The operator wave function provides a fine-grained description of quantum chaos and of the irreversible growth of simple operators into increasingly complex ones. Remarkably, at finite temperature, this wave function can acquire a phase that increases linearly with the operator's size, a phenomenon called size winding. Although size winding occurs naturally in a holographic setting, the emergence of a coherent phase in a scrambled operator remains mysterious from the standpoint of a thermalizing quantum many-body system. In this work, we elucidate this phenomenon by introducing the related concept of Krylov winding, whereby the operator wave function acquires a phase that winds linearly with the Krylov index. We show that Krylov winding is a generic feature of quantum chaotic systems and is a direct consequence of the universal operator growth bound hypothesis. It gives rise to size winding under two additional conditions: (i) a low-rank mapping between the Krylov and size bases, which ensures phase alignment among operators of the same size, and (ii) the saturation of the "chaos-operator growth" bound λ_{L}≤2α (with λ_{L} the Lyapunov exponent and α the growth rate), which ensures a linear phase dependence on size. For systems that do not saturate this bound, with h=λ_{L}/2α<1, the winding with Pauli size ℓ becomes superlinear, behaving as ℓ^{1/h}. We illustrate these results with two classes of microscopic models: the Sachdev-Ye-Kitaev (SYK) model and its variants, and a disordered k-local spin model.
The growth of simple operators is essential for the emergence of chaotic dynamics and quantum thermalization. Recent studies have proposed different measures, including the out-of-time-order correlator and Krylov complexity. It is established that the out-of-time-order correlator serves as the signature of quantum many-body chaos, while the Krylov complexity provides its upper bound. However, there exist nonchaotic systems in which Krylov complexity grows exponentially, indicating that the Krylov complexity itself is not a witness of many-body chaos. In this Letter, we introduce the missing ingredient called the Krylov metric K_{mn}, which probes the size of the Krylov basis. We propose that the universal criteria for fast scramblers include (i) the exponential growth of Krylov complexity, (ii) the diagonal elements K_{nn}∼n^{h} with h∈(0,1], and (iii) the negligibility of off-diagonal elements K_{mn} with m≠n. We further show that h=ϰ/2α is a ratio between the quantum Lyapunov exponent ϰ and the Krylov exponent α. This proposal is supported by both generic arguments and explicit examples, including solvable Sachdev-Ye-Kitaev models, Luttinger liquids, and many-body localized systems. Our results provide a refined understanding of how chaotic dynamics emerge from the Krylov space perspective.
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.
A hypothesis is presented for the universal properties of operators evolving under Hamiltonian dynamics in many-body systems. The hypothesis states that successive Lanczos coefficients in the continued fraction expansion of the Green's functions grow linearly with rate
alpha in generic systems, with an extra logarithmic correction in 1d. The rate
alpha --- an experimental observable --- governs the exponential growth of operator complexity in a sense that is made precise. This exponential growth prevails beyond semiclassical or large-
N limits. Moreover,
alpha upper bounds a large class of operator complexity measures, including the out-of-time-order correlator. As a result, a sharp bound is obtained on Lyapunov exponents
lambdaLleq2alpha, which complements and improves the known universal low-temperature bound
lambdaLleq2piT. Our results are illustrated in paradigmatic examples such as non-integrable spin chains, the Sachdev-Ye-Kitaev model, and classical models. Finally hypothesis is used in in conjunction with the recursion method to develop a technique for computing diffusion constants.
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.
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.
A bstract We study the operator growth in open quantum systems with dephasing dissipation terms, extending the Krylov complexity formalism of [1]. Our results are based on the study of the dissipative q -body Sachdev-Ye-Kitaev (SYK q ) model, governed by the Markovian dynamics. We introduce a notion of “operator size concentration” which allows a diagrammatic and combinatorial proof of the asymptotic linear behavior of the two sets of Lanczos coefficients ( a n and b n ) in the large q limit. Our results corroborate with the semi-analytics in finite q in the large N limit, and the numerical Arnoldi iteration in finite q and finite N limit. As a result, Krylov complexity exhibits exponential growth following a saturation at a time that grows logarithmically with the inverse dissipation strength. The growth of complexity is suppressed compared to the closed system results, yet it upper bounds the growth of the normalized out-of-time-ordered correlator (OTOC). We provide a plausible explanation of the results from the dual gravitational side.