Index theorem with Minimally Doubled Fermions in four space-time dimensions

TL;DR

This work extends the lattice verification of the Atiyah-Singer index theorem to four-dimensional Minimally Doubled Fermions (MDF), focusing on Karsten-Wilczek (KW) and Borici-Creutz (BC) formulations. By analyzing the spectral flow of $H(m)=\gamma_5(D+m)$ in background gauge fields with fixed topological charge $Q$ and on dynamical MILC lattices, the authors first observe index cancellations due to degenerate doublers. They resolve this with a flavor-dependent mass term $m C_{\text{flav}}$ and a modified chirality operator $\mathcal{X}$, achieving an index $\text{index}(D)=2Q$ that is corroborated by fermionic topological charge measurements. The results establish the universality of zero modes across backgrounds and demonstrate MDF as a promising, ultralocal chiral lattice fermion formulation for QCD studies, motivating further exploration in four dimensions.

Abstract

We determine the zero eigenmode spectrum of Minimally Doubled Fermions (MDF), namely in Karsten-Wilczek (KW) and Borici-Creutz (BC) formulations on the 4-dimensional space-time lattice. We employ background gauge fields with integer valued topological charges. The Atiyah-Singer index theorem is verified in the presence of two different background gauge fields, namely Smit-Vink [1] and cooled down MILC asqtad ensembles with $N_f=2+1$ dynamical flavors of quarks [2]. Using flavored mass terms [3,4], we find that the spectral flow of the eigenvalues detects the topology of the background gauge field. With the use of the modified chirality operator, we obtain chiralities of the zero eigenmodes and the fermionic topological charge.

Figures (15)

• Figure 1: Spectral flow with respect to degenerate bare masses $m$ for $\pm$ chiral states for both KW and BC fermions under $Q=-2$ and $\delta=0.05$ background $8^3\times8$ Smit-Vink lattice. Eigenvalues in left and right panels correspond to $\gamma_5(D_{\text{KW}}+m)$ and $\gamma_5(D_{\text{BC}}+m)$, respectively. The color coding is intended to guide the eye.
• Figure 2: Complex eigenvalue plots of KW and BC fermions for $Q=-2$ and $\delta=0.05$ background $8^3 \times 8$ Smit-Vink lattice with mass parameter $m=1$ for 5000 eigenvalues. The left and right panel correspond to $(D_\text{KW}+m)$ and $(D_\text{BC}+m)$.
• Figure 3: Complex eigenvalue plots of KW and BC fermions for free theory on $8^3\times8$ lattice with mass parameter $m=1$. The left and right panel correspond to $(D_{\text{KW}}+mC_{\text{sym}}\otimes 1)$ and $(D_{\text{BC}}+m(2C_{\text{sym}}-1)\otimes 1)$ operators, respectively.
• Figure 4: Same plot as Fig. \ref{['fig:complex_dirac_csym_eigval_free_th']} but with $Q=-2$ and $\delta=0.05$ background $8^3 \times 8$ Smit-Vink lattice.
• Figure 5: Spectral flow of the eigenvalues of $\mathcal{H}$ with respect to $m$ for $Q=-2$ and $\delta=0.05$ background Smit-Vink lattice. Four nearly overlapping lines (red in color) cross the zero eigenvalue line at $m=0$. The color coding is intended to guide the eye.