Abstract We analyze the ground state energy of N fermions in a two-dimensional box interacting with an impurity particle via two-body point interactions. We show that for weak coupling, the ground state energy is asymptotically described by the polaron energy, as proposed by F. Chevy in the physics literature. The polaron energy is the solution of a nonlinear equation involving the Green’s function of the free Fermi gas and the binding energy of the two-body point interaction. We provide quantitative error estimates that are uniform in the thermodynamic limit.
Abstract We analyze the ground state energy of N fermions in a two-dimensional box interacting with an impurity particle via two-body point interactions. We show that for weak coupling, the ground state energy is asymptotically described by the polaron energy, as proposed by F. Chevy in the physics literature. The polaron energy is the solution of a nonlinear equation involving the Green’s function of the free Fermi gas and the binding energy of the two-body point interaction. We provide quantitative error estimates that are uniform in the thermodynamic limit.
We analyze the ground state energy for N fermions in a two-dimensional box interacting with an impurity particle via two-body point interactions. We allow for mass ratios M > 1.225 between the impurity mass and the mass of a fermion and consider arbitrarily large box sizes while keeping the Fermi energy fixed. Our main result shows that the ground state energy in the limit of weak coupling is given by the polaron energy. The polaron energy is an energy estimate based on trial states up to first order in particle-hole expansion, which was proposed by Chevy in the physics literature. For the proof we apply a Birman-Schwinger principle that was recently obtained by Griesemer and Linden. One main new ingredient is a suitable localization of the polaron energy.
This thesis is concerned with the quantum mechanical system of a single particle interacting with an ideal gas of identical fermions by point interaction. In the physics literature this system is often referred to as Fermi polaron. We investigate the two-dimensional Fermi polaron. Unlike the one-dimensional case, point interactions in two or three dimensions cannot be implemented as perturbation of the quadratic form of the Laplacian. Either they are obtained as self-adjoint extensions of the Laplacian restricted to functions that vanish when the coordinates of two particles coincide, or they are constructed by a suitable limiting process. Choosing the second approach, a many-body operator with two-particle point interaction has firstly been rigorously defined by Dell'Antonio, Figari and Teta. We consider the Fermi polaron confined to a box with periodic boundary conditions and we identify a broad class of regularization schemes that approximate the Hamiltonian of the Fermi polaron as limit operator in the strong resolvent sense. The Hamiltonian is not given by a closed form, which could be conveniently used in standard variational principles. We establish a novel variational principle that characterizes all bound states, i.e. all energy eigenstates below the bottom of the spectrum of the kinetic energy. This variational principle turns out to be very useful for the following purposes. The ground state of the Fermi polaron is expected to be well approximated by the polaron and the molecule ansatz in the regime of weak and strong coupling between the impurity and the Fermi gas, respectively. In the physics literature, these two classes of trial states are used for variational computations with the (ultraviolet) regularized Hamiltonian. Although the implicit expressions for the minimal energy of both classes allow for the removal of the ultraviolet cutoff, it remains unclear whether the results are upper bounds to the ground state energy of the Fermi polaron. We show that the minimization of energy over polaron and molecule trial states can be reformulated in a natural way with the help of our variational principle. By doing so, the classes of trial states simplify considerably, and since there is no reference to regularized quantities, we can prove that the expressions for the polaron and the molecule energy in the physics literature are indeed upper bounds to the ground state energy of the Fermi polaron. As a further application of the variational principle, we prove analytically that to first order in a particle-hole expansion the molecule ansatz yields a better approximation to the ground state energy than the polaron ansatz if the coupling between the impurity and the Fermi gas is strong enough. So far, this had only been done numerically. The concluding chapter is devoted to the derivation of a lower bound to the ground state energy of the Fermi polaron in two-dimensional space. We show that the ground state energy can be bounded from below by a quantity that does not depend on the number of fermions in the Fermi gas. This result is correct under the assumption that the ratio of the mass of the impurity and the mass of a fermion exceeds 1.225. We also present a method which might yield a similar result for lower mass ratios. This method gives an estimate for the quadratic form of the regularized Hamiltonian in position space representation. In this connection, we present an inequality that bounds a singular potential of a two-dimensional Fermi gas depending only on the minimal distance between two fermions by the kinetic energy of the Fermi gas uniformly in the number of fermions. This inequality also applies to a potential with singularity 1/r^2, for which the Hardy inequality does not hold in two dimensions. Therefore, the full antisymmetry of the wave function has to be taken into account.
We investigate a single impurity interacting with a free two-dimensional atomic Fermi gas. The interaction between the impurity and the gas is characterized by an arbitrary attractive short-range potential, which, in two dimensions, always admits a two-particle bound state. We provide analytical expressions for the energy and the effective mass of the dressed impurity by including the two-body bound state, which is crucial for strong interactions, in the integral equation for the effective interaction. Using the same method, we also give the results for the polaron parameters in one and three dimensions and find good agreement with previous results. Thus, our relations can be used as a simple way to estimate the polaron parameters once the two-body bound state of the interaction potential is known.
The formalism of bilocal field theory was applied to the calculation of the ground-state energy and the effective mass of the large polaron. This calculation illustrates the bilocal field method using a relatively realistic model of a physical system. In bilocal theory the fermion field is dependent on two spatial points (nonrelativistically). Fermi particles can then be described in terms of a center of mass and internal wave function, and field operators can be expanded in functions related to the exact one-fermion Green's function, thereby eliminating self-energy terms from the S matrix and related operators. Similarities and differences between calculated large-polaron properties using the bilocal treatment and the customary local treatment are discussed. In particular, it is found that, within the approximations made, the bilocal and local methods give the same leading term in powers of the coupling constant for the ground-state energy and effective mass in the strong-coupling limit. For weak coupling, the bilocal energy is larger (more positive) and the effective mass smaller than in the local case due to an additional kinetic energy of internal motion.
We consider an impurity problem in a quasi-two-dimensional Fermi gas, where a spin-down impurity is immersed in a Fermi sea of N spin-up atoms. Using a variational approach and an effective two-channel model, we obtain the energy for a wide range of interaction strength and for various different mass ratios between the impurity and the background fermion in the context of heteronuclear mixture. We demonstrate that in a quasi-two-dimensional Fermi gas there exists a transition of the ground state from polaron in the weakly interacting region to molecule in the strongly interacting region. The critical interaction strength of the polaron–molecule transition is non-universal and depends on the particle density of the background Fermi sea. We also investigate the excited repulsive polaron state, and find similar non-universal behavior.
It has been commonly believed that a polaron-to-molecule transition occurs in three-dimensional (3D) and 2D Fermi polaron systems as the attraction between the single impurity and majority fermions gets stronger. The conclusion has been drawn from the separate treatment of polaron and molecule states and thus deserves a close reexamination. In this work, we explore the physics of polarons and molecules by utilizing a unified variational Ansatz with up to two particle-hole (ph) excitations (V-2ph). We confirm the existence of a first-order transition in 3D and 2D Fermi polarons, and we show that the nature of such a transition lies in an energy competition between systems with different momenta, $\mathbf{Q}=\mathbit{0}$ and ∣Q∣=kF; here Q is defined as the momentum of a Fermi polaron system with respect to the Fermi sea of majority fermions (with Fermi momentum kF). The literally proposed molecule Ansatz is identified as an asymptotic limit of the ∣Q∣=kF state in a strong-coupling regime, which implies a huge SO(3) (for 3D) or SO(2) (for 2D) ground-state degeneracy in this regime. The recognition of such degeneracy is crucially important for evaluating the molecule occupation in realistic systems with finite impurity density and at finite temperature. To compare with recent experiment of 3D Fermi polarons, we have calculated various physical quantities under the V-2ph framework, and we obtained results that are in good agreement with experimental data in the weak-coupling and near-resonance regime. Further, to check the validity of our conclusion in 2D, we have adopted a different variational method based on the Gaussian sample of high-order ph excitations (V-Gph), and we found the same conclusion on the nature of the polaron-molecule transition therein. For a 1D system, the V-2ph method predicts no sharp transition, and the ground state is always at the $\mathbf{Q}=\mathbit{0}$ sector, consistent with the exact Bethe-Ansatz solution. The presence/absence of a polaron-molecule transition is analyzed to be closely related to the interplay effect of Pauli-blocking and ph excitations in different dimensions.