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

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

5. Dimensional reduction to (2+1)-DA. Effective action of (2+1)-D insulators

• Coefficient in terms of Chern-Simons form

• Vanishing contributions from

• Theory of the QSHE

QHE QSHE

5. Dimensional reduction to (2+1)-DA. 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. Dimensional reduction to (2+1)-DA. Effective action of (2+1)-D insulators

• Continuum model for

• QSHE response

• 2D lattice Dirac model

• Adiabatic evolution

• Current response

5. Dimensional reduction to (2+1)-DA. Effective action of (2+1)-D insulators

• Adiabatic evolution (charge pumping)

• Current response

• Net charge flowing across

• Magneto-electric polarization

• Recall Dirac Hamiltonian

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

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

5. Dimensional reduction to (2+1)-DC. Physical properties of the Z2 nontrivial 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

5. Dimensional reduction to (2+1)-DC. Physical properties of the Z2 nontrivial insulators

• Distribution of 1D charge/current density

• Deep inside QSH/VAC:

• (1+1)-D edge theory

6. Unified theory of topological insulatorsA. Phase space Chern-Simons theories

• QHE action in “phase space”

6. Unified theory of topological insulatorsA. Phase space Chern-Simons theories

• QHE action in (2+1)-D

• Prescription for dimensional reduction

• Dimensional reduction to (1+1)-D

6. Unified theory of topological insulatorsA. Phase space Chern-Simons theories

• Explicit derivation of (0+1)-D action from (1+1)-D using prescription

• Dimensionally reduced action

6. Unified theory of topological insulatorsA. Phase space Chern-Simons theories

• Second family of topological insulators

• Phase space dimensional reduction prescription

• Prescription applied to (2+1)-D TRI insulator

6. Unified theory of topological insulatorsA. 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”

6. Unified theory of topological insulatorsA. Phase space Chern-Simons theories

• The “family tree”

6. Unified theory of topological insulatorsB. 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

6. Unified theory of topological insulatorsB. Z2 topological insulator in generic dimensions

• Failure of Z2 classification beyond 2nd descendent from stability of edge theory

• Generalizations to higher dimensions