Cooper pairing in two dimensions is analyzed with a set of renormalized equations to determine its binding energy for any fermion number density and all coupling assuming a generic pairwise residual interfermion interaction. Also considered are Cooper pairs ͑CP's͒ with nonzero center-of-mass momentum ͑CMM͒ and their binding energy is expanded analytically in powers of the CMM up to quadratic terms. A Fermi-sea-dependent linear term in the CMM dominates the pair excitation energy in weak coupling ͑also called the BCS regime͒ while the more familiar quadratic term prevails in strong coupling ͑the Bose regime͒. The crossover, though strictly unrelated to BCS theory per se, is studied numerically as it is expected to play a central role in a model of superconductivity as a Bose-Einstein condensation of CPs where the transition temperature vanishes for all dimensionality dр2 for quadratic dispersion, but is nonzero for all dу1 for linear dispersion.The original Cooper pair ͑CP͒ problem 1 in two ͑2D͒ and three ͑3D͒ dimensions possesses ultraviolet divergences in momentum space that are usually removed via interactions regularized with large-momentum cutoffs.2 One such regularized potential is the BCS model interaction which is of great practical use in studying Cooper pairing 1 and superconductivity.3 Although there are controversies over the precise pairing mechanism, and thus over the microscopic Hamiltonian appropriate for high-T c superconductors, some of the properties of these materials have been explained satisfactorily within a BCS-Bose crossover picture 4-7 via a renormalized BCS theory for a short-range interaction. In the weak-coupling limit of the BCS-Bose crossover description one recovers the pure mean-field BCS theory of weakly bound, severely overlapping CPs. For strong coupling ͑and/or low density͒ well separated, nonoverlapping ͑so-called ''local''͒ pairs appear 4 in what is known as the Bose regime. It is of interest to detail how renormalized Cooper pairing itself evolves independently of the BCS-Bose crossover picture in order to then discuss the possible BoseEinstein ͑BE͒ condensation ͑BEC͒ of such pairs. We address this here in a single-CP picture, while considering also the important case ͑generally neglected in BCS theory͒ of nonzero center-of-mass-momentum ͑CMM͒ CPs that are expected to play a significant role in BE condensates at higher temperatures.In this report we derive a renormalized Cooper equation for a pair of fermions interacting via either a zero-or a finiterange interaction. We find an analytic expression for the CP excitation energy up to terms quadratic in the CMM which is valid for any coupling. For weak coupling only the linear term dominates, as it also does for the BCS model interaction. 8 The linear term was mentioned for 3D as far back as 1964 ͑Ref. 9, p. 33͒. For strong coupling we now find that the quadratic term dominates and is just the kinetic energy of the strongly bound composite pair moving in vacuum.The CP dispersion relation enters into each summand in the BE dis...

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Using the Bethe-Salpeter (BS) equation, Cooper pairing can be generalized to include contributions from holes as well as particles from the ground state of either an ideal Fermi gas (IFG) or of a BCS many-fermion state. The BCS model interfermion interaction is employed throughout. In contrast to the better-known original Cooper pair problem for either two particles or two holes, the generalized Cooper equation in the IFG case has no real-energy solutions. Rather, it possesses two complex-conjugate solutions with purely imaginary energies. This implies that the IFG ground state is unstable when an attractive interaction is switched on. However, solving the BS equation for the BCS ground state reveals two types of real solutions: one describing moving (i.e., having nonzero total, or center-of-mass, momenta) Cooper pairs as resonances (or bound composite particles with a finite lifetime), and another exhibiting superconducting collective excitations analogous to Anderson-Bogoliubov-Higgs RPA modes. A Bose-Einstein-condensation-based picture of superconductivity is addressed.

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We have investigated the nucleation rate at which cavities are formed in He and He at negative pressures due to thermal fluctuations. To this end, we have used a density functional that reproduces the He liquid-gas interface along the coexistence line. The inclusion of thermal e8'ects in the calculation of the barrier against nucleation results in a sizable decrease of the absolute value of the tensile strength above 1.5 K. Theoretical investigations of liquid-helium properties at negative pressures have been prompted by recent experiments carried out by Nissen et al. s and by Xiong and Maris using ultrasonic waves. This method allows the study of cavitation in very small liquid volumes, considerably avoiding the possibility of heterogeneous nucleation at electron bubbles. Although the experimental results reported in Ref. 5 foi He at temperatures above 1.5 K seemed to be well reproduced by classical nucleation theory (C1NT), s the experixnent carried out in Ref. 6 appears to discard this possibility. A serious argument against the interpretation of the experimental findings of Ref. 5, already raised in Ref. 6 and confirmed in Ref. 4, is that the critical pressure P, at which liquid He becomes macroscopically unstable is bigger than the tensile strength yielded by C1NT and by the experiment reported in Ref. 5. (To avoid any possible misunderstanding, here we define the tensile strength as a negative quantity. )Xiong and Maris have found that the tensile strength for nucleation of bubbles in He for temperatures in the 0.8 -2 K range, is -3bars. To analyze their experimental results, they have resorted to a method that represents a considerable improvement over the ClNT. It is based on a density functional (DF) whose free parameters are fixed to yield the experimental velocity of sound propagation in the liquid as a function of the density p, and includes a gradient term A(V'p)~adjusted so as to reproduce the surface tension of 4He at T = 0 K. Using their revised nucleation theory, they have found a tensile strength that goes from -9 bars at T = 0 K, to -6.5 bars at T = 2 K, still lying in absolute value well above their experimental data. There may be several reasons for this disagreement. The first is the validity of their functional in the density domain corresponding to negative pressures. However, since other equations of state (see also below) yield very similar values for P, (around -9bars at T = 0 K), we do not believe this to be the cause of the disagreement. If the ultrasonic technique used in these experiments discards the possibility of heterogeneous nucleation, nucleation on vortice lines is likely the main origin of the discrepancy. Xiong and Maris have estimated the critical pressure P, " for nucleation due to vortices to be -6.5 bars at T = 0 K whereas Dalfovo, using a nonlocal DF, has obtained a value of -8bars, much closer to P,.In this work, we want to address the effect that a nonzero temperature has on the nucleation barrier. This has been overlooked in all previous calculations and is of relev...