We employ a general method, known as anomaly matching, to derive no-go theorems of fermionic lattice models. For our main result, we show that time-reversal invariant $(3+1)$-dimensional 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 multiplets or be fine tuned. This theorem holds for both noninteracting and interacting systems as long as the electromagnetic $U\left(1\right)$ symmetry is a normal subgroup of the microscopic symmetry group $G_{\mathrm{UV}}$, a condition that is ubiquitous in physical $U\left(1\right)$ 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 nononsite $\mathbb{R}$ symmetry (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.

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• Anomalies
• Dirac semimetal
• Weyl semimetal
• Symmetries
• Chiral symmetry
• Continuous symmetries in condensed matter
• Discrete symmetries in condensed matter