Renormalization Group Approaches to Higher Order Critical Points.

Critical points of higher order can exist in complex magnetic and fluid systems. By definition, at a critical point of order $\mathcal{O}$ , $\mathcal{O}$ phases become identical simultaneously. Here the Wilson renormalization-group method is generalized from ordinary critical points ( $\mathcal{O}=2$ ) and Gaussian tricritical points ( $\mathcal{O}=3$ ) to critical points of arbitrary order $\mathcal{O}$ . An expansion scheme in ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)\ensuremath{-}d}$ is proposed. The nontrivial fixed points and the critical exponents for $\mathcal{O}=3,4$ are calculated to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$. We present the relevant scaling fields and densities for $\mathcal{O}=3$ , and, in an appendix, justify the validity of the approximate recursion relation to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$.Critical points of higher order can exist in complex magnetic and fluid systems. By definition, at a critical point of order $\mathcal{O}$, $\mathcal{O}$ phases become identical simultaneously. Here the Wilson renormalization-group method is generalized from ordinary critical points ($\mathcal{O}=2$) and Gaussian tricritical points ($\mathcal{O}=3$) to critical points of arbitrary order $\mathcal{O}$. An expansion scheme in ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)\ensuremath{-}d}$ is proposed. The nontrivial fixed points and the critical exponents for $\mathcal{O}=3,4$ are calculated to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$. We present the relevant scaling fields and densities for $\mathcal{O}=3$, and, in an appendix, justify the validity of the approximate recursion relation to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$.

Physical review. B, Solid stateJun 1, 197463 citations

Phase Transitions and Renormalization Group: from Theory to Numbers

Jean Zinn‐Justin

The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].The large distance behaviour, near the critical temperature, of continuous phase transitions with short range interactions can be reproduced by an effective local field theory. Within this framework, the universal properties of the large distance can be determined by finding fixed points of general, functional renormalization group (RG) equations (Wilson, Wegner). For dimension D > 4, the large scale behavior is governed by the Gaussian fixed point (related to mean field theory), while for D < 4 the Gaussian fixed point is unstable. Near D = 4, the RG equations can be solved in the form of an ǫ = 4 − D expansion, a new stable fixed point is found and this leads to the famous Wilson-Fisher expansion for critical exponents and other universal quantities. Moreover, one realizes that, in this framework, the effective field theory can be reduced to a renormalizable field theory and, at leading order in the critical domain, the much simpler quantum field theory renormalization group can be used. A number of physicists, including the Saclay group (Brezin, Le Guillou, Zinn-Justin) have developed sophisticated field theory techniques, to prove general scaling laws and to calculate exponents to higher orders [1].

Birkhäuser Basel eBooksJan 1, 20032 citations

Critical phenomena and the renormalization group

Franz Wegner

Springer eBooksApr 6, 20086 citations

Critical Phenomena and the Renormalization Group

WORLD SCIENTIFIC eBooksMar 1, 20080 citations

Critical points of higher order can exist in complex magnetic and fluid systems. By definition, at a critical point of order

\mathcal{O}

\mathcal{O}

phases become identical simultaneously. Here the Wilson renormalization-group method is generalized from ordinary critical points (

\mathcal{O}=2

) and Gaussian tricritical points (

\mathcal{O}=3

) to critical points of arbitrary order

\mathcal{O}

. An expansion scheme in ${\ensuremath{\epsilon}}_{\mathcal{O}}\ensuremath{\equiv}\frac{2\mathcal{O}}{(\mathcal{O}\ensuremath{-}1)\ensuremath{-}d}$ is proposed. The nontrivial fixed points and the critical exponents for

\mathcal{O}=3,4

are calculated to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$. We present the relevant scaling fields and densities for

\mathcal{O}=3

, and, in an appendix, justify the validity of the approximate recursion relation to order ${\ensuremath{\epsilon}}_{\mathcal{O}}$.