  Ferroelectrics usually adopt a multi-domain state with domain walls
separating domains with polarization axes oriented differently. It has long
been recognized that domain walls can dramatically impact the properties of
ferroelectric materials. The enhancement of low-field
susceptibility/permittivity under subswitching conditions is usually attributed
to the reversible domain wall vibration. Recent experiments highlight the
stationary domain wall contribution to the dielectric susceptibility
irrespective of any lateral displacements or deformations of the wall. We study
the effects of domain walls on low-field permittivity of PbTiO$_3$ with density
functional theory and molecular dynamics simulations. The static dielectric
constant is calculated as a function of increasing domain wall density and
temperature. We find an increase of dielectric permittivity with increasing
domain wall density, which is expected to occur at low driving field where the
lateral motion of domain walls is forbidden. Real-space decomposition of
dielectric response reveals that frustrated dipoles within the finite width of
the domain walls are responsible for the enhanced low-field permittivity.
