Abstract As charge carrier of the macroscopic superconductivity, the Cooper pair is a composite particle of two paired electrons, which has both center-of-mass and inner-pair degrees of freedom. In most cases, these two different degrees of freedom can be well described by the macroscopic Ginzburg-Landau theory and the microscopic Bardeen-Cooper-Schrieffer (BCS) theory, respectively. Near the superconducting phase transition where the Cooper pair is fragile and unstable because of the small binding energy, there are non-trivial couplings between these two different degrees of freedom due to such as finite energy and/or momentum transfer. The non-trivial couplings make the original derivation of the Ginzburg-Landau theory from the BCS theory fail in principle as where these two different degrees of freedom should not be decoupled. In this article, we will present a renormalization formalism for an extended Ginzburg-Landau action for the superconducting phase transition where there is finite energy transfer between the center-of-mass and the inner-pair degrees of freedom of Cooper pairs. This renormalization formalism will provide a theoretical tool to study the unusual dynamical effects of the inner-pair time-retarded physics on the superconducting phase transition.
Starting from H. Fröhlich's second-quantized Hamiltonian for a
d-dimensional electron gas in interaction with lattice phonons describing the quantum vibrations of a metal, we present a rigorous mathematical derivation of the superconducting state, following the principles laid out originally in 1957 by J. Bardeen, L. Cooper and J. Schrieffer. As in the series of papers written on the subject in the 90es, of which the present paper is a continuation, the representation of ions as a uniform charge background allows for a
(1+d)-dimensional fermionic quantum-field theoretic reformulation of the model at equilibrium. For simplicity, we restrict in this article to
d=2 dimensions and zero temperature, and disregard effects due to electromagnetic interactions. Under these assumptions, we prove transition from a Fermi liquid state to a superconducting state made up of Cooper pairs of electrons at an energy level
Γϕ∼ℏωDe−π/mλ equal to the mass gap, expressed in terms of the Debye frequency
ωD, electron mass
m and coupling constant
λ. The dynamical
U(1)-symmetry breaking produces at energies lower than the energy gap
Γϕ a Goldstone boson, a non-massive particle described by an effective
(2+1)-dimensional non-linear sigma-model, whose parameters and correlations are computed. The proof relies on a mixture of general concepts and tools (multi-scale cluster expansions, Ward identities), adapted to this quantum many-body problem with its extended infra-red singularity located on the Fermi circle, and a specific
1/N-expansion giving the leading diagrams at intermediate energies. Ladder diagrams are proved to provide the leading behavior in the infra-red limit, in agreement with mean-field theory predictions.
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's) with nonzero center-of-mass momentum (CMM) 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 regime) while the more familiar quadratic term prevails in strong coupling (the Bose regime). 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
d< 2 for quadratic dispersion, but is nonzero for all
d> 1 for linear dispersion.
In this study, we present theoretical investigations of phase transitions and critical phenomena in materials through the lens of second-order Ginzburg-Landau theory, in conjunction with considerations of symmetry groups and thermal fluctuations. By addressing the residual effects after a renormalization process, a small number of macroscopic degrees of freedom can effectively replace the infinite number of microscopic degrees of freedom, emphasizing the significant role of dimensionality and the intrinsic characteristics of the system in understanding and analyzing transitions. We highlight several non-universal characteristics of continuous phase transitions near the transition temperature, including the non-monotonic relationship between the critical temperature and dimensionality, as well as the enhancement or disappearance of the specific heat jump in complex superconductors. While the resulting expressions for thermodynamic quantities are complex for one-dimensional systems, obeying Mermin-Wagner's theorem, they are considerably simplified for two-dimensional and three-dimensional systems.
We study a translation-invariant BCS functional of superconductivity for a system of spin-
1/2 fermions with spin-dependent chemical potentials,
μ+=μ−. Numerical work of Caldas and Mota and analytical work of Freiji, Hainzl, and Seiringer has suggested an unusual first-order phase transition in spin-imbalanced BCS models, in which the normal state becomes globally unstable while remaining locally stable. Here, we prove that a first-order phase transition occurs in such spin-imbalanced BCS functionals. For each sufficiently small fixed spin imbalance and sufficiently low temperature, there is a coexistence coupling
λfo strictly below the linear-stability threshold
λlin of the normal state at which the Cooper pair density exhibits a nonzero jump and the minimized energy has a kink. The methodological insight is a coercivity estimate obtained by decomposing the BCS energy minimization into two parts, where the ``inner'' problem minimizes over the one-particle density matrices at fixed pairing and can be exactly solved by a Legendre--Fenchel transform.
Using the Boson-Fermion formalism of superconductivity we calculate the condensation energy for several superconductors ranging from conventional to unconventional, or high temperature superconductors. It is calculated as the difference between the Helmholtz free energies of the superconducting and the normal state, which is a gas of N attractive electron gas, while the superconducting state is formed by the condensed Cooper pairs taken as composite bosons, coming from a fraction of the electrons inside the Debye shell, plus those electrons inside and outside the Debye shell that are unable to pair. After giving the analytic expressions for the internal energy U and the entropy S we obtain the Helmholtz free energy F = U -TS for both the superconducting and the normal states as functions of temperature. In the search for universalities, we calculate the ratio of the condensation energy at T=0 to the Sommerfeld constant (the normal state electronic specific heat over the temperature when T it's almost zero) using two different methods: the Boson-Fermion formalism developed here, as well as an analytical expression deduced from a combination of the BCS and Ginzburg-Landau theories. We find for the Boson-Fermion formalism
Econd/γ0=0.252Tc1.997, which is the same behavior described by the experimental fit of Kim et al
Econd/γ0=0.2Tc2.06 and by the one recently reported by Tallon et al for overdoped cuprate superconductors; while for the Ginzburg-Landau-BCS we get the expression
Econd/γ0=0.236Tc2, also in very good agreement with the partifirst method.
We formulate a new Bardeen-Cooper-Schrieffer (BCS)-type theory at finite temperature, by deriving a set of variational equations of the free energy after the particle-number projection. With its broad applicability, this theory can be a useful tool for investigating the pairing phase transition in finite systems with the particle-number conservation. This theory provides effects of the symmetry-restoring fluctuation (SRF) for the pairing phenomena in finite fermionic systems, distinctively from those of additional quantum fluctuations. It is shown by numerical calculations that the phase transition is compatible with the conservation in this theory, and that the SRF shifts up the critical temperature (
Tcr). This shift of
Tcr occurs due to reduction of degrees-of-freedom in canonical ensembles, and decreases only slowly as the particle-number increases (or as the level spacing narrows), in contrast to the conventional BCS theory.