5. Dimensional reduction to (2+1)-D

1. 5. Dimensional reduction to (2+1)-D A. Effective action of (2+1)-D insulators • Dimensionally reduced Dirac model in (2+1)-D • Replace gauge fields in the z and w directions: • Integrate out fermion fields • Coefficient in terms of Green’s functions

2. 5. Dimensional reduction to (2+1)-D A. Effective action of (2+1)-D insulators • Integrate out fermion fields • Coefficient in terms of Green’s functions • Coefficient satisfies the sum rule • Coefficient in terms of Chern-Simons form

3. 5. Dimensional reduction to (2+1)-D A. Effective action of (2+1)-D insulators • Coefficient in terms of Chern-Simons form • Vanishing contributions from • Theory of the QSHE QHE QSHE

4. 5. Dimensional reduction to (2+1)-D A. Effective action of (2+1)-D insulators • Hamiltonian of the (2+1)-D Dirac model • Compute the correlation functions • Consider slightly different lattice Dirac model • Continuum model for 2D version of Goldstone-Wilczek formula

5. 5. Dimensional reduction to (2+1)-D A. Effective action of (2+1)-D insulators • QSHE response • 2D lattice Dirac model • Adiabatic evolution • Continuum model for • Current response

6. 5. Dimensional reduction to (2+1)-D A. Effective action of (2+1)-D insulators • Adiabatic evolution (charge pumping) • Current response • Magneto-electric polarization • Net charge flowing across • Recall Dirac Hamiltonian

7. 5. Dimensional reduction to (2+1)-D A. Effective action of (2+1)-D insulators • Adiabatic evolution (e/2 domain wall) • QSHE response (charge density) • Charge density at the edge • Charge density at the corner

8. 5. Dimensional reduction to (2+1)-D B. Z2 topological classification of TRI insulators • Adiabatic interpolation between (2+1)-D Hamiltonians: • Recall the “relative second Chern parity” for a (3+1)-D insulator • Define “interpolation between interpolations”: • φ-component of Berry gauge field vanishes for both g’s at θ = 0 and π • Define equivalent Z2 quantity for (2+1)-D Hamiltonians

9. 5. Dimensional reduction to (2+1)-D C. Physical properties of the Z2nontrivial insulators • Interface between vacuum (h0) and QSHE (h1) • Two types of interpolations breaking time-reversal symmetry at the interface • Charge in the region area (A) enclosed in C • For interpolations between trivial/nontrivial (h0/h1): • Example: Magnetization domain wall at the interface

10. 5. Dimensional reduction to (2+1)-D C. Physical properties of the Z2nontrivial insulators • Distribution of 1D charge/current density • Deep inside QSH/VAC: • (1+1)-D edge theory

11. 6. Unified theory of topological insulators A. Phase space Chern-Simons theories • QHE action in “phase space”

12. 6. Unified theory of topological insulators A. Phase space Chern-Simons theories • Prescription for dimensional reduction • QHE action in (2+1)-D • Dimensional reduction to (1+1)-D

13. 6. Unified theory of topological insulators A. Phase space Chern-Simons theories • Explicit derivation of (0+1)-D action from (1+1)-D using prescription • Dimensionally reduced action

14. 6. Unified theory of topological insulators A. Phase space Chern-Simons theories • Second family of topological insulators • Phase space dimensional reduction prescription • Prescription applied to (2+1)-D TRI insulator

15. 6. Unified theory of topological insulators A. Phase space Chern-Simons theories • Phase space Chern-Simons effective theory in n dimensions • Phase space dimensional reduction prescription • Phase space Chern-Simons effective theory for the mth“descendant”

16. 6. Unified theory of topological insulators A. Phase space Chern-Simons theories • The “family tree”

17. 6. Unified theory of topological insulators B. Z2 topological insulator in generic dimensions • Effect of T and C on Aμ required by the invariance of Aμjμ • Can easily interchange • Transformation properties of the Chern-Simons Lagrangian • Recursive definition of Z2 classification • Interpolation of an interpolation fails

18. 6. Unified theory of topological insulators B. Z2 topological insulator in generic dimensions • Failure of Z2 classification beyond 2nd descendent from stability of edge theory • Generalizations to higher dimensions