Mobile impurity coupled to correlated lattice bosons

We investigate how the coherence and spatial dressing of a single impurity evolve in the two-dimensional Bose-Hubbard model when the impurity couples attractively to the bath. Using large-scale, sign-problem-free worm-algorithm quantum Monte Carlo, we measure the impurity winding, bath superfluid response and compressibility, and impurity-bath density correlations. In a compressible superfluid bath ($U_{\mathrm{b}}/t=13.3$), strengthening attraction drives an interaction-controlled \emph{winding-collapse self-trapping crossover}: a mobile light polaron evolves continuously into a heavy polaron and ultimately into a bound cluster with vanishing winding, while the bath remains globally superfluid. In incompressible Mott-insulating baths, by contrast, extended density rearrangements are suppressed and the dressing cloud collapses; we compare the resulting short-range deformation patterns for both attractive and repulsive couplings. Across the SF-MI transition at fixed moderate attraction ($U_{\mathrm{ib}}/t=-8.0$), the impurity crosses over from a mobile polaron with an extended deformation cloud to a nearly free defect with minimal dressing in the Mott background; for this coupling it does not lock into a fully self-trapped defect even deep in the insulator. Together with our companion Letter [Impurity Self-Trapping in Lattice Bose systems] on repulsive couplings, these results provide a unified microscopic picture of impurity self-trapping in correlated lattice bosons, connecting interaction-driven winding collapse in the superfluid to compressibility-controlled undressing across the SF-MI transition.

💡 Research Summary

This paper presents a comprehensive quantum‑Monte‑Carlo investigation of a single mobile impurity immersed in a two‑dimensional Bose‑Hubbard lattice. The impurity interacts attractively with the bath bosons (negative (U_{ib})), and the authors explore how the impurity’s coherence, effective mass, and spatial dressing evolve across three distinct parameter trajectories. The study employs a sign‑problem‑free multi‑species worm‑algorithm, allowing exact sampling of world‑line configurations for both impurity and bath species on lattices up to (L=20) with inverse temperature (\beta=L), ensuring ground‑state convergence.

The key observables are: (i) the impurity winding number squared (\langle W_{\text{imp}}^{2}\rangle), which directly measures the impurity’s ability to wind around periodic boundaries and therefore its coherent mobility; (ii) the bath superfluid density (\rho_{b}) obtained from the standard winding estimator, which signals the global superfluid–Mott‑insulator (SF–MI) transition; and (iii) impurity‑centered density correlators (C_{ib}(R)) and the cumulative deformation (\Delta N(R)), which quantify the real‑space redistribution of bath particles around the impurity.

Trajectory 1 – Interaction‑driven winding collapse in a compressible superfluid.
The bath interaction is fixed at (U_{b}/t=13.33), a point well inside the SF phase. The impurity‑bath coupling is tuned from weak attraction ((U_{ib}/t=-1)) to strong attraction ((U_{ib}/t=-40)). For weak coupling the impurity forms a light Bose polaron: (\langle W_{\text{imp}}^{2}\rangle) is finite, the effective mass is only modestly renormalized, and (\Delta N(R)) displays a broad, slowly decaying accumulation halo. As (|U_{ib}|) grows, the winding number continuously diminishes, signalling a crossover to a heavy polaron with a more localized dressing cloud. At the strongest couplings the winding collapses to zero, the impurity becomes self‑trapped, and the bath density around it forms a tightly bound cluster (a “bound cluster” state). Throughout this entire crossover the bulk superfluid density (\rho_{b}) remains essentially unchanged, confirming that the bath stays globally superfluid while the impurity undergoes a self‑trapping transition driven purely by the impurity‑bath interaction.

Trajectory 2 – Attractive vs. repulsive impurities in an incompressible Mott insulator.
Here the bath is deep in the MI regime (large (U_{b}/t) beyond the critical value). The authors compare two fixed couplings, (U_{ib}/t=+8) (repulsive) and (U_{ib}/t=-8) (attractive). Because the MI has vanishing compressibility, long‑range density fluctuations are frozen. Consequently, both types of impurity generate only short‑range distortions. For moderate (|U_{ib}|) the impurity behaves as a nearly free defect: (\langle W_{\text{imp}}^{2}\rangle) is very small but non‑zero, and (\Delta N(R)) is confined to a few lattice sites. When (|U_{ib}|) is increased further, the impurity binds an extra bath particle (attractive case) or expels a particle (repulsive case), forming a particle‑type or vacancy‑type defect, respectively. In both cases the winding number collapses to zero, indicating complete self‑trapping, and the deformation cloud becomes highly localized. This demonstrates that, unlike the SF case, the MI background suppresses the extended polaronic cloud and the self‑trapping mechanism is governed by the loss of compressibility rather than by interaction‑induced winding collapse.

Trajectory 3 – Compressibility‑controlled undressing across the SF–MI transition at fixed attraction.
The impurity‑bath coupling is held at a moderate attractive value (U_{ib}/t=-8) while the bath interaction (U_{b}/t) is swept from deep SF through the critical region into deep MI. In the SF regime the impurity is a light polaron with a sizable winding number and a broad accumulation halo. Approaching the critical point, both (\langle W_{\text{imp}}^{2}\rangle) and (\Delta N(R)) begin to shrink, reflecting the diminishing superfluid stiffness and compressibility of the bath. Once the system enters the MI phase, the winding is essentially zero and the deformation cloud collapses to a near‑flat profile, i.e., the impurity becomes a nearly free defect. If the attraction is further increased within the MI, the impurity can again bind a particle and become a localized particle‑defect, but the transition from polaron to defect is smooth and controlled by the bath’s compressibility.

Unified picture and broader implications.
Combining these results with the authors’ companion Letter on repulsive couplings, the paper establishes a global two‑route framework for impurity self‑trapping in the Bose‑Hubbard lattice: (i) an interaction‑driven winding‑collapse route operative within a compressible superfluid, and (ii) a compressibility‑controlled undressing route that governs the crossover when the bath is tuned across the SF–MI transition. The authors map these routes onto a (U_{b})–(U_{ib}) phase diagram that delineates light polarons, heavy polarons, saturated‑bubble or bound‑cluster states (SF side) and nearly free defects, particle/vacancy defects (MI side). Importantly, all diagnostics—winding numbers, superfluid density, and impurity‑centered real‑space correlators—are directly accessible in quantum‑gas‑microscope experiments, making the theoretical predictions experimentally testable.

In summary, the paper delivers a numerically exact, real‑space characterization of impurity dynamics across both compressible and incompressible backgrounds, clarifies the microscopic mechanisms behind interaction‑driven and compressibility‑driven self‑trapping, and provides concrete observables for future cold‑atom experiments investigating impurity physics in strongly correlated lattice bosons.