Math 2.0

Statistical mechanics, structure of matter Definition

Relevant and irrelevant operators

Builds on Fixed point of the renormalization groupJacobian matrixEigenvalue

Definition

Let 𝒦\mathcal K be an open subset of ℝn\R^n, a space of coupling constants. Let (Rb)b>1(R_b)_{b > 1} be a family of renormalization group maps Rb ⁣:𝒦→𝒦R_b \colon \mathcal K \to \mathcal K with Rb∘Rb′=Rbb′R_b \circ R_{b'} = R_{bb'}. Let K∗K^* be a common fixed point of all RbR_b, and assume each RbR_b is differentiable at K∗K^*. Let AbA_b be the Jacobian matrix of RbR_b at K∗K^*. Assume that the matrices AbA_b have a common basis of eigenvectors e1,…,ene_1, \dots, e_n with real eigenvalues λi(b)=byi\lambda_i(b) = b^{y_i}. The number yiy_i is the RG eigenvalue of the direction eie_i, and the perturbation K∗+heiK^* + h e_i, or the operator coupled to eie_i, is called:

  1. relevant if yi>0y_i > 0;
  2. irrelevant if yi<0y_i < 0;
  3. marginal if yi=0y_i = 0.

Remarks

The semigroup law gives AbAb′=Abb′A_b A_{b'} = A_{bb'}, so λi(b)λi(b′)=λi(bb′)\lambda_i(b) \lambda_i(b') = \lambda_i(bb'). A continuous solution of this equation with λi(b)>0\lambda_i(b) > 0 is a power byib^{y_i}, which explains the form of the eigenvalues. To first order, Rb(K∗+hei)=K∗+byiheiR_b(K^* + h e_i) = K^* + b^{y_i} h e_i, so a relevant component grows and an irrelevant one decays under iteration.

Example: at the critical fixed point of the two-dimensional Ising model, the thermal direction has yt=1y_t = 1 and the magnetic direction has yh=15/8y_h = 15/8. Both are relevant, and the correlation length exponent is ν=1/yt=1\nu = 1/y_t = 1.

Non-example: a direction with yi=0y_i = 0 is neither relevant nor irrelevant; its fate is decided by the higher-order terms of RbR_b, as for the marginal coupling in the Berezinskii–Kosterlitz–Thouless transition.

When there is no marginal direction, the relevant directions span the directions transverse to the critical surface at K∗K^*, and the irrelevant directions are tangent to it.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Eigenvalue, concept, no evidence levelEigenvalueconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFixed point of the renormalization group, concept, no evidence levelFixed point of the re…conceptFunction, concept, no evidence levelFunctionconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptJacobian matrix, concept, no evidence levelJacobian matrixconceptRelevant and irrelevant operators, definition, no evidence levelRelevant and irreleva…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Eigenvalue concept
  2. Euclidean space object
  3. Fixed point of the renormalization group concept
  4. Function concept
  5. Total derivative concept
  6. Function of several variables concept
  7. Partial derivative concept
  8. Jacobian matrix concept
  9. Relevant and irrelevant operators definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Renormalization_group
About this page
TagGAGWR57ZT0
URIm2://GAGWR57ZT0
Kinddefinition
WritingW0 stub
MSC82B28, 82B27
DefinesRelevant and irrelevant operators
UsesFixed point of the renormalization group, Jacobian matrix, Eigenvalue
Changed3 November 2026

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