We study a family of locally excited states in two-dimensional conformal field theories in different setups, defined on the plane, on the cylinder, or at finite temperature. These states are constructed by assigning independent chiral and anti-chiral regulators to a local primary operator, leading to excitations carrying net momentum. We analytically calculate the first few Lanczos coefficients and the early time behaviour of the spread complexity associated with these states. Then, using numerical methods, we obtain the coefficients reaching larger Krylov indices and find the associated spread complexity for a longer range of time evolution. We also compare the rate of change of the resulting spread complexity with the proposed holographic dual quantity, related to the proper momentum of an infalling particle in the dual bulk, but we observe a mismatch.
We study a family of locally excited states in two-dimensional conformal field theories in different setups, defined on the plane, on the cylinder, or at finite temperature. These states are constructed by assigning independent chiral and anti-chiral regulators to a local primary operator, leading to excitations carrying net momentum. We analytically calculate the first few Lanczos coefficients and the early time behaviour of the spread complexity associated with these states. Then, using numerical methods, we obtain the coefficients reaching larger Krylov indices and find the associated spread complexity for a longer range of time evolution. We also compare the rate of change of the resulting spread complexity with the proposed holographic dual quantity, related to the proper momentum of an infalling particle in the dual bulk, but we observe a mismatch.
We investigate the holographic dictionary relating point particles carrying longitudinal momentum or angular momentum in asymptotically AdS3 spacetimes to suitably regulated, time-evolved states created by local primary operators in the dual two-dimensional conformal field theory (2D CFT). We find that asymmetric left/right Euclidean smearing of local operators produces states carrying momentum, and the corresponding bulk excitation is a particle with conserved momentum. We compute the energy density and entanglement entropy in these states and in their dual back-reacted geometries, finding exact agreement between the CFT and gravity descriptions. We further extend this correspondence to particles with intrinsic spin, whose CFT duals are primary operators with unequal holomorphic and anti-holomorphic scaling dimensions. We again find a precise match between CFT and holographic calculations of energy densities and entanglement entropies. Finally, we explore applications of these setups beyond holography by deriving the evolution of Rényi entropies in 2D rational CFTs and introducing a new class of local quantum quench protocols with conserved longitudinal or angular momentum.
A bstract In this work, we investigate local quench dynamics in two-dimensional conformal field theories using Krylov space methods. We derive Lanczos coefficients, spread complexity, and Krylov entropies for local joining and splitting quenches in theories on an infinite line, a circle, a finite interval, and at finite temperature. We examine how these quantities depend on the central charge of the underlying conformal field theory and find that both spread complexity and Krylov entropy are proportional to it. Interestingly, Krylov entropies evolve logarithmically with time, mirroring standard entanglement entropies, making them useful for extracting the central charge. In the large central charge limit, using holography, we establish a connection between the rate of spread complexity and the proper momentum of the tip of the end-of-the world brane, which probes the bulk analogously to a point particle. Our results further demonstrate that spread complexity and Krylov entropies are powerful tools for probing non-equilibrium dynamics of interacting quantum systems.
We demonstrate a precise relation between the rate of complexity of quantum states excited by local operators in two-dimensional conformal field theories and the radial momentum of particles in three-dimensional anti–de Sitter spacetimes. Similar relations have been anticipated based on qualitative models for operator growth. Here, we make this correspondence sharp with two key ingredients: the precise definition of quantum complexity given by the spread complexity of states, and the match of its growth rate to the bulk momentum measured in the proper radial distance coordinate.
This thesis investigates quantum complexity in the AdS/CFT duality. We study a recently proposed measure of quantum complexity, the spread complexity. This is a measure of the number of orthonormal basis vectors needed to describe a given quantum state (its “spread” in the Hilbert space), naturally selecting a preferred basis adapted to the dynamics of the state which minimises this quantity. This thesis demonstrates that the spread complexity provides a quantum information- theoretic quantity which directly connects to bulk dynamics in holography. Quantum complexity has long been studied in the context of holography. One result of these investigations is a conjecture relating the rate of change of the complexity of certain states in the CFT to the radial momentum of a dual massive particle falling along a time-like geodesic in the bulk spacetime. We demonstrate that there is an exact match between the rate of change of the spread complexity of an excited 2D CFT state and the dual particle’s radial momentum, measured in appropriate coordinates. As a toy model of the localised bulk particle state, we show that a similar relationship exists between the spread complexity and momentum of coherent states of the quantum harmonic oscillator. The spread complexity of these CFT states can be expressed analytically by relating them to a discrete series representation of SL(2,R). Additionally, we find analytical results for the spread complexity when the excited CFT state is evolved by the modular Hamiltonian of an interval region in the CFT, and by the half-sided modular Hamiltonian associated with a further sub-interval region. Next, we study the spread complexity of CFT states which describe a particle on the BTZ black hole background. We also study the spread complexity of local states in the semi-classical bulk theory. This provides a direct bulk interpretation of the spread complexity to complement our CFT calculations. Finally, we explore the extensivity of the spread complexity of states and of the Krylov complexity of density matrix operators in non-interacting bipartite systems.
Fast scrambling and exponential growth of complexity are two distinctive features of quantum gravity, offering a unique window into the mechanisms that govern the semiclassical limit of the holographic duality. In this thesis, we explore these information-theoretic phenomena through two key diagnostics --- out-of-time ordered correlation functions (OTOCs) and spread complexity --- computed in the setting of two-dimensional holographic conformal field theories (CFTs). First, we focus on the large-c limit of Liouville theory, where the computation of OTOCs reveals distinctive insights into both the chaotic nature and the structural features of the theory. Remarkably, we find it exhibits a maximal Lyapunov exponent, despite lacking the identity operator in its spectrum, and a scrambling time logarithmic in the central charge. Using path integral techniques, we calculate heavy-light four-point functions, and uncover a conformal block expansion dominated by the exchange of an infinite tower of spinless double-trace operators. We attribute the chaotic behaviour of the OTOC to this exchange of operators, which can be thought of as dressed scramblons. Additionally, we observe that at the onset of scrambling, the dominance of conformal blocks shifts from heavy primaries at early times to light primaries at late times. Second, we turn to the diagnostic of spread complexity in locally excited holographic CFTs and argue that its time derivative is dual to the proper radial momentum of a massive point particle propagating in Anti-de Sitter spacetime (AdS). Exploiting the conformal symmetry and the emergence of an ``effective'' SL(2,R) group, we employ representation-theoretic techniques to compute return amplitudes, Lanczos coefficients, and spread complexities. We then verify our conjectured duality for three settings: the CFT on the line, dual to Poincaré AdS; the CFT on the circle, dual to global AdS; and the CFT at finite temperature, dual to a BTZ black hole. In the gravitational theory, the same SL(2,R) structure emerges as the isometry group of the infalling plane of the bulk particle. Third, we generalise the analysis of spread complexity to modular Hamiltonian evolution. Exploiting the rich algebraic structure of modular theory, we construct the modular Hamiltonians Kₘ and the half-sided modular translations G for the three example states above. Leveraging conformal symmetry, we once again compute modular return amplitudes, Lanczos coefficients, and modular spread complexities. In particular, we emphasise the role of ``conformal regularisation'' --- i.e. regulating with the operator that generates modular evolution - in preserving the effective SL(2,R) structure of the amplitudes. We find that evolution under Kₘ exhibits thermal spread complexity for β = 1 across all examples, while G-evolution universally leads to quadratic growth. Finally, we provide a formal explanation of these results, identifying the role of the Killing norm D of the evolution operator as the key factor governing the behaviour of complexity.
We study the time evolution of Renyi entanglement entropy for locally excited states in two dimensional large central charge CFTs. It generically shows a logarithmical growth and we compute the coefficient of log t term. Our analysis covers the entire parameter regions with respect to the replica number n and the conformal dimension h O of the primary operator which creates the excitation. We numerically analyse relevant vacuum conformal blocks by using Zamolodchikov’s recursion relation. We find that the behavior of the conformal blocks in two dimensional CFTs with a central charge c, drastically changes when the dimensions of external primary states reach the value c/32. In particular, when h O ≥ c/32 and n ≥ 2, we find a new universal formula ΔSA(n)≃24(n−1)nc log t. Our numerical results also confirm existing analytical results using the HHLL approximation.