Free Fermion Symmetry Breaking and a Long Exact Sequence in K-Theory*

We study symmetry breaking for free fermions. Specifically, we show how three processes

(1) producing Jackiw-Rebbi modes on a defect by imposing certain boundary conditions on a charged order parameter,

(2) counting zero modes localized to the origin of a free fermion Berry phase, and

(3) extending a lower-dimensional theory to a higher-dimensional theory with a mass term away from the origin

are mathematically described by Thom isomorphisms and by maps in a long exact sequence in equivariant K-theory. Using this framework, we can examine both the 10-fold way for free electron systems and anomaly matching in the spontaneous symmetry broken phase for more general free fermion phases.