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Theory of non-Abelian statistics: fusion space of topo. exc.
What are the most general properties of the topological
excitations? can be boson, can be fermion, can be semion, ...
Consider a state with quasiparticles |i1 , i2 , i3 , · · ·� at �x1 , �x2 , �x3 , · · ·,
which is a gapped ground state of
trap
trap
H + δHitrap
(�
x
)
+
δH
(�
x
)
+
δH
x3 ) + · · ·
1
2
i2
i3 (�
1
• The ground state subspace of the above Hamiltonian is the fusion
space V F (i1 , i2 , i3 , · · · ) of the quasiparticles i1 , i2 , i3 , · · ·.
• We assume the above ground state degeneracy is stable arbitary
purterbations around �x1 , �x2 , �x3 , · · · and the traped quasiparticles
are said to be simple.
• If the ground state subspace is not stable against any perturbations
δH(�x1 ) near �x1 , then the quasiparticle i1 at �x1 is composite.
• If i1 is composite, we can add δH(�x1 ) to split the ground state
subspace:
V F (i1 , i2 , i3 , · · · ) → V F (j1 , i2 , i3 , · · · ) ⊕ V F (k1 , i2 , i3 , · · · ) ⊕ · · ·
We denote i1 = j1 ⊕ k1 ⊕ · · ·.
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
Quantum entanglement, topological order, and tensor category
Fusion algebra of (non-Abelian) topological excitations
• For simplei, j, if we view (i, j) as one particle,
it may correspond to a composite particle:
˜ l1 , l 2 , · · · )
V F (i, j, l1 , l2 , · · · ) = ⊕˜ V F (k,
k
= ⊕k ⊕
⊕k Nkij k
Nkij
αijk =1
V Fij (k, l1 , l2 , · · · )
αk
(k2 , ..)
(k1 , ..)
→ the fusion algebra.
i ⊗j =
(i,j,...)
Associativity:
�
� jk in
ijk
ijk
ij mk
(i ⊗ j) ⊗ k = i ⊗ (j ⊗ k) = ⊕l Nl l, Nl = m Nm Nl = n Nn Nl
Quantum dimension and vector space fractionalization:
• In general, we cannot view V F (i, j, k, · · · ) as
V (i) ⊗ V (j) ⊗ V (k) ⊗ · · ·, and dim[V F (i, i, i, · · · )] �= din , di ∈ Z.
fractional degree freedom.
Quasiparticle i may carry
�
ii N m1 i · · · N mn−2 i = (Ni )n−1 ∼ d n
dim[V F (i, i, · · · , i)] = mi Nm
1
i
i1
1 m2
where the matrix (Ni )jk = Nkji , and di the largest eigenvalue of Ni .
• di is called the quantum dimension of the quasiparticle i.
Abelian particle → di = 1. Non-Abelian particle → di �= 1.
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
Quantum entanglement, topological order, and tensor category
Relation between fusion spaces and the F -matrix
• Two different ways to fuse i, j, k → l:
F
V (i, j, k, · · · ) =
ij
Nm
⊕m ⊕ ij
αm =1
= ⊕m ⊕
ij
Nm
αijm =1
i
⊕l ⊕
k
α ijm m mk
αl
l
VαFij (m, k, · · · )
m
Nlmk
αmk
l =1
j
i
j
F
α jkn
k
n α inl
l
VαFij ;αmk ,m (l, · · · )
m
l
ij
= ⊕l {|l; αm
, αlmk , m�} ⊗ V F (l, · · · )
F
V (i, j, k, · · · ) =
=
=
•
|l; αnjk , αlin , m�
=
Nnjk
⊕n ⊕ jk
αn =1
VαFjk (i, n, · · · )
n
Nlin
Nnjk
⊕n ⊕ jk ⊕l ⊕αin =1 VαFjk ;αin ,n (l, · · · )
αn =1
n
l
l
⊕l {|l; αnjk , αlin , n�} ⊗ V F (l, · · · )
�
in
n,αjk
n ,αl
F
ijk;m,αijm ,αmk
l
in
l;n,αjk
n ,αl
where Fijk
l is an unitary matrix.
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
|l; αnjk , αlin , n�
Quantum entanglement, topological order, and tensor category
ijk;mαβ
Consistent conditions for Fl;nχδ
and UFC
i
Two different ways of fusion
α
m
Φ
Φ
α
m
j
β
n
�i
k
l
=
χ
p
α
m
j
β
n
k
χ
p
l
�
=
=
�
�
i
mkl;nβχ
Φ
q,δ,� Fp;qδ�
ijk;mαβ
F
Φ
n;tηϕ
t,η,ϕ
�
β
n
two
� paths of F-moves:
� different
�
i
j
α
�i
j
l
i
and
χ
p
j
m
k
k
ε
δ
q
l
p
k
η
ϕt
n χ
p
l
�
�
=
=
�
j
k
δ
φ
γ q
s
p
q,δ,�;s,φ,γ
�
l
are related via
mkl;nβχ ijq;mα�
Fp;qδ�
Fp;sφγ Φ
ijk;mαβ itl;nϕχ
F
Fp;sκγ Φ
n;tηϕ
t,η,ϕ;s,κ,γ
i
ijk;mαβ itl;nϕχ jkl;tηκ 
F
Fp;sκγ Fs;qδφ Φ
n;tηϕ
t,η,κ;ϕ;s,κ,γ;q,δ,φ
j
k
γ
p
φ
s
δ
q
l

�i
�i
j
j
k
δ
φ
q
γ
s
p
k
η
t κ
γ s
p
l
l
�
,
�
.
The two paths should lead to the same unitary trans.:
� mkl;nβχ ijq;mα�
� ijk;mαβ
itl;nϕχ jkl;tηκ
Fn;tηϕ Fp;sκγ Fs;qδφ =
Fp;qδ� Fp;sφγ
t,η,ϕ,κ
�
Such a set of non-linear algebraic equations is the famous
pentagon identity. Moore-Seiberg 89
ijk;mαβ
Nkij , Fl;nχδ
→ Unitary fusion category (UFC)
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
Quantum entanglement, topological order, and tensor category
UFC and topological quasiparticles in different dimensions
• Topological excitations in 1+1D are described/classified by
(non-Abelian) UFC.
i
j
k
Consider topological excitations described by an arbitary
UFC, can we realize them via a 1+1D lattice model?
• Topological excitations in 2+1D (and beyond) are described by
Abelian (symmetric) UFC: Nkij = Nkji .
i
j
j
i
k
In higher dimension, topological excitations also have non-trivial
braiding properties.
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
Quantum entanglement, topological order, and tensor category
Braiding and R-matrix
i
• Two ways to fuse:
j
i
R
˜αF (k, · · · )
V (i, j, · · · ) = ⊕k,α V
F
α
= ⊕k {|k; α�� } ⊗ V F (k, · · · )
•
=
�
β
k
V F (i, j, · · · ) = ⊕k,β VβF (k, · · · )
|k, �
j
k
= ⊕k {|k; β�} ⊗ V F (k, · · · )
ij;α
R
β k;β |k, β�
ij;α
where Rk;β
is an unitray matrix.
• Relation to the spin θi = e i2πsi of the particle:
2π rotation of (i, j) = 2π rotation of k
2π rotation of (i, j) = 2π rotation
of i and j and exchange i, j twice
ij;γ ji;β
θi θj Rk;β
Rk;α = θk δγα
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
j
i
j
i
R
γ
k
j
i
R
β
k
α
k
Quantum entanglement, topological order, and tensor category
ij;α
Consistent conditions for Rk;β
and UMTC
j
i
k
j
i
R
i
j
k
λ
F
R
j
i
k
α
j
α
m
R
m
β
l
γ
k
j
i
k
F
m
α
l
η
l
i
γ
n
δ
l
l
k
F
p
λ
j
i
ε
φ
p
k
n
δ
χ
l
l
Hexagon identity:
� kij;pφλ mk;γ ijk;mαβ
ik;φ ikj;p�λ jk;η
Fl;mαγ Rl;β Fl;nχδ
Rp;� Fl;nηδ Rn;χ =
ijk;mαβ
ij;α
Nkij , Fl;nχδ
, Rk;β
mαβ
→ Unitary modular tensor category (UMTC)
which describes non-Abelian statistics of 2+1D topo. excitations.
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
Quantum entanglement, topological order, and tensor category
Boundary of topological order → gravitational anomaly
• Boundary of (some) topologically ordered states is gapless
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
Quantum entanglement, topological order, and tensor category
Boundary of topological order → gravitational anomaly
• Boundary of (some) topologically ordered states is gapless
• Boundary of topologically ordered states has gravitational anomaly
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
Quantum entanglement, topological order, and tensor category
Boundary of topological order → gravitational anomaly
• Boundary of (some) topologically ordered states is gapless
• Boundary of topologically ordered states has gravitational anomaly
There is an one-to-one
correspondence between
effective
d-dimensional topological
theory
Topologically
orders and d − 1-dimensional
with
ordered
gravitational anomalies
gravitational
state
anomaly
Example 1 (gapless):
• 1+1D chiral fermion L = i(ψ † ∂t ψ − ψ † ∂x ψ) → �(k) = vk.
Gravitational anomalous, cannot appear as low energy effective
theory of any well-definded local 1+1D lattice model.
• But the above chiral fermion theory cannot appear as low energy
effective theory for the boundary of a 2+1D topologically ordered
state – the ν = 1 IQH state (which has no topological excitations).
• The same bulk → many different boundary of the same
gravitational anomaly, e.g. 3 edge modes (v1 k, −v2 k, v3 k)
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
Quantum entanglement, topological order, and tensor category
Example 2 (gapless):
• 1+1D chiral boson (8 modes c = 8)
E
KIJ8
2π

2
1

0

0
=
0

0

0
0
1
2
1
0
0
0
1
0
0
1
2
1
0
0
0
0
0
0
1
2
1
0
0
0
0
0
0
1
2
1
0
0
0
0
0
0
1
2
0
0
0
1
0
0
0
0
2
1

0
0

0

0

0

0

1
2
L=
∂x φI ∂t φJ − VIJ ∂x φI ∂x φJ .
• Gravitational anomalous.
Realized as edge of
E8
K
8-layer �
bosonic QH state:
ΨE8 = (ziI − zjJ )KIJ
Filling fraction ν = 4
det(K E8 ) = 1 → no topo. exc.
Example 3 (gapped):
• 2+1D theory with excitations (1, e, m, �). Fusion:
e × e = m × m = � × � = 1, e × m = �. Braiding: e, m, � have mutual π
statistics, e, m are boson � is fermion.
• No gravitational anomaly. Can be realized by the toric code model.
Example 4 (gapped):
• 2+1D theory with excitations (1, e). e × e = 1. e is a boson.
• Grav. anomalous. Cannot be realized by any 2D lattice model.
But can be realized as the 2D boundary of 3+1D toric code model.
Example 5 (gapped):
• 2+1D theory with excitations (1, e). e × e = 1. e is a semion.
No grav. anomaly. Can be realized by ν = 1/2 bosonic Laughlin state.
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
Quantum entanglement, topological order, and tensor category
Entanglement = Geometry
• The boundary of topologically ordered states has gravitational
anomaly. Topological orders (patterns of long-range entanglement)
classify gravitational anomalies in one lower dimension.
long-range entanglement ↔ geometry
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
Quantum entanglement, topological order, and tensor category
Classify long-range entanglement and topological order
• 1+1D: there is no topological order
Verstraete-Cirac-Latorre 05
• 2+1D: Abelian topological order are classified by K -matrices
2+1D: topological orders are classified by (UMTC , c) = (T , S, c)?
2+1D: topo. order with gappable edge are classified by unitary
fusion categories (UFC): Z(UFC ) = UMTC Levin-Wen 05
Φ
�
i
α
m
j
k
β
l
�
=
�
ijk;mαβ
Φ
Fl;nχδ
�
i
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
j χk
δ n
l
�
Quantum entanglement, topological order, and tensor category
Classify long-range entanglement and topological order
• 1+1D: there is no topological order Verstraete-Cirac-Latorre 05
1+1D: anomalous topological order are classified by unitary fusion
categories (UFC). Lan-Wen 13 (anomalous topological order = gapped 2D edge)
• 2+1D: Abelian topological order are classified by K -matrices
2+1D: topological orders are classified by (UMTC , c) = (T , S, c)?
2+1D: topo. order with gappable edge are classified by unitary
fusion categories (UFC): Z(UFC ) = UMTC Levin-Wen 05
Φ
�
i
α
m
j
k
β
l
�
=
�
ijk;mαβ
Φ
Fl;nχδ
�
i
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
j χk
δ n
l
�
i
j
k
α m mk
αl
l
ij
m
i
j
F
α jkn
k
n α inl
l
Quantum entanglement, topological order, and tensor category
Classify long-range entanglement and topological order
• 1+1D: there is no topological order Verstraete-Cirac-Latorre 05
1+1D: anomalous topological order are classified by unitary fusion
categories (UFC). Lan-Wen 13 (anomalous topological order = gapped 2D edge)
• 2+1D: Abelian topological order are classified by K -matrices
2+1D: topological orders are classified by (UMTC , c) = (T , S, c)?
2+1D: topo. order with gappable edge are classified by unitary
fusion categories (UFC): Z(UFC ) = UMTC Levin-Wen 05
Φ
�
i
α
m
j
k
β
l
�
=
�
ijk;mαβ
Φ
Fl;nχδ
�
i
j χk
δ n
l
�
i
j
k
α m mk
αl
l
ij
m
• Topo. order with no non-trivial topo. excitations:
i
j
F
α jkn
k
n α inl
l
Kong-Wen 14
1 + 1D 2 + 1D 3 + 1D 4 + 1D 5 + 1D 6 + 1D
0
Z2
0
Z⊕Z
Boson:
0
Z E8
Z p+ip
?
?
?
?
Fermion:
Z2
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
Quantum entanglement, topological order, and tensor category
Volume-ind. partition function – Universal topo. inv.
• Assume the space-time = M � St1 (a fiber bundle over St1 ).
� ∈ MCG(M).
Such a fiber bundle is described an element in W
1
So we denote space-time = M �W
� St
• Volume-ind. (fixed-point) partition function Kong-Wen 14
1
1 −�grnd Vspace-time
Z (M �W
S
)
=
Z
(M
�
S
vol-ind
� t
� t )e
W
1
Zvol-ind (M �W
S
� t ) = Tr(W )
• Zvol-ind (M × St1 ) = the ground
state degeneracy on space M.
M
W−twist
t
Zvol-ind (S d × St1 ) = 1
Zvol-ind (S d−1 × S 1 × St1 ) = number of topological particle types.
Volume-ind. partition function, universal wave function
overlap, and non-Abelian geometric phases are the same
type of topological invariants for topologically ordered states
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
Quantum entanglement, topological order, and tensor category
Monoid and group structures of topological orders
• Let Cd = {a, b, c, · · · } be a set of topologically ordered phases in d
dimensions.
Stacking a-TO state and b-TO state → a c-TO state:
a � b = c, a, b, c ∈ Cd
a−TO
c−TO
b−TO
• � make Cd a monoid (a group without inverse).
Consider topological order a and topological order a∗
∗
a
1
a
1 ∗
Zvol-ind (M �W
� St ) = [Zvol-ind (M �W
� St )] , then
∗
∗
a�a
1
a
1
a
1
Zvol-ind (M �W
� St ) = Zvol-ind (M �W
� St )Zvol-ind (M �W
� St )
∗
a
1
a
1
∗
In general, Zvol-ind (M �W
� St )Zvol-ind (M �W
� St ) �= 1 → a � a is a
non trivial topological order, and a-TO has no inverse.
1
iθ
• A topological order is invertible iff its Zvol-ind (M �W
� St ) = e
A topological order is invertible iff it has no topological excitations.
Kong-Wen 14
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
Quantum entanglement, topological order, and tensor category
Classify invertible bosonic topo. order (with no topo. exc.)
In 2+1D:
1
�W
� St )
i
2πc
24
�
M�� St1
W
ω3 (gµν )
• Zvol-ind (M
=e
where ω3 is the
gravitational Chern-Simons term: dω3 = p1 and p1 is the first
Pontryagin class.
• �The quantization
term: c = 8 × int. → Z-class:
� of the topological
�
M ω3�(gµν ) = N,∂N=M p1 = N � ,∂N � =M p1 mod 3,
since Nclosed p1 = 0 mod 3.
2:
• Relation to gravitational
anomaly
on
the
boundary
B
�
�
2πc
bndry
(1) �
Z = e i B 2 Leff (gµν ) e i 24 M 3 ,∂M 3 =B 2 ω3 (gµν )
i 2πc
e �24 M 3 ,∂M 3 =B 2 ω3 (gµν� ) is not differomorphism invariant, but
i 2πc
i B 2 Lbndry
(g
)
µν
eff
e
e 24 M 3 ,∂M 3 =B 2 ω3 (gµν ) is.
W.
(2) Consider an 1+1D differomorphism W : B 2 → B 2 , gµν → gµν
�
�
�
2πc
bndry W
bndry
Leff (gµν ) −
Leff (gµν ) =
ω 3 (gµν )
24 B 2 �W S 1
B2
B2
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
Quantum entanglement, topological order, and tensor category
Classify invertible bosonic topo. order (with no topo. exc.)
In 4+1D:
iπ
1
�W
� St )
�
M�� St1
W
w2 w3
• Zvol-ind (M
=e
where�wi is the i th
Stiefel-Whitney class → Z2 -class. We find M� S 1 w2 w3 = 1 when
� t
W
M = CP 2 and W : CP 2 → (CP 2 )∗
• Global grav. anomaly: for M = CP 2 and W : CP 2 → (CP 2 )∗
�
�
�
bndry
W
)
−
Lbndry
(g
L
w2 w3
µν
eff
eff (gµν ) =
M
M�W S 1
M
In 6+1D:
• Two independent grav.� Chern-Simons terms:
�
2πi
�
M7
k1
ω
˜ 7 −2ω7
−2ω
˜ 7 +5ω7
+k
2
5
9
Zvol-ind (M 7 ) = e
where dω7 = p2 , d ω
˜ 7 = p1 p1 → Z ⊕ Z-class (k1 , k2 ).
Kong-Wen 14
1 + 1D 2 + 1D 3 + 1D 4 + 1D 5 + 1D 6 + 1D
Boson:
0
Z E8
0
Z2
0
Z⊕Z
Z p+ip
?
?
?
?
Fermion:
Z2
Xiao-Gang Wen, Perimeter/MIT ESI, Vienna, Aug., 2014
Quantum entanglement, topological order, and tensor category
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