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A bstract We study the mutual information between pairs of regions on the two asymptotic boundaries of maximally extended anisotropic black branes. This quantity characterizes the local pattern of entanglement of the thermofield double states which are dual to these geometries. We analyze the disruption of the mutual information in anisotropic shock wave geometries and show that the entanglement velocity plays an important role in this phenomenon. Moreover, we compute several chaos-related properties of this system, such as the entanglement velocity, the butterfly velocity, and the scrambling time. We find that the butterfly velocity and the entanglement velocity violate the upper bounds proposed in [1-3], but remain bounded by their corresponding values in the infrared effective theory.

We review recent developments encompassing the description of quantum chaos in holography. We discuss the characterization of quantum chaos based on the late time vanishing of out-of-time-order correlators and explain how this is realized in the dual gravitational description. We also review the connections of chaos with the spreading of quantum entanglement and diffusion phenomena.

A bstract We study a notion of operator growth known as Krylov complexity in free and interacting massive scalar quantum field theories in d -dimensions at finite temperature. We consider the effects of mass, one-loop self-energy due to perturbative interactions, and finite ultraviolet cutoffs in continuous momentum space. These deformations change the behavior of Lanczos coefficients and Krylov complexity and induce effects such as the “staggering” of the former into two families, a decrease in the exponential growth rate of the latter, and transitions in their asymptotic behavior. We also discuss the relation between the existence of a mass gap and the property of staggering, and the relation between our ultraviolet cutoffs in continuous theories and lattice theories.

A bstract We study out-of-time-order correlators (OTOCs) of rotating BTZ black holes using two different approaches: the elastic eikonal gravity approximation, and the Chern-Simons formulations of 3-dimensional gravity. Within both methods the OTOC is given as a sum of two contributions, corresponding to left and right moving modes. The contributions have different Lyapunov exponents, λ L ± = 2 π β 1 1 ∓ ℓ Ω , where Ω is the angular velocity and ℓ is the AdS radius. Since λ L − ≤ 2 π β ≤ λ L + , there is an apparent contradiction with the chaos bound. We discuss how the result can be made consistent with the chaos bound if one views the parameters β ± = β (1 ∓ ℓ Ω) as the effective inverse temperatures of the left and right moving modes.

A bstract We clarify general mathematical and physical properties of pole-skipping points. For this purpose, we analyse scalar and vector fields in hyperbolic space. This setup is chosen because it is simple enough to allow us to obtain analytical expressions for the Green’s function and check everything explicitly, while it contains all the essential features of pole-skipping points. We classify pole-skipping points in three types (type-I, II, III). Type-I and Type-II are distinguished by the (limiting) behavior of the Green’s function near the pole-skipping points. Type-III can arise at non-integer iω values, which is due to a specific UV condition, contrary to the types I and II, which are related to a non-unique near horizon boundary condition. We also clarify the relation between the pole-skipping structure of the Green’s function and the near horizon analysis. We point out that there are subtle cases where the near horizon analysis alone may not be able to capture the existence and properties of the pole-skipping points.

A bstract We study the scrambling properties of ( d + 1)-dimensional hyperbolic black holes. Using the eikonal approximation, we calculate out-of-time-order correlators (OTOCs) for a Rindler-AdS geometry with AdS radius ℓ , which is dual to a d -dimensional conformal field theory (CFT) in hyperbolic space with temperature T = 1 / (2 π ℓ ). We find agreement between our results for OTOCs and previously reported CFT calculations. For more generic hyperbolic black holes, we compute the butterfly velocity in two different ways, namely: from shock waves and from a pole-skipping analysis, finding perfect agreement between the two methods. The butterfly velocity v B ( T ) nicely interpolates between the Rindler-AdS result v B T = 1 2 π ℓ = 1 d − 1 and the planar result v B T ≫ 1 ℓ = d 2 d − 1 .

A bstract We explore spread and spectral complexity in quantum systems that exhibit a transition from integrability to chaos, namely the mixed-field Ising model and the next-to-nearest-neighbor deformation of the Heisenberg XXZ spin chain. We corroborate the observation that the presence of a peak in spread complexity before its saturation, is a characteristic feature in chaotic systems. We find that, in general, the saturation value of spread complexity post-peak depends not only on the spectral statistics of the Hamiltonian, but also on the specific state. However, there appears to be a maximal universal bound determined by the symmetries and dimension of the Hamiltonian, which is realized by the thermofield double state (TFD) at infinite temperature. We also find that the time scales at which the spread complexity and spectral form factor change their behaviour agree with each other and are independent of the chaotic properties of the systems. In the case of spectral complexity, we identify that the key factor determining its saturation value and timescale in chaotic systems is given by minimum energy difference in the theory’s spectrum. This explains observations made in the literature regarding its earlier saturation in chaotic systems compared to their integrable counterparts. We conclude by discussing the properties of the TFD which, we conjecture, make it suitable for probing signatures of chaos in quantum many-body systems.

A bstract We use holographic methods to study several chaotic properties of a super Yang-Mills theory at temperature T in the presence of a background magnetic field of constant strength B. The field theory we work on has a renormalization flow between a fixed point in the ultraviolet and another in the infrared, occurring in such a way that the energy at which the crossover takes place is a monotonically increasing function of the dimensionless ratio ℬ/ T 2 . By considering shock waves in the bulk of the dual gravitational theory, and varying ℬ/ T 2 , we study how several chaos-related properties of the system behave while the theory they live in follows the renormalization flow. In particular, we show that the entanglement and butterfly velocities generically increase in the infrared theory, violating the previously suggested upper bounds but never surpassing the speed of light. We also investigate the recent proposal relating the butterfly velocity with diffusion coefficients. We find that electric diffusion constants respect the lower bound proposed by Blake. All our results seem to consistently indicate that the global effect of the magnetic field is to strengthen the internal interaction of the system.

A bstract Motivated by the recent connection between pole-skipping phenomena of two point functions and four point out-of-time-order correlators (OTOCs), we study the pole structure of thermal two-point functions in d -dimensional conformal field theories (CFTs) in hyperbolic space. We derive the pole-skipping points of two-point functions of scalar and vector fields by three methods (one field theoretic and two holographic methods) and confirm that they agree. We show that the leading pole-skipping point of two point functions is related with the late time behavior of conformal blocks and shadow conformal blocks in four-point OTOCs.

A bstract Holographic theories with classical gravity duals are maximally chaotic: they saturate a set of bounds on the spread of quantum information. In this paper we question whether non-locality can affect such bounds. Specifically, we consider the gravity dual of a prototypical theory with non-local interactions, namely, N = 4 non-commutative super Yang Mills. We construct shock waves geometries that correspond to perturbations of the thermofield double state with definite momentum and study several chaos related properties of the theory, including the butterfly velocity, the entanglement velocity, the scrambling time and the maximal Lyapunov exponent. The latter two are unaffected by the non-commutative parameter θ , however, both the butterfly and entanglement velocities increase with the strength of the non-commutativity. This implies that non-local interactions can enhance the effective light-cone for the transfer of quantum information, eluding previously conjectured bounds encountered in the context of local quantum field theory. We comment on a possible limitation on the retrieval of quantum information imposed by non-locality.