Spectral analysis of two-dimensional Bose-Hubbard models

One-dimensional Bose-Hubbard models are well known to obey a transition from regular to quantum-chaotic spectral statistics. We are extending this concept to relatively simple two-dimensional many-body models. Also in two dimensions a transition from regular to chaotic spectral statistics is found and discussed. In particular, we analyze the dependence of the spectral properties on the bond number of the two-dimensional lattices and the applied boundary conditions. For maximal connectivity, the systems behave most regularly in agreement with the applicability of mean-field approaches in the limit of many nearest-neighbor couplings at each site.

Figure 1

Sketch of the lattice geometries used here. We study three basic settings of a $2\times 2$ (a), a $2\times 3$ (b), and a $3\times 3$ lattice (c). The minimal bonds are always present (solid lines), while the additional diagonal bonds (green dotted and red dashed lines) are gradually added. In (b) and (c) we allow also for periodic boundary conditions (represented by the blue dot-dashed lines).

Figure 2

Symmetries of our $2\times 3$ lattice system: $P_{ij}$ denotes the transposition (exchange of the occupation numbers) of the corresponding sites: (a) without diagonals, two discrete symmetries: The exchange of the upper and lower row $P_{14}{P}_{25}{P}_{36}$ and the exchange $P_{13}{P}_{46}$. Since each of theses symmetries is self-inverse, and therefore has the 2 eigenvalues $\pm 1$, the corresponding Hamiltonian decomposes in $2\times 2$ subblocks. (b) With two diagonals, only a point-symmetry $P_{16}{P}_{25}{P}_{34}$ giving rise to two subblocks. (c) This case has most symmetries: each of the permutations $P_{14}$, $P_{25}$, and $P_{36}$ is a symmetry on its own, allowing a decomposition into $2\times 2\times 2$ blocks. There exists another symmetry $P_{13}{P}_{46}$, which commutes with $P_{25}$ but not with the other two. It therefore only decomposes the blocks corresponding to equal eigenvalues for $P_{14}$ and $P_{36}$. Since these are 4 of our 8 subblocks, we end up with 12 subblocks after the whole decomposition.

Figure 3

Spectral analysis of 2D BHMs using the measures $P\left(r\right)$ [left panels: (a), (c), (e)] and $P\left(s\right)$ [right panels: (b), (d), (f)]. Each panel exemplifies one regular (blue/black dotted lines) and one chaotic case (red/gray solid lines). The lattice structures are $2\times 2$ (upper panels, $N=20$ with $U/J=2.42$ for regular and $U/J=0.87$ for chaotic case), $2\times 3$ (middle panels, $N=10$, $U/J=15.0$ and $U/J=1.51$), and $3\times 3$ (lower panels, $N=6$, $U/J=1000$, and $U/J=1.71$), all with open (Dirichlet) boundary conditions and without diagonal couplings. The RMT predictions are shown by the dashed lines by comparison, cf. Eqs. (3) and (4) (left) and (7) and (8) (right). The transition from regular to chaotic statistics is clearly visible. In the shown case it is controlled by the system parameter $U/J$. In panel (b), for the regular case, we do not have a fully regular spectrum for this parameter $U/J$. This fact is indicated by a Brody parameter $\beta \approx 0.3$ significantly larger than zero, cf. Eq. (5).

Figure 4

${\chi }^2$ statistical tests for the regular-to-chaotic transition for a $2\times 3$ lattice at $N=10$. Small ${\chi }_{\mathrm{Pois}}^2$ and ${\chi }_{\mathrm{WD}}^2$ values indicate a good correspondence with $P_{\mathrm{Pois}}\left(s\right)$ or $P_{\mathrm{Pois}}\left(s\right)$, respectively. (a, b) show results for no diagonal coupling in Fig. 1, (c, d) for the red dashed diagonal couplings, and (e, f) for a maximal number of diagonal couplings at open (a, c, e) and periodic (b, d, f) boundary conditions.

Figure 5

${\mathrm{\Sigma }}^2$ measure for the three cases: (a) $2\times 2$ lattice $N=20$; (b) $2\times 3$ lattice $N=10$; (c) $3\times 3$ lattice $N=6$. The control parameter $U/J$ is chosen to minimize ${\chi }_{WD}^2$. Displayed in each panel are data for: lattice without diagonal coupling (triangles) with $U/J=0.87$ (a), $U/J=1.57$ (b), and $U/J=1.71$ (c)—with one diagonal per plaquette, i.e., per square unit cell, (squares) at $U/J=0.955$ (a), $U/J=0.88$ (b), and $U/J=1.92$ (c)—maximal number of bonds at open boundary conditions (filled circles) at $U/J=1.385$ (a), $U/J=1.67$ (b), and $U/J=1.65$ (c)—and periodic boundary conditions without diagonals (filled diamonds) at $U/J=0.79$ (b) and $U/J=3.3$ (c). In (a) also a regular case with no diagonal bonds and $U/J=6.136$ (crosses) is presented for comparison.

Figure 6

Overview over the ${\chi }^2$ tests (left axes) and the analysis of long-range correlations with $L_{\mathrm{rel}}(q=1)$ from Eq. (13) (right axes). Lattices structures: $2\times 2$ with $N=20$ (upper panels), $2\times 3$ with $N=10$ (middle panels), and $3\times 3$ with $N=6$ (lower panels). The left panels are for the minimal couplings without any diagonal bonds in Fig. 1. The central column shows the cases of one diagonal bond (upper) and four diagonals with open boundary conditions (middle and lower panel). For the lower middle panel this corresponds to the red dashed lines in Fig 1. The right panels are for two diagonals in the case $2\times 2$ (upper), and no diagonals with periodic boundary conditions (middle and lower panel).