No-go theorem for single time-reversal invariant symmetry-protected Dirac fermions in 3+1d

We employ a general method, known as anomaly-matching, to derive new no-go theorems of fermionic lattice models. For our main result, we show that time-reversal invariant 3+1d lattice systems (such as Dirac and Weyl semimetals) can never admit a lone low-energy symmetry-protected Dirac fermion (or node), i.e., it must always come in higher muliplets or be fine-tuned. This theorem holds for both non-interacting and interacting systems as long as the electromagnetic $U(1)_{V,{\rm UV}}$ symmetry is a normal subgroup of the microscopic symmetry group $G_{\mathrm{UV}}$; a condition that is ubiquitous in physical $U(1)_{V,{\rm UV}}$ preserving lattice models. To show that our theorems are tight, we also explore both well-known and new systems that are converses of the no-go theorem, obtained by forfeiting certain assumptions such as a broken time-reversal symmetry (magnetic Weyl semimetal), a non-compact non-on-site $U(1)$ (almost local Dirac node model), no-symmetry protection (fine-tuned Dirac semimetal), or multiple low-energy Dirac nodes (time-reversal invariant Weyl and Dirac semimetals). We will also explicitly demonstrate that, while this theorem strictly prohibits single time-reversal invariant symmetry-protected Dirac node, it does allow for other odd numbers of Dirac nodes under certain circumstances, such as three Dirac nodes in the Fu-Kane-Mele diamond lattice model. This is akin to the Nielsen-Ninomiya theorem for an odd number of differently-charged chiral fermions, whose lattice realizations are allowed if certain anomaly cancellation conditions are met.