Dimensional crossover of bound complexes in a two-species Bose-Hubbard lattice: correlations and dynamics

TL;DR

This work analyzes a minimal four-particle mixture in a two-species Bose-Hubbard lattice to understand how geometry and interactions shape few-body bound states. Using exact diagonalization, the authors map equilibrium regimes from strictly 1D chains to coupled-chain arrays, identifying weakly and strongly correlated dimers as well as tetramers via binding energy $E_b$ and interspecies entanglement entropy $S_N$, supplemented by real-space two-body densities. They show that increasing transverse connectivity or interchain hopping $J_y$ expands the tetramer region and drives a transition from a degenerate dimer manifold to a localized tetramer ground state, with detailed correlation fingerprints distinguishing bound states. The paper also demonstrates dynamical preparation and interconversion of these complexes through interaction quenches and geometric quenches, achieving high fidelities and tracking entanglement growth, thereby providing a practical microscopic framework for engineering lattice-bound few-body bosonic states. These insights are relevant for deterministic preparation and site-resolved detection in optical lattices, offering pathways to explore few-body physics in tunable lattice geometries with potential extensions to imbalance, different mixtures, and long-range interactions.

Abstract

We study the equilibrium and nonequilibrium formation of four-particle complexes in a balanced two-species Bose-Hubbard model with repulsive intra- and attractive inter-species interactions. Using exact diagonalization, we characterize the transition from weakly- to strongly-correlated dimer and tetramer states along the one- to two-dimensional crossover in coupled-chain geometries by combining local correlation signatures with global diagnostics such as the binding energy and interspecies entanglement entropy. We show that transverse connectivity between chains qualitatively reshapes the phase diagram, substantially enlarging the tetramer region and, in particular, stabilizing weakly bound tetramers when compared to the one-dimensional chains. By tuning the interchain hopping, we identify a transition from a degenerate manifold of spatially separated dimers to a localized tetramer ground state, driven by the lifting of one-dimensional configurational degeneracies and an associated kinetic-energy gain. Finally, we demonstrate interaction and geometric quench protocols to dynamically prepare these complexes with high fidelity. Our results provide a microscopic framework for engineering and probing few-body bosonic bound states in tunable lattice geometries.

Figures (8)

• Figure 1: Ground-state few-body complexes and correlation signatures in a 1D lattice. (a) Binding energy, $E_b$, and (b) von Neumann entanglement entropy, $S_{\rm N}$, of the four-particle system as functions of $U_{AB}/U$ and $U/J_x$. (c) $S_{\rm N}$ (blue curve) and its derivative $\partial S_{\rm N}/\partial (U/J_x)$ (red curve) as a function of $U/J_x$ for a fixed ratio $U_{AB}/U = -0.35$. Panel (a) delineates the regions corresponding to the weakly- and strongly-correlated dimer, and tetramer states. The white dash-dotted line indicates the transition to the regime of negative binding energy, signaling the formation of a tetramer. The green line demarcates the low- and high-interspecies entanglement regimes ($S_{\mathrm{N}}^{\,\max} \approx 4.91$, for a maximally entangled state), and is identified from the peak of $\partial S_{\rm N}/\partial (U/J_x)$ [see, e.g., panel (c)]. (d1)-(d3) One-body density distributions of species $\sigma$, $\rho^{(1)}_{i;\sigma}$, for the tetramer (d1), weakly-correlated dimer (d2), and strongly-correlated dimer (d3), states, respectively. Markers at the top indicate the corresponding parameter values in panel (a). (e1)-(e3) Interspecies two-body densities $\rho^{(2)}_{ii',\sigma\overline{\sigma}}$ and (f1)-(f3) intraspecies two-body densities $\rho^{(2)}_{ii',\sigma\sigma}$, corresponding to the same parameter points marked in panel (a).
• Figure 2: Effect of lattice geometry. Binding energy $E_b$ as a function of $U_{AB}/U$ and $U/J_x$ for (a1) a 1D $L_x \times L_y = 8 \times 1$ lattice and (a2)–(a4) arrays of 1D lattice configurations with different $L_y$ (see legends), assuming isotropic hopping $J_y = J_x$. The tetramer phase exists over an increasingly larger region of the phase diagram as the $L_y$ increases. For large $U/J_x$, the tetramer eventually appears in two narrow strips of the phase diagram around, located $U_{AB}/U \approx -1$ and $\approx -0.1$. The former region consists of bound states formed from strongly-correlated $AB$ pairs, while the latter corresponds to weakly-correlated $AB$ pairs (see text). Shown also the behavior of the (b1) inter- and (b2) intra-species correlators between a reference site $(i', j') = (3, 3)$ in the bulk and its few neighboring sites as functions of $U_{AB}/U$ for an $8\times 8$ system at $U/J_x=20$ (dashed line in panel (a4))
• Figure 3: Effect of anisotropic hopping. (a1)–(a4) Binding energy $E_{b}$ as a function of $U_{AB}/U$ and $U/J_x$ on an $L_x \times L_y = 8 \times 5$ lattice for anisotropic hopping parameters with $J_y/J_x = 0$ [(a1)], $0.3$ [(a2)], $0.7$ [(a3)], and $1$ [(a4)]. (b) Energies of the lowest 25 eigenstates as a function of $J_y$ on an $8 \times 5$ geometry for $U_{AB}/U = -0.6$ and $U/J_x = 10$. For uncoupled chains ($J_y = 0$), the chosen parameters correspond to a dimer ground state with a ten-fold degeneracy, while for $J_y = J_x$ the ground state is a tetramer. The inset further highlights the degeneracy of the ground states at $J_y = 0$. Two-body (c1)–(c4), (d1)–(d4) inter-species density matrices $\rho^{(2)}_{(ij),(3,2);\sigma\bar{\sigma}}$, and (e1)–(e4), (f1)-(f4) intra-species density matrices $\rho^{(2)}_{(ij),(3,2);\sigma\sigma}$, where one particle is fixed at $(i', j') = (3, 2)$ for selected points indicated by the markers in panels (a1)–(a4). For $U_{AB}/U = -0.6$ and $U/J_x=10$, ground state switches from a dimer to a tetramer as $J_y$ increases (c1)-(c4), (e1)-(e4) while for $U_{AB}/U = -0.85$, $U/J_x=8$ ($E_b < 0$) the ground state remains a tetramer (d1)-(d4), (f1)-(f4).
• Figure 4: Dynamics following interaction quenches in a 1D lattice. (a1, a2) Time evolution of the one-body density, $\rho^{(1)}_{i;\sigma}$, for (a1) an intraspecies interaction quench from $U=2J_x$ to $U=25J_x$ at $U_{AB} = -0.35U$, representing a weakly-correlated to strongly-correlated dimer transition, and (a2) an interspecies interaction quench from $U_{AB} = -0.35U$ to $U_{AB} = -0.95U$ at $U=2J_x$, representing a weakly-correlated dimer to tetramer transition, for $\tau_Q=100$ . (b) Temporal evolution of the fidelity $\mathcal{O}(t)$. The inset shows the fidelity at the end of the quench $\mathcal{O}(t=\tau_Q)$ as a function of $\tau_Q$. (c) Time evolution of the interspecies entanglement entropy, $S_{\mathrm{N}}$ (left axis), and the binding energy, $E_b$ (right axis). Solid and dashed lines correspond to the weakly-correlated dimer to strongly-correlated dimer and tetramer quenches, respectively.
• Figure 5: Evolution of two-body correlations following interaction quenches. Instantaneous profiles of the two-body reduced density matrices for (a1)-(a4), (c1)-(c4) interspecies pairs $\rho^{(2)}_{ii';\sigma\bar{\sigma}}$ and (b1)-(b4), (d1)-(d4) intraspecies pairs $\rho^{(2)}_{ii';\sigma\sigma}$ ($\sigma = A, B$) . Panels (a1)–(a4) and (b1)–(b4) depict the dynamical formation of strongly-correlated dimers following a quench from $U/J = 2$ to $U/J_x = 25$ at $U_{AB} = -0.35U$. Panels (c1)–(c4) and (d1)–(d4) illustrate the formation of a tetramer state following a quench from $U_{AB} = -0.35U$ to $U_{AB} = -0.95U$ at $U=2J_x$. Both processes originate from a weakly-correlated dimer. The diagonal and off-diagonal features track the emergence of localized spatial correlations and the development of bound-state structures over time.