  In the domain-wall formulation of chiral fermion, the finite separation
between domain-walls ($L_s$) induces an effective quark mass ($m_{\rm eff}$)
which complicates the chiral limit. In this work, we study the size of the
effective mass as the 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 like in
the free case, $m_{\rm eff}$ 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.
