Bosonic Kondo-Hubbard model

We study, using quantum Monte Carlo simulations, the bosonic Kondo-Hubbard model in a two-dimensional square lattice. We explore the phase diagram and analyze the mobility of particles and magnetic properties. At unit filling, the transition from a paramagnetic Mott insulator to a ferromagnetic superfluid appears continuous, contrary to what was predicted with mean field. For double occupation per site, both the Mott insulating and superfluid phases are ferromagnetic and the transition is still continuous. Multiband tight-binding Hamiltonians can be realized in optical lattice experiments, which offer not only the possibility of tuning the different energy scales over wide ranges, but also the option of loading the system with either fermionic or bosonic atoms.

Figure 1

Schematic representation of the model. Spin-$\frac{1}{2}$ bosons (upper layer $a$) are moving through the lattice with hopping $t$ and are subject to onsite repulsion $U$. They interact antiferromagnetically with strength $V$ with the lower spin layer $b$. The lattice represented here is a one-dimensional (1D) chain; the studied system is a two-dimensional (2D) square lattice.

Figure 2

Phase diagram for several system sizes $L$. The insulating phase at $\rho =1$ is paramagnetic while the other phases are ferromagnetic. The dashed line indicates the cut in Fig. 10.

Figure 3

The one-body and the anticorrelated Green functions versus distance $R$ in the Mott insulator (a) and superfluid phase (b) for $\rho =1$. (a) All movement is suppressed as the particles form singlets. (b) Movement of individual particles as well as anticorrelated exchanges are observed.

Figure 4

The magnetic order that appears in the system. While the bosons and the spins form ferromagnetic layers, these two layers are coupled antiferromagnetically.

Figure 5

The one-body and the anticorrelated Green functions versus distance $R$ in the Mott insulator (a) and superfluid (b) phases for $\rho =2$. (a) Individual movements are suppressed but anticorrelated moves are possible in the Mott phase. (b) All kinds of movements are present in the superfluid phase.

Figure 6

Superfluid density ${\rho }_s$ versus $t/U$ in the first Mott lobe. We observe a continuous behavior for ${\rho }_s$. The blue curve with triangles represents a linear extrapolation to $L=\infty$ of the results obtained for the three sizes $L=6,8,10$. The curves for $L=10$ and the extrapolated one have been fitted with a power-law behavior ${\rho }_s\propto {(t/U-t/U_c)}^{\beta }$ (dashed lines). In both cases, we found $t/U_c\simeq 0.79$ and $\beta \simeq 0.33$.

Figure 7

Condensate fraction $\rho (k=0)$ versus $t/U$ in the first Mott lobe. The shape is very similar to Fig. 6, confirming the order of the transition.

Figure 8

Superfluid density ${\rho }_s$ versus $t/U$ in the first Mott lobe, for different ratio $V/U$. As we increase this ratio, ${\rho }_s$ becomes smoother, compared to Fig. 6 and the transition remains second order.

Figure 9

Superfluid density ${\rho }_s$ versus $t/U$ in the second Mott lobe. As for the $\rho =1$ case (Fig. 6), we do not observe any discontinuity in ${\rho }_s$ and the transition is second order.

Figure 10

Cut of Fig. 2 for $t/U=0.03$, showing the total density $\rho$ and the superfluid density ${\rho }_s$ as functions of $\mu /U$. All the transitions are continuous.

Figure 11

Magnetic correlations versus $t/U$ for $\rho =1$ and $V/U=0.05$. The dashed line marks the transition between the Mott and the superfluid phases. There is no magnetic order in the Mott phase where the $a$ and $b$ spins form singlets, as shown by the values of $S_{ab}\left(0\right)$. The superfluid phase shows the magnetic behavior depicted in Fig. 4: intraspecies ferromagnetic correlations and interspecies antiferromagnetic correlations. At large $t$, the magnetic correlations tend to their maximum values.

Figure 12

Magnetic correlations and ${\rho }_s$ versus $t/U$ for $\rho =1$ and $V/U=0.25$. The physical behavior is the same as in the $V/U=0.05$ case (Fig. 11), but it is more difficult to establish the magnetic correlations with this larger value of $V$.

Figure 13

Magnetic correlations versus $t/U$ for $\rho =2$ and $V/U=0.05$. The dashed line marks the transition between the Mott and the superfluid phases. The same magnetic order is present in the Mott and the superfluid phases although there are two different limiting regimes with stronger (for $t/U\gg 1$) or weaker (for $t/U\ll 1$) $b$ spin correlations. The magnetic behavior is the one sketched in Fig. 4. The total spin seems constant $S_{\mathrm{tot}}^2/L^4\simeq 1/4$.

Figure 14

Superfluid density ${\rho }_s$ versus $T$ for several sizes $L$ with $t/U=1$ and $V/U=0.05$. We calculate the intersection between the curves and $kT/\pi t$ to obtain $T_c\left(L\right)$. We then extrapolate linearly (see inset) the value of $T_c\left(L\right)$ to the large size limit to obtain the estimate of the transition temperature shown in Fig. 16.

Figure 15

The local density fluctuation $\kappa =\langle n_{a,i}^2\rangle -{\langle n_{a,i}\rangle }^2$ versus $kT$ for different temperatures. We see that $\kappa$ is almost constant at low $kT$ in the Mott region. We defined the crossover temperature between the Mott and the liquid behaviors as the temperature for which $\kappa$ departs from its low-$T$ value by more than 5%.

Figure 16

The phase diagrams for $\rho =1$ and 2 as a function of $t/U$ and $kT/U$ at $V/U=0.05$. There are three different regions in each diagram: a Mott region (MI), the normal Bose liquid (NBL), and the superfluid region (SF). The SF-NBL limit is a BKT transition. The MI-NBL limit is a crossover between weakly and highly compressible regimes.

Figure 17

Magnetization of the whole system ($S_{\mathrm{tot}}$) and sums over all distances of boson-boson ($S_{aa}$), spin-spin ($S_{bb}$), and boson-spin $(S_{ab}$) magnetic correlations for $\rho =1$ in the superfluid phase. The total magnetization $S_{\mathrm{tot}}$ is zero at $T=0$ but rises when $T$ is increased. As the bosons decouple from the spin, when $kT>V$, and no longer form singlets with those, they develop ferromagnetic correlations. Top panel: $\rho =1$, bottom panel: $\rho =2$.

Figure 18

Extrapolation of the magnetization as a function of the size. At high temperature ($kT=4.55$) the magnetization extrapolates to zero. At the temperatures where we observed the peaks of the moment in Fig. 17, the magnetization does not seem to extrapolate to zero. The data for $\rho =2,kT=0.9$ have been divided by 2 for better visibility.