Domain-wall induced quark masses in topologically nontrivial backgrounds

In the domain-wall formulation of chiral fermions, the finite separation between domain walls ${(L}_s)$ induces an effective quark mass ${(m}_{\mathrm{eff}})$ which complicates the chiral limit. In this work, we study the size of the effective mass as a function of $L_s$ and the domain-wall height $m_0$ by calculating the smallest eigenvalue of the Hermitian domain-wall Dirac operator in the topologically nontrivial background fields. We find that, just as in the free case, $m_{\mathrm{eff}}$ decreases exponentially in $L_s$ with a rate depending on $m_0.$ However, quantum fluctuations amplify the wall effects significantly. Our numerical result is consistent with a previous study of the effective mass from the Gell-Mann–Oakes–Renner relation.