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We study the critical behavior and the ground-state entanglement of a largeclass of $\mathrm{su}(1|1)$ supersymmetric spin chains with a general (notnecessarily monotonic) dispersion relation. We show that this class includesseveral relevant models, with both short- and long-range interactions of asimple form. We determine the low temperature behavior of the free energy perspin, and deduce that the models considered have a critical phase in theuniversality class of a $(1+1)$-dimensional conformal field theory (CFT), whosecentral charge coincides with the number of connected components of the Fermisea. We also study the R\'enyi entanglement entropy of the ground state,deriving its asymptotic behavior as the block size tends to infinity. Inparticular, we show that this entropy exhibits the logarithmic growthcharacteristic of $(1+1)$-dimensional CFTs and one-dimensional (fermionic)critical lattice models, with a central charge consistent with thelow-temperature behavior of the free energy. Our results confirm the widelybelieved conjecture that the critical behavior of fermionic lattice models iscompletely determined by the topology of their Fermi surface.

Supersymmetric spin chains with non-monotonic dispersion relation: criticality and entanglement entropy