Introduction to Renormalisation Group Methods

Abstract

The renormalisation group approach of WILSON, FISHER and KADANOFF [1,2,3] was developed to treat critical behaviour near continuous phase transitions. At such transitions, the correlations become of infinite range and the divergence of the correlation length introduces singularities in thermodynamic and other properties.These critical singularities are characterised by critical exponents α,β,…,defined as follows [4,5] for a simple magnet:

$${\matrix{{({\rm{i}})\quad \underline {{\rm{h}} = 0,} \;{{\underline {{\rm{T}} \sim {\rm{T}}} }_{\rm{C}}}:} & {{\rm{C}} \propto \left| {{\rm{T}} - {{\rm{T}}_{\rm{C}}}} \right|} \cr} ^{ - \alpha }}$$

((1))

$${\matrix{{} & {{\rm{M}} \propto ({{\rm{T}}_{\rm{C}}} - {\rm{T}})} \cr} ^\beta }\quad ({\rm{T}} < {{\rm{T}}_{\rm{C}}})$$

((2))

$${\matrix{{} & {\chi \, \propto \left| {{\rm{T}} - {{\rm{T}}_{\rm{C}}}} \right|} \cr} ^{ - \gamma }}$$

((3))

$$\begin{array}{*{20}{c}} {{{{\begin{array}{*{20}{c}} {({\text{ii}})} & {\underline {T = T} } \\ \end{array} }}_{c}},\underline {h \sim 0:} } & {M \propto |h| \frac{1}{\delta } } \\ \end{array}$$

(4)

$$\begin{array}{*{20}{c}} {({\text{iii)}} h = 0,} & {T \sim {{T}_{c}},r \to \infty :{{\Gamma }_{r}} \equiv \langle {{S}_{o}}{{S}_{r}}\rangle - \langle {{S}_{o}}\rangle \langle {{S}_{r}}\rangle } \\ \end{array} \sim \frac{{\exp ( - r/\xi )}}{{{{r}^{{d - 2 + \eta }}}}}$$

(5)

$$\matrix{{} & {\xi \; \propto {{\left| {{\rm{T}} - {{\rm{T}}_{\rm{C}}}} \right|}^{ - \nu }}} \cr}.$$

((6))

Here, C, M, χ, Γ_r, ξ are respectively specific heat, magnetisation, susceptibility, correlation function for spins S separated by distance r, and correlation length; h is a reduced magnetic field; T is temperature and T_c its critical value. These functions are not independent since, for example [4]

$$\chi = {\textstyle{{\partial {\rm{M}}} \over {\partial {\rm{h}}}}} = \mathop \Sigma \limits_{\rm{r}} \;{\Gamma _{\rm{r}}}.$$

((7))