Introduction to Renormalisation Group Methods

• R. B. Stinchcombe
Conference paper
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 48)

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, Γr, 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 Tc 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)

Keywords

Renormalisation Group Correlation Length Ising Model Critical Phenomenon Finite Size Scaling
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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