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We investigate the effects of the violation of the Ginsparg-Wilson (GW)relation in the M\"obius domain-wall fermion formulation on the lattice withfinite fifth dimension. Using a decomposion in terms of the eigenmodes of itsfour-dimensional effective Dirac operator, we isolate the GW-violating termsfor various physical quantities including the residual mass and the mesonsusceptibilities relevant for the effective restoration of the axial U(1)symmetry at finite temperature. Numerical result shows that the GW-violatingeffect is more significant, or even overwhelming, for the quantities that aredominated by the low-lying eigenmodes.

Violation of chirality of the M\"obius domain-wall Dirac operator from the eigenmodes