Phase diagram and pair Tomonaga-Luttinger liquid in a Bose-Hubbard model with flat bands

To explore superfluidity in flat-band systems, we consider a Bose-Hubbard model on a cross-linked ladder with $\pi$ flux, which has a flat band with a gap between the other band for noninteracting particles, where we study the effect of the on-site repulsion nonperturbatively. For low densities, we find exact degenerate ground states, each of which is a Wigner solid with nonoverlapping Wannier states on the flat band. At higher densities, the many-body system, when projected onto the lower flat band, can be mapped to a spin-chain model. This mapping enables us to reveal the existence of a Tomonaga-Luttinger liquid comprising pairs of bosons. Interestingly, the high- and low-density regions have an overlap, where the two phases coexist.

Figure 1

(a) Bose-Hubbard model on a cross-linked ladder with $\pi$ flux. Circles represent sites. $A$ (blue), $B$ (red), and $...,j-1,j,j+1,...$ are chain and rung indices, respectively. While interchain diagonal hopping is $-t$, intrachain hopping is $it$ ($-it$) for along (against) the direction of the arrow, respectively. (b) The same model with interchange of $A$ and $B$ sites on rungs with $j=\mathrm{even}$. (c) Dispersion relation of the present model for the noninteracting case. It has two flat bands at $E=\pm 2t$.

Figure 2

Schematic pictures of (a) Wigner solid, (b) pair TLL, (c) CDW, and (d) CDW with two single-boson sites (SBSs). Particles are localized in panels (a), (c), and (d) while they are mobile (arrows) in panel (b).

Figure 3

(a) The GS energy per site above its minimum, $e(\rho ,x)-e_{\min }$, against the fraction of Wigner solid phase $x$ in the density region $0.25\le \rho \le 0.35$. (b) System-size ($L$) dependence of the energy of the state with two adjacent SBSs ($E_{1\mathrm{g}}$) and that of the first excited state without SBSs ($E_{2\mathrm{g}}$) at $\rho =1/2$. Solid curves represent the extrapolation by fourth-order polynomials, while arrows are the analytically obtained values. The energy unit is $U=1$.

Figure 4

GS phase diagram as a function of the chemical potential $\mu$ or the density $\rho$. WS and FITLL stand for Wigner solid and field-induced TLL, respectively. The bottom axis is the corresponding phase diagram for the spin-1/2 $XXZ$ chain as a function of magnetization $M$.