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arXiv:2105.02841

Effective pair interaction between impurity particles induced by a dense Fermi gas

David Mitrouskas, Peter Pickl

Published May 6, 2021• 1 citation
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Abstract

We study the dynamics of a small number of impurity particles coupled to the ideal Fermi gas in a $d$-dimensional box. The impurities interact with the fermions via a two-body potential $λv(x)$ where $λ$ is a coupling constant and $v(x)$ for instance a screened Coulomb potential. After taking the large-volume limit at positive Fermi momentum $k_F$ we consider the regime of high density of the fermions, that is, $k_F$ large compared to one. For coupling constants that scale like $λ^2 \sim k_F^{(2-d)}$ we show that the impurity particles effectively decouple from the fermions but evolve with an attractive pair interaction among each other which is induced by fluctuations in the Fermi gas.

DOI
10.48550/arxiv.2105.02841

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We study the dynamics of a small number of impurity particles coupled to the ideal Fermi gas in a ddd-dimensional box. The impurities interact with the fermions via a two-body potential λv(x)λv(x)λv(x) where λλλ is a coupling constant and v(x)v(x)v(x) for instance a screened Coulomb potential. After taking the large-volume limit at positive Fermi momentum kFk_FkF​ we consider the regime of high density of the fermions, that is, kFk_FkF​ large compared to one. For coupling constants that scale like λ2∼kF(2−d)λ^2 \sim k_F^{(2-d)}λ2∼kF(2−d)​ we show that the impurity particles effectively decouple from the fermions but evolve with an attractive pair interaction among each other which is induced by fluctuations in the Fermi gas.
Understanding the behaviour of impurity particles in quantum many-body systems is fundamental to both mathematical and theoretical physics. Previous mathematical work established that impurities in a dense, non-interacting Fermi gas effectively decouple from the fermions while developing mediated interactions among themselves. This thesis extends these results to the physically more realistic case of weakly interacting fermions. The central difficulty is that fermion-fermion correlations introduce volume-enhanced terms that scale with positive powers of the system size L, potentially invalidating the effective theory in the thermodynamic limit. We identify the critical scaling \kappa^2 = L^{-2d} for the fermion-fermion coupling that suppresses these contributions. Using a perturbative resolvent method, we prove that under balanced scalings of the coupling constants, the impurities decouple at leading order and evolve according to an effective Hamiltonian with induced pairwise attractions, the same qualitative behavior as in the ideal Fermi gas, now rigorously established for interacting fermions.
Abstract We study the quantum dynamics of a homogeneous ideal Fermi gas coupled to an impurity particle on a three-dimensional box with periodic boundary condition. For large Fermi momentum kFk_{\text {F}}kF​ k F , we prove that the effective dynamics is generated by a Fröhlich-type polaron Hamiltonian, which linearly couples the impurity particle to an almost-bosonic excitation field. Moreover, we prove that the effective dynamics can be approximated by an explicit coupled coherent state. Our method is applicable to a range of interaction couplings, in particular including interaction couplings of order 1 and time scales of the order kF−1k_{\text {F}}^{-1}kF−1​ k F - 1 .
We study a one-dimensional system of two-component fermions in the limit of strong attractive particle-particle interactions. First, we analyze scattering in the corresponding few-body problem, which is analytically solvable via Bethe ansatz. This allows us to engineer effective interactions between the system's effective degrees of freedom: fermions and bosonic dimers (tightly bound pairs of fermions). We argue that, although these interactions are strong, the resulting effective problem can be mapped onto a weakly interacting one, paving the way for the use of perturbation theory. This finding simplifies studies of many-fermion systems under confinement that are beyond reach of state-of-the-art numerical methods. We illustrate this statement by considering an impurity atom in a Fermi gas.
We consider two identical impurities immersed in a Fermi sea for a broad range of masses and for both interacting and non-interacting impurities. The interaction between the particles is described through attractive zero-range potentials and the problem is solved in momentum space. The two impurities can attach to a fermion from the sea and form three-body bound states. The energy of these states increase as function of the Fermi momentum k F , leading to three-body bound states below the Fermi energy. The fate of the states depends highly on two- and three-body thresholds and we find evidence of medium-induced Borromean-like states in 2D. The corrections due to particle-hole fluctuations in the Fermi sea are considered in the three-body calculations and we show that in spite of the fact that they strongly affect both the two- and three-body systems, the correction to the point at which the three-body states cease to exist is small.
We study the properties of spin-less non-interacting fermions trapped in a confining potential in one dimension but in the presence of one or more impurities which are modelled by delta function potentials. We use a method based on the single particle Green's function. For a single impurity placed in the bulk, we compute the density of the Fermi gas near the impurity. Our results, in addition to recovering the Friedel oscillations at large distance from the impurity, allow the exact computation of the density at short distances. We also show how the density of the Fermi gas is modified when the impurity is placed near the edge of the trap in the region where the unperturbed system is described by the Airy gas. Our method also allows us to compute the effective potential felt by the impurity both in the bulk and at the edge. In the bulk this effective potential is shown to be a universal function only of the local Fermi wave vector, or equivalently of the local fermion density. When the impurity is placed near the edge of the Fermi gas, the effective potential can be expressed in terms of Airy functions. For an attractive impurity placed far outside the support of the fermion density, we show that an interesting transition occurs where a single fermion is pulled out of the Fermi sea and forms a bound state with the impurity. This is a quantum analogue of the well-known Baik-Ben Arous-Péché (BBP) transition, known in the theory of spiked random matrices. The density at the location of the impurity plays the role of an order parameter. We also consider the case of two impurities in the bulk and compute exactly the effective force between them mediated by the background Fermi gas.
We consider the dynamics of an impurity atom immersed in an ideal Fermi gas at zero temperature. We focus on the coherent quantum evolution of the impurity following a quench to strong impurity-fermion interactions, where the interactions are assumed to be short range like in cold-atom experiments. To approximately model the many-body time evolution, we use a truncated basis method, where at most two particle-hole excitations of the Fermi sea are included. When the system is initially noninteracting, we show that our method exactly captures the short-time dynamics following the quench, and we find that the overlap between initial and final states displays a universal nonanalytic dependence on time in this limit. We further demonstrate how our method can be used to compute the impurity spectral function, as well as describe many-body phenomena involving coupled impurity spin states, such as Rabi oscillations in a medium or highly engineered quantum quenches.