Summary. This is the first part of a series of two papers on using the conformal bootstrap to determine a variety of critical exponents for 3-dimensional field theories in the Ising universality class to a precision higher than ever before. This shows that the theoretical methods of operator-based quantum field theory are now developed to a very high level.
The first paper, reviewed below, provides background and formulas needed (and preliminary numerical results) for the numerical investigations described in the second paper, reviewed here.
Background. A universality class is a class of field theories with a critical point, whose members can be described asymptotically close to the critical point by a common conformal field theory.
The 3-dimensional Ising universality class is highly important because many realistic thermodynamic systems - among them all real fluids - are believed to belong to this class. (See, e.g., the discussion and references in my paper here.)
Although the Ising model (describing classically interacting binary spins on a 3-dimensional lattice) figures in the title, the body of the papers are not concerned with lattices. The title effectively refers to the name of the universality class, believed to also include the continuum limit of the 3-dimensional Ising model at its critical temperature (from which the name of the class derived).
No mathematically rigorous universality results in 3 space dimensions are known, but the physical evidence for universality is quite strong. For example, a variety of other methods ($\epsilon$-expansion of continuum field theories, high temperature expansions of lattice models, Monte Carlo studies with different lattices, and fits to experimental critical thermodynamic data of various fluids) were used in the past to approximate critical exponents and other critical data, and in spite of the different origins, the results were found to be consistent with each other. (Other universality classes are needed to model systems with different symmetries or different dimensions.)
Details. The setting is conformal quantum field theory on the 3-sphere, the 1-point compactification of 3-dimensional Euclidean space. The symmetry group is the Euclidean conformal group $SO(1,4)$ generated by translations, rotations, dilatations and inversions of 3-space. The conformal bootstrap refers to an approach that tries to reconstruct the properties of a conformal field theories from assumptions about the Wightman 4-point functions $\langle \phi_1(x_1)\phi_2(x_2)\phi_3(x_3) \phi_4(x_4)\rangle$ of the fields $\phi_j(x)$ (which may be the same or different).
The assumptions made are those believed to be valid for the scaling limit of field theories in the Ising universality class, namely:
- crossing symmetry, which in particular implies that for any scalar primary field $\sigma(x)$, the 4-point functions \[W(x_1,x_2,x_3,x_4):=\langle \sigma(x_1) \sigma(x_2) \sigma(x_3) \sigma(x_4)\rangle\] are invariant under a permutation of the factors;
- a convergent operator product expansion, whose associativity implies in particular that these 4-point functions can be written (using $x_{ik}:=x_i-x_k$) in terms of so-called conformal blocks $G_{\Delta,j}(u,v)$ as a sum \[W(x_1,x_2,x_3,x_4)=\sum_{\Delta,j} p_{\Delta,j}\frac{G_{\Delta,j}(u,v)}{x_{12}^{2\Delta_\sigma}x_{34}^{2\Delta_\sigma}}, ~~u=\frac{x_{12}^2x_{34}^2}{x_{13}^2x_{24}^2},~v=\frac{x_{14}^2x_{23}^2}{x_{13}^2x_{24}^2}\] over all scaling dimensions $\Delta$ and spins $j$;
- reflection positivity, which gives the field theory a Hilbert space structure;
- minimality of the central charge, the undetermined proportionality factor in the otherwise explictly known 2-point function of the stress tensor.
(Those interested in a rigorous background can find related material based on the conformal version of the Wightman axioms in N.M. Nikolov, K.H. Rehren and I.T. Todorov, Partial wave expansion and Wightman positivity in conformal field theory, Nuclear Physics B722 (2005), 266-296. It seems that the techniques there can be profitably combined with those in the present paper.)
The conformal blocks are explicitly known functions for which numerically efficient formulations are found. Reflection positivity implies that all $p_{\Delta,j}$ are nonnegative and gives restrictions on the possible pairs $(\Delta,j)$; in particular, all scaling dimensions are nonnegative. Crossing symmetry implies that
\[
g(u,v)=(u/v)^{\Delta_\sigma}g(v,u).
\]
One also has a normalization condition $p_{00}=1$.
These conditions together imply restrictions on the spectrum of the theory, i.e., the collection of pairs $(\Delta,j)$ that occur in the above sum with a positive coefficient, proving the presence of corresponding operators in the operator content of the theory.
The numerical techniques to probe this spectrum are tentatively developed in this paper and highly refined in the second paper; they will be discussed in the review there.
Conclusion. To realize that the Ising universality class is (numerically) characterized by the minimality of the central charge is the main novelty of the approach. That the numerical methods were developed to a point where (in the second paper) the accuracy significantly exceeds that of Monte Carlo lattice studies (the hitherto most accurate technique) makes the technique highly important.
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