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arXiv:2608.07258

Spectral Topology and Universal Krylov Dynamics

Jeff Murugan, Hendrik J. R. Van Zyl, Masataka Watanabe

Published Aug 7, 2026• 0 citations
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Abstract

The leading asymptotic growth of Lanczos coefficients is controlled by spectral tails and furnishes a coarse classification of Krylov dynamics. We show that the \textit{global topology} of the spectral measure, specifically the number of connected components, the gap structure, and the behaviour at gap-closing transitions, encodes a finer hierarchy of dynamical invariants invisible to tail-based arguments. Using the Riemann-Hilbert formulation of orthogonal polynomials and Deift-Zhou steepest descent, we recover the Freud growth laws $b_n\sim n^{1/β}$ for single-cut measures and determine their sub-leading corrections from endpoint data. Gapped spectra produce quasiperiodic Lanczos oscillations at a frequency fixed by the filling fraction of the spectral bands alone, and hence predictable from the band edges. We verify this in the SSH chain and its next-nearest-neighbour deformation. At a gap-closing transition the oscillation amplitude is governed by the Hastings-McLeod solution of Painlevé II, decaying as $n^{-1/3}$ at criticality and interpolating between the gapped and merged phases, so that the topology change of the spectral curve is realised as a Krylov phase transition. We also demonstrate that, while in the conformal limit of SYK the operator scaling dimension is invisible in the leading rate $α= πT$, it can be extracted from the subleading offset $b_0 = πT(Δ- \frac{1}{2})$. These results establish a refined notion of universality in operator growth, classified by spectral topology rather than spectral tails alone.

DOI
10.48550/arxiv.2608.07258

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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.
A bstract We study Krylov complexity in Schrödinger field theory in the grand canonical ensemble with chemical potential μ , with an emphasis on the qualitatively new features that arise for μ > 0. In this regime the fermionic Wightman power spectrum is effectively single-sided and sharply truncated at ω = μ , which induces a crossover in the Lanczos coefficients and signals a dynamical transition from a bulk-dominated regime to a spectral-edge-dominated regime: b n displays a two-stage linear growth (from an early-time slope π/β to an asymptotic slope 2 /β ), while a n bends from near-zero values to a linear descent with slope −4 /β . We provide analytic support for the resulting complexity growth from three complementary viewpoints: (i) using an SL(2 , ℝ) algebraic construction matched to the asymptotic Lanczos data, we show that the late-time Krylov complexity must grow quadratically, K ( t ) ∝ t 2 ; (ii) by analyzing engineered Wightman spectra with controlled decay and truncation, we identify single-sided exponential decay as the key spectral feature responsible for the quadratic asymptotics, while an approximately even two-sided exponential spectrum explains the early-time K ( t ) ~ sinh 2 ( πt/β ) behavior at large μ ; (iii) we formulate the problem in terms of orthogonal polynomials and estimate the crossover scale separating the early- and late-stage regimes. Overall, our results help clarify the role of chemical potential and spectral truncation in shaping operator growth and Krylov complexity in this non-relativistic quantum field theory setting.
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Krylov complexity is a powerful diagnostic of quantum dynamics, with clear connections to other measures of quantum chaos and operator growth. One such measure is the Loschmidt amplitude, defined as the overlap of initially identical states evolved under two slightly different Hamiltonians. Its decay in certain systems is controlled by the classical Lyapunov exponent. Using the algebraic properties of the Krylov complexity operator, we express Krylov complexity as the derivative of a Loschmidt amplitude whose perturbation is parameterized by an angular variable φφφ. This formulation allows us to define a spectral propagator that encodes the entire complexity distribution, which we characterize for specific types of systems. We study the two-dimensional quantum geometry spanned by time and φφφ where the original and deformed trajectories reside, demonstrating that Krylov complexity is upper-bounded by its volume. We also express the time derivative of Krylov complexity in terms of a distinct Loschmidt amplitude. Depending on the growth of the Lanczos coefficients, the perturbation term in this amplitude can be truncated. We propose that the strength of this perturbation provides a classification scheme for Krylov complexity dynamics and relate it to the φφφ-derivative of the spectral propagator. Using this analytical framework, we derive general relations between the time-dependence of the survival amplitude and Krylov space measures.
Krylov subspace methods quantify operator growth in quantum many-body systems through Lanczos coefficients that encode how operators spread under time evolution. Although these diagnostics were originally motivated by questions of chaos and integrability, quadratic fermionic Hamiltonians are often expected to exhibit trivial Lanczos structure. Here we show that, in the long-range Kitaev chain, Lanczos coefficients generated from local boundary operators sharply diagnose whether the lowest excitation gap is controlled by boundary-localized or bulk-extended modes. We introduce the KrylovKrylovKrylov staggeringstaggeringstaggering parameterparameterparameter for the Lanczos coefficients. In the short-range Kitaev chain with balanced hopping and pairing, we derive analytically for arbitrary system size (valid in the thermodynamic limit) and show that this quantity is exactly constant and its sign cleanly distinguishes the topological phase with Majorana edge modes from the trivial phase. Away from that limit, long-range couplings and pairing-hopping imbalance deform the simple flat structure and analytical control is lost, nevertheless, we show that the sign pattern of the diagnostic still tracks whether the lowest excitation gap is controlled by boundary modes or by bulk excitations. These results are enabled by an exact single-particle operator Lanczos algorithm, as derived in this work, which reduces the recursion from exponentially large operator space to a finite-dimensional linear problem and achieves machine precision for chains of hundreds of sites. Krylov diagnostics thus emerge as practical probes of boundary-versus-bulk low-energy physics in topological superconductors with broken U(1)U(1)U(1) symmetry and algebraically decaying couplings.
A bstract Quantum observables of generic many-body systems exhibit a universal pattern of growth in the Krylov space of operators. This pattern becomes particularly manifest in the Lanczos basis, where the evolution superoperator assumes the tridiagonal form. According to the universal operator growth hypothesis, the nonzero elements of the superoperator, known as Lanczos coefficients, grow asymptotically linearly. We introduce and explore broad families of Lanczos coefficients that are consistent with the universal operator growth and lead to the exactly solvable dynamics. Within these families, the subleading terms of asymptotic expansion of the Lanczos sequence can be controlled and fine-tuned to produce diverse dynamical patterns. For one of the families, the Krylov complexity is computed exactly.