Exact Chiral Symmetries of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>3</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mi mathvariant="normal">D</mml:mi> </mml:mrow> </mml:math> Hamiltonian Latt

We construct Hamiltonian models on a 3+1D cubic lattice for a single Weyl fermion and for a single Weyl doublet protected by exact (as opposed to emergent) chiral symmetries. In the former, we find a not-on-site, noncompact chiral symmetry which can be viewed as a Hamiltonian analog of the Ginsparg-Wilson symmetry in Euclidean lattice models of Weyl fermions. In the latter, we combine an on-site U(1) symmetry with a not-on-site U(1) symmetry, which together generate the SU(2) flavor symmetry of the doublet at low energies, while in the UV they generate an algebra known in integrability as the Onsager algebra. This latter model is in fact the celebrated magnetic Weyl semimetal, which is known to have a chiral anomaly from the action of U(1) and crystalline translation. This anomaly gives rise to an anomalous Hall response; however, when reinterpreted in our language, the model has two exact U(1) symmetries that give rise to the global SU(2) anomaly, which protects the gaplessness even when crystalline translations are broken. We also construct an exact symmetry-protected single Dirac cone in 2+1D with the U(1)⋊T parity anomaly. Our constructions evade both old and recently proven no-go theorems by using not-on-siteness in a crucial way, showing our results are sharp.