  Thermal properties of low-density neutron matter are investigated by
determinantal quantum Monte Carlo lattice calculations on 3+1 dimensional cubic
lattices. Nuclear effective field theory (EFT) is applied using the pionless
single- and two-parameter neutron-neutron interactions, determined from the
$^1S_0$ scattering length and effective range. The determination of the
interactions and the calculations of neutron matter are carried out
consistently by applying EFT power counting rules. The thermodynamic limit is
taken by the method of finite-size scaling, and the continuum limit is examined
in the vanishing lattice filling limit. The $^1S_0$ pairing gap at $T \approx
0$ is computed directly from the off-diagonal long-range order of the spin
pair-pair correlation function, and is found to be approximately 30% smaller
than BCS calculations with the conventional nucleon-nucleon potentials. The
critical temperature $T_c$ of the normal-to-superfluid phase transition and the
pairing temperature scale $T^\ast$ are determined, and the temperature-density
phase diagram is constructed. The physics of low-density neutron matter is
clearly identified as being a BCS-Bose-Einstein condensation crossover.
