Symmetries and broken symmetries have long been a source of fascination. The columns of the Parthenon were famous in the ancient world for the subtle way in which their symmetries were broken. Johannes Kepler had appealed to the five Platonic solids in seeking to explain just why the six planets were at a specified distance from the sun. He was fascinated by polyhedrons and their symmetries. Until the nineteenth century, symmetries were confined to purely geometric and aesthetic contexts. It’s only progressively that symmetries and broken symmetries have entered into physical theories. The homochirality of biological molecules, discovered by Louis Pasteur during his crystallographic studies of tartaric acids, may well be a clue to the origin of life.^{1} Symmetry has now become central to understanding the organization of the universe. In the Standard Model, local or gauge symmetries comprise the concept needed to unify the electromagnetic and nuclear forces.

Contemporary physics is constantly confronted with questions of broken symmetry. A phase transition, or broken symmetry, occurring directly after the Big Bang may well have been responsible for the period of cosmic inflation during which the universe grew exponentially. Quantum density fluctuations then led to the formation of the large-scale structure of the universe. After further cooling, this was followed by another phase transition in which quarks became confined. Many theorists believe that supersymmetry is needed to explain the world of elementary particles. Supersymmetry would endow the universe with a partner to every known particle. It is not broken at any of the energy scales that have been so far explored.

The subject of broken symmetries is fascinating because it encompasses a large number of different phenomena that occur at different scales. There are phase transitions in solid-state physics in which the organization of matter changes for a huge number of constituents. A broken symmetry is responsible for quark confinement, the Higgs transition, and the lightness of the pion. This is to say what happened, but not how. Symmetries break at a critical point. The onset symmetry breaking seems to represent a universal phenomenon. It was only when Kenneth Wilson introduced the renormalization group in the early 1970s that physicists began to understand what was going on and why.^{2}

## Spontaneously Broken

The solidification of water when the temperature falls below 0°C, if familiar, is also amazing. When ice crystals are formed in water, the disordered molecules of the liquid distribute themselves along a periodic spatial array. Nothing changes in the interactions between constituent molecules. The forces between them are independent of temperature. Nonetheless, a slight change in temperature has the effect of generating a new spatial ordering for a huge number of molecules. A tiny crystal of only a few micrometers contains more than a trillion molecules.

The idea of a new spatial *ordering* is somewhat misleading. A block of ice represents a broken symmetry. The liquid phase looks the same at every point and in every direction. Its symmetry is the full Euclidean group of all possible translations and rotations. When a crystal is formed, the remaining symmetry operations must be consistent with its spatial structure. These are the translations and rotations that preserve the lattice sites. The space group that results is a subgroup of the Euclidean group. The phase with the largest symmetry group is, in fact, the liquid. The phase of broken symmetry, invariant only under a subgroup of the symmetry operations, is the ordered phase. This symmetry breaking is spontaneous since it does not result from the action of an external agent. A solidification under pressure would not be called a spontaneous change of symmetry.

Spontaneously broken symmetries (SSB) manifest themselves in many physical systems. A sharp pencil placed vertically in equilibrium on its tip is symmetrical in every direction. A random perturbation serves to break the symmetry. A few other examples:

*Ferromagnetism*

Above the Curie temperature,*T*, there is no magnetization; below_{c}*T*, magnetization spontaneously appears. The spatial isotropy is broken._{c}*Superfluidity of helium*

Below 2.1K, the phase invariance of the wave function is broken.*Bardeen–Cooper–Schrieffer superconductivity (BCS)*^{3}

In a model with a fundamental spinor field, nucleons acquire mass and mesons remain massless.^{4}*Flavor symmetry*

Neglecting the masses of light quarks yields a chiral symmetry under SU(3) × SU(3). Breaking this symmetry explains why the octet of mesons is lighter than the octet of vector mesons.*Goldstone’s theorem*

If the Lagrangian has a continuous symmetry group, massless bosons appear in its spectrum.^{5}A continuous symmetry, if broken, is always accompanied by massless excitations.*Ginzburg–Landau theory of the Meissner effect*^{6}

This theory served as a model for the symmetry breaking required to give mass to the gauge bosons.*Rayleigh–Bénard convective instability*

If a fluid is enclosed between parallel planes and the lower plane is maintained at a higher temperature, the fluid develops convection cells, which break the Euclidean invariance of the liquid.

## Order and Disorder

At the end of the nineteenth century, Ludwig Boltzmann established the principles governing the thermal equilibrium of atoms and molecules. Two principles are paramount. First, the statistical weight of a configuration with total energy *E* is proportional to *e*^{−βE}, when β is proportional to the inverse of the absolute temperature. Thermodynamic properties are related to the partition function

$Z={\sum}_{C}{e}^{-\beta E\left(C\right)}$,

where the sum runs over all the configurations, *C*, of the particles. Second, the number of configurations with the same energy, *w*(*E*), is huge for a macroscopic system. Up to a constant *k*, the entropy of the system is defined as

*S*(*E*) = *k* log *w*(E).

*S*(*E*) is proportional to the number of constituents. The partition function may now be rewritten as

$Z=\sum _{E}w\left(E\right){e}^{-\beta E}$,

where the sum runs over all possible values of the energy. By the same token

$Z=\sum _{E}{e}^{-\beta \left(E-TS\left(E\right)\right)}$,

where *E* and *S*(*E*) are proportional to the number of particles. This sum is peaked around the energies that minimize the free energy *E* − *TS*(*E*).

The competition between order and disorder is manifest in the minimization of free energy. At low temperatures, the system is governed by low energy configurations and, in general, appears well ordered. At higher temperatures, higher energies play a role, leading to the appearance of disorder. Consider a periodic lattice of *N* sites. At each lattice site, there is one degree of freedom, and that is the spin ±1. Spins located at adjacent sites have an interaction energy −*J* if they are parallel, +*J* if antiparallel. The total energy of a configuration *C* of the lattice of *N* spins (σ_{1}, …, σ* _{N}*) is thus

$E\left(C\right)=-J\sum _{ \u27e8ij\u27e9}{\sigma}_{i}{\sigma}_{j}$.

At low temperatures, *E − TS*(*E*) leads to low energy configurations—parallel spins, and thus order. When the temperature increases, higher energy configurations are favored and disorder prevails. An ordered phase is the very manifestation of SSB.

## Phase Transitions

For many decades, physicists argued whether statistical physics was, by itself, adequate to describe phase transitions. In 1936, Rudolf Peierls proved that low temperatures provoked spontaneous magnetization in the two-dimensional Ising model.^{7} At high temperatures, the same model predicted vanishing magnetization in the absence of an external magnetic field. It is the thermodynamic limit that is the source of this SSB ferromagnetic phase. The exact solution of the Ising model by Lars Onsager confirmed that the free energy solution provided by statistical mechanics contained both a low-temperature ferromagnetic phase and a paramagnetic phase above a critical temperature.^{8}

How can one characterize ℤ_{2} symmetry breaking, where the underlying symmetry group is comprised of the identity element and an involution {1, *X*}? Assume that some tiny external magnetic field, *h*, favors up-spins, and consider the average spin at a given site

$m\left(h\right)= \u27e8si\u27e9 $.

If every spin changes sign, symmetry implies that

$\underset{h\to 0}{\mathrm{lim}}m\left(h\right)=0$.

Limits may be inverted if *N* goes to infinity and *h* to zero. Broken symmetry reveals itself whenever

${m}_{sp}=\underset{h\to 0}{\mathrm{lim}}\underset{N\to \infty}{\mathrm{lim}}m\left(h\right)$

does not vanish. Broken symmetries occur only in infinite systems, but if *N* is very large, finite systems look for all the world as if they were infinite. This analysis hinges on a vanishingly small, symmetry-breaking, external field. In cases of superconductivity or superfluidity, such fields are not physically accessible. There is another way of describing a broken symmetry, and that is to examine the correlation between spins at two distant points of the system. In a disordered phase, the correlation

$\underset{\left|j-i\right|\to 0}{\mathrm{lim}} \u27e8{s}_{i}{s}_{j}\u27e9 =0$

falls off exponentially and vanishes at long distances. If a phase transition does take place then

$\underset{\left|j-i\right|\to 0}{\mathrm{lim}} \u27e8{s}_{i}{s}_{j}\u27e9 \ne 0$,

and

$\underset{\left|j-i\right|\to 0}{\mathrm{lim}} \u27e8{s}_{i}{s}_{j}\u27e9 ={m}_{sp}^{2}$,

where the same spontaneous magnetization, *m _{sp}*, defined by an anomalous source, occurs again. The divergence of this correlation length at a critical point is the mark of collective behavior.

The Boltzmann–Gibbs principles of statistical physics allow for a description of phase transitions without any further assumptions. The situation is the same in quantum field theory, where the partition function is replaced by the Feynman path integral. The vacuum and its corresponding particles can break the symmetry of the underlying action. If the vacuum is not left invariant by symmetry operations, the number and nature of the particles are deeply modified.

## Continuous Symmetry

Although Jeffrey Goldstone formulated his eponymous theorem in the framework of relativistic quantum field theory, its consequences are also familiar in nonrelativistic statistical physics when a continuous symmetry is broken. In a spin system with vector spins ${\overrightarrow{S}}_{i}$, and a rotation invariant interaction between neighboring spins, spontaneous magnetization breaks the rotational invariance. The excitation of long wavelength modes in spin waves or magnons costs little by way of energy. In the liquid–solid transition, the broken translational and rotational symmetries produce Goldstone modes, or phonons.

In particle physics there is one massless scalar particle for each broken generator of the symmetry. A chiral SU(3) × SU(3) symmetry is present in quark models with light quarks of negligible mass. The occurrence of a vacuum condensate of pairs of left-right quarks breaks the symmetry of the diagonal subgroup SU(3). This generates eight massless pseudoscalar mesons, which transform as an octet of the residual SU(3).

Gauge symmetries require a more subtle analysis. Ginzburg–Landau theory is a good introduction to the way in which gauge fields are endowed with mass via the spontaneous symmetry breaking of a charge matter field. In superconductivity, the Meissner effect represents the expulsion of a magnetic field from a superconductor undergoing transition to a superconducting state. Vitaly Ginzburg and Lev Landau provided a remarkable phenomenological theory before the BCS era. They introduced a complex field, $\psi $, in the presence of an external magnetic field. The occurrence of a non-zero value for $\stackrel{-}{\psi}\psi $ at low temperatures provides mass to the electromagnetic field in the superconducting phase.

The same idea was applied to the gauge theory of electroweak interactions based on the gauge symmetry SU(2) × U(1) by Philip Anderson, Robert Brout, François Englert, and Peter Higgs.^{9} Without the Higgs field, the gauge bosons *W*^{±}, *Z*, and the photon, would all be massless. It was well known that the weak interactions carried by the gauge bosons have a very short range. They must be mediated by massive particles. Adding a mass term by hand spoils the consistency with quantum mechanics, because the model would not be renormalizable. The introduction of a Higgs field coupled to the gauge field allows for SSB, which, in turn, generates mass for the weak gauge fields. In breaking the U(2) symmetry, this mechanism would generate four massless Goldstone bosons. However, three of these four bosons are absorbed into transforming the three massless weak gauge bosons into massive spin-one particles. The photon remains massless. Symmetry breaking complete, the Higgs field provides the required mass for the weak gauge bosons. Leptons and quarks also acquire mass though their interactions with the Higgs field. The weak gauge bosons, *W* and *Z*, were discovered in 1983. The discovery of the Higgs particle in 2013 completed the Standard Model.

## The Renormalization Group

During the 1960s, hundreds of articles were devoted to the study of the neighborhoods of critical points. Consider a gas–liquid transition obtained by lowering the temperature of a gaseous substance. This transition is accompanied by a jump in density, one that decreases and even vanishes at the critical temperature *T _{c}*. There are many singularities in the vicinity of this critical point: divergences in correlation lengths; diverging responses to an infinitesimal symmetry-breaking field; and spontaneously occurring non-zero order parameters below

*T*. These early studies revealed amazing properties of scaling and universality.

_{c}^{10}Hundreds of different molecules have been studied near their respective critical points. Many substances have identical singularities, equations of state, and correlation functions. Numerical studies confirmed universality by varying the type of lattice structure, but offered no clue to its origin. Wilson provided the explanation in the early 1970s. The Wilson renormalization group

^{11}remains one of the most important tools in theoretical physics.

^{12}Starting with microscopic, interacting degrees of freedom, Wilson deduced their long-distance collective behavior by studying how the interaction parameters flow as length scales change. His formalism led to flow equations characterized by fixed points. Most interactions parameters flow towards fixed points, no matter their initial microscopic value. The remaining parameters are governed by the linearized flow near a fixed point. The universal critical exponents are the eigenvalues of this linearized flow. Universality is a consequence of the flow equations.

Wilson’s ideas went beyond the study of critical points. In the 1960s, very few theories were known to be renormalizable. In quantum field theories, divergent integrals lead to an ultraviolet cutoff. At tiny distances, the theory breaks down. In renormalizable theories, the cutoff disappears in the limit. Quantum electrodynamics (QED) is the great example. Wilson started from the opposite point of view. He explored experimentally short distances by means of very high energy collisions. Below some small distance, the true physical theory remains unknown. Wilson wished to consider an arbitrary model at short distances, and explore the flow equation generated by the much larger accessible distances. The ensuing fixed points provide an effective theory, valid at presently observable distance scales. Although very beautiful, QED is merely an effective theory that at shorter distances will inevitably be replaced.