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arXiv:1908.00530 · Physical Review A

Application of the renormalized random-phase approximation to polarized Fermi gases

David Durel, Mi­chael Urban

Published Jan 9, 2020• 8 citations
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Abstract

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 $T$-matrix approach or ladder approximation). The so-called renormalized RPA consists in computing the $T$ 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.

DOI
10.1103/physreva.101.013608

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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.
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.
The purpose of this thesis is the study of an attractive Fermi gas in the normal phase throughout the BCS-BEC crossover by means of diagrammatic approaches based on the t-matrix approximation. More specifically, the calculation is implemented by including different degrees of self-consistency in the t-matrix equations, from the non-self-consistent to the fully self-consistent (or Luttinger-Ward) approach. In the first part of the thesis, we perform a systematic and unbiased comparative study of all the five variants of the t-matrix approach with different degrees of selfconsistency considered thus far in the literature. The analysis is performed by calculating several thermodynamic and dynamical quantities of the spin-balanced Fermi gas in the normal phase within all the approaches. We also compare these results with the available quantum Monte Carlo (QMC), diagrammatic Monte Carlo (DMC) and experimental data. In the comparison, the fully self-consistent t-matrix approach turns up to be the best to describe thermodynamic quantities, however for dynamical quantities partially or non-self-consistent approaches appear to work better. In the second part of the thesis, we investigate pairing correlations in the normal phase of the spin-balanced system within the fully self-consistent t-matrix approach. We first calculate the pair correlation function g↑↓(ρ) and the pair coherence length ξpair for various couplings and temperatures. We then develop a new approach, based on the two-body Green’s function formalism, to calculate the number of preformed pairs in the normal phase. The approach is used to describe the experimental data on this quantity by the group of Prof. Denschlag in Ulm. In order to compare our calculation with the experiment, the fully self-consistent t-matrix approach is extended to consider a Fermi gas in a harmonic trapping potential. In general, very good agreement is found with experimental data. In the final part of the thesis, we present some preliminary results for the spinimbalanced Fermi gas, focusing on the temperature-coupling-polarization phase diagram and on T = 0 and finite T thermodynamic quantities. In this analysis, we show that the Luttinger theorem for an imbalanced Fermi gas at T = 0 is satisfied within the fully self-consistent t-matrix approach, and we present the extension of the calculation of the pair correlation function g↑↓(ρ) to the spin-imbalanced case.
The diagrammatic ttt-matrix approximation has often been adopted to describe a dilute Fermi gas. This approximation, originally considered by V. M. Galitskii (Zh. Eksp. Teor. Fiz. 34, 151 (1958) [Sov. Phys. JETP 7, 104 (1958)]) for a repulsive interparticle interaction, was later widely utilized for an attractive Fermi gas to describe the BCS-BEC crossover from strongly overlapping Cooper pairs in weak coupling to nonoverlapping composite bosons in strong coupling. Several variants of the ttt-matrix approximation have been considered in the literature, which are distinguished by the degree of self-consistency allowed in the building blocks of the diagrammatic structure. Here, we perform a systematic and comparative study of all possible variants on the degree of self-consistency for the ttt-matrix approximation in an attractive Fermi gas, which enables us to confront their outcomes for thermodynamic and dynamical quantities on the same footing in an unbiased way. For definiteness, only the normal phase above the superfluid critical temperature is considered. The dispute that can be raised in this context, about the adequateness of introducing progressive degrees of self-consistency over and above the non-self-consistent ttt-matrix approximation for an attractive Fermi gas, parallels the recent interest in the literature in assessing the importance of various degrees of self-consistency in the context of semiconductors and insulators.
A method is proposed to improve the treatment of the ground-state correlations in a finite Fermi system compared with the standard random phase approximation (RPA) or earlier suggested renormalized RPA. The correlations lead to nonzero quasiparticle occupancies in the ground state. The method employs modified quasiparticles obtained by a canonical transformation of usual quasiparticles explicitly involving the quasiparticle occupation numbers. A set of equations is derived, which allows one to determine these occupation numbers along with the RPA modes. The formalism is illustrated with the Lipkin-Meshkov-Glick model, and a model for superconducting pairing at a finite temperature. With the new approach, the ground-state correlations are significantly reduced, the energy of the first excited state becomes closer to the exact solution around the region where the RPA collapses, and the superconducting gap monotonously decreases instead of the sharp phase transition. We discuss the effective equivalence of the interaction effects and variation of temperature for the ground-state correlations.
The self-consistent random-phase approximation (SCRPA) is extended to nonzero temperature. Within the SCRPA, the collective dynamics of nucleon pairs is built on the occupation numbers that deviate from the Fermi-Dirac distribution as a result of temperature-dependent correlations beyond the RPA. The equations for single-particle occupation numbers are derived using the method of the double-time Green's functions. The approach is applied to the Richardson model of pairing. The results of numerical calculations for energies of states and occupation numbers, as well as for the thermodynamic quantities such as the excitation energy and heat capacity of the system, are compared with those obtained within the conventional RPA and the perturbative thermal SCRPA and with the exact results. We find that, overall, the SCRPA offers a better agreement with the exact results.