Abstract Using the variational method, we study the properties of a spin-down impurity immersed in a one-dimensional (1D) spin-up Fermi sea. With repulsive interactions between them, the impurity is dressed up by surrounding particles in Fermi sea and forms a polaron. We clearly calculate the binding energy, effective mass, momentum distribution, Tan contact, and pair correlation. Even in strong repulsive regimes, the results can agree with the exact Bethe Ansatz results. The repulsive polaron energy E + is below Fermi energy E F and no negative effective masses are found in whole interaction regimes, unequal masses polarons are also calculated. We show a clear momentum distribution and calculate the Tan contact from three different aspects. Furthermore, we explore the particle-hole excitation and find that the hole terms in Fermi sea have a great influence on the polaron energy and contact in repulsive regime. These results show that the variational method can still be used effectively in 1D repulsive polaron system .
We report a study of a spin-down impurity strongly coupled to a spin-up Fermi sea (a so-called Fermi polaron) with the diagrammatic Monte Carlo (DiagMC) technique. Conditions of zero temperature and three dimensions are considered for an ultracold atomic gas with resonant interactions in the zero-range limit. A Feynman diagrammatic series is developed for the one-body and two-body propagators providing information about the polaron and molecule channel, respectively. The DiagMC technique allows us to reach diagram orders that are high enough for extrapolation to infinite order. The robustness of the extracted results is examined by checking various resummation techniques and by running the simulations with various choices for the propagators and vertex functions. It turns out that dressing the lines in the diagrams as much as possible is not always the optimal choice. We also identify classes of dominant diagrams for the one-body and two-body self-energy in the region of strong interaction. These dominant diagrams turn out to be the leading processes of the strong-coupling limit. The quasiparticle energies and
Z factor are obtained as a function of the interaction strength. We find that the DiagMC results for the molecule and polaron properties are very similar to those obtained with a variational ansatz. Surprisingly, this variational ansatz gives very good predictions for the quasiparticle residue even when this residue is significantly less than 1.
We study the problem of a mobile impurity of mass
M interacting {\sl via} a s-wave broad or narrow Feshbach resonance with a Fermi sea of particles of mass
m. Truncating the Hilbert space to at most one pair of particle-hole excitations of the Fermi sea, we determine ground state properties of the polaronic branch other than its energy, namely the polaron quasiparticle residue
Z, and the impurity-to-fermion pair correlation function
G(x). We show that
G(x) deviates from unity at large distances as
−(A_4+B_4cos2k_Fx)/(k_Fx)4, where
k_F is the Fermi momentum; since
A_4>0 and
B_4>0, the polaron has a diverging rms radius and exhibits Friedel-like oscillations. In the weakly attractive limit, we obtain analytical results, that in particular detect the failure of the Hilbert space truncation for a diverging mass impurity, as expected from Anderson orthogonality catastrophe; at distances between
∼1/k_F and the asymptotic distance where the
1/x4 law applies, they reveal that
G(x) exhibits an intriguing multiscale structure.
A two-dimensional (2D) spin-up ideal Fermi gas interacting attractively with a spin-down impurity in the continuum undergoes, at zero temperature, a first-order phase transition from a polaron to a dimeron state. Here, we study a similar system on a square lattice, by considering the attractive 2D Fermi-Hubbard model with a single spin-down and a finite filling fraction of spin-up fermions. We study polaron and dimeron quasiparticle properties via variational ansatz up to one particle-hole excitation. Moreover, we develop a determinant diagrammatic Monte Carlo algorithm for this problem based on expansion in bare on-site coupling U . This algorithm turns out to be sign-problem-free at any filling of spin-up fermions, allowing one to sample very high diagram order (larger than 200 in our study) and to do simulations for large U / t (we go up to U / t = − 20 with t the hopping strength). Both methods give qualitatively consistent results. With variational ansatz we go to even larger on-site attraction. At very low spin-up filling fraction, we observe the polaron-to-dimeron transition, in agreement with the continuum case. Upon increasing the filling fraction, however, the transition shifts to higher values of | U | / t and the transition disappears beyond a filling fraction of about 20 % . In this region, the polaron state always gives a lower energy and has a finite quasiparticle residue. Our findings are directly relevant to cold atom experiments with 2D optical lattices: A small and finite density of spin-down impurities in the ground state will form a superfluid at strong coupling at low spin-up filling fraction. Above some critical spin-up filling fraction, the system is expected to remain a normal Fermi liquid, even in the strong-coupling limit.
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.
Using both the exact Bethe ansatz method and the variational method, we study properties of the one-dimensional Fermi polaron. We focus on the binding energy, effective mass, momentum distributions, Tan contact and correlation functions. As the attraction increases, the impurity is more tightly bound and correlated with the surrounding particles, and the size of formed polaron decreases. In addition, compared with the Bethe ansatz method, the variational method is totally qualified to study the one-dimensional Fermi polaron. The intrinsic reason is that the number of particle-hole excitations in a Fermi sea, caused by a single impurity, is always rather small. The variational method can be well extended to other impurity systems .