- The transitions between the solid, liquid, and gaseous phases (boiling, melting, sublimation, etc.)
- The transition between the ferromagnetic and paramagnetic phases of magnetic materials at the Curie point.
- The emergence of superconductivity in certain metals when cooled below a critical temperature.
- Quantum condensation of bosonic fluids, such as Bose-Einstein condensation and the superfluid transition in liquid helium.
- The breaking of symmetries in the laws of physics during the early history of the universe as its temperature cooled.

Table of contents |

2 Properties of phase transitions 3 Mean field theory of phase transitions 4 Fluctuations 5 Renormalization group theory of phase transitions |

The first attempt at classifying phase transitions was the **Ehrenfest classification scheme**, which grouped phase transitions based on the degree of non-analyticity involved. Though useful, Ehrenfest's classification is flawed, as we will discuss in the next section.

Under this scheme, phase transitions were labelled by the lowest derivative of the free energy that is discontinuous at the transition. **First-order phase transitions** exhibit a discontinuity in the first derivative of the free energy with a thermodynamic variable. The various solid/liquid/gas transitions are classified as first-order transitions, as the pressure, which is the first derivative of the free energy with volume, changes discontinuously across the transitions. **Second-order phase transitions** have a discontinuity in a second derivative of the free energy. These include the ferromagnetic phase transition in materials such as iron, where the magnetization, which is the first derivative of the free energy with the appplied magnetic field strength, increases continuously from zero as the temperature is lowered below the Curie temperature. The magnetic susceptibility, the second derivative of the free energy with the field, changes discontinuously. Under the Ehrenfest classication scheme, there could in principle be third, fourth, and higher-order phase transitions.

The **first-order phase transitions** are those that involve a latent heat. During such a transition, a system either absorbs or releases a fixed (and typically large) amount of energy. Because energy cannot be instantaneously transferred between the system and its environment, first-order transitions are associated with "mixed-phase regimes" in which some parts of the system have completed the transition and others have not. This phenomenon is familiar to anyone who has boiled a pot of water: the water does not instantly turn into gas, but forms a turbulent mixture of water and water vapor bubbles. Mixed-phase systems are difficult to study, because their dynamics are violent and hard to control. However, many important phase transitions fall in this category, including the solid/liquid/gas transitions.

The second class of phase transitions are the **continuous phase transitions**, also called **second-order phase transitions**. These have no associated latent heat. Examples of second-order phase transitions are the ferromagnetic transition, the superfluid transition, and Bose-Einstein condensation.

Several transitions are known as the **infinite-order phase transitions**.
They are continuous but break no symmetries (see **Symmetry** below). The most famous example is the Berezinsky-Kosterlitz-Thouless transition in the two-dimensional XY model. Many quantum phase transitions in
two-dimensional electron gases belong to this class.

Phase transitions often (but not always) take place between phases with different symmetry. Consider, for example, the transition between a fluid (i.e. liquid or gas) and a crystalline solid. A fluid, which is composed of atoms arranged in a disordered but homogenous manner, possesses continuous translational symmetry: each point inside the fluid has the same properties as any other point. A crystalline solid, on the other hand, is made up of atoms arranged in a regular lattice. Each point in the solid is *not* similar to other points, unless those points are displaced by an amount equal to some lattice spacing.

Generally, we may speak of one phase in a phase transition as being more symmetrical than the other. The transition from the more symmetrical phase to the less symmetrical one is a **symmetry-breaking** process. In the fluid-solid transition, for example, we say that continuous translation symmetry is broken.

The ferromagnetic transition is another example of a symmetry-breaking transition, in this case the symmetry under reversal of the direction of electric currents and magnetic field lines. This symmetry is referred to as "up-down symmetry" or "time-reversal symmetry". It is broken in the ferromagnetic phase due to the formation of magnetic domains containing aligned magnetic moments. Inside each domain, there is a magnetic field pointing in a fixed direction chosen spontaneously during the phase transition. The name "time-reversal symmetry" comes from the fact that electric currents reverse direction when the time coordinate is reversed.

The presence of symmetry-breaking (or nonbreaking) is important to the behavior of phase transitions. It was pointed out by Landau that, given any state of a system, one may unequivocally say whether or not it possesses a given symmetry. Therefore, it cannot be possible to analytically deform a state in one phase into a phase possessing a different symmetry. This means, for example, that it is impossible for the solid-liquid phase boundary to end in a critical point like the liquid-gas boundary. However, symmetry-breaking transitions can still be either first or second order.

Typically, the more symmetrical phase is on the high-temperature side of a phase transition, and the less symmetrical phase on the low-temperature side. This is certainly the case for the solid-fluid and ferromagnetic transitions. This happens because the Hamiltonian of a system usually exhibits all the possible symmetries of the system, whereas the low-energy states lack some of these symmetries (this phenomenon is known as spontaneous symmetry breaking.) At low temperatures, the system tends to be confined to the low-energy states. At higher temperatures, thermal fluctuations allow the system to access states in a broader range of energy, and thus more of the symmetries of the Hamiltonian.

When symmetry is broken, one needs to introduce one or more extra variables to describe the state of the system. For example, in the ferromagnetic phase one must provide the net magnetization, whose direction was spontaneously chosen when the system cooled below the Curie point. Such variables are instances of **order parameters**, which will be described later. However, note that order parameters can also be defined for symmetry-nonbreaking transitions.

Symmetry-breaking phase transitions play an important role in cosmology. It has been speculated that, in the hot early universe, the vacuum (i.e. the various quantum fields that fill space) possessed a large number of symmetries. As the universe expanded and cooled, the vacuum underwent a series of symmetry-breaking phase transitions. For example, the electroweak transition broke the SU(2)×U(1) symmetry of the electroweak field into the U(1) symmetry of the present-day electromagnetic field. This transition is important to understanding the asymmetry between the amount of matter and antimatter in the present-day universe (see electroweak baryogenesis.)

It turns out that continuous phase transitions can be characterized by parameters known as critical exponents. For instance, let us examine the behavior of the heat capacity near such a transition. We vary the temperature *T* of the system while keeping all the other thermodynamic variables fixed, and find that the transition occurs at some critical temperature *T _{c}*. When

Some model systems do not obey this power law behavior. For example, mean field theory predicts a finite discontinuity of the heat capacity at the transition temperature, and the two-dimensional Ising model has a logarithmic divergence. However, these systems are an exception to the rule. Real phase transitions exhibit power law behavior.

Several other critical exponents - β, γ, δ, ν, and η - are defined, examining the power law behavior of a measurable physical quantity near the phase transition.

It is a remarkable fact that phase transitions arising in different systems often possess the same set of critical exponents. This phenomenon is known as **universality**. For example, the critical exponents at the liquid-gas critical point have been found to be independent of the chemical composition of the fluid. More amazingly, they are an exact match for the critical exponents of the ferromagnetic phase transition in uniaxial magnets. Such systems are said to be in the same **universality class**. Universality is a prediction of the renormalization group theory of phase transitions, which states that the thermodynamic properties of a system near a phase transition depend only on a small number of features, such as dimensionality and symmetry, and is insensitive to the underlying microscopic properties of the system.

- Anderson, P.W.,
*Basic Notions of Condensed Matter Physics*, Perseus Publishing (1997). - Goldenfeld, N.,
*Lectures on Phase Transitions and the Renormalization Group*, Perseus Publishing (1992). - Landau, L.D. and Lifshitz, E.M.,
*Statistical Physics Part 1*, vol. 5 of*Course of Theoretical Physics*, Pargamon, 3rd Ed. (1994).