Brian McPeak, PhD
University of Pisa
This is a two-part post. Stay tuned for part two.
Hi all, I’m Brian—a recent PhD graduate of the University of Michigan. I wanted to write a post that connects some classic concepts in physics—phase transitions—with an important branch of research—the conformal bootstrap. The stuff at the beginning of this post is pretty introductory, but watch out: the end gets pretty technical, and a little knowledge of Quantum Field Theory would help too.
One of the exciting parts of theoretical physics is finding unexpected connections. Often we find that physical systems which look completely different at a superficial level are partially or entirely the same when you consider their dynamics in a more abstract way—for instance at the level of the organization of the states in the theory. Erin wrote a post discussing an example of this, which is both very surprising and very important in contemporary research: holography. One of the most important things that we’ve learned about fundamental physics in the last few decades is that quantum gravity in Anti-de Sitter space has a dual description in terms of the quantum dynamics on the boundary. Other dualities in high energy physics abound, such as those between different types of string theory, between different quantum field theories, or between the same theories in different regimes.
In this post, we will focus on a particular type of these coincidences that arise between different systems near certain kinds of phase transitions. The transitions that we most commonly encounter in everyday life are the freezing/melting transition between solid and liquids, and the evaporation/condensation transition between liquids and gases. However, many physical systems on earth undergo transitions, such as the formation of carbon into diamonds due to high temperatures and pressures in the earth’s mantle, or the onset of superconductivity when certain metals reach low enough temperatures. In general, phase transitions are defined as discontinuities in the behavior of a system when you change external thermodynamic quantities such as temperature or pressure. They may be broadly classified by where the discontinuity shows up. If the free energy of the system is a discontinuous function of the thermodynamic variables (typically temperature and pressure), it is called discontinuous or first order. This discontinuity in the free energy appears as a latent heat, which is thermal energy required for a phase transition but which does not change the temperature. Most of the transitions we encounter in day-to-day life are first order. For example, boiling water at atmospheric pressure has a latent heat of 40.65 kJ/mol. This means that even after you reach the boiling point, you still have to put in a lot of energy just to convert the water to gas– 40.65 kJ (or about 10 Calories) for every mole (about 18 grams) of water!
In this post, however, we will be interested in the other class of phase transitions, which are called continuous or second order. These are transitions where the free energy is continuous across the transition (but often its derivatives are not). In the case of continuous transitions, it is often useful to define an order parameter to describe the transition. These are quantities that are zero in one phase and non-zero in another. We’ll see a few examples below.
Another really fundamental feature of continuous phase transitions is that when physical systems are near them, parts of the systems which are far away from each other can still interact. These interactions are quantified by correlation functions, which describe how quantities in the system at different positions are related. These typically decay exponentially as the distance increases—that is, a variable like the spin at some point may be highly correlated with the spins at nearby points, but it will be almost completely uncorrelated with far away spins.
A simple example is a lattice of spins, where spin sits on a corner and can be up or down. In this simple example we might quantify this by
Here we’ve used ∼ to indicate that we are considering only the scaling, and might be ignoring constant and subleading pre-factors. This equation tells us that the correlation between the spins at site i and site j decays exponentially as the distance rij between i and j increases. Since we can’t put a dimensionful quantity like distance in an exponent, we are forced to introduce another distance scale ξ, which is called the correlation length. This variable ξ can depend on the external parameters of the system, such as temperature and pressure. However we find that as the system approaches a phase transition, the correlation length approaches infinity, implying that even greatly separated spins are now correlated.
Example 1: Boiling Water
Let’s illustrate this with an example. As we know, water boils at 100 degrees Celsius. Actually, that’s only true at atmospheric pressure. As you increase the pressure, the boiling point will increase—but not for- ever. Eventually, you reach a point, called a critical point, above which there is no sharp phase transition between liquid and gas. This is shown in the figure below, which is called a “phase diagram.” For water, this point occurs at 374 degrees Celsius and 218 atmospheres of pressure–this is far above the pressures we are used to in everyday life, which is only about one atmosphere! Below the critical point, the phase transition is discontinuous, which means that there is a latent heat. At the critical pressure, however, the latent heat disappears and the transition becomes continuous. The order parameter for this transition is the difference in density between the liquid and gas phase. We will denote this by ρ. Let’s consider the pressure to be fixed at its critical value. Then, if we were to measure ρ as we approach the critical temperature Tc, we would find a simple behavior for ρ:
The absolute values mean that there will be a discontinuity in the first derivative of ρ. β is a constant called a critical exponent that is intrinsic to the system. For the water / gas transition, we have
Recall also that the correlation length diverges as we approach the critical temperature. This is also characterized by a critical exponent:
β and ν are only two of a number of critical exponents characterizing the system.
Example 2: The Ising Model
Let’s turn to another example: the lattice of spins we mentioned earlier. This model has a particle at each corner i of a square lattice with spin σi. We’ll allow the spin at each site to be up or down, so σ = ±1. The spins in this model only interact with their nearest neighbor—the interaction energy is positive or negative according to whether the spins are the same or different. The total energy of the system depends on the configuration of spins and is given by:
Here ⟨ij⟩ means that we sum over all pairs of neighboring particles, while the sum over i is just a sum over every site. Therefore, J describes the interactions within the lattice and h models the effect of an external magnetic field. This is a very simplified model of a magnet—the spin of each particle comprising the magnet points up or down, and the resulting magnetic field is the net number of spins pointing up or down. This net field is called the magnetization, given by the net magnetic field averaged over the number of spins N:
The magnetization plays the role of the order parameter of this system, just as the density did for the liquid / gas transition. At high temperatures, the spins are oriented more or less randomly—each spin has an even chance of being up or down, so the magnetization is close to zero. But as the temperature drops, thermal fluctuations stop overriding the interaction energy, and nearby spins will be more and more correlated. At zero temperature, the system relaxes into its ground state. For J > 0, this is the state where all the spins point the same way—then the material is said to be ferromagnetic (J < 0 leads to a ground state with anti-aligned neighboring spins, a configuration called anti-ferromagnetic). With all the spins pointing the same way, the magnetization will approach one. There is a phase transition between the disordered, high-temperature phase and low-temperature, aligned phase. We can model this with a similar equation to the case of water:
and we can also consider the correlation length
This model can exist in any number of dimensions—in one dimensions the “lattice” is evenly spaced points on a line, in two dimensions it’s squares on a plane, and so on. The critical exponents β and ν are different for each number of dimensions. In two dimensions, we find:
while in three dimensions, we find
These three-dimensional values were the same ones that we found for the liquid/gas transition! The remarkable coincidence that we’ve been hinting at before is that completely different physical systems can be described by the same critical exponents! This phenomenon is called universality. It tells us that, near a critical point, the behavior of a system depends on the dimension and the symmetries of the problem, but not the underlying dynamics.
The Ising model has a long history. The one-dimensional model was solved by Ising himself in his PhD thesis way back in 1924. The two-dimensional model with no external magnetic field was solved by Onsager in 1944. The two-dimensional model with a magnetic field was only solved exactly in 1989, by Zamolodchikov. In four or more dimensions, the exponents can be computed using an approach called mean field theory. Therefore the three-dimensional model is of considerable theoretical interest, because it is notoriously difficult to study. Unlike the other dimensions, its exponents are not believed to be rational numbers*, and so far they do not have a closed form expression.
Conformal Field Theory
In this section we will introduce a formalism, called Conformal Field Theory (CFT), which can be used to study the behavior of the Ising model (and many other systems) at the critical point. In some cases, this formalism will allow us to fully solve the theory. This is the case for the two-dimensional Ising model, where CFT allows us to compute the exact values of the critical exponents. In other cases, such as the three-dimensional Ising model, the theory cannot be fully solved, but CFT gives an efficient set of tools for putting rigorous bounds on the exponents. Those tools are called the conformal bootstrap, and they will be the subject of the final section. However, for the sake of simplicity we will focus on the two-dimensional Ising model in what follows.
What is a CFT?
Let’s first go over what a conformal field theory (CFT) is. This is a huge field, and we won’t be able to do it justice in this post. Let’s briefly try to give a little of the flavor.
A CFT is a quantum field theory (QFT) for which an enlarged group of spacetime symmetries—the “conformal group”—act on the states. Typically, we study QFT in flat (rather than curved) spacetime, where the symmetries are translations, rotations, and Lorentz transformations. The conformal group, how- ever, includes these symmetries and adds two more—dilatations and special conformal transformations. A typical introduction to CFTs usually involves determining the commutation relations between the gener- ators of these symmetries and showing how they each act on the fields in the theory. Here, we’re going to skip to the important part: scaling transformations. These act on the coordinates as
which means the fields transform as
In a CFT, each field φ has a positive real number ∆ associated with it—this is called the scaling dimension of φ. It’s conceptually similar to the mass dimension in a normal quantum field theory**.
This extra symmetry may seem innocuous, but it affects the structure of the theory on a fundamental level. For one thing, scaling transformations mean that there is no notion of asymptotic states, because there is no real notion of particles getting “very far apart”. This means there is no way to define an S-matrix. Therefore, it is natural for the correlation functions to play the role of the primary observables in CFTs. We observe that two-point functions in a CFT transform in the following way under scaling:
The rotational and translational symmetry imply that these functions can only depend on the difference between x and y. Only one function of |x − y| satisfies the scaling relationship we’ve written above, so the two-point function in CFT is fixed to be
From Lattice Models to Continuous Fields
So much for our lightning outline of CFTs. As we’ve seen, they are basically collections of fields, and the physical content is described by the correlation functions of the fields, which transform a certain way under scaling. The next question is: what do these models have to do with the lattice models we’ve outlined above? They are quite different after all—the CFT is a quantum field theory and the fields take continuous values. In the lattice models, we also have correlation functions, but they are at discrete points in space, and the spin takes discrete values. But the remarkable fact is there exists a CFT whose correlation functions are the same as the correlation functions for the Ising model, in the limit of zero lattice spacing. The basic idea can be summarized succinctly as:
Here a is the spacing between sites, so the CFT operator at location x corresponds to a lattice spin at the site corresponding to the integer part of x/a. It is clear from this equation that the scaling transformation of the CFT correlation function holds for the right-hand side of the equation as long as a is scaled along with x and y. To get a feeling for the fields in this CFT, let’s consider the critical exponent ν of the lattice model, defined by:
where rij is the distance between the sites. We know from Onsager’s solution that
in the two-dimensional Ising model. Therefore, a continuum CFT describing the model must have a field σ(x) with ∆ = 8. Such a CFT does exist: it turns out to have three operators
In this case, I is the identity operator, which doesn’t do anything to the states it acts on. σ represents the local spin, and ε is the local energy density.
It is important to mention that this is not a proof that the models are the same. You might call it a “physicist’s proof”—if you find that enough quantities in two different-looking models are the same, you can convince yourself they are the same model. Nonetheless, proving they are the same is more difficult***.
To be continued....
in part two, the post will focus on the details of
the conformal bootstrap.
Thank you to Andrew Hanlon for several rounds of thorough editing, and thank you to the Theory Girls for the opportunity to write this post! Please stay tuned for part two.
Citations and Acknowledgements
The first section of this post was largely inspired by Henriette Elvang’s course on CFTs.
Paul Ginsparg’s introduction to 2D CFTs is at (9108028)
The modular bootstrap was introduced in (1608.06241).
The modern conformal bootstrap was introduced in (0807.0004). For a nice introduction, see David Simmons-Duffin’s TASI lectures, (1602.07982).
The bootstrap results for the three-dimensional Ising model and the liquid helium discrepancy were reported in (1603.04436).
*For a list of classes and their exponents, see wikipedia’s page on universality classes.
**Like the mass dimension, it is essentially determined by the dimension of the theory and spin of the field in a free theory, but can be changed by renormalization effects which may be drastic in strongly coupled theories.
***One approach involves a series of transformations between the two theories. First one must prove an equivalence between the two-dimensional (classical) Ising model we’ve described, and a 1D quantum model. Then the operators of the 1D model are related to fermionic operators via a Jordan-Wigner transformation. Finally, a theory of free Majorana fermions is obtained in the limit where the lattice spacing goes to zero, and this model has the three operators described above.
(1) This is from Project Runeberg book called The key to science In swedish., Public Domain, https://commons.wikimedia.org/w/index.php?curid=472708
(2) By Matthieumarechal, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=4623701