Krylov localization and suppression of complexity

Abstract

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 systems. We associate this behavior with a novel localization phenomenon on the Krylov chain by mapping the theory of complexity growth and spread to an Anderson localization hopping model with off-diagonal disorder, and find that localization is enhanced in the integrable case due to a stronger disorder in the hopping amplitudes, inducing an effective suppression of Krylov complexity. We demonstrate this behavior for an interacting integrable model, the XXZ spin chain, and show that the same behavior results from a phenomenological model that we define: this model captures the essential features of our analysis and is able to reproduce the behaviors we observe for chaotic and integrable systems via an adjustable disorder parameter.

Krylov complexity from integrability to chaos

“…In [5] K-complexity was computed for complex SYK 4 systems and it was shown numerically that its time-dependent profile fits the one expected from quantum computation as well as from holography [2,27]. K-complexity was computed in [7] for the XXZ model which is a strongly interacting many-body integrable system, and was shown to saturate at late times at values below those found for SYK 4 which is a maximally chaotic system [28][29][30][31]. This paper aims to bridge the gap between the integrable and the chaotic by introducing a Hamiltonian which interpolates between the two, and studying K-complexity for a fixed type of local operator.…”

“…In addition to the behavior of K-complexity for given individual quantum systems, such as the SYK model [3,5,6], 2D CFTs [17,18], and more general symmetry-based Hamiltonian systems [19,20], it is interesting and important to categorize the possible Krylov phenomenologies according to more universal criteria. One of the most interesting of these is clearly the behavior of K-complexity in the class of chaotic quantum systems as opposed to that of integrable ones, initiated in [7] for systems away from the thermodynamic limit. Quantum integrable systems, such as the strongly interacting XXZ chain [21] or the quadratic SYK model, are less efficient at exploring Krylov space, as evidenced for example by their reaching a lower saturation value of K-complexity at late times.…”

“…Quantum integrable systems, such as the strongly interacting XXZ chain [21] or the quadratic SYK model, are less efficient at exploring Krylov space, as evidenced for example by their reaching a lower saturation value of K-complexity at late times. By mapping the dynamics of operator spreading to an off-diagonal Anderson-like hopping problem on the Krylov chain this under-saturation is linked to the (partial) localization of the wave function on the Krylov chain [7]. Maximally chaotic systems feature a late-time complexity saturation value which is exponential in the number of degrees of freedom [5]; interacting integrable systems saturate at quantitatively lower values as compared to chaotic models due to localization effects in Krylov space [7], and free systems typically depict complexity saturation values at late times which are linear, or polynomial, in the number of degrees of freedom [5].…”

“…A mathematically precise definition of the complexity of time evolution under a given Hamiltonian is given by Krylov complexity [3][4][5] or 'K-complexity' for short. To date, several aspects of Krylov complexity have been studied in various setups and systems, for example [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

Krylov complexity and orthogonal polynomials

“…In contrast, Krylov complexity is suppressed in integrating integrable models of finite size [37], which has been called Krylov localization. A number of simple systems, which are symmetry generated and allow an exact treatment using generalized coherent states, were considered in [38,39].…”