Solvable model for quantum criticality between the Sachdev-Ye-Kitaev liquid and a disordered Fermi liquid

We propose a simple solvable variant of the Sachdev-Ye-Kitaev (SYK) model which displays a quantum phase transition from a fast-scrambling non-Fermi liquid to disordered Fermi liquid. Like the canonical SYK model, our variant involves a single species of Majorana fermions connected by all-to-all random four-fermion interactions. The phase transition is driven by a random two-fermion term added to the Hamiltonian whose structure is inspired by proposed solid-state realizations of the SYK model. Analytic expressions for the saddle-point solutions at a large number $N$ of fermions are obtained and show a characteristic scale-invariant ${\sim \left|\omega \right|}^{-1/2}$ behavior of the spectral function below the transition which is replaced by a ${\sim \left|\omega \right|}^{-1/3}$ singularity exactly at the critical point. These results are confirmed by numerical solutions of the saddle-point equations and discussed in the broader context of the field.

Figure 1

Evolution of the spectral function $A\left(\omega \right)$ across the quantum phase transition tuned by parameter $p=M/N$. Spectral functions are computed by a numerical iteration of the large-$N$ saddle-point equations in Keldysh representation given in Appendix pp2. Results for noninteracting ($J=0$, blue solid line) and interacting ($J=K$, red dashed line) cases are overlaid for each $p$. Note that the interactions broaden the zero modes inside the gap for $p<1/4$, resulting in a SYK non-Fermi liquid. Above the transition for $p>1/4$, interactions are irrelevant and do not significantly modify the spectral function.

Figure 2

Low-frequency scaling behavior of the spectral function $A\left(\omega \right)$ for various values of $p$. Other parameters $K=J$ and $T=0.0003J$ are fixed. Analytically obtained scaling forms are shown by dashed lines. For $p<1/4$ spectral function tends to $A\left(\omega \right)\sim {\omega }^{-1/2}$, characteristic of the SYK model. At the nFL/FL transition (green) where $p=1/4$ we confirm that $A\left(\omega \right)\sim {\omega }^{-1/3}$ in accord with the prediction. Above the transition $A\left(\omega \right)$ tends to a frequency-independent constant as expected for disordered Fermi liquid.