Editing Scale invariance (section)

====CFT description====
The fluctuations at temperature {{math|''T<sub>c</sub>''}} are scale-invariant, and so the Ising model at this phase transition is expected to be described by a scale-invariant statistical field theory. In fact, this theory is the '''Wilson–Fisher fixed point''', a particular scale-invariant [[scalar field (quantum field theory)|scalar field theory]].

In this context, {{math|''G''(''r'')}} is understood as a [[correlation function]] of scalar fields,
:<math>\langle\phi(0)\phi(r)\rangle\propto\frac{1}{r^{D-2+\eta}}.</math>
Now we can fit together a number of the ideas seen already.

From the above, one  sees that the critical exponent, {{mvar|η}}, for this phase transition, is also an '''anomalous dimension'''. This is because the classical dimension of the scalar field,
:<math>\Delta=\frac{D-2}{2}</math>
is modified to become
:<math>\Delta=\frac{D-2+\eta}{2},</math>
where {{mvar|D}} is the number of dimensions of the Ising model lattice.

So this '''anomalous dimension''' in the conformal field theory is the ''same'' as a particular critical exponent of the Ising model phase transition.

Note that for dimension {{math|''D'' ≡ 4−''ε''}}, {{mvar|η}} can be calculated approximately, using the '''epsilon expansion''', and one finds that
:<math>\eta=\frac{\epsilon^2}{54}+O(\epsilon^3)</math>.

In the physically interesting case of three spatial dimensions, we have {{mvar|ε}}=1, and so this expansion is not strictly reliable. However, a semi-quantitative prediction is that {{mvar|η}} is numerically small in three dimensions.

On the other hand, in the two-dimensional case the Ising model is exactly soluble. In particular, it is equivalent to one of the [[Minimal model (physics)|minimal models]], a family of well-understood CFTs, and it is possible to compute {{mvar|η}} (and the other critical exponents) exactly,
:<math>\eta_{_{D=2}}=\frac{1}{4}</math>.