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 paper presents the results of a functional-integral approach to the dynamics of a two-state system coupled to a dissipative environment. It is primarily an extended account of results obtained over the last four years by the authors; while they try to provide some background for orientation, it is emphatically not intended as a comprehensive review of the literature on the subject. Its contents include (1) an exact and general prescription for the reduction, under appropriate circumstances, of the problem of a system tunneling between two wells in the presence of a dissipative environment to the "spin-boson" problem; (2) the derivation of an exact formula for the dynamics of the latter problem; (3) the demonstration that there exists a simple approximation to this exact formula which is controlled, in the sense that we can put explicit bounds on the errors incurred in it, and that for almost all regions of the parameter space these errors are either very small in the limit of interest to us (the "slow-tunneling" limit) or can themselves be evaluated with satisfactory accuracy; (4) use of these results to obtain quantitative expressions for the dynamics of the system as a function of the spectral density $J(\ensuremath{\omega})$ of its coupling to the environment. If $J(\ensuremath{\omega})$ behaves as ${\ensuremath{\omega}}^{s}$ for frequencies of the order of the tunneling frequency or smaller, the authors find for the "unbiased" case the following results: For $sl1$ the system is localized at zero temperature, and at finite $T$ relaxes incoherently at a rate proportional to $\mathrm{exp}\ensuremath{-}{(\frac{{T}_{0}}{T})}^{1\ensuremath{-}s}$. For $sg2$ it undergoes underdamped coherent oscillations for all relevant temperatures, while for $1lsl2$ there is a crossover from coherent oscillation to overdamped relaxation as $T$ increases. Exact expressions for the oscillation and/or relaxation rates are presented in all these cases. For the "ohmic" case, $s=1$, the qualitative nature of the behavior depends critically on the dimensionless coupling strength $\ensuremath{\alpha}$ as well as the temperature $T$: over most of the ($\ensuremath{\alpha}$,$T$) plane (including the whole region $\ensuremath{\alpha}g1$) the behavior is an incoherent relaxation at a rate proportional to ${T}^{2\ensuremath{\alpha}\ensuremath{-}1}$, but for low $T$ and $0l\ensuremath{\alpha}l\frac{1}{2}$ the authors predict a combination of damped coherent oscillation and incoherent background which appears to disagree with the results of all previous approximations. The case of finite bias is also discussed.

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Dynamics of the dissipative two-state system

A canonical transformation is used to relate the Anderson model of a localized magnetic moment in a dilute alloy to that of Kondo. In the limit of small $s\ensuremath{-}d$ mixing, which is the most favorable case for the occurrence of a moment, the two models are shown to be equivalent. The Anderson model thus has low-temperature anomalies similar to those previously discussed for the Kondo model.

Relation between the Anderson and Kondo Hamiltonians

This article reviews recent developments in the theoretical understanding and the numerical implementation of variational renormalization group methods using matrix product states and projected entangled pair states.

http://univie.ac.at/qfp/publications3/pdffiles/0907.2796v1_verstraete.pdf

Matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems

This review discusses instabilities of the Fermi-liquid state of conduction electrons in metals with particular emphasis on magnetic quantum critical points. Both the existing theoretical concepts and experimental data on selected materials are presented; with the aim of assessing the validity of presently available theory. After briefly recalling the fundamentals of Fermi-liquid theory, the local Fermi-liquid state in quantum impurity models and their lattice versions is described. Next, the scaling concepts applicable to quantum phase transitions are presented. The Hertz-Millis-Moriya theory of quantum phase transitions is described in detail. The breakdown of the latter is analyzed in several examples. In the final part experimental data on heavy-fermion materials and transition-metal alloys are reviewed and confronted with existing theory.

http://export.arxiv.org/pdf/cond-mat/0606317

Fermi-liquid instabilities at magnetic quantum phase transitions

Based on the s-d interaction model for dilute magnetic alloys we have calculated the scattering probability of the conduction electrons to the second Born approximatism. Because of the dynamical character of the localized spin system, the Pauli principle should be taken into account in the intermediate states of the second order terms. Thus the effect of the Fermi sphere is involved in the scattering probability and gives rise to a singular term in the resistivity which involves clog T as a factor, where c is the concentration of impurity atoms. When combin:::d with the lattice resistivity, this gives rise to a resistance min~mum, provided the s-d exchan:~e integral J is negative. The temperature at which the minimum cccurs is proportional to c 15 and the depth of the minimum to c, as is observed. The predicted log T dependence is tested with available experiments and is confirmed. The value of J to have fit with experimmts is about -0.2 ev, which is of reasonable magnitude. Our conclusion is that J should be negative in alloys which show a resistance minimum. It is argued that the resistance minimum is a result of the sharp Fermi surface.

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Resistance Minimum in Dilute Magnetic Alloys

The one-dimensional Heisenberg model with S= 1/2 is treated with the use of the two­ time Green's functions. The hierarchy of the equations of motion of the Green's functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, <Sz), is set to zero, because the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn=4<S 0zSnz). The Green's function is essentially of the form representing undamped spin waves, whose spectrum depends on ct, c2 and one more parameter. They are determined by the requirement that c 1 and c2 should be self-consistent and that co should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numel:ically in the whole range of the temperature. Thermodynamic quantities have been calculated using these ·solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by en= (Jj4kBT)n. In the case of ferromagnetic coupling, the correlation function en at T=O is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T=O also agrees with the correct one. At T4:;JfkB, we find that the specific heat goes as T 112 and the suscepti­ bility goes as T-2• The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find cl=-0.55407 and c2=0.16100 at T=O, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.

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Green's-Function Formalism of the One-Dimensional Heisenberg Spin System

The intensity of the X-ray diffuse scattering from an electron-phonon system is calculated correct up to the second order of the electron-phonon interaction. The result consists of two terms: one is described as arising from softening of the phonon frequency due to the interaction, and the other as the shift of the center of the phonon coordinate. The former dominates when the electronic motion is faster than the phonon motion, and vice versa. It is argued that for a strongly Coulomb-correlated electron system, such as TTF-TCNQ, the electronic motion is slow enough to make the latter contribution important. A simple model for TTF-TCNQ is presented, which neglects the electron transfer entirely and for which the latter term is expressed in terms of the density correlation function of the electron system. This is calculated by applying the Monte Carlo method to a finite linear chain and also by using the hypernetted chain equation. We find a peak in the correlation function at q =4 k F , which explains the ...

Density Correlation of Classical 1-d Electron Gas with Reference to the 4kFAnomaly in TTF-TCNQ

Fermi Surface Effects

Available for the first time in English, this classic text by Jun Kondo describes the Kondo effect thoroughly and intuitively Its clear and concise treatment makes this book of interest to graduate students and researchers in condensed matter physics The first half of the book describes the rudiments of the theory of metals at a level that is accessible for undergraduate students The second half discusses key developments in the Kondo problem, covering topics including magnetic impurities in metals, the resistance minimum phenomenon, infrared divergence in metals and scaling theory, including Wilson's renormalization group treatment and the exact solution by the Bethe ansatz A new chapter has been added covering advances made since the Japanese edition was published, such as the quantum dot and heavy fermion systems

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Jun Kondo

Papers

Resistance Minimum in Dilute Magnetic Alloys

Green's-Function Formalism of the One-Dimensional Heisenberg Spin System

Density Correlation of Classical 1-d Electron Gas with Reference to the 4kFAnomaly in TTF-TCNQ

Fermi Surface Effects

The Physics of Dilute Magnetic Alloys