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Physical Review A

Liquid - impurity correlations for Fermi systems

Oscar Hipólito, R. Lobo

Published Jun 1, 1974• 3 citations
Publisher

Abstract

An attractive $\ensuremath{\delta}$-function potential is introduced in a Fermi gas. The contribution to the liquid-impurity correlation function of bound states is calculated exactly in a Green's-function formalism. The exact results are compared with those obtained in a self-consistent approximation and the range of validity of the latter is discussed.

DOI
10.1103/physreva.9.2783

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•Feb 19, 2020•0 citations
An attractive \ensuremathδ\ensuremath{\delta}\ensuremathδ-function potential is introduced in a Fermi gas. The contribution to the liquid-impurity correlation function of bound states is calculated exactly in a Green's-function formalism. The exact results are compared with those obtained in a self-consistent approximation and the range of validity of the latter is discussed.
We study the properties of spin-less non-interacting fermions trapped in a confining potential in one dimension but in the presence of one or more impurities which are modelled by delta function potentials. We use a method based on the single particle Green's function. For a single impurity placed in the bulk, we compute the density of the Fermi gas near the impurity. Our results, in addition to recovering the Friedel oscillations at large distance from the impurity, allow the exact computation of the density at short distances. We also show how the density of the Fermi gas is modified when the impurity is placed near the edge of the trap in the region where the unperturbed system is described by the Airy gas. Our method also allows us to compute the effective potential felt by the impurity both in the bulk and at the edge. In the bulk this effective potential is shown to be a universal function only of the local Fermi wave vector, or equivalently of the local fermion density. When the impurity is placed near the edge of the Fermi gas, the effective potential can be expressed in terms of Airy functions. For an attractive impurity placed far outside the support of the fermion density, we show that an interesting transition occurs where a single fermion is pulled out of the Fermi sea and forms a bound state with the impurity. This is a quantum analogue of the well-known Baik-Ben Arous-Péché (BBP) transition, known in the theory of spiked random matrices. The density at the location of the impurity plays the role of an order parameter. We also consider the case of two impurities in the bulk and compute exactly the effective force between them mediated by the background Fermi gas.
We consider an electron gas, both in two (2D) and three (3D) dimensions, interacting with quenched impurities and phonons within leading order finite-temperature many-body perturbation theories, calculating the electron self-energies, spectral functions, and momentum distribution functions at finite temperatures. The resultant spectral function is in general highly non-Lorentzian, indicating that the system is not a Fermi liquid in the usual sense. The calculated momentum distribution function cannot be approximated by a Fermi function at any temperature, providing a rather simple example of a non-Fermi liquid with well-understood properties.
We summarize results on the asymptotics of the two-particle Green functions of interacting electrons in one dimension.Below a critical value of the chemical potential the Fermi surface vanishes, and the system can no longer be described as a Luttinger liquid.Instead, the non-relativistic Fermi gas with infinite point-like repulsion becomes the universal model for the long-wavelength, low temperature physics of the one-dimensional electrons.This model, which we call the impenetrable electron gas, allows for a rigorous mathematical treatment.In particular, a so-called determinant representation for the two particle Green function could be derived.This determinant representation is related to an integrable classical evolution equation and to a Riemann-Hilbert problem, that enable the exact calculation of the asymptotics of the two-particle Green functions.
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.