Dangerously irrelevant operator: Difference between revisions

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== Critical phenomena ==

In the theory of critical phenomena, free energy of a system near the critical point depends analytically on the coefficients of generic (not dangerous) irrelevant operators, while the dependence on the coefficients of dangerously irrelevant operators is non-analytic (<ref name=":0">{{Cite book|last=Cardy|first=John|title=Scaling and Renormalization in Statistical Physics|publisher=Cambridge University Press|year=1996|isbn=|___location=|pages=}}</ref> p.&nbsp;49).

The presence of dangerously irrelevant operators leads to the violation of the hyperscaling relation <math>\alpha=2-d\nu</math> between the critical exponents <math>\alpha</math> and <math>\nu</math> in <math>d</math> dimensions. The simplest example (<ref name=":0" /> p.&nbsp;93) is the critical point of the Ising ferromagnet in <math>d\ge4</math> dimensions, which is a gaussian theory (free massless scalar <math>\phi</math>), but the leading irrelevant perturbation <math>\phi^4</math> is dangerously irrelevant. Another example occurs for the Ising model with random-field disorder, where the fixed point occurs at zero temperature, and the temperature perturbation is dangerously irrelevant (<ref name=":0" /> p.&nbsp;164).