To the editors:
In his essay, Édouard Brézin ably describes the role that symmetries and their spontaneous breaking have played in modern theoretical physics. His essay also illustrates how these ideas apply to phenomena on all scales, from subatomic physics through condensed matter to cosmology. The joint paradigms of symmetry (and its breaking) and the renormalization group have, in recent times, played such an important role in both high-energy physics and statistical physics that they have effectively unified the two, traditionally separate fields of study. These days a graduate student in one field would be at a considerable disadvantage if they did not also study the other. Brézin was one of the pioneers of this interdisciplinary approach, applying concepts of statistical physics to gauge theories of particle physics, as well as to the large random matrices that appear in problems ranging from nuclear physics to disordered solid state systems.
Doubtless for lack of space, Brézin does not discuss another important role that symmetries play in many branches of modern physics: they often imply the existence of localized topological defects which may, in some circumstances, play a crucial role in explaining the macroscopic behavior of such systems. It was for the application of these ideas in condensed-matter physics that the 2016 Nobel Prize in physics was awarded jointly to Duncan Haldane, J. Michael Kosterlitz, and David Thouless. The contributions of Haldane and, in part, those of Thouless involve more advanced mathematical tools, but the joint work of Kosterlitz and Thouless, as well as Vadim Berezinskii, on the classical XY model is simpler to describe and is an excellent example of the role of symmetry and topology in modern physics.
The XY model is like the Ising spin model described in Brézin’s article, but instead of taking only two values ±1, each spin may be thought of as the large hand on a clock, pointing anywhere on the circumference of a circle, so that it is specified by an angle between 0 and 360 degrees. The symmetry group of this model is thus that of rotations in this internal 2-dimensional space, called O(2). The XY model not only describes so-called easy-plane magnets, but also the transition to a superfluid in helium and to a superconducting state in metals, where the angle is the phase of the macroscopic wave function.
In spatial dimensions greater than two, the XY model exhibits spontaneous symmetry breaking at low temperatures, and because the symmetry is continuous, this gives rise to Goldstone bosons as low-energy excitations, as described in Brézin’s essay. However, there is a problem in two space dimensions. A theorem, resulting from David Mermin and Herbert Wagner’s work in statistical physics and Sidney Coleman’s work in high energy physics, in fact forbids spontaneous breaking of a continuous symmetry, and thereby the consequent Goldstone bosons. Yet there was ample evidence that the 2-dimensional XY model and the physical systems it describes do exhibit a phase transition from a disordered phase at high temperatures.
This paradox was resolved by the work of Berezinskii, Kosterlitz, and Thouless, who recognized the role played by topological excitations called vortices. A vortex is a configuration in which the hand of the clock, in traversing a circle around a given point in space, similarly describes a circle in the internal space. Each vortex has a certain energy E and entropy S, but, as Brézin describes, the important quantity is its free energy, F = E – TS (where T is the temperature), and free energy should be minimized. At low temperature F = E > 0, so vortices are suppressed. In that case it is easy to show that the correlations between the XY spins decay as a power law with their separation. This implies that there is indeed no spontaneous symmetry breaking, but instead a low-temperature phase exhibiting so-called quasi-long-range order, intermediate between the long-range order of spontaneous symmetry breaking and the disordered phase. At temperatures T > E/S, on the other hand, vortices should proliferate and act to disorder the XY spins. This picture is oversimplified since the vortices interact, but Kosterlitz, using renormalization group methods, was able to take this into account and give precise predictions for the critical behavior. These predictions were subsequently spectacularly verified in experiments on helium films carried out by John Reppy and others.
Similar excitations or defects occur in many situations in condensed-matter and high-energy physics. They are classified by the branch of mathematics known as topology, the study of geometric properties and spatial relations unaffected by a continuous change of shape or size. This robustness helps make them stable, and quantized vortices in 2-dimensional layers of electrons in strong magnetic fields have been proposed as a mechanism for error-free quantum computing. Another important example includes magnetic monopoles, which Gerard ’t Hooft and Alexander Polyakov have argued arise in many grand unified theories of particle physics. Although magnetic monopoles have not yet been observed experimentally, as Paul Dirac pointed out long ago, their mere existence leads to the observed discrete quantization of electric charge. Then there are cosmic strings, one-dimensional defects that could have formed during a phase transition in the early universe, of the type Brézin mentions. Unfortunately, these would be hard to detect apart from their effect of bending the light from distant galaxies, although if one were to pass close by, the effects might be more spectacular.
In all, there is much more to symmetry than simply breaking it in a uniform manner.
John Cardy
Édouard Brézin replies:
I would like to thank our colleagues for their comments, which indeed point to other relevant, significant contributions to the topic of spontaneous symmetry breaking. In my short article, I wanted to illustrate how much this concept was ubiquitous; from condensed matter to cosmology and particle physics, I have tried to point out how our understanding of many phenomena is based on spontaneous symmetry breaking.
John Cardy rightly points out that I should not have left aside the discussion of the interplay between symmetries and topological defects. Indeed, many modern topics on topological phases have enriched considerably the realm of spontaneous symmetry breaking: the phase structure of the 2-dimensional XY model, the dislocation melting of 2-dimensional solids, and the prediction of magnetic monopoles in grand unified gauge theories, are well-known examples of a still rapidly developing subject. His point is well taken.
