Probing quantum scars and weak ergodicity-breaking through quantum complexity

TL;DR

This paper investigates quantum scars and ergodicity-breaking by analyzing the complexity of states in the PXP model, revealing revivals linked to approximate SU(2) symmetry and providing analytic insights into the dynamics of special eigenstates.

Contribution

It introduces a Krylov complexity framework for quantum scars, deriving analytic expressions for Lanczos coefficients and demonstrating complexity revivals due to approximate SU(2) symmetry.

Findings

Neel state complexity revives approximately over time.

Generic ETH states show increasing complexity without revival.

Deformed Hamiltonian maintains near-perfect revivals for the Neel state.

Abstract

Scar states are special many-body eigenstates that weakly violate the eigenstate thermalization hypothesis (ETH). Using the explicit formalism of the Lanczos algorithm, usually known as the forward scattering approximation in this context, we compute the Krylov state (spread) complexity of typical states generated by the time evolution of the PXP Hamiltonian, hosting such states. We show that the complexity for the Neel state revives in an approximate sense, while complexity for the generic ETH-obeying state always increases. This can be attributed to the approximate SU(2) structure of the corresponding generators of the Hamiltonian. We quantify such ''closeness'' by the q-deformed SU(2) algebra and provide an analytic expression of Lanczos coefficients for the Neel state within the approximate Krylov subspace. We intuitively explain the results in terms of a tight-binding model. We further consider a deformation of the PXP Hamiltonian and compute the corresponding Lanczos coefficients and the complexity. We find that complexity for the Neel state shows nearly perfect revival while the same does not hold for a generic ETH-obeying state.