Conformal bootstrap signatures of the tricritical Ising universality class

Abstract

We study the tricritical Ising universality class using conformal bootstrap techniques. By studying bootstrap constraints originating from multiple correlators on the conformal field theory (CFT) data of multiple operator product expansions (OPEs), we are able to determine the scaling dimension of the spin field ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\sigma}}$ in various noninteger dimensions $2\ensuremath{\le}d\ensuremath{\le}3$. Here, ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\sigma}}$ is connected to the critical exponent $\ensuremath{\eta}$ that governs the (tri)critical behavior of the two-point function via the relation $\ensuremath{\eta}=2\ensuremath{-}d+2{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\sigma}}$. Our results for ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\sigma}}$ match with the exactly known values in two and three dimensions and are a conjecture for noninteger dimensions. We also compare our CFT results for ${\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\sigma}}$ with $\ensuremath{\epsilon}$-expansion results, available up to ${\ensuremath{\epsilon}}^{3}$ order. Our techniques can be naturally extended to study higher-order multicritical points.