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arXiv · 2509.19320

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

Frank Schindler
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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.

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Frank Schindler. 2025-09-12. A pedestrian's guide to the topological phases of free fermions. https://doi.org/10.21468/scipostphyslectnotes.122

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