Entanglement Spectrum of the Two-Dimensional Bose-Hubbard Model

We study the entanglement spectrum (ES) of the Bose-Hubbard model on the two-dimensional square lattice at unit filling, both in the Mott insulating and in the superfluid phase. In the Mott phase, we demonstrate that the ES is dominated by the physics at the boundary between the two subsystems. On top of the boundary-local (perturbative) structure, the ES exhibits substructures arising from one-dimensional dispersions along the boundary. In the superfluid phase, the structure of the ES is qualitatively different, and reflects the spontaneously broken $U(1)$ symmetry of the phase. We attribute the basic low-lying structure to the “tower of states” Hamiltonian of the model. We then discuss how these characteristic structures evolve across the superfluid to Mott insulator transition and their influence on the behavior of the entanglement entropies. We briefly outline the implications of the ES structure on the efficiency of matrix-product-state based algorithms in two dimensions.

Figure 1

ES in the Mott phase. (left) Perturbative structure, DMRG data for equally bipartitioned cylinder, $W=8$, $L=32$, $U=100$. Multiplicities are indicated near each group of states The dotted lines highlight the ES envelope. (center bottom) Bosonic configurations corresponding to leading order configurations for some $\delta N_A\ge 0$ levels. Only occupancies on sites nearer the boundary are shown (omitted sites are singly occupied). The arrows represent hopping (perturbative) processes to create a given configuration. In (d), to save space we show only one type of contributing configuration. The shown and omitted configurations contribute respectively $(1/2)W(W-1)(W-2)$ and $W$ to the multiplicity. In (f) we show the effective single particle (boundary) excitation corresponding to multiplet (b). (right) Entanglement dispersions. Data points are DMRG ($U=100$); black lines are perturbative results. (i) Lowest ES level ${\xi }_0$ as a function of $W$. (ii) Entanglement dispersion for the lowest ES multiplet at $\delta N_A=1$. $k$ denotes the momentum along the boundary.

Figure 2

(a) ES in the superfluid phase (DMRG data at $U=2$ for size $L/4=W$). (b) The same ES as in (a) but after subtracting the contribution of the envelope. (c) Spacing $\delta$ (defined in text and Fig. 3) plotted versus $1/W$; various $U$. Dotted lines are fits to $A/W+B/W^2$. (d) $\delta$ as function of $U$. Dotted lines are fits to $\delta \sim \sqrt{U}$ (expected for $U\to 0$).

Figure 3

(top) Restructuring of ES across the Mott-superfluid transition (DMRG data for $W=8=L/4$, equal bipartitions). The arrows explain the definitions of the quantities $\Delta$ and $\delta$. (bottom) $\Delta$ and $\delta$ plotted versus $U$ for several values of the boundary length $W$. The vertical line denotes the critical value $U_c$.

Figure 4

(a), (b) Renyi entropy $S_A^{(2)}$ (DMRG data). In (b) the data of (a) is rescaled by boundary length $W$, and the continuous line is the perturbative result. (c) Contributions to $\mathrm{Tr}{\rho }_A^2$ from the envelope ES levels (rhombi) and from the higher levels (circles). Vertical lines in each panel marks the transition point.