In this paper, we revisit the problem of dirty superconductors using a replica-symmetric BCS (RS-BCS) theory. We apply the RS-BCS theory to dirty superconductor grains of size $L^d$, where $L$ is the localization length and $d$ is the dimension of the system, assuming that the macroscopic system is composed of weakly coupled grains. Besides disordered potential, We also consider the case where regions with opposite signs of interaction exist in the grains, with net attractive interaction between electrons. Within the RS-BCS mean field theory, the system phase diagram, single-particle tunneling density of states and the superfluid density are computed within the RS-BCS theory for different strengths of disorder, We show that our result agrees qualitatively with previous numerical studies in the case with pure attractive interaction, and that a Cooper-pair-glass state may exist when regions with opposite sign of interaction exists and the net interaction between electrons is close to zero. The plausible relevance of our result to the anomalous metal state is discussed.
In this paper, we revisit the problem of dirty superconductors using a replica-symmetric BCS (RS-BCS) theory. We apply the RS-BCS theory to dirty superconductor grains of size Ld, where L is the localization length and d is the dimension of the system, assuming that the macroscopic system is composed of weakly coupled grains. Besides disordered potential, We also consider the case where regions with opposite signs of interaction exist in the grains, with net attractive interaction between electrons. Within the RS-BCS mean field theory, the system phase diagram, single-particle tunneling density of states and the superfluid density are computed within the RS-BCS theory for different strengths of disorder, We show that our result agrees qualitatively with previous numerical studies in the case with pure attractive interaction, and that a Cooper-pair-glass state may exist when regions with opposite sign of interaction exists and the net interaction between electrons is close to zero. The plausible relevance of our result to the anomalous metal state is discussed.
Motivated by the experimental observations of the anomalous metal states in many superconducting thin films systems, this thesis studies the interplay of disorder and interactions in superconductivity. We first demonstrate that the Fermi glass ground state is unstable in first-order perturbation theory under weak, short-ranged interaction, provided all states are localized. This finding elucidates why methods like the Hartree-Fock approximation are adequate to explain local moment formation in repulsive interactions. We then revisit the problem of dirty superconductors using a replica-symmetric BCS (RS-BCS) theory. We apply RS-BCS theory to disordered superconducting grains within the localization length, assuming the macroscopic system consists of weakly coupled grains. Besides disordered potential, We also consider regions with opposite signs of interaction exist in the grains, with net attractive interaction between electrons. Using RS-BCS mean field theory, we compute the phase diagram, tunneling density of states, and superfluid density across disorder strengths. Our results qualitatively agree with prior numerical studies for purely attractive interactions, and suggest a possible Cooper-pair-glass state when interactions of opposite signs nearly cancel.
Analytic expressions are derived for the root-mean-square (rms) radius of a pair of fermions in a BCS many-fermion state in one, two, and three dimensions, in terms of the BCS gap energy and the associated chemical potential. These expressions are valid for any coupling strength of any pair interaction model implying a momentum-independent gap energy. The latter holds, e.g., for an attractive \ensuremath{\delta} pair potential examined in the one-dimensional (1D) case (whose N-fermion ground state can be determined exactly) or for the BCS (electron-phonon) model interaction in any dimension. Weak-coupling and/or high-density limits for the rms radius are identical in 1D, 2D, and 3D, and reduce to the familiar well-known Pippard result to within a factor of order unity. In contrast, strong-coupling and/or low-density limits coincide in 1D and 3D, but differ by a factor of order unity in the 2D limit, and in each case are essentially the size of a single, isolated pair. The 1D \ensuremath{\delta} interaction McGuire-Yang-Gaudin many-fermion model is studied in detail. The interaction renormalization scheme of Miyake and of Randeria, Duan, and Shieh, and the BCS interaction model, both in 2D, are employed to analyze cuprate superconductor empirical results. Reasonable agreement between theoretical rms radii with experimental coherence lengths suggests that cuprates can be described moderately well as weakly coupled superconductors within the BCS-Bose formalism.
BCS superconductivity is explained by a simple Hamiltonian describing an attractive pairing interaction between pairs of electrons. The Hamiltonian may be treated using a mean-field method, which is adequate to study equilibrium properties and a variety of nonequilibrium effects. Nevertheless, in certain nonequilibrium situations, even in a macroscopic rather than a microscopic superconductor, the application of mean-field theory may not be valid. In such cases, one may resort to the full solution of the Hamiltonian, as given by Richardson in the 1960s. The relevance of Richardson's solution to macroscopic nonequilibrium superconductors was pointed out recently based on the existence of quantum instabilities out of equilibrium. It is then of interest to obtain analytical expressions for expectation values between exact eigenvalues of the pairing Hamiltonian within the Richardson approach for macroscopic systems. We undertake this task in the current paper. It should be noted that Richardson's approach yields the full set of eigenvalues of the Hamiltonian, while BCS theory yields only a subset. The results obtained here, then, generalize the familiar BCS expressions (e.g., for the electron occupation or pairing correlations) to cases where the spectrum of excitations diverges from BCS theory (e.g., in cases where the spectrum exhibits multiple gaps).
Bosonization of Cooper pairs and novel Bose-liquid superconductivity and superfluidity in high-Tc cuprates and other systems
The criteria for bosonization of Cooper pairs and novel Bose-liquid superconductivity and superfluidity in high-Tc cuprates and other pseudogap matters are formulated by using the uncertainty principle and the boson mean field theory. We argue that the BCS-type s- or d-wave superconductivity occurring in the fermionic limit of Cooper pairs (which exist in ordinary metals and heavily overdoped cuprates with large Fermi energies) is not characteristic of underdoped to overdoped cuprates with low Fermi energies. The superconducting order parameter in high-Tc cuprates and other pseudogap matters cannot be determined as the BCS-like (s- or d-wave) gap. We show that the unconventional superconductivity/superfluidity occurring in the bosonic limit of Cooper pairs exists in low Fermi energy systems where the bosonic Cooper pairs are formed at a pseudogap temperature T* above the uperconducting/superfluid transition temperature Tc and then such bosons condenses into a Bose superfluid at Tc. The pair condensation of attracting bosons occurs at Tc and then their single particle condensation sets in at Tc∗ lower than Tc (in three dimensions (3D)) or at T=0 (in two dimensions (2D)). The coherent single particle and pair condensates of bosons exist as two distinct superfluid phases below Tc. By solving the mean field equations for high-Tc cuprates, the novel superconducting states (i.e., a vortex-like state existing below the temperature Tv=Tc2D lower than T* but higher than Tc=Tc3D and two superconducting phases below Tc) and their properties characterized by the boson superfluid stiffness are determined and compared with the key experimental findings. The novel Bose-liquid superconductivity and superfluidity exists also in other exotic superconductors/superfluids, including quantum liquids, atomic Fermi gases and low-density nuclear matter.
I discuss the mean-field theory of superconductivity in a strongly disordered system of fermions with short-range attraction. It is argued that in this limit the effective theory at low energies is equivalent to the disordered Bose–Hubbard model, and I consider both the infinite-range and the "nearest-neighbor" hopping of bosons between the localized states. In the infinite-range case the mean-field theory is exact, and the superconducting gap is uniform in space, while in the latter case, the gap becomes highly non-uniform in space, but, surprisingly, is finite everywhere at T < T MF . I find that the mean-field transition temperature T MF > 0 always, and argue that the superconductor–insulator transition at T = 0 in models with net attraction between fermions is in the universality class of "dirty-bosons".
Abstract Chapter 6 introduces the BCS theory of superconductivity, proposed in 1957 by Bardeen, Cooper, and Schrieffer and which is based on the idea that electrons form a coherent state of electron pairs with opposite spins and opposite momenta. It then briefly explains how, along the Fröhlich idea, the electron-phonon interaction can lead to the ion-mediated attractive potential for Cooper pairing. Next, it introduces the reduced BCS potential, which acts between electrons having opposite spins and opposite momenta. Finally, this chapter discusses the drastic consequence of the reduced BCS potential on Cooper pair interaction and explains why Cooper pairs can strongly overlap without dissociating.