Theoretical exploration of competing phases of lattice Bose gases in a cavity

We consider bosonic atoms loaded into optical lattices with cavity-mediated infinite-range interactions. Competing short- and global-range interactions cultivate a rich phase diagram. With a systematic field-theoretical perspective, we present an analytical construction of a global ground-state phase diagram. We find that the infinite-range interaction enhances the fluctuation of the number density. In the strong-coupling regime, we find four branches of elementary excitations, with two being “particlelike” and two being “holelike,” and that the excitation gap becomes soft at the phase boundary between compressible phases and incompressible phases. We derive an effective theory describing compressible superfluid and supersolid states. To complement this perturbative study, we construct a self-consistent mean-field theory and find numerical results consistent with our theoretical analysis. We map out the phase diagram and find that a charge density wave may undergo a structure phase transition to a different charge density wave before it finally enters into the supersolid phase driven by increasing the hopping amplitude.

Figure 1

Global ground-state phase diagram spanned by $\mu /U$ and $K/U$ at the atomic limit ($t_{ij}/U=0$). The phase diagram can be loosely divided into three regimes depending on the strength of the infinite-range interaction: $\left(1\right)K/U\in (0,0.5)$, where the system is either in a Mott-insulating (MI) phase with $n_e=n_o$ or in a partially polarized charge density wave (CDW) phase with $n_e-n_o=1$, where we always assume $n_e\ge n_o$ as the system enjoys an Ising-type ${\mathbb{Z}}_2$ symmetry; $\left(2\right)K/U\in (0.5,1)$, where the system is in a fully polarized CDW phase with $n_o=0$; $\left(3\right)K/U>1$, where the system is unstable toward collapse.

Figure 2

Temperature dependence of (a) the variation of particle number $\delta n$ on one supercell (with one even site and one odd site) and (b) isothermal compressibility ${\kappa }_T$ for the $M{I}_{(1,1)}$ phase at different infinite-range interaction strengths $K/U=0$, 0.1, 0.2, and 0.3. Here, $\mu /U=0.60$ and $v_0$ is the volume of one supercell.

Figure 3

Four branches of excitation spectrum ${\omega }_i$ ($i=1,...,4$) at momentum $\mathbf{k}=(\vec{0},0)$ as a function of tunneling parameter $zt/U$ for (a) the $M{I}_{(1,1)}$ phase with $\mu /U=0.50$ and $K/U=0.25$, and (b) the $CD{W}_{(2,1)}$ phase with $\mu /U=1.0$ and $K/U=0.40$. The upper two branches are particle excitations and the lower two branches are hole excitations. For both insulating phases, there exists an energy gap for a particle-hole excitation, manifesting their noncompressible nature. Here, $z=4$ is the coordination number for square lattices.

Figure 4

The phase boundary separating compressible (SF or SS) and noncompressible (MI or CDW) phases for two typical infinite-range interaction strengths: (a) $K/U=0.10$ and (b) $K/U=0.25$. Increasing $K/U$ leads to the broadening of the region of $CDW$ phases and the shrinking of the region of $MI$ phases. Here a SF phase stands for a superfluid phase which has off-diagonal long-range order, and a SS phase stands for a supersolid phase which has both diagonal and off-diagonal long-range orders. Here the unlabeled small lobes are CDW phases generated by the infinite-range interaction.

Figure 5

Self-consistent mean-field calculation at zero temperature for $K/U=0.4$ and $\mu /U=1.0$. (a) The magnitude of the order parameters $|{\psi }_e|$ and $|{\psi }_o|$ in the supersolid phase as a function of $zt/U$. (b) The density at even site $n_e$ and odd site $n_o$ as a function of $zt/U$.

Figure 6

Self-consistent mean-field calculation at zero temperature for $K/U=0.7$ and $\mu /U=1.0$. (a) The magnitude of the order parameters $|{\psi }_e|$ and $|{\psi }_o|$ in the supersolid phase as a function of $zt/U$. (b) The density at even site $n_e$ and odd site $n_o$ as a function of $zt/U$.

Figure 7

Phase diagram spanned by $zt/U$ and $\mu /U$ from the self-consistent mean-field calculation for $K/U=0.4$.