Krylov complexity has emerged as an important tool in the description of quantum information and, in particular, quantum chaos. Here we formulate Krylov complexity $K(t)$ for quantum mechanical systems as a path integral, and argue that at large times, for classical chaotic systems with at least two minima of the potential, that have a plateau for $K(t)$, the value of the plateau is described by quantum mechanical instantons, as is the case for standard transition amplitudes. We explain and test these ideas in a simple toy model.
Krylov complexity has emerged as an important tool in the description of quantum information and, in particular, quantum chaos. Here we formulate Krylov complexity K(t) for quantum mechanical systems as a path integral, and argue that at large times, for classical chaotic systems with at least two minima of the potential, that have a plateau for K(t), the value of the plateau is described by quantum mechanical instantons, as is the case for standard transition amplitudes. We explain and test these ideas in a simple toy model.
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.
Abstract In this work, we explore in detail, the time evolution of Krylov complexity. We demonstrate, through analytical computations, that in finite many-body systems, while ramp and plateau are two generic features of Krylov complexity, the manner in which complexity saturates reveals the chaotic nature of the system. In particular, we show that the dynamics towards saturation precisely distinguish between chaotic and integrable systems. For chaotic models, the saturation value of complexity reaches its infinite time average at a finite saturation time. In this case, depending on the initial state, it may also exhibit a peak before saturation. In contrast, in integrable models, complexity approaches the infinite time average value from below at a much longer timescale. We confirm this distinction using numerical results for specific spin models.
Abstract Krylov complexity measures the spread of the wavefunction in the Krylov basis, which is constructed using the Hamiltonian and an initial state. We investigate the evolution of the maximally entangled state in the Krylov basis for both chaotic and non-chaotic systems. For this purpose, we derive an Ehrenfest theorem for the Krylov complexity, which reveals its close relation to the spectrum. Our findings suggest that neither the linear growth nor the saturation of Krylov complexity is necessarily associated with chaos. However, for chaotic systems, we observe a universal rise-slope-ramp-plateau behavior in the transition probability from the initial state to one of the Krylov basis states. Moreover, a long ramp in the transition probability is a signal for spectral rigidity, characterizing quantum chaos. Also, this ramp is directly responsible for the late-time peak of Krylov complexity observed in the literature. On the other hand, for non-chaotic systems, this long ramp is absent. Therefore, our results help to clarify which features of the wave function time evolution in Krylov space characterize chaos. We exemplify this by considering the Sachdev-Ye-Kitaev model with two-body or four-body interactions.
Krylov state complexity, or spread complexity, has emerged as a sharp and versatile diagnostic of quantum chaos, information spreading, and many-body dynamics. Built from the Lanczos algorithm and grounded in the optimal-basis theorem, Krylov complexity thereby provides a robust spectroscopic window into quantum dynamics. A central theme is the characteristic overshoot observed in chaotic systems: a complexity peak in which chaotic evolution drives the state deeper into the Krylov chain than in integrable systems before relaxing to equilibrium. This behavior, tied to random-matrix universality classes of spectral statistics, is illustrated across a broad range of models, including quantum billiards, quantum spin chains, and variants of the SYK model. We also discuss proposed holographic descriptions of Krylov complexity in Einstein gravity, and conclude by outlining future directions and open problems, including time-dependent systems and quantum-field-theoretic formulations. A Mathematica notebook is provided for numerical exploration of Krylov complexity and spectral statistics across models.
Krylov complexity, as a novel measure of operator complexity under Heisenberg evolution, exhibits many interesting universal behaviors and also bounds many other complexity measures. In this work, we study Krylov complexity K(t) in Random Matrix Theory (RMT). In large N limit: (1) For infinite temperature, we analytically show that the Lanczos coefficient {bn} saturate to constant plateau n→∞limbn=b, rendering a linear growing complexity K(t)∼t, in contrast to the exponential-in-time growth in chaotic local systems in thermodynamic limit. After numerically comparing this plateau value b to a large class of chaotic local quantum systems, we find that up to small fluctuations, it actually bounds the {bn} in chaotic local quantum systems. Therefore we conjecture that in chaotic local quantum systems after scrambling time, the speed of linear growth of Krylov complexity cannot be larger than that in RMT. (2) For low temperature, we analytically show that bn will first exhibit linear growth with n, whose slope saturates the famous chaos bound. After hitting the same plateau b, bn will then remain constant. This indicates K(t)∼e2πt/β before scrambling time t∗∼O(βlogβ), and after that it will grow linearly in time, with the same speed as in infinite temperature. We finally remark on the effect of finite N corrections.
Quantum materials permeate the modern world - these are systems that exhibit quantum mechanical properties, like topological phases and superconductivity, at macroscopic scales. The principles governing quantum materials include entanglement and coherence -phenomena which are fundamentally quantum mechanical in nature, without classical counterparts. The semiconductors in mobile phones, computers, and solar cells; light emitting diodes (LEDs); sensors in medical equipment and other precision devices; Maglev trains and particle accelerators - all utilize quantum materials in one form or another. The trajectory of modernization over the past century has been significantly shaped by developments in quantum materials, underscoring the importance of studying their phases in detail. This thesis focuses on one aspect of the research program to improve the understanding of quantum materials: mathematical probing for the presence of quantum phases (QPs) of matter and the transitions (QPTs) among them. Specifically, it proposes that Krylov Complexity can be utilized to detect QPs and QPTs. Krylov Complexity is a quantity that has been recently proposed in the physics literature as a measure of chaotic nature of a quantum system i.e. it encodes the information transport properties of a system - exponential signatures in the Krylov Complexity typically characterize a chaotic quantum system. A priori, one may not expect Krylov Complexity to be sensitive to the presence of QPs and QPTs. This thesis gives evidence contrary to this expectation. The results demonstrate that Krylov Complexity exhibits distinctive signatures at the boundaries of QPs, such as sharp peaks or discontinuities, which correspond to the quantum critical points. Numerous techniques have been developed to study quantum materials both theoretically and experimentally: Tensor Network Methods, Renormalization Group Theory, Scanning Tunnel Microscopy, Hall Effect Measurement etc. This thesis highlights the simplicity and effectiveness of Krylov Complexity, which utilizes known information from studying Hamiltonians of many- body quantum systems with minimal additional computation. In summary, through a comprehensive review of the theoretical framework underpinning Krylov Complexity, this thesis provides compelling evidence that it is a simple yet effective tool for probing QPs and QPTs, thereby opening new avenues for understanding quantum materials, their phases, and the transitions among these phases.