PHASE TRANSITIONS

Most substances exist in three distinct phases: solid, liquid, and vapor. From everyday experience, in particular with water, it is observed that a substance in a given phase may transition into another phase as a response to some imposed change, for example, of temperature or pressure. These phase transitions occur at very specific temperatures and pressures for different substances. At sea level water transitions from a liquid to a vapor phase—it boils—at 100° Celsius; at an altitude of 3,000 meters, due to the lower atmospheric pressure, water boils at only 90° Celsius.

Historically, the study of phase transitions has been of interest mostly to condensed-matter and statistical physicists. However, since the early 1960s, it has become clear that several parallels can be drawn between the physics of phase transitions and the changes in the properties of elementary particles of matter and their interactions at different energies. The theories describing elementary particles and their interactions can also be understood as having different phases, which reveal themselves at different energies or temperatures. The physics of these phase transitions can be probed in two different arenas: high-energy collisions between subatomic particles in particle accelerators, and during the early stages of the universe's history when, according to the prevailing Big Bang theory of cosmology, the temperatures were high enough to promote these changes.

Phase Transitions and Symmetry

At the microscopic level, phase transitions can be understood as a spatial rearrangement of the molecules of a given substance or a mixture of substances resulting from externally imposed changes in temperature, pressure, and, in some cases, magnetic field. In general, the phase of a substance is determined by a competition between the chemical bonding of its molecules and their thermal agitation. In the vapor phase, which occurs at high enough temperatures, thermal agitation is the dominant factor, causing the molecules to move freely, colliding with one another but never forming large clusters. At lower temperatures, the chemical bonding between the molecules starts to counterbalance their thermal agitation, and a liquid phase sets in, where the molecules are held closely together but still in a disorderly fashion. At sufficiently low temperatures, the further loss of thermal energy facilitates the arrangement of the molecules in rigid clusters characterizing the solid phase. The phase of a given substance is defined by the spatial ordering of its molecules.

A phase change can also result from structural changes within different solid phases of the same substance—known as solid-solid transitions, or, in the case of magnetic materials, from the rearrangement of magnetic fields within a collection of atoms. As with ordinary phase transitions, it is often possible to associate these structural rearrangements with changes in the underlying spatial symmetry of the substance. When water is in its liquid phase, the probability of finding a molecule anywhere within a given volume is approximately the same, as all molecules are on average equally spaced; one can say that liquid water has a large spatial symmetry. However, as water freezes, its molecules rearrange themselves in a crystal lattice, and the probability of finding a molecule is no longer approximately the same everywhere. The spatial symmetry of liquid water is lost when it freezes. In other words, a drop in temperature may decrease the amount of symmetry in a substance.

As a second example, magnetic materials also exhibit a phase transition, related to the spatial ordering of its atoms. Each atom can be considered as a small magnet, which interacts with its neighbor. At high temperatures, the small magnets point at random directions, and the net magnetization of a sample of the material is zero. This is called the paramagnetic phase. However, as the temperature drops, the neighboring magnets tend to align in the same direction in order to minimize their interaction energy. Below a temperature known as the critical temperature the material separates into domains with a net magnetization, as indicated in Figure 1. If an external magnetic field is applied, the magnetization of the separate domains aligns in the same direction of the magnetic field. This is referred to as the ferromagnetic phase. Again, the symmetry that exists at high temperatures, when all directions are equivalent,

FIGURE 1

is broken below the critical temperature, when only one direction prevails.

Symmetry Breaking in Particle Physics

According to the Standard Model of particle physics, there are four fundamental forces—or interactions—in nature: the long-range gravitational and electromagnetic forces, and the strong and weak nuclear forces. The two latter forces act only within the atomic nucleus; thus, they are very short-range forces. These four forces describe the interactions between the twelve elementary particles of matter, which, according to the Standard Model, can be divided into two groups: six quarks and six leptons. The quarks make up particles that interact via the strong nuclear force, such as protons and neutrons, whereas the leptons participate in the weak interactions, responsible, for example, for radioactive decay. Each of the forces has an associated symmetry, related to quantities that are conserved during the interactions. For example, the associated conserved quantity of electromagnetic interactions is the electric charge.

The crucial link between particle physics and the physics of phase transitions is that the nature of the four fundamental interactions, and thus their symmetries, change with energy. At high enough energies, the interactions start to behave in similar ways: at energies greater than 100 times the mass of the proton (times the square of the speed of light, as required by the E = mc² relation, where c is the speed of light), the weak interaction becomes long range and is indistinguishable from the electromagnetic interaction. Thus, above these energies, the interactions between matter particles can be described in terms of three fundamental forces and not four: gravity, strong, and electroweak. At much higher energies (on the order of a thousand trillion proton masses times the square of the speed of light), it is expected that the strong interactions join the electroweak force to become the grand unified force. If this proves to be correct, at these enormous energies nature can be described in terms of two forces: gravity and the grand unified force. Some theories presently under investigation, such as superstring theories, attempt to include gravity in the unification scheme. This high-energy unification of the fundamental interactions comes with an increase in the underlying symmetry, very much as the increase in symmetry in a solid-to-liquid transition or when a ferromagnetic material becomes paramagnetic at high temperatures.

To summarize, each of the four fundamental interactions has an associated symmetry. At high energies, the interactions can be described in a unified way, and their underlying symmetry increases. In other words, at extremely high energies nature is highly symmetric. As the interactions between particles are probed at lower energies, this large symmetry is progressively broken, until we reach the current Standard Model description in terms of four fundamental forces.

Phase Transitions in Particle Accelerators and in the Early Universe

There are two ways to test the prediction of increased symmetry at high energies: at particle accelerators and during the early stages of the universe's history. The unification of the electromagnetic and weak interactions was verified at the European Laboratory for Particle Physics (CERN) in 1983. The theory of strong interactions, known as quantum chromodynamics (QCD) predicts that above a certain energy protons, neutrons, and other composite particles known as hadrons break into a plasma of free quarks and gluons, the particles that promote their interactions. This prediction, which is also interpreted as a phase transition, is presently being tested at the Relativist Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory in New York.

The early universe offers the best laboratory to test unification ideas, as temperatures were high enough to probe the larger symmetry regimes predicted by particle physics. As the universe expanded and cooled from its extremely hot and dense initial state, it may have undergone a succession of phase transitions related to the breaking of the initial large unified symmetry into smaller ones associated with the cited four interactions (see Figure 2). Each of these phase transitions may have left imprints and remnants that could be observed today.

As is true with the transition from water to ice or from a paramagnet to a ferromagnet, it is possible that these cosmological phase transitions also

FIGURE 2

developed domains and other imperfections, which carry the particular signature of the symmetry breaking process. One possibility of great interest is that the observed excess of matter over antimatter, a puzzle still unsolved in particle physics, could be explained due to the particular details of the electroweak phase transition, predicted to have occurred when the universe was one trillionth of a second old. This transition, which has some similarities with the vapor-water transition, may have produced the conditions necessary to generate a small excess of particles of matter over particles of antimatter, which eventually led to the existence of complex material structures such as galaxies, stars, and people.