Statistical mechanics, structure of matter Definition

Fixed point of the renormalization group

Builds on Renormalization group Block spin transformation Correlation function

Definition

Let $\mathcal K$ be a space of coupling constants, each $K \in \mathcal K$ describing a Hamiltonian $H_K$ of a lattice system. Let $b > 1$ and let $R_b \colon \mathcal K \to \mathcal K$ be a renormalization group transformation with scale factor $b$ , such as a block spin transformation. A fixed point of $R_b$ is a point $K^* \in \mathcal K$ with $R_b(K^*) = K^*.$ A fixed point is critical if the correlation length $\xi(K^*)$ of $H_{K^*}$ , the decay length of its correlation function, is infinite, and trivial if $\xi(K^*) = 0$ .

Remarks

The transformation rescales lengths by $b$ , so $\xi(R_b K) = \xi(K)/b$ . At a fixed point this gives $\xi(K^*) = \xi(K^*)/b$ , so $\xi(K^*)$ is $0$ or $\infty$ .

Example: for the Ising model with coupling $K$ , the point $K = 0$ of independent spins is a trivial fixed point of the majority-rule block spin transformation. Check: let the spins be independent and uniform. The block spins are functions of disjoint sets of spins, so they are again independent. Each block spin is uniform by the symmetry $\sigma \mapsto -\sigma$ .

Non-example: for the two-dimensional Ising model at a coupling $K$ with $0 < K < K_c$ , the correlation length is finite and positive, so $K$ is not a fixed point. Under repeated transformations it flows toward the high-temperature trivial fixed point.

The set of couplings that flow to a critical fixed point under iteration is its critical surface.

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Block spin transformation concept
Correlation function concept
Renormalization group concept
Fixed point of the renormalization group definition

Tag	`KMA1HCE104`
URI	`m2://KMA1HCE104`
Kind	definition
Writing	W0 stub
MSC	82B28, 82B27
Defines	Fixed point of the renormalization group
Uses	Renormalization group, Block spin transformation, Correlation function
Changed	3 November 2026

Fixed point of the renormalization group

Definition

Remarks

Reading path

Further reading

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