Emergence of fermion-mediated interactions in Bose-Fermi mixtures

TL;DR

The paper provides a rigorous derivation of fermion-mediated interactions in Bose–Fermi mixtures by proving that the low-energy spectrum of the interspecies many-body Hamiltonian converges to that of an effective bosonic Hamiltonian with a mediated two-body attraction given by $W^{\rm eff}=W-V*V$. This is achieved via a Born–Oppenheimer–type adiabatic decoupling, implemented through a particle–hole transformation that yields an excitation Hamiltonian, and a careful upper/lower bound analysis that yields explicit error control in the large $k_{\rm F}$ limit. In the GP scaling, the mediated interaction shifts the bosonic energy landscape, and the authors prove a stability–instability transition: for small coupling the bosonic subsystem has bounded energy per particle, while beyond a critical coupling it collapses. Additionally, the work clarifies the ground-state structure, showing approximate factorization into an effective Bose ground state and a Fermi sea state, and discusses extensions to small mass ratios. Overall, this work provides a mathematically rigorous bridge from microscopic Bose–Fermi models to an effective bosonic theory with controlled mediated interactions and a concrete phase diagram for stability.

Abstract

This work is inspired by recent experimental observations in ultracold atomic Bose-Fermi mixtures [DeSalvo et al., Nature 568 (2019)]. These experiments reveal the emergence of an attractive fermion-mediated interaction between bosons, as well as a stability-instability transition. We give the first mathematical demonstration of this transition by studying the low-energy spectrum of a many-body interspecies Hamiltonian. More precisely, we show the convergence of its eigenvalues towards those of an effective Bose Hamiltonian, which includes fermion-mediated effects. Applying this result to a model with short-range potentials, we derive a stability-instability transition in the bosonic subsystem, driven by the Bose-Fermi coupling strength $g$. For small $|g|$, the bosons form a stable Bose-Einstein condensate with the energy per particle uniformly bounded from below. For large $|g|$, the energy per particle is no longer uniformly bounded from below, signaling the collapse of the condensate.

Figures (2)

• Figure 1: Visualization of the effective interaction: (a) illustrates a coherent three-particle process (in blue) involving two bosons and one fermion, which generates the effective interaction (in red). (b) and (c) show the same process in momentum space, where the first boson excites a fermion from the Fermi ball and the second boson subsequently annihilates this excitation.
• Figure 2: Sketch of the phase diagram: For $w_g\geqslant 0$, the energy satisfies $\mathcal{E}(g) = 4\pi \mathfrak a(g)$. In the intermediate stable regime $g_0\leqslant g \leqslant g_c$, only monotonicity of $\mathcal{E}(g)$ is known. In the unstable regime, $\mathcal{E}(g)=-\infty$.

Theorems & Definitions (31)

• Theorem 1
• Corollary 1.1
• Theorem 2
• proof : Proof of Theorem \ref{['thm1']}
• proof : Proof of Theorem \ref{['cor']}
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Remark 4.1