Why are Critical Exponents simple non-integer powers?

Question

I'm reading Baxter's Exactly Solved Models in Statistical Physics, and he claims that for $$t=\frac{T-T_c}{T_c}$$ which is just a change of variable in temperature to centre and normalise w.r.t. the critical temperature, we expect the singularities of thermodynamic functions at $t=0$ to be simple non-integer powers. As far as I can tell, this means that for some thermodynamic function $f(t)$ we have $$f(t)\sim t^\alpha$$ where $X\sim Y$ means $X/Y$ tends to a non-zero limit, where $\alpha$ is some rational number. $\alpha$ is the critical exponent. What I don't understand is

1. Does a "simple non-integer power" mean "rational power"?
2. Why don't we expect an integer power?

Answer 1

In two dimensions, we know from conformal field theory that many critical exponents are rational numbers. In three or higher dimensions it is not known whether this remains so. In particular, in field-theory calculations, transcendental numbers such as $\zeta(3)$ appear as coefficients in the epsilon expansion.