Recent observations show that supergiant fast X-ray transients (SFXTs) spend most of their lifetime at an intermediate level luminosity of ∼1033–1034 erg−1, and, when a blackbody model for the spectra is adopted, the resulting radii of the emission region are always only a few hundred metres, supporting the idea that during the intermediate state SFXTs are accreting matter from the companion star. From these observational phenomena we derive possible constraints on the magnetic field strengths of the neutron stars in four SFXTs with known spin periods. While IGR J11215−5952, IGR J16465−4507 and IGR J18483−0311 may have magnetic fields (up to a few 1011–1012 G) similar to those of normal X-ray pulsars, the magnetic field of AX J1841.0−0536 is considerably low (≲1010 G). The high-mass companion stars in SFXTs imply that the neutron stars are relatively young objects, with age less than ∼107 yr. Analysis of the spin evolution shows that neutron stars like AX J1841.0−0536 should be born with relatively long spin periods (≲1 s). Considering the fact that among the four SFXTs only AX J1841.0−0536 is a ‘proper’ one, and the other three are either ‘intermediate’ SFXTs or have peculiar characteristics, we suggest that the neutron stars in some of the SFXTs may have similar characteristics as several young central compact objects in supernova remnants called ‘anti-magnetars’. These features, combining with accretion from clumpy winds, could make them distinct from standard supergiant X-ray binaries – the low fields and relatively long spin periods guarantee accretion at a very low level, resulting in a large dynamic range (104–105) of X-ray luminosity.