Strongly Interacting Fermions Are Nontrivial yet Nonglassy

Random spin systems at low temperatures are glassy and feature computational hardness in finding low-energy states. We study the random all-to-all interacting fermionic Sachdev-Ye-Kitaev model and prove that, in contrast, the low-energy states have polynomial circuit depth, yet the annealed and quenched free energies agree to polynomially inverse low temperatures, ruling out a glassy phase transition in this sense. These results are derived by showing that fermionic and spin systems significantly differ in their "commutation index," which quantifies the noncommutativity of Hamiltonian terms. Our results suggest that low-temperature strongly interacting fermions, unlike spins, belong in a classically nontrivial yet quantumly easy phase.