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arXiv:1109.5585 · Physical Review A

Interaction between polarons and analogous effects in polarized Fermi gases

S. Giraud, R. Combescot

Published Jan 4, 2012• 13 citations
arXivHTMLPDF

Abstract

We consider an imbalanced mixture of two different ultracold Fermi gases, which are strongly interacting. Calling spin-down the minority component and spin-up the majority component, the limit of small relative density $x={n}_{\ensuremath{\downarrow}}/{n}_{\ensuremath{\uparrow}}$ is usually considered as a gas of noninteracting polarons. This allows us to calculate, in the expansion of the total energy of the system in powers of $x$, the terms proportional to $x$ (corresponding to the binding energy of the polaron) and to ${x}^{5/3}$ (corresponding to the kinetic energy of the polaron Fermi sea). We investigate in this paper terms physically due to an interaction between polarons and which are proportional to ${x}^{2}$ and ${x}^{7/3}$. We find three such terms. The first one corresponds to the overlap between the clouds dressing two polarons. The two other ones are due to the modification of the single polaron binding energy caused by the nonzero density of polarons. The second term is due to the restriction of the polaron momentum by the Fermi sea formed by the other polarons. The last one results from the modification of the spin-up Fermi sea brought by the other polarons. The calculation of all these terms is made at the simplest level of a single particle-hole excitation. It is performed for all the possible interaction strengths within the stability range of the polaron. At unitarity the last two terms give a fairly weak contribution while the first one is strong and leads to a marked disagreement with Monte Carlo results. The possible origins of this discrepancy are discussed.

DOI
10.1103/physreva.85.013605

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We consider an imbalanced mixture of two different ultracold Fermi gases, which are strongly interacting. Calling spin-down the minority component and spin-up the majority component, the limit of small relative density x=n\ensuremath↓/n\ensuremath↑x={n}_{\ensuremath{\downarrow}}/{n}_{\ensuremath{\uparrow}}x=n\ensuremath↓​/n\ensuremath↑​ is usually considered as a gas of noninteracting polarons. This allows us to calculate, in the expansion of the total energy of the system in powers of xxx, the terms proportional to xxx (corresponding to the binding energy of the polaron) and to x5/3{x}^{5/3}x5/3 (corresponding to the kinetic energy of the polaron Fermi sea). We investigate in this paper terms physically due to an interaction between polarons and which are proportional to x2{x}^{2}x2 and x7/3{x}^{7/3}x7/3. We find three such terms. The first one corresponds to the overlap between the clouds dressing two polarons. The two other ones are due to the modification of the single polaron binding energy caused by the nonzero density of polarons. The second term is due to the restriction of the polaron momentum by the Fermi sea formed by the other polarons. The last one results from the modification of the spin-up Fermi sea brought by the other polarons. The calculation of all these terms is made at the simplest level of a single particle-hole excitation. It is performed for all the possible interaction strengths within the stability range of the polaron. At unitarity the last two terms give a fairly weak contribution while the first one is strong and leads to a marked disagreement with Monte Carlo results. The possible origins of this discrepancy are discussed.
This thesis presents a theoretical study of strongly-interacting Fermi systems with population imbalance, which is motivated by some differences in cold atoms experiments. We calculate the energy of a single fermion interacting resonantly with a Fermi sea of different species fermions in anisotropic traps, and show that finite particle numbers and the trap geometry impact the phase structure and the critical polarization, the limit of resonance superfluidity in traps. Our findings contribute to understanding some experimental discrepancies as finite-size and confinement effects. For an imbalanced gas in the uniform system, we calculate the energy of adding an impurity, and construct the equation of state of the partially-polarized normal Fermi liquid. Finally, we study the properties of a spin-down polaron in a trapped gas containing arbitrary numbers of spin-up and spin-down fermions, and derive a self-consistent equation for the polaron energy.
The Fermi polaron problem, which describes a mobile impurity that interacts with a spin-polarized Fermi sea, is a paradigmatic system in quantum many-body physics and has been challenging to address quantitatively in its strong coupling regime. We present the first controlled thermodynamic calculations for the Fermi polaron at strong coupling using finite-temperature auxiliary-field quantum Monte Carlo (AFMC) methods in the framework of the canonical ensemble. Modeled as a spin-imbalanced system, the Fermi polaron has a Monte Carlo sign problem, but we show that it is moderate over a wide range of temperatures and coupling strengths beyond the unitary limit of the BCS-BEC crossover. We calculate the contact, a quantity which measures the strength of the short-range correlations, as a function of temperature at unitarity and as a function of the coupling strength at fixed temperature and find good agreement with a variational approach based on one particle-hole excitation of the Fermi sea. We compare our results for the contact with recent experiments and find good agreement at unitarity (within error bars) but discrepancies away from unitarity on the BEC side of the crossover. We also calculate the thermal energy gap at unitarity as a function of temperature.
An extremely polarized mixture of an ultracold Fermi gas is expected to reduce to a Fermi polaron system, which consists of a single impurity immersed in the Fermi sea of majority atoms. By developing a many-body T-matrix theory, we investigate spectral properties of the polarized mixture in experimentally relevant regimes in which the system of finite impurity concentration at nonzero temperature is concerned. We explicitly demonstrate presence of polaron physics in the polarized limit and discuss effects of many polarons in an intermediate regime in a selfconsistent manner. By analyzing the spectral function at finite impurity concentration, we extract the attractive and repulsive polaron energies. We find that a renormalization of majority atoms via an interaction with minority atoms is of significance to depict the many-polaron regime.
I study the properties of a spin-down neutron impurity immersed in a low-density free Fermi gas of spin-up neutrons. In particular, I analyze its energy (${E}_{\ensuremath{\downarrow}}$), effective mass (${m}_{\ensuremath{\downarrow}}^{*}$), and quasiparticle residue (${Z}_{\ensuremath{\downarrow}}$). Results are compared with those of state-of-the-art quantum Monte Carlo calculations of the attractive Fermi polaron realized in ultracold atomic gases experiments, and with those of previous studies of the neutron polaron. Calculations are performed within the Brueckner-Hartree-Fock approach using the chiral two-body nucleon-nucleon interaction of Entem and Machleidt at N3LO{\mathrm{N}}^{3}\mathrm{LO}N3LO with a 500 MeV cut-off and the Argonne V18 phenomenological potential. Only contributions from the 1S0{}^{1}{S}_{0}1S0​ partial wave, which is the dominant one in the low-density region considered, are included. Contributions from three-nucleon forces are expected to be irrelevant at these densities and, therefore, are neglected in the calculation. It is shown that for Fermi momenta between $\ensuremath{\approx}0.25$ and $\ensuremath{\approx}0.45\phantom{\rule{4pt}{0ex}}{\mathrm{fm}}^{\ensuremath{-}1}$ the energy, effective mass, and quasiparticle residue of the impurity vary only slightly, respectively, in the ranges $\ensuremath{-}0.604\phantom{\rule{0.16em}{0ex}}{E}_{F}<{E}_{\ensuremath{\downarrow}}<\ensuremath{-}0.635\phantom{\rule{0.16em}{0ex}}{E}_{F}$, $1.300\phantom{\rule{0.16em}{0ex}}m<{m}_{\ensuremath{\downarrow}}^{*}<1.085\phantom{\rule{0.16em}{0ex}}m$, and $0.741<{Z}_{\ensuremath{\downarrow}}<0.836$ in the case of the chiral interaction, and $\ensuremath{-}0.621\phantom{\rule{0.16em}{0ex}}{E}_{F}<{E}_{\ensuremath{\downarrow}}<\ensuremath{-}0.643\phantom{\rule{0.16em}{0ex}}{E}_{F}$, $1.310\phantom{\rule{0.16em}{0ex}}m<{m}_{\ensuremath{\downarrow}}^{*}<1.089\phantom{\rule{0.16em}{0ex}}m$, and $0.739<{Z}_{\ensuremath{\downarrow}}<0.832$ when using the Argonne V18 potential. These results are compatible with those derived from ultracold atoms and show that a spin-down neutron impurity in a free Fermi gas of spin-up neutrons with a Fermi momentum in the range $0.25\ensuremath{\lesssim}{k}_{F}\ensuremath{\lesssim}0.45\phantom{\rule{4pt}{0ex}}{\mathrm{fm}}^{\ensuremath{-}1}$ exhibits properties very similar to those of an attractive Fermi polaron in the unitary limit.
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 consider the highly spin-imbalanced limit of a two-component Fermi gas, where there is a small density of $\ensuremath{\downarrow}$ impurities attractively interacting with a sea of $\ensuremath{\uparrow}$ fermions. In the single-impurity limit at zero temperature, there exists the so-called polaron-molecule transition, where the impurity sharply changes its character by binding a $\ensuremath{\uparrow}$ fermion at sufficiently strong attraction. Using a recently developed variational approach, we calculate the thermodynamic properties of the impurity, and we show that the transition becomes a smooth crossover at finite temperature due to the thermal occupation of excited states in the impurity spectral function. However, remnants of the single-impurity transition are apparent in the momentum-resolved spectral function, which can in principle be probed with Raman spectroscopy. We furthermore show that the Tan contact exhibits a characteristic nonmonotonic dependence on temperature that provides a signature of the zero-temperature polaron-molecule transition. For a finite impurity density, we argue that descriptions purely based on the behavior of the Fermi polaron are invalid near the polaron-molecule transition, since correlations between impurities cannot be ignored. In particular, we show that the spin-imbalanced system undergoes phase separation at low temperatures due to the strong attraction between $\ensuremath{\uparrow}\ensuremath{\downarrow}$ molecules induced by the Fermi sea. Thus, we find that the impurity spectrum and the induced impurity-impurity interactions are key to understanding the phase diagram of the spin-imbalanced Fermi gas.