Local criticality, diffusion and chaos in generalized Sachdev-Ye-Kitaev models

TL;DR

The paper extends the solvable SYK framework to higher spatial dimensions by constructing a lattice of SYK sites with random inter-site couplings, preserving local criticality and extensive zero-temperature entropy while enabling diffusive energy transport and chaotic propagation. Through a large-N saddle-point analysis with bilocal fields, the authors derive a conformal, locally critical two-point function and compute four-point functions via a ladder kernel, revealing a diffusive energy mode governed by a reparameterization field and a butterfly velocity that sets the speed of chaos spreading. A key result is the universal relation D = v_B^2/(2πT) linking diffusion and chaos, analogous to holographic incoherent metals, and the formalism extends to higher dimensions, graphs, and generalized q-body interactions. These solvable, chaotic lattice models provide a controlled platform to study thermalization, hydrodynamics, and potential holographic duals in strongly correlated systems.

Abstract

The Sachdev-Ye-Kitaev model is a $(0+1)$-dimensional model describing Majorana fermions or complex fermions with random interactions. This model has various interesting properties such as approximate local criticality (power law correlation in time), zero temperature entropy, and quantum chaos. In this article, we propose a higher dimensional generalization of the Sachdev-Ye-Kitaev model, which is a lattice model with $N$ Majorana fermions at each site and random interactions between them. Our model can be defined on arbitrary lattices in arbitrary spatial dimensions. In the large $N$ limit, the higher dimensional model preserves many properties of the Sachdev-Ye-Kitaev model such as local criticality in two-point functions, zero temperature entropy and chaos measured by the out-of-time-ordered correlation functions. In addition, we obtain new properties unique to higher dimensions such as diffusive energy transport and a "butterfly velocity" describing the propagation of chaos in space. We mainly present results for a $(1+1)$-dimensional example, and discuss the general case near the end.

Figures (14)

• Figure 1: A chain of coupled SYK sites: each site contains $N\gg 1$ fermion with SYK interaction. The coupling between nearest neighbor sites are four fermion interaction with two from each site.
• Figure 2: The leading order diagrams only connect fermions with same flavor and spatial coordinate under random average of disorder fields (dashed line).
• Figure 3: Replicon diagonal v.s. off-diagonal contributions to the partition function $\overline{Z^n}$: solid lines connects same replica index. Different replica indices can be connected only by dashed lines, which are disorder fields. The replicon off-diagonal diagram on the right is suppressed by $1/N^3$ compared to the diagonal diagram on the left.
• Figure 4: (a) Green's function $G_x(\tau_1,\tau_2)$ is a function of two imaginary time variables, each defined on the imaginary time circle. It transforms covariantly under the reparametrization field $f_x\in \operatorname{Diff}(S^1)$. (b) The space-time picture for the chain model. The Schwinger-Dyson equation at conformal limit has global reparametrization symmetry, but the conformal solution spontaneously breaks $\operatorname{Diff}(S^1)$ to $\operatorname{PSL}_2(\mathbb{R})$. Moreover, the UV term $-i\omega$ in (\ref{['eqn: chain Dyson']}) breaks the emergent reparametrization and lifts the Goldstone modes to pseudo-Goldstone modes. (c) The situation in the SYK chain model is similar to a ferromagnetic spin chain with a small pinning field, where the $\operatorname{SU}(2)$ symmetry is "almost spontaneously" broken to $\operatorname{U}(1)$, leading to a pseudo-Goldstone mode.
• Figure 5: Two regions of the four-point function are illustrated. Factorization in the configuration at left gives the propagating bilocal fields. The configuration at right can be continued to the OTOC which diagnoses chaos.