Cooper pair dispersion relation for weak to strong coupling

Abstract

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...

One-dimensional Cooper pairing

“…It appears to be a precursor of the smooth "maxon-like" hump in 2D (Ref. [40], Figs. 1 and 3) and less pronounced in 3D (Ref.…”

“…Here we focus on some novel and unique properties of CPs in 1D, as compared with the 2D [40] and 3D [41] cases. We start from a nonlocal, separable interaction given by …”

“…In this work we study electron pairing in a 1D Fermi gas under zero-and finite-range, two-fermion interactions to address some novel and unique properties of CPs in 1D as compared with the 2D [40] and 3D [41] cases. In Section 2 we review the Cooper pairing mechanism in a simplified model where two particles near a static Fermi level interact at T = 0.…”

Specific heat of underdoped cuprate superconductors from a phenomenological layered Boson–Fermion model

“…In the framework of the most basic Boson-Fermion model of superconductivity [11,[26][27][28], we assume that Cooper pairs are composite-spin-zero-bosons with either zero or nonzero center-of-mass momenta (CMM), coexisting with a fermion fluid of the unpaired electrons. To include the effect of the layered structure of cuprates in the Boson-Fermion model, we calculate the BEC critical temperature and the thermodynamic properties for a system of non-interacting bosons immersed in a periodic multilayer array [29,30], simulated by an external Dirac comb potential along the perpendicular direction of the CuO 2 planes, while the Cooper pairs are allowed to move freely within the planes, with a linear energy-momentum dispersion relation [28].…”

“…To include the effect of the layered structure of cuprates in the Boson-Fermion model, we calculate the BEC critical temperature and the thermodynamic properties for a system of non-interacting bosons immersed in a periodic multilayer array [29,30], simulated by an external Dirac comb potential along the perpendicular direction of the CuO 2 planes, while the Cooper pairs are allowed to move freely within the planes, with a linear energy-momentum dispersion relation [28]. The fermion counterpart is treated in a similar way [31] as the boson gas, subject to the same external potential.…”