Crossover to Sachdev-Ye-Kitaev criticality in an infinite-range quantum Heisenberg spin glass

We study the equilibrium dynamics of an infinite-range quantum Heisenberg model with random couplings, in which local magnetic moments arise from $\mathcal{N}_f$ flavors of spinful fermions. We employ an expansion in $\mathcal{N}_f$, which controls the strength of quantum fluctuations, and self-consistently include $1/\mathcal{N}_f$ corrections to the Luttinger-Ward functional. In the large-$\mathcal{N}_f$ limit, where quantum fluctuations are weak, the high- and low-temperature phases are respectively paramagnetic and spin glass ordered, with a transition temperature independent of $\mathcal{N}_f$. For small numbers of fermionic flavors, however, quantum fluctuations substantially suppress the ordering temperature. We show that this behavior reflects the proximity of the system to a Sachdev-Ye-Kitaev (SYK) phase, where both fermionic and spin spectral densities display critical behavior over a broad range of finite frequencies, with the latter exhibiting the scale-invariant form $\chi''(\omega)\sim \operatorname{sgn}(\omega)$. At the lowest energies and temperatures, spin-glass dynamics ultimately take over, producing a universal sub-Ohmic dynamical spin susceptibility $\chi''(\omega)\sim \operatorname{sgn}(\omega)\sqrt{|\omega|}$. Our results establish a minimal framework for understanding dynamical crossovers between SYK criticality and spin-glass ordering.