Aspects of Holography and Quantum complexity

Abstract

We begin with the Principle of holography and the AdS/CFT correspondence. We begin our investigations by considering the Principle of Holography, at the level of codimension-1 screens. We argue that the mechanism of encoding the holographic data in terms of sources, condensates, and correlators is fairly general and has generalizations to a large class of spacetimes. We formulate the holographic correspondence in terms of bulk sources localized on a screen, instead of boundary values of bulk fields. We discuss the extension of the familiar notion of normalizable and non-normalizable modes to moderately general settings beyond AdS. We discuss a simple map between this prescription and the usual AdS/CFT correspondence, as well as work out explicit correlators via this prescription in flat space. Finally, we discuss the general utility of this approach of using sources to describe dynamics.

We then focus on the widely successful AdS/CFT correspondence. Specifically, we study the Bulk Reconstruction program in AdS/CFT. We discuss various aspects of the HKLL bulk reconstruction in AdS. We construct the space-like Kernel for the non-normalizable mode as a mode sum and via a Green's function approach (in even dimensions). This puts the normalizable and non-normalizable modes on equal footing. In Poincaré AdS, we delve into the technical details of this construction. We propose a spatial complexification and discuss an antipodal identification as crucial steps in obtaining a space-like Kernel in terms of the chordal distance, for certain values of scaling dimension. We also note some interesting features of this construction inside the Brietienlohner-Freedman bound, where both the normalizable and non-normalizable modes have equivalent interpretations.

At this stage, we shift gears and start addressing some problems in quantum chaos via quantum complexity. In view of the Operator Growth Hypothesis for chaotic and intergable systems, we study classically integrable systems with unstable saddle points. We find that Krylov complexity (of operators) is a hypersensitive probe of chaos. We discuss some features of Krylov complexity and autocorrelation functions. We supplement our study by numerical calculations using the Lipkin-Meshkov-Glick and Feigngold-Peres models.

Using Krylov complexity, we also study quantum many-body scars. We utilize the Lanczos mechanism to study special eigenstates for the (chaotic) PXP model which behave as states with ``low chaos'' as compared to the other eigenstates. The presence of these states indicate the emergence of some underlying symmetry, which we characterize by $q-$ deformed SU(2) algebra. Finally, we study the nature Krylov spread complexity for the scar states and compare them to the generic states. We conclude by discussing a ``tight-binding'' interpretation of Krylov spread complexity.