We construct an effective field theory (EFT) that captures the universal behavior of out-of-time-order correlators (OTOCs) at late times in generic quantum many-body systems with conservation laws. The EFT hinges on a generalization of the strong-to-weak spontaneous symmetry breaking pattern adapted to out-of-time-order observables, and reduces to conventional fluctuating hydrodynamics when time-ordered observables are probed. We use the EFT to explain different power-law behavior observed in OTOCs at late times, and show that many OTOCs are entirely fixed by conventional transport data at leading order. Nevertheless, we show that a specific combination of OTOCs is sensitive to novel transport parameters, not visible in regular time-ordered correlators. We test our predictions in Hamiltonian and Floquet spin chains in one spatial dimension.
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11+9 pages, 13 figures, (v2: fixed Fig. 6, added appendix and references, v3: published version)
We construct an effective field theory (EFT) that captures the universal behavior of out-of-time-order correlators (OTOCs) at late times in generic quantum many-body systems with conservation laws. The EFT hinges on a generalization of the strong-to-weak spontaneous symmetry breaking pattern adapted to out-of-time-order observables, and reduces to conventional fluctuating hydrodynamics when time-ordered observables are probed. We use the EFT to explain different power-law behavior observed in OTOCs at late times, and show that many OTOCs are entirely fixed by conventional transport data at leading order. Nevertheless, we show that a specific combination of OTOCs is sensitive to novel transport parameters, not visible in regular time-ordered correlators. We test our predictions in Hamiltonian and Floquet spin chains in one spatial dimension.
Hydrodynamics as an effective field theory (EFT) has been immensely successful for predicting the behaviour of time-ordered correlation functions in the long-time, large-distance regime. In this paper, we analyse the predictions of hydrodynamics for scrambling as probed by out-of-time order correlators (OTOCs). To do so, we consider an all-derivative order generalisation of the Schwinger-Keldysh EFT of hydrodynamics extended to a four-fold time contour required to capture OTOCs. A crucial ingredient of the construction is a local SL(2,R) symmetry, which has previously been suggested as a mechanism for scrambling. We also derive this exact EFT from the microscopic formulation of the maximally chaotic low-temperature limit of the SYK lattice, thereby determining all transport coefficients for this model. As a novel application, we compute the OTOC of energy currents, find that its exponential growth saturates the chaos bound, and match it to expectations based on a shockwave (or scramblon) effective description. Our work shows that the universality of hydrodynamics degrees of freedom extends beyond traditional time-ordered correlators to more complex observables such as OTOCs.
Fluctuating hydrodynamics emerges in essentially any local many-body system at nonzero temperature. Effective field theory (EFT) approaches enable the quantitative study of this emergence, providing a controlled framework to capture late-time observables. These lectures introduce the organizing principles behind equilibrium and out-of-equilibrium dynamics in these thermalizing systems. A central focus is the modern construction of these EFTs, which frames fluctuating hydrodynamics through the lens of "strong-to-weak" spontaneous symmetry breaking. Drawing examples from both high-energy and condensed matter physics, we show how this paradigm adapts to systems ranging from spin chains to relativistic quantum field theories, including models with generalized symmetries or symmetries with 't Hooft anomalies. Finally, we discuss UV/IR constraints on transport parameters -- viewed as the Wilson coefficients of hydrodynamic EFTs -- both in continuum and on the lattice.
Abstract In non-maximally quantum chaotic systems, the exponential behavior of out-of-time-ordered correlators (OTOCs) results from summing over exchanges of an infinite tower of higher “spin” operators. We construct an effective field theory (EFT) to capture these exchanges in (0 + 1) dimensions. The EFT generalizes the one for maximally chaotic systems, and reduces to it in the limit of maximal chaos. The theory predicts the general structure of OTOCs both at leading order in the 1/N expansion (N is the number of degrees of freedom), and after resuming over an infinite number of higher order 1/N corrections. These general results agree with those previously explicitly obtained in specific models. We also show that the general structure of the EFT can be extracted from the large q SYK model.
Operator scrambling is commonly diagnosed by the growth of out-of-time-ordered correlators (OTOCs), yet a general symmetry principle underlying their effective dynamics has remained elusive. For Brownian or short-time-correlated large-N Majorana systems, we develop a symmetry-based effective field theory for operator scrambling, organized by a strong-to-weak U(1) symmetry breaking in operator space. The key observation is that, in the noninteracting fermion limit, the four-fold Keldysh contour representation of an OTOC admits an emergent strong U(1) symmetry in a doubled Hilbert-space description, even when the original system has no ordinary conserved quantity. The associated slow mode is the phase of the strong-charge creation operator, whose conjugate density is identified with the local operator size. Generic interactions explicitly break the strong symmetry and generate a mass term at lowest order for the would-be Goldstone mode, thereby converting diffusive operator spreading into chaotic growth. We further show that higher-order symmetry breaking terms are tightly constrained by an emergent duality that combines time reversal with contour permutation. This duality fixes the effective action up to quadratic order in the response field, relates the multiplicative noise strength directly to the Lyapunov exponent, and makes the positivity of the Lyapunov exponent a consequence of convergence of the real-time path integral. The resulting OTOC dynamics is governed by a noisy FKPP equation, which captures within a unified framework the early-time exponential growth, ballistic propagation, nonlinear saturation, and stochastic front broadening of operator scrambling. We verify this construction in a Brownian SYK chain, where a direct saddle-point expansion reproduces the symmetry-based effective action. Our results reveal a symmetry origin of operator-size hydrodynamics and scrambling.
Out-of-time-order correlators (OTOCs) are central probes of quantum scrambling, and their generalizations have recently become key primitives for both benchmarking quantum advantage and learning the structure of Hamiltonians. Yet their behavior has lacked a unified algorithmic interpretation. We show that higher-order OTOCs naturally fit within the framework of quantum signal processing (QSP): each OTOC(k) measures the 2k-th Fourier component of the phase distribution associated with the singular values of a spatially resolved truncated propagator. This explains the contrasting sensitivities of time-ordered correlators (TOCs) and higher-order OTOCs to causal-cone structure and to chaotic, integrable, or localized dynamics. Based on this understanding, we further generalize higher-order OTOCs by polynomial transformation of the singular values of the spatially resolved truncated propagator. The resultant signal allows us to construct frequency-selective filters, which we call \emph{OTOC spectroscopy}. This extends conventional OTOCs into a mode-resolved tool for probing scrambling and spectral structure of quantum many-body dynamics.
The scrambling of quantum information in closed many-body systems, as measured by out-of-timeordered correlation functions (OTOCs), has received considerable attention lately.Recently, a hydrodynamical description of OTOCs has emerged from considering random local circuits.Numerical work suggests that aspects of this description are universal to ergodic many-body systems, even without randomness; a conjectured explanation for this is that while the random circuits have noise built into them, deterministic quantum systems, much like classically chaotic ones, "generate their own noise" and look effectively random on sufficient length scales and timescales.In this paper, we extend this approach to systems with locally conserved quantities (e.g., energy).We do this by considering local random unitary circuits with a conserved U(1) charge and argue, with numerical and analytical evidence, that the presence of a conservation law slows relaxation in both time-ordered and out-of-time-ordered correlation functions; both can have a diffusively relaxing component or "hydrodynamic tail" at late times.We verify the presence of such tails also in a deterministic, periodically driven system.We show that for OTOCs, the combination of diffusive and ballistic components leads to a wave front with a specific asymmetric shape, decaying as a power law behind the front.These results also explain existing numerical investigations in non-noisy ergodic systems with energy conservation.Moreover, we consider OTOCs in Gibbs states, parametrized by a chemical potential μ, and apply perturbative arguments to show that for μ ≫ 1 the ballistic front of information spreading can only develop at times exponentially large in μ-with the information traveling diffusively at earlier times.We also develop a new formalism for describing OTOCs and operator spreading, which allows us to interpret the saturation of OTOCs as a form of thermalization on the Hilbert space of operators.