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

TL;DR

This paper demonstrates a lattice-regulated realization of chiral symmetry for Kogut–Susskind fermions in (3+1) dimensions by introducing a unitary diagonal shift operator $\Gamma$ that acts as a lattice chiral operator. It defines a non-onsite axial charge $Q_A$ whose Hermitian part is conserved and commutes with the vector charge, and shows that $Q_A$ fits within an Onsager-algebra framework as a central charge, distinct from quantized charges in the continuum. In the presence of a $U(1)$ gauge field, the authors identify specific link configurations for which $[H,\Gamma^U]=0$, enabling lattice calculations that reproduce the continuum axial anomaly, with numerical results validating the anomaly relation for a two-flavor theory under adiabatic evolution. The work moreover confirms that the lattice diagonal shift construction reproduces known continuum results and discusses the potential to construct explicit lattice QED Hamiltonians with chiral symmetry, advancing numerical access to theta-terms and related phenomena.

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 $Γ= i S_1 S_2 S_3$ 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.

Figures (4)

• Figure 1: Energy spectrum represented by the argument (phase) of $\Gamma^U$ at $N=8$, $m=0$, and $t=0$. The left and right panels correspond to $n = -1$ and $n = 2$, respectively.
• Figure 2: Time evolution of the energy spectrum at $N=8$ and $m=0$. The left and right panels correspond to $n = -1$ and $n = 2$, respectively. The color gradation represents the eigenvalues of $\Gamma^U$: the positive and negative imaginary parts are shown in red and blue, respectively, while the real part is encoded in the brightness.
• Figure 3: Left panel: Time evolution of the expectation value of the chiral charge density $j_A^0$ under a magnetic field in the diagonal direction and an electric field applied along the $x^3$-axis. Filled symbols indicate spatially averaged values, with error bars showing spatial variations. Solid lines represent the fitting functions defined in Eq. \ref{['eq: fitting function qA']}. The white circle marks the discontinuity. Right panel: The same plot for $\tilde{j}^0$, where the fitting function is defined in Eq. \ref{['eq: fitting function tq']}. We fix $N=8$ and $m=0$, and choose $n=-1$ and $n=2$.
• Figure 4: Same plots as in Fig. \ref{['fig: q5_3D']}, but with both the magnetic and electric fields aligned along the $x^3$-axis.