Probing quantum scars and weak ergodicity breaking through quantum complexity
Budhaditya Bhattacharjee, Samudra Sur, Pratik Nandy
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 N\'eel 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 N\'eel 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 N\'eel state shows nearly perfect revival while the same does not hold for a generic ETH-obeying state.