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arXiv:1102.5534 · Physical Review A

Polaron-molecule transitions in a two-dimensional Fermi gas

Meera M. Parish

Published May 11, 2011• 100 citations
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Abstract

We address the problem of a single ``spin-down'' impurity atom interacting attractively with a spin-up Fermi gas in two dimensions (2D). We consider the case where the mass of the impurity is greater than or equal to the mass of a spin-up fermion. Using a variational approach, we resolve the questions raised by previous studies and show that there is, in fact, a transition between polaron and molecule (dimer) ground states in 2D. For the molecule state, we use a variational wave function with a single particle-hole excitation on the Fermi sea and we find that its energy matches that of the exact solution in the limit of infinite impurity mass. Thus, we expect the variational approach to provide a reliable tool for investigating 2D systems.

DOI
10.1103/physreva.83.051603

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We address the problem of a single ``spin-down'' impurity atom interacting attractively with a spin-up Fermi gas in two dimensions (2D). We consider the case where the mass of the impurity is greater than or equal to the mass of a spin-up fermion. Using a variational approach, we resolve the questions raised by previous studies and show that there is, in fact, a transition between polaron and molecule (dimer) ground states in 2D. For the molecule state, we use a variational wave function with a single particle-hole excitation on the Fermi sea and we find that its energy matches that of the exact solution in the limit of infinite impurity mass. Thus, we expect the variational approach to provide a reliable tool for investigating 2D systems.
We study an impurity atom in a two-dimensional Fermi gas using variational wave functions for (i) an impurity dressed by particle-hole excitations (polaron) and (ii) a dimer consisting of the impurity and a majority atom. In contrast to three dimensions, where similar calculations predict a sharp transition to a dimer state with increasing interspecies attraction, we show that the polaron Ansatz always gives a lower energy. However, the exact solution for a heavy impurity reveals that both a two-body bound state and distortions of the Fermi sea are crucial. This reflects the importance of particle-hole pairs in lower dimensions and makes simple variational calculations unreliable. We show that the energy of an impurity gives important information about its dressing cloud, for which both Ans\"atze give inaccurate results.
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.
We address the problem of a single impurity atom moving in a two-dimensional (2D) layer immersed in a 3D Fermi gas. Using a variational approach, we show that in this mixed-dimensional (MD) system, there is a transition between polaron and molecule ground states, similar to the case of pure 3D. Moreover, we find that the attractive polaron energy in MD is higher than that in 3D, while molecular energy in MD is lower than that in 3D, which leads to a shift of the critical interaction strength of transition. Further analysis shows that the energy difference between 3D and MD systems is attributed to the increment of the effective mass of the impurity, which is induced by the spatial constraint on the impurity.
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|\mathbf{Q}|={k}_{F}∣Q∣=kF​; here Q\mathbf{Q}Q is defined as the momentum of a Fermi polaron system with respect to the Fermi sea of majority fermions (with Fermi momentum kF{k}_{F}kF​). The literally proposed molecule Ansatz is identified as an asymptotic limit of the ∣Q∣=kF|\mathbf{Q}|={k}_{F}∣Q∣=kF​ state in a strong-coupling regime, which implies a huge SO(3)\text{SO}(3)SO(3) (for 3D) or SO(2)\text{SO}(2)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.
Using exact Bethe ansatz solutions, we show that a spin-down fermion immersed into a fully polarized spin-up Fermi sea with a weak attraction is dressed by the surrounding spin-up fermions to form the one-dimensional analog of a polaron. As the attraction becomes strong, the spin-down fermion binds with one spin-up fermion to form a tightly bound molecule. Throughout the whole interaction regime, a crossover from the polaron to a molecule state is fully demonstrated through exact results of the excitation spectrum, the effective mass, binding energy and kinetic energy. Furthermore, a clear distinction between the polaron and molecule is conceived by the probability distribution, single-particle reduced density matrix, and density-density correlations, which are calculated directly from the Bethe ansatz wave function. Such a polaron-molecule crossover presents a universal nature of an impurity immersed into a fermionic medium with an attraction in one dimension.