Role of impurity statistics and medium constraints in polaron-polaron interactions

TL;DR

This work develops a unified, perturbative framework for polaron-polaron interactions in quantum mixtures, showing that both impurity statistics (bosons, fermions, or distinguishable) and the thermodynamic constraint on the medium (fixed density or fixed chemical potential) decisively shape mediated interactions. By constructing two-impurity wave functions in Bose or Fermi backgrounds and expanding to second order in impurity-medium coupling, the authors derive explicit expressions for the induced polaron interactions and reveal an exact thermodynamic relation connecting the fixed-density and fixed-chemical-potential descriptions: $F_{\mu,\sigma\sigma'} = F_{n,\sigma\sigma'} - (\Delta N_\sigma \Delta N_{\sigma'})/\mathcal{N}$. The results reconcile apparent discrepancies across platforms (cold atoms and 2D semiconductors) and predict novel phenomena such as medium-enhanced repulsion for degenerate bosons and distinct cross-species interactions for distinguishable impurities. These insights provide a rigorous baseline for interpreting current experiments and for building strong-coupling theories that include exchange, Hartree-type, and phase-space–filling effects. The framework is general and extensible to low-dimensional systems, bipolaron formation, and lattice settings.

Abstract

We consider the behavior of a small density of mobile impurities (polarons) immersed in a quantum gas, a generic scenario that can be realized in cold atomic gases, liquid helium mixtures and doped semiconductors. We present a unified theoretical framework for understanding polaron quasiparticles beyond the single-impurity limit, and we identify two key factors that control the polaron-polaron interactions: (i) the statistics of the impurities, including whether or not they are degenerate, and (ii) the constraints on the medium response, i.e., whether the medium density or chemical potential is held fixed. By constructing wave functions for two bosonic, fermionic, or distinguishable impurities immersed in a Bose or Fermi gas, we derive rigorous results for the polaron interactions in the limit of weak impurity-medium coupling. We furthermore obtain an exact relationship between the polaron interactions at fixed medium density and at fixed chemical potential, a result which is valid for arbitrary interaction strength. Our work provides an important guide for understanding experiments, and it acts as a starting point for future strong-coupling theories of polaron interactions that capture all of the effects identified in this work.

Figures (7)

• Figure 1: Sketch of the different constraints on the medium (blue particles): either its chemical potential (a) or its density (b) is fixed. These can impact the strength of the interactions between polarons. For instance, for identical and degenerate bosonic impurities (red), the interactions can be attractive at fixed chemical potential and repulsive at fixed density.
• Figure 2: Sketch of the different measurement protocols carried out in a trapped system. In an equilibrium measurement (a), such as one that probes the shape of the impurity cloud, the medium chemical potential is kept fixed across the trap. On the other hand, in an injection measurement (b) carried out at a time scale short compared with the (inverse) trap frequency, the medium chemical potential is deformed locally while the density is kept constant.
• Figure 3: (a) Exchange and (b) Hartree diagrams for quasiparticle interactions between Bose polarons at lowest order in perturbation theory. The black lines denote impurity propagators (fermionic, bosonic, or distinguishable), while the blue solid and dotted lines are Bogoliubov excitations and condensate lines of the majority Bose gas, respectively. The squares are the impurity-medium interaction constants, which can in principle be different for distinguishable impurities. The exchange term only exists for fermionic or non-degenerate (${\bf p}_1\neq{\bf p}_2$) bosonic impurities, while the Hartree term only contributes to the quasiparticle interactions at fixed medium chemical potential.
• Figure 4: (a) Bare impurity interaction (circle) and (b,c) contributions where the medium enhances the bare interactions. Lines and squares as in Fig. \ref{['fig:hartreefockBose']}. Note that, in the case of distinguishable impurities ($\sigma \neq \sigma'$), diagram (c) does not include processes involving $g_{\sigma\sigma'}g^2_{b\sigma}$ since these would correspond to a self-energy insertion on the $\sigma$ impurity propagator, and such a self-energy insertion only contributes to the polaron energy at fixed density $n$ and not to the effective interactions.
• Figure 5: (a) Exchange and (b) Hartree diagrams for quasiparticle interactions between Fermi polarons at lowest order in perturbation theory. The black lines denote fermionic, bosonic, or distinguishable impurities, while the blue lines denote majority particle propagators. The squares are the impurity-medium interaction constants, and can in principle be different for distinguishable impurities. As in Fig. \ref{['fig:hartreefockBose']}, the exchange term only exists for fermionic or non-degenerate bosonic impurities. The Hartree term only contributes to the quasiparticle interactions at fixed medium chemical potential, not at fixed medium density.