Evolution of a Single Impurity Across the Superfluid-Mott insulator Transition in the Bose-Hubbard Model

TL;DR

This work presents a microscopic, real-space depiction of how a single impurity localizes in the two-dimensional Bose–Hubbard model as the bath transitions between SF and MI phases. Using sign-problem-free, two-component worm-algorithm QMC, it tracks impurity winding, bath superfluid response, and impurity-centered density deformations across attractive and repulsive couplings. In a compressible SF bath, localization proceeds via an interaction-driven winding-collapse, moving from a mobile light polaron to a heavy polaron and then to a bound cluster, while the bath stays superfluid. In an incompressible MI bath, localization occurs through quantized defect formation, with a vacancy for repulsive and a particle defect for attractive couplings; a compressibility-driven pathway is also shown by fixing $U_{ ext{ib}}/t$ and sweeping $U_{ ext{b}}/t$ across the SF–MI transition. Together with a companion Letter on repulsive couplings, these results provide a unified microscopic picture of impurity localization that can be probed with quantum gas microscopes in current ultracold-atom experiments.

Abstract

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 quantum Monte Carlo simulations based on the worm algorithm, we track the impurity winding number, the bath superfluid density and compressibility, and impurity--bath density correlations. First, by fixing the bath interaction at $U_{\mathrm{b}}/t=13.3$ in the superfluid regime and tuning the attractive impurity-bath coupling from $U_{\mathrm{ib}}/t=-1.0$ to $-40.0$, we uncover an interaction-driven winding-collapse localization: the impurity evolves from a mobile light polaron with finite winding to a heavy polaron and, finally, to a bound cluster with vanishing winding, while the bath remains globally superfluid. Second, we analyze impurity in Mott-insulating baths for both attractive and repulsive impurity--bath couplings, contrasting the resulting deformation clouds and localization patterns. Third, for a moderate attractive impurity-bath coupling $U_{\mathrm{ib}}/t=-8.0$, we tune the bath interaction $U_{\mathrm{b}}/t$ across the superfluid--Mott-insulator transition and find a compressibility-controlled localization crossover of coherent impurity motion. Finally, together with our companion Letter [\emph{The Fate of a Single Impurity in the Bose--Hubbard Model}], which focused on repulsive impurity-bath couplings, these results provide a unified microscopic picture of impurity localization in the Bose--Hubbard model, connecting interaction-driven and compressibility-controlled mechanisms across both attractive and repulsive regimes.

Figures (15)

• Figure 1: Schematic impurity behavior in superfluid (SF) and Mott-insulating (MI) baths. The bath undergoes a SF--MI transition at $U_{\mathrm{b}}/t \simeq 16.7$. (a) In a compressible SF bath, bosons (blue wavy background) possess long-range phase coherence and support extended density fluctuations. A repulsive impurity (red disk) creates a broad depletion cloud, whereas an attractive impurity induces a smooth accumulation halo. Because the bath is compressible, these density modulations spread over many sites and the impurity retains coherent motion with finite winding, forming a light polaron at weak coupling and an extended heavy state at stronger coupling. (b) In an incompressible MI bath, density fluctuations are frozen and only short-range particle--hole excitations are allowed. For both repulsive and attractive impurity--bath couplings, the impurity behaves as a nearly free defect at moderate $U_{\mathrm{b}}/t$, producing only weak local distortion. At larger $U_{\mathrm{b}}/t$ the impurity becomes fully localized: a repulsive impurity expels one bath particle and forms a quantized vacancy defect, while an attractive impurity binds extra one boson into a particle defect. Both MI-side defects exhibit short-range density distortion and vanishing impurity winding, in stark contrast to the extended polaronic dressing in the SF regime.
• Figure 2: Quasiparticle properties of an attractive impurity in a superfluid bath. (a) the polaron ground-state energy $E_{\mathrm{p}}/t$, (b) the effective-mass ratio $m^{*}/m_0$ with $m_0$ the bare particle mass, and (c) the quasiparticle residue $Z_0$ as functions of the attractive impurity--bath coupling $|U_{\mathrm{ib}}|/t$ at fixed bath interaction $U_{\mathrm{b}}/t=13.3$, deep in the superfluid regime. Error bars denote statistical uncertainties and are smaller than the symbols where not visible.
• Figure 3: Real-space impurity dressing for attractive impurity in a superfluid bath at $U_{\mathrm{b}}/t = 13.3$. (a) Cumulative bath-density enhancement $\Delta N(R)$ for various attractive impurity--bath couplings $U_{\mathrm{ib}}/t<0$, showing the evolution from a broad, weak dressing cloud (light polaron) to a sharply localized multi-particle cluster. (b) Radius of the maximum $R(\Delta N_{\max})$, demonstrating the monotonic inward compression of the dressing cloud as $|U_{\mathrm{ib}}|/t$ increases. (c) Magnitude of maximum $\Delta N_{\max}(R)$, which increases without saturation, reflecting the grand-canonical nature of the bath: stronger attraction continuously recruits additional bosons into the impurity vicinity. Unlike the repulsive saturated-bubble case where $\Delta N(R)$ saturates at $-1$, the attractive branch exhibits unbounded growth due to unrestricted local accumulation. These results demonstrate a continuous crossover: light polaron $\rightarrow$ heavy polaron $\rightarrow$ bound cluster, while the bath remains globally superfluid.
• Figure 4: Real-space impurity-centered correlator $C_{\mathrm{ib}}(R)$ in the superfluid bath at $U_{\mathrm{b}}/t=13.3$ for various attractive couplings $U_{\mathrm{ib}}/t<0$. For weak attraction ($|U_{\mathrm{ib}}|/t\lesssim13.3$, red curves), $C_{\mathrm{ib}}(R)$ is small and spatially broad, indicating an extended, compressible dressing cloud characteristic of a mobile light polaron. In the intermediate regime ($13.3< |U_{\mathrm{ib}}|/t \lesssim 20.0$, yellow curves), the response becomes stronger and short-ranged, signaling a compact heavy polaron with reduced coherent motion. For strong attraction ($|U_{\mathrm{ib}}|/t > 20.0$, blue curves), the correlation collapses to a sharply peaked on-site form, forming a self-trapped, multi-particle bound cluster. The inset shows that the on-site value $C_{\mathrm{ib}}(0)$ grows monotonically with $|U_{\mathrm{ib}}|/t$ and does not saturate, highlighting that in a grand-canonical superfluid bath, a deeper attractive impurity continuously recruits additional bath bosons without intrinsic upper bound. This demonstrates an interaction-driven winding-collapse crossover in a compressible superfluid host.
• Figure 5: Interaction-controlled winding-collapse localization of an attractive impurity in a superfluid bath at fixed $U_{\mathrm{b}}/t=13.3$. (a) Impurity winding number $\langle W_{\mathrm{imp}}^{2}\rangle$ as a function of attractive coupling $U_{\mathrm{ib}}/t<0$. A smooth suppression followed by a rapid collapse at $|U_{\mathrm{ib}}|/t\gtrsim10$ signals a transition from a mobile light polaron to a localized bound cluster. (b) Bath superfluid density $\rho_{\mathrm{b}}$ remains essentially unchanged across the entire coupling range, confirming that the impurity localization is not induced by the loss of global coherence in the bath. This establishes an interaction-driven winding-collapse localization mechanism: the impurity becomes localized even while the bath remains fully superfluid and compressible.