We propose a simple analytic representation of the correlation energy ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$ for a uniform electron gas, as a function of density parameter ${\mathit{r}}_{\mathit{s}}$ and relative spin polarization \ensuremath{\zeta}. Within the random-phase approximation (RPA), this representation allows for the ${\mathit{r}}_{\mathit{s}}^{\mathrm{\ensuremath{-}}3/4}$ behavior as ${\mathit{r}}_{\mathit{s}}$\ensuremath{\rightarrow}\ensuremath{\infty}. Close agreement with numerical RPA values for ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$(${\mathit{r}}_{\mathit{s}}$,0), ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$(${\mathit{r}}_{\mathit{s}}$,1), and the spin stiffness ${\mathrm{\ensuremath{\alpha}}}_{\mathit{c}}$(${\mathit{r}}_{\mathit{s}}$)=${\mathrm{\ensuremath{\partial}}}^{2}$${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$(${\mathit{r}}_{\mathit{s}}$, \ensuremath{\zeta}=0)/\ensuremath{\delta}${\mathrm{\ensuremath{\zeta}}}^{2}$, and recovery of the correct ${\mathit{r}}_{\mathit{s}}$ln${\mathit{r}}_{\mathit{s}}$ term for ${\mathit{r}}_{\mathit{s}}$\ensuremath{\rightarrow}0, indicate the appropriateness of the chosen analytic form. Beyond RPA, different parameters for the same analytic form are found by fitting to the Green's-function Monte Carlo data of Ceperley and Alder [Phys. Rev. Lett. 45, 566 (1980)], taking into account data uncertainties that have been ignored in earlier fits by Vosko, Wilk, and Nusair (VWN) [Can. J. Phys. 58, 1200 (1980)] or by Perdew and Zunger (PZ) [Phys. Rev. B 23, 5048 (1981)]. While we confirm the practical accuracy of the VWN and PZ representations, we eliminate some minor problems with these forms. We study the \ensuremath{\zeta}-dependent coefficients in the high- and low-density expansions, and the ${\mathit{r}}_{\mathit{s}}$-dependent spin susceptibility. We also present a conjecture for the exact low-density limit. The correlation potential ${\mathrm{\ensuremath{\mu}}}_{\mathit{c}}^{\mathrm{\ensuremath{\sigma}}}$(${\mathit{r}}_{\mathit{s}}$,\ensuremath{\zeta}) is evaluated for use in self-consistent density-functional calculations.
This article reviews the physics of high-temperature superconductors from the point of view of the doping of a Mott insulator. The basic electronic structure of cuprates is reviewed, emphasizing the physics of strong correlation and establishing the model of a doped Mott insulator as a starting point. A variety of experiments are discussed, focusing on the region of the phase diagram close to the Mott insulator (the underdoped region) where the behavior is most anomalous. The normal state in this region exhibits pseudogap phenomenon. In contrast, the quasiparticles in the superconducting state are well defined and behave according to theory. This review introduces Anderson's idea of the resonating valence bond and argues that it gives a qualitative account of the data. The importance of phase fluctuations is discussed, leading to a theory of the transition temperature, which is driven by phase fluctuations and the thermal excitation of quasiparticles. However, an argument is made that phase fluctuations can only explain pseudogap phenomenology over a limited temperature range, and some additional physics is needed to explain the onset of singlet formation at very high temperatures. A description of the numerical method of the projected wave function is presented, which turns out to be a very useful technique for implementing the strong correlation constraint and leads to a number of predictions which are in agreement with experiments. The remainder of the paper deals with an analytic treatment of the $t\text{\ensuremath{-}}J$ model, with the goal of putting the resonating valence bond idea on a more formal footing. The slave boson is introduced to enforce the constraint againt double occupation and it is shown that the implementation of this local constraint leads naturally to gauge theories. This review follows the historical order by first examining the U(1) formulation of the gauge theory. Some inadequacies of this formulation for underdoping are discussed, leading to the SU(2) formulation. Here follows a rather thorough discussion of the role of gauge theory in describing the spin-liquid phase of the undoped Mott insulator. The difference between the high-energy gauge group in the formulation of the problem versus the low-energy gauge group, which is an emergent phenomenon, is emphasized. Several possible routes to deconfinement based on different emergent gauge groups are discussed, which leads to the physics of fractionalization and spin-charge separation. Next the extension of the SU(2) formulation to nonzero doping is described with a focus on a part of the mean-field phase diagram called the staggered flux liquid phase. It will be shown that inclusion of the gauge fluctuation provides a reasonable description of the pseudogap phase. It is emphasized that $d$-wave superconductivity can be considered as evolving from a stable U(1) spin liquid. These ideas are applied to the high-${T}_{c}$ cuprates, and their implications for the vortex structure and the phase diagram are discussed. A possible test of the topological structure of the pseudogap phase is described.
The fundamental physical properties of doped ${\mathrm{LaMnO}}_{3},$ generically termed ``manganites,'' and much of the underlying physics, were known more than 40 years ago. This article first reviews progress made at that time, the concept of double exchange in particular, and points out the missing elements that have led to a massive resurgence of interest in these and related materials. More recent research is then described, treating first the ground states that emerge as divalent atoms are substituted for trivalent La. A wide range of ground states appear, including ferromagnetic metals, orbital- and charge-ordered antiferromagnets, and more complex stripe and spin-glass states. Because of the interest in so-called colossal magnetoresistance that occurs in the ferromagnetic/metallic composition range, a section is devoted to reviewing the atypical properties of that phase. Next the high-temperature phase is examined, in particular, evidence for the formation of self-trapped small polarons and the importance of Jahn-Teller coupling in this process. The transitions between the high-temperature polaronic phase and the ferromagnetic and charge-ordered states are treated in a fourth section. In each section, the authors stress the competition among charge, spin, and lattice coupling and review the current state of theoretical understanding. They conclude with some comments on the impact that research on these materials has on our understanding of doped oxides and other strongly correlated electronic materials.
