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

Ian Richard C Buckley

Published May 6, 2026• 0 citations
Publisher

Abstract

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 Concept Definition SYK Model Sachdev-Ye-Kitaev model; a solvable large-$N$ quantum mechanics exhibiting maximal chaos Fano Plane The smallest projective plane $\text{PG}(2,2)$; 7 points, 7 lines, incidence-governed OTOC Out-of-time-order correlator; measures operator scrambling and information spreading MSS Bound Maldacena-Shenker-Stanford upper limit on Lyapunov exponent $\lambda_L \leq 2\pi k_B T/\hbar$ Pre-thermal Early-time dynamics before equilibration; regime where scrambling dominates

DOI
10.5281/zenodo.20057808

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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-NNN infrared limit. This paper introduces the Fano-SYK Model, in which the random Gaussian interaction tensor of the standard SYK4\text{SYK}_4SYK4​ model is replaced by a deterministic, sparse tensor governed by the incidence geometry of the Fano plane PG(2,2)\text{PG}(2,2)PG(2,2) — the combinatorial skeleton of the G2G_2G2​ exceptional Lie group. Results Through exact diagonalisation of Majorana systems at N=7,14,21,N = 7, 14, 21,N=7,14,21, and 282828 (Hilbert space dimension up to 214=16,3842^{14} = 16,384214=16,384), we extract out-of-time-order correlators (OTOCs) to measure the pre-thermal scrambling rate. We define the suppression ratio R(N)=λRandomλFanoR(N) = \frac{\lambda_{\text{Random}}}{\lambda_{\text{Fano}}}R(N)=λFano​λRandom​​ and find R(N)∈7.0,3.7,3.4,2.85R(N) \in {7.0, 3.7, 3.4, 2.85}R(N)∈7.0,3.7,3.4,2.85 across the four system sizes. A power-law fit R(N)=R∞+aNbR(N) = R_\infty + \frac{a}{N^b}R(N)=R∞​+Nba​ yields R∞=2.723±0.489R_\infty = 2.723 \pm 0.489R∞​=2.723±0.489, with the lower 1σ1\sigma1σ bound R∞>2.23R_\infty > 2.23R∞​>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 ppp-adic Bruhat-Tits geometry in holographic boundary scattering. Working paper in the Adelic Simplicial Architecture (ASA) series, Portfolio A. Key Concepts Concept Definition SYK Model Sachdev-Ye-Kitaev model; a solvable large-NNN quantum mechanics exhibiting maximal chaos Fano Plane The smallest projective plane PG(2,2)\text{PG}(2,2)PG(2,2); 7 points, 7 lines, incidence-governed OTOC Out-of-time-order correlator; measures operator scrambling and information spreading MSS Bound Maldacena-Shenker-Stanford upper limit on Lyapunov exponent λL≤2πkBT/ℏ\lambda_L \leq 2\pi k_B T/\hbarλL​≤2πkB​T/ℏ Pre-thermal Early-time dynamics before equilibration; regime where scrambling dominates
The complex Sachdev-Ye-Kitaev (SYK) model, featuring fermions with all-to-all interactions, serves as a dual paradigm for understanding non-Fermi liquid behavior and the holographic nature of charged black holes. Two defining characteristics of the standard SYK model are its maximal chaos (Lyapunov exponent λL=2πTλ_{\mathrm{L}}=2πTλL​=2πT at temperature TTT), and its finite zero-temperature residual entropy. While previous studies have largely focused on couplings drawn from a zero-mean Gaussian distribution, we investigate a generalized model with a finite mean-to-standard-deviation ratio, g≡J0/δJg\equiv J_{0}/δJg≡J0​/δJ of the coupling distribution in order to get deeper insight into the evolution of chaos. We find that increasing ggg yields the following effects: (i) The system remains a fast scrambler with λL=A Tλ_{\mathrm{L}}=A~TλL​=A T, but with a suppressed coefficient A<2πA<2πA<2π. (ii) In the limit g→∞g\to \inftyg→∞, out-of-time-ordered correlators (OTOCs) no longer exhibit exponential growth with λL≃0λ_{\mathrm{L}}\simeq 0λL​≃0. (iii) The spectral correlations indicative of late-time chaos maintain Wigner-Dyson level spacing statistics for all values of ggg. (iv) The system preserves a finite residual entropy, albeit with reduced magnitude, for all ggg values. We conclude that in this generalized SYK model, there is a chaotic to non-chaotic crossover. Moreover different measures of chaos decouple, demonstrating that the presence of finite residual entropy does not strictly imply maximal chaos.
We study the original Sachdev-Ye (SY) model in its Majorana fermion representation which can be called the two indices Sachdev-Ye-Kitaev (SYK) model. Its advantage over the original SY model in the $ SU(M) $ complex fermion representation is that it need no local constraints, so a $1/M $ expansion can be more easily performed. Its advantage over the 4 indices SYK model is that it has only two site indices $ J_{ij} $ instead of four indices $ J_{ijkl} $, so it may fit the bulk string theory better. By performing a $1/M $ expansion at $ N=\infty $, we show that a quantum spin liquid (QSL) state remains stable at a finite $ M $. The $ 1/M $ corrections are exactly marginal, so the system remains conformably invariant at any finite $ M $. The 4-point out of time correlation ( OTOC ) shows quantum chaos neither at $ N=\infty $ at any finite $ M $, nor at $ M=\infty $ at any finite $ N $. By looking at the replica off-diagonal channel, we find there is a quantum spin glass (QSG) instability at an exponentially suppressed temperature in $ M $. We work out a criterion for the two large numbers $ N $ and $ M $ to satisfy so that the QSG instability may be avoided. We speculate that at any finite $ N $, the quantum chaos appears at the order of $ 1/M^{0} $, which is the subleading order in the $ 1/M $ expansion. When the $ 1/N $ quantum fluctuations at any finite $ M $ are considered, from a general reparametrization symmetry breaking point of view, we argue that the eThis work may motivate future works to study the possible new gravity dual of the 2 indices SYK model.ffective action should still be described by the Schwarzian one, the OTOC shows maximal quantum chaos.
Abstract We study the generalization of the Sachdev-Ye-Kitaev (SYK) model to a 1 + 1 dimensional chiral SYK model of N flavors of right-moving chiral Majorana fermions with all-to-all random 4-fermion interactions. The interactions in this model are exactly marginal, leading to an exact scaling symmetry. We show the Schwinger-Dyson equation of this model in the large N limit is exactly solvable. In addition, we show this model is integrable for small N ≤ 6 by bosonization. Surprisingly, the two point function in the large N limit has exactly the same form as that for N = 4, although the four point functions of the two cases are quite different. The ground state entropy in the large N limit is the same as that of N free chiral Majorana fermions, leading to a zero ground state entropy density. The OTOC of the model in the large N limit exhibits a non-trivial spacetime structure reminscent of that found by Gu and Kitaev [1] for generic SYK-like models. Specifically we find a Lyapunov regime inside an asymmetric butterfly cone, which are signatures of quantum chaos, and that the maximal velocity dependent Lyapunov exponent approaches the chaos bound 2π/β as the interaction strength approaches its physical upper bound. Finally, the model is integrable for (at least) N ≤ 6 but chaotic in the large N limit, leading us to conjecture that there is a transition from integrability to chaos as N increases past a critical value.
In this thesis, we discuss the Sachdev-Ye-Kitaev (SYK) model and tensor models with similar properties. The SYK model is a quantum field theoretical model describing N interacting fermions, whose coupling constants are drawn from a Gaussian ensemble. Noteworthy properties of the SYK model include that it is analytically solvable in the large N limit, that it exhibits conformal symmetry at low energies and that it is maximally chaotic. These properties are remarkably similar to those of a 1 + 1 dimensional Schwarzschild black hole. It has been conjectured the SYK model is a holographic dual to the black hole. We introduce a set of Feynman rules for the SYK model. Using these rules, we show that in the large N limit the diagrams that contribute to the two-point function are all so-called iterated melonic diagrams. This allows us to derive a Schwinger-Dyson equation for the two-point function, which, in turn, can be solved exactly in the infrared limit. We also consider the four-point function. In the large N limit, the leading-order correction to the four-point function is given by so-called ladder diagrams. This allows us to derive an explicit expression for the four-point function. The SYK model can be generalized in a few different ways. In this thesis, we consider the generalization where the fermions act through q-fold interactions instead of quartic interactions present in the original SYK model. In particular, considerable simplifications can be achieved in the q → ∞ limit or q = 2 case, which we study. While the SYK model has many interesting properties, its random couplings limit its usability especially as a dual to a Schwarzschild black hole. We therefore also consider tensor models which do not have this drawback but manage to preserve the interesting properties of the SYK model. In the last chapter, we briefly inspect the chaotic behaviour of the SYK and tensor models and derive Lyapunov exponent for them. It can be shown that the expression saturates an upper bound for Lyapunov exponents of a large class of quantum systems, including large N systems.
The growth of information scrambling, captured by out-of-time-order correlation functions (OTOCs), is a central indicator of the nature of many-body quantum dynamics. Here, we compute analytically the complete time dependence of the OTOC for an integrable Sachdev-Ye-Kitaev (SYK) model, N Majoranas with random two-body interactions of infinite range, coupled to a Markovian bath at finite temperature. In the limit of no coupling to the bath, the time evolution of scrambling experiences different stages. For t≲sqrt[N], after an initial polynomial growth, the OTOC approaches saturation in a power-law fashion with oscillations superimposed. At t∼sqrt[N], the OTOC reverses trend and starts to decrease linearly in time. The reason for this linear decrease is that, despite being a subleading 1/N effect, the OTOC in this region is governed by the spectral form factor of the antisymmetric couplings of the SYK model. The linear decrease stops at t∼2N, the Heisenberg time, where saturation occurs. The effect of the environment is an overall exponential decay of the OTOC for times longer than the inverse of the coupling strength to the bath. The oscillations at t≲sqrt[N] indicate lack of thermalization-a desired feature for better performance of quantum information devices.
The Sachdev-Ye-Kitaev (SYK) model is a quantum-mechanical model of NNN Majorana fermions which displays a number of appealing features---solvability in the strong coupling regime, near-conformal invariance and maximal chaos---which make it a suitable model for black holes in the context of the AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT holography. In this paper, we show for the colored SYK model and several of its tensor model cousins that the next-to-leading order in the large-NNN expansion preserves the conformal invariance of the two-point function in the strong-coupling regime, up to the contribution of the pseudo-Goldstone bosons due to the explicit breaking of the symmetry which are already seen in the leading-order four-point function. We also comment on the composite field approach for computing correlation functions in colored tensor models.