Topological Insulators and Beyond

Kai SunUniversity of Maryland, College Park

Outline

• Topological state of matter• Topological nontrivial structure and

topological index• Anomalous quantum Hall state and the Chern

number• Z2 topological insulator with time-reversal

symmetry• Summary

Definition

• Many• A state of matter whose ground state wave-

function has certain nontrivial topological structure– the property of a state – Hamiltonian and excitations are of little

importance

Family tree

Resonating Valence Bond State•Frustrated spin system•Orbital motion of ultracold dipole molecule on a special lattice

Quantum Hall StateFraction Quantum Hall

Anomalous Quantum Hall

Quantum Spin Hall

Anomalous Quantum Spin Hall

Topological superconductors

Family tree

Resonating Valence Bond State•Frustrated spin system•Orbital motion of ultracold dipole molecule on a special lattice

Quantum Hall StateFraction Quantum Hall

Anomalous Quantum Hall

Quantum Spin Hall

Anomalous Quantum Spin Hall

Topological superconductors

Topological insulators

Magnetic Monopole

Gauge Transformation

Vector potential cannot be defined globally

Matter field

wave-function on each semi-sphere is single valued

Magnetic flux for a compact surface:

2D noninteracting fermions

• Hamiltonian:

• A gauge-like symmetry:

• “Gauge” field: (Berry connection)

• “Magnetic” field: (Berry phase)

• Compact manifold: (to define flux) Brillouin zone: T2

• Only for insulators: no Fermi surfaces• Quantized flux (Chern number)

Haldane, PRL 93, 206602 (2004).

Two-band model (one “gauge” field)Hamiltonian:

Kernel:

withDispersion relation:

with

With i=x, y or z

For insulators:

Topological index for 2D insulators :

Implications

• Theoretical: – wavefunction and the “gauge field” cannot be

defined globally– Chern number change sign under time-reversal– Time-reversal symmetry is broken

• Experimentally– Integer Hall conductivity (without a magnetic field)

– (chiral) Edge states• Stable against impurites (no localization)

Interactions

• Ward identity:

• Hall Conductivity:

3D Anomalous Hall states?

• No corresponding topological index available in 3D (4D has)

• No Quantum Hall insulators in 3D (4D has)• But, it is possible to have stacked 2D layers of

QHI

Time-reversal symmetry preserved insulator with topological ordering?

• Idea: Spin up and spin down electrons are both in a (anomalous) quantum Hall state and have opposite Hall conductivity (opposite Chern number)

• Result:– Hall conductivity cancels – Under time-reversal transformation

• Spin up and down exchange• Chern number change sign• Whole system remains invariant

Naïve picture

• Described by an integer topological index• Hall conductivity being zero• No chiral charge edge current• Have a chiral spin edge currentHowever, life is not always so simple• Spin is not a conserved quantity

Time-reversal symmetry for fermions and Kramers pair

• For spin-1/2 particles, T2=-1

– T flip spin:– T2 flip spin twice– Fermions: change sign if the spin is rotated one

circle.• Every state has a degenerate partner (Kramers

pair)

1D Edge of a 2D insulator (Z2 Topological classification)

Topological protected edge states

Z2 topological index

• Bands appears in pairs (Kramers pair)– Total Chern number for each pair is zero

• For the occupied bands: select one band from each pair and calculate the sum of all Chern numbers.

• This number is an integer.• But due to the ambiguous of selecting the

bands, the integer is well defined up to mod 2.

Another approach

• T symmetry need only half the BZ

• However, half the BZ is not a compact manifold.• Need to be extended (add two lids for the

cylinder)• The arbitrary of how to extending cylinder into a

closed manifold has ambiguity of mod 2.

4-band model with inversion symmetry

• 4=2 (bands)x 2 (spin)• Assumptions:• High symmetry points in the BZ: invariant under k to –k• Two possible situations– P is identity: trivial insulator– P is not identity:

• Parity at high symmetry points:• Topological index:

3D system

• 8 high symmetry points– 1 center+1 corner+3 face center+3 bond center

• Strong topological index• Three weak-topological indices (stacks of 2D

topologycal insulators)

Interaction and topological gauge field theory

• Starting by Fermions + Gauge field• Integrate out Fermions– For insulators, fermions are gapped– Integrate out a gapped mode the provide a well-

defined-local gauge field• What is left? Gauge field

• Insulators can be described by the gauge field only

Gauge field

• Original gauge theory:• 2+1D (anomalous) Quantum Hall state

• 3D time-reversal symmetry preserved

Summary

• 3D Magnetic Monopole: – integer topological index: monopole charge

• 2D Quantum Hall insulator– integer topological: integer: Berry phase– Quantized Hall conductivity and a chiral edge state

• 2D/3D Quantum Spin Hall insulator (with T symmetry)

– Z2 topological index (+/-1 or say 0 and 1)– Chiral spin edge/surface state

• Superconductor can be classified in a similar way (not same due to an extra particle-hole symmetry)