Superconductivity is a phenomenon where electrical current flows without friction. The current standard theory for it is the BCS (Bardeen-Cooper-Schrieffer) theory, which explains it as due to the energy gap formation by the electron-pairing, and the key ingredient for the supercurrent generation is the gauge symmetry breaking brought about by it. It was thought that superconductivity was fully understood by this standard theory; however, the discovery of superconductivity in cuprates in 1986 changed this situation, showing a number of experimental results that contradict the standard theory. It is also notable that the standard theory contradicts in the supecurrent carrier mass in the London moment; the predicted mass is an effective mass, while the experimental value is the free electron mass. The above contradictions suggest the necessity for a fundamental revision for the theory of superconductivity. Here we show that the required revision may be achieved by using the Berry phase formalism. It was developed after the establishment of the BCS theory, and provides a way to detect emergent gauge fields. A self-referencing phase factor on the wave function detected by the Berry phase formalism explains the supercurrent generation in the conventional and cuprate superconductors. It gives rise to a gauge field that enables the gauge symmetry breaking in the standard theory interpretation.
Superconductivity is a phenomenon where electrical current flows without friction. The current standard theory for it is the BCS (Bardeen-Cooper-Schrieffer) theory, which explains it as due to the energy gap formation by the electron-pairing, and the key ingredient for the supercurrent generation is the gauge symmetry breaking brought about by it. It was thought that superconductivity was fully understood by this standard theory; however, the discovery of superconductivity in cuprates in 1986 changed this situation, showing a number of experimental results that contradict the standard theory. It is also notable that the standard theory contradicts in the supecurrent carrier mass in the London moment; the predicted mass is an effective mass, while the experimental value is the free electron mass. The above contradictions suggest the necessity for a fundamental revision for the theory of superconductivity. Here we show that the required revision may be achieved by using the Berry phase formalism. It was developed after the establishment of the BCS theory, and provides a way to detect emergent gauge fields. A self-referencing phase factor on the wave function detected by the Berry phase formalism explains the supercurrent generation in the conventional and cuprate superconductors. It gives rise to a gauge field that enables the gauge symmetry breaking in the standard theory interpretation.
Abstract Superconductivity is reformulated as a phenomenon in which a stable velocity field is created by a U (1) phase neglected by Dirac in the Schrödinger representation of quantum mechanics. The neglected phase gives rise to a U (1) gauge field expressed as the Berry connection from many-body wave functions. The inclusion of this gauge field transforms the standard particle-number non-conserving formalism of superconductivity to a particle-number conserving one with many results of the former unaltered. In other words, the new formalism indicates that the current standard one is an approximation that effectively takes into account this neglected U (1) gauge field by employing the particle-number non-conserving formalism. Since the standard and new formalisms are physically different, conflicting results are predicted in some cases. We reexamine the Josephson relation and show that a capacitance contribution of the Josephson junction to the U (1) phase is missing in the standard formalism, and inclusion of it indicates that the standard theory actually does not agree with the experiment while the new one does. It is also shown that the dissipative quantum phase transition in Josephson junctions predicted in the standard theory does not exist in the new one in accordance with a recent experiment (Murani et al 2020 Phys. Rev. X 10 021003).
Neglected U(1) phase in the Schroedinger representation of quantum mechanics and particle number conserving formalisms for superconductivity
Superconductivity is reformulated as a phenomenon in which a stable velocity field is created by a U(1) phase neglected by Dirac in the Schroedinger representation of quantum mechanics. The neglected phase gives rise to a U(1) gauge field expressed as the Berry connection from many-body wave functions. The inclusion of this gauge field transforms the standard particle-number non-conserving formalism of superconductivity to a particle-number conserving one with many results of the former unaltered. In other words, the new formalism indicates that the current standard one is an approximation that effectively takes into account this neglected U(1) gauge field by employing the particle-number non-conserving formalism. Since the standard and new formalisms are physically different, conflicting results are predicted in some cases. We reexamine the Josephson relation and show that a capacitance contribution of the Josephson junction to the U(1) phase is missing in the standard formalism, and inclusion of it indicates that the standard theory actually does one agree with the experiment while the new one does. It is also shown that the dissipative quantum phase transition in Josephson junctions predicted in the standard theory does not exit in the new one in accordance with the recent experimental result.
Recently, a new theory of superconductivity has been put forward that attributes the origin of superconductivity to the appearance of a non-trivial Berry connection from many-electron wave functions. This theory reproduces the major results of the BCS theory with conserving the particle number, and predicts the single-electron supercurrent tunneling across the Josephson junction with keeping the correct Josephson relation. We re-examine the quantization of superconducting qubit circuits by taking into account the above development, and show that the dynamical variables used in the standard theory, the flux nodes relating to the voltage, should be replaced by those relating to the electromagnetic vector potential. The fact that the Josephson junction tunneling allows the single-electron supercurrent tunneling is the reason for the existence of excited single electrons in superconducting qubits with Josephson junctions. We predict that it will be avoided by weakening the coupling between two superconductors in the Josephson junction.
When Schrödinger solved the Schrödinger equation for the hydrogen atom, he assumed the single-valuedness of the electronic wave function. Thereafter, this assumption has been one of the fundamental postulates of quantum mechanics. When wave functions are multi-component, however, the imposing of the single-valued condition may become nontrivial. The spin-degree-of-freedom of electron makes electronic wave functions two-component. When spin-vortices are created by the conduction electrons and they move in the self-consistent field with the spin-vortices, the twisting of the spin basis occurs; then, the imposing of the single-valued condition becomes nontrivial, and a vector potential is induced. As a consequence, the effective vector potential becomes the sum of the vector potential from the induced one and that originates from the electric current. This effective vector potential is gauge invariant and the persistent current is generated by it. In the present work, we argue that if interactions that are omitted in the BCS reduced Hamiltonian are included, spin-vortices may be generated upon the application of a magnetic field. Then, the vector potential is induced and provides with the phase variable, θ, of the electron pair amplitude. The appearance of the spin-vortex provides with a new origin of θ; it originates from the induced gauge potential. This origin is compatible with the superselection rule for charge in contrast to the currently-accepted origin.
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.
Abstract One of the important steps leading to the successful formulation of the microscopic theory of superconductivity was the model calculation performed by Cooper (1956) which showed that two electrons, interacting above a filled Fermi sea, had a bound state in the presence of an arbitrarily weak attractive interaction. This chapter first examines Cooper’s calculation. The success of the Cooper model calculation in producing a bound state, with its implication that a proper many-body theory would contain a gap in the excitation spectrum, strongly suggests that the ground state wave function should be constructed from pairs of electrons. However, it is difficult to perform calculations using the many-body pair wave function from Cooper’s model. The chapter then introduces the wave function proposed by Bardeen, Cooper, and Schrieffer (BCS) as an alternative.