Quantum liquid of repulsively bound pairs of particles in a lattice

Repulsively interacting particles in a periodic potential can form bound composite objects, whose dissociation is suppressed by a band gap. Nearly pure samples of such repulsively bound pairs of cold atoms—“dimers”—have recently been prepared by Winkler et al. [Nature (London) 441, 853 (2006)]. We here derive an effective Hamiltonian for a lattice loaded with dimers only and discuss its implications for the many-body dynamics of the system. We find that the dimer-dimer interaction includes strong on-site repulsion and nearest-neighbor attraction which always dominates over the dimer kinetic energy at low temperatures. The dimers then form incompressible, minimal-surface “droplets” of a quantum lattice liquid. For low lattice filling, the effective Hamiltonian can be mapped onto the spin-$1∕2$ $XXZ$ model with fixed total magnetization which exhibits a first-order phase transition from the droplet to a gas phase. This opens the door to studying first-order phase transitions using highly controllable ultracold atoms.

Figure 1

Energy level diagram and tunnel couplings employed in the adiabatic elimination of nonresonant states with odd occupation numbers. $\cdots \mid n_i⟩\mid n_j⟩\cdots$ denotes a state with $n_i=2m_i$ bosons at site $i$ and $n_j=2m_j$ bosons at site $j=i+1$, etc., while the energy is $E(n_i,n_j)=(U∕2)[n_i(n_i-1)+n_j(n_j-1)]+C$, where $C$ contains the energy of all the other sites.

Figure 2

(Color online) Dynamics of one dimer (a) and (b) and two dimers (c) and (d), in a 1D lattice of 20 sites, for $J∕U=0.1$. (a) and (c) are numerical solutions of the Schrödinger equation with the Bose-Hubbard Hamiltonian (1), while (b) and (d) are obtained with the effective Hamiltonian (2) [or (6)]. Inset in (a) shows the time evolution of ${\sum }_j{\mathbf{\mid }⟨2_j\mid \Psi ⟩\mathbf{\mid }}^2$, where $\mid \Psi \left(t\right)⟩$ is the system wave function. Time is in units of $J^{-1}$.

Figure 3

(Color online) Energy per dimer $E∕M$ versus the number of dimers $M$ forming a cluster in a 2D square lattice. As seen, $E∕M$ abruptly drops once a square droplet with the dimension $\sqrt{M}\times \sqrt{M}$ is formed, since the addition of the last dimer results in the formation of two “bonds.”

Figure 4

(Color online) Phase diagram of the grand canonical ensemble obtained from exact diagonalization of Eq. (7). The Hilbert space is restricted by five sites (periodic boundary conditions), with each site occupation number in the range of $0⩽m⩽4$. The areas of integer filling are tightly adjoined to each other, with no significant extent of fractional filling phase.

Figure 5

Temperature phase diagram of the 2D Ising model. In the shaded area, the ferromagnetic spin domains are formed. As temperature is increased, for $⟨{\sigma }^z⟩\ne 0$, the system undergoes a first-order phase transition to the gas phase, while at $⟨{\sigma }^z⟩=0$ it is a second-order phase transition. See text for more details.