Tunable breakdown of the polaron picture for mobile impurities in a topological semimetal

Mobile impurities in cold atomic gases constitute a new platform for investigating polaron physics. Here we show that when impurity atoms interact with a two-dimensional Fermi gas with quadratic band touching, the polaron picture may either hold or break down depending on the particle-hole asymmetry of the band structure. If the hole band has a smaller effective mass than the particle band, the quasiparticle is stable and its diffusion coefficient varies with temperature as $D\left(T\right)\propto {ln}^{2}T$. If the hole band has larger mass, the quasiparticle weight vanishes at low energies due to an emergent orthogonality catastrophe. In this case we map the problem onto a set of one-dimensional channels and use conformal field theory techniques to obtain $D\left(T\right)\propto T^{\nu }$ with an interaction-dependent exponent $\nu$. The different regimes can be detected in the nonequilibrium expansion dynamics of an initially confined impurity.

Figure 1

(a) Checkerboard lattice. Red sites belong to the A sublattice and green sites to the B sublattice. There is a hopping parameter $t$ (solid line) between nearest neighbors in different sublattices and direction-dependent hopping $t_1$ (dashed line) or $t_2$ (dotted line) between second neighbors. (b) Band structure for $t_1/t=0.7$ and $t_2/t=-0.3$, showing a QBCP in the regime $m_{+}<m_{-}$.

Figure 2

Renormalization of the impurity-fermion interaction. (a) Diagrams that contribute at $O\left({\gamma }^2\right)$. The solid line represents fermion propagators and the dashed line the impurity propagator. (b)–(d) illustrate the mechanism. (b) For $\gamma >0$, the fermion density (blue shaded region) is depleted near the impurity. Conversely, the impurity experiences an effective potential that attracts it to the region of lower density. If the impurity is localized, the eigenstates of the Hamiltonian are the scattering states of a static potential. If the impurity mass is large but finite, the impurity can create particle-hole pairs as it moves slowly through the system. (c) For $m_{+}<m_{-}$, the particle wave packet (with density $\delta {\rho }_{+}$) will diffuse faster and have a larger width than the hole wave packet (with density $\delta {\rho }_{-}$) after some short time $\delta t$. The net density $\delta \rho =\delta {\rho }_{+}+\delta {\rho }_{-}$ is then a nonzero function of position with a minimum at the center. (d) This leads to a further depletion of the total density at the instantaneous position of the impurity and an enhancement of the effective potential. For $m_{+}>m_{-}$, the sign of $\delta \rho$ is reversed and the effective potential becomes shallower, implying that the interaction flows to weak coupling.

Figure 3

Sketch of the temperature dependence of the diffusion coefficient for $m_{+}>m_{-}$ (blue solid line) and $m_{+}<m_{-}$ (red dashed line). The dotted line indicates the scaling result $D\left(T\right)=\text{const}$ expected at higher $T$.