Quantum Critical Behavior of Entanglement in Lattice Bosons with Cavity-Mediated Long-Range Interactions

We analyze the ground-state entanglement entropy of the extended Bose-Hubbard model with infinite-range interactions. This model describes the low-energy dynamics of ultracold bosons tightly bound to an optical lattice and dispersively coupled to a cavity mode. The competition between on-site repulsion and global cavity-induced interactions leads to a rich phase diagram, which exhibits superfluid, supersolid, and insulating (Mott and checkerboard) phases. We use a slave-boson treatment of harmonic quantum fluctuations around the mean-field solution and calculate the entanglement entropy across the phase transitions. At commensurate filling, the insulator-superfluid transition is signaled by a singularity in the area-law scaling coefficient of the entanglement entropy, which is similar to the one reported for the standard Bose-Hubbard model. Remarkably, at the continuous ${\mathbb{Z}}_2$ superfluid-to-supersolid transition we find a critical logarithmic term, regardless of the filling. This behavior originates from the appearance of a roton mode in the excitation and entanglement spectrum, becoming gapless at the critical point, and it is characteristic of collective models.

Figure 1

The Bose-Hubbard model with competing short and global interactions can be realized with atoms tightly bound by an optical lattice that coherently scatters laser photons ($\mathrm{\Omega }$) into the mode of a high-finesse cavity [26]. (a) The picture shows the geometry of the $\mathbb{A}/\mathbb{B}$ bipartition considered in this Letter. (b) Illustration of competing processes of the Hamiltonian: the nearest-neighbor tunneling (with amplitude $t$), the on site repulsion ($U_0$), and the global density-density interactions ($U_{\mathrm{LR}}$), which are attractive here.

Figure 2

Color plot of the half-system EE $S$ [Eq. (1)] [see the partition in Fig. 1] for (a) $U_{\mathrm{LR}}/U_0=0.3$ and (b) $U_{\mathrm{LR}}/U_0=0.6$ as functions of the tunneling $t$ and the chemical potential $\mu$ in units of $U_0$ for a $L\times L$ square lattice with $L=40$. The nonanalyticities of $S$ coincide with the phase transition lines predicted by mean-field theory (not indicated here). The lower panels show $S/L$ as a function of $t/U_0$ (c) at fixed density $\rho$ for $U_{\mathrm{LR}}=0.3U_0$; and (d) at fixed chemical potential $\mu /U_0$ for $U_{\mathrm{LR}}=0.6U_0$. Here the system the size is $L=60$.

Figure 3

(a) The roton-mode frequencies ${\lambda }_{\mathrm{rot}}={\lambda }_{\pi ,\overline{m}}$ ($\overline{m}$ being the index of the rotonlike mode) and ${\omega }_{\mathrm{rot}}$ in the entanglement and physical spectrum (respectively) as a function of $4t/U_0$ across the SS-SF phase transition. Both frequencies vanish at the SS-SF transition. (b) Entanglement spectrum for $4t=0.55U_0$, $U_{\mathrm{LR}}=0.6U_0$, $\mu =-0.05U_0$, and $L=60$ as functions of $k_y$. The roton mode in the entanglement spectrum is highlighted by the red cross.

Figure 4

The half-system EE as a function of the tunneling rate $t$ in units of $U_0$ across the transition from SS-SF, for $U_{\mathrm{LR}}=0.6U_0$ and constant $\mu =-0.05U_0$. (b) Scaling of the $S/L$ values at the maximum (“o” symbols), for $4t/U_0=0.56$ (“x” symbols), and for $4t/U_0=0.53$ (“$+$” symbols) for different $L$. The coefficients $A$ and $B$ are obtained by fitting Eq. (5) to $S$ vs $L$ and are given in the table. For all the data in this figure the scaling exponent of the regularizing field $h(L)$ has been chosen as $\kappa =4$.