  Recent Chandra and XMM-Newton observations of a number of X-ray ``dim''
pulsating neutron stars revealed quite unexpected features in the emission from
these sources. Their soft thermal spectrum, believed to originate directly from
the star surface, shows evidence for a phase-varying absorption line at some
hundred eVs. The pulse modulation is relatively large (pulsed fractions in the
range ~12%-35%), the pulse shape is often non-sinusoidal, and the hard X-ray
color appears to be anti-correlated in phase with the total emission. Moreover,
the prototype of this class, RX J0720.4-3125, has been found to undergo rather
sensible changes both in its spectral and timing properties over a timescale of
a few years. All these new findings seem difficult to reconcile with the
standard picture of a cooling neutron star endowed with a purely dipolar
magnetic field, at least if surface emission is produced in an atmosphere on
top of the crust. In this paper we explore how a dipolar+quadrupolar
star-centered field influence the properties of the observed lightcurves. The
phase-resolved spectrum has been evaluated accounting for both radiative
transfer in a magnetized atmosphere and general relativistic ray-bending. We
computed over 78000 lightcurves varying the quadrupolar components and the
viewing geometry. A comparison of the data with our model indicate that higher
order multipoles are required to reproduce the observations.
