By projecting the BCS ground state of superconducting electron condensate on the N-electron Hilbert space, a real-space equation-of-motion is obtained for the electron pair function [Formula: see text] at absolute zero temperature (T = 0):[Formula: see text]where ρN−2 denotes electron density of the (N – 2)-electron condensate given as[Formula: see text]Since the exchange-correlation potential is given as an explicit functional of electron density, this equation represents the fundamental working equation for the new density functional theory of superconductivity. The 2nd-order density matrix ΓN(1, 2|1′, 2′) projected on the N-electron Hilbert space satisfies[Formula: see text]so that asymptotically[Formula: see text]where [Formula: see text] denotes the center-of-mass coordinate of electrons e1and e2; this is considered the ODLRO (off-diagonal long-range order) at T = 0 projected on the N-electron Hilbert space. A new attractive potential analysis for the two-electron scattering problem (A. Tachibana, Bull. Chem. Soc. Jpn. 66, 3319 (1993); Int. J. Quantum Chem. 49, 625 (1994)) is straightforwardly applicable to the present equation-of-motion, and we can also plug in the vibronic interaction for the enhancement of the attractive force. Our approach is purely mathematical and basic, restricted merely at T = 0, but proves to serve as a real-space analysis of the pair function itself. Key words: equation-of-motion of electron pair, BCS theory, superconductivity, electron pair function, density functional theory.
By projecting the BCS ground state of superconducting electron condensate on the N-electron Hilbert space, a real-space equation-of-motion is obtained for the electron pair function [Formula: see text] at absolute zero temperature (T = 0):[Formula: see text]where ρN−2 denotes electron density of the (N – 2)-electron condensate given as[Formula: see text]Since the exchange-correlation potential is given as an explicit functional of electron density, this equation represents the fundamental working equation for the new density functional theory of superconductivity. The 2nd-order density matrix ΓN(1, 2|1′, 2′) projected on the N-electron Hilbert space satisfies[Formula: see text]so that asymptotically[Formula: see text]where [Formula: see text] denotes the center-of-mass coordinate of electrons e1and e2; this is considered the ODLRO (off-diagonal long-range order) at T = 0 projected on the N-electron Hilbert space. A new attractive potential analysis for the two-electron scattering problem (A. Tachibana, Bull. Chem. Soc. Jpn. 66, 3319 (1993); Int. J. Quantum Chem. 49, 625 (1994)) is straightforwardly applicable to the present equation-of-motion, and we can also plug in the vibronic interaction for the enhancement of the attractive force. Our approach is purely mathematical and basic, restricted merely at T = 0, but proves to serve as a real-space analysis of the pair function itself. Key words: equation-of-motion of electron pair, BCS theory, superconductivity, electron pair function, density functional theory.
The main objective of this paper is to introduce a new quantum mechanism of condensates and superconductivity based on a new interpretation of quantum mechanical wavefunctions, and on recent developments in quantum physics and statistical physics. First, we postulate that the wavefunction [Formula: see text]e[Formula: see text] is the common wavefunction for all particles in the same class determined by the external potential V(x), [Formula: see text](x)[Formula: see text] represents the distribution density of the particles, and [Formula: see text] is the velocity field of the particles. Although the new interpretation does not alter the basic theories of quantum mechanics, it is an entirely different interpretation from the classical Bohr interpretation, removes all absurdities and offers new insights for quantum physics and for condensed matter physics. Second, we show that the key for condensation of bosonic particles is that their interaction is sufficiently weak to ensure that a large collection of boson particles are in a state governed by the same condensation wavefunction field [Formula: see text] under the same bounding potential V. For superconductivity, the formation of superconductivity comes down to conditions for the formation of electron pairs, and for the electron pairs to share a common wavefunction. Thanks to the recently developed principle of interaction dynamics (PID) interaction potential of electrons and the average-energy level formula of temperature, these conditions for superconductivity are explicitly derived. Furthermore, we obtain both microscopic and macroscopic formulas for the critical temperature. The field and topological phase transition equations for condensates are also derived.
We show that in the translation invariant case and in the antiferromagnetic phase, the reduced density matrix ρ2 has no off-diagonal long-range order of on-site electron pairs for the single-band Hubbard model on a cubic lattice away from half filling at finite temperatures both for the positive coupling and for the negative coupling. In these cases the model can not give a mechanism for the superconductivity caused by the condensation of on-site electron pairs and the nearestneighbor electron pairs. PACS numbers: 71.10.+x, 74.20.-z, 05.30.-d. 1 It was proposed the possibility that the Hubbard model can give a mechanism to characterize high temperature superconductivity[1]. It has been shown that the η pairing states for the single band Hubbard model with constant hopping matrix element for nearest neighbor sites have ODLRO[2]. It has also been shown that in the ground state the model with negative coupling has off-diagonal long-range order (ODLRO) on some bipartite lattice[3]. However, as the temperature T> 0 whether the reduced density matrix ρ2 has ODLRO is inconclusive. The purpose of this work is to study the ODLRO of ρ2 in the single-band Hubbard model with constant hopping matrix element for nearest neighbor pairs on a cubic lattice at finite temperatures. In the translation invariant case we obtain an equation ( (10) in the following) which is satisfied by the long-range correlation functions of on-site electron pairs. From which we find that there is no ODLRO of on-site electron pairs. Then we consider the case with an antiferromagnetic background. In this case we will show that there is still no ODLRO of on-site electron pairs. The Hamiltonian of the model is H = −t ∑ (a † rar ′ + b † rb ∑
Abstract Chapter 13 addresses Bose condensation in superfluids (and superconductors), which involves the field operator ψ having a c-number component (<ψ(x,t)>≠0), challenging number conservation. The nonlinear Gross-Pitaevskii equation is derived for this condensate wave function <ψ>=ψ−ψ˜, facilitating identification of the coherence length and the core region of vortex motion. The noncondensate Green’s function G˜1(1,1′)=−i<(ψ˜(1)ψ˜+(1′))+> and the nonvanishing anomalous correlation function F˜∗(2,1′)=−i<(ψ˜+(2)ψ˜+(1′))+> describe the dynamics and elementary excitations of the non-condensate states and are discussed in conjunction with Landau’s criterion for viscosity. Associated concepts of off-diagonal long-range order and the interpretation of <ψ> as a superfluid order parameter are also introduced. Anderson’s Bose-condensed state, as a phase-coherent wave packet superposition of number states, resolves issues of number conservation. Superconductivity involves bound Cooper pairs of electrons capable of Bose condensation and superfluid behavior. Correspondingly, the two-particle Green’s function has a term involving a product of anomalous bound-Cooper-pair condensate wave functions of the type F(1,2)=−i<(ψ(1)ψ(2))+>≠0, such that G2(1,2;1′,2′)=F(1,2)F+(1′,2′)+G˜2(1,2;1′,2′). Here, G˜2 describes the dynamics/excitations of the non-superfluid-condensate states, while nonvanishing F,F+ represent a phase-coherent wave packet superposition of Cooper-pair number states and off-diagonal long range order. Employing this form of G2 in the G1-equation couples the condensed state with the non-condensate excitations. Taken jointly with the dynamical equation for F(1,2), this leads to the Gorkov equations, encompassing the Bardeen–Cooper–Schrieffer (BCS) energy gap, critical temperature, and Bogoliubov-de Gennes eigenfunction Bogoliubons. Superconductor thermodynamics and critical magnetic field are discussed. For a weak magnetic field, the Gorkov-equations lead to Ginzburg–Landau theory and a nonlinear Schrödinger-like equation for the pair wave function and the associated supercurrent, along with identification of the Cooper pair density. Furthermore, Chapter 13 addresses the apparent lack of gauge invariance of London theory with an elegant variational analysis involving re-gauging the potentials, yielding a manifestly gauge invariant generalization of the London equation. Consistency with the equation of continuity implies the existence of Anderson’s acoustic normal mode, which is supplanted by the plasmon for Coulomb interaction. Type II superconductors and the penetration (and interaction) of quantized magnetic flux lines are also discussed. Finally, Chapter 13 addresses Josephson tunneling between superconductors.
We investigate the condensate density and the condensate fraction of conduction electrons in weak-coupling superconductors by using the BCS theory and the concept of off-diagonal-long-range-order. We discuss the analytical formula of the zero-temperature condensate density of Cooper pairs as a function of Debye frequency and energy gap, and calculate the condensate fraction for some metals. We study the density of Cooper pairs also at finite temperature showing its connection with the gap order parameter and the effects of the electron-phonon coupling. Finally, we analyze similarities and differences between superconductors and ultracold Fermi atoms in the determination of their condensate density by using the BCS theory.
We show that the eigenvalues of the number operator [Formula: see text] in the pair state (k1, k2), where c's and c†'s are fermion operators, are limited to 0 or 1, while the eigenvalues of the pair-number operator with the net momentum q≡ k1+k2 fixed, [Formula: see text], are 0, 1, 2,…. Both fermionic and bosonic natures of the Cooper pairs (pairons) must be used in the superconductivity theory. The Bose–Einstein condensation (BEC) of the pairons moving with a linear energy–momentum relation occurs at [Formula: see text] (3D), [Formula: see text] (2D), where n0 is the pairon density and v F the Fermi velocity. The connection between this [Formula: see text] and the BCS superconducting temperature is discussed.
A original microscopic theory of superfluidity and superconductivity driven by the single particle (SPC) and pair condensation (PC) of attracting bosons both in Fermi and in Bose systems is developed. This theory (as distinct from the existing theories) for Fermi systems contains two order parameters Δ F and Δ B characterizing the attracting fermion pairs and boson pairs, respectively. In such systems superconducting (SC) phase transition is accompanied, as a rule, by the formation of k-space composite bosons (e.g. Cooper pairs and bipolarons) with their subsequent transition to the superfluid (SF) state by attractive SPC and PC. A novel Fermi-liquid and SF Bose-liquid theories are elaborated for description this two-stage Fermi–Bose-liquid (FBL) scenario of SC (or SF) transition. The crossover from k- to real (r)-space pairing regime for BCS-like coupling constants γ F ≃ 0.7-0.9 and the irrelevance of r-space pairs to the superconductivity are shown. The developed SF Bose-liquid theory predicts the first-order phase transition SPC ↔ PC of attracting 3d-bosons with the kink-like behaviors of all SC (SF) parameters near [Formula: see text] in accordance with the observations in 4 He , 3 He and superconductors. It is argued that the coexistence of the order parameters Δ F and Δ B leads to the superconductivity by two FBL scenarios. One of these scenarios is realized in the so-called fermion (type I, II and III) superconductors (FSC) (where formation of k-space composite bosons and their condensation occur at the same temperature) and the other in the boson (type II and III) superconductors (BSC) (where BCS-like pairing take place in the normal state with manifesting of the second-order phase transition and opening of the pseudogap at T=T F > T c ). There the gapless superfluidity (superconductivity) is caused by the gapless excitation spectrum of bosons at [Formula: see text] and not by the presence of point or line nodes of the BCS-like gap Δ F assumed in some s-, p- and d-pairing models. The 3D- and 2D-insulator–metal–superconductor phase diagrams are presented. The necessary and sufficient microscopic criterions for superfluidity is formulated. The theory proposed are in close agreement with the observations in 4 He , 3 He , superconductors, nuclear and neutron star matter, cosmology, etc.