We investigate the Krylov complexity of thermofield double states in systems with mixed phase space, uncovering a direct correlation with the Brody distribution, which interpolates between Poisson and Wigner statistics. Our analysis spans two-dimensional random matrix models featuring (I) GOE-Poisson and (II) GUE-Poisson transitions and extends to higher-dimensional cases, including a stringy matrix model (GOE-Poisson) and the mass-deformed SYK model (GUE-Poisson). Krylov complexity consistently emerges as a reliable marker of quantum chaos, displaying a characteristic peak in the chaotic regime that gradually diminishes as the Brody parameter approaches zero, signaling a shift toward integrability. These results establish Krylov complexity as a powerful diagnostic of quantum chaos and highlight its interplay with eigenvalue statistics in mixed phase systems.
We investigate the Krylov complexity of thermofield double states in systems with mixed phase space, uncovering a direct correlation with the Brody distribution, which interpolates between Poisson and Wigner statistics. Our analysis spans two-dimensional random matrix models featuring (I) GOE-Poisson and (II) GUE-Poisson transitions and extends to higher-dimensional cases, including a stringy matrix model (GOE-Poisson) and the mass-deformed SYK model (GUE-Poisson). Krylov complexity consistently emerges as a reliable marker of quantum chaos, displaying a characteristic peak in the chaotic regime that gradually diminishes as the Brody parameter approaches zero, signaling a shift toward integrability. These results establish Krylov complexity as a powerful diagnostic of quantum chaos and highlight its interplay with eigenvalue statistics in mixed phase systems.
Krylov complexity has recently emerged as a new paradigm to characterize quantum chaos in many-body systems. However, which features of Krylov complexity are a prerogative of quantum chaotic systems and how they relate to more standard probes, such as spectral statistics or out-of-time-order correlators (OTOCs), remain open questions. Recent insights have revealed that in quantum chaotic systems Krylov state complexity exhibits a distinct peak during time evolution before settling into a well-understood late-time plateau. In this work we propose that this Krylov complexity peak (KCP) is a hallmark of quantum chaotic systems and suggest that its height could serve as an order parameter for quantum chaos. We demonstrate that the KCP effectively identifies chaotic-integrable transitions in two representative quantum-mechanical models at both infinite and finite temperature: the mass-deformed Sachdev-Ye-Kitaev model and the sparse Sachdev-Ye-Kitaev model. Our findings align with established results from spectral statistics and OTOCs while introducing an operator-independent diagnostic for quantum chaos, offering more universal insights and a deeper understanding of the general properties of quantum chaotic systems.
The Yukawa-Sachdev-Ye-Kitaev (YSYK) model, a solvable model of non-Fermi liquids with a proposed cavity-QED realization, couples N complex fermions to M bosons of frequency ω0 through random Yukawa couplings of strength g, and its dynamics depends only on R≡ω0/g2/3. For R≪1 the bosons are slow and act as a static random background in which the fermions are free, while for R≫1 they are only virtually excited and generate a random four-fermion interaction of rank M, which is generically chaotic for M≥2. Using exact diagonalization at infinite temperature, we follow the emergence of random-matrix statistics between these limits with unfolded spectral statistics and the Krylov complexity of the thermofield-double state. For complex couplings the level-spacing distribution evolves from Poisson to Gaussian-unitary-ensemble (GUE) statistics, the spectral form factor develops a correlation hole and a ramp, and the Krylov complexity evaluated on the unfolded spectrum develops a peak above its late-time plateau whose height reaches the GUE value. All diagnostics locate the onset around R≈0.3, and adding boson modes moves it to smaller R. This delineates the parameter regime in which a cavity-QED realization would display random-matrix spectral correlations. On the other hand, real couplings give the same crossover towards Gaussian-orthogonal-ensemble (GOE) statistics, and along an interpolation between the two the peak height follows the mean gap ratio, so the peak identifies the symmetry class as well as the onset of chaos. Without unfolding the peak stays about 20% below its random-matrix value in both classes, which we trace to the non-semicircular density of states, and at larger R the boson-number bands slow down the spreading of the state in Krylov space.
A bstract In quantum many-body systems, time-evolved states typically remain confined to a smaller region of the Hilbert space known as the Krylov subspace . The time evolution can be mapped onto a one-dimensional problem of a particle moving on a chain, where the average position 〈 n 〉 defines Krylov state complexity or spread complexity. Generalized spread complexities, associated with higher-order moments 〈 n p 〉 for p > 1, provide finer insights into the dynamics. We investigate the time evolution of generalized spread complexities following a quantum quench in random matrix theory. The quench is implemented by transitioning from an initial random Hamiltonian to a post-quench Hamiltonian obtained by dividing it into four blocks and flipping the sign of the off-diagonal blocks. This setup captures universal features of chaotic quantum quenches. When the initial state is the thermofield double state of the post-quench Hamiltonian, a peak in spread complexity preceding equilibration signals level repulsion, a hallmark of quantum chaos. We examine the robustness of this peak for other initial states, such as the ground state or the thermofield double state of the pre-quench Hamiltonian. To quantify this behavior, we introduce a measure based on the peak height relative to the late-time saturation value. In the continuous limit, higher-order complexities show increased sensitivity to the peak, supported by numerical simulations for finite-size random matrices.
We employ machine learning techniques to estimate the amount of information contained in three Krylov-space observables: the spread complexity C(t), the discrete Wigner negativity N(t), and the normalized negativity χ(t)=N(t)/∣S(t)∣, where S(t) is the survival amplitude. We study thermofield-double evolutions on GUE, GOE and Poisson spectra, and on the integrable SL(2,R) Krylov chain of a two-dimensional CFT primary, with Lanczos coefficients bn=αn(n+2h−1). To introduce chaos we add a GUE random matrix to the integrable chain, H(ε)=HSL(2,R)+εR0WGUE, where R0 is its spectral half-width. Either C(t) or N(t) alone determines the thermofield temperature, with R2≃0.999. However, neither can reconstruct the fine structure of the spectral form factor. In the integrable sector all three observables carry the same information. When we turn on chaos an asymmetry appears: χ predicts C better than C predicts χ. This gap grows from +0.33 to +0.77, while the same gap for the raw negativity decays to zero. The extra information comes from the survival amplitude, and χ(t) carries it while C(t) can still be recovered from χ(t). We derive a lower bound on this gap, set by the fraction of the variance of χ that comes from S.
Krylov complexity has recently been proposed as a quantum probe of chaos. The Krylov exponent characterizing the exponential growth of Krylov complexity is conjectured to upper-bound the Lyapunov exponent. We compute the Krylov and the Lyapunov exponents in the Sachdev-Ye-Kitaev model and in some of its deformations. We do this analysis both at infinite and finite temperatures, in models where the number of fermionic interactions is both finite and infinite. We consider deformations that interpolate between two regions of near-maximal chaos and deformations that become nearly integrable at low temperatures. In all cases, we find that the Krylov exponent upper-bounds the Lyapunov one. However, we find that while the Lyapunov exponent can have nonmonotonic behavior as a function of temperature, in all studied examples the Krylov exponent behaves monotonically. For instance, we find models where the Lyapunov exponent goes to zero at low temperatures, while the Krylov exponent saturates to its maximal bound. We speculate on the possibility that this monotonicity might be a generic feature of the Krylov exponent in quantum systems evolving under unitary evolution.
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.