Krylov complexity from integrability to chaos

Abstract

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.

Entanglement and geometry from subalgebras of the Virasoro

“…19 See [38,39,68,[70][71][72][73][74][75][76] for some recent developments pedagogical reviews. 20 The local k = 1 case was analyzed in [77].…”

Probing quantum scars and weak ergodicity breaking through quantum complexity

“…This is a somewhat different formalism as compared to the formalism of Krylov complexity for describing operator growth [50]. This notion finds various applications in the study of chaos, scrambling, and integrability in many-body quantum and semiclassical systems [51,52,[54][55][56][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78].…”

A Tale of Two Hungarians: Tridiagonalizing Random Matrices

“…Note first that the probability distribution in ( 25) is proportional to exp −N/4 Tr(H 2 ) with a symmetric H and hence is independently Gaussian in each entry of H up to the symmetricity constraint. Thus, since the entry a 0 is unchanged after the transformation (32), it continues to be Gaussian distributed with the same mean and variance as the GOE, namely 2/N in our normalization since it belongs to the diagonal. The norm ||x|| 2 is then the square root of the sum of uncorrelated Gaussian random variables with zero mean and variance equal to the off-diagonal entries of the GOE, which is 1/N .…”

“…In this context Ref. [20,32] use RMT techniques to analyze certain aspects of Krylov complexity. The Lanczos approach has also been used to compute out-of-time-ordered four-point functions and Lyapunov exponents [34].…”