We consider the McGuire model of a one-dimensional gas of free fermions interacting with a single impurity. We compute the static one-body function and momentum distribution of the impurity at finite temperatures. The results involve averages over Fredholm determinants that we further analyze using the effective form factors approach. With this approach, we derive the large-distance behavior of the one-body function, which takes the form of an averaged exponential decay. This method allows us to study an experimentally important regime of small momenta of the impurity's momentum distribution. We also consider the one-body function at short distances and compute finite-temperature Tan's contact.
We study in three dimensions the problem of a spatially homogeneous zero-temperature ideal Fermi gas of spin-polarized particles of mass
m perturbed by the presence of a single distinguishable impurity of mass
M. The interaction between the impurity and the fermions involves only the partial
s wave through the scattering length
a and has negligible range
b compared to the inverse Fermi wave number
1/kF of the gas. Through the interactions with the Fermi gas the impurity gives birth to a quasiparticle, which will be here a Fermi polaron (or more precisely a monomeron). We consider the general case of an impurity moving with wave vector $\mathbf{K}\ensuremath{\ne}\mathbf{0}$: Then the quasiparticle acquires a finite lifetime in its initial momentum channel because it can radiate particle-hole pairs in the Fermi sea. A description of the system using a variational approach, based on a finite number of particle-hole excitations of the Fermi sea, then becomes inappropriate around
K=0. We rely thus upon perturbation theory, where the small and negative parameter ${k}_{\mathrm{F}}a\ensuremath{\rightarrow}{0}^{\ensuremath{-}}$ excludes any branches other than the monomeronic one in the ground state (as, e.g., the dimeronic one), and allows us a systematic study of the system. We calculate the impurity self-energy ${\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$ up to second order included in
a. Remarkably, we obtain an analytical explicit expression for ${\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$, allowing us to study its derivatives in the plane $(K,\ensuremath{\omega})$. These present interesting singularities, which in general appear in the third-order derivatives ${\ensuremath{\partial}}^{3}{\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$. In the special case of equal masses,
M=m, singularities appear already in the physically more accessible second-order derivatives ${\ensuremath{\partial}}^{2}{\ensuremath{\Sigma}}^{(2)}(\mathbf{K},\ensuremath{\omega})$; using a self-consistent heuristic approach based on ${\ensuremath{\Sigma}}^{(2)}$ we then regularize the divergence of the second-order derivative ${\ensuremath{\partial}}_{K}^{2}\ensuremath{\Delta}E(\mathbf{K})$ of the complex energy of the quasiparticle found in Trefzger and Castin [Europhys. Lett. 104, 50005 (2013)] at
K=kF, and we predict an interesting scaling law in the neighborhood of
K=kF. As a by product of our theory we have access to all moments of the momentum of the particle-hole pair emitted by the impurity while damping its motion in the Fermi sea at the level of Fermi's golden rule.
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
kF we consider the regime of high density of the fermions, that is,
kF large compared to one. For coupling constants that scale like
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.
We define the three-body scattering hypervolume
DF for identical spin-polarized fermions in two dimensions by considering the wave function of three such fermions colliding at zero energy and zero orbital angular momentum. We derive the asymptotic expansions of such a wave function when three fermions are far apart or one pair and the third fermion are far apart, and
DF appears in the coefficients of such expansions. For weak-interaction potentials, we derive an approximate formula of
DF by using the Born expansion. We then study the shift of energy of three such fermions in a large periodic area due to
DF. This shift is proportional to
DF times the square of the area of the triangle formed by the momenta of the fermions. We also calculate the shifts of energy and of pressure of spin-polarized two-dimensional Fermi gases due to a nonzero
DF and the three-body recombination rate of spin-polarized ultracold atomic Fermi gases in two dimensions.
We investigate the momentum distribution function of a single distinguishable impurity particle which formed a polaron state in a gas of either free fermions or Tonks-Girardeau bosons in one spatial dimension. We obtain a Fredholm determinant representation of the distribution function for the Bethe ansatz solvable model of an impurity-gas \delta δ -function interaction potential at zero temperature, in both repulsive and attractive regimes. We deduce from this representation the fourth power decay at a large momentum, and a weakly divergent (quasi-condensate) peak at a finite momentum. We also demonstrate that the momentum distribution function in the limiting case of infinitely strong interaction can be expressed through a correlation function of the one-dimensional impenetrable anyons.
We investigate the highly polarized limit of a two-dimensional Fermi gas, where we effectively have a single spin-down impurity atom immersed in a spin-up Fermi sea. By constructing variational wave functions for the impurity, we map out the ground state phase diagram as a function of mass ratio ${m}_{\ensuremath{\uparrow}}/{m}_{\ensuremath{\downarrow}}$ and interaction strength. In particular, we determine when it is favorable for the dressed impurity (polaron) to bind particles from the Fermi sea to form a dimer, trimer, or even larger clusters. Similarly to three dimensions, we find that the Fermi sea favors the trimer state so that it exists for ${m}_{\ensuremath{\uparrow}}/{m}_{\ensuremath{\downarrow}}$ less than the critical mass ratio for trimer formation in the vacuum. We also find a region where dimers have finite momentum in the ground state, a scenario that corresponds to the Fulde-Ferrell-Larkin-Ovchinnikov superfluid state in the limit of large spin imbalance. For equal masses (${m}_{\ensuremath{\uparrow}}={m}_{\ensuremath{\downarrow}}$), we compute rigorous bounds on the polaron-dimer transition, and we show that the polaron energy and residue is well captured by the variational approach, with the former quantity being in good agreement with experiment. When there is a finite density of impurities, we find that this polaron-dimer transition is preempted by a first-order superfluid-normal transition at zero temperature, but it remains an open question what happens at finite temperature.