We review the dynamical mean-field theory of strongly correlated electron systems which is based on a mapping of lattice models onto quantum impurity models subject to a self-consistency condition. This mapping is exact for models of correlated electrons in the limit of large lattice coordination (or infinite spatial dimensions). It extends the standard mean-field construction from classical statistical mechanics to quantum problems. We discuss the physical ideas underlying this theory and its mathematical derivation. Various analytic and numerical techniques that have been developed recently in order to analyze and solve the dynamical mean-field equations are reviewed and compared to each other. The method can be used for the determination of phase diagrams (by comparing the stability of various types of long-range order), and the calculation of thermodynamic properties, one-particle Green's functions, and response functions. We review in detail the recent progress in understanding the Hubbard model and the Mott metal-insulator transition within this approach, including some comparison to experiments on three-dimensional transition-metal oxides. We present an overview of the rapidly developing field of applications of this method to other systems. The present limitations of the approach, and possible extensions of the formalism are finally discussed. Computer programs for the numerical implementation of this method are also provided with this article.

Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions

The calculation of rate coefficients is a discipline of nonlinear science of importance to much of physics, chemistry, engineering, and biology. Fifty years after Kramers' seminal paper on thermally activated barrier crossing, the authors report, extend, and interpret much of our current understanding relating to theories of noise-activated escape, for which many of the notable contributions are originating from the communities both of physics and of physical chemistry. Theoretical as well as numerical approaches are discussed for single- and many-dimensional metastable systems (including fields) in gases and condensed phases. The role of many-dimensional transition-state theory is contrasted with Kramers' reaction-rate theory for moderate-to-strong friction; the authors emphasize the physical situation and the close connection between unimolecular rate theory and Kramers' work for weakly damped systems. The rate theory accounting for memory friction is presented, together with a unifying theoretical approach which covers the whole regime of weak-to-moderate-to-strong friction on the same basis (turnover theory). The peculiarities of noise-activated escape in a variety of physically different metastable potential configurations is elucidated in terms of the mean-first-passage-time technique. Moreover, the role and the complexity of escape in driven systems exhibiting possibly multiple, metastable stationary nonequilibrium states is identified. At lower temperatures, quantum tunneling effects start to dominate the rate mechanism. The early quantum approaches as well as the latest quantum versions of Kramers' theory are discussed, thereby providing a description of dissipative escape events at all temperatures. In addition, an attempt is made to discuss prominent experimental work as it relates to Kramers' reaction-rate theory and to indicate the most important areas for future research in theory and experiment.

Reaction-rate theory: fifty years after Kramers

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.

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Doping a Mott insulator: Physics of high-temperature superconductivity

The appealing electronic properties of the monolayer semiconductor molybdenum disulphide make it a candidate material for electronic devices. The observation of tightly bound trions in this system—which have no analogue in conventional semiconductors—opens up possibilities for controlling these quasiparticles in future optoelectronic applications.

Tightly bound trions in monolayer MoS2

This article reviews the effort to understand the physics of high temperature superconductors from the point of view of doping a Mott insulator. The basic electronic structure of the 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. We introduce Anderson's idea of the resonating valence bond (RVB) and argue that it gives a qualitative account of the data. The importance of phase fluctuation is discussed, leading to a theory of the transition temperature which is driven by phase fluctuation and thermal excitation of quasiparticles. We then describe the numerical method of projected wavefunction which turns out to be a very useful technique to implement 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-J model, with the goal of putting the RVB idea on a more formal footing. The slave-boson is introduced to enforce the constraint of no double occupation. The implementation of the local constraint leads naturally to gauge theories. We give a rather thorough discussion of the role of gauge theory in describing the spin liquid phase of the undoped Mott insulator. We next describe the extension of the SU(2) formulation to nonzero doping. We show that inclusion of gauge fluctuation provides a reasonable description of the pseudogap phase.

Doping a Mott Insulator: Physics of High Temperature Superconductivity

We propose a new functional integral representation of the Hubbarda and Anderson models of lattice fermions. The simplest saddle-point approximation leads, at zero temperature, to the results derived from the Gutzwiller variational wave function. This approach uncovers the limitations of the Gutzwiller approximation and clarifies its connection to the "auxiliary-boson" mean-field theory of the Anderson model. This formulation leads to a novel strong-coupling mean-field theory which allows for a unified treatment of antiferromagnetism and ferromagnetism, metal-to-insulator transition, and Kondo compensation effects.

New functional integral approach to strongly correlated Fermi systems: The Gutzwiller approximation as a saddle point.

By use of an effective Hamiltonian which takes into account the constraints on the motion of a hole in a quantum antiferromagnet, the spectral function of the hole is calculated. For small exchange and away from the antiferromagnetic zone boundary, it is found to be dominated by incoherent multiplespin-wave processes. The dispersion of the quasiparticle part and the possible implications for disordering of the quantum antiferromagnet are discussed.

Spectral Function of Holes in a Quantum Antiferromagnet

We develop the simplest mean-field theory of an extended Hubbard model in the limit of a large intrasite Coulomb interaction, concentrating on the possibility of superconductivity induced by the superexchange interaction and weakened by the intersite Coulomb repulsion. We calculate the critical temperature and the coherence length as a function of filling, as well as the temperature dependence of the magnetic susceptibility and specific heat. Finally, we comment on the physics of the insulating state at half filling, and mention the probable effects of fluctuations.

Mean-field theory of high-T c superconductivity: The superexchange mechanism

We discuss the many-body aspects of the optical spectra of doped semiconductors, for both bulk and low-dimensional systems. At low doping concentrations the spectra are dominated by excitonic effects, consistent with the single-particle band structure of these systems. With increasing doping atomic excitons lose their identity and eventually unbind, while the spectral weight moves continuously to the Fermi level. Depending on the electron-mass to hole-mass ratio, the spectra display a broadened singularity as well as Auger-like indirect transitions. We calculate the onset of absorption in this regime to second order in the effective interaction. We stress that a calculation of the spectra near the Fermi level is outside the scope of conventional perturbation theory and we comment on the validity of various previously proposed calculational schemes. We conclude by discussing the problem in terms of Fadeev-like equations, which describe up to three-particle correlations exactly.

Many-body aspects of the optical spectra of bulk and low-dimensional doped semiconductors

We investigate a mean-field theory of the Hubbard model in the large on-site correlation limit, allowing for the possibility of a commensurate spin-density wave together with s- and d-wave-like superconductivity. As a function of t/U and band filling, the resulting phase diagram exhibits phases which are simultaneously superconducting and antiferromagnetic. We show that the superconducting order parameter goes to zero in the limit of a 1/2-filled Hubbard band. We comment on the possible relevance of these results to the newly discovered high-${T}_{c}$ superconductors.

Andrei E. Ruckenstein

Papers

New functional integral approach to strongly correlated Fermi systems: The Gutzwiller approximation as a saddle point.

Spectral Function of Holes in a Quantum Antiferromagnet

Mean-field theory of high-T c superconductivity: The superexchange mechanism

Many-body aspects of the optical spectra of bulk and low-dimensional doped semiconductors

Coexistence of antiferromagnetism and superconductivity in a mean-field theory of high-T c superconductors