Chiral anomaly of Kogut-Susskind fermions 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 = − 1 S 1 S 2 S 3 can define an axial charge as Q A = ( 1 / 2 ) ∑ x χ † ( x ) ( Γ + Γ † ) χ ( x ) , which is non–on site, nonquantized, and commutative with the vector charge, analogous to Q ˜ A = ( 1 / 2 ) ∑ n ( χ n † χ n + 1 + χ 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.