We study the statistical properties of Lanczos coefficients over an ensemble of random initial operators generating the Krylov space. We propose two statistical quantities that are important in characterizing the complexity: the average correlation matrix $\langle x_{i} x_{j}\rangle$ of Lanczos coefficients and the resulting distribution of the variance of Lanczos coefficients. Their resulting statistics are the Wishart distribution and the (rescaled) chi-square distribution respectively, which are independent of the distributions of initial operators and become the normal distribution in the case of large matrix size. As a numerical example, we use the typical billiard system with an integrability-breaking term and choose samples of random initial operators from given probability distributions (GOE, GUE and the uniform distribution). It agrees with the phenomenological analysis and further interesting behaviors are obtained, which indicates a consistent connection between RMT, Anderson localization and Krylov complexity.
Quantum chaos and late-time equipartition of symmetry-resolved Krylov complexity
Saha Institute of Nuclear Physics（萨哈核物理研究所）；Homi Bhabha National Institute（霍米·巴巴国立研究所）；Instituto de Física Teórica UAM/CSIC（马德里自治大学/西班牙国家科学研究委员会理论物理研究所）；Universidad Autónoma de Madrid（马德里自治大学）
Quantum Chaos and Spread of States in Krylov Subspace: A Topical Review
Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Pohang University of Science and Technology（浦项科技大学）；Instituto de Física Teórica UAM/CSIC（马德里自治大学/西班牙国家科学研究委员会理论物理研究所）
Krylov complexity is a novel approach to study how an operator spreads over a specific basis. Recently, it has been stated that this quantity has a long-time saturation that depends on the amount of chaos in the system. Since this quantity not only depends on the Hamiltonian but also on the chosen operator, in this work we study the level of generality of this hypothesis by studying how the saturation value varies in the integrability to chaos transition when different operators are expanded. To do this, we work with an Ising chain with a transverse-longitudinal magnetic field and compare the …
We apply a notion of quantum complexity, called "Krylov complexity", to study the evolution of systems from integrability to chaos. For this purpose we investigate the integrable XXZ spin chain, enriched with an integrability breaking deformation that allows one to interpolate between integrable and chaotic behavior. K-complexity can act as a probe of the integrable or chaotic nature of the underlying system via its late-time saturation value that is suppressed in the integrable phase and increases as the system is driven to the chaotic phase. We furthermore ascribe the (under-)saturation of t…
Universidad de Buenos Aires（布宜诺斯艾利斯大学）；CONICET - Universidad de Buenos Aires, Instituto de Física de Buenos Aires (IFIBA)（阿根廷国家科学研究委员会-布宜诺斯艾利斯大学，布宜诺斯艾利斯物理研究所）
Recently, the out-of-time-ordered correlator(OTOC) and Krylov complexity have been studied actively as a measure of operator growth. OTOC is known to exhibit exponential growth in chaotic systems, which was confirmed in many previous works. However, in some non-chaotic systems, it was observed that OTOC shows chaotic behavior and cannot distinguish saddle-dominated scrambling from chaotic systems. For K-complexity, in the universal operator growth hypothesis, it was stated that Lanczos coefficients show linear growth in chaotic systems, which is the fastest. But recently, it appeared that Lanc…
Unfolded Krylov complexity: universal chaotic dynamics without false positives
Julius-Maximilians-Universität Würzburg（维尔茨堡大学）；Shanghai Jiao Tong University（上海交通大学）；Pohang University of Science and Technology（韩国科学技术院）；Instituto de Física Teórica UAM/CSIC（马德里自治大学/西班牙国家科学研究委员会理论物理研究所）
Quantum complexity, suitably defined, has been suggested as an important probe of late-time dynamics of black holes, particularly in the context of AdS/CFT. A notion of quantum complexity can be effectively captured by quantifying the spread of an operator in Krylov space as a consequence of time evolution. Complexity is expected to behave differently in chaotic many-body systems, as compared to integrable ones. In this paper we investigate Krylov complexity for the case of interacting integrable models at finite size and find that complexity saturation is suppressed as compared to chaotic sys…
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 al…
Quantum Chaos Diagnostics for non-Hermitian Systems from Bi-Lanczos Krylov Dynamics
Shanghai Jiao Tong University（上海交通大学）；Shanghai Research Center for Quantum Sciences（上海量子科学研究中心）；Instituto de Física Teórica UAM/CSIC（UAM/CSIC理论物理研究所）；Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Pohang University of Science and Technology（韩国科学技术院）；Technical University of Munich（慕尼黑工业大学）；Ludwig-Maximilians-Universität München（慕尼黑大学）；Gwangju Institute of Science and Technology（光州科学技术研究院）
In semi-classical systems, the exponential growth of the out-of-timeorder correlator (OTOC) is believed to be the hallmark of quantum chaos. However,on several occasions, it has been argued that, even in integrable systems, OTOC can grow exponentially due to the presence of unstable saddle points in the phase space. In this work, we probe such an integrable system exhibiting saddle dominated scrambling through Krylov complexity and the associated Lanczos coefficients. In the realm of the universal operator growth hypothesis, we demonstrate that the Lanczos coefficients follow the linear growth…
We study the spectral properties of two classes of random matrix models: non-Gaussian RMT with quartic and sextic potentials, and RMT with Gaussian noise. We compute and analyze the quantum Krylov complexity and the spectral form factor for both of these models. We find that both models show suppression of the spectral form factor at short times due to decoherence effects, but they differ in their long-time behavior. In particular, we show that the Krylov complexity for the non-Gaussian RMT and RMT with noise deviates from that of a Gaussian RMT. We discuss the implications and limitations of …
Probing the Chaos to Integrability Transition in Double-Scaled SYK
Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Julius-Maximilians-Universität Würzburg（维尔茨堡大学）；Weizmann Institute of Science（魏茨曼科学研究所）；Massachusetts Institute of Technology（麻省理工学院）；Freie Universität Berlin（柏林自由大学）
Analytic Spread Complexity from Level Statistics: From Chaos to Integrability
Mandelstam Institute for Theoretical Physics, School of Physics, University of the Witwatersrand（曼德尔斯坦理论物理研究所，物理学系，威特沃特斯兰德大学）；Instituto de Física Teórica UAM/CSIC（UAM/CSIC理论物理研究所）
Inspired by the universal operator growth hypothesis, we extend the formalism of Krylov construction in dissipative open quantum systems connected to a Markovian bath. Our construction is based upon the modification of the Liouvillian superoperator by the appropriate Lindbladian, thereby following the vectorized Lanczos algorithm and the Arnoldi iteration. This is well justified due to the incorporation of non-Hermitian effects due to the environment. We study the growth of Lanczos coefficients in the transverse field Ising model (integrable and chaotic limits) for boundary amplitude damping a…
