Lattice Polarons across the Superfluid to Mott Insulator Transition

We study the physics of a mobile impurity confined in a two-dimensional lattice, moving within a Bose-Hubbard bath at zero temperature. Exploiting the quantum Gutzwiller formalism, we develop a beyond-Fröhlich model of the bath-impurity interaction to describe the properties of the polaronic quasiparticle formed by the dressing of the impurity by quantum fluctuations of the bath. We find a stable and well-defined polaron throughout the entire phase diagram of the bath, except for the very low tunneling limit of the hard-core superfluid. The polaron properties are highly sensitive to the different universality classes of the quantum phase transition between the superfluid and Mott insulating phases, providing an unambiguous probe of correlations and collective modes in a quantum critical many-body environment.

Figure 1

(a) Mean-field Gutzwiller phase diagram of a Bose-Hubbard bath on a $d$-dimensional lattice around the $⟨\widehat{n}⟩=1$ MI lobe showing constant $⟨\widehat{n}⟩$ (blue, black, green) and $\mu /U$ (red, blue) lines. The SF to MI transition can be crossed at the tip at constant integer density ($O(2)$) transition or on the edges via the commensurate-incommensurate (CI) transition. For increasing interaction strength $U$, noninteger filling lines connect deep and hard-core (HC) regimes of the superfluid. (b) A mobile impurity with hopping $J$ and coupling $U_{12}$ to a BH bath is dressed by a cloud of excitations, producing a Bose lattice polaron.

Figure 2

Diagrammatic representation of the beyond mean-field contributions to the interacting impurity Green’s function to second order in $U_{12}$ within the zero-temperature QGW approach. (a) and (b) depict the one- and two-particle diagrams with QGW vertex functions $N_{\alpha ,\mathit{q}}$ and $W_{\alpha \mathit{q},\beta {\mathit{q}}^{\prime }}$, respectively. Full lines represent the bare impurity Green’s function $G^{(0)}$, while dashed lines correspond to bare Green’s functions $D_{\alpha }^{(0)}$ of the collective modes of the BH bath.

Figure 3

(a),(c),(e) Polaron properties for fixed $\mu /U$ across the $O(2)$ (blue) and CI (red) transitions, with the nonanalytical nature of the latter shown in the inset of panel (c). (b),(d),(f) Polaron properties at (black) integer and (green, blue) noninteger filling of the bath $⟨\widehat{n}⟩$. Dots indicate the quantum critical points shown in Fig. 1.

Figure 4

Behavior of the polaron decay rate [see Eq. (8)] for (a) fixed $\mu /U$ across the $O(2)$ (blue) and CI (red) transitions; (b) fixed $⟨\widehat{n}⟩$ at integer filling (black) across the $O(2)$ transition and noninteger fillings (blue, green). Dots indicate quantum critical points, with the Landau-critical values of the impurity momentum $k_c$ shown in the insets.