Krylov complexity of anyons

Ergodicity and thermalization in quantum many-body systems are fundamental concepts in statistical mechanics, typically governed by the eigenstate thermalization hypothesis. However, systems can exhibit weak ergodicity breaking through phenomena such as quantum many-body scars, where certain initial states fail to thermalize. While standard quantum statistics distinguish sharply between bosons and fermions, anyons obey fractional statistics that interpolate between these two limits. This paper explores how these fractional statistical phases and on-site interactions jointly govern quantum-state complexity, specifically examining the spreading of quantum states in Krylov space within the one-dimensional anyon-Hubbard model.

The authors analyze the one-dimensional anyon-Hubbard model, which incorporates a density-dependent gauge field encoding fractional statistics via a statistical phase. Using exact diagonalization, analytical constructions, and the Lanczos algorithm, they compute entanglement entropy, fidelity, and Krylov complexity. They construct exact many-body scar eigenstates as towers of states and investigate their finite-size scaling. Furthermore, they simulate sudden statistical-phase quenches across both insulating and superfluid regimes to observe how initial-state and post-quench parameters influence the temporal growth and saturation of Krylov complexity.

The analysis reveals that the anyon-Hubbard model harbors exact many-body scar states whose entanglement entropy exhibits logarithmic scaling rather than volume-law scaling, confirming weak ergodicity breaking. In the absence of quenches, these scar states display perfect periodic revivals of fidelity and Krylov complexity that are entirely independent of the statistical phase. Following a sudden quench of the statistical phase, the resulting Krylov complexity dynamics depend heavily on both the interaction regime and the magnitude of the phase change. In the insulating regime, complexity growth is slow and yields long-lived approximate scars, whereas in the superfluid regime, the complexity rapidly grows and saturates at a high plateau. Short-time growth universally exhibits quadratic behavior, while intermediate-time behavior transitions toward linear scaling in the superfluid regime.