Geometry-induced pair condensation

We study a one-dimensional model of interacting bosons on a lattice with two flat bands. Regular condensation is suppressed due to the absence of a well-defined minimum in the single-particle spectrum. We find that interactions stabilize a number of nontrivial phases such as a pair (quasi)condensate, a supersolid at incommensurable fillings, and valence bond crystals at commensurability. We support our analytical calculations with numerical simulations using the density matrix renormalization group technique. Implications for cold atoms are discussed.

Figure 1

Creutz ladder. Illustration of the hopping amplitudes on the cross-linked Creutz ladder. The hopping amplitudes along the legs are such that the particles pick up a phase $\pi$ when going around a square plaquette. The blue (gray) box indicates the Wannier function in the flat band limit for the two bands ($\pm$); the numbers correspond to the respective amplitudes.

Figure 2

Processes. Illustration of the microscopic processes responsible for the different phases. Crosses stand for empty sites, circles denote a particle. See the text for a detailed description.

Figure 3

Phase diagram. (Top panel) Phase diagram as a function of the density $n$, the hopping amplitudes $m$ (horizontal axis to the left), and $\epsilon$ (horizontal axis to the right). The bars at $n=1/4$ and $1/2$ indicate the valence bond crystal (VBC) and the pair VBC, respectively. The circles mark the points in the phase diagram where we display the correlation functions in the bottom panels. (Bottom panels) Correlation functions at the respective points (A–D) in the phase diagram. On top: The structure factor $S\left(q\right)$ [normalized to $S\left(0\right)$] indicating either a pair TLL with two power-law peaks (A), perfect long-range order with a $\delta$-function peak at $q=\pi$ (C and D), or the absence of density order in B. The Green's function $G\left(i\right)$ (green circles) and the pair correlation function $P\left(i\right)$ (black diamonds) differentiate between a pair TLL (A), a regular (quasi)condensate (B), a supersolid (C), and the gapped crystalline phase (D).