Edge states of Z2

Edge states of Z

topological phase

Spin conserved case

chiral edge states to helical ones Kramers degeneracy

Z2 characterization by the edge states

Z2 edge states

What’s this

Ener gy

k

⇡

⇡

Z2 edge states

Edge states are not chiral, but helical

2D Quantum Spin Hall state

H(k) ¹ ⇠= H( k)

+k k

Not independent Correlated

Generically TR is broken in momentum space

TR is OK

at special momentum

⇡ ⁰

H(0) = H( 0) H(⇡) = H( ⇡)

and

Energy

k ⇡

Identification of edge states (QSHE)

spin conserved case

Z2 edge states

Energy

k ⇡

Identification of edge states (QSHE)

spin conserved case

Z2 edge states

decompose into up & down

Energy

k ⇡

k_y

Energy

⇡

Identification of edge states (QSHE)

spin conserved case

Z2 edge states

upspin

decompose into up & down

Energy

k_y ⇡

Energy

k ⇡

k_y

Energy

⇡

Identification of edge states (QSHE)

spin conserved case

Z2 edge states

upspin down spin

decompose into up & down

Energy

k_y ⇡

Energy

k ⇡

k_y

Energy

⇡

Identification of edge states (QSHE)

spin conserved case

R L L

L L

L R R

Z2 edge states

upspin down spin

decompose into up & down

Energy

k_y ⇡

Energy

k ⇡

k_y

Energy

⇡

Identification of edge states (QSHE)

spin conserved case

R L L

L L

L R R

Z2 edge states

upspin down spin

decompose into up & down

I₁ = 1 I₂ = 2

Energy

k_y ⇡

Energy

k ⇡

k_y

Energy

⇡

Identification of edge states (QSHE)

L R

L L R

L R L

spin conserved case

R L L

L L

L R R

Z2 edge states

upspin down spin

decompose into up & down

I₁ = 1 I₂ = 2

Energy

k_y ⇡

Energy

k ⇡

k_y

Energy

⇡

Identification of edge states (QSHE)

L R

L L R

L R L

spin conserved case

R L L

L L

L R R

Z2 edge states

upspin down spin

decompose into up & down

I₁ = 1

I₂ = 2 ^{I2 = 2}

I₁ = 1

Energy

k_y ⇡

Energy

k ⇡

k_y

Energy

⇡

Identification of edge states (QSHE)

L R

L L R

L R L

R

L L

R R L

L L

spin conserved case

R L L

L L

L R R

Z2 edge states

upspin down spin

decompose into up & down

I₁ = 1

I₂ = 2 ^{I2 = 2}

I₁ = 1

Energy

k_y ⇡

Energy

k ⇡

k_y

Energy

⇡

Identification of edge states (QSHE)

L R

L L R

L R L

R

L L

R R L

L L

R L

L R R

L L

L

spin conserved case

R L L

L L

L R R

Z2 edge states

upspin down spin

decompose into up & down

I₁ = 1

I₂ = 2 ^{I2 = 2}

I₁ = 1

Spin Conserved

R R

L L

L L

R

R R R

L L

L L L

Energy L

k_y ⇡

⇡

Z2 edge states

Identification of edge states (QSHE)

I₁ = 1 I₂ = 2 I₂ = 2

I₁ = 1

Spin Conserved

R R

L L

L L

R

R R R

L L

L L L

L

With Spin Orbit

Energy

k_y ⇡

⇡

Z2 edge states

Identification of edge states (QSHE)

I₁ = 1 I₂ = 2 I₂ = 2

I₁ = 1

⇡

Energy

k_y

⇡

Spin Conserved

R R

L L

L L

R

R R R

L L

L L L

L

With Spin Orbit

Energy

k_y ⇡

⇡

Z2 edge states

Identification of edge states (QSHE)

I₁ = 1 I₂ = 2 I₂ = 2

I₁ = 1

⇡

Energy

k_y

⇡

Spin Conserved

R R

L L

L L

R

R R R

L L

L L L

L

With Spin Orbit

L L

L L L L

L R R L

R

R R

R

Energy

k_y ⇡

⇡

Z2 edge states

Identification of edge states (QSHE)

I₁ = 1 I₂ = 2 I₂ = 2

I₁ = 1

Adiabatic modifications

⇡

Energy

k_y

⇡

Spin Conserved

R R

L L

L L

R

R R R

L L

L L L

L

With Spin Orbit

L L

L L L L

L R R L

R

R R

R

Energy

k_y ⇡

⇡

Z2 edge states

Identification of edge states (QSHE)

I₁ = 1 I₂ = 2 I₂ = 2

I₁ = 1

Adiabatic modifications

Any topological protection ?

⇡

Energy

k_y

⇡

Spin Conserved

R R

L L

L L

R

R R R

L L

L L L

L

With Spin Orbit

L L

L L L L

L R R L

R

R R

R

Energy

k_y ⇡

⇡

Z2 edge states

Identification of edge states (QSHE)

I₁ = 1 I₂ = 2 I₂ = 2

I₁ = 1

Adiabatic modifications

Kramers degeneracy at TR invariant momenta

⇡

Energy

k_y

⇡

Spin Conserved

R R

L L

L L

R

R R R

L L

L L L

L

With Spin Orbit

L L

L L L L

L R R L

R

R R

R

Energy

k_y ⇡

⇡

Z2 edge states

Identification of edge states (QSHE)

I₁ = 1 I₂ = 2 I₂ = 2

I₁ = 1

Stable

Adiabatic modifications

Kramers degeneracy at TR invariant momenta

⇡

Energy

k_y

⇡

Spin Conserved

R R

L L

L L

R

R R R

L L

L L L

L

With Spin Orbit

L L

L L L L

L R R L

R

R R

R

Energy

k_y ⇡

⇡

Z2 edge states

Identification of edge states (QSHE)

I₁ = 1 I₂ = 2 I₂ = 2

I₁ = 1

Stable Unstable

Adiabatic modifications

Kramers degeneracy at TR invariant momenta

L

L L

R L

R R L

With Spin Orbit

k ⇡

Identification of edge states (QSHE)

L

L L

R L

R R L

With Spin Orbit

k ⇡

Identification of edge states (QSHE)

L

L L

L

Stable Unstable

L

⇡

k_y

Kramers degeneracy

Kramers degeneracy

L edge

L

L L

R L

R R L

With Spin Orbit

k ⇡

R

R R

R edge

Stable Unstable

⇡

k

Kramers degeneracy

Kramers degeneracy

Identification of edge states (QSHE)

L

L L

L

Stable Unstable

L

⇡

k_y

Kramers degeneracy

Kramers degeneracy

L edge

L

L L

R L

R R L

With Spin Orbit

k ⇡

R

R R

R edge

Stable Unstable

⇡

k

Kramers degeneracy

Kramers degeneracy

Identification of edge states (QSHE)

L

L L

L

Stable Unstable

L

⇡

k_y

Kramers degeneracy

Kramers degeneracy

L edge

3D

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Ener gy Ener gy Featureless Bulk Featureless Bulk

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Ener gy Ener gy

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Ener gy Ener gy

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Trivial phase

Ener gy Ener gy

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Trivial phase

Ener gy Ener gy

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Trivial phase Non-trivial phase

Ener gy Ener gy

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Trivial phase Non-trivial phase

Clearly two choices

Ener gy Ener gy

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Trivial phase Non-trivial phase

Clearly two choices

Ener gy Ener gy

Z

topological phase

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Trivial phase Non-trivial phase

Clearly two choices

Ener gy Ener gy

Z

topological phase

Z2 edge states

Edges characterize the featureless bulk

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Trivial phase Non-trivial phase

Clearly two choices

Ener gy Ener gy

Z

topological phase

Z2 edge states

Edges characterize the featureless bulk

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Ener gy Ener gy Featureless Bulk Featureless Bulk

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Ener gy Ener gy

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Ener gy Ener gy

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Trivial phase

Ener gy Ener gy

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Trivial phase

Ener gy Ener gy

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Trivial phase Non-trivial phase

Ener gy Ener gy

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Trivial phase Non-trivial phase

Clearly two choices

Ener gy Ener gy

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Trivial phase Non-trivial phase

Clearly two choices

Ener gy Ener gy

Z

topological phase

Z2 edge states

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Trivial phase Non-trivial phase

Clearly two choices

Ener gy Ener gy

Z

topological phase

Z2 edge states

Edges characterize the featureless bulk

Hasan-Kane, RMP (2010)

Topologically protected edge states

0 k

⇡

TR invariant Point

TR invariant Point

0 k

⇡

TR invariant Point

TR invariant Point

Trivial phase Non-trivial phase

Clearly two choices

Ener gy Ener gy

Z

topological phase

Z2 edge states

Edges characterize the featureless bulk

Hasan-Kane, RMP (2010)

Spin Hall edge states

Konig, Wiedmann, Brüne, Roth, Hartmut Buhmann, Molenkamp,Qi and Zhang, Science 318, 776 (2007)

2D

3D

....

Z2 edge states