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 short-range, one-band model for electron correlations in a narrow energy band is solved exactly in the one-dimensional case. The ground-state energy, wave function, and the chemical potentials are obtained, and it is found that the ground state exhibits no conductor-insulator transition as the correlation strength is increased.

Absence of Mott Transition in an Exact Solution of the Short-Range, One-Band Model in One Dimension

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

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 total spin S is a good quantum number in problems of interacting spins. We have shown that for rather general antiferromagnetic or ferrimagnetic Hamiltonians, which need not exhibit translational invariance, the lowest energy eigenvalue for each value of S [denoted E(S)] is ordered in a natural way. In antiferromagnetism, E(S + 1) > E(S) for S > O. In ferrimagnetism, E(S + 1) > E(S) for S > S, and in addition the ground state belongs to S < S. S is defined as follows: Let the maximum spin of the A sublattice be S A and of the B sublattice S B; then S ≡ S A − S B. Antiferromagnetism is treated as the special case of S = O. We also briefly discuss the structure of the lowest eigen-functions in an external magnetic field.

Ordering Energy Levels of Interacting Spin Systems

Consider a system of N electrons in one dimension subject to an arbitrary symmetric potential, V(x1,...,x N ),and let E(S) be the lowest energy belonging to the total spin value S. We have proved the following theorem: E(S) < E(S’) if S < S’. Hence, the ground state is unmagnetized. The theorem also holds in two or three dimensions (although it is possible to have degeneracies) provided V (x1,y1,x1, …; x N ,y N ,z N ) is separately symmetric in the x i , y i , and z i . The potential need not be separable, however. Our theorem has strong implications in the theory of ferromagnetism because it is generally assumed that for certain repulsive potentials, the ground state is magnetized. If such be the case, it is a very delicate matter, for a plausible theory must not be so general as to give ferromagnetism in one dimension, nor in three dimensions with a separately symmetric potential.

Theory of Ferromagnetism and the Ordering of Electronic Energy Levels

We have investigated the magnetic ground state of metals using an idealized theory of magnetism based on the Ruderman-Kittel-Yosida indirect exchange interaction. The preliminary, but suggestive, results reported here are for simple cubic structures and spherical Fermi surface. We find that as the number of electrons is increased, ferromagnetism is replaced by two different antiferromagnetic structures before the N\'eel state is finally obtained. The first transition occurs at ${k}_{F}a\ensuremath{\sim}0.5\ensuremath{\pi}$ (${k}_{F}=\mathrm{Fermi}\mathrm{wave}\mathrm{vector}$, $a=\mathrm{lattice}\mathrm{constant}$) at which value the rapid changeover occurs, from ferromagnetism to an antiferromagnetic structure of planes of uniform magnetization, which point along alternating directions. These planes lie perpendicular to the (1, 0, 0) axis at first, then abruptly change over to the (1, 1, 0) axis at ${k}_{F}a\ensuremath{\sim}0.7\ensuremath{\pi}$. The new configuration remains the ground state until ${k}_{F}a$ is further increased to $0.85\ensuremath{\pi}$. But, at that value a final transition occurs to the N\'eel state. The N\'eel state (where every spin is surrounded by antiparallel nearest neighbors) then becomes increasingly stable and reaches maximum stability when the Fermi surface touches the zone boundary at ${k}_{F}a=\ensuremath{\pi}$. It appears from our calculations that the above-mentioned states are the only stable spiral configurations in the simple cubic lattice without the introduction of anisotropy or nonlinearity into the theory.

Daniel C. Mattis

Papers

Two soluble models of an antiferromagnetic chain

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

Ordering Energy Levels of Interacting Spin Systems

Theory of Ferromagnetism and the Ordering of Electronic Energy Levels

Role of Fermi Surface and Crystal Structure in Theory of Magnetic Metals