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arXiv:1903.01960 · Physical review. B./Physical review. B

Fermi gas throughout the BCS-BEC crossover: Comparative study of t -matrix approaches with various degrees of self-consistency

Michele Pini, P. Pieri, Giancarlo Calvanese Strinati

Published Mar 4, 2019• 50 citations
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Abstract

The diagrammatic $t$-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 $t$-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 $t$-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 $t$-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.

DOI
10.1103/physrevb.99.094502

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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.
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.
In the latest several years, diagrammatic theories based on the t-matrix approximation with different \ndegrees of self-consistency have been developed to describe a Fermi gas through the BCS-BEC crossover. A comprehensive comparison between the results obtained by the different \napproaches for both thermodynamical and dynamical quantities is, however, still lacking. In the present \nwork, we consider all the possible ways of implementing self-consistency in the t-matrix approximation and \ncompare the outcomes of the different approaches between each other (and with the available experimental \nand Quantum Monte Carlo data) through the whole BCS-BEC crossover. In this way, we expect to provide \na guideline on the virtues and the drawbacks of each scheme that can help future research in this field.
We consider the problem of the crossover from BCS superconductivity to Bose-Einstein condensation in three dimensions for a system of fermions with a mutual attractive interaction, for which we adopt the simplifying assumption of a suitably regularized point-contact interaction. We examine in a critical way the fermionic (self-consistent) T-matrix approximation, which has been widely utilized in the literature to describe this crossover above the superconducting critical temperature, and show that it fails to yield the correct behavior of the system in the strong-coupling limit, where composite bosons form as tightly bound fermion pairs. We then set up the correct approximation for a ``dilute'' system of composite bosons and show that a class of diagrams has to be considered in the place of the fermionic T-matrix approximation for the self-energy. This class of diagrams correctly describes both the weak- and strong-coupling limits, and consequently results in an improved interpolation scheme for the intermediate (crossover) region. In this context, we provide also a systematic mapping between the corresponding diagrammatic theories for the composite bosons and the constituent fermions. As a preliminary result to demonstrate the numerical effect of our class of diagrams on physical quantities, we calculate the value of the scattering length for composite bosons in the strong-coupling limit and show that it is considerably modified with respect to the result obtained within the self-consistent fermionic T-matrix approximation.
We present a self-consistent theory for the thermodynamics of the BCS-BEC crossover in the normal and superfluid phase which is both conserving and gapless. It is based on the variational many-body formalism developed by Luttinger and Ward and by DeDominicis and Martin. Truncating the exact functional for the entropy to that obtained within a ladder approximation, the resulting self-consistent integral equations for the normal and anomalous Green functions are solved numerically for arbitrary coupling. The critical temperature, the equation of state, and the entropy are determined as a function of the dimensionless parameter 1∕kFa1∕{k}_{F}a1∕kF​a, which controls the crossover from the BCS regime of extended pairs to the BEC regime of tightly bound molecules. The tightly bound pairs turn out to be described by a Popov-type approximation for a dilute, repulsive Bose gas. Even though our approximation does not capture the critical behavior near the continuous superfluid transition, our results provide a consistent picture for the complete crossover thermodynamics which compares well with recent numerical and field-theoretic approaches at the unitarity point.
PACS. 03.75.Hh – Static properties of condensates; thermodynamical, statistical and structural properties. PACS. 03.75.Ss – Degenerate Fermi gases. PACS. 05.30.Fk – Fermion systems and electron gas. Abstract. – We present a theory for a superfluid Fermi gas near the BCS-BEC crossover, including pairing fluctuation contributions to the free energy similar to that considered by Nozières and Schmitt-Rink for the normal phase. In the strong coupling limit, our theory is able to recover the Bogoliubov theory of a weakly interacting Bose gas with a molecular scattering length very close to the known exact result. We compare our results with recent Quantum Monte Carlo simulations both for the ground state and at finite temperature. Excellent agreement is found for all interaction strengths where simulation results are available. The recent experimental realization of strongly interacting Fermi gases of 6 Li and 40 K atoms near a Feshbach resonance has opened up the exciting possibility of investigating the crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein condensate (BEC) [1–5]. In these systems, the inter-atomic interaction strength can be varied by tuning
We investigate magnetic properties and effects of pairing fluctuations in the Bardeen-Cooper-Schrieffer--Bose-Einstein condensation (BCS-BEC) crossover regime of an ultracold Fermi gas. Recently, Liu and Hu [Europhys. Lett. 75, 364 (2006)], and Parish et al. [Nat. Phys. 3, 124 (2007)], pointed out that the strong-coupling theory developed by Nozi\`eres and Schmitt-Rink (NSR), which has been extensively used to successfully clarify various physical properties of cold Fermi gases, unphysically gives negative spin susceptibility in the BCS-BEC crossover region. The same problem also exists in the ordinary non-self-consistent TTT-matrix approximation. In this paper, we clarify that this serious problem comes from incomplete treatment in term of pseudogap phenomena originating from strong pairing fluctuations, as well as effects of spin fluctuations on the spin susceptibility. Including these two key issues, we construct an extended TTT-matrix theory which can overcome this problem. The resulting positive spin susceptibility agrees well with the recent experiment on a 6{}^{6}6Li Fermi gas done by Sanner et al. [Phys. Rev. Lett. 106, 010402 (2011)]. We also apply our theory to a polarized Fermi gas to examine the superfluid phase transition temperature Tc{T}_{\mathrm{c}}Tc​, as a function of the polarization rate. Since the spin susceptibility is an important physical quantity, especially in singlet Fermi superfluids, our results would be useful in considering how singlet pairs appear above and below Tc{T}_{\mathrm{c}}Tc​ in the BCS-BEC crossover regime of cold Fermi gases.