1. Topological Insulators and Beyond
Kai Sun
University of Maryland, College Park

2. 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

3. 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

4. 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

5. 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

6. 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:

7. 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, (2004).

8. 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 :

9. 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)

10. Interactions
Ward identity:
Hall Conductivity:

11. 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

12. 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

13. 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

14. 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)

15. 1D Edge of a 2D insulator (Z2 Topological classification)
Topological protected edge states

16. 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.

17. 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.

18. 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:

19. 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)

20. 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

21. Gauge field Original gauge theory: 2+1D (anomalous) Quantum Hall state
3D time-reversal symmetry preserved

22. 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)