Abstract We study the distance of two wave functions under chaotic time evolution. The two initial states differ only by a local perturbation. To be entitled ‘chaos’ the distance should have a rapid growth afterwards. Instead of focusing on the entire wave function, we measure the distance d 2 ( t ) by investigating the difference of two reduced density matrices of the subsystem A that is spatially separated from the local perturbation. This distance d 2 ( t ) grows with time and eventually saturates to a small constant. We interpret the distance growth in terms of operator scrambling picture, which relates d 2 ( t ) to the square of commutator (out-of-time-order correlator) and shows that both these quantities measure the area of the operator wave front in subsystem A . Among various one-dimensional spin- models, we numerically show that the models with non-local power-law interaction can have an exponentially growing regime in d 2 ( t ) when the local perturbation and subsystem A are well separated. This so-called Lyapunov regime is absent in the spin- chain with local interaction only. After sufficiently long time evolution, d 2 ( t ) relaxes to a small constant, which decays exponentially as we increase the system size and is consistent with eigenstate thermalization hypothesis. Based on these results, we demonstrate that d 2 ( t ) is a useful quantity to characterize both quantum chaos and quantum thermalization in many-body wave functions.
We investigate operator growth in a Brownian spin Sachdev--Ye--Kitaev (SYK) model with random all-to-all interactions, focusing on the full operator-size distribution. For Hamiltonians containing interactions of order two up to
L, we derive a closed master equation for the Pauli-string expansion coefficients and recast their dynamics into a generating-function formulation suitable for the large-
N limit. This approach allows us to diagonalize the leading-order evolution operator explicitly and obtain exact solutions for arbitrary initial operator distributions, including the effects of decoherence. Going beyond leading order, we develop a systematic
1/N expansion that captures higher-order corrections to the operator-size dynamics and the late-time behavior. Our results demonstrate that higher-order effects play a crucial role in operator scrambling and that the full operator-size distribution provides a more refined probe of quantum chaos in Brownian and open quantum systems.
The Ising spin chain with longitudinal and transverse magnetic fields is often used in studies of quantum chaos, displaying both chaotic and integrable regions in its parameter space. However, even at a strongly chaotic point this model does not exhibit Lyapunov growth of the commutator squared of spin operators, as this observable saturates before exponential growth can manifest itself (even in situations where a spatial suppression factor makes the initial commutator small). We extend this model from the spin 1/2 Ising model to higher spins, demonstrate numerically that a window of exponential growth opens up for sufficiently large spin, and extract a quantity which corresponds to a notion of a Lyapunov exponent. In the classical infinite-spin limit, we identify and compute the appropriate classical analog of the commutator squared, and show that the corresponding exponent agrees with the infinite-spin limit extracted from the quantum spin chain.
Out-of-Time-Ordered Commutators (OTOCs), representing a key diagnostic for scrambling as a facet of short-time quantum chaos, have attracted wide-ranging interest, from many-body physics to quantum gravity. By means of a suitable form of the Wigner-Moyal expansion, and invoking ensemble equivalence in statistical physics, we provide a consistent approach to the growth rate of the OTOC for many-body systems with chaotic classical limit where both the classical Lyapunov exponent and the quantum nature of the density of states enter. Applying this construction to quantized high-dimensional hyperbolic motion, i.e., a quantum chaotic system that exhibits gravity-like correlation functions in the late-time regime, we compute the OTOC growth rate
Λ as a function of the number of degrees of freedom,
f, and inverse temperature,
β.
We show that the scaled growth rate,
Λ/f, can be described by a universal function of
fβ and displays a cross-over from classical to quantum behavior as we increase
f and/or lower the temperature. In the deep quantum regime of infinite
f, we find maximally fast scrambling in the sense of the Maldacena-Shenker-Stanford bound on chaos. This elucidates the non-perturbative mechanism underlying the saturation of the bound via quantum contributions to the mean density of states, and it provides further support for this dynamical system as a dual to two-dimensional quantum gravity. In this way, we present first evidence of maximally fast scrambling in a quantum chaotic system with a well-defined classical Hamiltonian limit, without invoking any external mechanism such as (disorder) averaging.
Operator scrambling is a crucial ingredient of quantum chaos. Specifically, in the quantum chaotic system, a simple operator can become increasingly complicated under unitary time evolution. This can be diagnosed by various measures such as square of the commutator (out-of-time-ordered correlator), operator entanglement entropy etc. In this paper, we discuss operator scrambling in three representative models: a chaotic spin-
1/2 chain with spatially local interactions, a 2-local spin model and the quantum linear map. In the first two examples, although the speeds of scrambling are quite different, a simple Pauli spin operator can eventually approach a "highly entangled" operator with operator entanglement entropy taking a volume law value (close to the Page value). Meanwhile, the spectrum of the operator reduced density matrix develops a universal spectral correlation which can be characterized by the Wishart random matrix ensemble. In the second example, we further connect the 2-local model into a one dimensional chain and briefly discuss the operator scrambling there. In contrast, in the quantum linear map, although the square of commutator can increase exponentially with time, a simple operator does not scramble but performs chaotic motion in the operator basis space determined by the classical linear map. We show that once we modify the quantum linear map such that operator can mix in the operator basis, the operator entanglement entropy can grow and eventually saturate to its Page value, thus making it a truly quantum chaotic model.
We give a detailed theory for the leading coarse-grained dynamics of entanglement entropy of states and of operators in generic short-range interacting quantum many-body systems. This includes operators spreading under Heisenberg time evolution, which we find are much less entangled than "typical" operators of the same spatial support. Extending previous conjectures based on random circuit dynamics, we provide evidence that the leading-order entanglement dynamics of a given chaotic system are determined by a function
E(v), which is model-dependent, but which we argue satisfies certain general constraints. In a minimal membrane picture,
E(v) is the "surface tension" of the membrane and is a function of the membrane's orientation
v in spacetime. For one-dimensional (1D) systems this surface tension is related by a Legendre transformation to an entanglement entropy growth rate
Γ(∂S/∂x) which depends on the spatial "gradient" of the entanglement entropy
S(x,t) across the cut at position
x. We show how to extract the entanglement growth functions numerically in 1D at infinite temperature using the concept of the operator entanglement of the time evolution operator, and we discuss possible universality of
E at low temperatures. Our theoretical ideas are tested against and informed by numerical results for a quantum-chaotic 1D spin Hamiltonian. These results are relevant to the broad class of chaotic many-particle systems or field theories with spatially local interactions, both in 1D and above.