We perform a systematic holographic study of Krylov complexity for a wide class of confining quantum field theories. Using the geometric prescription that identifies the time derivative of the complexity with the proper momentum of a massive probe, we analyse radial geodesics in several top-down gravity duals exhibiting confinement and a mass gap. In all geometries with a smooth infrared end-of-space we uncover a robust and universal qualitative feature: Krylov complexity exhibits oscillatory behaviour. The oscillation frequency is controlled by the confinement scale, while the amplitude depends on both the ultraviolet cutoff and the infrared scale. Additional conserved charges modify these patterns without altering their qualitative structure. We further compare our results with the Krylov complexity of the longitudinally perturbed Ising model. The qualitative agreement suggests that oscillatory behaviour of Krylov complexity constitutes a universal signature of confinement and provides a sensitive probe of infrared reorganisation in strongly coupled quantum field theories.
We perform a systematic holographic study of Krylov complexity for a wide class of confining quantum field theories. Using the geometric prescription that identifies the time derivative of the complexity with the proper momentum of a massive probe, we analyse radial geodesics in several top-down gravity duals exhibiting confinement and a mass gap. In all geometries with a smooth infrared end-of-space we uncover a robust and universal qualitative feature: Krylov complexity exhibits oscillatory behaviour. The oscillation frequency is controlled by the confinement scale, while the amplitude depends on both the ultraviolet cutoff and the infrared scale. Additional conserved charges modify these patterns without altering their qualitative structure. We further compare our results with the Krylov complexity of the longitudinally perturbed Ising model. The qualitative agreement suggests that oscillatory behaviour of Krylov complexity constitutes a universal signature of confinement and provides a sensitive probe of infrared reorganisation in strongly coupled quantum field theories.
We investigate holographic Krylov complexity in bottom-up holographic QCD models describing the confined phase, using the recently proposed correspondence between the growth rate of Krylov complexity and the proper radial momentum of probes, such as particles and strings, falling in the bulk spacetime. As representative confining backgrounds, we consider two horizonless geometries, namely the dilaton-deformed AdS-soliton and the deformed thermal-AdS spacetime, both arising as solutions of an Einstein-dilaton gravity system. We find that the Krylov complexity associated with a massive particle exhibits qualitatively distinct behavior in these two backgrounds when analyzed in the Einstein frame. In the deformed AdS-soliton geometry, the complexity undergoes periodic oscillations, with both the oscillation period and amplitude depending on the parameters of the model. In contrast, in the deformed thermal-AdS geometry, the complexity grows exponentially at late times and does not display oscillatory behavior. We then investigate string dynamics in the string frame and find that the Krylov complexity exhibits oscillatory behavior in both confining backgrounds. This behavior is traced to the emergence of an additional turning point in the deep infrared of the string-frame geometry. Motivated by this observation, we further analyze massive-particle dynamics in the string frame, where we find qualitatively similar behavior in the two confining backgrounds. Our results demonstrate that the Krylov complexity of confined holographic systems depends sensitively on the choice of probe and frame, while also revealing a closer correspondence between the two confining geometries in the string-frame description.
We investigate the interplay between confinement and Krylov complexity in the long-range transverse-field Ising chain, where power-law interactions ∼1/rα generate an effective confining potential between domain walls without an explicit symmetry-breaking field. Using the time-averaged Krylov state complexity after a quench as a diagnostic, we find that the complexity is strongly suppressed and only weakly system size-dependent at small α and small transverse field hx, where confinement is strongest, and crosses over to rapid growth at larger α, where it increases with system size. Within the confined regime, the oscillation frequencies of the complexity agree with the meson masses of an effective two-kink model, with the dominant frequency growing linearly with the lightest meson mass and the time-averaged complexity decreasing approximately as its inverse square. We further study operator Krylov complexity at finite temperature using the Wightman inner product. At low temperature, it exhibits bounded oscillations that directly encode the meson spectrum, whereas this structure is progressively washed out as the temperature increases. These results identify Krylov complexity as a spectroscopic probe of interaction-induced confinement and reproduce, in a microscopic lattice model, the hallmarks of Krylov complexity identified in confining field theories.
Following the holographic proposal that identifies the growth rate of Krylov complexity with the proper radial momentum of an infalling massive probe, we study Krylov complexity in Lifshitz and hyperscaling-violating backgrounds. For pure Lifshitz geometries, we find quadratic complexity growth for all values of the dynamical exponent. For hyperscaling-violating backgrounds, we show that the hyperscaling-violating exponent directly controls the late-time growth exponent, with a special limiting case exhibiting oscillatory behavior with a logarithmic envelope. We also extend our analysis to top-down string constructions, confirming the robustness of the scaling behaviors obtained in the bottom-up approach. Our analysis establishes that the momentum-Krylov correspondence extends naturally to non-relativistic holographic settings and remains well-defined despite the causal pathologies of Lifshitz spacetimes.
In high-energy physics, confinement denotes the tendency of fundamental particles to remain bound together, preventing their observation as free, isolated entities. Interestingly, analogous confinement behavior emerges in certain condensed matter systems, for instance, in the Ising model with both transverse and longitudinal fields, where domain walls become confined into meson-like bound states as a result of a longitudinal field-induced linear potential. In this work, we employ the Ising model to demonstrate that Krylov state complexity--a measure quantifying the spread of quantum information under the repeated action of the Hamiltonian on a quantum state--serves as a sensitive and quantitative probe of confinement. We show that confinement manifests as a pronounced suppression of Krylov complexity growth following quenches within the ferromagnetic phase in the presence of a longitudinal field, reflecting slow correlation dynamics. In contrast, while quenches within the paramagnetic phase exhibit enhanced complexity with increasing longitudinal field, reflecting the absence of confinement, those crossing the critical point to the ferromagnetic phase reveal a distinct regime characterized by orders-of-magnitude larger complexity and display trends of weak confinement. Notably, in the confining regime, the complexity oscillates at frequencies corresponding to the meson masses, with its power-spectrum peaks closely matching the semiclassical predictions.
This Thesis explores the notion of Krylov complexity as a probe of quantum chaos and as a candidate for holographic complexity. The first Part is devoted to presenting the fundamental notions required to conduct research in this area. Namely, an extensive introduction to the Lanczos algorithm, its properties and associated algebraic structures, as well as technical details related to its practical implementation, is given. Subsequently, an overview of the seminal references and the main debates regarding Krylov complexity and its relation to chaos and holography is provided. The text throughout this first Part combines review material with original analyses which either intend to contextualize, compare and criticize results in the literature, or are the fruit of the investigations leading to the publications on which this Thesis is based. These research projects are the subject of the second Part of the manuscript. In them, methods for the efficient implementation of the Lanczos algorithm in finite many-body systems were developed, allowing to compute numerically the Krylov complexity of models like SYK or the XXZ spin chain up to time scales exponentially large in system size. It was observed that the operator Krylov complexity profile in SYK, a paradigmatic low-dimensional chaotic system with a holographic dual, agrees with holographic expectations, while in the case of integrable models like XXZ complexity is affected by a novel localization effect in the so-called Krylov space which hinders its growth. Finally, an exact, analytical, correspondence between the Krylov complexity of the infinite-temperature thermofield double state in the low-energy regime of the double-scaled SYK model and bulk length in the theory of JT gravity is established.
A bstract In this study, we analyze Krylov Complexity in two-dimensional conformal field theories subjected to deformed SL(2 , ℝ) Hamiltonians. In the Vacuum state, we find that the K-Complexity exhibits a universal phase structure. The phase structure involves the K-Complexity exhibiting an oscillatory behaviour in the non-heating phase, which contrasts with the exponential growth observed in the heating phase, while it displays polynomial growth at the phase boundary. Furthermore, we extend our analysis to compute the K-Complexity of a light operator in excited states, considering both large-c CFT and free field theory. In the free field theory, we find a state-independent phase structure of K-Complexity. However, in the large-c CFT, the behaviour varies, with the K-Complexity once again displaying exponential growth in the heating phase and polynomial growth at the phase boundary. Notably, the precise exponent governing this growth depends on the heaviness of the state under examination. In the non-heating phase, we observe a transition in K-Complexity behaviour from oscillatory to exponential growth, akin to findings in [1], as it represents a special case within the non-heating phase.