An SYK-Like Model Without Disorder

TL;DR

This work constructs a disorder-free, tensor-model variant of the SYK model that preserves the same leading large-$N$ physics as SYK. By arranging $q=D+1$ real fermions into a highly symmetric tensor structure and scaling the coupling as $j=J/n^{D(D-1)/4}$, the model's melonic (leading) diagrams reproduce SYK correlation functions and thermodynamics without quenched randomness. The analysis uses cyclic-order fatgraph reductions and a degree function $\omega(\mathcal{G})$ to prove that diagrams with $\omega=0$ dominate, mirroring SYK's diagrammatics, while describing how $1/N$ corrections differ. This approach provides a concrete path toward holographic probes and black-hole physics in a disorder-free quantum system.

Abstract

Making use of known facts about "tensor models," it is possible to construct a quantum system without quenched disorder that has the same large $n$ limit for its correlation functions and thermodynamics as the SYK model. This might be useful in further probes of this approach to holographic duality.

Figures (7)

• Figure 1: The iterative procedure that generates the leading diagrams of the SYK model for large $N$. Figures are drawn for $q=4$ (quartic vertices). At each step of the process, one replaces a propagator (part (a)) with the two-loop diagram of part (b). For example, after another iteration, one can generate the four-loop diagram of part (c).
• Figure 2: Two views of the basic Feynman vertex of the theory. Here and later, figures are drawn for quartic vertices ($q=4$ or $D=3$). In (a), the vertex is drawn as a simple quartic vertex with external lines labeled $0,1,2$, or 3. In (b), each line is resolved into three "strands," representing how the "indices" of a field transform under a symmetry group. For example, the field $\psi_0$ is a trifundamental of $G_{01}\times G_{02}\times G_{03}$, so it is represented with three strands labeled $01,$$02$, and $03$. The vertex is constructed by connecting these strands in the only way consistent with their labeling. It can be visualized as a tetrahedron.
• Figure 3: A typical diagram that survives (a) or does not survive (b) in the large $n$ limit.
• Figure 4: (a) The vacuum amplitude in free fermion theory is represented by this one-loop diagram (the loop may carry any label 0,1,2, or 3). (b) The lowest order nontrivial contribution to the vacuum amplitude, obtained by applying the iterative step in fig. \ref{['Iteration']} to the one-loop diagram in (a).
• Figure 5: The next step in the iteration can generate this more complicated vacuum diagram, which also survives in the large $N$ limit.