Chiral Anomaly of Kogut-Susskind Fermion in the (3+1)-dimensional Hamiltonian formalism

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 $Γ= i S_1 S_2 S_3$ can define an axial charge as $Q_A = (1/2)\sum_x χ^\dagger(x) \left(Γ+Γ^\dagger \right)χ(x)$, which is non-onsite, nonquantized, and commutative with the vector charge, analogous to $\tilde{Q}A = (1/2) \sum_n ( χ^\dagger_n χ{n+1} + χ^\dagger_{n+1} χ_{n} )$ 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.

💡 Research Summary

The paper investigates the chiral symmetry and its associated anomaly for Kogut‑Susskind (staggered) fermions formulated in a (3+1)‑dimensional Hamiltonian framework. Starting from the free staggered‑fermion Hamiltonian, the authors emphasize that the massless theory possesses three independent shift symmetries (S_k) (k = 1,2,3) acting on the lattice. The product of the three shifts, modulo a two‑site translation, defines a unitary discrete chiral transformation. Its kernel (\Gamma = i S_1 S_2 S_3) is not Hermitian; its Hermitian part (\Gamma+\Gamma^\dagger) is used to construct a non‑onsite axial charge \