  We study the critical behavior and the ground-state entanglement of a large
class of $\mathrm{su}(1|1)$ supersymmetric spin chains with a general (not
necessarily monotonic) dispersion relation. We show that this class includes
several relevant models, with both short- and long-range interactions of a
simple form. We determine the low temperature behavior of the free energy per
spin, and deduce that the models considered have a critical phase in the
universality class of a $(1+1)$-dimensional conformal field theory (CFT), whose
central charge coincides with the number of connected components of the Fermi
sea. We also study the R\'enyi entanglement entropy of the ground state,
deriving its asymptotic behavior as the block size tends to infinity. In
particular, we show that this entropy exhibits the logarithmic growth
characteristic of $(1+1)$-dimensional CFTs and one-dimensional (fermionic)
critical lattice models, with a central charge consistent with the
low-temperature behavior of the free energy. Our results confirm the widely
believed conjecture that the critical behavior of fermionic lattice models is
completely determined by the topology of their Fermi surface.
