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A bstract Heisenberg time evolution under a chaotic many-body Hamiltonian H transforms an initially simple operator into an increasingly complex one, as it spreads over Hilbert space. Krylov complexity, or ‘K-complexity’, quantifies this growth with respect to a special basis, generated by H by successive nested commutators with the operator. In this work we study the evolution of K-complexity in finite-entropy systems for time scales greater than the scrambling time t s > log( S ). We prove rigorous bounds on K-complexity as well as the associated Lanczos sequence and, using refined parallelized algorithms, we undertake a detailed numerical study of these quantities in the SYK 4 model, which is maximally chaotic, and compare the results with the SYK 2 model, which is integrable. While the former saturates the bound, the latter stays exponentially below it. We discuss to what extent this is a generic feature distinguishing between chaotic vs. integrable systems.

A bstract We apply a notion of quantum complexity, called “Krylov complexity”, to study the evolution of systems from integrability to chaos. For this purpose we investigate the integrable XXZ spin chain, enriched with an integrability breaking deformation that allows one to interpolate between integrable and chaotic behavior. K-complexity can act as a probe of the integrable or chaotic nature of the underlying system via its late-time saturation value that is suppressed in the integrable phase and increases as the system is driven to the chaotic phase. We furthermore ascribe the (under-)saturation of the late-time bound to the amount of disorder present in the Lanczos sequence, by mapping the complexity evolution to an auxiliary off-diagonal Anderson hopping model. We compare the late-time saturation of K-complexity in the chaotic phase with that of random matrix ensembles and find that the chaotic system indeed approaches the RMT behavior in the appropriate symmetry class. We investigate the dependence of the results on the two key ingredients of K-complexity: the dynamics of the Hamiltonian and the character of the operator whose time dependence is followed.

A bstract There are various definitions of the concept of complexity in Quantum Field Theory as well as for finite quantum systems. For several of them there are conjectured holographic bulk duals. In this work we establish an entry in the AdS/CFT dictionary for one such class of complexity, namely Krylov or K-complexity. For this purpose we work in the double-scaled SYK model which is dual in a certain limit to JT gravity, a theory of gravity in AdS 2 . In particular, states on the boundary have a clear geometrical definition in the bulk. We use this result to show that Krylov complexity of the infinite-temperature thermofield double state on the boundary of AdS 2 has a precise bulk description in JT gravity, namely the length of the two-sided wormhole. We do this by showing that the Krylov basis elements, which are eigenstates of the Krylov complexity operator, are mapped to length eigenstates in the bulk theory by subjecting K-complexity to the bulk-boundary map identifying the bulk/boundary Hilbert spaces. Our result makes extensive use of chord diagram techniques and identifies the Krylov basis of the boundary quantum system with fixed chord number states building the bulk gravitational Hilbert space.

A bstract Quantum complexity, suitably defined, has been suggested as an important probe of late-time dynamics of black holes, particularly in the context of AdS/CFT. A notion of quantum complexity can be effectively captured by quantifying the spread of an operator in Krylov space as a consequence of time evolution. Complexity is expected to behave differently in chaotic many-body systems, as compared to integrable ones. In this paper we investigate Krylov complexity for the case of interacting integrable models at finite size and find that complexity saturation is suppressed as compared to chaotic systems. We associate this behavior with a novel localization phenomenon on the Krylov chain by mapping the theory of complexity growth and spread to an Anderson localization hopping model with off-diagonal disorder, and find that localization is enhanced in the integrable case due to a stronger disorder in the hopping amplitudes, inducing an effective suppression of Krylov complexity. We demonstrate this behavior for an interacting integrable model, the XXZ spin chain, and show that the same behavior results from a phenomenological model that we define: this model captures the essential features of our analysis and is able to reproduce the behaviors we observe for chaotic and integrable systems via an adjustable disorder parameter.

A bstract We study operator complexity on various time scales with emphasis on those much larger than the scrambling period. We use, for systems with a large but finite number of degrees of freedom, the notion of K-complexity employed in [ 1 ] for infinite systems. We present evidence that K-complexity of ETH operators has indeed the character associated with the bulk time evolution of extremal volumes and actions. Namely, after a period of exponential growth during the scrambling period the K-complexity increases only linearly with time for exponentially long times in terms of the entropy, and it eventually saturates at a constant value also exponential in terms of the entropy. This constant value depends on the Hamiltonian and the operator but not on any extrinsic tolerance parameter. Thus K-complexity deserves to be an entry in the AdS/CFT dictionary. Invoking a concept of K-entropy and some numerical examples we also discuss the extent to which the long period of linear complexity growth entails an efficient randomization of operators.

A bstract We study superstring theory in three dimensional Anti-de Sitter spacetime with NS-NS flux, focusing on the case where the radius of curvature is equal to the string length. This corresponds to the critical level k = 1 in the formulation as a Wess-Zumino-Witten model. Previously, it was argued that a transition takes place at this special radius, from a phase dominated by black holes at larger radius to one dominated by long strings at smaller radius. We argue that the infinite tower of modes that become massless at k = 1 is a signal of this transition. We propose a simple two-dimensional conformal field theory as the holographic dual to superstring theory at k = 1. As evidence for our conjecture, we demonstrate that our putative dual exactly reproduces the full spectrum of the long strings of the weakly coupled string theory, including states unprotected by supersymmetry.

A bstract In this paper we study the notion of complexity under time evolution in chaotic quantum systems with holographic duals. Continuing on from our previous work, we turn our attention to the issue of Krylov complexity upon the insertion of a class of single-particle operators in the double-scaled SYK model. Such an operator is described by a matter-chord insertion, which splits the theory into left/right sectors, allowing us, via chord-diagram technology, to compute two different notions of complexity associated to the operator insertion: first a Krylov operator complexity, and second the Krylov complexity of a state obtained by an operator acting on the thermofield double state. We will provide both an analytic proof and detailed numerical evidence, that both Krylov complexities arise from a recursively defined basis of states characterized by a constant total chord number. As a consequence, in all cases we are able to establish that Krylov complexity is given by the expectation value of a length operator acting on the Hilbert space of the theory, expressed in terms of basis states, organized by left and right chord number. We find analytic expressions for the semiclassical limit of K-complexity, and study how the size of the operator encodes the scrambling dynamics upon the matter insertion in Krylov language. We furthermore determine the effective Hamiltonian governing the evolution of K-complexity, showing that evolution on the Krylov chain can equivalently be understood as a particle moving in a Morse potential. A particular type of triple scaling limit allows to access the gravitational sector of the theory, in which the geometrical nature of K-complexity is assured by virtue of being a total chord length, in an analogous fashion to what was found in [ 1 ] for the K-complexity of the thermofield double state.

A bstract We discuss some aspects of critical electric and magnetic fields in a field theory with holographic dual description. We extend the analysis of [1], which finds a critical electric field at which the Schwinger pair production barrier drops to zero, to the case of magnetic fields. We first find that, unlike ordinary weakly coupled theories, the magnetic field is not subject to any perturbative instability originating from the presence of a tachyonic ground state in the W-boson spectrum. This follows from the large value of the ’t Hooft coupling λ, which prevents the Zeeman interaction term to overcome the particle mass at high B . Consequently, we study the next possible B-field instability, i.e. monopole pair production, which is the S-dual version of the Schwinger effect. Also in this case a critical magnetic field is expected when the tunneling barrier drops to zero. These Schwinger-type criticalities are the holographic duals, in the bulk, to the fields E or B reaching the tension of F1 or D1 strings respectively. We then discuss how this effect is modified when electric and magnetic fields are present simultaneously and dyonic states in the spectrum can be pair produced by a generic E − B background. Finally, we analyze finite temperature effects on Schwinger criticalities, i.e. in the AdS-Schwarzshild black hole background.

A bstract We consider string pair production in non homogeneous electric backgrounds. We study several particular configurations which can be addressed with the Euclidean world-sheet instanton technique, the analogue of the world-line instanton for particles. In the first case the string is suspended between two D-branes in flat space-time, in the second case the string lives in AdS and terminates on one D-brane (this realizes the holographic Schwinger effect). In some regions of parameter space the result is well approximated by the known analytical formulas, either the particle pair production in non-homogeneous background or the string pair production in homogeneous background. In other cases we see effects which are intrinsically stringy and related to the non-homogeneity of the background. The pair production is enhanced already for particles in time dependent electric field backgrounds. The string nature enhances this even further. For spacial varying electrical background fields the string pair production is less suppressed than the rate of particle pair production. We discuss in some detail how the critical field is affected by the non-homogeneity, for both time and space dependent electric field backgrouds. We also comment on what could be an interesting new prediction for the small field limit. The third case we consider is pair production in holographic confining backgrounds with homogeneous and non-homogeneous fields.

We study superstring theory in three dimensional Anti-de Sitter spacetime with NS-NS flux, focusing on the case where the radius of curvature is equal to the string length. This corresponds to the critical level k = 1 in the formulation as a Wess-Zumino-Witten model. Previously, it was argued that a transition takes place at this special radius, from a phase dominated by black holes at larger radius to one dominated by long strings at smaller radius. We argue that the infinite tower of modes that become massless at k = 1 is a signal of this transition. We propose a simple two-dimensional conformal field theory as the holographic dual to superstring theory at k = 1. As evidence for our conjecture, we demonstrate that our putative dual exactly reproduces the full spectrum of the long strings of the weakly coupled string theory, including states unprotected by supersymmetry.