Magnetar is a kind of pulsar powered by magnetic field energy. Part of the X-ray luminosities of magnetars in quiescence have a thermal origin and can be fitted by a blackbody spectrum with temperature kT ～ 0.2－0.6 keV, much higher than the typical values for rotation-powered pulsars. The observation and theoretical study of magnetar are one of hot topics in the field of pulsar research. The activity and emission characteristics of magnetar can be attributed to internal superhigh magnetic field. According to the work of WGW19 and combining with the equation of state, we first calculate the electric conductivity of the crust under a strong magnetic field, and then calculate the toroidal magnetic field decay rate and magnetic energy decay rate by using an eigenvalue equation of toroidal magnetic field decay and considering the effect of general relativity. We reinvestigate the L_X-L_rot relationship of 22 magnetars with persistent soft X-ray luminosities and obtain two new fitting formulas on L_X-L_rot. We find that for the magnetars with L_X < L_rot, the soft X-ray radiations may originate from their rotational energy loss rate, or from magneto-sphere flow and particle wind heating. For the magnetars with L_X > L_rot, the Ohmic decay of crustal toroidal magnetic fields can provide their observed isotropic soft X-ray radiation and maintain higher thermal temperature.

As for the initial dipole magnetic fields of magnetars, we mainly refer to the rersearch by Viganò et al. (Viganò D, Rea N, Pons J A, Perna R, Aguilera D N, Miralles J A 2013 Mon. Not. R. Astron. Soc. 434 123), because they first proposed the up-dated neutron star magneto-thermal evolution model, which can successfully explain the X-ray radiation and cooling mechanism of young pulsars including magnetars and high-magnetic field pulsars. Objectively speaking, as to the decay of toroidal magnetic fields, there are some differences between our theoretical calculations of magnetic energy release rates and the actual situation of magnetic field decay in magnetars, this is because the estimate of initial dipolar magnetic field, true age and the thickness of inner crust of a magnetar are somewhat uncertain. In addition, due to the interstellar-medium’s absorptions to soft X-ray and the uncertainties of distance estimations, the observed soft X-ray luminosities of magnetars have certain deviations. With the continuous improvement of observation, equipment and methods, as well as the in-depth development of theoretical research, our model will be further improved, and the theoretical results are better accordant with the high-energy observation of magnetars.

We also discuss other possible anisotropy origins of soft X-ray fluxes of magnetars, such as the formation of magnetic spots and thermoplastic flow wave heating in the polar cap. Although anisotropic heating mechanisms are different from Ohmic decay, all of them require that there exist strong toroidal magnetic fields inside a magnetar. However, the anisotropic heating mechanisms require higher toroidal multipole fields inside a magnetar (such as magnetic octupole field) and are related to complex Hall drift: these may be our research subjects in the future.