A Generalization of Sachdev-Ye-Kitaev

TL;DR

This work introduces a broad generalization of the SYK model to f flavors of fermions with flavor-specific site counts N_a and q_a-body interactions, showing that an infrared fixed point persists generically. It derives coupled Schwinger-Dyson equations for the flavor-resolved two-point functions, determines IR dimensions Δ_k (with a compact large-q limit yielding explicit expressions), and analyzes the spectrum of singlet bilinear operators, proving the universal presence of a dimension-two operator that drives conformal symmetry breaking and maximal chaos in the IR four-point function. Post-disorder averaging, the symmetry reduces to O(N_1)×…×O(N_f), enriching the operator content with flavor-tensor structures and multiple operator towers. The paper also solves q=2 SYK at finite N and discusses the equal-q, equal-κ special case, where the four-point function decomposes into symmetric and antisymmetric flavor sectors with distinct conformal data. Overall, the results demonstrate that SYK-like chaos and conformal symmetry breaking survive in a significantly broader class of large-N quantum mechanical models, offering new avenues for understanding holography and non-Fermi liquid behavior.

Abstract

The SYK model: fermions with a $q$-body, Gaussian-random, all-to-all interaction, is the first of a fascinating new class of solvable large $N$ models. We generalize SYK to include $f$ flavors of fermions, each occupying $N_a$ sites and appearing with a $q_a$ order in the interaction. Like SYK, this entire class of models generically has an infrared fixed point. We compute the infrared dimensions of the fermions, and the spectrum of singlet bilinear operators. We show that there is always a dimension-two operator in the spectrum, which implies that, like in SYK, there is breaking of conformal invariance and maximal chaos in the infrared four-point function of the generalized model. After a disorder average, the generalized model has a global $O(N_1) \times O(N_2) \times \ldots\times O(N_f)$ symmetry: a subgroup of the $O(N)$ symmetry of SYK; thereby giving a richer spectrum. We also elucidate aspects of the large $q$ limit and the OPE, and solve $q=2$ SYK at finite $N$.

Figures (8)

• Figure 1: The self-energy (\ref{['SD2']}) for a fermion in SYK (\ref{['SYK']}). The figure is for $q=6$. The solid line with a filled circle is the two-point function. The dashed line is the disorder.
• Figure 2: The self-energy (\ref{['SigGen']}) for a fermion of flavor $k$ in the generalized model (\ref{['GFM']}). The figure is for two flavors with $q_1 = q_2 = 3$. This is, for instance, the self-energy for fermion of flavor $1$. The black line with a filled circle is the two-point function for the fermion of flavor $1$, while the blue wavy line with a filled square is the two-point function for the flavor $2$ fermion.
• Figure 3: The two-point function for SYK at large $q$ consists only of diagrams that split into two trees under a vertical cut, such as the one shown in (a). Diagrams like (b) are suppressed by factors of $1/q$. In (c) we show the recursion relation for the self-energy, represented by a filled triangle. Note that the solid lines are the (free) fermion propagators, and we have suppressed the disorder lines.
• Figure 4: (a) The diagrams being summed to compute the three-point function $\langle\chi_i(\tau_1) \chi_j(\tau_2) \mathcal{O}(\tau_0)\rangle$ for $q=6$ SYK. This can be done iteratively, as shown in (b) (see Eq. \ref{['EIG1']}), with the kernel shown in (c) adding rungs to the ladder. In the IR we can simplify (b) to get (d).
• Figure 5: The eigenvalues $g(h)$ (\ref{['ghSYK']}) of the SYK kernel (\ref{['K1flav']}) for $q=6$ as a function of dimension $h$. The $h$ for which $g(h)=1$ are the IR dimensions of the fermion bilinear operators (\ref{['O2n1']}).