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

In narrow-band systems electrons can interact with each other via a short-range nonretarded attractive potential. The origin of such an effective local attraction can be polaronic or it can be due to a coupling between electrons and excitons or plasmons. It can also result from purely chemical (electronic) mechanisms, especially in compounds with elements favoring disproportionation of valent states. These mechanisms are discussed and an exhaustive list of materials in which such local electron pairing occurs is given. The authors review the thermodynamic and electromagnetic properties of such systems in several limiting scenarios: (i) Systems with on-site pairing which can be described by the extended negative-$U$ Hubbard model. The strong-attraction limit of this model, at which it reduces to a system of tightly bound electron pairs (bipolarons) on a lattice, is extensively discussed. These electron pairs behaving as hard-core charged bosons can exhibit a superconducting state analogous to that of superfluid $^{4}\mathrm{He}$ II. The change-over from weak-attraction BCS-like superconductivity to the superfluidity of charged hard-core bosons is examined. (ii) Systems with intersite pairing described by an extended Hubbard model with $Ug0$ and nearest-neighbor attraction and/or nearest-neighbor spin exchange as well as correlated hopping. (iii) A mixture of local pairs and itinerant electrons interacting via a charge-exchange mechanism giving rise to a mutually induced superconductivity in both subsystems. The authors discuss to what extent the picture of local pairing, and in particular superfluidity of hard-core charged bosons on a lattice, can be an explanation for the superconducting and normal-state properties of the high-${T}_{c}$ oxides: doped BaBi${\mathrm{O}}_{3}$ and the cuprates.

Superconductivity in narrow-band systems with local nonretarded attractive interactions

It is remarkable how the Frohlich polaron, one of the simplest examples of a Quantum Field Theoretical problem, as it basically consists of a single fermion interacting with a scalar Bose field of ion displacements, has resisted full analytical or numerical solution at all coupling since ~1950, when its Hamiltonian was first written. The field has been a testing ground for analytical, semi-analytical and numerical techniques, such as path integrals, strong-coupling perturbation expansion, advanced variational, exact diagonalization (ED) and quantum Monte Carlo (QMC) techniques. This paper reviews recent developments in the field of continuum and discrete (lattice) Frohlich (bi)polarons starting with the basics and covering a number of active directions of research.

https://iopscience.iop.org/article/10.1088/0034-4885/72/6/066501/pdf

Fröhlich polaron and bipolaron: recent developments

It is remarkable how the Froehlich polaron, one of the simplest examples of a Quantum Field Theoretical problem, as it basically consists of a single fermion interacting with a scalar Bose field of ion displacements, has resisted full analytical or numerical solution at all coupling since 1950, when its Hamiltonian was first written. The field has been a testing ground for analytical, semi-analytical, and numerical techniques, such as path integrals, strong-coupling perturbation expansion, advanced variational, exact diagonalisation (ED), and quantum Monte Carlo (QMC) techniques. This article reviews recent developments in the field of continuum and discrete (lattice) Froehlich (bi)polarons starting with the basics and covering a number of active directions of research.

Froehlich Polaron and Bipolaron: Recent Developments

One‐Dimensional Inorganic Complexes

: The authors consider a narrow one dimensional tight binding chain with intra-atomic Coulomb repulsions described in the Hubbard model. The electrons at each site are furthermore assumed coupled to a high frequency local oscillator. In the limit, where the oscillator frequency is large compared to the bandwidth and in the extreme band narrowing regime, for various electron densities, the authors found that: half-filled band ground state is an antiferromagnetic insulator; one-third filled band is a diamagnetic insulator; one-quarter filled band is a weakly coupled system of bound singlet pairs somewhat reminiscent of superconductivity. (Modified author abstract)

Low-temperature properties of the one-dimensional polaron band. I. Extreme-band-narrowing regime

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Low-temperature properties of the one-dimensional polaron band. I. Extreme-band-narrowing regime