Operator delocalization in disordered spin chains via exact MPO marginals

TL;DR

This work introduces operator length as a complementary diagnostic to operator mass for studying scrambling in disordered quantum many-body systems. By encoding time-evolved operators as matrix-product-operators in Pauli space, the authors show that both mass and length can be computed exactly and efficiently, yielding full probability distributions without stochastic sampling. An in-depth study of the disordered XXZ spin chain reveals a sharp dichotomy: Anderson localization yields rapid saturation of mass, length, and operator entanglement, while many-body localization induces robust logarithmic growth in these quantities, consistent with a logarithmic light cone and an effective $\ell$-bit model. The results demonstrate a tight link between operator spreading and entanglement dynamics, and they outline practical experimental strategies using Choi-state representations and classical shadows to access these diagnostics in quantum simulators.

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.

Figures (5)

• Figure 1: The average operator length $\overline{h}(t)$ as a function of the time $t$, for the model of Eq. \ref{['hmbl:eqn']}, for a system with $L=12$ sites, as obtained by means of ED computations up to $t=10^5$. Here we fix $\Delta = 1$, while the various data sets are for different values of $W$ (see legend). Averages are performed over $N_{\rm r} = 48$ disorder realizations. Notice the logarithmic scale on the horizontal axis and the logarithmic growth in time over several decades.
• Figure 2: MPS dynamics of the integrated operator entanglement (a), the operator length $\overline{h}(t)$ (b), and the operator mass $\overline{m}(t)$ (c), for $L=12$ and $L=20$, up to $t=500$. Notice the logarithmic increase occurring only for the interacting case $\Delta=1$ (dashed lines are guide for the eyes). Averages have been taken over $N_{\rm r} = 200$ disorder realizations.
• Figure 3: Same as in Fig. \ref{['fig:mps_ent_h_m']}, but for fixed disorder strength $W = 6.5$ and varying interaction strength $\Delta$. In the non-interacting limit ($\Delta = 0$), all quantities rapidly saturate, signaling the absence of operator spreading. Upon turning on interactions, even weakly, a long-time logarithmic growth emerges abruptly and in a non-perturbative manner, characteristic of the interacting MBL regime.
• Figure 4: Time-averaged operator length $\overline{h}(L,W)$ and mass $\overline{m}(L,W)$, obtained by integrating $h(t,W)$ and $m(t,W)$ over the time window $[0,L]$. Panel (a): dependence on the system size $L$, for different fixed disorder strengths. Panel (b): dependence on the disorder $W$, for fixed system size. Panel (c): rescaled data $\overline{h}(L,W)/\ln L$ and $\overline{m}(L,W)/\ln L$, showing the expected collapse in the many-body localized regime, consistent with the logarithmic light-cone $h(t)\sim \xi(W)\ln t$.
• Figure A1: $\overline{h}(t)$ versus $t$ for the $\ell$-bit model of Eq. \ref{['lbit:eqn']} with $h_j=0$, for different values of the exponential decay rates $\kappa$ of the couplings $J_{jl}$, as defined in Eq. \ref{['lbit:eqn']} (see legend). Data are averaged over $N_{\rm r} = 192$ disorder realizations and we consider $W=1$. Notice the logarithmic scale on the horizontal axis. The logarithmic increase in time is visible until saturation to the finite-size boundary value ($h\leq L$) occurs.