Abstract
We study the tricritical Ising universality class using conformal bootstrap techniques. By studying bootstrap constraints originating from multiple correlators on the CFT data of multiple OPEs, we are able to determine the scaling dimension of the spin field $\Delta_\sigma$ in various non-integer dimensions $2 \le d \le 3$. $\Delta_{\sigma}$ is connected to the critical exponent $\eta$ that governs the (tri-)critical behaviour of the two point function via the relation, $\eta = 2 - d + 2 \Delta_{\sigma}$. Our results for $\Delta_\sigma$ match with the exactly known values in two and three dimensions and are a conjecture for non-integer dimensions. We also compare our CFT results for $\Delta_\sigma$ with $\epsilon$-expansion results, available up to $\epsilon^3$ order. Our techniques can be naturally extended to study higher-order multi-critical points.
Highlights
The consequences of the conformal hypothesis which posits conformal invariance to the behavior of physical systems at criticality, in addition to scale invariance, are most far reaching in two dimensions, where the conformal symmetry is the infinite-dimensional Virasoro algebra
Ε0 is the lowest scalar in one operator product expansions (OPEs) and the next-to-lowest scalar in another. This is the problem we study here using the numerical conformal bootstrap—conformal field theory (CFT) with a low-lying spectrum and OPEs given by (5) [33]—and we find that the bootstrap constraints reflect many aspects of tricritical phenomena
We show that in two dimensions, the tricritical Ising CFT seems to be characterized by the property that ε00 becomes irrelevant as a function of Δε0 while keeping both Δσ and Δε fixed—much like the Ising CFT [11] which is characterized by the property that ε0 goes from being relevant to irrelevant, as a function of Δε, while keeping Δσ fixed
Summary
The consequences of the conformal hypothesis which posits conformal invariance to the behavior of physical systems at criticality, in addition to scale invariance, are most far reaching in two dimensions, where the conformal symmetry is the infinite-dimensional Virasoro algebra. Each of these models describes a universality class, and the exact knowledge of the scaling dimensions of the operators amounts to a derivation of the critical exponents purely from conformal invariance. The restrictions imposed by conformal invariance and found that the conformal field theory corresponding to the Ising universality class sits on the boundary of the allowed region at a kinklike point in the space of scaling dimensions of the only two relevant operators. Other critical points (tricritical and multicritical) have conformal symmetry, and here we use conformal field theory techniques to study tricritical points.
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