D-commuting SYK model: building quantum chaos from integrable blocks

TL;DR

The paper introduces the dcSYK model by summing $d$ commuting SYK blocks into $\tilde{H}=\frac{1}{\sqrt{d}}\sum_{a=1}^d\tilde{H}_a$, producing a tunable integrability-to-chaos transition as $d$ increases. Using multi-color chord diagrams and free probability at $q=0$, the authors obtain exact spectral insights and show a noncompact spectrum for finite $d$, recovering a compact, double-scaled SYK-like spectrum as $d\to\infty$. In the Schwarzian regime ($q\to1$), a coarse-grained path integral yields a 1-loop spectrum with a Schwarzian edge and a nonperturbative exponential tail characterized by a critical temperature $T_c$ that decreases with $d$, indicating a chaos-dominated regime above $T_c$. Numerical diagonalization and spectral-form-factor analyses corroborate the analytic findings, revealing a two-phase spectral structure and a chaotic signature above $T_c$. The framework provides a practical diagnostic for chaos in generic local Hamiltonians via decomposition into commuting blocks and offers a route to exploring holographic-like dynamics in quantum simulations.

Abstract

We construct a new family of quantum chaotic models by combining multiple copies of integrable commuting SYK models. As each copy of the commuting SYK model does not commute with others, this construction breaks the integrability of each commuting SYK and the family of models demonstrates the emergence of quantum chaos. We study the spectrum of this model analytically in the double-scaled limit. As the number of copies tends to infinity, the spectrum becomes compact and equivalent to the regular SYK model. For finite $d$ copies, the spectrum is close to the regular SYK model in UV but has an exponential tail $e^{E/T_c}$ in the IR. We identify the reciprocal of the exponent in the tail as a critical temperature $T_c$, above which the model should be quantum chaotic. $T_c$ monotonically decreases as $d$ increases, which expands the chaotic regime over the non-chaotic regime. We propose the existence of a new phase around $T_c$, and the dynamics should be very different in two phases. We further carry out numeric analysis at finite $d$, which supports our proposal. Given any finite dimensional local Hamiltonian, by decomposing it into $d$ groups, in which all terms in one group commute with each other but terms from different groups may not, our analysis can give an estimate of the critical temperature for quantum chaos based on the decomposition. We also comment on the implication of the critical temperature to future quantum simulations of quantum chaos and quantum gravity.

Figures (16)

• Figure 1: (a) A generic chord diagram within $M_{10}$, where each end of a chord represents a $\tilde{H}_a$ in the trace, and each chord represents two contracted $\tilde{H}_a$ with identical index $a$. Different $a$ are labeled by different colors. (b) The three chord diagrams for $M_4$.
• Figure 2: Cut open a chord diagram. To generate a $n$-chord diagram, we need $2n$ steps.
• Figure 3: The spectrum for $q=0$. In the plot, blue, yellow, green, red and gray are for $d=1,3,6,15,\infty$ respectively.
• Figure 4: The edge of the spectrum for $q=0$. In each figure, the solid line is the exact result, the dashed line is the large $d$ perturbative solution with square-root law, and the dotted line is the non-perturbative asymptotic solution with exponential suppresion. The figures with blue, yellow, green, red and purple curves are for $d=2,5,10,20,40$ respectively.
• Figure 5: (a) The boundary of a $d=3$ color chord diagram is split into $s$ segments. (b) For the open chords from the $i$-th segment, there are three steps before landing on the $j$-th and $k$-th segments: generate, split and reorder.

Theorems & Definitions (11)

• Definition
• Definition
• Definition
• Definition
• Theorem 1
• Corollary 1.1
• Theorem 2
• Lemma 3
• Lemma 4
• Proposition 5