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arXiv:2312.17416

Operator Krylov complexity in random matrix theory

Tang Haifeng

Published Dec 29, 2023• 1 citation
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Abstract

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 $\mathcal{K}(t)$ in Random Matrix Theory (RMT). In large $N$ limit: (1) For infinite temperature, we analytically show that the Lanczos coefficient $\{b_n\}$ saturate to constant plateau $\lim\limits_{n\rightarrow\infty}b_n=b$, rendering a linear growing complexity $\mathcal{K}(t)\sim t$, 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 $\{b_n\}$ 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 $b_n$ will first exhibit linear growth with $n$, whose slope saturates the famous chaos bound. After hitting the same plateau $b$, $b_n$ will then remain constant. This indicates $\mathcal{K}(t)\sim e^{2πt/β}$ before scrambling time $t_*\sim O(β\logβ)$, 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.

DOI
10.48550/arxiv.2312.17416

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Recently, a novel measure for the complexity of operator growth is proposed based on Lanczos algorithm and Krylov recursion method. We study this Krylov complexity in quantum mechanical systems derived from some well-known local toric Calabi–Yau geometries, as well as some nonrelativistic models. We find that for the Calabi–Yau models, the Lanczos coefficients grow slower than linearly for small [Formula: see text]’s, consistent with the behavior of integrable models. On the other hand, for the nonrelativistic models, the Lanczos coefficients initially grow linearly for small [Formula: see text]’s, then reach a plateau. Although this looks like the behavior of a chaotic system, it is mostly likely due to saddle-dominated scrambling effects instead, as argued in the literature. In our cases, the slopes of linearly growing Lanczos coefficients almost saturate a bound by the temperature. During our study, we also provide an alternative general derivation of the bound for the slope.
We continue the analysis of the Krylov complexity in the IP matrix model. In a previous paper, for a fundamental operator, it was shown that at zero temperature, the Krylov complexity oscillates and does not grow, but in the infinite temperature limit, the Krylov complexity grows exponentially in time as ∼exp⁡(O(t))\sim \exp\left( {\mathcal{O}\left( {\sqrt{t}}\right)} \right)∼exp(O(t​)). We study how the Krylov complexity changes from a zero-temperature oscillation to an infinite-temperature exponential growth. At low temperatures, the spectral density is approximated as collections of infinite Wigner semicircles. We showed that this infinite collection of branch cuts yields linear growth to the Lanczos coefficients and gives exponential growth of the Krylov complexity. Thus the IP model for any nonzero temperature shows exponential growth for the Krylov complexity even though the Green function decays by a power law in time. We also study the Lanczos coefficients and the Krylov complexity in the IOP matrix model taking into account the 1/N21/N^21/N2 corrections. There, the Lanczos coefficients are constants and the Krylov complexity does not grow exponentially as expected.
A bstract 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 the late-time bound to the amount of disorder present in the Lanczos sequence, by mapping the complexity evolution to an auxiliary off-diagonal Anderson hopping model. We compare the late-time saturation of K-complexity in the chaotic phase with that of random matrix ensembles and find that the chaotic system indeed approaches the RMT behavior in the appropriate symmetry class. We investigate the dependence of the results on the two key ingredients of K-complexity: the dynamics of the Hamiltonian and the character of the operator whose time dependence is followed.
Quantum chaotic systems are conjectured to display a spectrum whose fine-grained features (gaps and correlations) are well described by random matrix theory (RMT). We propose and develop a complementary version of this conjecture: quantum chaotic systems display a Lanczos spectrum whose local means and covariances are well described by RMT. To support this proposal, we first demonstrate its validity in examples of chaotic and integrable systems. We then show that for Haar-random initial states in RMTs the mean and covariance of the Lanczos spectrum suffice to produce the full long-time behavior of general survival probabilities including the spectral form factor, as well as the spread complexity. In addition, for initial states with continuous overlap with energy eigenstates, we analytically find the long-time averages of the probabilities of Krylov basis elements in terms of the mean Lanczos spectrum. This analysis suggests a notion of eigenstate complexity, the statistics of which differentiate integrable systems and classes of quantum chaos. Finally, we clarify the relation between spread complexity and the universality classes of RMT by exploring various values of the Dyson index and Poisson distributed spectra.
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 the operator growth problem and its complexity in the many-body localization (MBL) system from the Lanczos algorithm perspective. Using the Krylov basis, the operator growth problem can be viewed as a single-particle hopping problem on a semi-infinite chain with the hopping amplitudes given by the Lanczos coefficients. We find that, in the MBL systems, the Lanczos coefficients scale as \sim n/\ln(n) ∼ n / ln ( n ) asymptotically, same as in the ergodic systems, but with an additional even-odd alteration and an effective randomness. We use a simple linear extrapolation scheme as an attempt to extrapolate the Lanczos coefficients to the thermodynamic limit. With the original and extrapolated Lanczos coefficients, we study the properties of the emergent single-particle hopping problem via its spectral function, integrals of motion, Krylov complexity, wavefunction profile and return probability. Our numerical results of the above quantities suggest that the emergent single-particle hopping problem in the MBL system is localized when initialized on the first site. We also study the operator growth in the MBL phenomenological model, whose Lanczos coefficients also have an even-odd alteration, but approach constants asymptotically. The Krylov complexity grows linearly in time in this case.