Single-impurity polarons in hard-core lattice bosons at low and intermediate fillings

We extract momentum-space quasiparticle properties of the impurity from its single-particle Green’s function,

where $\mathbf{r}$ denotes the lattice displacement (equivalently, the site index $i$ used in Sec. II under periodic boundary conditions), and $\tau\in[0,\beta]$ is the imaginary-time separation. Within the multi-species worm algorithm, $G_{\mathrm{imp}}(\mathbf{r},\tau)$ is measured from the statistics of the impurity worm-end separation in the off-diagonal sector Prokof’ev et al. 1998; Prokof’ev et al. 1998; Capogrosso-Sansone et al. 2010; Lingua et al. 2018.

The momentum-resolved Green’s function is obtained by a discrete Fourier transform,

with lattice momenta $\mathbf{k}=(2\pi n_{x}/L,\,2\pi n_{y}/L)$ with $n_{x}$ and $n_{y}$ integers.

At sufficiently low temperature, the asymptotic behavior is dominated by the lowest polaron state at momentum $\mathbf{k}$,

This defines the polaron dispersion $E_{\mathrm{p}}(\mathbf{k})$ and the quasiparticle residue $Z_{\mathbf{k}}$.

A central goal of this work is to quantify how strongly and over what spatial extent the bath is distorted by a single mobile impurity. We therefore define an impurity-conditioned bath-density profile in real space and construct all real-space diagnostics from it.

We begin by characterizing the impurity through its momentum-space quasiparticle properties. Figure 2 summarizes the evolution of the ground-state energy $E_{p}(0)$, effective mass $m^{*}/m_{0}$, and quasiparticle residue $Z_{0}$ as functions of the bath filling $n_{\mathrm{b}}$ for different impurity–bath coupling strengths. For $t_{\rm imp}=t_{\rm b}$, the quasiparticle datasets extend up to $n_{\rm b}\simeq 0.96$, $0.90$, $0.72$, and $0.39$ for $U_{\rm ib}/t_{\rm b}=1$, $5$, $10$, and $\infty$, respectively.

As shown in Fig. 2(a), the impurity ground-state energy $E_{p}(0)$ increases monotonically with bath filling for all $U_{\rm ib}/t_{\rm b}$. This trend reflects the increasing cost of creating a local distortion in a denser bath subject to the hard-core constraint. Importantly, $E_{p}(0)$ evolves smoothly with $n_{\mathrm{b}}$ over the entire filling range studied.

Figure 2(b) shows the effective mass ratio $m^{*}/m_{0}$ extracted from the low-momentum dispersion of the impurity Green’s function. With increasing $n_{\mathrm{b}}$, the effective mass grows, signaling reduced impurity mobility due to enhanced dressing by bath density fluctuations. This effect is particularly pronounced in the hard-core limit, where the absence of double occupancy strongly constrains bath rearrangements around the impurity. Nevertheless, even in this extreme case the mass enhancement remains finite at the highest fillings accessible in the hard-core dataset, consistent with a heavy but mobile polaron.

The quasiparticle residue $Z_{0}$, shown in Fig. 2(c), provides a complementary measure of impurity dressing. As either $n_{\mathrm{b}}$ or $U_{\rm ib}/t_{\rm b}$ is increased, $Z_{0}$ decreases continuously. Notably, $Z_{0}$ remains nonzero for all fillings and impurity-bath coupling strengths studied here, including the hard-core-limit case, demonstrating that within the explored range the impurity retains coherent quasiparticle character.

Taken together, Figs. 2(a)–(c) show a smooth evolution from weaker to stronger dressing as the bath density increases. Over the filling range investigated at each coupling, the extracted mass remains finite and the residue remains nonzero. These data support a coherent polaron over the explored range.

To complement the momentum-space quasiparticle diagnostics, we now resolve the microscopic structure of impurity dressing in real space. Our central object is the impurity-centered correlator $C_{\rm ib}(\mathbf{r})$; we then perform radial averaging to obtain $C_{\rm ib}(R)$, from which all derived measures below are constructed. Figure 3 summarizes the results at strong repulsion $U_{\rm ib}/t_{\rm b}=10.0$ as a function of bath filling $n_{\mathrm{b}}$.

The cumulative bath-density deformation $\Delta N(R)$ in Fig. 3(a) exhibits a pronounced negative minimum for all fillings, demonstrating the formation of a depletion cloud around the impurity. With increasing $n_{\mathrm{b}}$ the minimum depth $\Delta N_{\min}$ becomes progressively more negative, i.e., the impurity expels more bath weight from its vicinity. At large distances, $\Delta N(R)$ returns smoothly to zero by construction, as required by the background-subtraction sum rule discussed in Sec. III.2.

The radially averaged impurity-centered correlator $C_{\rm ib}(R)$ in Fig. 3(b) shows a small negative response at small $R$ and decays to zero within $R\sim 7$–$8$. The magnitude of the near-origin depletion increases monotonically with $n_{\mathrm{b}}$, reflecting stronger local exclusion in a denser hard-core bath. The decay indicates that the dressing cloud is spatially compact at $U_{\rm ib}/t_{\rm b}=10.0$, while its overall strength can still vary substantially.

To summarize the deformation profile in Fig. 3(a), we track two robust features extracted from $\Delta N(R)$: the radius of its minimum $R(\Delta N_{\min})$ and the minimum value $\Delta N_{\min}$ at this $R$ [Figs. 3(c,d)]. A notable trend is that $R(\Delta N_{\min})$ stays stable across $0.06\leq n_{\mathrm{b}}\leq 0.72$. Thus, increasing $n_{\mathrm{b}}$ does not substantially shift the characteristic radius at which $\Delta N(R)$ is most negative. Instead, the dominant effect of filling is on the amplitude of the local response: $\Delta N_{\min}$ becomes more negative as $n_{\mathrm{b}}$ increases. This is consistent with the behavior of the on-site contrast $C_{\rm ib}(0)$ [Fig. 3(e)], whose magnitude also grows with filling. Although $C_{\rm ib}(0)$ remains numerically small, its monotonic trend shows that the extra dressing acquired at larger $n_{\mathrm{b}}$ is concentrated in the near-core region rather than in a broad expansion of the cloud.

Within the investigated interval $0.06\leq n_{\mathrm{b}}\leq 0.72$, $\Delta N_{\min}$ and $C_{\rm ib}(0)$ evolve smoothly and monotonically, without a plateau or abrupt change. The real-space response therefore shows a gradual increase of the short-range dressing amplitude. This trend accompanies the smooth mass and residue data in Sec. IV.1.

We finally consider the two-component hard-core limit, where both the impurity and the bath satisfy the on-site constraint $n_{i}\in\{0,1\}$ and the impurity–bath repulsion is effectively infinite, $U_{\rm ib}/t_{\rm b}\to\infty$.

Figure 4(a) shows the cumulative bath-density deformation $\Delta N(R)$ for several bath fillings. For all fillings considered, $\Delta N(R)$ develops a clear negative minimum at a finite radius and then recovers smoothly toward zero at large $R$, as required by the same background-subtraction sum rule.

Over $0.09\leq n_{\mathrm{b}}\leq 0.39$, the large-$R$ portions of the $\Delta N(R)$ curves overlap closely and therefore exhibit weak filling dependence at the available resolution. By contrast, increasing $n_{\mathrm{b}}$ changes the depth of $\Delta N_{\min}$ and the short-range magnitude of $C_{\rm ib}(R)$. This comparison indicates that the observed filling dependence is concentrated mainly in the short-distance response.

To quantify the dressing profile we plot the position of the minimum, $R(\Delta N_{\min})$, and its depth $\Delta N_{\min}$ in Fig. 4(c,d). We find that $R(\Delta N_{\min})$ depends only weakly on $n_{\mathrm{b}}$, whereas $\Delta N_{\min}$ becomes more negative with increasing $n_{\mathrm{b}}$.

Figure 4(e) plots $C_{\rm ib}(0)$ as a function of $n_{\mathrm{b}}$. As defined in Sec. III.2, this is an imaginary-time-averaged impurity-centered correlator, not an equal-time double-occupancy correlator. It can therefore be small and nonzero even in the two-component hard-core limit. Its magnitude increases from roughly $0.02$ to $0.04$ across the plotted fillings, consistent with the stronger near-core response in the other panels.

Momentum-space quantities such as the effective mass and quasiparticle residue directly characterize the impurity mobility and coherence, but they do not explicitly show how the bath is reorganized in real space. A complementary geometric summary is provided by the impurity-centered correlator. Since our primary real-space observable is the radially averaged profile $C_{\rm ib}(R)$, it is useful to compress this profile into two scalar measures that characterize its typical amplitude and spatial extent. The definitions of the shell-averaged dressing strength $\mathcal{D}_{\rm pol}$ and the shell-based polaron radius $\xi_{\rm pol}$ are given in Sec. III.2; here we focus on their physical interpretation and their evolution with bath filling.

Within this framework, $\mathcal{D}_{\rm pol}$ measures the overall magnitude of the radial response across distance shells: larger values indicate that the impurity induces a stronger local rearrangement of the bath, independent of its sign. The length scale $\xi_{\rm pol}$ measures where this shell-averaged weight is concentrated. A decrease of $\xi_{\rm pol}$ therefore indicates that the dominant response is pulled closer to the impurity rather than distributed over a broader range of $R$.

Figure 5(a) shows that $\mathcal{D}_{\rm pol}$ increases monotonically with $n_{\mathrm{b}}$ for all couplings considered. Physically, as the bath becomes denser, more bath weight is available near the impurity and the hard-core constraint makes local exclusion more effective, so the short-range depletion sharpens and the shell-averaged amplitude grows.

In contrast, Fig. 5(b) shows that the shell-based polaron radius $\xi_{\rm pol}$ decreases with increasing $n_{\mathrm{b}}$. Below unit filling the bath remains compressible, increasing density primarily enhances short-range exclusion and local rearrangement, which naturally compresses the effective dressing length scale.

The opposite trends of $\mathcal{D}_{\rm pol}$ and $\xi_{\rm pol}$ reveal a clear separation between the magnitude and the spatial extent of polaronic dressing: increasing $n_{\mathrm{b}}$ amplifies the shell-averaged deformation amplitude while shifting the dominant response toward shorter distances. Across the investigated two-component hard-core fillings, $\mathcal{D}_{\rm pol}$ increases with $n_{\mathrm{b}}$ while $\xi_{\rm pol}$ decreases, indicating a stronger but more spatially concentrated response around the impurity.

So far we have focused on an impurity with hopping $t_{\mathrm{imp}}=1$. To isolate the role of impurity mobility, we now reduce the hopping to $t_{\mathrm{imp}}=0.5$ while keeping the bath hard-core and at incommensurate fillings $n_{\rm b}<1$. We consider this reduced-mobility impurity in two representative impurity–bath coupling regimes: a strong but finite repulsion $U_{\rm ib}/t_{\rm b}=10.0$ and the two-component hard-core limit $U_{\rm ib}/t_{\rm b}\to\infty$. Decreasing $t_{\mathrm{imp}}$ doubles the bare lattice mass $m_{0}=(2t_{\mathrm{imp}})^{-1}$ and lowers the impurity kinetic-energy scale, thereby amplifying polaronic dressing at fixed interaction strength. Importantly, for $n_{\rm b}<1$ the bath remains compressible and its Hamiltonian is unchanged; thus varying $t_{\mathrm{imp}}$ primarily tunes the impurity mobility. This provides a controlled knob to disentangle kinetic effects from impurity–bath coupling and to quantify the competition between impurity motion and local impurity–bath exclusion.

Figure 6 summarizes the impurity quasiparticle properties for reduced impurity hopping $t_{\mathrm{imp}}=0.5$ at strong impurity–bath repulsion, comparing finite coupling $U_{\rm ib}/t_{\rm b}=10.0$ with the two-component hard-core limit $U_{\rm ib}/t_{\rm b}\rightarrow\infty$. In addition, the $t_{\mathrm{imp}}=0.5$ results can be contrasted with our baseline $t_{\mathrm{imp}}=1.0$ data (Sec. IV.1) for the mobility- and coherence-sensitive observables: reducing $t_{\mathrm{imp}}$ doubles the bare lattice mass $m_{0}=(2t_{\mathrm{imp}})^{-1}$ and lowers the impurity kinetic scale, thereby enhancing bath-induced dressing at fixed bath parameters and fixed $U_{\rm ib}/t_{\rm b}$.

The polaron ground-state energy $E_{\mathrm{p}}(0)$ [Fig. 6(a)] evolves smoothly with bath filling for both coupling limits. For $t_{\mathrm{imp}}=0.5$, $E_{\mathrm{p}}(0)$ increases monotonically with $n_{\mathrm{b}}$, reflecting the growing energetic cost of propagating an impurity through a denser background.

The effective-mass renormalization $m^{*}/m_{0}$ [Fig. 6(b)] exhibits a pronounced enhancement with increasing filling, demonstrating progressive suppression of impurity mobility by bath-induced dressing. At fixed $t_{\mathrm{imp}}$, the mass enhancement is stronger in the two-component hard-core limit than at $U_{\rm ib}/t_{\rm b}=10.0$, reflecting the most restrictive local constraint. Crucially, relative to the $t_{\mathrm{imp}}=1.0$ case the mass renormalization is amplified across the entire filling range studied: lowering $t_{\mathrm{imp}}$ provides an efficient route to enhance polaronic dressing without modifying the bath itself. Nevertheless, in both interaction limits $m^{*}/m_{0}$ varies continuously with $n_{\mathrm{b}}$.

Consistently, the quasiparticle residue $Z_{0}$ [Fig. 6(c)] decreases as $n_{\mathrm{b}}$ increases, signaling a gradual loss of coherent impurity motion. The suppression is more pronounced in the hard-core limit, again reflecting stronger short-range impurity–bath couplings. The extracted $Z_{0}$ remains finite throughout the investigated parameter range, supporting a coherent dressed quasiparticle within that range.

Overall, at fixed strong repulsion the two knobs—strengthening the local constraint (finite $U_{\rm ib}$ versus hard-core) and reducing the impurity mobility ($t_{\mathrm{imp}}=1.0$ versus $0.5$)—act in the same qualitative direction: they enhance the mass renormalization and suppress the residue, i.e., they make the polaron heavier and less coherent. At the same time, all quasiparticle observables remain smooth functions of $n_{\mathrm{b}}$ over the respective investigated ranges, indicating a continuous strengthening of polaronic dressing over the parameter range resolved here.

The real-space consequences of reduced impurity mobility are summarized in Fig. 7 in terms of the shell-averaged dressing strength $\mathcal{D}_{\mathrm{pol}}$ and the shell-based polaron radius $\xi_{\mathrm{pol}}$ for reduced impurity mobility $t_{\mathrm{imp}}=0.5$. For $t_{\mathrm{imp}}=0.5$, $\mathcal{D}_{\mathrm{pol}}$ increases with bath filling while $\xi_{\mathrm{pol}}$ decreases, for both the strong finite-coupling and hard-core results shown. The added dressing therefore accumulates mainly in the near-core region, again indicating a response that becomes stronger without spreading further out.

Comparing with the baseline $t_{\mathrm{imp}}=1.0$ results clarifies the role of reduced mobility. Lowering $t_{\mathrm{imp}}$ makes the impurity slower, so the induced deformation is less able to propagate over many shells. Taken together, these results show that lowering $t_{\mathrm{imp}}$ primarily reshuffles the balance between core strength and spatial extent: the response becomes more localized in real space while remaining smooth as a function of bath filling.

The setting considered here can be approached with a dilute second atomic component immersed in a two-dimensional bosonic gas in a square optical lattice. A large bath interaction-to-hopping ratio suppresses doublons and approximates the hard-core regime, while a local spin flip or dilute doping can prepare a single impurity. State-dependent lattices and interspecies Feshbach resonances provide control of $t_{\rm imp}/t_{\rm b}$ and $U_{\rm ib}/t_{\rm b}$ Fukuhara et al. 2013; Jørgensen et al. 2016; Hu et al. 2016. Radio-frequency spectroscopy can probe the polaron energy and the coherent spectral weight associated with $Z_{0}$, while wave-packet expansion or momentum-resolved probes can constrain the effective mass. Species-resolved quantum-gas microscopy can measure the equal-time bath-density profile conditioned on the simultaneously observed impurity position. Centering and averaging repeated snapshots on that position provides an experimental analogue of the real-space dressing cloud, although this observable is not identical to the imaginary-time-averaged $C_{\rm ib}(\mathbf{r})$ defined here. The hard-core and $U_{\rm ib}/t_{\rm b}\!\to\!\infty$ cases correspond experimentally to large but finite interaction-to-hopping ratios with strongly suppressed double occupancy.