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arXiv:1407.2069 · Physical Review A

Occupation numbers in strongly polarized Fermi gases and the Luttinger theorem

Mi­chael Urban, Peter Schuck

Published Aug 27, 2014• 10 citations
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Abstract

We study a two-component Fermi gas that is so strongly polarized that it remains normal fluid at zero temperature. We calculate the occupation numbers within the particle-particle random-phase approximation, which is similar to the Nozi\`eres-Schmitt-Rink approach. We show that the Luttinger theorem is fulfilled in this approach. We also study the change of the chemical potentials which allows us to extract, in the limit of extreme polarization, the polaron energy.

DOI
10.1103/physreva.90.023632

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We study a two-component Fermi gas that is so strongly polarized that it remains normal fluid at zero temperature. We calculate the occupation numbers within the particle-particle random-phase approximation, which is similar to the Nozi`eres-Schmitt-Rink approach. We show that the Luttinger theorem is fulfilled in this approach. We also study the change of the chemical potentials which allows us to extract, in the limit of extreme polarization, the polaron energy.
Boson-fermion pair correlations in a mixture are considered at zero temperature in the TTT-matrix approximation. Special attention is paid to the Luttinger theorem. In a strict random-phase approximation variant of the Nozi\`eres-Schmitt-Rink approach, it is shown that this theorem is respected also in the homogeneous infinite matter case. We calculate the corresponding occupation numbers of fermions and bosons and the condensate depletion. We also show that in the limit of very small boson density, our results are in good agreement with the results found in the literature for the Fermi polaron in strongly imbalanced Fermi-Fermi mixtures.
We consider a polarized Fermi gas in the BCS-BEC crossover region above the critical temperature within a T-matrix formalism. By treating the mean-field-like shift of the quasiparticle energies in a self-consistent manner, we avoid the known pathological behavior of the standard Nozi\`eres-Schmitt-Rink approach in the polarized case, i.e., the polarization has the right sign and the spin polarizability is positive. The momentum distributions of the correlated system are computed and it is shown that, in the zero-temperature limit, they satisfy the Luttinger theorem. Results for the phase diagram, the spin susceptibility, and the compressibility are discussed.
We consider a spin-imbalanced Fermi gas at zero temperature in the normal phase on the BCS side of the BCS-BEC crossover and around unitarity. We compute the critical polarization for pairing, the correlated occupation numbers, and the contact in an extension of particle-particle random-phase approximation (RPA) (also called non-self-consistent TTT-matrix approach or ladder approximation). The so-called renormalized RPA consists in computing the TTT matrix with self-consistently determined occupation numbers. The occupation numbers are determined either by keeping the self-energy only to first order or by resumming the Dyson equation. In this way, the result for the critical polarization, strongly overestimated in standard RPA, is clearly improved. We also discuss some problems of this approach.
The equation of state of the partially polarized two component Fermi gas at zero temperature in the unitary limit is computed by the ab initio auxiliary field Monte Carlo method. From this, we obtain the critical ratio of the chemical potentials ${\ensuremath{\mu}}_{\ensuremath{\downarrow}}∕{\ensuremath{\mu}}_{\ensuremath{\uparrow}}$ at the phase transitions. The value of ${\ensuremath{\mu}}_{\ensuremath{\downarrow}}∕{\ensuremath{\mu}}_{\ensuremath{\uparrow}}$ at the transition between the fully paired superfluid and the partially polarized phases is 0.11, while the critical value at the phase transition between the partially polarized phase and the fully polarized normal fluid is $\ensuremath{-}0.59$. We also determine the radial boundaries of the phase transitions of the Fermi gas in the harmonic trap as a function of the total polarization. We find that beyond the critical polarization 0.65, the fully paired superfluid core disappears in the trapped Fermi gas.
We investigate the phase diagram of asymmetric two-component Fermi gases at zero temperature as a function of polarization and interaction strength. The equations of state of the uniform superfluid and normal phase are determined using quantum Monte Carlo simulations. We find three different mixed states, where the superfluid and the normal phase coexist in equilibrium, corresponding to phase separation between (a) the polarized superfluid and the fully polarized normal gas, (b) the polarized superfluid and the partially polarized normal gas, and (c) the unpolarized superfluid and the partially polarized normal gas.
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.