Impurity Problem in a Bilayer System of Dipoles

We consider a bilayer geometry where a single impurity moves in a two-dimensional plane and is coupled, via dipolar interactions, to a two-dimensional system of fermions residing in the second layer. Dipoles in both layers point in the same direction oriented by an external field perpendicular to the plane of motion. We use quantum Monte Carlo methods to calculate the binding energy and the effective mass of the impurity at zero temperature as a function of the distance between layers as well as of the in-plane interaction strength. In the regime where the fermionic dipoles form a Wigner crystal, the physics of the impurity can be described in terms of a polaron coupled to the bath of lattice phonons. By reducing the distance between layers this polaron exhibits a crossover from a free-moving to a tightly bound regime where its effective mass is orders of magnitude larger than the bare mass.

Figure 1

Binding energy of the impurity as a function of the distance between layers for different values of the in-plane interaction strength $k_F{r}_0$. Circles refer to the FL and squares to the WC phase. Dashed lines correspond to the binding energy of the two-body problem calculated from the Schrödinger equation. Solid lines are the results of perturbation theory holding when $k_F\lambda$ is large. The red line refers to a weakly interacting FL state, while the black line corresponds to Eq. (6) in the WC phase.

Figure 2

Inverse effective mass of the impurity as a function of the distance between layers for different values of the in-plane interaction strength $k_F{r}_0$. Circles refer to the FL and squares to the WC phase. The stars correspond to $m/m^{*}$ when only the static periodic potential $U({\mathbf{r}}_a)$ is considered and phonons are frozen. Lines connecting the symbols are a guide to the eye. The solid black line corresponds to Eq. (7) in the WC phase.

Figure 3

Mean square displacement, in units of the lattice spacing $a$, of the impurity coupled to the WC phase at $k_F{r}_0=35$: lowermost and uppermost curves correspond to $k_F\lambda =1$ and $k_F\lambda =2$, respectively. The imaginary time $\tau$ is in units of $m{r}_0^2/\hslash$. The dashed line corresponds to the free diffusion $m/m^{*}=1$.

Figure 4

Particle-impurity distribution function in terms of the in-plane distance. WC phase at $k_F{r}_0=35$ [panel (a)] and FL phase at $k_F{r}_0=2.5$ [panel (b)]. In both cases the peak at $r=0$ is higher for $k_F\lambda$ smaller. The particle-particle correlation function of the clean system is also shown for the two cases (dashed line).