We discuss what phase transitions are, why they are of interest in many different areas of physics and how they facilitate an understanding of the interplay between simple microscopic laws and complex macroscopic behaviour. We focus on fundamental concepts accompanied by a few simple examples.
Introduction
Phase transitions come in many shapes. What unites them is some discontinuous change in the behaviour of a macroscopic system as it undergoes a change in external parameters. This is both exciting and rare – Nature seems to behave smoothly in most cases. What is even more puzzling is that such unexpected behaviour can arise even if the constituents of the system themselves do behave smoothly. How can they then give rise to violent transformations in macroscopic systems?
Think about the water you boil, one minute happily sitting in a pot, getting hotter and hotter, a seemingly quiet and boring1 system where physical quantities such as its density ρ only slowly change. The next minute it suddenly – and loudly – rises to the ceiling with your pot shaking from the violent outburst of a sudden change of state. Now there are two systems consisting of the same molecules but with either of the two densities ρliquid<ρgas. As the boiling continues the denser system disappears and the pot empties. At no time will you find water with a density between ρgas and ρliquid.
Phase transitions are often sudden and unexpected, they arise from a competition between different states and they break with traditions in physics. Additionally, while only occuring in macroscopic systems phase transitions are ultimately determined by microscopic properties of those systems. This dependence is highly non-trivial and it took physicists well into the 20th century to understand2. Ultimately, many are only explained by means of the Renormalisation Group (RG). Historically, phase transitions led K. Wilson to formulate the modern description of RG leading to a better understanding of Quantum Field Theory (QFT).
Phase transitions connect to some of the most fundamental principles in physics. They break symmetries and connect physics on different scales so they have every reason to be interesting!
The liquid-gas transition for water.
What Is A Phase Transition And Why Is It Relevant?
Let us start by classifying what a phase transition is. To do so, we first need to revise some statistical physics in order to understand what a phase is.
Statistical Mechanics
The goal of Statistical Physics is to describe the collective behaviour of a system that is made up of some very large number of constituent objects. Typical examples include liquids and gases (consisting of very many individual molecules) and magnets (consisting of interacting spins, usually on a crystal lattice).
An estimate for the number of constituents in some macroscopic piece of matter is given by the Avogadro constant NA:
NA≈6.022×1023(1)
Macroscopic objects of relevance to our everyday life are composed of upwards of ≈1023 constituents. Describing all relevant degrees of freedom (DOF) would be neither useful nor needed for a description on macroscopic scales. Instead we usually employ thermodynamics. A generic system is then described by a few parameters which often correspond to averages taken over all constituents. The exact configuration is not mentioned. Statistical Mechanics can describe the emergence of such an averaged description from microscopic states (micro-states). In the following we use units with kB≡1.
Systems And Couplings
A statistical system can take on an enormous number of (micro-) states. Many of these micro-states realize the same macroscopic (“macro”) state which is characterized by a few quantities such as pressure p, volume V, temperature T, entropy S, energy E, chemical potential μ, particle number N, ... They are connected to the energy by the first law of thermodynamics:
E=TS−pV+μN(2)
Let us assume for the moment that these are all the relevant macroscopically accessible parameters.
The entire physics, i.e. all thermodynamic properties, is contained in the partition function Z. For a system with fixed V, N and temperature T it can be calculated by a weighted integral over phase space,
Z(T,V,N)=e−βF=∫ΓV,Ne−βH,(3)
with β=T1, the thermodynamic free energy F and the Hamilton function H assigning to each state its energy. The integral is taken over ΓV,N, the N-particle phase space with a restriction to the accessible volume V.
As all (thermodynamic) properties can be derived from Z we might ask ourselves how they behave as we vary some of the parameters that were previously fixed: e.g. T,V or N.
More generally, we can think of the partition function as dependent on some set of external parameters{Ki:i=1,...,n}, e.g. {T,V,N,...}. Not all quantities we could choose would be independent but let us assume we have chosen an independent set {Ki}.3 A generic physical quantity (an observable) is given by a combination of l derivatives of logZ,
∂K1⋯∂Kl∂llogZ({Ki}).(4)
Examples include (specific) heat capacities, expansion coefficients or densities. Let us call the space of {Ki}-values the parameter space.
In large parts of the parameter space all thermodynamic quantities change smoothly. However, there are some lines and (hyper-) surfaces in the n-dimensional parameter space across which physical observables change abruptly. These transitions are called phase transitions and they appear with different (co-)dimensions, meaning they can trace lines, surfaces etc. in parameter space. Their existence divides the so-called phase diagram, a plot of parameter space with phase transitions indicated, into different parts.
In short, phase transitions describe non-analytic behaviour of thermodynamic systems as a function of some parameter.
Attempts At Classification
A first attempt to classify phases of matter might be declaring two phases to be different if trajectories through the phase space that lead from one to the other cross a phase transition.
However, this raises questions: Do all such trajectories have to cross a certain phase transition? If not, then which trajectory do we use for classification? The liquid-gas transition of water, a common example of two phases of the same substance, are separated by a phase transition – but not for all values of p and T (which we can take as our {Ki}). This is confusing – does it mean they are in fact the same phase and our intuition was wrong? Or is do we need a better classification? Also the way in which quantities change across phase transitions is highly different for across systems – and even for the same system at different parameter values. This calls for a better understanding and classification of transitions.
Ehrenfest Classification
A widely known way to classify phase transitions was proposed and introduced by Ehrenfest.
Phase transitions come with more or less abrupt, non-analytic changes. Since all physical quantities of interest can be derived from the free energy F=−TlogZ, Ehrenfest proposed to classify phase transitions by the degree of non-analyticity of F. If the first discontinuity appears for an m-th derivative of F the transition is called “of order m”.
According to the Ehrenfest classification, many of the most prominent examples are phase transitions of first order. When a liquid boils its density jumps. Since the density can be written as a first derivative of the free energy (density), the transition is of 1st order. It is also for these transitions that we observe latent heat, a discontinuity in the energy across a transition – which is a first derivative of F. Often, one simplifies things and only distinguishes between 1st-order transitions and continuous transitions, meaning all higher order-transitions.
Despite being useful the Ehrenfest classification suffers from several problems. The most obvious becomes apparent in a system such as ordinary water. There exists a phase transition (of 1st order) between the liquid and gaseous phase, however it is also possible to transition from one to the other without crossing the phase boundary by simply passing through a high-temperature regime, see Figure 2. More generally speaking, just because there exists as path between the two points in phase space with non-smooth free energy along it does not mean this is true for all paths. There also is no canonical path we could consider when classifying a potential phase transition. So does a phase transition necessarily separate two different phases?
A sketch of the phase diagram of water which is in part accessible by everyday experience. TP marks the triple point where liquid gaseous and solid phases can (co-)exist and CP refers to the critical point. Here, the discontinuous changes across the transition line become continuous and the phase transition is of 2nd order. At the critical point density fluctuations become scale-invariant, a phenomenon explained by RG. Taken from [2].
A more general sketch of the simplified phase diagram of water. Source: Cmglee, https://commons.wikimedia.org/w/index.php?lang=und&title=File%3APhase_diagram_of_water_simplified.svg (CC BY-SA 3.0).
Classification By Symmetry
A different classification of phases was introduced by Landau. He made the observation that phase transitions correspond to breaking (or introducing) a symmetry of the underlying microscopic system. A crystalline (i.e. regular) solid that melts has its lattice structure broken: before, the molecules were aligned along a lattice with a certain rotational symmetry (e.g. invariant under 90° rotations around the lattice axes). After the transition they are much less ordered as a liquid state is fully symmetric under rotations (isotropic). Since a symmetry cannot be introduced “slowly”, two phases with different symmetry group G must be separated by a phase transition. A change from one symmetry to another will necessarily involve some drastic change. It is therefore believed that any material with solid and liquid phase will exhibit a phase transition between these two phases. Symmetry-based arguments do generally not give the kind of phase transition according to the Ehrenfest classification.
Both gases and liquid have the same symmetry group consisting of translations and rotations since both are unordered and assumed to be translationally irrelevant.4
GsolidGgas=R3⋊Grot=Gliquid=R3⋊SO(3).(5)
Here R3 denotes the translation symmetry and for a crystalline material Grot is the crystals corresponding discrete rotation group.5
Classification by symmetry explains why the transition line between liquid and gas in Figure 2 is allowed to end for high T. Both “phases” share the same symmetry so there need not be a phase transition between them. It then makes sense to define phases of matter as different exactly if they have a different symmetry group.
This clashes with our intuition not only in the case of the liquid-gas transition. Many metals can exist in different solid states with different symmetry groups and it is not entirely clear yet how certain quantum effects that give rise to rare symmetries should be classified. Yet, the Landau classification is very useful. All states can be sorted into one of the many phases depending on their symmetry and often this symmetry alone explains partly the low temperature-behaviour.
A detailed phase diagram of water. As we can see there are in fact many different solid phases that usually go by the name “ice”. They are distuingished by their internal symmetry and phase transitions between them exist. Source: Cmglee, https://en.wikipedia.org/wiki/File:Phase_diagram_of_water.svg (CC BY-SA 3.0).
Why can the Liquid-Gas transition exist?
We have established that the liquid and gaseous phases can be regarded as the same. But then why is there a phase transition in between them? The answer is subtle and slightly disappointing. In short terms: for certain values of T and P (between the points TP and CP in Figure 2) a Z2-symmetry emerges. Its breaking explains why in the Landau classification a phase transition between two parts of the same phase can be possible. We have seen that distinguishing between liquid and gas only makes sense for a certain regime of (P,T)-values. More precisely, talking about the two different “phases” only really makes sense on the line of transition. Here, there is a clear distinction between two competing (ground) states of the system with different macroscopic properties. One can use the density as an order parameter6 which jumps between ρgas and ρliquid across the transition. There is an emerging Z2-symmetry that is broken and allows a Landau classification. Precisely on the line of transition both states have the same energy and can coexist. The symmetry is not present for high temperature where the two “phases” mix in a continuous matter and F is smooth. The Z2-symmetry is also the reason why the liquid-gas transition is in many regards identical to the Ising model’s phase transition with an intrinsic Z2-symmetry, see the second part of this article and [1].
Why Do Phase Transitions Exist?
Having built some intuition we can attempt an explanation of the exact mechanism of a pase transition.
Since in practically all cases one of the parameters {Ki} is the temperature T all thermodynamic properties can be derived from the free energy F(T,V,N,...). There are heuristically speaking two different contributions to F namely energetic and entropic. While the system would in principle like to take on the lowest energy-state (ground state) at positive temperature there are more possible states with higher energy and so by this entropic argument the resulting ground state is a compromise between both contributions. We can now imagine two ways in which a phase transition might take place.
The first would be a ground state-change: what was the lowest energy macrostate might not be the lowest energy state with an infinitesimally changed parameter Ki (e.g. the ground state symmetry might change). Despite there being thermal fluctuations around that ground state (entropy) such a change would trigger a macroscopic transition. Secondly a lowering of the temperature would weaken the fluctuations (entropic contributions to F) and could shift the balance towards an ordering forming. This argument explains why many materials from crystalline solids at low temperatures.
From a thermodynamic, i.e. averaged, description any phase transition comes as a surprise as microscopic argument involving ground state-symmetries and entropy are not accessible. Phase transitions usually take place very quickly relative to macroscopic time scales. The transition from the old (macro-) state to the state now favoured by F happens via thermal fluctuations. At T>0 there is fluctuation among the microstates realizing the macrostate and these fluctations bring the systen from the old to the new microstate.
Liquid water has very little structure. It is translationally invariant and isotropic, i.e. has no preferred direction – even on smaller scales than shown here. Source: Francesco Ungaro via www.pexels.com.
When water molecules are cooled down they start forming a hexagonal spatial structure, here in the form of a snow flake. Under normal conditions this arrrangement would have minimised their energy even at higher temperatures but the thermal fluctuations (entropy) disallowed the formation of regular crystals. Hexagonal ice Ih is not the only crystalline phase solid water can take but it is the only one of relevance to our everyday lifes. Source: Charles Schmitt, https://en.wikipedia.org/wiki/Snowflake#/media/File:Snowflake_Detail.jpg (CC BY-SA 4.0).
Why Can Phase Transitions Even Exist?
Phase transitions seem very natural to us from everyday experience. Yet, starting from a microscopic model it is not clear at all why a phase transition can even take place. In fact, for a long time it was disputed whether the free energy and partition function of some bulk material is able to describe more than one phase of some material, see [2].
Any substance we might handle is made up of a finite number N of individual parts, atoms or molecules.7 In general, there will be an energy function that assigns to constituent i an associated energy, including a kinetic and a potential part,
Hi=Ekin,i+Vi(1,...,N)(6)
where the potential part might depend on all other parts j=i, too. Either way, the energy function is generally assumed to be smooth in all quantities describing the state of i (its position, momentum, spin, ...) and all external parameters {Ki} (volume, temperature, magnetic field, ...).
The overall energy function is then simply given by
H=i=1∑NHi(7)
and the partition function is calculated by Eq. 3, possibly with a sum over states replacing the integral. Both are smooth as a finite sum of smooth functions. But if Z is smooth how can we use the non-smoothness of the free energy F=−Tlog(Z) to classify phase transitions (Ehrenfest)?
The answer is given by the thermodynamic limit. It consists in taking N→∞ in the construction as any macroscopic piece of matter has so many constituent parts that finite-size effects will only give a small correction to the behaviour we expect for an infinitely large sample. An infinite sum over smooth functions is not bound to be smooth anymore and can show the type of discontinuous behavior referred to by the Ehrenfest classification. That the thermodynamic limit works is non-trivial and some mathematical care must be taken when taking the limit.
The N→∞-limit is a very useful theoretical tool. Yet, the fundamental statement remains: there is no true discontinuity in a finite system. So whenever we observe a discontinuous change in finite-size matter what we really see is a continuous change that – far beyond any experimental precision – looks like a discontinuous change!
Now that we have covered some basic notions on phase transitions we will discuss topics such as models, the Ising model and more details on the liquid-gas transition in the second part to this article.