The fate of a single impurity in the Bose-Hubbard model

Abstract

We map out the global phase diagram of a single mobile impurity in the two-dimensional Bose-Hubbard model, spanning the bath evolution from a compressible superfluid (SF) to an incompressible Mott insulator (MI) and the full range of impurity-bath coupling. Using sign-problem-free worm-algorithm quantum Monte Carlo method, we identify two sharply distinct localization mechanisms that organize the entire diagram. In the compressible SF, increasing impurity-bath coupling $|U_{\mathrm{ib}}|$ drives an \emph{interaction-driven winding-collapse crossover}: a light, extended polaron with finite winding evolves continuously into a heavy polaron and ultimately a self-trapped state -- a repulsive \emph{saturated bubble} or an attractive \emph{bound cluster} -- even while the bath remains globally superfluid, establishing localization without any bath phase transition. By contrast, upon tuning the bath across the SF-MI transition at fixed impurity-bath coupling $U_{\mathrm{ib}}$, localization becomes \emph{compressibility controlled}: the vanishing bath compressibility quenches polaronic dressing and collapses the impurity-centered density response, converting the polaron into an almost free defect and, deep in the MI, into a fully localized vacancy or particle defect. In the incompressible regime, localization proceeds via \emph{quantized defect formation}, manifested by discrete changes in the total bath occupation. Together, our results provide a unified microscopic picture of impurity localization in correlated lattice bosons, governed by winding collapse in the SF and compressibility-driven localization across the SF-MI transition.

Figures (5)

• Figure 1: Phase diagram of a single impurity in the two-dimensional Bose--Hubbard model in the $(U_{\mathrm{ib}}/t,\, U_{\mathrm{b}}/t)$ plane. The vertical dashed line separates repulsive ($U_{\mathrm{ib}}/t>0$) and attractive ($U_{\mathrm{ib}}/t<0$) impurity–bath couplings, while the horizontal dashed line marks the superfluid–Mott transition of the bath ($U_{\mathrm{b}}/t\simeq16.7$). In the compressible superfluid region ($U_{\mathrm{b}}/t<16.7$), increasing $|U_{\mathrm{ib}}|/t$ drives an interaction-driven, winding-collapse polaron localization: a mobile light polaron continuously evolves into a heavy polaron and, for sufficiently strong coupling, into a saturated bubble (repulsive side) or a bound cluster (attractive side). In the incompressible Mott-insulating regime ($U_{\mathrm{b}}/t\gtrsim16.7$), the impurity localizes by quantized defect formation. Upon increasing $|U_{\mathrm{ib}}|/t$ beyond a threshold, the impurity changes abruptly from a nearly free defect to a pinned vacancy (repulsive) or particle (attractive) defect, signaled by an integer-valued bath occupation change. For intermediate couplings, the impurity remains only weakly dressed and essentially free. Along vertical cuts at fixed $U_{\mathrm{ib}}/t$, crossing the SF–MI boundary thus realizes compressibility-controlled localization. Black dots mark QMC data points; solid white lines are guides to the eye highlighting continuous crossover boundaries in the superfluid, while dotted lines indicate sharp transitions between distinct defect regimes in the Mott phase.
• Figure 2: Impurity quasiparticle properties in a superfluid bath ($U_{\mathrm{b}}/t = 13.3$). (a) Ground-state energy $E_{\mathrm{p}}/t$, (b) effective mass $m^*/m_0$ (with $m_0=1/2t=1/2$ the bare particle mass), and (c) quasiparticle residue $Z_0$ as functions of repulsive impurity-bath coupling $U_{\mathrm{ib}}/t$. All data correspond to finite impurity winding $\langle W_{\mathrm{imp}}^2\rangle>0$, where a mobile light polaron is well defined. Increasing $U_{\mathrm{ib}}$ continuously enhances mass renormalization and suppresses $Z_0$, reflecting stronger impurity–bath correlations. Beyond $U_{\mathrm{ib}}/t\gtrsim13.3$, the winding number vanishes (shown in Figure \ref{['fig:realspace']}(e)), and the quasiparticle description breaks down, marking the boundary of the light-polaron regime. The ground-state energy changes sommthly throughout, confirming a continuous interaction-driven crossover rather than a sharp transition.
• Figure 3: Real-space crossover from a light polaron to a saturated bubble in the superfluid bath ($U_{\mathrm{b}}/t=13.3$). (a) Cumulative bath-density response $\Delta N(R)$ for increasing impurity--bath coupling $U_{\mathrm{ib}}/t$. Red curves denote the light-polaron regime, the single orange curve highlights the heavy-polaron regime, and blue curves correspond to the saturated-bubble regime where the impurity becomes localized. (b) Impurity--centered correlation $C_{\mathrm{ib}}(R)$ and its on-site contrast $C_{\mathrm{ib}}(0)$ (inset). (c,d) Position $R_{\min}$ and depth $\Delta N_{\min}$ of the depletion minimum extracted from $\Delta N(R)$. (e) Impurity winding number $\langle W_{\mathrm{imp}}^2\rangle$, exhibiting the continuous loss of coherent motion. (f) Bath superfluid density $\rho_{\mathrm{b}}$, which remains nearly constant across the crossover. The color evolution illustrates the continuous crossover from a mobile light polaron (red) to a compact heavy polaron (orange), and finally to a fully localized saturated bubble (blue).
• Figure 4: Compressibility-controlled collapse of polaronic dressing at a fixed repulsive impurity bath coupling $U_{\mathrm{ib}}/t = 8.0$. (a) Ground-state energy $E_{\mathrm{p}}/t$, (b) effective-mass ratio $m^*/m_0$, and (c) quasiparticle residue $Z_0$ as functions of bath interaction $U_{\mathrm{b}}/t$. The shaded region marks the superfluid--MI transition of the bath ($U_{\mathrm{b}}/t\simeq 16.7$). In the compressible superfluid regime ($U_{\mathrm{b}}/t<16.7$), the impurity exhibits a weakly renormalized light-polaron behavior with finite dressing cloud. As the bath becomes increasingly less compressible, $m^*/m_0$ decreases while $Z_0$ rises, indicating a collapse of polaronic dressing. Once the bath enters the incompressible MI regime ($U_{\mathrm{b}}/t\gtrsim16.7$), $m^*/m_0$ approaches unity and $Z_0\to1$, confirming the emergence of a nearly-free.
• Figure 5: Compressibility-controlled collapse of polaronic dressing at fixed repulsive impurity bath coupling $U_{\mathrm{ib}}/t=8.0$. (a) Cumulative bath-density depletion $\Delta N(R)$ and (b) impurity--centered correlator $C_{\mathrm{ib}}(R)$ for increasing bath interaction $U_{\mathrm{b}}/t$, color-coded from deep superfluid (dark red: $U_{\mathrm{b}}/t=8.0$) through intermediate (light red/orange: $U_{\mathrm{b}}/t=13.3,15.0$) to near the transition (light blue: $U_{\mathrm{b}}/t=17.0$) and deep Mott regime (dark blue: $U_{\mathrm{b}}/t=20.0, 22.0 ,24.0$). (c,d) Peak position $R(\Delta N_{\min})$ and depth $\Delta N_{\min}(R)$, showing the loss of extended bath correlations. (e) Impurity winding number $\langle W_{\mathrm{imp}}^{2}\rangle$ increases sharply once the bath loses compressibility, signaling the loss of coherent propagation. (f) Bath superfluid density $\rho_{\mathrm{b}}$ confirms that the collapse is driven by the vanishing bath compressibility. Together these observables reveal a compressibility-controlled crossover from a mobile polaron to a nearly free defect, and finally to a localized particle-defect in the MI background.