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

Controlling chaos

Preface. Part I: Gathering the Tools. 1. Introduction to Quantum Physics. 2. Quantum Tests. 3. Complex Vector Space. 4. Continuous Variables. Part II: Cryptodeterminism and Quantum Inseparability. 5. Composite Systems. 6. Bell's Theorem. 7. Contextuality. Part III: Quantum Dynamics and Information. 8. Spacetime Symmetries. 9. Information and Thermodynamics. 10. Semiclassical Methods. 11. Chaos and Irreversibility. 12. The Measuring Process. Author Index. Subject Index.

Quantum Theory: Concepts and Methods

We show that a bounded, isolated quantum system of many particles in a specific initial state will approach thermal equilibrium if the energy eigenfunctions which are superposed to form that state obey Berry's conjecture. Berry's conjecture is expected to hold only if the corresponding classical system is chaotic, and essentially states that the energy eigenfunctions behave as if they were Gaussian random variables. We review the existing evidence, and show that previously neglected effects substantially strengthen the case for Berry's conjecture. We study a rarefied hard-sphere gas as an explicit example of a many-body system which is known to be classically chaotic, and show that an energy eigenstate which obeys Berry's conjecture predicts a Maxwell-Boltzmann, Bose-Einstein, or Fermi-Dirac distribution for the momentum of each constituent particle, depending on whether the wave functions are taken to be nonsymmetric, completely symmetric, or completely antisymmetric functions of the positions of the particles. We call this phenomenon eigenstate thermalization. We show that a generic initial state will approach thermal equilibrium at least as fast as O(\ensuremath{\Elzxh}/\ensuremath{\Delta})${\mathit{t}}^{\mathrm{\ensuremath{-}}1}$, where \ensuremath{\Delta} is the uncertainty in the total energy of the gas. This result holds for an individual initial state; in contrast to the classical theory, no averaging over an ensemble of initial states is needed. We argue that these results constitute a sound foundation for quantum statistical mechanics.

https://arxiv.org/pdf/cond-mat/9403051.pdf

Chaos and quantum thermalization

Contents: Introduction- The Mechanics of Lagrange- The Mechanics of Hamilton and Jacobi- Integrable Systems- The Three-Body Problem: Moon-Earth-Sun- Three Methods of Section- Periodic Orbits- The Surface of Solution- Models of the Galaxy and of Small Molecules- Soft Chaos and the KAM Theorem- Entropy and Other Measures of Chaos- The Anisotropic Kepler Problem- The Transition From Classical to Quantum Mechanics- The New World of Quantum Mechanics- The Quantization of Integrable Systems- Wave Functions in Classically Chaotic Systems- The Energy Spectrum of a Classically Chaotic System- The Trace Formula- The Diamagnetic Kepler Problem- Motion on a Surface of Constant Negative Curvature- Scattering Problems, Coding and Multifractal Invariant Measures- References- Index

Chaos in classical and quantum mechanics

The relation between the solutions of the time‐independent Schrodinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability angle times ℏ. Unstable periodic orbits give a series of broadened peaks whose half‐width equals the stability exponent times ℏ, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well‐defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropicgermanium.

Periodic orbits and classical quantization conditions

The wave function for the electrons is investigated when a set of narrow bands (valence states) has its energies within a wide band (conduction states). The valence states are linear combinations of localized states which are attached to each lattice site. The intra-atomic Coulomb and exchange integrals for the localized states are much larger than the bandwidths of the valence states. Some of the narrow bands are neither completely empty nor completely filled. The wave function is therefore expected to be correlated, because it is disadvantageous for the electrons to crowd into the same lattice site, or take up some configuration contrary to Hund's rule. This correlation is important in trasition metals, where it is considered to be the cause of ferromagnetism. The correlated wave function is obtained by applying to the uncorrelated antisymmetrized product of Bloch functions an operator which provides each configuration of localized valence states with an appropriate amplitude and phase factor. The procedure is worked out in detail for the case of few particles (electrons or holes) in the narrow bands with the help of a diagram analysis. The localized orbits of different lattice sites do not have to be orthogonal to on another, and the computational rules are actually simplified thereby. The example of a twofold degenerate band such as the upper part of the $3d$ band in Ni is treated, and the conditions for the occurrence of ferromagnetism are stated in the case of few $3d$ holes per lattice site.

Effect of Correlation on the Ferromagnetism of Transition Metals

The ground-state wave function for the electrons in a narrow $s$ band is investigated for arbitrary density of electrons and arbitrary strength of interaction. An approximation is proposed which limits all the calculations to counting certain types of configurations and attaching the proper weights. The expectation values of the one-particle and two-particle density matrix are computed for the ferromagnetic and for the non-ferromagnetic case. The ground-state energy is obtained under the assumption that only the intra-atomic Coulomb interaction is of importance. Ferromagnetism is found to occur if the density of states is large at the band edges rather than in the center, and if the intra-atomic Coulomb repulsion is sufficiently strong. The relation of this approximation to certain exact results for one-dimensional models is discussed.

Correlation of Electrons in a Narrow s Band

The phase integral approximation of the Green's function in momentum space is investigated for an electron of negative energy (corresponding to a bound state) which moves in a spherically symmetric potential. If the propagator rather than the wavefunction is considered, all classical orbits enter into the formulas, rather than only the ones which satisfy certain quantum conditions, and the separation of variables can be avoided. The distinction between classically accessible and classically inaccessible regions does not arise in momentum space, because any two momenta can be connected by a classical trajectory of given negative energy for a typical atomic potential. Three approaches are discussed: the Fourier transform of the phase integral approximation in coordinate space, the approximate solution of Schrodinger's equation in momentum space by a WKB ansatz, and taking the limit of small Planck's quantum in the Feynman‐type functional integral which was recently proposed by Garrod for the energy‐momentum representation. In particular, the last procedure is used to obtain the phase jumps of π/2 which occur every time neighboring classical trajectories cross one another. These extra phase factors are directly related to the signature of the second variation for the action function, and provide a physical application of Morse's calculus of variation in the large. The phase integral approximation in momentum space is then applied to the Coulomb potential. The location of the poles on the negative energy axis gives the Bohr formula for the bound‐state energies, and the residues of the approximate Green's function are shown to yield all the exact wavefunctions for the bound states of the hydrogen atom.

Martin C. Gutzwiller

Papers

Chaos in classical and quantum mechanics

Periodic orbits and classical quantization conditions

Effect of Correlation on the Ferromagnetism of Transition Metals

Correlation of Electrons in a Narrow s Band

Phase-Integral Approximation in Momentum Space and the Bound States of an Atom