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 dangeorously 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 (,[1] 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 (,[1] p. 164).

Quantum field theory

Let us suppose there is a field with a potential depending upon two parameters, and .

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

.

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]

gollark: Okay, very hacky but technically workable: have an XTMF metadata block of a fixed size, and after the actual JSON data, instead of just ending it with a `}`, have enough spaces to fill up the remaining space then a `}`.
gollark: XTMF was not really designed for this use case, so it'll be quite hacky. What you can do is leave a space at the start of the tape of a fixed size, and stick the metadata at the start of that fixed-size region; the main problem is that start/end locations are relative to the end of the metadata, not the start of the tape, so you'll have to recalculate the offsets each time the metadata changes size. Unfortunately, I just realized now that the size of the metadata can be affected by what the offset is.
gollark: The advantage of XTMF is that your tapes would be playable by any compliant program for playback, and your thing would be able to read tapes from another program.
gollark: Tape Shuffler would be okay with it, Tape Jockey doesn't have the same old-format parsing fallbacks and its JSON handling likely won't like trailing nuls, no idea what tako's program thinks.
gollark: Although I think some parsers might *technically* be okay with you reserving 8190 bytes for metadata but then ending it with a null byte early, and handle the offsets accordingly, I would not rely on it.

References

  1. Cardy, John (1996). Scaling and Renormalization in Statistical Physics. Cambridge University Press.
  2. Gukov, Sergei (2016-01-05). "Counting RG flows". Journal of High Energy Physics. 2016 (1): 20. arXiv:1503.01474. doi:10.1007/JHEP01(2016)020. ISSN 1029-8479.
  3. Amit, Daniel J; Peliti, Luca (1982). "On dangerous irrelevant operators". Annals of Physics. 140 (2): 207–231. doi:10.1016/0003-4916(82)90159-2.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.