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Cambridge University Press eBooks

A Brief Excursus to the BCS–BEC Crossover

Giancarlo Calvanese Strinati

Published Feb 6, 2026• 0 citations
Publisher

Abstract

This chapter provides a concise account of the salient features of the BCS–BEC crossover. After a brief historical review of the topic, the key features of the BCS–BEC crossover are recalled. In particular, the BCS wave function for the ground state is shown to contain the BEC state of composite bosons as a limiting situation, and the special role played in this context by the chemical potential is pointed out. The need for pairing fluctuations beyond mean field is also emphasized, together with the occurrence of two (coupling and temperature dependent) lengths, which measure the correlation either within a pair of fermions with opposite spins or among different pairs. The limiting physical situations corresponding to the Ginzburg–Landau and Gross–Pitaevskii equations are finally considered.

DOI
10.1017/9781009411509.028

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arXiv:cond-mat/0406099•Physical Review B•Sep 17, 2004•136 citations
This chapter provides a concise account of the salient features of the BCS–BEC crossover. After a brief historical review of the topic, the key features of the BCS–BEC crossover are recalled. In particular, the BCS wave function for the ground state is shown to contain the BEC state of composite bosons as a limiting situation, and the special role played in this context by the chemical potential is pointed out. The need for pairing fluctuations beyond mean field is also emphasized, together with the occurrence of two (coupling and temperature dependent) lengths, which measure the correlation either within a pair of fermions with opposite spins or among different pairs. The limiting physical situations corresponding to the Ginzburg–Landau and Gross–Pitaevskii equations are finally considered.
This chapter derives from first principles the time-dependent Gross–Pitaevskii equation, which describes the time-dependent behavior of the condensate wave function associated with the composite bosons that form on the BEC side of the BCS–BEC crossover at sufficiently low temperature. The derivation relies on the Green’s functions method for nonequilibrium problems developed before and explores the assumption that the fermionic chemical potential, associated with the initial preparation of the system at thermodynamic equilibrium, is the largest energy scale in the problem. The relation between the scattering length for composite bosons and the scattering length for the constituent fermions is also discussed.
Abstract This is the first of a series of chapters devoted to interacting Fermi gases, with special focus here on the effects of superfluidity. It starts with a brief discussion of the ideal Fermi gas and then focuses on the properties of dilute interacting Fermi gases. Topics include the weakly repulsive Fermi gas; the gas of composite bosons; the BCS limit of a weakly interacting Fermi gas; the strongly interacting, but still dilute, unitary Fermi gas where the scattering length is much larger than the average interatomic distance; the BCS–BEC crossover; and the Bogoliubov–de Gennes approach. A discussion of the main physical quantities predicted by mean-field theory and by quantum Monte Carlo simulations, including, in particular, the equation of state, the momentum distribution, and the condensation of pairs, is also presented.
This chapter derives the time-dependent Ginzburg–Landau equation from first principles, by relying on the same formal approach that has led to the time-dependent Gross–Pitaevskii equation of Chapter 28. Specifically, the time-dependent Ginzburg–Landau equation holds close to the critical temperature of the initial equilibrium preparation and in the (extreme) BCS limit of the BCS–BEC crossover, when the Cooper pairs are largely overlapping with each other. Care has to be exerted when dealing with the analytic properties in the wave-vector and frequency space of the normal and anomalous particle–particle bubbles.
Abstract The pioneering works by Eagles, Leggett, and Nozieres/Schmitt‐Rink [Eagles, Phys Rev, 1969, 186, 456; Leggett, In: Modern Trends in the Theory of Condensed Matter, Springer‐Verlag: Berlin, 1980; Nozieres and Schmitt‐Rink, Low Temp Phys, 1985, 59, 980] (reviewed and augmented by Randeria [Randeria, Bose‐Einstein Condensation, Cambridge University Press, Cambridge, 1995, p 255]) emphasize that in the limits of the models studied both at T = 0 and T ≠ 0, the “cross‐over” from a BCS‐type to a BEC‐type superconductor is continuous. The BCS and BEC “end points” seem to be well‐established. However, in the intermediate region—home to fulleride and high temperature superconductors—considerable extrapolation of the models must be done as there still is no exact theory. Yet, considerable current literature is devoted to what appears to be more “singular‐type phenomena” such as quantum critical points, “stripe” formation, insulator to superconductor phase transitions, loss of validity of the Fermi liquid theory, etc. Using a connection we have made with “cold” atom fermion‐boson crossover theory [Chen, Stajic, Levin, 2005, e‐print cond‐mat/0508603], we can establish that the resonance previously discussed [Squire and March, Int J Quant Chem, 2006, 106, 3343] is a result of the crossing of the fermion band by the boson band. While the ground state appears to remain continuous, the paired energy gap becomes transformed. We discuss features of the resonance and the experimentally observed pre‐formed “BEC Cooper pair” formation, essential to the boson‐fermion resonance theory. In addition some of the various singular phenomena discussed above can be put more into perspective. Finally, in both limits the relation of characteristic lengths to the inverse Fermi momentum is strongly emphasized, as is the role of the chemical potential near the pseudogap regime. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007
A functional integral formulation, used previously to calculate Tc{\mathrm{T}}_{\mathrm{c}}Tc​ and describe normal state properties of the BCS-Bose crossover, is extended to TTc{\mathrm{T}}_{\mathrm{c}}Tc​. The saddle point approximation is shown to be qualitatively correct for T\ensuremath{\ll}Tc{\mathrm{T}}_{\mathrm{c}}Tc​ for all couplings, in contrast to the situation above Tc{\mathrm{T}}_{\mathrm{c}}Tc​. Several features of the crossover are described. The difference between the T=0 ``pair size'' and the (prefactor of the T dependent) Ginzburg-Landau coherence length is pointed out: the two quantities are the same only in the BCS limit. The evolution of the collective modes from the BCS to the Bose regime is discussed together with the mixing of the amplitude and phase in the absence of a particle-hole symmetry.
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.