅x ) = e br [⌅x ]e
�
in fi
� D
(�)
�x(�)=�x
braids ⇥ in
�
NA quantum statistics Anyon School
T. H. Hansson Berlin, 2013
tisdag 24 september 13 5
Why are two dimensions special ?
≡ d > 2
d = 2
≡|
So there are no braids for d>2, and we are back to
just having the permutation symmetry, and thus
only bosons and fermions!
NA quantum statistics Anyon School
T. H. Hansson Berlin, 2013
tisdag 24 september 13 6
Wave functions for identical particles
• “Pin down” the particles
• Move them around
• Calculate the statistical phase
We can pin down a particle
by putting it in a box:
2
p
− − !
H = V (!x R)
box
2m
!
The particle can now be moved by
R
!
R
changing .
NA quantum statistics Anyon School
T. H. Hansson Berlin, 2013
tisdag 24 september 13 7
Question:
What happens when two
particles are exchanged,
?
or alternatively,
one encirles the other:
Answer:
The w.f. “picks up” a phase factor:
→ iθ
Ψ(!x,x! ) e Ψ(!x,x! )
Abelian
i j j i
a
→ iθ T
Ψ (!x,x! ) e a αβ Ψ (!x,x! ) Non-Abelian
α i j β j i
NA quantum statistics Anyon School
T. H. Hansson Berlin, 2013
tisdag 24 september 13 8
But how do we calculate?
Michael Berry’s phase:
Assume: H(R )� (⇥r; R ) = E (R )� (⇥r; R )
i n i n i n i
How will � (⇥r; R ) evolve under adiabatic time evolution
n i
if the parameters become timedependent: R R (t)?
i i
�
i T
eiγ dtE (R(t))
Nbuott � (⌥r, T ; R ) = B e n � (⌥r, 0; R )
n i � � 0 n i
R
T
∂
˙
where the Berry phase is given by, = a
γ i dt R !ψ | |ψ "
B n n
! ∂R
0 a
can be calculated from the wave function!
NA quantum statistics Anyon School
T. H. Hansson Berlin, 2013
tisdag 24 september 13 9
For a closed path, in parameter space Berry’s phase
is a geometric property - i.e. it does not depend on
|ψ (R )!
any arbitrary choices of phase of .
n i
Example: Particle in a magnetic field
!
B
Φ
BA
= =
γ A
B
φ φ
0 0
¯
h h
= 2 =
where φ π is the elementary flux quantum.
0
e e
This result is independent of gauge choice!!
NA quantum statistics Anyon School
T. H. Hansson Berlin, 2013
tisdag 24 september 13 10