We consider Kogut-Susskind fermions (also known as staggered fermions) in a ($3+1$)-dimensional Hamiltonian formalism and examine a chiral transformation and its associated chiral anomaly. The Hamiltonian of the massless Kogut-Susskind fermion has symmetry under the shift transformations in each space direction $S_k$ ($k=1,2,3$), and the product of the three shift transformations in particular (the odd shifts in general) may be regarded as a unitary discrete chiral transformation, modulo two-site translations. The Hermitian part of the transformation kernel $\mathrm{\Gamma }=\sqrt{-1}{S}_1{S}_2{S}_3$ can define an axial charge as $Q_A=(1/2){\sum }_x{\chi }^{†}(x)(\mathrm{\Gamma }+{\mathrm{\Gamma }}^{†})\chi (x)$, which is non–on site, nonquantized, and commutative with the vector charge, analogous to ${\tilde{Q}}_A=(1/2){\sum }_n({\chi }_n^{†}{\chi }_{n+1}+{\chi }_{n+1}^{†}{\chi }_n)$ for the ($1+1$)-dimensional Kogut-Susskind fermion. However, our $Q_A$ cannot be expressed in terms of any quantized charges in a generalized Onsager algebra. Although $Q_A$ does not commute with the fermion Hamiltonian in general when coupled to background link gauge fields, we show that they become commutative for a class of $U(1)$ link configurations carrying nontrivial magnetic and electric fields. We then verify numerically that the vacuum expectation value of $Q_A$ satisfies the anomalous conservation law of axial charge in the continuum two-flavor theory under an adiabatic evolution of the link gauge field.

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• Hamiltonian systems
• Lattice gauge theory