Universal chaotic dynamics from Krylov space

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.

Integrability and complexity in quantum spin chains

“…There is a related branch of research that appeals to Krylov complexity [7,8] in an attempt to quantify how many different operators/states one needs to approximate well the time evolution starting from a given operator/state. 1 While in all of these approaches, some difference has been observed between complexities of integrable and chaotic evolutions [5,6,17], no direct relation between the integrability structure (the presence of a large number of analytic conservation laws) and complexity reduction has been presented (see also [18]). Our aim in this article is to close this gap.…”

“…The discrete minimization problem (18) is solved at any time t by choosing k n such that E ′ n ∈ [−π, π[. Geometrically, one can think of the real vector Et extending as time evolves with a constant slope through a D-dimensional hypercubic lattice of spacing equal to 2π.…”

“…Geometrically, one can think of the real vector Et extending as time evolves with a constant slope through a D-dimensional hypercubic lattice of spacing equal to 2π. The minimization in (18) is then straightforwardly solved at any instant of time by projecting the vector Et onto the closest lattice site in (2π ) D . This can be achieved by rounding each component E n t/(2π) to the nearest integer…”

“…Since the manifold of unitaries is compact, the optimal distance between any two unitary operators, and hence the complexity, is bounded from above. Within our Ansatz, (18) is manifestly bounded from above by Dπ since every term in the sum can be chosen to lie between −π and π using the optimal k n given by (19). However, this rough upper bound typically overestimates the saturation value of the complexity [6].…”

“…Such a structure allows some of the integer vectors k in (26) to point in null directions of the Q-matrix, hence contributing very little to the complexity. This provides a way to reduce the coefficients E n t by combinations of integer subtractions (up to factors of 2π) almost as efficiently as in the bi-invariant metric (18). Combining many subtractions in purely local directions, when possible, will generally be more effective than forcing the E ′ n to lie exactly between −π and π using (19), since this latter approach generally induces large penalties due to contributions from nonlocal directions.…”