Krylov complexity in ergodically constrained nonintegrable transverse-field Ising model

TL;DR

This paper investigates how spatial inhomogeneity in the transverse-field Ising model suppresses ergodic behavior, affecting operator growth, entanglement, and spectral properties, thus providing a minimal route to ergodicity breaking.

Contribution

It introduces a simple inhomogeneous variant of the transverse-field Ising model and analyzes its ergodic-to-nonergodic crossover using multiple diagnostics.

Findings

Inhomogeneity suppresses ergodic behavior in the model.

Operator growth in Krylov space is hindered by inhomogeneity.

Spectral statistics indicate a transition from ergodic to constrained dynamics.

Abstract

The nonintegrable transverse-field Ising model is a common platform for studying ergodic quantum dynamics. In this work, we introduce a simple variant of the model in which this ergodic behaviour is suppressed by introducing a spatial inhomogeneity in the interaction strengths. For this we partition the chain into two equal segments within which the spins interact with different coupling strengths. The ratio of these couplings defines an inhomogeneity parameter, whose variation away from unity leads to constrained dynamics. We characterize this crossover using multiple diagnostics, such as the long-time saturation of out-of-time-ordered correlators, level-spacing statistics, and the spectral form factor. We further examine the consequences for operator growth in Krylov space and for entanglement generation in the system's eigenstates. Together, these results demonstrate that introducing a macroscopic inhomogeneity in coupling strengths provides a minimal, disorder-free route to breaking ergodicity in this specific model of interacting spins.