The Schwarzschild-Black String AdS Soliton: Instability and Holographic Heat Transport

We present a calculation of two-point correlation functions of the stress-energy tensor in the strongly-coupled, confining gauge theory which is holographically dual to the AdS soliton geometry. The fact that the AdS soliton smoothly caps off at a certain point along the holographic direction, ensures that these correlators are dominated by quasinormal mode contributions and thus show an exponential decay in position space. In order to study such a field theory on a curved spacetime, we foliate the six-dimensional AdS soliton with a Schwarzschild black hole. Via gauge/gravity duality, this new geometry describes a confining field theory with supersymmetry breaking boundary conditions on a non-dynamical Schwarzschild black hole background. We also calculate stress-energy correlators for this setting, thus demonstrating exponentially damped heat transport. This analysis is valid in the confined phase. We model a deconfinement transition by explicitly demonstrating a classical instability of Gregory-Laflamme-type of this bulk spacetime.Comment: 26 pages, 3 figure

Fine-Grained Chaos in AdS2AdS_2 Gravity

Quantum chaos can be characterized by an exponential growth of the thermal out-of-time-order four-point function up to a scrambling time $\widehat{u}_*$. We discuss generalizations of this statement for certain higher-point correlation functions. For concreteness, we study the Schwarzian theory of a one-dimensional time reparametrization mode, which describes $AdS_2$ gravity and the low-energy dynamics of the SYK model. We identify a particular set of $2k$-point functions, characterized as being both "maximally braided" and "k-OTO", which exhibit exponential growth until progressively longer timescales $\widehat{u}^{(k)}_* = (k-1)\widehat{u}_*$. We suggest an interpretation as scrambling of increasingly fine-grained measures of quantum information, which correspondingly take progressively longer time to reach their thermal values.Comment: 8 pages; v2: minor clarifications, typos, added ref

Effective Field Theory for Chaotic CFTs

We derive an effective field theory for general chaotic two-dimensional conformal field theories with a large central charge. The theory is a specific and calculable instance of a more general framework recently proposed in [1]. We discuss the gauge symmetries of the model and how they relate to the Lyapunov behaviour of certain correlators. We calculate the out-of-time-ordered correlators diagnosing quantum chaos, as well as certain more fine-grained higher-point generalizations, using our Lorentzian effective field theory. We comment on potential future applications of the effective theory to real-time thermal physics and conformal field theory.Comment: 33 pages, 4 figures; v2: minor improvements, added paragraph on higher spin exchanges; v3: minor improvements, added reference, published versio

Schwinger-Keldysh formalism II: Thermal equivariant cohomology

Causally ordered correlation functions of local operators in near-thermal quantum systems computed using the Schwinger-Keldysh formalism obey a set of Ward identities. These can be understood rather simply as the consequence of a topological (BRST) algebra, called the universal Schwinger-Keldysh superalgebra, as explained in our companion paper arXiv:1610.01940. In the present paper we provide a mathematical discussion of this topological algebra. In particular, we argue that the structures can be understood in the language of extended equivariant cohomology. To keep the discussion self-contained, we provide a basic review of the algebraic construction of equivariant cohomology and explain how it can be understood in familiar terms as a superspace gauge algebra. We demonstrate how the Schwinger-Keldysh construction can be succinctly encoded in terms a thermal equivariant cohomology algebra which naturally acts on the operator (super)-algebra of the quantum system. The main rationale behind this exploration is to extract symmetry statements which are robust under renormalization group flow and can hence be used to understand low-energy effective field theory of near-thermal physics. To illustrate the general principles, we focus on Langevin dynamics of a Brownian particle, rephrasing some known results in terms of thermal equivariant cohomology. As described elsewhere, the general framework enables construction of effective actions for dissipative hydrodynamics and could potentially illumine our understanding of black holes.Comment: 72 pages; v2: fixed typos. v3: minor clarifications and improvements to non-equilbirum work relations discussion. v4: typos fixed. published versio

The eightfold way to dissipation

We provide a complete characterization of hydrodynamic transport consistent with the second law of thermodynamics at arbitrary orders in the gradient expansion. A key ingredient in facilitating this analysis is the notion of adiabatic hydrodynamics, which enables isolation of the genuinely dissipative parts of transport. We demonstrate that most transport is adiabatic. Furthermore, of the dissipative part, only terms at the leading order in gradient expansion are constrained to be sign-definite by the second law (as has been derived before).Comment: 5 pages, 1 figure. v2: minor clarifications. v3: minor changes. title in published version differ

The quantum pp-spin glass model: A user manual for holographers

We study a large-$N$ bosonic quantum mechanical sigma-model with a spherical target space subject to disordered interactions, more colloquially known as the $p$-spin spherical model. Replica symmetry is broken at low temperatures and for sufficiently weak quantum fluctuations, which drives the system into a spin glass phase. The first half of this paper is dedicated to a discussion of this model's thermodynamics, with particular emphasis on the marginally stable spin glass. This phase exhibits an emergent conformal symmetry in the strong coupling regime, which dictates its thermodynamic properties. It is associated with an extensive number of nearby states in the free energy landscape. We discuss in detail an elegant approximate solution to the spin glass equations, which interpolates between the conformal regime and an ultraviolet-complete short distance solution. In the second half of this paper we explore the real-time dynamics of the model and uncover quantum chaos as measured by out-of-time-order four-point functions, both numerically and analytically. We find exponential Lyapunov growth, which intricately depends on the model's couplings and becomes strongest in the quantum critical regime. We emphasize that the spin glass phase also exhibits quantum chaos, albeit with parametrically smaller Lyapunov exponent than in the replica symmetric phase. An analytical calculation in the marginal spin glass phase suggests that this Lyapunov exponent vanishes in a particular infinite coupling limit. We comment on the potential meaning of these observations from the perspective of holography.Comment: 66 (+39) pages, 19 (+2) figures; v2: added references and minor comments (published version

Collisions of localized shocks and quantum circuits

We study collisions between localized shockwaves inside a black hole interior. We give a holographic boundary description of this process in terms of the overlap of two growing perturbations in a shared quantum circuit. The perturbations grow both exponentially as well as ballistically. Due to a competition between different physical effects, the circuit analysis shows dependence on the transverse locations and exhibits four regimes of qualitatively different behaviors. On the gravity side we study properties of the post-collision geometry, using exact calculations in simple setups and estimations in more general circumstances. We show that the circuit analysis offers intuitive and surprisingly accurate predictions about gravity computations involving non-linear features of general relativity.Comment: v1: 26+11 pages, 17 figures; v2: published version in JHE

Operator growth and black hole formation

When two particles collide in an asymptotically AdS spacetime with high enough energy and small enough impact parameter, they can form a black hole. Motivated by dual quantum circuit considerations, we propose a threshold condition for black hole formation. Intuitively the condition can be understood as the onset of overlap of the butterfly cones describing the ballistic spread of the effect of the perturbations on the boundary systems. We verify the correctness of the condition in three bulk dimensions. We describe a six-point correlation function that can diagnose this condition and compute it in two-dimensional CFTs using eikonal resummation.Comment: 20+9 pages, 10 figures. v2: discussions added in sections 5 and