Sparse SYK Model: Chaos & Holography

• Sparse SYK model is a variant of the SYK framework that dilutes q-body interactions to O(N) terms while retaining key features like emergent conformal symmetry and quantum chaos.
• The model exhibits a phase transition at connectivity k ≳ 1, where level statistics transition to universal random matrix behavior and the Lyapunov exponent reaches the chaos bound.
• Experimental implementations and quantum simulation techniques for sparse SYK enable scalable studies, linking strange metal behavior with holographic duality and quantum gravity analogs.

The sparse Sachdev-Ye-Kitaev (SYK) model is a variant of the canonical SYK model, constructed to capture strong quantum correlations and maximal chaos in systems with a reduced, sub-extensive set of random interactions. Unlike the classical SYK model—which assumes all-to-all random q-body couplings among N Majorana fermions—the sparse SYK model restricts the number of nonzero Hamiltonian terms, often to $\mathcal{O}(N)$, while preserving key features such as emergent conformal symmetry, spectral correlations, and many-body quantum chaos. This paradigm has opened new directions in quantum many-body physics, quantum simulation, and the paper of holographic duality, driven by its theoretical flexibility and experimental feasibility.

1. Construction and Definition

The sparse SYK model generalizes the original SYK framework by introducing controlled dilution of Hamiltonian terms. The canonical $q$-body SYK Hamiltonian is: $$H_{\mathrm{SYK}} = i^{q/2} \sum_{1 \leq i_1 < \cdots < i_q \leq N} J_{i_1\cdots i_q} \psi_{i_1} \cdots \psi_{i_q}$$ where $\psi_i$ are Majorana fermions and $J_{i_1\cdots i_q}$ are all-to-all independent random couplings with variance scaling as $J_q^2 (q-1)! / N^{q-1}$. In the sparse variant, the sum is restricted by projectors $p_{i_1\cdots i_q} \in \{0,1\}$ such that

$$H_{\mathrm{sparse}} = i^{q/2} \sum_{i_1<...<i_q} p_{i_1...i_q} J_{i_1...i_q} \psi_{i_1} ... \psi_{i_q}$$

The sparseness is quantified by $$\langle p_{i_1...i_q}\rangle = p \sim kN / {N\choose q}$$, where $k$ becomes the tunable “connectivity” parameter (García-García et al., 2020, Xu et al., 2020, Caceres et al., 2021). Typical constructions include:

• Random pruning: Randomly select a fixed number $\sim kN$ of $q$-body terms per realization.
• Regular hypergraph: Each fermion participates in exactly $k q$ $q$-body terms, constructing a $(kq, q)$-regular random hypergraph of interactions.
• Binary coupling variant: The nonzero couplings $J_{i_1\cdots i_q}$ are set to $\pm 1$ with equal probability, further simplifying the disorder structure (Tezuka et al., 2022).

The coupling variance for each retained term is rescaled to preserve the “melonic” planar diagrammatics and the correct large-$N$ limit: $$\langle J_{i_1...i_q}^2 \rangle \sim \frac{(q-1)! J^2}{N^{q-1} p}$$

2. Emergence of Quantum Chaos and Minimum Connectivity

A central question in the sparse SYK model is identifying conditions for the emergence of quantum chaos, as diagnosed by random matrix theory (RMT) statistics of the spectrum and out-of-time-order correlators (OTOCs). It is established that quantum chaos is present for $k \gtrsim 1$, where $k$ is the normalized number of terms per fermion (García-García et al., 2020, Xu et al., 2020, García-García et al., 2023):

• For $k$ below unity, level statistics deviate from Wigner-Dyson distributions, degeneracies proliferate, and the system exhibits features of integrable or fragmented behavior.
• For $k \gtrsim 1$, short-range level correlations conform to universal RMT classes and the many-body Lyapunov exponent $\lambda_L$ (extracted from the exponential decay of OTOCs) saturates the chaos bound, $\lambda_L = 2\pi T$ in the low-temperature limit, just as in the dense SYK model (García-García et al., 2023).

The connectivity threshold thus demarcates a quantum phase transition: below $k_c$ the system behaves as a disordered Fermi liquid with Poisson level statistics; above $k_c$, it realizes the maximally chaotic, non-Fermi liquid regime with Schwarzian low-energy dynamics, preserving essential features needed for a gravity dual.

3. Spectral Properties, Global Symmetries, and Quantum Gravity Dual

The spectral density of the sparse SYK model, for $k \gtrsim 1$, matches the dense SYK's Q-Hermite form in the infrared regime: after appropriate renormalization (e.g., $\eta(k) = \eta + 3/(kN)$), it exhibits the same low-energy Schwarzian tail, signaling proximity to nearly AdS$_2$ Jackiw-Teitelboim gravity (García-García et al., 2020). Key points:

• Leading spectral moments are unaltered; subleading corrections appear at $\mathcal{O}(1/(kN))$ and can be related exactly to corrections in high-dimensional lattice gauge theories (García-García et al., 2020).
• When $k$ approaches unity, emergent global symmetries (“accidental” commuting operators), and with them $2^m$-fold degeneracies (e.g., $m$ as high as 8 for $N=26$) may arise in many disorder realizations, increasing macroscopic spectral fluctuations and occasionally altering global level statistics (García-García et al., 2020).

For all $k \gtrsim 1$, the system retains a robust quantum gravity dual, as indicated by:

• Schwarzian-like low-energy spectral density.
• Maximal Lyapunov exponent and fast scrambling.
• Gapped spectrum for coupled twin-SYK setups (modeling traversable wormholes) where the ground state approximates a thermofield double (Caceres et al., 2021).

4. Analytical and Numerical Techniques

The main theoretical treatment uses large-$N$ disorder averaging and the path integral formalism, incorporating both the replica trick (to compute entanglement and disorder averages) and diagrammatic analysis:

• Replica-diagonal saddle points in the path integral capture the universal low-energy conformal physics, with the same dynamical exponents and soft Schwarzian mode.
• In sparse models, fluctuations about the saddle may have additional $1/k$ corrections, but numerical results up to $N=52$ show strong agreement with dense SYK for $k \gtrsim 4$ (Xu et al., 2020).
• Corrections due to sparse connectivity in observables (e.g., entanglement entropy, level statistics) scale as powers of $1/k$, with deviations vanishing in the dense limit.

Numerical diagonalization is greatly facilitated by order-of-magnitude reduction in Hamiltonian size: only $\sim kN$ terms need to be stored and summed, permitting studies of much larger $N$ (Xu et al., 2020, Caceres et al., 2021). Krylov subspace/time-evolution methods, GPU-accelerated matrix-free algorithms, and advanced quantum simulation protocols (e.g., TETRIS, randomized sampling, mirror circuit benchmarking) are actively employed to simulate sparse SYK dynamics on both classical and quantum hardware (Granet et al., 10 Jul 2025, García-García et al., 2023).

5. Physical Realizations and Experimental Proposals

Sparse SYK models are designed with experimental implementation in mind. Realistic proposals and platforms include:

• Majorana Wire Arrays + Quantum Dot: An N-wire array with only one Majorana mode per pair hybridized to a common disordered dot generates the required random couplings; time-reversal symmetry suppresses bilinear terms, and quantum dot size determines the effective $k$ (Chew et al., 2017).
• Quantum Dots in Topological Insulators: Topological insulator flakes under inhomogeneous magnetic field and Coulomb interactions yield the required random coupling of surface zero modes, modeling both standard and spinful/supersymmetric SYK (Lantagne-Hurtubise et al., 2020).
• Multimode and Single-Mode Cavity QED: Arrays of ultracold atoms in optical cavities, with photon-mediated all-to-all coupling and fast time-dependent disorder cycling (Trotterization) densify effective sparse couplings, enabling scalable and programmable realization of SYK-type Hamiltonians (Uhrich et al., 2023, Baumgartner et al., 26 Nov 2024).
• Bosonic Spin Chains: Kitaev spin chains with perturbations generating random four-fermion interactions among nonlocal Majorana zero modes, mapping the effective bosonic dynamics onto the SYK model (Zuo et al., 12 Dec 2024).

Discretization or binarization of the couplings (i.e., allowing only a small finite set of values, even just $\pm 1$) does not destroy the chaotic phase; it can, in fact, stabilize RMT correlations more efficiently and simplify the requirements for quantum simulation (Tezuka et al., 2022, Cao et al., 2020).

6. Information-theoretic Diagnostics and Quantum Simulation Algorithms

Sophisticated diagnostics quantify the convergence of sparse to dense SYK variants by evaluating information-theoretic distance between coupling distributions (Kullback-Leibler divergence), with convergence scaling $D_{\mathrm{KL}} \sim 1/R^2$ for $R$ independent disorder realizations under Trotterization (Baumgartner et al., 26 Nov 2024). Quantum algorithms tailored to the sparse SYK setting include:

• TETRIS (Time Evolution Through Random Independent Sampling): Randomized application of sparse Pauli-string exponentials, where the statistical average over random circuits reproduces the target time–evolution operator, allowing simulation of large-$N$ sparse SYK on trapped ion processors (Granet et al., 10 Jul 2025).
• Asymmetric Qubitization: Efficient quantum simulation via sampling random circuit unitaries and controlled application of Pauli strings, reducing gate complexity to $$O(N^{7/2} t + N^{5/2} t\ \mathrm{polylog}(N/\epsilon))$$ (Babbush et al., 2018).
• Mirror-on-average Benchmarks: Circuit fidelity assessed using randomized twin circuits, with scalable verification schemes more sensitive to local operator errors relevant for dynamics and scrambling (Granet et al., 10 Jul 2025).

These methods are critical for overcoming classical intractability in simulating quantum chaos and for benchmarking the reliability of quantum hardware in the regime of maximal entropy production and information scrambling.

7. Significance for Strange Metal Behavior, Quantum Gravity, and Outlook

Sparse SYK models preserve the non-quasiparticle, strange metal phenomenology (Planckian dissipation, universal linear-$T$ resistivity, extensive zero-temperature entropy) that endows the dense model with theoretical significance in both condensed matter and gravitational contexts (Jha, 9 Jul 2025, Sachdev, 27 Feb 2024). The robustness of maximal chaos, Schwarzian low-energy dynamics, and emergent gravitational physics even in the sparsified regime demonstrates:

• The universality of holographic correspondence and quantum gravity analogues beyond fully connected models.
• New insights into the transition from chaos to integrability as system connectivity ($k$) decreases, with direct control over quantum phase transitions and emergence of symmetries/degeneracies in physical systems (Lantagne-Hurtubise et al., 2018, García-García et al., 2020).
• Scalability for quantum simulation, with prospects for laboratory realization of black hole-like physics and traversable wormholes in tunable, many-body quantum platforms (Caceres et al., 2021).

Sparse SYK thus provides a general framework connecting solvable quantum chaos, holography, and quantum simulation on both theoretical and experimental fronts. The detailed understanding of how sparsity-controlled connectivity modulates chaos, thermalization, and emergent symmetries continues to drive advancements in both foundational and applied quantum research.