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Phase Transitions II

Published onFeb 05, 2024
Phase Transitions II
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Abstract

In the first part to this article we have covered the basics of phase transitions. Let us continue the discussion here.

Examples And Models

After reviewing some general classification of phases and transitions, it is time to turn to models. Let us first discuss why it makes sense to look at models.

Why Do Models Work?

In a model a physical system is broken down using a crude description that neglects most of the messy details. For example a solid might become an infinitely large collection of atoms on a regular lattice that only interact with their nearest neighbour. One thus neglects long-range interactions, lattice defects, etc. A gas might be described by a large number of hard balls in a box and one often assumes their density is low and they only interact through collisions (i.e. the gas is ideal). There are many more examples and in light of our knowledge of the details of any such system it is valid to raise the question of why crude models even yield any insight at all. Should the many details we ignored not change just about anything?

A short answer would be the following: Since we are concerned with the systems behaviour on large scales (roughly the size of the entire system which is much larger than the individual atoms) many of the details “average out”. Most microscopic details of a system become irrelevant on large scales - this is one of the key takeaways from the Renormalisation Group which provides a more detailed answer. Without going into detail, it often suffices to regard the simplest interactions on a microscopic level to understand the general structure of a systems phase diagram. In a sense one has to leave out the right details.

Ising Model

One of the most common examples of a phase transition model is the (ferromagnetic) Ising model. It serves as a starting point to describe real magnetism since it is simple yet shows a transition between a ferromagnetic state at low temperatures and a paramagnetic state at high temperatures.

Let us start with the simple case of spins1 {Si=±1:i=1,...,N}\{S_{i}=\pm 1: i=1,...,N\} on a cubic lattice in dd-dimensions with NLN_{L} sites in each direction and N=NLdN=N_{L}^{d} sites in total. Let us also assume periodic boundary conditions which will not affect the qualitative properties of a macroscopically-sized piece of matter.

The system’s Hamiltonian in an external magnetic field BB is given by

H({Si})=Bi=1NSiijJijSiSj(1)\begin{aligned} H(\{S_{i}\})=-B\sum_{i=1}^{N} S_{i} - \sum_{\langle ij\rangle} J_{ij}S_{i}S_{j}-\cdots \end{aligned} \tag{1}

where the coupling Jij>0J_{ij}>0 weights next-neighbour interactions ij\langle ij\rangle between spins on neighbouring sites. Interactions between more distant spins are not spelled out and can be ignored for the moment – they are in fact not relevant to our discussion as long as they are not too long-ranged. At fixed T>0T>0 the partition function is again given by a sum over all possible configurations of the spins {Si}\{S_{i}\},

Z={Si}eβH({Si}),(2)\begin{aligned} Z=\sum_{\{S_{i}\}} e^{-\beta H(\{S_{i}\})}, \end{aligned} \tag{2}

and the free energy by F=1βlogZ.F=-\frac{1}{\beta}\log Z. An important thermodynamic parameter is the magnetisation

M=1Ni=1NSi[1,+1](3)\begin{aligned} M=\frac{1}{N}\sum_{i=1}^{N}S_{i}\in [-1,+1] \end{aligned} \tag{3}

as it couples to the thermodynamic energy by

U=TSBM.(4)\begin{aligned} U=TS-BM. \end{aligned} \tag{4}

It is easy to see that at temperature T=0T=0 and B>0B>0 the system will be in a specific ground state, namely the state Si=+1S_{i}=+1 i=1,,N\forall i=1,\dots,N. For B<0B<0 all spins take on Si=1S_{i}=-1. The amplitude B\lvert B\rvert does not influence ground state at T=0T=0. That means that as the external magnetic field changes sign at zero temperature the system must jump from one ground state to another at B=0B=0. In fact it can be shown - for example using Mean Field Theory – that in dimensions d>1d>1 this behaviour qualitatively persists for temperatures below a non-zero critical temperature TCT_{C} called the Curie temperature. Below TCT_{C} it is energetically favourable for the spins to align even with no external field, B=0B=0. Above TCT_{C} the thermal fluctuations and the increased entropy mean an external field B0B\neq 0 is needed to reach a net magnetisation M0M\neq 0. In the latter case there is a smooth transition in the thermodynamic properties of the system as BB changes sign, which is not present in the former case.

Since M=dFdBM=-\frac{dF}{dB} is a 1st1^{\text{st}}-order derivative of the free energy and discontinuous at the phase transition (0T<TC0\leq T<T_{C}) it is a first order by transition by the Ehrenfest classification. It turns out that at the critical temperature it becomes a second order transition.

The two ground states for B>0B>0 and B<0B<0 are connected by a spin-flip operation SiSi,BBS_{i}\mapsto -S_{i}, B\mapsto -B. This symmetry is inherent to the Hamiltonian H({Si})H(\{S_{i}\}) and its ground state for T>TCT>T_{C} but broken below the critical temperature making the symmetry broken by the transition Z2\mathbb{Z}_{2}.

The qualitative dependence of M on T with no external field B = 0. We have argued that M =  ± 1 at T = 0 and it turns out the phase transition persists for positive T < TC in dimensions d > 1. Which of the legs of the graph the sytem chooses depends on boundary conditions or the way in which B → 0. Either way below TC the system shows a net magnetisation M ≠ 0 which flips sign as B does. The qualitative function here is derived in the Mean Field approximation. Taken from Tong, David (2017). Statistical Field Theory. https://www.damtp.cam.ac.uk/user/tong/sft.html.

The phase diagram for d > 1-dim. Ising Model. For transitions from B > 0 to B < 0 and vice versa there is a phase transition below the critical temperature TC. As one goes from T > TC to T < TC at B = 0 the system will spontaneously choose a ground state with positive or negative m. The point at B = 0, T = TC is a second order phase transition. From Tong, David (2017). Statistical Field Theory. https://www.damtp.cam.ac.uk/user/tong/sft.html.

Liquid-Gas Transition

Let us take a closer look at the liquid-gas transition for e.g. water. We focus on the region of the phase diagram close to the critical point.

There is an obvious similarity with the Ising model’s phase diagram if we replace BpB\leftrightarrow p. The only difference lies in the curved line of phase transitions. In fact the two transitions are basically “the same”, meaning that the critical points behave identically in the way quantities diverge and the symmetry that is broken by the phase transition is the same. We can make this more plausible by looking at a very crude model of a gas (water vapour).
Let us assume the same lattice as for the Ising model to describe the possible positions of gas atoms/molecules. Assuming discrete spatial positions seems a crude approximation but with a lattice spacing that is much smaller than the observational scales (so NL1N_{L}\gg 1) it becomes plausible. Label the particle number at site ii by nin_{i}. We assume no two particles can take the same site (“hard-core lattice”) so ni=0,1n_{i}=0,1. Then the Hamiltonian would certainly include a term coupling the particle number with the chemical potential μ\mu,i.e. μN=μi=1Nni\mu N=\mu\sum_{i=1}^{N}n_{i}. Assuming neighbouring particles attract each other – as they do in e.g. polar solutions – we can add a next-neighbour interaction coupled by J>0J'>0. The simple model Hamiltonian reads

H({ni})=Jijninjμi=1Nni.(5)\begin{aligned} H(\{n_{i}\})=-J'\sum_{\langle ij\rangle}n_{i}n_{j}-\mu\sum_{i=1}^{N}n_{i}. \end{aligned} \tag{5}

This is just the Ising model Hamiltonian (restricted to nearest neighbour-interactions) with the identification Si=2ni1S_{i}=2n_{i}-1 and a rescaling of 4J=J4J'=J. As such it is less surprising to find similarities between the two phase transitions and it is encouraging to find the principles described in section 4.1 realized in a sample model.

Critical Opalescence

We have established that many liquid-gas transitions behave very similarly to the Ising model and both types of systems undergo a 2nd2^{\text{nd}}-order phase transition at the critical point. There the fluctuations become self-similar meaning that fluctuations on small and large scales are statistically the same. Such scale-invariant configuration is best described by Renormalisation Group methods. Without going into detail here we can note one very interesting effect: critical opalescence.

In the liquid-gas system light passing through is absorbed and refracted differently depending on its wavelength. Very shortly speaking only light with wavelengths that roughly correspond to length scales in the system (particle size, size of fluctuations in the density distribution, ...) can be scattered. If at the critical point density fluctuations on all length scales (up to the size of the sample) exist then light of all colours will be scattered and the system becomes opaque.

Below TCT_{C} there will usually be a liquid phase and above TCT_{C} liquid and gas can coexist so there exist two separate phases.

Example for an experimental realisation of critical opalescence. Some see-through liquid at T < TC (left) is heated to the critical temperature TC (middle) and becomes opalescent. It is then heated above TC (right) and the liquid and gaseous phase start to separate. Both of those are see-through again. Cropped from M. Buklesk, V. Petruševski on www.scielo.org.mx/scielo.php?script=sci_arttext&pid=S0187-893X2019000100111 (accessed 2023-12-03) under CC BY-NC-ND 4.0 Deed.

Quantum Phase Transitions

The phase transitions we have discussed so far are referred to as thermal phase transitions. We have neglected quantum fluctuations completely. This is reasonable since at positive temperature they are heavily suppressed relative to thermal fluctuations. While it is not theoretically impossible to find quantum fluctuations on small scales their effect on long-range order (that is created or destroyed by a phase transition) vanishes in the thermodynamic limit NN\rightarrow \infty.

There is however a class of transitions at T=0T=0 called Quantum Phase Transitions (QPTs). They arise as some external parameter, usually the pressure pp, is varied at T=0T=0 and they are interesting for many reasons. Firstly, a QPT at T=0T=0 cannot be measured experimentally. However in the vicinity of such a QPT its effects can be measured as power law scaling of quantities. The state is given by the ground state relative to quantum fluctuations (at T=0T=0) with thermal fluctuations added on top. This behaviour is cut off by high temperatures but it is believed that in many systems the presence of a QPT at T=0T=0 influence and possibly explains uncommon effects up to relative high temperatures. For more on QPTs see [1].

Closing Remarks

We mentioned why phase transitions are important from both practical and theoretical considerations and discussed the Ehrenfest and Landau classifications. Both are sensible approaches to classifying phase transitions that also come with problems. We have discussed the liquid-gas transition in water and the Ising model phase transition in detail. We have also hinted at how microscopic behaviour drives macroscopic, unexpected behaviour and how phase transitions connect to Quantum Field Theories. There is much more to phase transitions than water and the Ising model. Let us mention as one area of current research the exact structure of the phase diagram of Quantum Chronodynamics (QCD)-matter. Let us close on the following remark: Despite being motivated by everyday experience, our investigation of phase transitions quickly leads to deep theoretical insights and connects many areas of physics.

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