Impact of geometry on many-body localization

The impact of geometry on many-body localization (MBL) is studied on simple, exemplary systems amenable to exact diagonalization treatment. The crossover between ergodic and MBL phase for uniform as well as quasirandom disorder is analyzed using energy-level statistics. It is observed that the transition to many-body localized phase is correlated with the number of nearest coupled neighbors. The crossover from extended to localized systems is approximately described by the so called plasma model.

Figure 1

Geometries considered: (a) one-dimensional chain, (b) 2D ladderlike model with eight sites in two columns, (c) a square lattice, and (d) a triangular lattice. We assume periodic boundary conditions (PBCs); formulating them for a triangular lattice is nontrivial. The letters in (d) indicate which links are to be identified, i.e., “$a$” with “$a$,” “$b$” with “$b$,” etc.

Figure 2

Color maps of the averaged gap ratio $\overline{r}$ as a function of disorder amplitude and relative energy $\varepsilon$ (9). Observe that the crossover from GOE to Poisson statistics depends on the system geometry. Results are obtained for random uniform disorder with PBC for configurations: $2\times 8$, $4\times 4$, and the triangular configuration.

Figure 3

The average gap ratio $\overline{r}$ (5) averaged over 800–1500 realizations of uniform disorder or quasidisorder for eigenvalues included in $\varepsilon \in (0.49,0.51)$ is presented for the mentioned lattices. It shows directly that the statistics of eigenvalues changes from GOE to Poisson distribution during the crossover. The top row corresponds to details of the 1D model: $XXX$, $XXZ$ with uniform disorder, and $XXX$ with quasidisorder. The bottom row presents a comparison of averaged $\overline{r}$ for geometries and disorders mentioned in the article.

Figure 4

Finite-size scaling for the average gap ratio $\overline{r}$ (5) for uniform disorder [eigenvalues within $\varepsilon \in (0.49,0.51)$ range are included; number of disorder realizations varied between 250 and 100 000 depending on the matrix size]. The upper panel shows results for the ladder system, while the lower panel shows the $3\times L$ rectangular lattice. Critical disorder values (indicated in the figure) $W_c$ are obtained from crossing points between curves forming original data shown in the insets.

Figure 5

Distributions $P\left(s\right)$ and $P\left(\overline{r}\right)$ for a $2\times 8$ lattice based on formulas (8) and (7) with fitted parameters $\beta$ and $\gamma$. Results were calculated for uniform disorder and quasidisorder.

Figure 6

Distributions $P\left(s\right)$ and $P\left(\overline{r}\right)$ for a $4\times 4$ lattice based on formulas (8) and (7) with fitted parameters $\beta$ and $\gamma$. Results were calculated for uniform disorder and quasidisorder.

Figure 7

Distributions $P\left(s\right)$ and $P\left(\overline{r}\right)$ for a triangular lattice based on formulas (8) and (7) with fitted parameters $\beta$ and $\gamma$. Results were calculated for uniform disorder and quasidisorder.

Figure 8

Average gap ratio $\overline{r}$ (5) averaged over 800–1500 realizations of quasidisorder for eigenvalues included in $\varepsilon \in (0.49,0.51)$. The localized to extended state transition shifts gradually from a 1D to a 2D value when the vertical coupling between horizontal 1D chains is increased.