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arXiv:2511.20769 · Journal of High Energy Physics

Krylov complexity of supersymmetric SYK models

James Chryssanthacopoulos, David Vegh

Published Aug 24, 2026• 0 citations
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Abstract

A bstract We study the effect of supersymmetry breaking on Krylov complexity in the $$ \mathcal{N}=2 $$ N = 2 SYK model under irrelevant and mass deformations of the Hamiltonian. The irrelevant deformation breaks $$ \mathcal{N}=2 $$ N = 2 supersymmetry down to $$ \mathcal{N}=1 $$ N = 1 , while the mass deformation breaks supersymmetry completely. Using Krylov subspace methods, we analyze the Lanczos sequence, Krylov dimension, complexity, and entropy of the undeformed model and both deformations at finite system size. Both deformations enlarge the Krylov space and raise the saturation complexity as the BPS degeneracy is lifted. For the system sizes explored, the irrelevant deformation drives the saturation complexity to roughly half the Krylov dimension, while the mass deformation decreases it over an intermediate range of deformation strengths as the system drifts toward integrability. These distinct behaviors reveal how the mechanism of supersymmetry breaking leaves an imprint on quantum complexity.

DOI
10.1007/jhep08(2026)178

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A bstract We consider a perturbative expansion of the Lanczos coefficients and the Krylov complexity for two-dimensional conformal field theories under integrable deformations. Specifically, we explore the consequences of TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ , JT‾ \textrm{J}\overline{\textrm{T}} JT J T ¯ , and JJ‾ \textrm{J}\overline{\textrm{J}} JJ J J ¯ deformations, focusing on first-order corrections in the deformation parameter. Under TT‾ \textrm{T}\overline{\textrm{T}} TT T T ¯ deformation, we demonstrate that the Lanczos coefficients b n exhibit unexpected behavior, deviating from linear growth within the valid perturbative regime. Notably, the Krylov exponent characterizing the rate of exponential growth of complexity surpasses that of the undeformed theory for positive value of deformation parameter, suggesting a potential violation of the conjectured operator growth bound within the realm of perturbative analysis. One may attribute this to the existence of logarithmic branch points along with higher order poles in the autocorrelation function compared to the undeformed case. In contrast to this, both JJ‾ \textrm{J}\overline{\textrm{J}} JJ J J ¯ and JT‾ \textrm{J}\overline{\textrm{T}} JT J T ¯ deformations induce no first order correction to either the linear growth of Lanczos coefficients at large- n or the Krylov exponent and hence the results for these two deformations align with those of the undeformed theory.
The quantum dynamics of a complex system can be efficiently described in Krylov space, the minimal subspace in which the dynamics unfolds. We apply the Krylov subspace method for Hamiltonian deformations, which provides a systematic way of constructing solvable models from known instances. In doing so, we relate the evolution of deformed and undeformed theories and investigate their complexity. For a certain class of deformations, the resulting Krylov subspace is unchanged, and we observe time evolutions with a reorganized basis. The tridiagonal form of the generator in the Krylov space is maintained, and we obtain generalized Toda equations as a function of the deformation parameters. The imaginary-time-like evolutions can be described by real-time unitary ones. As possible applications, we discuss coherent Gibbs states for thermodynamic systems, for which we analyze the survival probability, spread complexity, Krylov entropy, and associated time-averaged quantities. We further discuss the statistical properties of random matrices and supersymmetric systems for quadratic deformations.
In this work we study the relationship between quantum random walks on graphs and Krylov/spread complexity. We show that the latter's definition naturally emerges through a canonical method of reducing a graph to a chain, on which we can identify the usual Krylov structure. We use this identification to construct families of graphs corresponding to special classes of systems with known complexity features and conversely, to compute Krylov complexity for graphs of physical interest. The two main outcomes are the analytic computation of the Lanczos coefficients for the SYK model for an arbitrary number qqq of interacting fermions and the complete characterization of Krylov complexity for the hypercube graph in any number of dimensions. The latter serves as the starting point for an in-depth comparison between Krylov and circuit complexities as they purportedly arise in the context of black holes. We find that while under certain conditions Krylov complexity follows the growth and saturation pattern ascribed to such systems, the timescale at which saturation happens can generally be shorter than what is predicted by random unitary circuits, due to the effects of quantum speed-ups commonly occurring when comparing quantum and classical random walks.
A bstract Heisenberg time evolution under a chaotic many-body Hamiltonian H transforms an initially simple operator into an increasingly complex one, as it spreads over Hilbert space. Krylov complexity, or ‘K-complexity’, quantifies this growth with respect to a special basis, generated by H by successive nested commutators with the operator. In this work we study the evolution of K-complexity in finite-entropy systems for time scales greater than the scrambling time t s > log( S ). We prove rigorous bounds on K-complexity as well as the associated Lanczos sequence and, using refined parallelized algorithms, we undertake a detailed numerical study of these quantities in the SYK 4 model, which is maximally chaotic, and compare the results with the SYK 2 model, which is integrable. While the former saturates the bound, the latter stays exponentially below it. We discuss to what extent this is a generic feature distinguishing between chaotic vs. integrable systems.
We study Krylov complexity in BMN Plane Wave Matrix Model at large mass deformation. We consider various consistent reductions of the matrix model that allow us to perform a Hamiltonian analysis which leads to different notions of the Krylov complexity. In the first part of the paper, we study the Krylov state complexity considering systematic reduction of N=3N=3N=3 and N=4N=4N=4 representations of the matrix model, which reveals a universal characteristic scaling for the Lanczos coefficients and fix them completely in terms of the mass deformation parameter. In the second part of the paper, we study the Krylov operator growth in the matrix model and compute the corresponding Lanczos coefficients. In both cases, we observe a \emph{linear} scaling of Lanczos coefficients with the mass parameter. The early time growth in Krylov complexity receives quadratic correction due to the presence of the massive deformation in the matrix model. Our analysis reveals that such massive corrections appear at same order in time for both the notion of the Krylov complexity.
A bstract Considering the large q expansion of the Sachdev-Ye-Kitaev (SYK) model in the two-stage limit, we compute the Lanczos coefficients, Krylov complexity, and the higher Krylov cumulants in subleading order, along with the t/q effects. The Krylov complexity naturally describes the “size” of the distribution while the higher cumulants encode richer information. We further consider the double-scaled limit of SYK q at infinite temperature, where q ~ N \sqrt{N} N​ N . In such a limit, we find that the scrambling time shrinks to zero, and the Lanczos coefficients diverge. The growth of Krylov complexity appears to be “hyperfast”, which is previously conjectured to be associated with scrambling in de Sitter space.