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Dimensional Reduction to Effective Actions in (2+1)-D Topological Insulators

This work explores the dimensional reduction of effective actions in (2+1)-D topological insulators by integrating out fermion fields and analyzing their properties in the context of the quantum spin Hall effect (QSHE) and quantum Hall effect (QHE). We delve into the Chern-Simons formulation and the physical implications of Z2 classification for time-reversal invariant insulators, depicting charge responses, current densities, and adiabatic evolution dynamics. The discussion covers various theoretical frameworks and the impact of dimensionality on topological characteristics.

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Dimensional Reduction to Effective Actions in (2+1)-D Topological Insulators

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  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

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