  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.
