To say that this is the best book on the quantum theory of fields is no praise, since to my knowledge it is the only book on this subject But it is a very good and most useful book The original was written in German and appeared in 1942 This is a translation with some minor changes A few remarks have been added, concerning meson theory and nuclear forces, also footnotes referring to modern work in this field, and finally an appendix on the symmetrization of the energy momentum tensor according to Belinfante Quantum Theory of Fields Prof Gregor Wentzel Translated from the German by Charlotte Houtermans and J M Jauch Pp ix + 224, (New York and London: Interscience Publishers, Inc, 1949) 36s

Quantum Theory of Fields

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

We give a general introduction to quantum phase transitions in strongly-correlated electron systems. These transitions which occur at zero temperature when a non-thermal parameter $g$ like pressure, chemical composition or magnetic field is tuned to a critical value are characterized by a dynamic exponent $z$ related to the energy and length scales $\Delta$ and $\xi$. Simple arguments based on an expansion to first order in the effective interaction allow to define an upper-critical dimension $D_{C}=4$ (where $D=d+z$ and $d$ is the spatial dimension) below which mean-field description is no longer valid. We emphasize the role of pertubative renormalization group (RG) approaches and self-consistent renormalized spin fluctuation (SCR-SF) theories to understand the quantum-classical crossover in the vicinity of the quantum critical point with generalization to the Kondo effect in heavy-fermion systems. Finally we quote some recent inelastic neutron scattering experiments performed on heavy-fermions which lead to unusual scaling law in $\omega /T$ for the dynamical spin susceptibility revealing critical local modes beyond the itinerant magnetism scheme and mention new attempts to describe this local quantum critical point.

https://arxiv.org/pdf/cond-mat/0102119.pdf

Quantum Phase Transitions

Quantum impurity models describe an atom or molecule embedded in a host material with which it can exchange electrons. They are basic to nanoscience as representations of quantum dots and molecular conductors and play an increasingly important role in the theory of "correlated electron" materials as auxiliary problems whose solution gives the "dynamical mean field" approximation to the self energy and local correlation functions. These applications require a method of solution which provides access to both high and low energy scales and is effective for wide classes of physically realistic models. The continuous-time quantum Monte Carlo algorithms reviewed in this article meet this challenge. We present derivations and descriptions of the algorithms in enough detail to allow other workers to write their own implementations, discuss the strengths and weaknesses of the methods, summarize the problems to which the new methods have been successfully applied and outline prospects for future applications.

Continuous-time Monte Carlo methods for quantum impurity models

The discovery of materials has often introduced new physical paradigms and enabled the development of novel devices. Two-dimensional magnetism, which is associated with strong intrinsic spin fluctuations, has long been the focus of fundamental questions in condensed matter physics regarding our understanding and control of new phases. Here we discuss magnetic van der Waals materials: two-dimensional atomic crystals that contain magnetic elements and thus exhibit intrinsic magnetic properties. These cleavable materials provide the ideal platform for exploring magnetism in the two-dimensional limit, where new physical phenomena are expected, and represent a substantial shift in our ability to control and investigate nanoscale phases. We present the theoretical background and motivation for investigating this class of crystals, describe the material landscape and the current experimental status of measurement techniques as well as devices, and discuss promising future directions for the study of magnetic van der Waals materials.

Magnetism in two-dimensional van der Waals materials.

We develop a diagrammatic approach with local and nonlocal self-energy diagrams, constructed from the local irreducible vertex. This approach includes the local correlations of dynamical mean-field theory and long-range correlations beyond. It allows us, for example, to describe (para)magnons and weak localization effects---in strongly correlated systems. As a first application, we study the interplay between nonlocal antiferromagnetic correlations and the strong local correlations emerging in the vicinity of a Mott-Hubbard transition.

/pdf/dynamical-vertex-approximation-a-step-beyond-dynamical-mean-31lp43o57g.pdf

Dynamical vertex approximation : A step beyond dynamical mean-field theory

We consider the fulfillment of conservation laws and Ward identities in the one- and two-loop functional renormalization group approach. It is shown that in a one-particle irreducible scheme of this approach Ward identities are fulfilled only with the accuracy of the neglected two-loop terms $O({V}_{\ensuremath{\Lambda}}^{3})$ at one-loop order, and with the accuracy $O({V}_{\ensuremath{\Lambda}}^{4})$ at two-loop order (${V}_{\ensuremath{\Lambda}}$ is the effective interaction vertex at scale $\ensuremath{\Lambda}$). The one-particle self-consistent version of the two-loop renormalization group equations which leads to smaller errors in Ward identities due to the absence of the terms with nonoverlapping loops, is proposed. In particular, these modified equations exactly satisfy Ward identities in the ladder approximation.

Fulfillment of Ward identities in the functional renormalization group approach

We use the dynamical vertex approximation $(\text{D}\ensuremath{\Gamma}\text{A})$ with a Moriyaesque $\ensuremath{\lambda}$ correction for studying the impact of antiferromagnetic fluctuations on the spectral function of the Hubbard model in two and three dimensions Our results show the suppression of the quasiparticle weight in three dimensions and dramatically stronger impact of spin fluctuations in two dimensions where the pseudogap is formed at low enough temperatures Even in the presence of the Hubbard subbands, the origin of the pseudogap at weak-to-intermediate coupling is in the splitting of the quasiparticle peak At stronger coupling (closer to the insulating phase) the splitting of Hubbard subbands is expected instead The $\mathbf{k}$ dependence of the self-energy appears to be also much more pronounced in two dimensions as can be observed in the $\mathbf{k}$-resolved $\text{D}\ensuremath{\Gamma}\text{A}$ spectra, experimentally accessible by angular resolved photoemission spectroscopy in layered correlated systems

Comparing pertinent effects of antiferromagnetic fluctuations in the two- and three-dimensional Hubbard model

We consider the ground-state magnetic phase diagram of the two-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping in terms of electronic density and interaction strength. We treat commensurate ferromagnetic and antiferromagnetic as well as incommensurate (spiral) magnetic phases. The first-order magnetic transitions with changing chemical potential, resulting in a phase separation (PS) in terms of density, are found between ferromagnetic, antiferromagnetic, and spiral magnetic phases. We argue that PS has a dramatic influence on the phase diagram in the vicinity of half-filling. The results imply possible interpretation of the unusual behavior of magnetic properties of single-layer cuprates in terms of PS between collinear and spiral magnetic phases. The relevance of our results to the magnetic properties of the ruthenates is also discussed.

Incommensurate magnetic order and phase separation in the two-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping

We present a novel scheme for an unbiased, nonperturbative treatment of strongly correlated fermions. The proposed approach combines two of the most successful many-body methods, the dynamical mean field theory and the functional renormalization group. Physically, this allows for a systematic inclusion of nonlocal correlations via the functional renormalization group flow equations, after the local correlations are taken into account nonperturbatively by the dynamical mean field theory. To demonstrate the feasibility of the approach, we present numerical results for the two-dimensional Hubbard model at half filling.

/pdf/from-infinite-to-two-dimensions-through-the-functional-3sof246c48.pdf

Andrey A. Katanin

Papers

Dynamical vertex approximation : A step beyond dynamical mean-field theory

Fulfillment of Ward identities in the functional renormalization group approach

Comparing pertinent effects of antiferromagnetic fluctuations in the two- and three-dimensional Hubbard model

Incommensurate magnetic order and phase separation in the two-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping

From infinite to two dimensions through the functional renormalization group.