As we noted in the preface, this and the following chapter are at a more advanced level, and some readers may wish to skip ahead to Chapter 8.

In Chapter 5 we studied the O(N) quantum rotor model in the large-N limit for a number of values of the spatial dimensionality, including d = 1. We noted that the results provided an adequate description of the static properties in d = 1 for N ≥ 3. This will be justified in the present chapter where we will obtain a number of exact results for the same static observables. We also noted that the large-N limit did a very poor job of describing dynamical properties at nonzero temperatures. This will be repaired in this chapter by simple physical arguments that lead to a fairly complete (and believed exact) description of the long-time behavior. Some of the discussion in this chapter will be specialized to the O(N = 3) model, which is also the case of greatest physical importance; the properties of the O(N > 3) models are very similar, and many of our results will be quoted for general N. Of the remaining cases, the d = 1, N = 1 model has been already considered in Chapter 4, and study of the d = 1, N = 2 model is postponed to Section 14.3.

The physical picture of the T = 0, N = 3 state that emerged in Chapter 5 was very simple.

This chapter turns to the models obtained by the quantum–classical mapping QC on the D-dimensional, N-component, classical ferromagnets in (3.3) with N ≥ 2. These are the O(N) quantum rotor models in d = D − 1 dimensions, originally written down in (1.23).

The quantum Ising model studied previously had a discrete Z2 symmetry. An important new ingredient in the rotor models will be the presence of a continuous symmetry: The physics is invariant under a uniform, global O(N) transformation on the orientation of the rotors, which is broken in the magnetically ordered state. We will introduce the important concept of the spin stiffness, which characterizes the rigidity of the ordered state and determines the dispersion spectrum of the low energy “spin-wave” excitations. Apart from this, much of the technology and the physical ideas introduced earlier for the d = 1 Ising chain will generalize straightforwardly, although we will no longer be able to obtain exact results for crossover functions. The characterization of the physics in terms of three regions separated by smooth crossovers, the high-T and the two low-T regions on either side of the quantum critical point, will continue to be extremely useful and will again be the basis of our discussion. Because we will consider models in spatial dimensions d > 1, it will be possible to have a thermodynamic phase transition at a nonzero temperature. We shall be particularly interested in the interplay between the critical singularities of the finite-temperature transition and those of the quantum critical point.

Symmetries and orthoscheme tetrahedra

Group generated by reflections. The characteristic triangle

Let us consider a triangle ABC, with angles α, β, γ, located on any surface, either Euclidean, spherical or hyperbolic, and let a, b, c denote the simple reflections in the sides BC, AC and AB (du Val 1964). In the following we shall embed the spherical geometry in R3 as the standard sphere S2. ABC becomes in this case a spherical triangle, whose sides are great circle arcs; a, b and c are then the simple reflections in the planes defined by these great circles. In the hyperbolic space case, we shall use the unit disc conformal model (appendix A1). The triangle sides are now arcs of circles orthogonal to the absolute (these lines can degenerate into straight parts of a diameter when the geodesies run through the origin), and reflections in the sides are replaced by inversion with respect to these circles (figure A4.1).

Let us first briefly recall what a circle inversion is. Consider a circle of centre O and radius r in the Euclidean plane and a point M anywhere in that plane. The inverse M′ of M with respect to the circle is located on the line OM at a position such that OM.OM′ = r2.

The three types of triangle are represented in figure A4.2.

It is possible to generate rotations by combining the reflections. For example the symmetry operations obtained as the products b.c, c.a and a.b are rotations of angles 2α, 2β and 2γ about A, B and C respectively.

The three geometries

Spherical, Euclidean and hyperbolic geometries are the three types of homo-geneous geometries on which a two-dimensional manifold can be locally modelled. They have a constant Gaussian curvature which is positive in spherical space, null in Euclidean space and negative in hyperbolic space. In three dimensions these three geometries have an analogue, but there are five other homogeneous geometries (Thurston 1982, Scott 1983). In this section, we briefly present Euclidean and hyperbolic geometries, while the next section is devoted to spherical spaces. Section A1.3 is concerned with the more exotic cylindrical spaces and §A1.4 presents the different notions of curvature.

Euclidean geometry

Two- and three-dimensional Euclidean spaces do not require any long description. The main transformations which let the space be globally invariant are translations (which let no points be invariant), rotations (with one point – in two dimensions – and one line – in three dimensions – kept invariant) and mirror inversion (with one line – in two dimensions – and one plane – in three dimensions – kept invariant). Note that any rotation can be viewed as the combination of two mirror inversions, property which will be illustrated later with the groups generated by reflections, like in a kaleidoscope (appendix A4).

Hyperbolic geometry

Hyperbolic geometry differs from the Euclidean geometry by the basic axiom concerning parallelism. Now ‘through a point not on a line, there can be more than one line parallel to the given line’. This is quite unusual and we shall now give a quick description of what has also been called the Lobachevski–Bolyai geometry, with reference to its two independent discoverers.

From cubism to icosahedrism

Among the many scientific breakthroughs which have brought solid state physics to be considered as a distinct field of physics, with its own concepts and methods, the most important one was probably the discovery, by Max Von Laue, Walter Friedrich and Paul Knipping in 1912, of the diffraction of X-rays by a crystal. It proved that crystals were made of a periodic array of atoms or molecules, an idea which was already supported by the work of abbé René-Juste Haüy at the end of the eighteenth century. Indeed, the latter proposed a periodic microscopic structure for crystals, based on the observation of the regular facets of crystal grains at a macroscopic level.

The mathematicians of the nineteenth century contributed to this story by inventing a very important tool, the concept of a transformation group, which would prove useful in almost every field of physics. As far as the groups of space symmetry operations are concerned, the complete classification of the spatial groups was fulfilled by the end of the nineteenth century for the three-dimensional Euclidean space. A very important result which follows this classification is the crucial restriction on the compatibility between rotations and translations: only rotations of order 2, 3, 4, 6 can let a crystal be invariant. An important ingredient in the description of an ordered structure is its point group, which enumerates the symmetry operations which leave a point fixed.

A unified approach to very different materials

It is not a priori surprising that an approach whose nature is mainly geometrical can describe very different materials, like metallic and covalent solids, amphiphilic molecules, cholesteric blue phases, etc …. As far as crystals are concerned, ‘mathematical’ crystallography takes this role of classifying all the possible structures compatible with periodicity in space, without regard to the chemical nature of the atoms. The physicist enters the field in order to relate an experimental pattern, usually the result of a diffraction experiment, to one of the possible periodic arrangements. This is already an idealization in the sense that most of the studied materials are only imperfect crystals, containing many types of defect. A perfect crystal is impossible for the first reason that it is necessarily limited in size. As is well known now, atoms close to the surface rearrange for the sake of energy minimization. A second reason is temperature: when T is finite, vacancies can contribute to configurational entropy. But standard, bulk diffraction is usually blind to these alterations of the periodic order. A natural and unavoidable obstacle to a precise structural determination is the experimental precision. The wave probes (X-rays, electrons, neutrons) have limited wave vectors, which introduce cut-like effects in the passage from reciprocal to real space.

This may not be important for simple crystals, but can prevent a precise determination for crystals with large unit cells.

Disclinations

A disclination is a defect involving a rotation operation, as opposed to the more familiar dislocation, which is associated with a translation given by its Burgers vector (Friedel 1964). For this reason, this defect, introduced by Volterra at the beginning of the twentieth century in his description of a continuous solid medium, is sometimes called a rotation-dislocation. A disclination can be generated by a so-called ‘Volterra’ process, by cutting the structure along a line and adding (or removing) a sector of material between the two lips of the cut. In two dimensions, this defect is point-like, while it is linear in three dimensions. The two lips of the sector should be equivalent under a rotation belonging to the structure symmetry group in order to get a pure topological defect confined near the apex of the cut (Kléman 1983).

A simple example of disclinations: wedge disclinations in two dimensions

It is possible to describe this defect, and the induced deformation, as a concentration of curvature (figure 4.1). This will be argued, in §4.4, from a differential geometry analysis, but it is possible, in two dimensions, to describe this relation more simply. Let us first do the Volterra construction with a sheet of paper. We first cut it along a straight segment up to its centre. Then, upon rotating around this centre, we can either add or remove a sector, and then glue again along the lips of the cut.

Introduction

We now consider quasiperiodic structures derived from the eight-dimensional lattice E8. Indeed, using the cut and projection method, it is possible to generate a four-dimensional quasicrystal having the symmetry of polytope {3, 3, 5} (Elser and Sloane 1987). We present here a modified version of this method. The network is foliated into successive shells surrounding a vertex. These shells belong to S7 spheres. We then take advantage of the Hopf fibration of S7, with S3 fibres, to split the E8 sites into symmetrically disposed sets of 24 sites in the S3 fibres. This method has two advantages: first, in the selection process, it is a full fibre which is either selected or rejected, which simplifies this algorithmic step; more important, this selection process eventually amounts to a simple arithmetic criterion. We then get a shell-by-shell analysis of the four-dimensional structure, which recalls in some respects the 2d–1d algorithm used to generate the Fibonacci chain. The number of points on these shells is exactly given. Note finally that, making sections in this quasicrystal, lower-dimensional quasicrystals are obtained, for instance structures with icosahedral or tetrahedral symmetries in three dimensions. Here we will only summarize these different results, the detailed results being found in the reference (Sadoc and Mosseri 1993).

The E8 lattice

The E8 lattice is known to provide the densest sphere packing in eight dimensions; it belongs to the family of ‘laminated’ lattices, and is sometimes denoted Λ8 in this context.

Quasicrystals: the spectacular appearance of quasiperiodic order in solid state physics

As has been emphasized at length in this book, frustration most often leads to the presence of various complex structures. We have so far concentrated on essentially two such families of structures, the disordered amorphous ones, and the large cell crystals. But Nature recently proved once again that she can use all possibilities to fill space that mathematics allows. We already knew that almost all the three-dimensional space groups describe at least one real structure. But, more than ten years ago, solid state physicists received a great surprise (if not a great shock for some): the experimental result that certain metallic alloys, which were quickly called quasicrystals, adopt a long range icosahedral order at the atomic level (Shechtman et al. 1984), whose signature is their diffraction spectra presenting peaks – indicating long range order – displaying icosahedral order – a forbidden symmetry for standard crystallography. A new and very active field of research was born, which could take advantage of the fact that, on the one hand, and rather rapidly, thermodynamically stable materials of high quality have been synthesized; and, on the other hand, from the theoretical point of view, that quasiperiodic plane tilings, proposed in the mid-1970s by Roger Penrose, quickly gave an approximate image of the atomic arrangement in these alloys, through their generalization in three dimensions.

We do not aim at giving here a review of quasicrystal physics.

Fibrations

A space can be considered as a fibre bundle if there is a sub-space (the fibre) which can be reproduced by a displacement so that any point of the space is on a fibre, and only one. For example the Euclidean space R3 can be considered as a fibre bundle of parallel straight lines, all perpendicular to the same plane.

What is a fibration?

From a mathematical point of view, a fibred space E is defined by a mapping p from E onto the so-called ‘base’, B, any point of a given fibre being mapped onto the same base point. A fibre is therefore the full pre-image of one base point under the mapping p. In a three-dimensional fibred space with one-dimensional fibres, the base is a two-dimensional manifold. In the above simple R3 example, the two-dimensional base space is just the plane orthogonal to the fibres; in this case the base is a sub-space of the whole space, and the fibration is called ‘trivial’. But this latter property is not general, the base may not be embedded in the fibre bundle space, as will be seen below with the Hopf fibration.

Fibration of S2 × R1

When a space is the direct product between two spaces, it can trivially be considered as a fibre bundle. If fibres are straight lines in S2 × R1 then the base is the sphere S2.