Abstract Mean-field theory is usually taken as a first step toward understanding critical phenomena, providing an overview that reveals qualitative behaviour of physical quantities. However, it is necessary to proceed beyond the mean-field theory to better understand the situation, both qualitatively and quantitatively, when fluctuations play vital roles leading to exponents that cannot be explained by dimensional analysis, thus introducing anomalous dimensions. The present chapter explains the basic concepts of the renormalization group and scaling theory, which allow us to analyze critical phenomena with fluctuations systematically taken into account. The essential step in a renormalization group calculation consists of establishing recursion relations between the parameters defining the Hamiltonian of the system. These recursion or renormalization group equations define a flow with well-defined fixed points. Details other than the values of the relevant operators have no influence on the critical exponents and this represents universality.Abstract Mean-field theory is usually taken as a first step toward understanding critical phenomena, providing an overview that reveals qualitative behaviour of physical quantities. However, it is necessary to proceed beyond the mean-field theory to better understand the situation, both qualitatively and quantitatively, when fluctuations play vital roles leading to exponents that cannot be explained by dimensional analysis, thus introducing anomalous dimensions. The present chapter explains the basic concepts of the renormalization group and scaling theory, which allow us to analyze critical phenomena with fluctuations systematically taken into account. The essential step in a renormalization group calculation consists of establishing recursion relations between the parameters defining the Hamiltonian of the system. These recursion or renormalization group equations define a flow with well-defined fixed points. Details other than the values of the relevant operators have no influence on the critical exponents and this represents universality.

Dec 2, 20100 citations

Universal critical phenomena of the cloud --> crystal phase transition in the Paul trap: Powerlaws

Daniel Weiss, Yunseong Nam, R. Blümel

APSJan 1, 20160 citations

Critical Phenomena: field theoretical approach

Jean Zinn‐Justin

ScholarpediaJan 1, 201050 citations

Universal critical phenomena of the cloud $\rightarrow$ crystal phase transition in the Paul trap: PowerlawsUniversal critical phenomena of the cloud $\rightarrow$ crystal phase transition in the Paul trap: Powerlaws

Daniel Weiss, Yunseong Nam, R. Blümel

Bulletin of the American Physical SocietyMar 18, 20160 citations

Critical Phenomena for Field Theorists

Steven Weinberg

Springer eBooksJan 1, 1978206 citations

Universal critical phenomena of the cloud

\rightarrow

crystal phase transition in the Paul trap: Powerlaws