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

Abstract

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