Operator delocalization in disordered spin chains via exact MPO marginals

TL;DR

This paper introduces a novel method to analyze operator delocalization in disordered spin chains using exact MPO marginals, revealing distinct behaviors in localized and many-body localized regimes.

Contribution

It develops an efficient, exact MPO-based approach to compute operator complexity measures, enabling detailed study of operator dynamics in disordered quantum systems.

Findings

Operator mass and length saturate in Anderson localization, indicating no scrambling.

In MBL, operator measures grow logarithmically, showing slow information spreading.

The approach accurately captures operator dynamics across different regimes and system sizes.

Abstract

We investigate operator delocalization in disordered one-dimensional spin chains by introducing -- besides the already known operator mass -- a complementary measure of operator complexity: the operator length. Like the operator nonstabilizerness, both these quantities are defined from the expansion of time-evolved operators in the Pauli basis. They characterize, respectively, the number of sites on which an operator acts nontrivially and the spatial extent of its support. We show that both the operator mass and length can be computed efficiently and exactly within a matrix-product-operator (MPS) framework, providing direct access to their full probability distributions, without resorting to stochastic sampling. Applying this approach to the disordered XXZ spin-1/2 chain, we find sharply distinct behaviors in non-interacting and interacting regimes. In the Anderson-localized case, operator mass, length, and operator entanglement entropy rapidly saturate, signaling the absence of scrambling. By contrast, in the many-body localized (MBL) regime, for arbitrarily weak interactions, all quantities exhibit a robust logarithmic growth in time, consistent with the known logarithmic light cone of quantum-correlation propagation in MBL. We demonstrate that this behavior is quantitatively captured by an effective $\ell$-bit model and persists across system sizes accessible via tensor-network simulations.