Classification of interacting Dirac semimetals

Topological band theory predicts a <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi mathvariant="double-struck">Z</a:mi></a:math> classification of three-dimensional (3D) Dirac semimetals (DSMs) at the single-particle level. Namely, an arbitrary number of identical bulk Dirac nodes will always remain locally stable and gapless in the single-particle band spectrum, as long as the protecting symmetry is preserved. In this work we find that this single-particle classification for <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:msub><c:mi>C</c:mi><c:mi>n</c:mi></c:msub></c:math>-symmetric DSMs will break down to <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:msub><d:mi mathvariant="double-struck">Z</d:mi><d:mrow><d:mi>n</d:mi><d:mo>/</d:mo><d:mtext>gcd</d:mtext><d:mo>(</d:mo><d:mn>2</d:mn><d:mo>,</d:mo><d:mi>n</d:mi><d:mo>)</d:mo></d:mrow></d:msub></d:math> in the presence of symmetry-preserving electron interactions. Our theory is based on a dimensional reduction strategy which reduces a 3D Dirac fermions to one-dimensional building blocks, i.e., vortex-line modes, while respecting all the key symmetries. Using bosonization technique, we find that there exists a minimal number <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi>N</f:mi><f:mo>=</f:mo><f:mi>n</f:mi><f:mo>/</f:mo><f:mtext>gcd</f:mtext><f:mo>(</f:mo><f:mn>2</f:mn><f:mo>,</f:mo><f:mi>n</f:mi><f:mo>)</f:mo></f:mrow></f:math> such that the collection of vortex-line modes in <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mi>N</g:mi></g:math> copies of DSMs can be symmetrically eliminated via four-fermion interactions. While this gapping mechanism does not have any free-fermion counterpart, it yields an intuitive “electron-trion coupling” picture. By developing a topological field theory for DSMs and further checking the anomaly-free condition, we independently arrive at the same classification results. Our theory paves the way for understanding topological crystalline semimetallic phases in the strongly correlated regime. Published by the American Physical Society 2024