It is the purpose of this paper to explore the theory of high temperature superconductivity. Much of the motivation for this comes from the study of cuprate high temperature superconductors. However, we do not focus in great detail on the remarkable and exciting physics that has been discovered in these materials. Rather, we focus on the core theoretical issues associated with the mechanism of high temperature superconductivity. Although our discussions of theoretical issues in a strongly correlated superconductor are intended to be self contained and pedagogically complete, our discussions of experiments in the cuprates are, unfortunately, considerably more truncated and impressionistic.Our primary focus is on physics at intermediate temperature scales of order T c (as well as the somewhat larger "pseudogap" temperature) and energies of order the gap maximum, ∆ 0 . Consequently (and reluctantly) we have omitted any detailed discussion of a number of fascinating topics in cuprate superconductivity, including the low energy physics associated with nodal quasiparticles, the properties of the vortex matter which results from the application of a magnetic field, the effects of disorder, and a host of material specific issues. This paper is long enough as it is! 13.3 Our view of the phase diagram-Reprise 13.3.1 Pseudogap scales 13.3.1 Dimensional crossovers 13.3.2 The cuprates as quasi-1D superconductors 13.3.3 Inherent competition 13.4 Some open questions 13.4.1 Are stripes universal in the cuprate superconductors? 13.4.2 Are stripes an unimportant low temperature complication? 13.4.3 Are the length and time scales reasonable? 13.4.4 Are stripes conducting or insulating? 13.4.5 Are stripes good or bad for superconductivity? 13.4.6 Do stripes produce pairing? 13.4.7 Do stripes really make the electronic structure quasi-1D? 13.4.8 What about overdoping? 13.4.9 How large is the regime of substantial fluctuation superconductivity? 13.4.10 What about phonons? 13.4.11 What are the effects of quenched disorder? List of Symbols 6 Quasi-1D Physics in a Dynamical Stripe Array As mentioned before, in the simplest microscopic realizations of the 1DEG Competition between CDW and SS is key in quasi-1D systems.with repulsive interactions, 0 < K c < 1 and hence the CDW susceptibility is the most divergent as T → 0 (See Eq. (32).) This seemingly implies that the typical fate of a quasi-one dimensional system with a spin gap is to wind up a CDW insulator in which CDW modulations on neighboring chains phase

The one-dimensional electron gas exhibits spin-charge separation and power-law spectral responses to many experimentally relevant probes. Ordering in a quasi-one-dimensional system is necessarily associated with a dimensional crossover, at which sharp quasiparticle peaks, with small spectral weight, emerge from the incoherent background. Using methods of Abelian bosonization, we derive asymptotically correct expressions for the spectral changes induced by this crossover. Comparison is made with experiments on the high-temperature superconductors, which are electronically quasi-one-dimensional on a local scale.

We calculate within a mean-field theory the spectral signatures of various striped d-wave superconducting phases. We consider both in-phase and anti-phase modulations of the superconducting order across a stripe boundary, and the effects of coexisting inhomogeneous orders, including spin stripes, charge stripes, and modulated d-density-wave. We find that the anti-phase modulated d-wave superconductor exhibits zero-energy spectral weight, primarily along extended arcs in momentum space. Concomitantly, a Fermi surface appears and typically includes both open segments and closed pockets. When weak homogeneous superconductivity is also present the Fermi surface collapses onto nodal points. Among them are the nodal points of the homogeneous d-wave superconductor, but others typically exist at positions which depend on the details of the modulation and the band structure. Upon increasing the amplitude of the constant component these additional points move towards the edges of the reduced Brillouin zone where they eventually disappear. The above signatures are also manifested in the density of states of the clean, and the disordered system. While the presence of coexisting orders changes some details of the spectral function, we find that the evolution of the Fermi-surface and the distribution of the low-energy spectral weight are largely unaffected by them.

The superconducting transition temperature T(c) of bilayers comprising underdoped La(2-x)Sr(x)CuO(4) films capped by a thin heavily overdoped metallic La(1.65)Sr(0.35)CuO(4) layer, is found to increase with respect to T(c) of the bare underdoped films. The highest T(c) is achieved for x=0.12, close to the "anomalous" 1/8 doping level, and exceeds that of the optimally doped bare film. Our data suggest that the enhanced superconductivity is confined to the interface between the layers. We attribute the effect to a combination of the high pairing scale in the underdoped layer with an enhanced phase stiffness induced by the overdoped film.

PrefaceIt is the purpose of this paper to explore the theory of high temperature superconductivity. Much of the motivation for this comes from the study of cuprate high temperature superconductors. However, we do not focus in great detail on the remarkable and exciting physics that has been discovered in these materials. Rather, we focus on the core theoretical issues associated with the mechanism of high temperature superconductivity. Although our discussions of theoretical issues in a strongly correlated superconductor are intended to be self contained and pedagogically complete, our discussions of experiments in the cuprates are, unfortunately, considerably more truncated and impressionistic.Our primary focus is on physics at intermediate temperature scales of order T c (as well as the somewhat larger "pseudogap" temperature) and energies of order the gap maximum, ∆ 0 . Consequently (and reluctantly) we have omitted any detailed discussion of a number of fascinating topics in cuprate superconductivity, including the low energy physics associated with nodal quasiparticles, the properties of the vortex matter which results from the application of a magnetic field, the effects of disorder, and a host of material specific issues. This paper is long enough as it is!

Apparently, some form of local superconducting pairing persists to temperatures well above the maximum observed Tc in underdoped cuprates, i.e. Tc is suppressed due to the small phase stiffness. With this in mind, we consider the following question -Given a system with a high pairing scale ∆0 but with Tc reduced by phase fluctuations, can one design a composite system in which Tc approaches its mean-field value, Tc → TMF ≈ ∆0/2 ? Here, we study a simple two component model in which a "metallic layer" with ∆0 = 0 is coupled by single-particle tunneling to a "pairing layer" with ∆0 > 0 but zero phase stiffness. We show that in the limit that the bandwidth of the metal is much larger than ∆0, Tc of the composite system can reach the upper limit Tc ≈ ∆0/2.

In the normal state of the high temperature superconductors Bi(2)Sr(2)CaCu(2)O(8+delta) and La2(-x)Sr(x)CuO4, and in the related "stripe ordered" material, La(1.25)Nd(0.6)Sr(0.15)CuO4, there is sharp structure in the measured single hole spectral function, A<(k-->,omega), considered as a function of k--> at fixed small binding energy omega. At the same time, as a function of omega at fixed k--> on much of the putative Fermi surface, any structure in A<(k-->,omega), other than the Fermi cutoff, is very broad. This is characteristic of the situation in which there are no stable excitations with the quantum numbers of the electron, as is the case in the one-dimensional electron gas.

We study the statistics of quasiparticle and quasihole levels in small interacting disordered systems within the Hartree-Fock approximation. The distribution of the inverse compressibility, given according to Koopmans' theorem by the distance between the two levels across the Fermi energy, evolves from a Wigner distribution in the non-interacting limit to a shifted Gaussian for strong interactions. On the other hand the nature of the distribution of spacings between neighboring levels on the same side of the Fermi energy (corresponding to energy differences between excited states of the system with one missing or one extra electron) is not affected by the interaction and follows Wigner-Dyson statistics. These results are derived analytically by isolating and solving the appropriate Hartree-Fock equations for the two levels. They are substantiated by numerical simulations of the full set of Hartree-Fock equations for a disordered quantum dot with Coulomb interactions. We find enhanced fluctuations of the inverse compressibility compared to the prediction of Random Matrix Theory, possibly due to localization of the wavefunctions around the edge of the dot. The distribution of the inverse compressibility calculated from the discrete second derivative with respect to the number of particles of the Hartree-Fock ground state energy deviates from the distribution of the level spacing across the Fermi energy. The two distributions have similar shapes but are shifted with respect to each other. The deviation increases with the strength of the interaction thus indicating the breakdown of Koopmans' theorem in the strongly interacting limit.