Superfluid and Mott-insulator phases of one-dimensional Bose-Fermi mixtures

We study the ground state phases of Bose-Fermi mixtures in one-dimensional optical lattices with quantum Monte Carlo simulations using the canonical worm algorithm. Depending on the filling of bosons and fermions, and the on-site intra- and interspecies interaction, different kinds of incompressible and superfluid phases appear. On the compressible side, correlations between bosons and fermions can lead to a distinctive behavior of the bosonic superfluid density and the fermionic stiffness, as well as of the equal-time Green functions, which allow one to identify regions where the two species exhibit anticorrelated flow. We present here complete phase diagrams for these systems at different fillings and as a function of the interaction parameters.

Figure 1

(Color online) ${\rho }_B$ as a function of chemical potential ${\mu }_B$. The fermion density is ${\rho }_F=1∕4$ and the interaction strengths are fixed at $U_{\mathit{BB}}=10$ and $U_{\mathit{BF}}=16$. There are Mott plateaus at ${\rho }_B=1-{\rho }_F=3∕4$ and ${\rho }_B=1$ as predicted by the $t_B=t_F=0$ analysis. The positions of the Mott lobes coincide for different lattice sizes $L=36,44$ and temperatures $\beta =32,48$ to within our error bars, which are smaller than the symbol size. The dependence of ${\rho }_B$ on ${\mu }_B$ in the absence of fermions is given for comparison.

Figure 2

(Color online) Top panel: Phase diagram in the ${\mu }_B\text{−}{U}_{\mathit{BF}}$ plane for $U_{\mathit{BB}}=10$ and ${\rho }_F=1∕4$ obtained by a sequence of plots such as that in Fig. 1. The vertical line at $U_{\mathit{BF}}=16$ corresponds to the coupling value in Fig. 1. Within the regions labeled ${\rho }_B=0.75$ and ${\rho }_B=1$, the boson density is frozen even though the chemical potential varies. The ${\rho }_B=1$ lobe is pinched off at $U_{\mathit{BF}}\approx 2U_{\mathit{BB}}$. The labeling of the phases is I, superfluid (${\rho }_B^s\ne 0$, ${\rho }_F^s\ne 0$); II, anticorrelated phase (${\rho }_B^s\ne 0$, ${\rho }_F^s\ne 0$, ${\rho }_B^s={\rho }_F^s$); III, anticorrelated phase/relay superfluid (${\rho }_B^s\ne 0$, ${\rho }_F^s\ne 0$) (see the text for a detailed explanation); IV, Mott insulator/Luttinger liquid (${\rho }_B^s=0$, ${\rho }_F^s\ne 0$); V, insulator, ${\rho }_B^s=0$, ${\rho }_F^s=0$; VI, phase separation. Bottom panel: Phase diagram in the ${\mu }_B\text{−}{U}_{\mathit{BB}}$ plane for $U_{\mathit{BF}}=10$ and ${\rho }_F=1∕4$. The upper boundary of phase II is defined by the increase of energy when a boson is added at filling ${\rho }_B+{\rho }_F=1$; this is approximately ${\mu }_B=U_{\mathit{BF}}$ (for $U_{\mathit{BF}}<2U_{\mathit{BB}}$), a line of slope 1 in the ${\mu }_B\text{−}{U}_{\mathit{BF}}$ plane and slope 0 in the ${\mu }_B\text{−}{U}_{\mathit{BB}}$ plane. A similar strong coupling analysis applies for the other boundaries. (See also Fig. 13 and [8].)

Figure 3

(Color online) “Horizontal” sweep, which is fixed densities ${\rho }_B=3∕4$ and ${\rho }_F=1∕4$ and varying $U_{\mathit{BF}}$, through the phase diagram of Fig. 2. As in Figs. 1, 2, $U_{\mathit{BB}}=10$. Top panel: Both the bosonic and fermionic species exhibit a finite stiffness at weak coupling (region II of the phase diagram), which decays as $U_{\mathit{BF}}$ increases until insulating behavior occurs (region V of the phase diagram). Bottom panel: Near $U_{\mathit{BF}}=0$ both ${\rho }_c^s\ne 0$ and ${\rho }_a^s\ne 0$ are very similar. However, quickly after turning on $U_{\mathit{BF}}$, the correlated and anticorrelated stiffness show that the bosons and fermions propagate in opposite directions ${\rho }_a^s\ne 0$, while ${\rho }_c^s=0$.

Figure 4

(Color online) Same “horizontal cut” as Fig. 3 except at commensurate density for the boson species alone ${\rho }_B=1$ and ${\rho }_F=1∕4$. Top panel: The bosons are insulating at weak $U_{\mathit{BF}}$ within this Mott lobe of commensurate bosonic filling (region IV of the phase diagram). However, the fermions are free to flow on the uniform boson background and have nonzero stiffness. Upon emerging from the lobe, at $U_{\mathit{BF}}\approx 2U_{\mathit{BB}}$, ${\rho }_B^s$ becomes nonzero in a window where the two repulsions work against each other. After the peak, the system becomes phase separated and ${\rho }_B^s$ and ${\rho }_F^s$ go to zero. Bottom panel: The correlated and anticorrelated stiffnesses are essentially equal throughout the weak coupling because the flow is dominated by fermions. However, in the window the anticorrelated stiffness increases beyond the correlated stiffness in a weak simulacrum of phase II (as discussed in the text).

Figure 5

(Color online) “Vertical” sweep across ${\rho }_B$ at ${\rho }_F=1∕4$, $U_{\mathit{BB}}=10$, and $U_{\mathit{BF}}=16$. Top panel: The boson superfluid density changes much less as $\beta$ is increased from $\beta =32$ to $\beta =108$ in the superfluid phase I at ${\rho }_B<3∕4$ and ${\rho }_B>1$ than for phase III $3∕4<{\rho }_B<1$. This is a hallmark of the “relay” superfluid discussed in the text. Bottom panel: The correlated winding decreases to zero at ${\rho }_B=3∕4$ while the anticorrelated winding remains finite. As ${\rho }_B$ is increased beyond $3∕4$ the correlated winding increases and overtakes the anticorrelated winding. This is another sign of the relay superfluid.

Figure 6

(Color online) Vertical sweep across ${\rho }_B$ at ${\rho }_F=1∕4$, $U_{\mathit{BB}}=10$, and $U_{\mathit{BF}}=24$. Unlike the case for the weaker coupling $U_{\mathit{BF}}=16$ in Fig. 5, both superfluid densities vanish in the insulating region of the Mott lobe at commensurate total filling.

Figure 7

(Color online) I, $U_{\mathit{BF}}=16$, $U_{\mathit{BB}}=10$, $\beta =108$, and $N_B=20$; momentum distributions for bosons and fermions, and Fourier transform of the anticorrelated pairing $\left[n_a\left(k\right)\right]$ Green function. The sharp peak in bosonic momentum distribution indicates the presence of a quasicondensate, while fermions have a plateau indicating Luttinger liquidlike behavior with a clear Fermi momentum, a property that is also shared by the composite fermions described by the anticorrelated pairing.

Figure 8

(Color online) II, $U_{\mathit{BF}}=16$, $U_{\mathit{BB}}=10$, $\beta =108$, and $N_B=27$. Bosons do not have a peak at $k=0$ and the Fermi momentum is washed out, both reflecting the onset of short range one-particle correlations. On the other hand, the plateau in the Fourier transform of the anticorrelated pairing shows that the composite fermions formed by pairing a fermion and a boson have a well defined Fermi momentum [11].

Figure 9

(Color online) III, $U_{\mathit{BF}}=16$, $U_{\mathit{BB}}=10$, $\beta =108$, and $N_B=32$. Qualitatively, this picture is similar to Fig. 7—we have a peak in the bosonic momentum distribution and a plateau in the fermionic and anticorrelated pairing momentum distributions, all indicating power-law decaying correlations of their corresponding real space Green functions.

Figure 10

(Color online) IV, $U_{\mathit{BF}}=16$, $U_{\mathit{BB}}=10$, $\beta =108$, and $N_B=36$. There is no sharp peak in $n_B\left(k\right)$ and a plateau in $n_F\left(k\right)$ is present. This phase is a Mott insulator for bosons and Luttinger liquid behavior for the fermions. In this case the composite fermions do not exhibit a Fermi momentum.

Figure 11

(Color online) V, $U_{\mathit{BF}}=30$, $U_{\mathit{BB}}=10$, $\beta =108$, and $N_B=27$. The momentum distribution functions in this case exhibit the behavior expected from an insulator, i.e., no sharp peak in $n_B\left(k\right)$, no plateau in $n_F\left(k\right)$, and no Fermi edge in $n_a\left(k\right)$.

Figure 12

(Color online) VI, $U_{\mathit{BF}}=28$, $U_{\mathit{BB}}=10$, $\beta =108$, and $N_B=36$. This is phase separation. The boson momentum distribution function has a peak indicating that there may be a kind of superflow in their separate area. Fermions, on the other hand, behave as an insulator. The $n_a\left(k\right)$ curve indicates that the coupling between bosons and fermions is weak, as we would expect in phase separation.

Figure 13

(Color online) A comparison of our phase diagram (Fig. 2) with the strong coupling boundaries. Symbols and dashed lines are the results of the present QMC work, while the solid lines are for $t_B=t_F=0$. Unsubscripted Roman symbols denote our phases while subscripted Roman symbols are the labeling of Lewenstein et al. [8].