Dangerously irrelevant operator

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 ([1] p. 49).

The presence of dangerously irrelevant operators leads to the violation of the hyperscaling relation ${\displaystyle \alpha =2-d\nu }$ between the critical exponents ${\displaystyle \alpha }$ and ${\displaystyle \nu }$ in ${\displaystyle d}$ dimensions. The simplest example ([1] p. 93) is the critical point of the Ising ferromagnet in ${\displaystyle d\geq 4}$ dimensions, which is a gaussian theory (free massless scalar ${\displaystyle \phi }$), but the leading irrelevant perturbation ${\displaystyle \phi ^{4}}$ 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 ([1] p. 164).

Let us suppose there is a field ${\displaystyle \phi }$ with a potential depending upon two parameters, ${\displaystyle a}$ and ${\displaystyle b}$.

${\displaystyle V\left(\phi \right)=-a\phi ^{\alpha }+b\phi ^{\beta }}$

Let us also suppose that ${\displaystyle a}$ is positive and nonzero and ${\displaystyle \beta }$ > ${\displaystyle \alpha }$. If ${\displaystyle b}$ is zero, there is no stable equilibrium. If the scaling dimension of ${\displaystyle \phi }$ is ${\displaystyle c}$, then the scaling dimension of ${\displaystyle b}$ is ${\displaystyle d-\beta c}$ where ${\displaystyle d}$ is the number of dimensions. It is clear that if the scaling dimension of ${\displaystyle b}$ is negative, ${\displaystyle b}$ is an irrelevant parameter. However, the crucial point is, that the VEV

${\displaystyle \langle \phi \rangle =\left({\frac {a\alpha }{b\beta }}\right)^{\frac {1}{\beta -\alpha }}=\left({\frac {a\alpha }{\beta }}\right)^{\frac {1}{\beta -\alpha }}b^{-{\frac {1}{\beta -\alpha }}}}$.

depends very sensitively upon ${\displaystyle b}$, at least for small values of ${\displaystyle b}$. Because the nature of infrared physics also depends upon the VEV, it looks very different even for a tiny change in ${\displaystyle b}$ not because the physics in the vicinity of ${\displaystyle \phi =0}$ changes much — it hardly changes at all — but because the VEV we are expanding about has changed enormously.

Supersymmetric models with a modulus can often have dangerously irrelevant parameters.

Consider a renormalization group (RG) flow triggered at short distances by a relevant perturbation of an ultra-violet (UV) fixed point, and flowing at long distances to an infra-red (IR) fixed point. It may be possible (e.g. in perturbation theory) to monitor how dimensions of UV operators change along the RG flow. In such a situation, one sometimes[2] calls dangerously irrelevant a UV operator whose scaling dimension, while irrelevant at short distances: ${\displaystyle \Delta _{\rm {UV}}>d}$ , receives a negative correction along a renormalization group flow, so that the operator becomes relevant at long distances: ${\displaystyle \Delta _{\rm {IR}}<d}$. This usage of the term is different from the one originally introduced in statistical physics.[3]