The Renormalization group and the epsilon expansion

Recent interest in aspects common to quantum information and condensed matter has prompted a flurry of activity at the border of these disciplines that were far distant until a few years ago. Numerous interesting questions have been addressed so far. Here an important part of this field, the properties of the entanglement in many-body systems, are reviewed. The zero and finite temperature properties of entanglement in interacting spin, fermion, and boson model systems are discussed. Both bipartite and multipartite entanglement will be considered. In equilibrium entanglement is shown tightly connected to the characteristics of the phase diagram. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium entangled states are generated and manipulated by means of many-body Hamiltonians.

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Entanglement in many-body systems

Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.

Colloquium: Area laws for the entanglement entropy

The theory of critical phenomena in systems at equilibrium is reviewed at an introductory level with special emphasis on the values of the critical point exponents α, β, γ,..., and their interrelations. The experimental observations are surveyed and the analogies between different physical systems - fluids, magnets, superfluids, binary alloys, etc. - are developed phenomenologically. An exact theoretical basis for the analogies follows from the equivalence between classical and quantal `lattice gases' and the Ising and Heisenberg-Ising magnetic models. General rigorous inequalities for critical exponents at and below Tc are derived. The nature and validity of the `classical' (phenomenological and mean field) theories are discussed, their predictions being contrasted with the exact results for plane Ising models, which are summarized concisely. Pade approximant and ratio techniques applied to appropriate series expansions lead to precise critical-point estimates for the three-dimensional Heisenberg and Ising models (tables of data are presented). With this background a critique is presented of recent theoretical ideas: namely, the `droplet' picture of the critical point and the `homogeneity' and `scaling' hypotheses. These lead to a `law of corresponding states' near a critical point and to relations between the various exponents which suggest that perhaps only two or three exponents might be algebraically independent for any system.

https://iopscience.iop.org/article/10.1088/0034-4885/30/2/306/pdf

The theory of equilibrium critical phenomena

1. Introduction This paper has three parts. The first part is a simplified presentation of the basic ideas of the renormalization group and the ε expansion applied to critical phenomena , following roughly a summary exposition given in 1972 1. The second part is an account of the history (as I remember it) of work leading up to the papers in I971-1972 on the renormalization group. Finally, some of the developments since 197 1 will be summarized, and an assessment for the future given.

The renormalization group and critical phenomena

Two soluble models of an antiferromagnetic chain

The two-dimensional Ising model for a system of interacting spins (or for the ordering of an AB alloy) on a square lattice is one of the very few nontrivial many-body problems that is exactly soluble and shows a phase transition. Although the exact solution in the absence of an external magnetic field was first given almost twenty years ago in a famous paper by Onsager1 using the theory of Lie algebras, the flow of papers on both approximate and exact methods has remained strong to this day.2 One reason for this has been the interest in testing approximate methods on an exactly soluble problem. A second reason, no doubt, has been the considerable formidability of the Onsager method. The simplification achieved by Bruria Kaufman3 using the theory of spinor representations has diminished, but not removed, the reputation of the Onsager approach for incomprehensibility, while the subsequent application of this method by Yang4 to the calculation of the spontaneous magnetization has, if anything, helped to restore this reputation.

Two-Dimensional Ising Model as a Soluble Problem of Many Fermions

The thermal expansivity of a single crystal of EuO was determined in the temperature range 25 to 250\ifmmode^\circ\else\textdegree\fi{}K by a differential-strain-gauge method. The temperature of the peak in the $\ensuremath{\lambda}$ curve of expansivity is 69.2\ifmmode^\circ\else\textdegree\fi{}K, in agreement with the specific-heat measurements. After correcting for the normal lattice expansivity using the Gr\"uneisen theory, we observe that the resulting magnetoelastic component of expansivity ${\ensuremath{\alpha}}_{\mathrm{me}}$ obeys a magnetic Gr\"uneisen law, being proportional to the magnetic specific heat ${C}_{m}$ over wide ranges of temperature both above and below the $\ensuremath{\lambda}$ transition. Europium oxide can therefore be characterized by a temperature-independent "magnetic" Gr\"uneisen constant, $\frac{\ensuremath{\partial}\mathrm{ln}{U}_{m}}{\ensuremath{\partial}\mathrm{ln}V}=\ensuremath{-}5.3$, given by $\ensuremath{-}3{\ensuremath{\alpha}}_{\mathrm{me}}{{C}_{m}}^{\ensuremath{-}1}{B}_{T}$, where the isothermal bulk modulus ${B}_{T}=1.07\ifmmode\times\else\texttimes\fi{}{10}^{+12}$ dyn/${\mathrm{cm}}^{2}$. The lattice Gr\"uneisen constant, $\frac{\ensuremath{\partial}\mathrm{ln}{U}_{l}}{\ensuremath{\partial}\mathrm{ln}V}=1.9$, was similarly derived from the data at temperatures well above the $\ensuremath{\lambda}$ anomaly. For ${U}_{m}$, the internal magnetic energy, we also derive the variation with temperature $\frac{{U}_{m}(T)}{{U}_{m}(0)}$, the variation with pressure $\frac{\ensuremath{\partial}\mathrm{ln}{U}_{m}}{\ensuremath{\partial}P}=4.9\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}12}$ ${\mathrm{dyn}}^{\ensuremath{-}1}$ ${\mathrm{cm}}^{2}$, and the value ${U}_{m}(0)=\ensuremath{-}4.9\ifmmode\times\else\texttimes\fi{}{10}^{8}$ erg/${\mathrm{cm}}^{3}$ at 0\ifmmode^\circ\else\textdegree\fi{}K. Comparison with results of other experiments and with theories based on the Heisenberg Hamiltonian is also presented. The model of D. C. Mattis and T. D. Schultz and of E. Pytte is consistent with the observed proportionality between ${C}_{m}$ and ${\ensuremath{\alpha}}_{\mathrm{me}}$. A more general model proposed by E. R. Callen and H. B. Callen includes magnetoelastic coupling of unequal strengths to first- and second-nearest neighbors. When the second-neighbor interaction is weaker than the first, this model is also consistent with a single effective magnetic Gr\"uneisen constant not only because the model then differs only slightly from a special case of that of Mattis and Schultz and of Pytte, but also because the spin correlation functions ${〈\mathrm{S}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}^{\ensuremath{'}}〉}_{1\mathrm{s}\mathrm{t}\phantom{\rule{0ex}{0ex}}\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{r}}$ and ${〈\mathrm{S}\ifmmode\cdot\else\textperiodcentered\fi{}{\mathrm{S}}^{\ensuremath{'}}〉}_{2\mathrm{n}\mathrm{d}\phantom{\rule{0ex}{0ex}}\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{r}}$ appear to be nearly proportional to each other over a wide temperature range.

Magnetoelastic Behavior of Single-Crystal Europium Oxide. I. Thermal Expansion Anomaly

A study is made of the interaction of an electron bound to an impurity center in a semiconductor with the phonon field of the material. In particular we study the shift and shape of optical absorption lines of donor impurities in semiconductors when the excitation energy lies close to the energy of an optical phonon branch of the vibrational spectrum of the crystal. The results exhibit a variety of phenomena. The optical absorption lines may be split or broadened either symmetrically or asymmetrically. The different possibilities depend primarily on the dispersion of the phonon bands in the vicinity of the electronic excitation energy. If the phonon energy is approximately independent of the wave number, a splitting of the line arises. Particular emphasis is given to the case of bismuth donors in silicon interacting with transverse optical phonons.

Effects of Resonant Phonon Interactions on Shapes of Impurity Absorption Lines

Effects of isostructural doping of the donor and acceptor stacks on the phase transitions and magnetic and electrical properties of tetrathiafulvalene-tetracyanoquinodimethane

T. D. Schultz

Papers

Two soluble models of an antiferromagnetic chain

Two-Dimensional Ising Model as a Soluble Problem of Many Fermions

Magnetoelastic Behavior of Single-Crystal Europium Oxide. I. Thermal Expansion Anomaly

Effects of Resonant Phonon Interactions on Shapes of Impurity Absorption Lines

Effects of isostructural doping of the donor and acceptor stacks on the phase transitions and magnetic and electrical properties of tetrathiafulvalene-tetracyanoquinodimethane