The tricritical Ising CFT and conformal bootstrap

The tricritical Ising CFT is the IR fixed-point of $λϕ^6$ theory. It can be seen as a one-parameter family of CFTs connecting between an $\varepsilon$-expansion near the upper critical dimension 3 and the exactly solved minimal model in $d=2$. We review what is known about the tricritical Ising CFT, and study it with the numerical conformal bootstrap for various dimensions. Using a mixed system with three external operators ${ϕ\simσ,ϕ^2\sim ε,ϕ^3\simσ’}$, we find three-dimensional “bootstrap islands” in $d=2.75$ and $d=2.5$ dimensions consistent with interpolations between the perturbative estimates and the 2d exact values. In $d=2$ and $d=2.25$ the setup is not strong enough to isolate the theory. This paper also contains a survey of the perturbative spectrum and a review of results from the literature.

💡 Research Summary

This paper presents a comprehensive study of the tricritical Ising conformal field theory (CFT) using a combination of analytical review and state-of-the-art numerical conformal bootstrap techniques. The tricritical Ising CFT is the infrared fixed point of the λϕ^6 theory and is physically realized at multicritical points in systems with Z2 symmetry, such as the Blume-Capel model. It is conjectured to form a one-parameter family of CFTs, continuously connecting the perturbative ε-expansion near its upper critical dimension d=3 and the exactly solvable minimal model M(5,4) in d=2.

The author first provides a thorough review of known results, including the six-loop ε-expansion for scaling dimensions, exact values in d=2, and the physical context of tricritical phenomena. Some new order-ε anomalous dimensions are also computed using the one-loop dilation operator method.

The core of the work lies in the numerical bootstrap investigation. The strategy involves studying a mixed correlator system comprising all four-point functions of the three lowest-lying Z2-odd and even scalar operators: {ϕ, ϕ^2, ϕ^3} (denoted {σ, ε, σ’} in 2d). To isolate the theory, physically motivated gap assumptions are imposed, notably a large gap in the Z2-odd sector above ϕ^3, reflecting the absence of ϕ^5 in the expected spectrum.

Employing the modern “navigator” minimization approach within the Simpleboot framework, the author conducts searches in fractional dimensions. The main success is the discovery of closed, three-dimensional “bootstrap islands” rigorously bounding the scaling dimensions (Δ_ϕ, Δ_(ϕ^2), Δ_(ϕ^3)) in d=2.75 and d=2.5. These islands are consistent with smooth interpolations between Padé approximants of the ε-expansion and the exact 2d values, providing strong non-perturbative evidence for the continuous family of CFTs.

However, with the current setup and derivative order (Λ=19), similar islands could not be isolated in d=2 and d=2.25. Navigator searches in these dimensions did not converge to isolated regions but instead drifted, indicating the presence of a “peninsula” rather than an island. This suggests that either higher numerical precision (increased Λ) or an expanded bootstrap system (e.g., including more external operators or scanning over OPE coefficients) is required to fully pin down the theory closer to two dimensions.

The paper also extracts estimates for a few low-lying operator dimensions in d=2.5 using the extremal spectrum method and discusses comparisons with earlier results. Finally, the author outlines several promising directions for future work, including strategies to find islands in d=2 and d=2.25, the need for further perturbative data to guide bootstrap studies, and the potential extension of this methodology to tricritical O(n) models and general multicritical theories. Overall, this work establishes a robust bootstrap pipeline for studying the tricritical Ising CFT and paves the way for exploring more complex non-unitary critical phenomena.