Numerical Evidence for Many-Body Localization in Two and Three Dimensions

Disorder and interactions can lead to the breakdown of statistical mechanics in certain quantum systems, a phenomenon known as many-body localization (MBL). Much of the phenomenology of MBL emerges from the existence of $\ell \mathrm{bits}$, a set of conserved quantities that are quasilocal and binary (i.e., possess only $\pm 1$ eigenvalues). While MBL and $\ell \mathrm{bits}$ are known to exist in one-dimensional systems, their existence in dimensions greater than one is a key open question. To tackle this question, we develop an algorithm that can find approximate binary $\ell \mathrm{bits}$ in arbitrary dimensions by adaptively generating a basis of operators in which to represent the $\ell \mathrm{bit}$. We use the algorithm to study four models: the one-, two-, and three-dimensional disordered Heisenberg models and the two-dimensional disordered hard-core Bose-Hubbard model. For all four of the models studied, our algorithm finds high-quality $\ell \mathrm{bits}$ at large disorder strength and rapid qualitative changes in the distributions of $\ell \mathrm{bits}$ in particular ranges of disorder strengths, suggesting the existence of MBL transitions. These transitions in the one-dimensional Heisenberg model and two-dimensional Bose-Hubbard model coincide well with past estimates of the critical disorder strengths in these models, which further validates the evidence of MBL phenomenology in the other two- and three-dimensional models we examine. In addition to finding MBL behavior in higher dimensions, our algorithm can be used to probe MBL in various geometries and dimensionality.

Figure 1

Typical weights $w_{\mathbf{r}}$ of random ${\tau }_i^z$ for the (a) 1D, (b) 2D, and (c) 3D disordered Heisenberg models at different disorder strengths.

Figure 2

The average and median commutator norms $\parallel {[H,{\tau }_i^z]\parallel }^2$ and binarities $\parallel {({\tau }_i^z{)}^2-I\parallel }^2$ [only for (a)] of our optimized ${\tau }_i^z$ operators for the disordered (a) 1D Heisenberg model and (b) 2D and 3D Heisenberg models and 2D hard-core Bose-Hubbard model. The average commutator norms obtained by Ref. [35] using shallow 2D tensor networks for the 2D Bose-Hubbard model are also shown. Note that the method of Ref. [35] finds all ${\tau }_i^z$ in a $10\times 10$ lattice, while our method finds only a single ${\tau }_i^z$.

Figure 3

Interpolated histograms of $|⟨{\tau }_i^z,{\sigma }_i^z⟩{|}^2$ at different disorder strengths. The histograms are made of 50 evenly spaced bins (25 for 2D Bose-Hubbard) and are normalized so that at a fixed disorder strength the maximum of the histogram is at a value of 1. The black lines are contour lines corresponding to normalized histogram values of 0.2, 0.4, 0.6, and 0.8.

Figure 4

The average correlation lengths of our ${\tau }_i^z$ operators versus disorder strength. For comparison, we show average correlation lengths of $\ell \mathrm{bits}$ obtained by Ref. [46] for the 1D model and by Ref. [35] for the 2D Bose-Hubbard model. Horizontal dashed lines are drawn at (a) $\xi =1/\ln (4)$ and (b) $\xi =1/\ln (4^2)$; shading indicates our estimates of the transition regions (see Fig. 3 and the Supplemental Material [63]).