Editing Scale invariance (section)

==Phase transitions==
In [[statistical mechanics]], as a system undergoes a [[phase transition]], its fluctuations are described by a scale-invariant [[statistical field theory]]. For a system in equilibrium (i.e. time-independent) in {{mvar|D}} spatial dimensions, the corresponding statistical field theory is formally similar to a {{mvar|D}}-dimensional CFT. The scaling dimensions in such problems are usually referred to as [[critical exponent]]s, and one can in principle compute these exponents in the appropriate CFT.

===The Ising model===
An example that links together many of the ideas in this article is the phase transition of the [[Ising model]], a simple model of [[ferromagnet]]ic substances. This is a statistical mechanics model, which also has a description in terms of conformal field theory. The system consists of an array of lattice sites, which form a {{mvar|D}}-dimensional periodic lattice. Associated with each lattice site is a [[magnetic moment]], or [[Spin (physics)|spin]], and this spin can take either the value +1 or −1. (These states are also called up and down, respectively.)

The key point is that the Ising model has a spin-spin interaction, making it energetically favourable for two adjacent spins to be aligned. On the other hand, thermal fluctuations typically introduce a randomness into the alignment of spins. At some critical temperature, {{math|''T<sub>c</sub>''}} , [[spontaneous magnetization]] is said to occur. This means that below {{math|''T<sub>c</sub>''}} the spin-spin interaction will begin to dominate, and there is some net alignment of spins in one of the two directions.

An example of the kind of physical quantities one would like to calculate at this critical temperature is the correlation between spins separated by a distance {{mvar|r}}. This has the generic behaviour:
:<math>G(r)\propto\frac{1}{r^{D-2+\eta}},</math>
for some particular value of <math>\eta</math>, which is an example of a critical exponent.

====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>.

===Schramm–Loewner evolution===
The anomalous dimensions in certain two-dimensional CFTs can be related to the typical [[fractal dimension]]s of random walks, where the random walks are defined via [[Schramm–Loewner evolution]] (SLE). As we have seen above, CFTs describe the physics of phase transitions, and so one can relate the critical exponents of certain phase transitions to these fractal dimensions. Examples include the 2''d'' critical Ising model and the more general 2''d'' critical [[Potts model]]. Relating other 2''d'' CFTs to SLE is an active area of research.