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...

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 obtain the thermodynamic properties for a non-interacting Bose gas constrained on multilayers modeled by a periodic Kronig-Penney delta potential in one direction and allowed to be free in the other two directions. We report Bose-Einstein condensation (BEC) critical temperatures, chemical potential, internal energy, specific heat, and entropy for different values of a dimensionless impenetrability P 0 between layers. The BEC critical temperature Tc coincides with the ideal gas BEC critical temperature T0 when P = 0 and rapidly goes to zero as P increases to infinity for any finite interlayer separation. The specific heat CV vs T for finite P and plane separation a exhibits one minimum and one or two maxima in addition to the BEC, for temperatures larger than Tc which highlights the effects due to particle confinement. Then we discuss a distinctive dimensional crossover of the system through the specific heat behavior driven by the magnitude of P . For T < Tc the crossover is revealed by the change in the slope of log CV (T ) and when T > Tc, it is evidenced by a broad minimum in CV (T ).

The critical BEC temperature Tc of a non interacting boson gas in a layered structure like those of cuprate superconductors is shown to have a minimum Tc,m, at a characteristic separation between planes am. It is shown that for a < am, Tc increases monotonically back up to the ideal Bose gas T0 suggesting that a reduction in the separation between planes, as happens when one increases the pressure in a cuprate, leads to an increase in the critical temperature. For finite plane separation and penetrability the specific heat as a function of temperature shows two novel crests connected by a ridge in addition to the well-known BEC peak at Tc associated with the 3D behavior of the gas. For completely impenetrable planes the model reduces to many disconnected infinite slabs for which just one hump survives becoming a peak only when the slab widths are infinite.Since London first suggested 1 that superfluidity in liquid 4 He might well be a manifestation of Bose-Einstein condensation (BEC) of the helium atoms before interatomic interactions are "switched on," BEC in layered systems began to be studied to understand helium films 2,3,4 . The discovery of high-T c superconductivity stimulated renewed interest in compounds with layered structures 5 in which a BEC mechanism seems to be an essential feature to explain high critical temperatures 6 . The so-called "Uemura plot" 7 of data from muon-spin relaxation (µSR), neutron and Raman scattering, and angle-resolved photoemission (ARPES) measurements exhibits T c vs Fermi temperatures T F ≡ E F /k B where E F the Fermi energy and k B the Boltzmann constant. Empirical T c s of many cuprates straddle a line parallel to the Uemura-plot diagonal line associated with the simple BEC formula T 0 ≃ 3.31 2 n 2/3 B /mk B ≃ 0.218T F corresponding to an ideal gas of bosons of mass m = 2m * and number density n B = n s /2 where m * is the individualcharge-carrier effective mass and n s their number density. The parallel line of data is shifted down from T 0 by a factor of 4-5. This has been judged 8 a "fundamental importance of the BEC concept in cuprates." In addition, the possibility of creating BECs or superfluidity of ultracold fermions 9 in optical lattices 10 , along with the expected observation of BEC of excitons (electron-hole pairs) in semiconductors 11 , have further revived theoretical and experimental 12,13 efforts to better understand the behavior of quantum gases in layered geometries.Most models based on layered structures 6,15 simulating quasi-2D high-T c superconductors, or other to study BEC 16,17,18 , rely on a single-boson hopping interaction term producing nearest-interlayer couplings in one spatial dimension while moving freely in the other two directions. The energy spectrum is typically of the form ǫ k = 2 (k 2 x + k 2 y )/2m + ǫ kz with ǫ kz = ( 2 /M a 2 )(1 − cos k z a) where a is the plane separation and the constant 2 /M a 2 is a measure of the bosonic Cooper pair hopping probability between planes. In the case of CuO 2 planes in cuprate superconductors a bos...