Dangerously irrelevant operator

In statistical mechanics and quantum field theory, a dangerously irrelevant operator (or dangerous irrelevant operator) is an operator which is irrelevant at a renormalization group fixed point, yet affects the infrared (IR) physics significantly (e.g. because the vacuum expectation value (VEV) of some field depends sensitively upon the coefficient of this operator).
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 ( p. 49).
The presence of dangerously irrelevant operators leads to the violation of the hyperscaling relation formula_1 between the critical exponents formula_2 and formula_3 in formula_4 dimensions. The simplest example ( p. 93) is the critical point of the Ising ferromagnet in formula_5 dimensions, which is a gaussian theory (free massless scalar formula_6), but the leading irrelevant perturbation formula_7 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 ( p. 164).
Quantum field theory.
Let us suppose there is a field formula_6 with a potential depending upon two parameters, formula_9 and formula_10.
Let us also suppose that formula_9 is positive and nonzero and formula_13 > formula_2. If formula_10 is zero, there is no stable equilibrium. If the scaling dimension of formula_6 is formula_17, then the scaling dimension of formula_10 is formula_19 where formula_4 is the number of dimensions. It is clear that if the scaling dimension of formula_10 is negative, formula_10 is an irrelevant parameter. However, the crucial point is, that the VEV
depends very sensitively upon formula_10, at least for small values of formula_10. Because the nature of infrared physics also depends upon the VEV, it looks very different even for a tiny change in formula_10 not because the physics in the vicinity of formula_27 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.
Other uses of the term.
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 calls dangerously irrelevant a UV operator whose scaling dimension, while irrelevant at short distances: formula_28 , receives a negative correction along a renormalization group flow, so that the operator becomes relevant at long distances: formula_29. This usage of the term is different from the one originally introduced in statistical physics.