Ground-state properties of multicomponent bosonic mixtures: A Gutzwiller mean-field study

Using the single-site Gutzwiller method, we theoretically study the ground state and the interspecies entanglement properties of interexchange symmetric multicomponent (two- and three-component) bosonic mixtures in an optical lattice, and the results are generalized to an $n$-component $(n=2,3,4,...)$ system. We compute the mean-field phase diagram, the interspecies entanglement entropy, and the ground-state spectral decomposition. Three phases—namely, the $n$-component superfluid state (nSF), the $n$-component Mott insulator state (nMI), and the supercounterfluid state (SCF)—are observed. Interestingly, we find that there are $n-1$ SCF lobes to separate every two neighboring nMI lobes in the phase diagram. More importantly, we derive the exact general expression of the interspecies entanglement entropy for the SCF phase. In addition, we also investigate the demixing effect of an $n$-component mixture and demonstrate that the mixing-demixing critical point is independent of $n$.

Figure 1

Ground-state properties of a two-component bosonic mixture. The model parameters are set as $U_a=U_b=U$ and $U_{ab}/U=0.5$. (a) The $zJ/U\text{−}\mu /U$ phase diagram (in units of $U$). We use an index $m=1,2,...$ to label the first, second, ... 2MI and SCF lobes. Once $m$ is determined, the ground state of the corresponding lobe is addressed. (b) and (c) The order parameters and the single-site interspecies EE as a function of $\mu /U$ for a fixed value $zJ/U=0.04$. In (c), the white regions (uncolored) indicate the 2SF phase. (d)–(f) $|c_{m_a,m_b}{|}^2$ for three distinct phases. $N_a=N_b=10$ is the maximum boson occupation number of components $a$ and $b$, respectively. The color bar reflects the value of $|c_{m_a,m_b}{|}^2$. (d) The spectral decompositions of the third 2MI lobe ($m=3$) for $(zJ/U,\mu /U)=(0.04,4.0)$. (e) The spectral decompositions of the fourth SCF lobe ($m=4$) for $(zJ/U,\mu /U)=(0.016,4.8)$. (f) The spectral decompositions of the 2SF phase for $(zJ/U,\mu /U)=(0.4,4.0)$. In (d), a single peak is observed at (0.3,0.3), representing that the 2MI ground state is $|3,3\rangle$. In (e), the two peaks are present at (0.3,0.4) and (0.4,0.3), and its ground state is thus $\left(\right|4,3\rangle +|3,4\rangle )/\sqrt{2}$. In (f), there are several peaks that reveal a 2SF phase. Also, we use an interpolation algorithm to make the data more smooth in (d)–(f).

Figure 2

Demixing effect of the 2SF phase for a two-component bosonic mixture under the parameters $zJ/U=0.4$ and $\mu /U=4.0$. (a) and (b) The averaged particle numbers and the order parameters as a function of $U_{ab}/U$. (c) The interspecies EE as a function of $U_{ab}/U$. (d)–(f) The spectral decompositions for three different values of $U_{ab}/U$. Here, we set $U$ as the energy unit. $N_a=N_b=10$ is the maximum boson occupation number of components $a$ and $b$, respectively. The color bar in (d)–(f) reflects the value of $|c_{m_a,m_b}{|}^2$.

Figure 3

Ground-state properties of a three-component bosonic mixture. The model parameters are set as $U_a=U_b=U_c=U$ and $U_{ab}/U=U_{bc}/U=U_{ac}/U=U^{\prime }/U=0.5$. (a) The $zJ/U\text{−}\mu /U$ phase diagram (in units of $U$). We use an index $m$ to label the first, second, ... 3MI lobes and use $(m,k)$ to label the SCF lobes ($m=1,2,3,...$ and $k=1,2$). In the SCF-I, II, III, and IV lobes, $(m,k)$ is (1,1), (1,2), (2,1), and (2,2), respectively. Once the indices are determined, the ground state of the corresponding lobe is addressed. (b) and (c) The order parameters and the interspecies EE as a function of $\mu /U$ for a fixed value $zJ/U=0.04$. In (c), the white regions (uncolored) indicate the 3SF phase. (d)–(f) $|c_{m_a,m_b,m_c}{|}^2$ for three different phases. In (d)–(f), the parallel coordinates plot is used to display the spectral decompositions, which means the variables $m_a$, $m_b$, $m_c$, and $|c_{m_a,m_b,m_c}{|}^2$ are represented by a vertical axis, respectively. A line is drawn connecting the values of each variable for each data point. (d) The spectral decompositions of the second 3MI lobe $(m=2)$ for $(zJ/U,\mu /U)=(0.04,3.5)$. The ground state of this lobe is known as $|2,2,2\rangle$, and it is characterized by a single peak in the panel. (e) The spectral decompositions of the SCF-IV $\left[\right(m,k)=(2,2\left)\right]$ lobe for $(zJ/U,\mu /U)=(0.02,2.7)$. Its ground state is $\left(\right|1,2,2\rangle +|2,1,2\rangle +|2,2,1\rangle )/\sqrt{3}$, symbolized by three dashed lines in the panel. (f) The spectral decompositions of the 3SF phase for $(zJ/U,\mu /U)=(0.4,3.5)$. The ground state in this phase is a linear combination of several Fock states, which is why multiple lines can be observed.

Figure 4

Demixing effect of the 3SF phase for a three-component bosonic mixture under the parameters $zJ/U=0.2$ and $\mu /U=3.5$ (in units of $U$). (a) and (b) show the averaged particle numbers and the order parameters as a function of the interspecies interaction strength $U^{\prime }/U$, respectively. (c) The interspecies EE as a function of $U^{\prime }/U$. (d) and (e) The spectral decompositions of the ground-state wave function for two different values of $U^{\prime }/U$. In (d), a 3SF phase is formed, and its ground state is a linear combination of several Fock states. In (e), a demixed phase is established and only boson $a$ is observed.

Figure 5

A sketch phase diagram of the $n$-component ($n\ge 2$) bosonic mixture. Here, we only show the lowest $n-1$ SCF lobes and the first nMI lobe.

Figure 6

Details of Fig. 2 at $U_{ab}/U>1$. It clearly shows that if $\langle n_a\rangle$ ($\langle n_b\rangle$) is nonzero, then $\langle n_b\rangle$ ($\langle n_a\rangle$) must be zero, implying that the mixture is in a demixed state. (a) and (b) represent the outcomes of running the self-consistent process twice.