topological phases and topological insulators manuel...

Topological Phases and TopologicalInsulators

Manuel AsoreyUniversidad de Zaragoza

IFWGP 2015

Zaragoza, September 2015

Physics and Topology

G. Gamow

Biography of Physics:The great physicists from Galileo to Einstein

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.




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


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


L =1

2m x2



Topological limit: m → 0

L = e A · x

Topological Phases


L =1

2m x2



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

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


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

)


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:

Topological Phases

Then quantize:

• Prequantization condition

e

2π

∫

T

F = k

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|

Semiclasical arguments

# quantum states ⇔ volume of phase space

Semiclasical arguments

# quantum states ⇔ volume of phase space+ topological correction

Band structure

Periodic perturbation:

H ′′= H ′

+ V0 sin k φ1 sin k φ2

Hall effect with boundaries


Boundary effects:

B


Boundary effects: Time reversal symmetry?

B


Boundary effects: Time reversal symmetry

B


Boundary effects:Time reversal symmetry breaking


B


Semiclassical argument :

Finite size effects

M. A., A. P. Balachandran, J.M. Perez-Prado, JHEP 2013and [arXiv:1505.03461]

Finite size effects

Band structure in Brillouin zone

ψ(k) = eik·ruk(r)

Boundary Insulators

Boundary interactions ⇒Anderson localization

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

Quantum Spin Hall

Spin-Orbit interaction is TR invariant

HI = gL · S

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

Topological Insulators:Z2 Index

normal insulator topological insulator

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


M. Koenig et al. Science (2007)

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





• Robust under impurities. Time reversal symmetry

• Applications to Spintronics and QuantumComputation

topological phases and topological insulators manuel...

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