Phase transitions in the inner crust of neutron stars within the superfluid band theory: Competition between <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mi>S</mml:mi> <mml:mn>0</mml:mn> <mml:none/> <mml:mprescri

Background: Phase transitions of matter under changes of external environment such as temperature and magnetic field have attracted great interests to various quantum many-body systems. Several phase transitions must have occurred in neutron stars as well such as transitions from normal to superfluid and superconducting phases and crust formation. While the temperature of a protoneutron star is as high as 10 MeV ($\ensuremath{\approx}{10}^{11}\phantom{\rule{0.28em}{0ex}}\mathrm{K}$) or higher, which are above critical temperatures for the emergence of superfluidity and crust formation, it cools rapidly down to 0.1 keV ($\ensuremath{\approx}{10}^{6}\phantom{\rule{0.28em}{0ex}}\mathrm{K}$) already after hundreds of years. While ordinary neutron stars have surface magnetic field strength of around ${10}^{12}\phantom{\rule{0.28em}{0ex}}\mathrm{G}$, those having higher magnetic field strength of ${10}^{14--15}\phantom{\rule{0.28em}{0ex}}\mathrm{G}$ or higher, so-called magnetars, have been observed. To uncover detailed evolution of neutron stars from their birth to later years, it is desired to develop fully microscopic approaches that take into account effects of superfluidity and superconductivity, finite temperature and magnetic field, on the same footing.Purpose: The main purpose of this work is twofold: (1) to extend the formalism of the fully self-consistent superfluid nuclear band theory, developed in our previous work [K. Yoshimura and K. Sekizawa, Phys. Rev. C 109, 065804 (2024)], for finite-temperature and finite-magnetic-field systems, and (2) to explore possible phase transitions of nuclear matter by varying temperature and magnetic field.Methods: We employ the superfluid band theory which is based on the Kohn-Sham density functional theory for superfluid systems with a local treatment of paring, known as superfluid local density approximation, subjected to the Bloch boundary conditions. We assume periodic spatial variation along $z$ direction with uniform distribution along $xy$ direction, allowing us to describe the slab phase as well as uniform nuclear matter. The finite-temperature extension is achieved in a similar manner as a finite-temperature Hartree-Fock-Bogoliubov calculation. Magnetic field effects are introduced taking into account both the Landau levels formation of relativistic electrons and the couplings of the magnetic field with nucleons' magnetic moments.Results: We have performed superfluid band theory calculations for the slab phase of neutron star matter at ${n}_{B}=0.04,\phantom{\rule{0.28em}{0ex}}0.05,\phantom{\rule{0.28em}{0ex}}0.06$, and $0.07\phantom{\rule{0.28em}{0ex}}{\phantom{\rule{0.16em}{0ex}}\mathrm{fm}}^{\ensuremath{-}3}$ under various sets of temperature and magnetic field. From the results without magnetic field ($B=0$), we find that the superfluidity of neutrons disappears at around ${k}_{B}T=0.6--0.9\phantom{\rule{0.28em}{0ex}}\mathrm{MeV}$, and ``melting'' of nuclear slabs, that is, a structural change into the uniform matter, takes place at around ${k}_{B}T=2.5--4.5\phantom{\rule{0.28em}{0ex}}\mathrm{MeV}$. By turning on the magnetic field, we find that protons' spin gets polarized at around $B={10}^{16}\phantom{\rule{0.28em}{0ex}}\mathrm{G}$, whereas neutrons' spin is kept unpolarized on average up to around $B={10}^{17}\phantom{\rule{0.28em}{0ex}}\mathrm{G}$. Intriguingly, our microscopic calculations reveal that neutrons' spin is actually polarized locally inside and outside of the slab already at $B\ensuremath{\approx}{10}^{16}\phantom{\rule{0.28em}{0ex}}\mathrm{G}$, while keeping the system unpolarized in total. We show that the local polarization of neutrons' spin is caused by an interplay of ${}^{1}{S}_{0}$ pairing among neutrons and spin-dependent interactions between neutron and protons.Conclusions: We have demonstrated validity and usefulness of the fully self-consistent superfluid nuclear band theory for describing neutron star matter under arbitrary temperature and magnetic field. Critical temperatures and magnetic fields have been predicted for (1) superfluid to normal transition, (2) crust formation, and (3) spin polarization, under conditions relevant to realistic neutron star environments.