Brian McPeak, PhD
University of Pisa
This is a two-part post. Be sure to check out part one.
Hi all, I’m Brian—a postdoc working at the University of Pisa. 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 part one is pretty introductory, but watch out: the end gets pretty technical, and a little knowledge of Quantum Field Theory would help too.
The Conformal Bootstrap
So now we know that we can describe the critical points of statistical systems with CFTs, but what does that get us? The answer, it turns out, is “a lot”. We’ve seen in part one of this post that one of the critical exponents, ν, can be directly computed from the scaling dimension of the operator σ(x). In fact, for the two-dimensional Ising model, all the critical exponents are determined by the dimensions of the σ(x) and ε(x) operators. This is related to the fact that the content of the theory is encoded in the correlation functions, and the simplest correlation functions (those with two and three operators) are determined by the scaling dimensions.* The two-dimensional model is a little special—conformal symmetry provides enough constraints to allow us to fully solve the theory. In three-dimensions the Ising model is not exactly solvable. However, conformal field theory still helps—it gives an efficient method of numerically computing the critical exponents. This method is called the conformal bootstrap, and it includes a large set of related techniques that apply to different aspects of conformal field theory. In this section, we will describe how conformal bootstrap methods allow us to put a lower bound on the energy gap between the ground state and the first excited state in two-dimensional conformal field theories.
The Modular Bootstrap
The specific form of the bootstrap we’ll focus on is called the modular bootstrap. It’s going to get a little technical, so hang on. The first object we’ll need is the partition function. In statistical mechanics, the partition function is the sum over all the states in the theory, weighted by their “Boltzmann factor”:
Here we sum over states with distinct energies E—degeneracy is accounted for by N(E), which counts the number of states with energy E. β is the inverse temperature, 1/T, so this function adds up each state, exponentially damped by their energy over the temperature (we use k_B = 1, so temperature and energy have the same units).
In a two-dimensional theory, the partition function is similar, but two-dimensional Lorentz invariance means that states have additional charges—in addition to their energy, they have a spin. Therefore we have two temperature-like variables that show up in the partition function—we will call them τ1 and τ2. For a two-dimensional CFT, the energy of a state is equal to its scaling dimension plus a Casimir energy equal to −c/12 (this real number, c, is called the central charge of the theory, and it shows up in a lot of places, but we won’t review that here). The partition function is
Now we sum over all possible spins s and dimensions ∆. This object is actually the partition function when the background manifold of the theory is the torus. This can be seen by computing the quantum path integral on the torus. We tend to think of a torus as a cylinder with its ends sewn together to look like a donut. This is partially accurate, but it misses the fact that there are actually many tori—the two ends can be sewn together with an offset, thereby twisting the torus. A twisted torus has a different periodicity condition. If you go straight forward along one direction for the length of the torus, you won’t come back to where you started—you’ll be shifted to the left or right. This is shown in the above picture, which was borrowed from the excellent review “Applied Conformal Field Theory,” by Paul Ginsparg. The complex torus is defined by three real numbers—two periodicities (one in each direction) and the amount of twisting. However, in conformal field theory, scale invariance means that the physics doesn’t depend on the total size of the spacetime manifold. As is customary, we set the length of the spatial dimension to 1. Then the two remaining parameters, the time-periodicity (τ2) and the twisting of the torus (τ1), are combined into one complex parameter τ = τ1 + iτ2, which is called the modular parameter. These are exactly the parameters which appear in the partition function above.
It is sensible that the partition function might depend on the manifold the theory lives on, but it may seem a little mysterious that the periodicity of the manifold shows up in the partition function in the same place as the inverse temperature in the statistical example. This is actually a general feature, and is related to the fact that QFT at finite temperature is periodic in imaginary time. That means that partition functions with temperature T are the same as Euclidean (imaginary time) path integrals with periodicity β, which is a fact that can be derived from the path integral.
Now let’s think a little harder about this modular parameter τ, which describes the length and twisting of our torus. First of all, we can’t twist too much. If we twist the torus a little bit, then going around the torus in the temporal direction will move us a little bit to the left or right of where we started in the spatial direction. But if we keep twisting more and more, eventually the amount of twisting will be exactly the size of the spatial circle—that is, 1, and we will end up back where we started. So it turns out that
There’s a further transformation that doesn’t change the partition function, which essentially swaps the space and time directions. This is actually the same thing as a duality that shows up in the Ising model called the Kramers-Wannier duality, which relates the theory at high temperatures to the theory at low temperatures. Mathematically, it acts by τ → −1/τ. So we have a second invariance:
These two transformations that don’t change the partition function are called modular transformations. Conformal invariance is required to ensure that the τ → −1/τ is a symmetry, so general partition functions are not invariant under modular transformations. We will now use them to derive something very non-trivial about the theory. First we rewrite the second invariance as a “crossing equation”:
Recalling the form we used above, with ∆ and s, we find
This equation is true for any value of τ1 and τ2, so we may also take τ derivatives of this equation, and it will still hold. Let’s define the one-derivative and three-derivative operators
and let’s make another definition for convenience:
Now we can act these operators on our crossing equation. We need one more fact first: the point (τ1, τ2) = (0, 1) is invariant under the second modular transformation (23). This simplifies the crossing equation greatly, essentially because it means that the two partition functions in the “crossing equation” will be equal term-by-term (rather than only being equal after the sum, which is the case for generic values of τ). The result is the following:
I’ve skipped a few steps—if you’re interested in how this works, you may want to fill them in yourself. This equation is telling us that a bunch of terms sum up to zero. The idea of the bootstrap is this: unless every term is zero, this implies there must be positive and negative terms. In particular, because N and e−2π∆∗ must be positive, it means there is a term which satisfies
This is a constraint on the spectrum. In fact, it is a useless constraint—it is satisfied by the ground state, for which
That is why we need the three-derivative operator, from which we obtain:
The trick is going to be finding a linear combination which vanishes on the ground state. This will let us put a bound on the first excited state.** To do this, we define the new operator
which is constructed to be zero on the ground state and negative on the high-∆ states. When we apply it to the crossing equation, it will imply that there must exist a low-∆ excited state! Specifically, when we act on the crossing equation, we find
The ground state has ∆⋆ = −∆0, so this equation implies that the summand vanishes on the ground state. For states with very large values of ∆⋆, the ∆3⋆ term dominates, and the summand is negative. The crossing equation requires positive and negative terms, so there must be a term where the summand is positive. If we do a little algebra, we will find that the summand is negative if ∆⋆ < ∆0. Therefore, the spectrum of the two-dimensional CFT must include a state which has
So to recap, we found that the one-derivative crossing rule (28) implies there is a state with ∆ < c . This is satisfied by the ground state. Then we found a three-derivative crossing rule, (33). Since it vanished on the ground state, it implies there must be an excited state with ∆ < 6c .
This is a very non-trivial bound on the gap between the ground state and the first excited state of the theory. This bound may be improved by including higher-derivative terms in the linear functionals, and it may be generalized by considering global symmetries, or including τ1 derivatives, or in a number of other ways. All of these methods comprise the modular bootstrap program.
The Conformal Bootstrap
The modular bootstrap is one of the simplest bootstrap methods in conformal field theory. Unfortunately, it is not known how to apply it in higher dimensions, because there is no modular invariance to constrain the partition function (there are some known versions of modular invariance in higher dimensions but they fail for various reasons). There is a more general technique which goes by the name of the “conformal bootstrap,” which uses the conformal symmetry to constrain four-point functions rather than the partition function. The idea is the same though—take the difference between the four-point function and the four- point function which has been transformed by crossing symmetry. Then apply various derivative operators. If the result is always positive, you can rule out that theory from being consistent. This approach is more technically complicated because the four-point functions take a very complicated form, so we won’t go into it here. But we should point out that it was through this method that the best theoretical prediction of the three-dimensional Ising critical exponents, some of which we encountered above, have been obtained. The results agree with those obtained by Monte-Carlo simulations but are significantly more precise. Interestingly, however, these methods have led to a large (8σ!) discrepancy between these theoretical predictions and experiment.
In 1992, an experiment aboard the space shuttle STS-52 took advantage of the low pressure environment to measure a number of properties of the superfluid phase transition of liquid helium. The results for one of the critical exponents—the divergence of the specific heat—disagree significantly with both the Monte-Carlo simulations and the conformal bootstrap calculations. Further calculations are required to fully understand this disagreement. These calculations will certainly be done, so we will have to wait to find out the answer to this puzzle.
And that concludes part two!
And, thank you once again to Andrew Hanlon for several rounds of thorough editing, and thank you to the Theory Girls for the opportunity to write this post!
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).
*CFTs do have information beyond the scaling dimensions—these include the couplings between different operators, and operators’ charges under the various symmetries of the theory, which includes the spin, and potentially their electric / magnetic charges if there are global symmetries.
**Actually, I’ve done something a little not-okay here by using this form of the partition function instead of the Virasoro characters, but this leads to the same bound.
Image Credits:
(1) Liquid Helium: wiki https://en.wikipedia.org/wiki/Liquid_helium#/media/File:Liquid_helium_Rollin_film.jpg