arXiv subjects

Hendrik J. R. van Zyl

Publications and source records attributed to Hendrik J. R. van Zyl.

9 recordsLinked to original sources

arXiv2026

A Schwinger-Keldysh Formulation of Semiclassical Operator Dynamics

In this work we develop a real-time Schwinger-Keldysh formulation of Krylov dynamics that treats Krylov complexity as an in-in observable generated by a closed time contour path integral. The resulting generating functional exposes an emergent phase-space description in which the Lanczos coefficients define an effective Hamiltonian governing operator motion along the Krylov chain. In the semiclassical limit, exponential complexity growth arises from hyperbolic trajectories, and asymptotically linear Lanczos growth appears as a universal chaotic fixed point, with sub-leading deformations classified as irrelevant, marginal or relevant. Going beyond the saddle, the Schwinger-Keldysh framework provides controlled access to fluctuations and large deviations of Krylov complexity, revealing sharp signatures of integrability-chaos crossovers that are invisible at the level of the mean. This formulation reorganises Krylov complexity into a dynamical field-theoretic framework and identifies new fluctuation diagnostics of operator growth in closed quantum systems.

quant-ph↗

arXiv2026

Krylov complexity from a simple quantum mechanical model for a radiating black hole

We investigate Krylov complexity in a simple quantum mechanical model describing a black hole coupled to its radiation. The model is constructed as a simplified ``mini-BMN" matrix system inspired by a recent proposal of Maldacena. Our aim is not to reproduce the full dynamics of the BMN matrix model, but rather to isolate a tractable setting in which the information-theoretic behaviour of a radiating black hole can be studied explicitly. We analyze both the early- and late-time behaviour of Krylov complexity and the associated Krylov entropy. At early times, perturbative and numerical analyses reveal the expected growth characteristic of chaotic quantum dynamics. At late times, however, the dynamics saturates to a plateau, consistent with equilibration between the black hole and its radiation and with general expectations from finite-entropy quantum systems. We argue that this plateau behaviour admits a semiclassical interpretation in terms of Euclidean instanton contributions in an effective path-integral. The toy model studied here offers a controlled framework in which these features can be investigated analytically and numerically.

hep-th↗

arXiv2026

Superadditivity of Krylov Complexity for Tensor Products

We study Krylov complexity for quantum systems whose Hamiltonians factorise as tensor products. We prove that complexity is superadditive under tensor products, $C_{12}\ge C_1+C_2$, and identify a positive operator that quantifies the resulting excess complexity. The underlying mechanism is made transparent by introducing a Krylov graph representation in which tensor products generate a higher-dimensional lattice whose diagonal shells encode operator growth and binomial path multiplicities. In the continuum limit, Krylov dynamics reduces to diffusion on this graph, with superadditivity arising from geometric broadening across shells. Explicit examples illustrate how deviations from synchronous evolution generate bounded, oscillatory excess complexity.

hep-th↗

arXiv2025

(A)Symmetric Complexity and the Quantum Mpemba Effect

The Quantum Mpemba Effect (QME) -- the counter-intuitive phenomenon where states further from equilibrium can relax faster than those closer to it -- challenges standard expectations of quantum thermalization. In this work, we introduce Krylov complexity as a sensitive diagnostic for the QME. We show that Krylov spread complexity encodes the asymmetry essential to the effect, and we define a new class of projective (a)symmetric complexities that sharpen this connection. Strikingly, the structure of these projective complexities at the initial moment ($t=0$) already carries predictive power for the onset of Mpemba-like inversions, obviating the need for explicit time evolution. Our results suggest that the geometry of states in Krylov space captures deep information about non-monotonic relaxation and provides a powerful framework for diagnosing and anticipating anomalous thermalization phenomena in quantum systems.

hep-th↗

arXiv2024

On Complexity and Duality

We explore the relationship between complexity and duality in quantum systems, focusing on how local and non-local operators evolve under time evolution. We find that non-local operators, which are dual to local operators under specific mappings, exhibit behavior that mimics the growth of their local counterparts, particularly when considering state complexity. For the open transverse Ising model this leads to a neat organisation of the operator dynamics on either side of the duality, both consistent with growth expected in a quadratic fermion model like the Kitaev chain. When examing periodic chains, however, the mapping of boundary terms provides access to multiple branches of highly complex operators. These give rise to much larger saturation values of complexity for parity-mixing operators and are in contrast to what one would expect for a quadratic Hamiltonian. Our results shed light on the intricate relationship between non-locality, complexity growth, and duality in quantum systems.

hep-th↗

arXiv2023

Spread complexity as classical dilaton solutions

We demonstrate a relation between Nielsen's approach towards circuit complexity and Krylov complexity through a particular construction of quantum state space geometry. We start by associating Kähler structures on the full projective Hilbert space of low rank algebras. This geometric structure of the states in the Hilbert space ensures that every unitary transformation of the associated algebras leave the metric and the symplectic forms invariant. We further associate a classical matter free Jackiw-Teitelboim (JT) gravity model with these state manifolds and show that the dilaton can be interpreted as the quantum mechanical expectation values of the symmetry generators. On the other hand we identify the dilaton with the spread complexity over a Krylov basis thereby proposing a geometric perspective connecting two different notions of complexity.

hep-th↗

arXiv2018

Semiclassical SL(2) Strings on LLM backgrounds

We study semiclassical string solutions that live on white regions of the LLM plane for a generic LLM geometry. These string excitations are labelled by conserved charges E, J and S and are thus holographically dual to operators in the SL(2) sector of N = 4 super-Yang Mills made up of covariant derivatives acting on complex scalar fields Z. On the other hand, the LLM geometry itself is dual to an operator consisting of O(N^2) Z-fields so that the operators dual to our solutions, containing both the stringy excitation and background, are non-planar. In an appropriate short string limit we argue that the string solution we find should be dual to a localised SL(2) excitation in the gauge theory language. This allows us to perform a non-trivial check of the recent proposal that the dynamics of localised excitations should be identical, up to a rescaling of the 't Hooft coupling, to the dynamics of those same excitations in the AdS5xS5 background

hep-th↗

arXiv2017

Inelastic Magnon Scattering

We study the worldsheet S-matrix of a string attached to a D-brane in AdS$_5\times$S$^5$. The D-brane is either a giant graviton or a dual giant graviton. In the gauge theory, the operators we consider belong to the $su(2|3)$ sector of the theory. Magnon excitations of open strings can exhibit both elastic (when magnons in the bulk of the string scatter) and inelastic (when magnons at the endpoint of an open string participate) scattering. Both of these $S$-matrices are determined (up to an overall phase) by the $su(2|2)^2$ global symmetry of the theory. In this note we study the $S$-matrix for inelastic scattering. We show that it exhibits poles corresponding to boundstates of bulk and boundary magnons. A crossing equation is derived for the overall phase. It reproduces the crossing equation for maximal giant gravitons, in the appropriate limit. Finally, scattering in the $su(2)$ sector is computed to two loops. This two loop result, which determines the overall phase to two loops, will be useful when a unique solution to the crossing equation is to be selected.

hep-th↗

arXiv2016

LLM Magnons

We consider excitations of LLM geometries described by coloring the LLM plane with concentric black rings. Certain closed string excitations are localized at the edges of these rings. The string theory predictions for the energies of magnon excitations of these strings depends on the radii of the edges of the rings. In this article we construct the operators dual to these closed string excitations and show how to reproduce the string theory predictions for magnon energies by computing one loop anomalous dimensions. These operators are linear combinations of restricted Schur polynomials. The distinction between what is the background and what is the excitation is accomplished in the choice of the subgroup and the representations used to construct the operator.

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