Anomalous Diffusion and Localization in a Positionally Disordered Quantum Spin Array

Disorder in quantum systems can lead to the disruption of long-range order in the ground state and to the localization of the elementary excitations. Here we exhibit an alternative paradigm, by which disorder preserves long-range order in the ground state, while it localizes the elementary excitations above it, introducing a stark dichotomy between static properties—mostly sensitive to the density of states of excitations—and nonequilibrium dynamical properties—sensitive to the spatial structure of excitations. We exemplify this paradigm with a positionally disordered $2d$ quantum Ising model with $r^{-6}$ interactions, capturing the internal-state physics of Rydberg-atom arrays. Disorder is found to lead to multifractality and localization of the spin-wave excitations above a ferromagnetic ground state; as a result, the spreading of entanglement and correlations starting from a factorized state exhibits anomalous diffusion with a continuously varying dynamical exponent, interpolating between ballistic and arrested transport. Our findings are directly relevant for the low-energy dynamics in quantum simulators of quantum Ising models with power-law decaying interactions.

Figure 1

Positionally disordered 2D spin array. (a) Sketch of the system geometry; the blue dashed lines indicate the couplings (limited to nearest neighbors for illustration purposes), and the spin orientations offer a sketch of the mean-field ground state. (b) Inverse participation ratio $I_2$ of the harmonic eigenmodes as a function of disorder and normalized eigenmode energy $\epsilon$ (see text) for a system size of $L=40$. (c) Scaling of $I_2$ of the eigenmodes with $\epsilon =0.5$ for various disorder strengths [referred to by points with same colors in panel (b)]—error bars are smaller than the symbol size. (d) Effective fractal dimension $D_q$ of the $\epsilon =0.5$ eigenmodes.

Figure 2

Dynamics in the positionally disordered spin array. Entanglement entropy (EE) of the half torus $L/2\times L$ for $\mathrm{\Delta }=3\times {10}^{-2}$ and various system sizes, showing consistently a power-law scaling $E_{L/2}\sim t^{1/z}$ with $z\approx 2.8$ (dashed line). (b) EE growth for different disorder strengths in the anomalously diffusive regime—here $L=30$. (c) Spreading of correlations for the $x$ spin components, $C^{xx}=⟨S_i^x{S}_j^x⟩-⟨S_i^x⟩⟨S_j^x⟩$, multiplied $r^{\beta }$, where $\beta \approx 1.1$ is the exponent of the asymptotic power-law decay—here $\mathrm{\Delta }=3\times {10}^{-2}$ and $z$ as in (a). (d) Scaling of the generalized IPR ($n_q$) for the longtime excess quasiparticle distribution (see inset) induced by a localized spin flip, compared with the same quantity ($I_q$) for the Bogoliubov modes at $\epsilon =0.5$; here $q=2.5$ and $\mathrm{\Delta }=3\times {10}^{-2}$. In panels (a),(b),(d), whenever not shown, error bars are smaller than the symbol size.