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A pedestrian's guide to the topological phases of free fermions

Frank Schindler

Abstract

These lecture notes explain the classification of some simple fermionic topological phases of matter in a pedestrian manner, with an aim to be maximally pedagogical = doing things in excruciating detail. We focus on a many-body perspective, even if many of the models we work with are non-interacting. We start out with symmetry protected topological (SPT) phases of free fermions that are protected by U(1) symmetry = topological insulators. We then look at fermion topological phases that don't even need a symmetry = topological superconductors, and explain how their classification changes in presence of spinless time-reversal symmetry. We close by perturbatively checking which of the 1D topological phases we had found are stable to interactions.

A pedestrian's guide to the topological phases of free fermions

Abstract

These lecture notes explain the classification of some simple fermionic topological phases of matter in a pedestrian manner, with an aim to be maximally pedagogical = doing things in excruciating detail. We focus on a many-body perspective, even if many of the models we work with are non-interacting. We start out with symmetry protected topological (SPT) phases of free fermions that are protected by U(1) symmetry = topological insulators. We then look at fermion topological phases that don't even need a symmetry = topological superconductors, and explain how their classification changes in presence of spinless time-reversal symmetry. We close by perturbatively checking which of the 1D topological phases we had found are stable to interactions.

Paper Structure

This paper contains 30 sections, 121 equations, 3 figures.

Table of Contents

  1. Introduction
  2. What is a SPT phase?
  3. Notation
  4. Free fermion SPTs with charge conservation symmetry
  5. 0D
  6. 1D
  7. Bloch Hamiltonian and band structure
  8. Dirac theory of gap closing
  9. Topology of the space of possible Dirac mass terms
  10. Broken translational symmetry
  11. 2D
  12. 3D and higher
  13. Free fermion topological phases without symmetry
  14. Majorana fermions and fermion parity
  15. Topological classification in 0D and 1D
  16. ...and 15 more sections

Figures (3)

  • Figure 1: The OBC Kitaev chain Hamiltonian $H_1$ from Eq. \ref{['eq: H1 OBC']}. Each circle represents a Majorana operator $\gamma_\alpha$. Each bar represents a coupling term of the form $\mathrm{i} \gamma_\alpha \gamma_\beta$ that appears in the Hamiltonian.
  • Figure 2: Two copies of $H_1$.
  • Figure 3: Two copies of $H_1$ with local edge couplings.