Higher-dimensional Sachdev-Ye-Kitaev non-Fermi liquids at Lifshitz transitions

We address the key open problem of a higher-dimensional generalization of the Sachdev-Ye-Kitaev (SYK) model. We construct a model on a lattice of SYK dots with nonrandom intersite hopping. The crucial feature of the resulting band dispersion is the presence of a Lifshitz point where two bands touch with a tunable power-law divergent density of states (DOS). For a certain regime of the power-law exponent, we obtain a class of interaction-dominated non-Fermi-liquid (NFL) states, which exhibits exciting features such as a zero-temperature scaling symmetry, an emergent (approximate) time reparameterization invariance, a power-law entropy-temperature relationship, and a fermion dimension that depends continuously on the DOS exponent. Notably, we further demonstrate that these NFL states are fast scramblers with a Lyapunov exponent ${\lambda }_L\propto T$, although they do not saturate the upper bound of chaos, rendering them truly unique.

Figure 1

Model: A $d$-dimensional hypercubic lattice of SYK dots of two colors—R (red) and B (blue), each with $N$ flavors (indexed by $i,j$). The flavors within a dot interact via $q$-body color-preserving interactions (with strength $J^{R,x}, J^{B,x}$). Translationally invariant hoppings (green arrows) preserve the flavor but flip color. Figure shown for $d=2$ and $q=3$.

Figure 2

Numerical results ($q=2, J=\mathrm{\Lambda }=1$): (a) Numerical spectral function (points) at $T=0.0005$ with power-law (${\left|\omega \right|}^{-\alpha }$) fits (lines); $\alpha =0.097\pm 0.0003,0.28\pm 0.0006$ for $\gamma =0.1,0.3$, respectively. (b) Numerical spectral function (points) at $T=0.0005$ for $\gamma =0.5,0.7,0.9$ compared with analytical Eq. (12) (no fitting parameters, lines). (c) Entropy ($S$) vs temperature ($T$) shown in log-log scale. (d) $S-T$ exponent $\zeta$ (points) vs $\gamma$ compared with theory (lines) in the LFL ($\gamma <{\gamma }_c$) and NFL ($\gamma >{\gamma }_c$) [Eq. (13)] regimes [here ${\gamma }_c=1/3$, Eq. (10)].

Figure 3

Zero-mode Lyapunov exponent (${\lambda }_{\text{L}}^{\text{(M)}}$) for $q=2, J=\mathrm{\Lambda }=1$. (a) ${\lambda }_{\text{L}}^{\text{(M)}}/2\pi T$ vs temperature ($T$) for $\gamma =0.1-0.95$, bold red line is for zero-dim SYK. (b) Same plot in log-log scale, showing the change from a chaotic(NFL) to a nonchaotic (LFL) fixed point.