We study a simple model of a two-dimensional s-wave superconductor in the presence of a random potential in the regime of large disorder. We first use the Bogoliubov-de Gennes (BdG) approach to show that, with increasing disorder the pairing amplitude becomes spatially inhomogeneous, and the system cannot be described within conventional approaches for studying disordered superconductors which assume a uniform order parameter. In the high disorder regime, we find that the system breaks up into superconducting islands (with large pairing amplitude) separated by an insulating sea. We show that this inhomogeneity has important implications for the physical properties of this system, such as superfluid density and the density of states. We find that a finite spectral gap persists in the density of states for all values of disorder and we provide a detailed understanding of this remarkable result. We next generalize Anderson's idea of the pairing of exact eigenstates to include an inhomogeneous pairing amplitude, and show that it is able to qualitatively capture many of the nontrivial features of the full BdG analysis. Finally, we study the transition to a gapped insulating state driven by the quantum phase fluctuations about the inhomogeneous superconducting state.PACS numbers:

The effect of nonmagnetic impurities on 2D s-wave superconductors is studied beyond the weak disorder regime. Within the Bogoliubov -de Gennes (BdG) framework, the local pairing amplitude develops a broad distribution with significant weight near zero with increasing disorder. Surprisingly, the density of states continues to show a finite spectral gap. The persistence of the spectral gap at large disorder is shown to arise from the breakup of the system into superconducting "islands." Superfluid density and off-diagonal correlations show a substantial reduction at high disorder. A simple analysis of phase fluctuations about the highly inhomogeneous BdG state is shown to lead to a transition to a nonsuperconducting state. [S0031-9007(98)07517-6] PACS numbers: 74.20.-z, 74.40.+ k, 74.80.-g The effect of strong disorder on superconductivity has been a subject of considerable interest, both theoretically [1,2] and experimentally [3,4], for a long time. A generally accepted physical picture of how the superconducting (SC) state is destroyed and the nature of the non-SC state has not yet emerged. Much of the theoretical work ("pairing of exact eigenstates" [1,5] or diagrammatics [2,6]) assumes that the pairing amplitude D͑r͒ is uniform in space (r independent) even for a highly disordered SC; see, however, [7,8]. Recent work on universal properties at the SC-insulator transition [9] has also ignored amplitude fluctuations, since phase fluctuations are presumably responsible for critical properties.In this paper we consider a simple model of a 2D s-wave superconductor at T 0 in a random potential, defined by Eq. (1) below, and analyze it in detail within a Bogoliubov-de Gennes (BdG) framework [10]. Our goal is to see how the local pairing amplitude D͑r͒ varies spatially in the presence of disorder, and the effect of this inhomogeneity on physically relevant correlation functions. Our results can be summarized as follows.(1) With increasing disorder, the distribution P͑D͒ of the local pairing amplitude D͑r͒ becomes very broad, eventually developing considerable weight near D ഠ 0.(2) The spectral gap in the one-particle density of states persists even at high disorder in spite of the growing number of sites with D͑r͒ ഠ 0. A detailed understanding of this surprising effect emerges from a study of the spatial variation of D͑r͒ and of the BdG eigenfunctions.(3) There is substantial reduction in the superfluid stiffness and off-diagonal correlations with increasing disorder; however, the amplitude fluctuations by themselves cannot destroy the superconductivity.(4) Phase fluctuations about the inhomogeneous BdG state are described by a quantum XY model whose parameters, compressibility and phase stiffness, are obtained from the BdG results. A simple analysis of this effective model within a self-consistent harmonic approximation leads to a transition to a non-SC state.We conclude with some comments on the implications of our results for experiments on disordered films.We model the 2D disordered s-wave SC by an attractive Hu...

Properties of the 'electron gas'-in which conduction electrons interact by means of Coulomb forces but ionic potentials are neglected-change dramatically depending on the balance between kinetic energy and Coulomb repulsion. The limits are well understood 1 . For very weak interactions (high density), the system behaves as a Fermi liquid, with delocalized electrons. In contrast, in the strongly interacting limit (low density), the electrons localize and order into a Wigner crystal phase. The physics at intermediate densities, however, remains a subject of fundamental research 2-8 . Here, we study the intermediate-density electron gas confined to a circular disc, where the degree of confinement can be tuned to control the density. Using accurate quantum Monte Carlo techniques 9 , we show that the electron-electron correlation induced by an increase of the interaction first smoothly causes rings, and then angular modulation, without any signature of a sharp transition in this density range. This suggests that inhomogeneities in a confined system, which exist even without interactions, are significantly enhanced by correlations.Quantum dots 10 -a nanoscale island containing a puddle of electrons-provide a highly tunable and simple setting to study the effects of large Coulomb interaction. They introduce level quantization and quantum interference in a controlled way, and can, in principle, be made in the very-low-density regime, where correlation effects are strong 11 . In addition, there are natural parallels between quantum dots and other confined systems of interacting particles, such as cold atoms in traps.Therefore, we consider a model quantum dot consisting of electrons moving in a two-dimensional (2D) plane, with kinetic energy (−(1/2) i ∇ with n being the density of electrons. For our confined system in which n(r) varies, we define r s in the same way using the mean densityn ≡ n 2 (r)dr/N. We have studied this system up to N = 20 electrons. The spring constant ω makes the oscillator potential narrow (for large ω) or shallow (for small ω); it thereby tunes the average density of electrons between high and low values, thus controlling r s . For example, for N = 20, varying ω between 3 and 0.0075 changes r s from 0.4 to 17.7. The radius of the dot grows significantly as r s increases, in an approximately linear fashion (see Fig. 1).In the bulk 2D electron gas, numerical work suggests a transition from a Fermi-liquid state to a Wigner crystal around r c s ≈ 30-35 (refs 2-4,8). On the other hand, experiments on the 2D electron gas (which include, of course, disorder and residual effects of the ions) show more-complex behaviour, including evidence for a metal-insulator transition 5 . Circular quantum dots have been studied previously using a variety of methods, yielding a largely inconclusive scenario. Many studies 12-14 have used density functional theory or the HartreeFock method. These typically predict charge or spin-density-wave order even for modest r s (unless the symmetry is restored after the fact 14 ), ...

We study the development of electron-electron correlations in circular quantum dots as the density is decreased. We consider a wide range of both electron number, N ≤ 20, and electron gas parameter, rs < ∼ 18, using the diffusion quantum Monte Carlo technique. Features associated with correlation appear to develop very differently in quantum dots than in bulk. The main reason is that translational symmetry is necessarily broken in a dot, leading to density modulation and inhomogeneity. Electron-electron interactions act to enhance this modulation ultimately leading to localization. This process appears to be completely smooth and occurs over a wide range of density. Thus there is a broad regime of "incipient" Wigner crystallization in these quantum dots. Our specific conclusions are: (i) The density develops sharp rings while the pair density shows both radial and angular inhomogeneity. (ii) The spin of the ground state is consistent with Hund's (first) rule throughout our entire range of rs for all 4 ≤ N ≤ 20. (iii) The addition energy curve first becomes smoother as interactions strengthen -the mesoscopic fluctuations are damped by correlation -and then starts to show features characteristic of the classical addition energy. (iv) Localization effects are stronger for a smaller number of electrons. (v) Finally, the gap to certain spin excitations becomes small at the strong interaction (large rs) side of our regime.

The d-wave vortex lattice state is studied within the framework of Bogoliubov-de Gennes (BdG) mean field theory. We allow antiferromagnetic (AFM) order to develop selfconsistently along with d-wave singlet superconducting (dSC) order in response to an external magnetic field that generates vortices. The resulting AFM order has strong peaks at the vortex centers, and changes sign, creating domain walls along lines where ∇×js ≈ 0. The length scale for decay of this AFM order is found to be much larger than the bare d-wave coherence length, ξ. Coexistence of dSC and AFM order in this system is shown to induce π-triplet superconducting order. Competition between different orders is found to suppress the local density of states at the vortex center and comparison to recent experimental findings is discussed.

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The extreme variability of observables across the phase diagram of the cuprate high-temperature superconductors has remained a profound mystery, with no convincing explanation for the superconducting dome. Although much attention has been paid to the underdoped regime of the hole-doped cuprates because of its proximity to a complex Mott insulating phase, little attention has been paid to the overdoped regime. Experiments are beginning to reveal that the phenomenology of the overdoped regime is just as puzzling. For example, the electrons appear to form a Landau Fermi liquid, but this interpretation is problematic; any trace of Mott phenomena, as signified by incommensurate antiferromagnetic fluctuations, is absent, and the uniform spin susceptibility shows a ferromagnetic upturn. Here, we show and justify that many of these puzzles can be resolved if we assume that competing ferromagnetic fluctuations are simultaneously present with superconductivity, and the termination of the superconducting dome in the overdoped regime marks a quantum critical point beyond which there should be a genuine ferromagnetic phase at zero temperature. We propose experiments and make predictions to test our theory and suggest that an effort must be mounted to elucidate the nature of the overdoped regime, if the problem of high-temperature superconductivity is to be solved. Our approach places competing order as the root of the complexity of the cuprate phase diagram.quantum-phase transition ͉ non-Fermi liquid ͉ broken symmetry ͉ quantum order ͉ criticality T he superconducting dome (see Fig. 1), that is, the shape of the superconducting transition temperature T c as a function of doping (added charge carriers), x, is a clue that the high-T c superconductors are unconventional. Conventional superconductors, explained so beautifully by Bardeen, Cooper, and Schrieffer (52), have a unique ground state that is not naturally separated by any nonsuperconducting states. The electronphonon mechanism leads to superconductivity for arbitrarily weak attraction between electrons. To destroy a superconducting state requires a magnetic field or strong material disorder. In the absence of disorder or magnetic field, it is difficult to explain the sharp cutoffs at x 1 and x 2 within the Bardeen, Cooper, and Schrieffer theory. For high-T c superconductors, there is considerable evidence that competing order parameters are the underlying reason. Thus, x 1 and x 2 signify quantum phase transitions, most likely quantum critical points (QCPs) (1). Understanding high-T c superconductors therefore requires an understanding of possible competing orders (2, 3). The QCP at x 1 has been extensively studied (4-8), but little is known about x 2 . Recent work has also emphasized the importance of the maximum of the uniform susceptibility in defining the pseudogap line T* in Fig. 1 that ends at another QCP at x c (8). Here, we attempt, instead, at gaining insight from the possible existence of a QCP at x 2 .Complex materials (cuprates, heavy fermions, and organics)...

Free nodal fermionic excitations are simple but interesting examples of fermionic quantum criticality in which the dynamic critical exponent $z=1$, and the quasiparticles are well defined. They arise in a number of physical contexts. We derive the scaling form of the diamagnetic susceptibility, $\chi$, at finite temperatures and for finite chemical potential. From measurements in graphene, or in $\mathrm{Bi_{1-x}Sb_{x}}$ ($x=0.04$), one may be able to infer the striking Landau diamagnetic susceptibility of the system at the quantum critical point. Although the quasiparticles in the mean field description of the proposed $d$-density wave (DDW) condensate in high temperature superconductors is another example of nodal quasiparticles, the crossover from the high temperature behavior to the quantum critical behavior takes place at a far lower temperature due to the reduction of the velocity scale from the fermi velocity $v_{F}$ in graphene to $\sqrt{v_{F}v_{\mathrm{DDW}}}$, where $v_{\mathrm{DDW}}$ is the velocity in the direction orthogonal to the nodal direction at the Fermi point of the spectra of the DDW condensate.Comment: 11 pages, 2 eps figure