Dangerously irrelevant operator explained
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 ([1] p. 49).
The presence of dangerously irrelevant operators leads to the violation of the hyperscaling relation
between the critical exponents
and
in
dimensions. The simplest example (p. 93) is the critical point of the Ising ferromagnet in
dimensions, which is a gaussian theory (free massless scalar
), but the leading irrelevant perturbation
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
with a
potential depending upon two parameters,
and
.
V\left(\phi\right)=-a\phi\alpha+b\phi\beta
Let us also suppose that
is positive and nonzero and
>
. If
is zero, there is no stable equilibrium. If the
scaling dimension of
is
, then the scaling dimension of
is
where
is the number of dimensions. It is clear that if the scaling dimension of
is negative,
is an irrelevant parameter. However, the crucial point is, that the VEV
| \langle\phi\rangle=\left( | a\alpha |
| b\beta |
.
depends very sensitively upon
, at least for small values of
. Because the nature of infrared physics also depends upon the VEV, it looks very different even for a tiny change in
not because the physics in the vicinity of
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[2] calls dangerously irrelevant a UV operator whose scaling dimension, while irrelevant at short distances:
, receives a negative correction along a renormalization group flow, so that the operator becomes relevant at long distances:
. This usage of the term is different from the one originally introduced in statistical physics.
[3] Notes and References
- Book: Cardy, John. Scaling and Renormalization in Statistical Physics. Cambridge University Press. 1996.
- Gukov. Sergei. 2016-01-05. Counting RG flows. Journal of High Energy Physics. en. 2016. 1. 20. 1503.01474. 10.1007/JHEP01(2016)020. 2016JHEP...01..020G . 23582290. 1029-8479.
- Amit. Daniel J. Peliti. Luca. 1982. On dangerous irrelevant operators. Annals of Physics. en. 140. 2. 207–231. 10.1016/0003-4916(82)90159-2. 1982AnPhy.140..207A .
