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Topological Phases and TopologicalInsulators
Manuel AsoreyUniversidad de Zaragoza
IFWGP 2015
Zaragoza, September 2015
The future of physics:In working up toward a dramatic conclusion of thisvolume
• Einstein:Gravity ⇒ Riemannian geometry
• Heisenberg:Quantum physics ⇒ Non-commutative algebra
The future of physics:In working up toward a dramatic conclusion of thisvolume
• Einstein:Gravity ⇒ Riemannian geometry
• Heisenberg:Quantum physics ⇒ Non-commutative algebra
Only number theory and topology still remain as pure
mathematical disciplines without any physicalapplication.
The future of physics:In working up toward a dramatic conclusion of thisvolume
• Einstein:Gravity ⇒ Riemannian geometry
• Heisenberg:Quantum physics ⇒ Non-commutative algebra
Only number theory and topology still remain as pure
mathematical disciplines without any physicalapplication. Could it be that they will be called tohelp in our understanding of the riddles of nature?.
Current Physics
• Number Theory
– Integrable systems (Potts models)– Casimir effect (Riemann zeta function)– Riemann hypothesis (Berry-Keating model)
Current Physics
• Number Theory
– Integrable systems (Potts models)– Casimir effect (Riemann zeta function)– Riemann hypothesis (Berry-Keating model)
• Topology
– Dirac Monopoles– Aharonov-Bohm effect– Chern-Simons theory (knot theory)– Topological Insulators
Topological Insulators
Zhang, Haldane, Kane
Topological insulators: New materialsInsulators in the bulk, but conductors on the boundary
Topological Phases
Particle in Magnetic Field in R3
L =1
2m x2
+ e A.xm mass of the particlee electric charge
A magnetic vector potential B = ∇× A
Topological Phases
Particle in Magnetic Field in R3
L =1
2m x2
+ e A.xm mass of the particlee electric charge
A magnetic vector potential B = ∇× A
Topological limit: m → 0
L = e A · x
Topological Phases
Particle in Magnetic Field in R3
L =1
2m x2
+ e A.xm mass of the particlee electric charge
A magnetic vector potential B = ∇× A
Topological limit: m → 0
L = e A · x
• metric independent
• constrained system p = e A
Topological Phases
Canonical formalism (T ∗R3,ω0)
H =1
2m(p − e A)2
Non-canonical transformation: (T ∗R3,ω0) ⇒ (T ∗R3,ω)
p → p′= p − e A
ω0 ⇒ ω = ω0 + e π∗0 dA = ω0 + e π∗0 F
H ⇒ H ′=
1
2mp′2
Topological Phases
Constraints analysis in the topological phase reduceto a contact phase ( p′ = 0 )
(T ∗R
3,ω) ⇒ (R3, e F)
and
H ′= 0
Topological Phases
Constraints analysis in the topological phase reduceto a contact phase ( p′ = 0 )
(T ∗R
3,ω) ⇒ (R3, e F)
and
H ′= 0
Singular limit:
limm→0
1
2mp′2 ⇒ H ′
= 0
Topological Phases
Generalization for arbitrary Riemannian manifolds (M,g)
Constraints analysis reduce in the topological phase to
(T ∗M,ω) ⇒ (M, e F)
and
H = H ′= 0
Topological Phases
Generalization for arbitrary Riemannian manifolds (M,g)
Constraints analysis reduce in the topological phase to
(T ∗M,ω) ⇒ (M, e F)
and
H = H ′= 0
Singular limit:
limm→0
1
2m(p′,p′)g ⇒ H = H ′
= 0
Quantization
If M is even dimensional and F is regular (M, e F) is asymplectic manifold and no further reductions areneeded
Quantization requires that
[ e
2π
]
F ∈ H 2(M,Z)
Non-trivial topologies induce quantization ofmagnetic fluxes
Quantum states are sections of a line bundle E(M ,C)
with a connection A such that π∗F = dA is thecurvature of A by π : E → M .
Quantization
If M is an oriented Riemannian manifold the quantumHamiltonian is
IH = − 1
2md∗
AdA = − 1
2mΔA, (1)
Quantization
If M is an oriented Riemannian manifold the quantumHamiltonian is
IH = − 1
2md∗
AdA = − 1
2mΔA, (2)
S2 Sphere and Magnetic Monopole
e
2π
∫
S2
F = k ∈ Z
B = gx
||x||3 k = 2ge
Quantization
In complex coordinates S2 = CP1
z = aeiφ tan θ/2
z = ae−iφ tanθ/2
the Hamiltonian is
H = − 1
2m
[
(
1 +zz
a2
)2
∂∂ +k
2a2
(
1 +zz
a2
)
(z∂ − z∂) − k2
4a4zz
]
Quantization
In complex coordinates S2 = CP1
z = aeiφ tan θ/2
z = ae−iφ tanθ/2
the Hamiltonian is
H = − 1
2m
[
(
1 +zz
a2
)2
∂∂ +k
2a2
(
1 +zz
a2
)
(z∂ − z∂) − k2
4a4zz
]
Energy levels (degeneracy 2l + |k|+ 1 )
El =1
2ma2
[
|k|(l + 12) + l(l + 1)
]
l = 0, 1, 2, . . .
Eigenfunctions
ψlj (z, z)=
(
1+zz
a2
)−k/2
zjP(j,|k|−j)l
(
a2 − zz
a2 + zz
)
l = 0, 1, 2, . . .
j = −l ,−l + 1, . . . , l + |k|
Hall Effect in 2D Torus T
e
2π
∫
T
F = k ∈ Z k = eB/2π
In complex coordinates: z = x1 + ix2, z = x1 − ix2
IH = − 1
2m
[
4∂∂ + eB(z∂ − z∂) − e2B2
4zz
]
Energy levels (degeneracy: |k|)
En =2π|k|
m
(
n +1
2
)
Hall Effect in 2D Torus T
e
2π
∫
T
F = k ∈ Z k = eB/2π
Ground State Eigenfunctions (degeneracy: |k|) :
Holomorphic sections of E(T 2,C)
ψj(z, z) = ekπz(z+z)/2Θ
[
j/|k|0
]
(|k|z, i|k|)
= ekπz2/2∑
l∈Z+j/|k|
e−π|k|l2+i2π|k|lz
j = 0, 1, 2 . . . , |k| − 1.
Topological Phases
The massless limit m → 0 can be analysed in two(equivalent) ways
• First constrain and then quantize
• First quantize and then constrain
Topological Phases
The massless limit m → 0 can be analysed in two(equivalent) ways
• First constrain and then quantize
• First quantize and then constrain
First constrain:
Constraints: p′ = 0
Reduced symplectic space: (T, e F)
Hamiltonian: H ′ = 0
Topological Phases
Then quantize:
• Prequantization condition
e
2π
∫
T
F = k
• Holomorphic quantization:Quantum states ≈ holomorphic sections of a linebundle E(T 2,C) with Chern class number c1(E) = k
H0k = {ξ : T 2 → E ; ξ is holomorphic}
Topological Phases
Riemann-Roch theorem
dimH0k =
1
8π
∫
Σ
√gR +
1
2π
∫
Σ
F + dimH02g−2−|k|
• S2 sphere dimH0k = |k|+ 1
• T torus dimH0k = |k|
• Σg Riemann surface of genus g
dimH0k = |k| − g + 1 + dimH0
2g−2−|k|
if |k| − 2g + 2 > 0 dimH02g−2−|k|
= 0
(Kodaira’s vanishing theorem)
Topological Phases
First quantize:
IH = − 1
2md∗
AdA = − 1
2mΔA, (3)
and then constrain:
The Hilbert space reduces to ground states dimH0k
• S2 sphere dimH0k = |k|+ 1
• T torus dimH0k = |k|
• Σg Riemann surface of genus g
dimH0k = |k| − g + 1 + dimH0
2g−2−|k|
Magnetic flux dependence
IH = − 12m
(
∂θ − i eφ2π
)2En =
12m(n − ε)2
1/2 1
ε
nE
n=1 n=0
n=2n=-1
ε = eφ/2π
Time reversal invariance at ε = 0, 12
Magnetic flux dependence
IH = − 12m
(
∂θ − i eφ2π
)2+ V0(1 − cos θ)
ε = eφ/2π
degeneracy is not robust
Time Reversal and Kramers degeneracy
s = 12
spin systems
Θψ = eiπSyψ∗
Θ2= −I
Kramers theorem:For a time reversal invariant Hamiltonian all energylevels are double degenerated
Θψ = λψ, Θ2ψ = |λ|2ψ = −ψFor a non-degenerate energy level ψ
Time Reversal protection
Time reversal invariant interactions HI
Θ|k, ↑>= | − k, ↓>,Θ| − k, ↓>= −|k, ↑>HIΘ = ΘHI
do no mix Kramers doublet states
< k, ↑ |HI | − k, ↓> =< k, ↑ |HIΘ|k, ↑>
=< k, ↑ |ΘHI |k, ↑>= − < −k, ↓ |HI |k, ↑>= 0
Z2 Index
Time reversal matrix
wmn(k) =< um(k)|Θ|un(−k) > |un(k) > filled states
wmn(k) = −wnm(−k)
For TR invariant ka the matrix w(ka) is antisymmetricZ2 invariant ν is defined by
(−1)ν =∏
a
Pf (w(ka))
det w(ka)= ±1
TOPOLOGICAL INSULATORS
• 2D topological insulators discovered in 2007
• 3D topological insulators discovered in 2008.Chern-Simons index
TOPOLOGICAL INSULATORS
• 2D topological insulators discovered in 2007
• 3D topological insulators discovered in 2008.Chern-Simons index
• Bulk insulators and edge conductors
TOPOLOGICAL INSULATORS
• 2D topological insulators discovered in 2007
• 3D topological insulators discovered in 2008.Chern-Simons index
• Bulk insulators and edge conductors
• Robust under impurities. Time reversal symmetry
TOPOLOGICAL INSULATORS
• 2D topological insulators discovered in 2007
• 3D topological insulators discovered in 2008.Chern-Simons index
• Bulk insulators and edge conductors
• Robust under impurities. Time reversal symmetry
• Applications to Spintronics and QuantumComputation
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