Few-body bound states of bosonic mixtures in two-dimensional optical lattices

TL;DR

This work analyzes bound-state formation in a binary bosonic mixture confined to small two-dimensional optical lattices by exactly diagonalizing the two-component Bose-Hubbard model. For balanced populations with $N_A=N_B=2$ and $N_A=N_B=3$, bound tetramers and hexamers emerge at intermediate interspecies attraction $U_{AB}$ and repulsive intraspecies interactions $U$, with the binding energy $\epsilon_b$ showing a non-monotonic dependence on $U_{AB}/U$ at small tunneling $t$. Entanglement between species, quantified by the von Neumann entropy $S_E$, peaks near the binding-energy minimum, indicating strong interspecies correlations, while inter- and intra-species distances reveal the cluster structure. The results highlight lattice-geometry–driven binding mechanisms that differ from 1D chains and continuum cases, and point to bound clusters as precursors to many-body droplets in higher dimensions. The study provides a framework for exploring few-body bound states in lattice bosonic mixtures and guides experimental efforts in quasi-2D optical lattices.

Abstract

We study the formation of bound states in a binary mixture of a few bosons in small square optical lattices. Using the exact diagonalization method, we find that bound clusters of all available bosons can form. We provide a comprehensive numerical examination of these bound states for a wide range of repulsive intraspecies and attractive interspecies interactions. In contrast to binary mixtures in one-dimensional chains, we reveal that the binding energy of the clusters shows a non-monotonic dependence on the interspecies interaction strengths for small tunneling rates, developing a local minimum for intermediate attractive interactions. The findings of this work highlight the difference between the binding mechanisms of binary bosonic mixtures in one- and higher-dimensional lattices.

Figures (10)

• Figure 1: Illustration of the system under consideration, where a few bosons of species $A$ (red circles) and $B$ (blue circles) are trapped in a square optical lattice.
• Figure 2: Ground-state binding energy $\epsilon_{b}$ for the AABB [(a) and (c)] and AAABBB [(b) and (d)] systems. The top panels show heatmaps of $\epsilon_{b}$ as a function of $U_{AB}/U$ and $U/t$ for lattices with $7\times 7$ (a) and $5\times 5$ (b) sites. The dashed black lines indicate the minimum of $\epsilon_{b}$. The bottom panels show $\epsilon_{b}$ as a function of $U_{AB}/U$ for $U/t=40$ (orange lines) and $U/t=60$ (blue lines) for different lattice sizes as indicated by the legends.
• Figure 3: Magnitude of the local minimum of the binding energy $\epsilon_{b}^*$ (a) and the interaction strength at which it appears $U_{AB}^*/U$ (b) for the AABB system as a function of $U/t$. The minima are located with a golden-section search algorithm, and thus the error bars indicate the associated error. Results for different lattice sizes are reported, as indicated by the legends.
• Figure 4: Average distances for the AABB [(a) and (c)] and AAABBB [(b) and (d)] systems as a function of $U_{AB}/U$ and $U/t$. The top panels show distances between the two different species $r_{AB}$, while the bottom panels show distances between the same species $r_{\sigma\sigma}$. The four- and six-body systems consider lattices with $7\times 7$ and $5\times 5$ sites, respectively. The average distance between two non-interacting bosons, $r_0$, is $\approx 2.859d$ for the $7\times 7$ lattice, while it is $\approx 2.009d$ for the $5\times 5$ lattice. The dashed white lines indicate the position of the minimum of $\epsilon_{b}$.
• Figure 5: Interspecies distance $r_{AB}$ at $U_{AB}^*$ for the AABB system as a function of $U/t$. The minima are located with a golden-section search algorithm, and thus the error bars indicate the associated error. Results for different lattice sizes are reported, as indicated by the legend.