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 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.
The boundary quantity that captures the growth of black-hole interiors quantified by holographic complexity remains unknown beyond 2d dilaton gravity. We provide a critical analysis of a partition-function construction of Krylov spread complexity for thermofield-double states that provides a dimension-independent boundary reconstruction from semiclassical holographic partition functions, while developing the present dynamical and bulk construction for the BTZ saddle. In the double-scaled Sachdev-Ye-Kitaev model, where exact and semiclassical results can be compared, we show that the classical limit is reliable only when taken after the complexity has been reconstructed; taking this limit at the level of individual Lanczos coefficients discards essential information. Applying the construction to a large-central-charge two-dimensional conformal field theory above the Hawking-Page temperature dual to a Bañados-Teitelboim-Zanelli black hole, we find an intermediate departure from early-time quadratic growth followed by behavior compatible with a return toward asymptotically quadratic growth, rather than the linear late-time behavior of the volume and the standard finite-functional complexity = anything class. We then match this boundary behavior to a generalized complexity = anything bulk object built from an infinite series of extrinsic-curvature invariants. The construction provides a systematic route from black-hole thermodynamics to Krylov dynamics and can naturally be extended to higher-dimensional holographic black holes.
Abstract We study Krylov complexity in Lifshitz-type Dirac field theories with a generic dynamical critical exponent z . By computing the Lanczos coefficients for massless and massive cases, we analyze the growth and saturation behavior of Krylov complexity in different regimes. We incorporate a hard UV cutoff and investigate the effects of lattice discretization, revealing fundamental differences between continuum and lattice models. In the presence of a UV cutoff, Krylov complexity exhibits an initial exponential growth followed by a linear regime, with saturation values dictated by the cutoff scale. For the lattice model, we find a fundamental departure from the continuum case: due to the finite Krylov basis, Krylov complexity saturates rather than growing indefinitely. Our findings suggest that Lifshitz scaling influences operator growth and information spreading in quantum systems. We further find that increasing the Lifshitz exponent z suppresses Krylov complexity, entropy, and Lanczos growth in both massless and massive cases, while enhancing K-variance. This trend reverses under a hard UV cutoff, where complexity and entropy increase with z . In lattice models, early-time complexity and
bn b n decay shift with z , echoing the continuum behavior of massive and massless regimes.
We holographically study quantum chaos in hyperscaling-violating Lifshitz (HVL) theories (with charge). Specifically, we present a detailed computation of the out-of-time ordered correlator (OTOC) via shockwave analysis in the bulk HVL geometry with a planar horizon topology. We also compute the butterfly velocity (
vB) using the entanglement wedge reconstruction and find that it matches the result obtained from the shockwave analysis. Using a recently developed thermodynamic dictionary for HVL theories, we express
vB purely in terms of boundary thermodynamic variables. Furthermore, we analyze in detail the behavior of
vB with respect to the dynamical critical exponent (
z), hyperscaling-violating parameter (
θ), entropy (more precisely, the ratio of entropy to the central charge,
S~), and charge (more precisely, the ratio of charge to the central charge,
Q~). Interestingly,
vB varies non-monotonically with
z for
S~<1, whereas it increases monotonically with
z for
S~≥1. Additionally,
vB varies non-monotonically with
θ for non-zero charge. Moreover,
vB monotonically increases with
S~ and decreases with
Q~ for all allowed values of
z and
θ. All these features are reported for combinations /{
z,
θ,
S~,
Q~/} for which the temperature is positive, the null energy condition is satisfied, and
vB is not superluminal. Unpacking the non-monotonicities in
vB can offer interesting insights into these theories.
Motivated by exploring the thermalization process in relativistic and nonrelativistic holographic field theories after a nonlocal quench, we investigate some features in the time evolution of the entanglement wedge cross section (EWCS). This quantity is a possible holographic dual to some nonlocal information measures such as entanglement of purification. In particular, we focus on the time dependence of the EWCS during black hole formation in (
D+2)-dimensional anti--de Sitter spacetime as well as geometries with Lifshitz and hyperscaling-violating exponents. A combination of analytic and numerical results for large symmetric strip shaped boundary subregions shows that the scaling of the EWCS at early times only depends on the Lifshitz exponent. In addition, this early-growth regime is followed by a linear-growth regime whose velocity depends on the dimensions of spacetime, the Lifshitz exponent, and the hyperscaling parameter. This velocity is the same as the entanglement velocity and for nontrivial dynamical exponent depends on the temperature of the final equilibrium state.