Numerical Bootstrap Results for 3D Ising and O(N) Models

1. The 3D Ising Model: Harvesting the Island

In the previous lesson, we saw the multi-correlator bootstrap analysis for a theory with $\mathbb{Z}_2$ symmetry shrink the vast allowed parameter space for the scaling dimensions $(\Delta_\sigma, \Delta_\epsilon)$ into a miniscule, isolated region. The 3D Ising CFT must live inside this island. By pushing the computational parameter $\Lambda$ (the number of derivatives used in the functional) to its limits, the bootstrap collaboration has been able to determine the location of this island with astonishing precision.

This precision on CFT data ($\Delta$s and OPE coefficients) translates directly into the most accurate theoretical predictions for the physical critical exponents of the 3D Ising universality class.

Let's dissect the key results from this reading:

• The Island: The tiny island in the $(\Delta_\sigma, \Delta_\epsilon)$ plane, shown in Figure 3 of the paper, represents the pinnacle of the multi-correlator bootstrap for this system. The coordinates of this island give us the values:

  • $\Delta_\sigma \approx 0.5181489(10)$
  • $\Delta_\epsilon \approx 1.412625(10)$

• Critical Exponents $\eta$ and $\nu$: As we discussed, these dimensions determine the two primary critical exponents for the Ising model:

  • $\eta = 2\Delta_\sigma - 1 \approx 0.036298(2)$
  • $\nu = 1/(3 - \Delta_\epsilon) \approx 0.629971(4)$

• Correction-to-Scaling Exponent $\omega$: The bootstrap can also determine properties of less-relevant operators. The leading irrelevant $\mathbb{Z}_2$-even scalar operator, let's call it $\epsilon'$, has a dimension $\Delta_{\epsilon'}$. This dimension controls the leading correction to the universal scaling behavior near the critical point. The associated exponent $\omega$ is given by:

  $$\omega ={\Delta }_{{ϵ}^{\prime }}-d={\Delta }_{{ϵ}^{\prime }}-3$$

  The bootstrap determines $\omega \approx 0.82968(23)$. This is a notoriously difficult quantity to calculate with other methods.

As you saw in Table 1, the bootstrap results are not merely consistent with other methods; they are significantly more precise. This is a profound statement about the power of using only the most basic axioms of QFT—symmetry and unitarity—to constrain dynamics.

2. Generalizing to O(N) Vector Models

The $\mathbb{Z}_2$ symmetry of the Ising model is the simplest non-trivial global symmetry. A natural and physically important generalization is to consider theories with a continuous $O(N)$ global symmetry. These are described by a vector of $N$ real scalar fields, $\vec{\phi} = (\phi_1, ..., \phi_N)$, whose interactions are invariant under rotations in this $N$-dimensional internal space.

These "O(N) vector models" describe a wide array of critical phenomena:

• N=1: This is just a single scalar field with a $\phi \to -\phi$ symmetry, which is the Ising model ($O(1) \cong \mathbb{Z}_2$).
• N=2: This model describes the superfluid transition in liquid Helium-4 and certain quantum critical points. The $O(2)$ symmetry is the rotation in the complex plane describing the phase of the superfluid wavefunction.
• N=3: This model describes the critical behavior of isotropic ferromagnets, where the order parameter is a 3D vector representing the direction of magnetization.
• Large N: In the limit $N \to \infty$, these models become exactly solvable via a $1/N$ expansion. This provides a crucial theoretical checkpoint for any non-perturbative method.

Given your interest in AdS/CFT, the large $N$ limit is a theme that will become central in the next module. Here, it serves as a valuable analytical handle on a class of interacting QFTs.

3. Bootstrap Results for O(N) Models

To apply the bootstrap to O(N) models, we consider the four-point function of the fundamental field, $\langle \phi_i(x_1) \phi_j(x_2) \phi_k(x_3) \phi_l(x_4) \rangle$. The operator product expansion $\phi_i \times \phi_j$ contains operators that transform in different representations of the $O(N)$ group. Much like decomposing a product of angular momentum states in quantum mechanics, we have:

This means the four-point function's crossing equation decomposes into several separate equations, one for each representation exchanged in the OPE, providing even more constraining power.

By studying the system of correlators for the $O(N)$ vector $\phi_i$ and the leading $O(N)$-singlet scalar $s$ (analogous to $\epsilon$ in the Ising model), the bootstrap can produce islands for each value of N.

Here are the key results for the O(N) models:

• Bounds for all N: The bootstrap can produce rigorous bounds on $(\Delta_\phi, \Delta_s)$ that are valid for any $N$ and become stronger for specific integer $N$. The "3D O(N) Bounds" plot you viewed shows a series of kinks, each corresponding to a different $O(N)$ model. As $N \to \infty$, the kink approaches the known analytic value, providing a strong consistency check.

• The O(N) Archipelago: Just as with the Ising model, combining correlators (e.g., $\langle\phi\phi\phi\phi\rangle$ and $\langle\phi\phi s s\rangle$ etc.) crushes the allowed regions into a series of islands, one for each $N$, often called the "O(N) archipelago". This allows for high-precision determinations of the critical exponents for each of these universality classes.

• The Lambda Point Puzzle (O(2) Model): One of the most celebrated results concerns the O(2) model, which describes the $\lambda$-point (superfluid) transition in Helium-4. For years, there was a persistent $8\sigma$ discrepancy between the most precise space shuttle experiment and the best Monte Carlo simulations for the critical exponent $\alpha$ (related to $\Delta_s$). The conformal bootstrap provided a new, independent, and ultra-precise theoretical value:

  • Experiment: $\alpha = -0.0127(3)$
  • Bootstrap: $\alpha = -0.01527(21)$
  • Monte Carlo: $\alpha = -0.01507(21)$
    The bootstrap's agreement with Monte Carlo provided strong theoretical evidence that the discrepancy likely originates from subtle systematic effects in the experiment or its theoretical modeling, a major clarification in condensed matter physics.

Conclusion

In this lesson, you have seen the tangible outputs of the modern conformal bootstrap. By turning the axioms of CFT into a large-scale numerical optimization problem, this method delivers some of the most precise predictions in all of theoretical physics.

Key Takeaways:

• 3D Ising Model: The multi-correlator bootstrap isolates a tiny island in the space of CFT data, yielding values for critical exponents ($\eta, \nu, \omega$ etc.) that are more precise than any other method, including Monte Carlo simulations and experiments.
• O(N) Models: The bootstrap framework generalizes elegantly to theories with continuous symmetries. It produces an "archipelago" of allowed islands, one for each O(N) model, and its predictions match analytic results in the large N limit.
• Impact on Physics: The bootstrap's results are not just numerical curiosities. They provide sharp, falsifiable predictions and have played a key role in resolving long-standing discrepancies between theory and experiment, as exemplified by the O(2) model and the Helium superfluid transition.

Next Lesson Preview:

We have reached the summit of our module on the modern bootstrap. We now pivot to an entirely different, yet deeply connected, frontier: Holography and the AdS/CFT Correspondence. You will learn about the crucial role of the large N expansion, which we briefly touched upon, in dramatically simplifying certain quantum field theories. This will set the stage for the holographic principle and the revolutionary idea that a d-dimensional CFT can be dual to a theory of quantum gravity in a (d+1)-dimensional universe. The bootstrap provides essential, non-perturbative data that serves as a lamppost in the often-dark landscape of quantum gravity.