What about the magnetic field strength inside of an old accreting neutron star? As the magnetic field strength outside an old accreting neutron star is thought to be small (about $10^8-10^{10}$ Gauss), what about it inside the star? Because of the superconductivity and degeneration, can the strength reach $10^{17}$ Gauss or even more? Is there any article that describe such problem?
This turns out to be tricky to answer definitively and is a topic of contemporary research.
The interior fields may be quite high ($\sim 10^{12}$ G) or approaching zero, depending on details of the interior microphysics.
The argument that fields might be close to zero is based on the probable presence of a superconducting, superfluid of protons. The interior of a neutron star cannot be just neutrons, because of the weak decay mechanism into protons, electons and (anti)neutrinos. Neutron star interiors are therefore expected to have an equilibrium concentration of $\sim 1$% protons (and electrons). If temperatures are low enough, and by that we mean $&lt; \sim 10^9$ K, then a long-range interaction between the protons can create bosonic pairs in much the same way that electrons pair up in a low-temperature superconductor. A similar pairing mechanism can create a superfluid of neutrons in the interior.
In neutron stars, despite neutrino cooling being strongly inhibited by the degeneracy of neutrons, protons and electrons, the cooling rate is very rapid because the same degeneracy means neutron stars have an extremely low heat capacities. This means the core cools on timescales of tens of seconds after the supernova, slowing to decades after a year or two. The result is that the interior temperature falls below $10^9$ K and superfluidity (with accompanying superconductivity in the case of protons) will probably set in on timescales of hundreds of years.
When materials make a phase tranition to a superconducting state it is possible for the magnetic flux that threads them to be expelled. This is known as the Meissner effect.
On that basis, one might expect the magnetic field to be expelled from the interior of the neutron star once the protons become superconducting.
However, the superconductive state in a neutron star may be akin to a type II superconductor. This does allow the penetration of magnetic flux tubes (or &quot;fluxoids&quot;) into the interior (Baym, Pethick &amp; Pines 1969 - sorry, can't find a free version) if the magnetic field is greater than some critical value.
There seems to be healthy debate about exactly what state the interior of a neutron star is in as a function of time and whether the timescale for magnetic flux expulsion is short enough for the Meissner effect to take place or whether there is effectively a &quot;frozen in&quot; field in older neutron stars (e.g. Wynn et al. 2017).