Single-particle spectra are calculated in the superconducting state for a fermionic system with an\nattractive interaction, as functions of temperature and coupling strength from weak to strong. The\nfermionic system is described by a single-particle self-energy that includes pairing-fluctuation effects\nin the superconducting state. The theory reduces to the ordinary BCS approximation in weak coupling\nand to the Bogoliubov approximation for the composite bosons in strong coupling. Several features of\nthe single-particle spectral function are shown to compare favorably with experimental data for cuprate\nsuperconductors.
The BCS-BEC crossover is studied in a systematic way in the broken-symmetry phase between zero temperature and the critical temperature. This study bridges two regimes where quantum and thermal fluctuations are, respectively, important. The theory is implemented on physical grounds, by adopting a fermionic self-energy in the broken-symmetry phase that represents fermions coupled to superconducting fluctuations in weak coupling and to bosons described by the Bogoliubov theory in strong coupling. This extension of the theory beyond mean field proves important at finite temperature, to connect with the results in the normal phase. The order parameter, the chemical potential, and the single-particle spectral function are calculated numerically for a wide range of coupling and temperature. This enables us to assess the quantitative importance of superconducting fluctuations in the broken-symmetry phase over the whole BCS-BEC crossover. Our results are relevant to the possible realizations of this crossover with high-temperature cuprate superconductors and with ultracold fermionic atoms in a trap.
The two-particle spectrum in a dense Fermion system is treated using a thermodynamic Green function approach. A self-consistent description of possible bound states and a superfluid condensate in a correlated medium is given. A detailed understanding of superfluidity and superconductivity in correlated Fermion systems, especially the transition from the Cooper-paired state (weak-coupling limit) to the Bose condensed state of tightly bound pairs of Fermions (strong coupling limit) [1], is of great interest for very different physical systems. The problem of a unified treatment of Bose–Einstein condensation (BEC) and the Bardeen–Cooper–Schrieffer (BCS) phase arises not only in describing the electron structure of strongly correlated electron superfluids such as superconductors [2], the electronhole system in semiconductors [3], spin-polarized hydrogen and liquid He [4], but also in the theory of nuclear matter [5] and quark–gluon systems [6]. Recently, there have been several new approaches to this stimulating problem. A Monte-Carlo simulation of a finite He system at zero temperature has been performed in Ref. [7]. The microscopic theory of strongly coupled quantum fluids has been treated within the Jastrow approximation (cf. Ref. [8]) to obtain the ground state and low-lying excited states of strongly correlated boson quantum fluids. The crossover from weak to strong coupling superconductivity has been considered using a functional integral representation [2] (see also Ref. [7]).
A strongly coupled many-body system (a system of fermions confined in a shell surrounding the Fermi surface for which the variation in the single-particle kinetic energy is neglected) is defined. Some general properties of this system are obtained, in particular a criterion for permissible approximations. It is shown that a set of diagrams in which an electron of momentum $\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ and of spin \ensuremath{\downarrow} interacts with many electrons of momentum $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}+\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$ and spin \ensuremath{\uparrow}, where $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$ does not grow too large, can be consistently extracted from the totality of graphs generated by this strongly coupled Hamiltonian. When analyzed, these lead to a ground-state and single-particle spectrum displaying the important qualitative features of a BCS superconductor.
The crossover from weak to strong coupling for a three-dimensional continuum model of fermions interacting via an attractive contact potential is studied above the superconducting critical temperature
Tc. The pair-fluctuation propagator, the one-loop self-energy, and the spectral function are investigated in a systematic way from the superconducting fluctuation regime (weak coupling) to the bosonic regime (strong coupling). Analytic and numerical results are reported. In the strong-coupling regime, where the pair fluctuation propagator has bosonic character, two quite different peaks appear in the spectral function at a given wave vector, a broad one at negative frequencies and a narrow one at positive frequencies. The broad peak is asymmetric about its maximum, with its spectral weight decreasing by increasing coupling and temperature. In this regime, two crossover temperatures
T1∗ (at which the two peaks in the spectral function merge in one peak) and
T0∗ (at which the maximum of the lower peak crosses zero frequency) can be identified, with ${T}_{c}\ensuremath{\ll}{T}_{0}^{*}<{T}_{1}^{*}.$ By decreasing coupling, the two-peak structure evolves smoothly. In the weak-coupling regime, where the fluctuation propagator has diffusive Ginzburg-Landau character, the overall line shape of the spectral function is more symmetric and the two crossover temperatures approach
Tc. The analysis of the spectral function identifies specific features which allow one to distinguish by ARPES whether a system is in the weak- or strong-coupling regime. Connection of the results of our analysis with the phenomenology of cuprate superconductors is also attempted.
We consider the current correlation function for a three-dimensional system of fermions embedded in a homogeneous background and mutually interacting via an attractive short-range potential, below the (superconducting) critical temperature. Diagrammatic contributions in the broken-symmetry phase are identified, which yield for the (wave-vector and frequency-dependent) current correlation function the fermionic BCS approximation in the weak-coupling limit and the bosonic Bogoliubov approximation in the strong-coupling limit (whereby composite bosons form as bound-fermion pairs). The temperature dependence of the superfluid density (from the BCS exponential behavior at weak coupling to a power-law behavior at strong coupling) and the form of the Pippard-like kernel at zero temperature are explicitly obtained from weak to strong coupling. Quite generally, it is shown that the Pippard-like kernel is the sum of a local (London-like) term and of a nonlocal component, the local term being dominant in the strong-coupling limit and the nonlocal component in the BCS (weak-coupling) limit. It is also shown that the range of the nonlocal component is determined by the coherence length measuring the spatial correlations of the amplitude of the order parameter, namely, the correlations among different Cooper pairs (or composite bosons), rather than between the fermionic partners of a given pair. In addition, a prescription is provided for mapping the fermionic onto the bosonic diagrammatic theories in the broken-symmetry phase, thus complementing what has been already done in the normal phase.