Conformal field theory and the web of quantum chaos diagnostics

TL;DR

The paper develops a torus-partition-function–based dictionary linking three key chaos diagnostics in 2D CFTs: the spectral form factor, out-of-time-ordered correlators, and operator entanglement. By performing different analytic continuations and leveraging modular data, it classifies scrambling behavior across free theories, rational and irrational CFTs, compactified bosons, and holographic CFTs, highlighting how the diagnostics diverge or align as integrability breaks down toward maximal chaos. It provides explicit late-time limits and scaling relations for OTOCs and TOMI in RCFTs (e.g., minimal models and SU(2)$_k$), shows how irrational theories generically exhibit infinite recurrence times and distinct decay patterns, and analyzes holographic CFTs where TOMI signals extensive scrambling yet may not saturate universal bounds. The work clarifies the complementary information captured by SFF, OTOC, and TOMI, offering a unified framework that connects spectral statistics, operator dynamics, and information scrambling in QFT and gravity duals with practical implications for diagnosing chaos in strongly correlated quantum systems.

Abstract

We study three prominent diagnostics of chaos and scrambling in the context of two-dimensional conformal field theory: the spectral form factor, out-of-time-ordered correlators, and unitary operator entanglement. With the observation that all three quantities may be obtained by different analytic continuations of the torus partition function, we address the connections and distinctions between the information that each quantity provides us. In this process, we study the emergence of irrationality from "large-N" limits of rational conformal field theories (RCFTs) as well as the explicit breakdown of rationality for theories with central charges greater than the number of their conserved currents. Our analysis begins to elucidate the intermediate dynamical behavior of theories that bridge the gap between integrable RCFTs and maximally chaotic holographic CFTs.

Figures (13)

• Figure 1: We lay out a web of chaos diagnostics. In this paper, we focus on the three highlighted in red. The numbered connections have been studied as follows: (1-3) Related in this paper through analytic continuations of the torus partition function. (1) Higher point OTOCs were related to higher-point spectral form factors in Ref. 2017JHEP...11..048C and averaged OTOCs were further equated with the SFF in quantum field theory in Refs. 2019PhLB..795..183D2019arXiv190704289M. (3) In spin systems, the Rényi operator mutual information is also directly related to the average OTOC 2016JHEP...02..004H. (4,6-7) Rényi entropy of disjoint intervals after a global quantum quench may also be computed by a particular analytic continuation of the torus partition function 2015JHEP...09..110A, though we do not focus on this in this paper. (5) The OTOC was equated with the thermally averaged Loschmidt echo in Refs. 2019arXiv190302651Y2019JHEP...07..107R. This was further explored in Ref. 2019arXiv190901894B. (8) The relative entropy of perturbed thermal states was related to OTOC in Ref. 2018JHEP...07..002N. (9) $2k$-point OTOCs were equated with $k^{th}$ frame potentials in Ref. 2017JHEP...04..121R. (10-12) The rate of growth of Lanczos coefficients, a notion of operator complexity was shown to bound the Lyapunov exponent of OTOCs in Ref. 2018arXiv181208657P. Both the bounds on this rate and the Lyapunov exponent were shown to follow from the eigenstate thermalization hypothesis in Ref. 2019arXiv190610808M. (13) The circuit complexity was suggested to be related to the logarithm of the Loschmidt echo in Ref. 2019arXiv190901894B. (14) The frame potentials were shown to bound the circuit complexity in Ref. 2017JHEP...11..048C. (15-16) The entanglement content of a Heisenberg time-evolved local operator may be shown to be related to relative entropy of excited states and the OTOC of the local operator with twist fields 2020arXiv200514243K
• Figure 2: General setup of operator mutual information. $\mathcal{H}_{1,2}$ are the spatially 1-dimensional Hilbert spaces represented by the horizontal black lines. The subsystems are represented by the blue lines. The grey area in between the two Hilbert spaces indicates time evolution dictated by $U_\epsilon (t)$, with the arrow of time $t$ pointing from $\mathcal{H}_{\hbox{\tiny input}}$ to $\mathcal{H}_{\hbox{\tiny output}}$. (a) Bi-partite case, (b) Tri-partite case. For TOMI, we generally take $B_1$ and $B_2$ to be a bipartition of the entire output Hilbert space.
• Figure 3: Spectral form factor for a single free fermion with $\beta = .5$. The function is clearly periodic and does not demonstrate anything close to random matrix statistics. The recurrence time is $2\pi$.
• Figure 4: A cartoon of the quasi-particle picture for operator entanglement. At $t = 0$, the quasi-particles (yellow) are emitted from region $A$ and move at the speed of light. The quasi-particle picture for operator entanglement dictates that the operator mutual information between input region $A$ and output region $B$ is the number of quasi-particles located within region $B$ that originated from $A$ (the overlap section between region B and the yellow area). On the left, we display a sketch of the operator mutual information between $A$ and $B$ for the partially overlapping configuration: after a certain time $I^{(2)}(A,B)$ starts to decrease due to the quasi-particles leaving region $B$.
• Figure 5: The spectral form factor (blue) and its progressive time-averaged version (yellow) for the unitary minimal models with $m = 5$ (left) and $m=50$ (right). We see the emergence of the dip, ramp, and plateau as we increase $m$. While the peaks in the non-averaged function grow linearly, the time-averaged function is sublinear. This can be seen by the green lines which are linear in $t$. The insets show the functions on linearly scaled axes.