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

The density-matrix renormalization group (DMRG) is a numerical algorithm for the efficient truncation of the Hilbert space of low-dimensional strongly correlated quantum systems based on a rather general decimation prescription. This algorithm has achieved unprecedented precision in the description of one-dimensional quantum systems. It has therefore quickly become the method of choice for numerical studies of such systems. Its applications to the calculation of static, dynamic, and thermodynamic quantities in these systems are reviewed here. The potential of DMRG applications in the fields of two-dimensional quantum systems, quantum chemistry, three-dimensional small grains, nuclear physics, equilibrium and nonequilibrium statistical physics, and time-dependent phenomena is also discussed. This review additionally considers the theoretical foundations of the method, examining its relationship to matrix-product states and the quantum information content of the density matrices generated by the DMRG.

http://www.physics.udel.edu/~bnikolic/QTTG/NOTES/DMRG/SCHOLLWOCK=RMP_dmrg.pdf

The density-matrix renormalization group

To make the transition from the quasi-long-range order in a chain of antiferromagnetically coupled S = 1/2 spins to the true long-range order that occurs in a plane, one can assemble chains to make ladders of increasing width. Surprisingly, this crossover between one and two dimensions is not at all smooth. Ladders with an even number of legs have purely short-range magnetic order and a finite energy gap to all magnetic excitations. Predictions of this ground state have now been verified experimentally. Holes doped into these ladders are predicted to pair and possibly superconduct.

https://science.sciencemag.org/content/sci/271/5249/618.full.pdf

Surprises on the Way from One- to Two-Dimensional Quantum Magnets: The Ladder Materials

The physics of one-dimensional interacting bosonic systems is reviewed. Beginning with results from exactly solvable models and computational approaches, the concept of bosonic Tomonaga-Luttinger liquids relevant for one-dimensional Bose fluids is introduced, and compared with Bose-Einstein condensates existing in dimensions higher than one. The effects of various perturbations on the Tomonaga-Luttinger liquid state are discussed as well as extensions to multicomponent and out of equilibrium situations. Finally, the experimental systems that can be described in terms of models of interacting bosons in one dimension are discussed.

/pdf/one-dimensional-bosons-from-condensed-matter-systems-to-4vp39jxn82.pdf

One dimensional bosons: From condensed matter systems to ultracold gases

Recently it has been shown that many traditional models used for a description of dilute magnetic alloys are completely integrable and may be solved exactly without any approximation. In this article we summarize the results which have been obtained in this way. The main part of the article is devoted to the consistent and detailed account of the Bethe-Ansatz technique for the s-d exchange (Kondo) model with arbitrary impurity spin, s-d model with anisotropic exchange, degenerate exchange model and for the canonical Anderson model. The thermodynamic properties of a magnetic impurity in a non-magnetic metal host obtained by the Bethe method are considered in detail. Mainly attention is paid to the analysis of singularities associated with the formation of a localized moment, Kondo effect and mixedvalence phenomenon, which can be treated analytically. In the introductory part of the article we discuss the applicability of the models which we study in the paper to the real alloys and consider some m...

Exact results in the theory of magnetic alloys

The ground state of the alternating spin 1/2 Heisenberg chain with couplings J(> 0) and J′ is studied. This model interpolates the S=1 and S=1/2 antiferromagnetic Heisenberg chains continuously. The behavior of the string order and energy gap indicates that the ground state over the whole range −∞ ≤ J′< J can be regarded as a single phase. The physical picture of the ground state and the first excited state is also discussed.

Crossover between the Haldane-gap phase and the dimer phase in the spin-1/2 alternating Heisenberg chain.

One-dimensional $S=1$ $\mathrm{XXZ}$ chains with uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite-size systems. The numerical data are analyzed using conformal field theory, the level spectroscopy, phenomenological renormalization-group, and finite-size scaling method. We thus present a quantitatively reliable ground-state phase diagram of this model. The ground states of this model contain the Haldane phase, large-D phase, N\'eel phase, two $\mathrm{XY}$ phases, and the ferromagnetic phase. There are four different types of transitions between these phases: the Brezinskii-Kosterlitz-Thouless-type transitions, the Gaussian-type transitions, the Ising-type transitions, and the first-order transitions. The locations of these critical lines are accurately determined.

https://arxiv.org/pdf/cond-mat/0209403.pdf

Ground-state phase diagram of S = 1 XXZ chains with uniaxial single-ion-type anisotropy

The magnetic properties of the ferromagnetic-ferromagnetic-antiferromagnetic trimerized spin-1/2 Heisenberg chain are studied theoretically. The high temperature susceptibilty and the ground state saturation magnetic field are calculated and the exchange energies of the trimer compound 3CuCl 2 ·2dx are determined. The magnetization curve is obtained by numerical diagonalization of finite size systems. The result explains the low temperature magnetization data for 3CuCl 2 ·2dx with the exchange energies obtained as above. It is predicted that the magnetization curve has a plateau at 1/3 of the saturation magnetization if the ferromagnetic exchange energy is comparable to or smaller than the antiferromagnetic exchange energy.

Magnetic Properties of the Spin-1/2 Ferromagnetic-Ferromagnetic-Antiferromagnetic Trimerized Heisenberg Chain

The magnetization process of the S =1 and 1/2 kagome Heisenberg antiferromagnet is studied by means of the numerical exact diagonalization method. It is found that the magnetization curve at zero temperature has a plateau at 1/3 of the full magnetization. In the presence of $\sqrt{3} \times \sqrt{3}$ lattice distortion, this plateau is enhanced and eventually the ferrimagnetic state is realized. There also appear the minor plateaux above the main plateau. The physical origin of these phenomena is discussed.

Magnetization Process of the S=1 and 1/2 Uniform and Distorted Kagomé Heisenberg Antiferromagnets

The ground state of the ferromagnetic-antiferromagnetic alternating spin-1/2 Heisenberg chain is studied with Ising-type anisotropy on the ferromagnetic bond. This model tends to the spin-1 antiferromagnetic Heisenberg chain with on-site anisotropy in the limit of strong ferromagnetic bond. It is found that the Haldane-gap phase, the large-D phase, and the two types of N\'eel phases appear as the ground state of this model. The duality relations between these four phases are proved for several limiting cases. The string order parameter characterizing the large-D phase is introduced based on these duality relations. The ground-state phase diagram is determined by the numerical diagonalization of the finite-size system.

Kazuo Hida

Papers

Crossover between the Haldane-gap phase and the dimer phase in the spin-1/2 alternating Heisenberg chain.

Ground-state phase diagram of S = 1 XXZ chains with uniaxial single-ion-type anisotropy

Magnetic Properties of the Spin-1/2 Ferromagnetic-Ferromagnetic-Antiferromagnetic Trimerized Heisenberg Chain

Magnetization Process of the S=1 and 1/2 Uniform and Distorted Kagomé Heisenberg Antiferromagnets

Ground-state phase diagram of the spin-1/2 ferromagnetic-antiferromagnetic alternating Heisenberg chain with anisotropy