Competing supersolids of Bose-Bose mixtures in a triangular lattice

We study the ground-state properties of a frustrated two-species mixture of hard-core bosons on a triangular lattice, as a function of tunable amplitudes for tunneling and interactions. By combining three different methods, a self-consistent cluster mean-field, exact diagonalization, and effective theories, we unravel a very rich and complex phase diagram. More specifically, we discuss the existence of three original mixture supersolids: (i) a commensurate with frozen densities and supersolidity in spin degrees of freedom, in a regime of strong interspecies interactions; and (ii) when this interaction is weaker, two mutually competing incommensurate supersolids. Finally, we show how these phases can be stabilized by a quantum fluctuation enhancement of peculiar insulating parent states.

Figure 1

Triangular lattice hosting bosons of two species labeled $a$ (blue) and $b$ (brown). The bosons can hope from two neighboring sites with the energy $-t$, and interact via an intraspecies interaction $V$ and a point contact interspecies interaction $U$.

Figure 2

Clusters considered in our CMFT analysis with $N=3$, 6, and 12 sites. Internal and external bonds correspond, respectively, to continuous and dashed lines. For each cluster, the cluster scaling parameter $\lambda$ is given.

Figure 3

Phase diagrams as obtained from CMFT calculations on a $N=3$ cluster at $t=0.15$ and $V=1$. (Top) Wide range of ${\mu }^{*}$ showing two-spin superfluids, 2/3 and 4/3 plateaus, $XXZ$ physics domain and original collective two-spin physics (white rectangle). (Bottom) Zoom of the white rectangle and all new two-spin phases (see text for acronyms). Solid (dashed) lines are phase boundaries subsisting (vanishing) in the thermodynamic limit (TL).

Figure 4

Strong $U$ regime characterized using either $N=12$ [ED, (a)] or $N=6$ [CMFT, (b)] clusters. (a) $n$, $S_{ab}^d$, and $S_{ab}^{od}$ as a function of ${\mu }^{*}/V$ for $U=2V$ and $t=0.15V$. The $n=1$ plateau and both finite $S_{ab}^d$ and $S_{ab}^{od}$ evidence the SSS; (b) $n$ and $\varphi ={\varphi }_a+{\varphi }_b$ as a function of ${\mu }^{*}/U$ distinguishing the SSF ($V=0$) and the SSS ($V=0.2U$) regimes.

Figure 5

Strong $U$ regime characterized using either $N=6$ [CMFT, (a)] or $N=12$ [ED, (b)] clusters. (a) Species-averaged order parameter $M$ and superfluid fraction $\Phi$, evidencing the quantum fluctuation enhancement from the SSS to the BPB phase at $V=1$ and $t=0.15$; (b) comparison of the kinetic energy gain ${\epsilon }^{*}\simeq -k{t}^2/U$ for $H_{ab}$ (filled) and $H_{XXZ}$ (empty) for three ($t,V$) sets.

Figure 6

(Color online) Species-resolved observables ($n_{\alpha },M_{\alpha },{\varphi }_{\alpha }$). Each species corresponds to either continuous or dashed line. All data come from a $N=6$ CMFT with fixed $t/V=0.15$ and $U=0.5V$. The arrow points out a tiny region which disappears in the TL (see text). The shaded regions correspond to the different phases of Fig. 3.

Figure 7

Same as Fig. 6, but for $U=2V$.

Figure 8

Examples of two-spin supersolids with 3-sublattice structure obtained by CMFT density maps on the 12-site cluster (a)–(c) and by QMC correlations for $36\times 36$ sites cluster (d) and (e). (a) The $n=1$ spin supersolid (SSS) at ${\mu }^{*}=0$ and $U=2V$; (b) and (c), respectively, the bosonic pinball (BPB) at ${\mu }^{*}=-0.15V$ and the 3-supersolid (3SS) at ${\mu }^{*}=-0.25V$, for $t=0.15V$ and $U=0.5V$. $n_a-n_b$ is 0 for the 3SS and finite for the BPB for all CMFT clusters (Fig. 9 for the TL study). (d) and (e) are, respectively, QMC results for the BPB at ${\mu }^{*}=-0.12V$ and the 3SS at $-0.72V$. Note that the 3SS and BPB have distinct symmetries (under $\pi /6$ rotations).

Figure 9

$M$ and $\varphi$ as a function of ${\mu }^{*}$ for $t=0.15V$ and $U=0.5V$ obtained by CMFT on clusters of $N=3$ (continuous), 6 (dashed), and 12 (symbols) site. See Fig. 2 for the cluster shapes. Cluster scalings as a function of $\lambda$ (see text) are shown for (b) $n_a-n_b$ with different possible fits giving either zero (continuous) or finite (dashed) TL values (see text) and (c) $M$ (filled symbols) and $\varphi /2$ (empty symbols) within both the BPB at $|{\mu }^{*}|=t$ (squares) and the 3SS at $|{\mu }^{*}|=3t$ (circles).