  We study the effect of vacuum polarization on the atmosphere structure and
radiation spectra of neutron stars with surface magnetic fields B=10^14-10^15
G, as appropriate for magnetars. Vacuum polarization modifies the dielectric
property of the medium and gives rise to a resonance feature in the opacity;
this feature is narrow and occurs at a photon energy that depends on the plasma
density. Vacuum polarization can also induce resonant conversion of photon
modes via a mechanism analogous to the MSW mechanism for neutrino oscillation.
We construct atmosphere models in radiative equilibrium with an effective
temperature of a few \times 10^6 K by solving the full radiative transfer
equations for both polarization modes in a fully ionized hydrogen plasma. We
discuss the subtleties in treating the vacuum polarization effects in the
atmosphere models and present approximate solutions to the radiative transfer
problem which bracket the true answer. We show from both analytic
considerations and numerical calculations that vacuum polarization produces a
broad depression in the X-ray flux at high energies (a few keV \la E \la a few
tens of keV) as compared to models without vacuum polarization; this arises
from the density dependence of the vacuum resonance feature and the large
density gradient present in the atmosphere. Thus the vacuum polarization effect
softens the high energy tail of the thermal spectrum, although the atmospheric
emission is still harder than the blackbody spectrum because of the non-grey
opacities. We also show that the depression of continuum flux strongly
suppresses the equivalent width of the ion cyclotron line and therefore makes
the line more difficult to observe.
