User:Sylvain Ribault/Bootstat2021/Rychkov

Discussion on the spectrum of the 3d Ising model.

The model is numerically found in the plane $(\Delta _{\sigma },\Delta _{\epsilon })$ by bootstrapping three four-point functions of two fields $\sigma ,\epsilon$. Does this also fix the exchanged operators?

Yes, this was explored by Simmons-Duffin. He randomly picked 60 points inside the island. For each point, looked at the extremal functional and its zeros. Exchanged operators must be zeros of that functional. Working at high derivative order, the functional typically as many zeros as derivatives, so we can get about 100 zeros. Given such a spectrum we can solve crossing equations for the OPE coefficients. Doing it 60 times, we expect that operators vary little. Plot operators as fct of spin $\ell$ and twist $\tau =\Delta -\ell$. For a given spin, two operators are considered identical if their dimensions differ by at most 0.03. Then count how many of the 60 solutions have an operator with given parameters. We find a few dozen operators that are shared by all solutions, especially for low twist (less than about 4). On the other hand, we see scattered results at high twist.

From these results we could try to detect an extended symmetry in the model. But we do not see anything, for example no family of integer-spaced operators. Still, some family of operators sit on recognizable curves.

We expect that operators converge faster when they have large OPE coefficients. OPE coefficients decrease with the twist, which makes it difficult to converge towards such operators. Moreover, the density of operators grows with the twist, which makes them more difficult to resolve.

Lorentzian theory and the role of the twist

Need to think about the Lorentzian theory, which is not obvious when working on a statistical model. Some operators are most easily understood from Lorentzian viewpoint. In Euclidean space, the scaling dimension is the parameter that suppresses operators. Adding spin does not change the scaling, still controlled by the scaling dimension, with OPE coefficients of the type

${\frac {x^{\mu _{1}}\cdots x^{\mu _{\ell }}}{(x^{2})^{\frac {\ell }{2}}}}(x^{2})^{\frac {\Delta }{2}}$

In the Lorenzian, we have $x^{2}=x_{+}x_{-}$. We can choose indices $\mu _{i}$ such that numerator only involves $x_{+}$. We can take the limit $x_{-}\to 0$. The contribution in this limit is now controlled by the twist.

A theorem states that for large enough spin,

$\tau \left([O_{1}O_{2}]_{\ell ,n}\right){\underset {\ell \to \infty }{\sim }}\Delta _{1}+\Delta _{2}+2n$

with $n\in \mathbb {N}$. This works only for $d>2$. This relies on the identity being the only operator with zero twist. In $d=2$ there are many other such operators including the energy-momentum tensor. And actually the Ising model is an obvious counterexample.

Regge trajectories

Dimensions of some operators are functions on spins.

Unitarity bound is $\Delta \geq \ell +1$. From the stress tensor $(\ell ,\Delta )=(2,3)$, we have a Regge trajectory of operators with even spins. The trajectory $[\sigma \sigma ]_{n=0}$ has twists approaching $\tau \to 2\Delta _{\sigma }=1.04$, very close to the unitarity bound. Continuing this trajectory to $\ell <2$ (below unitarity bound) has some meaning.

The slopes of Regge trajectories go to one at large spin, but have varying behaviours at small spins. Do all operators fall on trajectories? This question is not very well-defined, and the answer seems to be no. If the answer was yes, we could parametrize the spectrum by Regge trajectories rather than by individual operators.

Correlation functions

We can now compute the three 4-point functions that involve only $\sigma ,\epsilon$ to very high precision. This can be used for learning about perturbed Ising models. For example, starting with several copies of the Ising CFT, we can add weakly relevant perturbations of the type $\epsilon _{i}\epsilon _{j}$. Beta function at one loop depends on three-point functions, at two loops it depends on four-point functions.