Mobile impurities and orthogonality catastrophe in two-dimensional vortex lattices

We investigate the properties of a neutral impurity atom coupled with the Tkachenko modes of a two-dimensional vortex lattice in a Bose-Einstein condensate. In contrast with polarons in homogeneous condensates, the marginal impurity-boson interaction in the vortex lattice leads to infrared singularities in perturbation theory and to the breakdown of the quasiparticle picture in the low-energy limit. These infrared singularities are interpreted in terms of a renormalization of the coupling constant, quasiparticle weight, and effective impurity mass. The divergence of the effective mass in the low-energy limit gives rise to a power-law singularity in the impurity spectral function and provides an example of an emergent orthogonality catastrophe in a bosonic system.

Figure 1

Impurity self-energy (a) and vertex correction Feynman diagram (b). The solid and wavy lines represent free impurity and Tkachenko mode (analogous to a phonon) propagators, respectively.

Figure 2

RG flow diagram. The starting points correspond to $Z=1$ but different values of the bare dimensionless parameters ${\lambda }_{0}M$ and $M/m_0$. The end points correspond to a line of low-energy fixed points with $Z\to 0,M/m\to 0$, but finite $\lambda M$.

Figure 3

(a) Lowest order diagram included in the approximation for the impurity Green's function in the limit $m\gg M$. The disconnected part with multiple wiggly lines represents the boson cloud propagator. (b, c) Connected cloud diagrams neglected in the approximation.

Figure 4

Impurity spectral function $A(k,\omega )$ in the low-energy regime for four different values of $k={\Lambda }_0{e}^{-\ell }$. As $\ell$ increases, the line shape crosses over from a Lorentzian to an approximate power-law singularity. The inset shows the RG flow of the dimensionless parameters; here we have set the bare values to $Z\left(0\right)=1,\tilde{m}\left(0\right)=0.6,\tilde{\lambda }\left(0\right)=0.8$. Here we set ${\Lambda }_0=M=1$. The spectral function is normalized so as to obey the sum rule [29] ${\int }_0^{K}d\omega A(k,\omega )=1$, with a cutoff on the high-frequency tail of Eq. (51) set by $K=k^2/\left[M\tilde{m}\left(0\right)\right]$.