In this work, we interpreted the high braking index of PSR J1640−4631 with a combination of the magnetodipole radiation and dipole magnetic field decay models. By introducing a mean rotation energy conversion coefficient z , the ratio of the total high-energy photon energy to the total rotation energy loss in the whole life of the pulsar, and combining the pulsar's high-energy and timing observations with a reliable nuclear equation of state, we estimate the pulsar's initial spin period, P 17 44 0~( -) ms, corresponding to the moment of inertia I 0.8 2.1 10 45( -) g cm 2 .Assuming that PSR J1640−4631 has experienced a long-term exponential decay of the dipole magnetic field, we calculate the true age t age , the effective magnetic field decay timescale D t , and the initial surface dipole magnetic field at the pole B 0 p ( ) of the pulsar to be 2900−3100 yr, 1.07 2 10 5 ( ) yr, and 1.84 4.20 10 13 ( -) G, respectively. The measured braking index of n 3.15 3 = ( ) for PSR J1640−4631 is attributed to its long-term dipole magnetic field decay and a low magnetic field decay rate, dB dt 1.66 3.85 10 p 8-( -) G yr −1 . Our model can be applied to both the high braking index (n 3 > ) and low braking index (n 3 < ) pulsars, tested by the future polarization, timing, and high-energy observations of PSR J1640−4631.

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By introducing Dirac's δ-function in superhigh magnetic fields, we deduce a general formula for the pressure of degenerate and relativistic electrons, P e , which is suitable for superhigh magnetic fields, discuss the quantization of Landau levels of electrons, and consider the quantum electrodynamic(QED) effects on the equations of states (EOSs) for different matter systems. The main conclusions are as follows: the stronger the magnetic field strength, the higher the electron pressure becomes; compared with a common radio pulsar, a magnetar could be a more compact oblate spheroid-like deformed neutron star due to the anisotropic total pressure; and an increase in the maximum mass of a magnetar is expected because of the positive contribution of the magnetic field energy to the EoS of the star. Since this is an original work in which some uncertainties could exist, modifications and improvements of our theory should be considered in our future studies.