Topological Insulators and Beyond Kai Sun University 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 Quantum Hall State Frustrated spin system Orbital motion of ultracold dipole molecule on a special lattice Quantum Hall State Fraction Quantum Hall Anomalous Quantum Hall Quantum Spin Hall Anomalous Quantum Spin Hall Topological superconductors

Family tree Resonating Valence Bond State Quantum Hall State Frustrated spin system Orbital motion of ultracold dipole molecule on a special lattice Quantum Hall State Fraction Quantum Hall Anomalous Quantum Hall Quantum Spin Hall Anomalous Quantum Spin Hall Topological insulators Topological superconductors

Magnetic Monopole Vector potential cannot be defined globally Gauge Transformation 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) with Hamiltonian: Kernel: With i=x, y or z Dispersion relation: with For insulators: Topological index for 2D insulators :

Implications Theoretical: Experimentally 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 current However, 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 Strong topological index 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: 2D Quantum Hall insulator 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)